Bose-Einstein Condensate Wavefunction Reconstruction Through Collisions with Optical Potentials

Christopher Ellenor

Abstract

A new technique for the interferometric measurement of an atomic wavefunction is introduced theoretically, which is able to extract phase and amplitude information in a single measurement. I focus on the application of this technique to the single-particle wavefunction of a Bose condensed cloud of rubidium atoms. The technique differs from existing techniques mainly in its simplicity, as it requires only a single laser beam to be added to a typical Bose-Einstein condensation apparatus. A second novel aspect is the consideration of condensate collisions with an optical potential in the low-intensity limit where the potential barrier may be viewed as a phase mask. The technique is then demonstrated experimentally. A related effect, the transient enhancement of momentum during a collision, first predicted by JG Muga et al., has also been demonstrated. Finally, significant redesign and construction of an apparatus to produce condensates of 87 Rb is documented. The main result of this work is the production of pure condensates of up to 150k atoms which can be repeated every 45s. A calibration technique is devised and demonstrated, whereby copies of the condensate are made, and the copies are used to reduce the centre-of-mass momentum uncertainty of the interacting cloud by a factor of five.

1 Background and Motivation

One of the more surprising aspects of quantum mechanics has been the requirement that particles under some circumstances should behave as waves. When an atom can exhibit wavelike behaviour, the results can be shockingly at odds with our classical intuition. Atoms moving towards an energetically forbidden region may appear on the other side through a process known as tunnelling. When being reflected by a barrier they may even be suddenly accelerated. Atoms headed for the same place may even cancel each other out and arrive somewhere altogether different. Historically these phenomena have happened in some abstract space, but with the recent achievement of macroscopic wavefunctions through Bose-Einstein condensation, we can now attempt to watch them happen.

1.1 Bose-Einstein Condensation

Bose-Einstein condensation is a quantum phase transition in which the single-particle ground state of a system is macroscopically occupied. In other words, a significant fraction - often approaching 1 - of the atoms occupy a single state at the bottom of the trap, meaning a single-particle wavefunction is occupied by thousands, millions, or even billions of atoms. The size of this wavefunction is set by details of the trapping potential, but is typically on the order of a few thousandths of a millimeter. Its large size and high occupation mean that measuring the wavefunction is as simple as taking a photograph with an off-the-shelf optical system (see Section 3.2.6↓ for details).
An estimate of the critical temperature, Tc, associated with the phase transition may be found by assuming that Bose-Einstein Condensation occurs when the thermal deBroglie wavelength becomes comparable to the inter-particle spacing, i.e. when the atoms begin to overlap in space and become indistinguishable. The thermal deBroglie wavelength is given by
(1.1) λdB = 2π2mkBT
where T is the temperature of the gas, m is the mass of the constituent atoms and kB is the Boltzmann constant. The challenge to the experimentalist is thus to create a cold, yet sufficiently dense gas. A more precise calculation 28 shows that in a box potential, the phase transition occurs when nλdB3≃2.6, where n is the peak density of the atom cloud, and nλdB3 is defined as the phase space density.
The theoretical prediction of Bose-Einstein Condensation in 1925 predated its observation in atomic vapour by 70 years, though it was identified with superfluidity in Helium-4, and later with superconducting metals. It is of historical interest that superfluidity in Helium-4 was discovered in 1938 by the University of Toronto’s own Allen and Meisner, as well as Kapitza in Moscow. A brief overview of this discovery and the events that followed is given by Allan Griffin in 57. Work by Satyendra Nath Bose to describe a gas of photons was extended by Einstein to include massive particles, and the result was Bose-Einstein statistics. As described by a grand-canonical ensemble, a Bose gas has a mean occupation number at energy ϵν of
(1.2) f0(ϵν) = 1z − 1eβϵν − 1 = 1e(ϵν − μ) ⁄ kBT − 1
where z is the fugacity, defined as z ≡ exp(μ ⁄ kBT), β = 1 ⁄ kBT, and μ is the chemical potential. Equation 1.2↑ is the Bose distribution function, and μ is determined by the requirement that the sum of occupations over all states equal the total number of particles, N. At high temperatures, occupation numbers are very low and μ is correspondingly much less than ϵmin, the energy of the lowest-lying state. As the temperature is lowered, μ rises as occupation numbers increase. Clearly this can only proceed until μ = ϵmin, beyond which Equation 1.2↑ would predict the unphysical situation of negative occupation of the minimum energy state. Upon reaching this threshold, the number of particles in the minimum energy state is macroscopic and can be arbitrarily large. The part of the cloud occupying this state is known as a Bose-Einstein condensate.
To determine the temperature, Tc, associated with this phase transition, consider the total number of particles in excited states
(1.3) Nex = 0 dϵ g(ϵ)f0(ϵ)
where g(ϵ) is the density of states at energy ϵ. The transition temperature is defined as the temperature at which the excited state levels can no longer accommodate all of the particles in the system. The quantity in Equation 1.3↑ is maximized when μ = ϵmin = 0, so it is possible to write
(1.4) N = Nex(Tc, μ = 0) = 0 dϵ g(ϵ)1eϵ ⁄ kBTc − 1
To evaluate this integral, consider the case of a three dimensional harmonic oscillator, which is the potential used in most BEC experiments, including this one. A more general treatment may be found in 28. In this case, the density of states is given by
(1.5) g(ϵ) = ϵ22ℏ3ωxωyωz
where ωi is the trap frequency in dimension i. Making the variable substitution x = ϵ ⁄ kBTc, Equation 1.4↑ becomes
(1.6) N = (kBTc)32ℏ3ωxωyωz0 dx x2ex − 1
The integral portion is equal to 2ζ(3), where ζ is the Riemann zeta function. An approximate value of Tc is thus given by
(1.7) kBTc ≈ 0.94ℏ(ωxωyωzN)1 ⁄ 3
In this experiment I will be dealing with atom numbers of a few hundred thousand, and trap frequencies of around 150  Hz, yielding transition temperatures of a few hundred nanokelvin. Condensation will typically be pursued at the highest trap frequencies possible, with the aim of maximizing collision rates and thus maximizing the efficiency of evaporative cooling (see Section 3.3.6↓). Once the transition has been achieved, the trap may be adiabatically weakened to lower the temperatures of both the cloud and the transition.
To have a system that resembles a single quantum wavefunction as closely as possible, it is desirable to have as many of the atoms in a single state as possible. After condensation, cooling continues and the excited states are able to accommodate fewer and fewer atoms. The condensate fraction is determined by again finding the number of atoms in excited states
(1.8) Nex(T) = (kBT)32ℏ3ωxωyωz0 dx x2ex − 1 = N(TTc)3
where Equation 1.6↑ has been used to write the result in terms of Tc. The number of condensed atoms can thus be expressed as
(1.9) N0 = N − Nex = N[1 − (TTc)3]

1.1.1 Interacting Bosons: The Gross-Pitaevskii Equation

In the description of Bose-Einstein condensation provided in the previous section, atoms were assumed to be non-interacting. In this description, the condensed fraction of atoms should occupy the ground state of the harmonic potential and have a Gaussian density profile. In a real system, however, atoms can interact with one another, and the ground state of the system is a many-body ground state. In the case of an ultracold, dilute atomic gas, it is sufficient to assume only binary s-wave scattering interactions, and a Hartree, or mean-field approach provides a good description of not only this state, but the dynamics of the gas.
This scattering interaction is well described by a contact interaction (see 28, Chapter 5) U0δ(r − r’) where r and r’ are the positions of two interacting particles and U0 is the effective interaction defined as
(1.10) U0 ≡ 4π2am
The interaction is thus characterized entirely by the scattering length, a ≈ 5.2  nm 29. The mean field approach allows inter-atomic interactions to be written as a pseudopotential, and the result is the Gross-Pitaevskii equation (GPE)
(1.11)  − 22m\triangledown2ψ(r) + V(r)ψ(r) + U0|ψ(r)|2ψ(r) = μψ(r)
where V(r) is the external trapping potential and ψ(r) is the condensate wavefunction.
Commonly in BEC experiments the atom number will be high enough that the ratio of kinetic energy to interaction energy is small. In this case, it is a good approximation to drop the kinetic energy term ( − 22m\triangledown2ψ(r)) from Equation 1.11↑. This is known as the Thomas-Fermi approximation. The GPE then becomes
(1.12) [V(r) + U0|ψ(r)|2]ψ(r) = μψ(r)
and the solution for the density is simply
(1.13) n(r) = |ψ(r)|2 = [μ − V(r)] ⁄ U0
and the cloud takes the shape of the potential, equalizing potential energy at all points. In a harmonic potential, the cloud density takes the form of a truncated parabola along each of the three axes, and the shape of the cloud can thus be characterized by three semi-axes, which denote the radii at which the cloud density reaches zero 28:
(1.14) Ri2 = 2μmωi2
where ωi is the trap frequency along axis i. Remarkably, a Thomas-Fermi condensate retains its parabolic shape after release from the confining potential 2, and the expanding cloud can be characterized by a time-dependent version of these semi-axes.

1.2 Quantum collisions: Tunnelling and Enhanced Momentum

With thousands or millions of particles occupying a single, macroscopic wavefunction, the Bose condensate is an ideal tool to study the quantum mechanical dynamics of a single particle. The aim of the experiments presented herein is to probe these dynamics through collisions of this quantum wavefuntion with different potentials. Potentials for collisions involving condensates of alkali atoms may be realized through use of the AC Stark effect. An off-resonant electric field can induce an electric dipole in a neutral atom which will then be attracted or repelled by regions of high electric field, depending on the sign of the detuning. Creating a potential suitable for a collision is then a matter of creating a spatially-dependent oscillating electric field, which can be as simple as focussing a laser beam where a potential is desired. Typical energy scales in a BEC experiment mean that the atomic wavefunction will be produced with a size on the order of optical wavelengths, and optical potentials may be custom built to vary significantly over relevant length scales. Moreover, optical potentials can be varied almost instantaneously with respect to condensate timescales. The remainder of this section describes experiments both planned and performed on this apparatus, which make use of such optical potentials.

1.2.1 Scattering in the non-asymptotic regime

In a 1998 paper 1, JG Muga et. al. suggested that if an ensemble of atoms is studied in the midst of a collision, nonclassical effects in the momentum distribution may be observed. Namely, in a classical model, atoms impinging on a repulsive potential in the absence of interactions are not permitted to increase their momentum. If the atoms are treated quantum mechanically, as waves, it can be shown that some of the atoms in the ensemble --- now the occupants of a single wavefunction --- will be accelerated during the collision. This phenomenon is most simply explained as follows. The presence of the potential within the atomic wavefunction causes phase to be accrued where the barrier is, and if the barrier is sufficiently narrow, removing the barrier during the collision leaves a discontinuity in phase between parts of the wavefunction that have and have not experienced the barrier. This sharp discontinuity in the spatial wavefunction comes with a corresponding broadening in the momentum wavefunction.
Scattering theory is usually concerned with the asymptotic regime, that is, with objects long before and long after the scattering event. If the collision can be interrupted however, the state of objects during the collision may be considered, and this is a scattering problem studied in the non-asymptotic regime. This is well-suited to a BEC experiment as the repulsive potential may be formed by a focused laser beam as described above. Such a potential can be changed or removed in tens of nanoseconds, whereas a collision typically lasts milliseconds. If the potential is quickly removed during the collision, the cloud may be allowed to freely expand. As a cloud expands during freefall, position and momentum become correlated and a measurement of the spatial density profile asymptotes to the momentum density distribution. It is by staging the collision of a BEC wavefunction with an optical potential, interrupting the collision and measuring this momentum distribution in the so-called non-asymptotic regime that nonclassical transient enhancement of momentum is to be observed. Of the proposed scattering experiments, this is the least technically demanding, and is the first to be attempted.
Interest in atom-optics with Bose-Einstein condensates has lead to the development of various types of scattering potentials aimed at the specular reflection of condensates. Condensates have been scattered using time-dependent or static magnetic potentials 58, evanescent waves 59, and in one case, a one-dimensional “sheet of light” similar to the one used in this thesis 60. This work is the first to consider a potential in the extremely weak, transmissive limit, and also the first to study collision dynamics by interrupting the collision.

1.2.2 Wavefunction Measurement

The quantum wavefunction is a complex quantity. While the magnitude of the wavefunction is normally easily accessible through a projective measurement, the phase is often somewhat more elusive. The recovery of this phase is a well-studied problem, particularly in the field of optics. As in optics, schemes to extract the phase, and thus the full wavefunction of matterwaves have typically relied on interferometric measurements. Also in a fashion similar to optics, to know the wavefunction as a function of one variable, it may be simpler to measure in the complementary space. This concept has become commonplace in the measurement of laser pulses, where electronics have no hope or resolving the envelopes of sub-picosecond pulses in the time-domain. Pulse spectra, however, are easily obtainable and efforts focus on recovering the phase of the spectral components composing the pulse, with which the time-domain pulse may be recovered via Fourier transform. Most methods hinge on the measurement of the pulse using a copy of itself, the best known of which is the pulse autocorrelation. An excellent review of laser pulse measurement can be found in 27.
The concept of autocorrelation has been applied as well to matterwave studies, where a wavepacket’s position and momentum are formally similar to the time and frequency domains of a laser pulse. The functional form of the phase of a Bose-Einstein condensate has been measured at NIST 4, and various proposals exist on techniques that will glean more detailed information 26,25. All of these techniques, however, are essentially tomographic and require the assembly of multiple measurements to yield the full wavefunction. An idea introduced in this thesis is the use of momentum-space interference between “enhanced” momenta created in a collision with pre-existing momenta unaffected by the collision to glean information about the quantum wavefunction of a condensate. This interference combined with information in the envelope of the interferogram gives simultaneous information about both the phase and the amplitude, in some cases obviating the need for multiple measurements.

1.2.3 Quantum tunnelling

The longer-term ambition of the experiment is to observe quantum tunnelling of condensate atoms through an optical potential, and to study them as they travel. The main question of interest is whether the time spent by an atom in a potential through which it is tunnelling can be measured. Such experiments are beyond the scope of this thesis, but two proposals are briefly mentioned below, as they provide some motivation for overall design considerations of the experiment.

1.2.3.1 Larmor interaction time

A gedanken experiment proposed by Baz’ in 1967 44 imagines a spin-polarized electron impinging on a potential barrier containing a localized magnetic field. The precession of the electron spin about that field is used as a clock to measure the time of interaction with the barrier, also known as the “Larmor” time, or the “dwell” time. A similar coupling between angular momentum states in a rubidium atom may be effected by optical fields. Due to the the ability to finely tune laser frequencies, transitions between magnetic sub-levels with well-defined Rabi frequencies may be localized to very small regions in space, namely the tunnelling region. Investigation of the magnetic spin state of the atom after tunnelling can reveal this interaction time.

1.2.3.2 Bttiker-Landauer Clock

In 1982, Bttiker and Landauer consider a time-dependent tunnelling potential 45. If the height of such a potential is varied slowly compared to the transit time of the atoms, the transmission of tunnelling particles will be correspondingly modulated. If the height of the potential is modulated very quickly, the atoms will see a time-averaged potential and a constant transmission amplitude. The transition between these two behaviours will mark the timescale of atom traversal. The performance of this experiment also hinges on the flexibility of optical potentials, as modulation of laser power on even nanosecond timescales is a simple matter with acousto-optics.

1.3 Thesis outline

This thesis presents a body of work aimed at observing the effects described in the previous sections in a Bose-Einstein condensation experiment. Chapter 2↓ expands on theoretical work presented by Muga et. al. 1,3, and considers its application to our experimental system. Predictions are made for transient momentum enhancement in view of experimental limitations, including nonlinear interactions in the atomic cloud, limitations on the design of an optical potential and limitations on the ability to measure momentum distributions. A first treatment of interference fringes produced in the process is given, and the technique to extract wavefunction information is developed. Chapter 3↓ discusses the apparatus employed, including a detailed look at components that were redesigned as a part of this thesis. Every major system in the experiment has undergone some amount of redesign and details on design and evaluation are presented. Redesigned systems include the ultra-high vacuum system, the optical system, the magnetic field systems and the electronics system. Some background theory is also provided covering relevant topics such as laser cooling, magnetic trapping and optical potentials. Chapter 4↓ describes the experimental process used to create BEC, and gives a step-by-step account of the transition from a thermal vapour of room-temperature 87 Rb to a 30  nK Bose-condensed cloud. Emphasis is placed on the improvement achieved over previous iterations of the experiment, namely in atom number, cycle duration and day-to-day stability. Chapter 5↓ presents experimental demonstrations of momentum-space interference, as well as transient momentum enhancement.

2 Momentum Enhancement and Wavefunction Measurement

In this chapter I will outline the theoretical predictions of transient momentum enhancement and momentum-space interference. I will discuss their measurement on a BEC apparatus, considering experimental limitations such as nonlinear interaction energy and finite optical resolutions. Most results are produced from one-dimensional numerical modeling, a summary of which is provided in Appendix B.1↓.

2.1 Momentum Enhancement

It was first predicted by Muga et al. 1 that particles colliding with a static, repulsive potential could exhibit a classically forbidden, transient enhancement of high momentum components during the collision. They consider an ensemble of classical particles impinging on some potential, V(x), which is bounded from below. Arguing from conservation of energy, a particle interacting with this potential may have its momentum increased by a maximum ofwhere m is the mass of the particle, and Vmin is the minimum of the potential V(x). Considering then an ensemble of particles which occupies some region of phase space, it is further argued that the accumulated probability of momenta above p + pν may at no time exceed the accumulated probability of momenta above p in the initial state. This result is formalized by defining G, the difference in these accumulated probabilities, which under classical conditions will always be less than or equal to zero:
(2.1) Gcl(p, t) = pP(p’ + pν, t) − P(p’, 0)} dp’ ≤ 0
where P(p, t) is the probability distribution as a function of momentum, p, and time, t. It is shown, however, that a quantum mechanical treatment of the ensemble of atoms can yield a different result. Equation 2.1↑ can be recast as
(2.2) Gq(p, t) = p|Ψ(p’ + pν, t)|2 − |Ψ(p’, 0)|2} dp
where Ψ is the wavefunction of a colliding particle, or the product of several such wavefunctions, and the phase space covered previously by a classical ensemble is now covered by one or several wavefunctions. The quantum mechanical equivalent, Gq, is not constrained to be less than or equal to zero, and a measured positive value would show the deficiency of a classical description. Figure 2.1↓ illustrates an example of the measurement of G for a classical and a quantum distribution of particles.
figure MugDemo_Ens_Top2_Mod.png
(a) An ensemble of classical particles before (black) and during (red) a collision with a repulsive potential. Inset is the a representation of the ensemble in phase space, where it is shown that the particles begin with a positive (moving to the right) momentum. G is found in this case by summing the part of both distibutions above the black dashed line, and subtracting the value after the collision from the value before the collision. If energy is conserved, momenta must decrease as they climb the potential and G must be less than or equal to zero.
figure MugaDemo_Quantum.png
(b) In the quantum picture, the ensemble of particles occupies a single wavefunction whose Wigner distributions before and during the collision are shown. G in this case is found by integrating above the solid black line, and taking the difference. A broadening of the momentum distribution during the collison allows for a positive value of G.
Figure 2.1 Cartoon illustrating the measurement of G in a classical (a) and quantum (b) picture.
This effect was first considered in collisions with an infinite wall, where it can be argued quite intuitively that a colliding wave packet will acquire sharp spatial features and thus, correspondingly broad features in its momentum distribution. A later paper 30 showed that the values of G > 0 could in fact be maximized through collision with a weak and narrow, almost entirely transmissive potential. This weak potential would be high enough that a wavefunction passing over it would accrue only a π phase shift. Halfway through the collision, only half of the wavefunction has received this phase shift, and the phase of the cloud’s wavefunction contains a step function in its spatial coordinates. The result is similar to what might be achieved with a phase mask. The resulting momentum distribution therefore falls off as 1 ⁄ k due to the spatial step function, and the probability of high momenta is increased. Different collision parameters were studied, and a maximum value of Gmax = 0.37 was reported, where an infinitely thin barrier was considered. Figure 2.2↓ shows the momentum distribution of a Gaussian cloud before, during and after such a collision. The inset shows the value of G given by Equation 2.2↑ as a function of the integration limit, p.
As was implied above, the appearance of enhanced high momentum components is transient, and occurs only during the time that the cloud is interacting with the potential. If, for example, the cloud is studied long after the collision, the described π phase shift is written across the entire cloud, and the momentum distribution is unchanged with respect to its pre-collision state. Scattering theory is normally concerned with scattering particles well before and well after the influence of the interaction is felt; the so-called asymptotic regime. In order to observe enhanced momentum, however, this experiment will depart from this paradigm and endeavour to measure the momentum spectrum during the collision, thus studying scattering in the non-asymptotic regime. A collision will be staged, and the potential removed effectively instantaneously 54 during the collision, thus ending the collisional redistribution of momentum, and enabling a subsequent measurement of momentum through time-of-flight.
By measuring the momentum distribution of a colliding Bose-Einstein condensate and demonstrating a positive value of Gq, it is possible to show that atoms in such a state defy classical description, and are indeed quantum mechanical objects.
figure Momentum Enhancement Model/MugaSample.png
Figure 2.2 Momentum distributions of a wavepacket before and during a collision with a weak potential. The inset shows a plot of G (Eq. 2.2↑) calculated as a function of integral limit.

2.1.1 Nonlinear interaction

A Bose-Einstein condensate is not simply a transform-limited Gaussian, but rather a many-body state with non-negligible interaction energy. Treating the mean-field energy as a repulsive potential, I calculate the associated momentum using Equation and recalculate the inequality, even though Vmin is usually associated with the colliding potential. The mean-field potential of a condensate in the Thomas-Fermi limit is a paraboloid with a maximum at the centre, which is equal to the chemical potential
(2.3) Umax = μ = 12ω(15Namω)
The chemical potential represents the energy required to add an atom to the cloud while trapped, or the energy to place an atom at the centre of the untrapped cloud. It therefore represents the maximum energy a single atom could obtain from the pseudopotential. In the Thomas-Fermi approximation, the energy per atom in a harmonic trap is 57μ, and the interaction energy per atom is 27μ 28. Figure 2.3↓ shows calculations of the value of Gmax expected for a Gaussian cloud in an anisotropic harmonic trap, where f = 125.5  Hz. Corrections for the mean-field pseudopotential are made by setting the value pν from Equation to correspond to the interaction energy in the condensate. One curve is shown where Vmin = μ, where μ is the chemical potential, and another corrects by the mean interaction energy, and Vmin = 27μ. While significant violation may still be achieved when subtracting the interaction energy, removing the interactions from the problem allows for a more elegant result, dispensing with the need for the theoretical overhead of modelling nonlinear dynamics. The expansion of the cloud before the collision in order to reduce density and therefore interaction energy will be discussed in Section 2.1.3↓.
figure Momentum Enhancement Model/NonlinearCorrection2.png
Figure 2.3 The mitigation of transitory enhancement of momentum through consideration of mean field pseudopotential as a classical potential. The term pν defined in Equation represents the momentum that may be gained by interaction with an attractive potential, and is zero in the case of a repulsive potential. Here I let this term represent the momentum that may be gained from the repulsive mean field pseudo-potential of a Bose-Einstein condensate. Momentum distributions are calculated using a linear model, and the Gmax is calculated according to Equation 2.2↑, substituting the nonlinear pseudopotential for Vmin of Equation . The lower curve uses Vmin = μ, where μ is the chemical potential, and the middle curve Vmin = 27μ where 27μ is the mean interaction energy per particle.

2.1.2 Finite Barrier Size

The repulsive potential is a focused laser beam, whose size is limited by the numerical aperture of the optical system used to produce it. The phase mask written will therefore not be an infinitely sharp step, but rather an integrated Gaussian.
(2.4) φ(x) ∝ erf(x ⁄ 2σB)
where φ(x) is the phase written on the spatial wavefunction, σB is the spatial rms width of the barrier, and erf is the error function. It is intuitively clear that to add momentum to the system, the spatial width of the barrier, must be small compared to the spatial width of the colliding wavepacket. Unlike the restriction imposed by the nonlinear interaction, a too-large barrier may suppress the effect entirely. Figure 2.4↓ shows the effect of a widening barrier on the momentum enhancement. The effect is almost completely extinguished when the barrier is half the size of the cloud.
figure Momentum Enhancement Model/BarrierWidths01.png
Figure 2.4 Momentum enhancement Gmax as function of barrier width, which is measured in rms widths of the cloud. Momentum enhancement is almost completely extinguished when the barrier is half the size of the cloud.

2.1.3 Chirped Wavepacket

In order to decrease the nonlinear interaction energy of the colliding wavepacket, the wavepacket can be allowed to undergo ballistic expansion before the collision, thus lowering the density. This disambiguates any observed higher momenta, and it will be shown in this section that it also has the effect of mitigating the effects of finite potential widths. An unexpected result of allowing this expansion is the appearance of “ripples” in the momentum spectrum. This effect is discussed in detail in Section 2.2↓.
Figure 2.5↓ shows the effect of ballistic expansion prior to the collision on the maximum value of G. Different curves correspond to different amounts of free expansion before the collision, and the upper curve is identical to that shown in Figure 2.4↑. In the case of a narrow barrier, momentum enhancement is reduced with increasing expansion before the collision. This is simply because the barrier interacts with a smaller fraction of the wavefunction. The benefit, however, is that the momentum enhancement falls off more slowly with increasing barrier widths for longer pre-collision expansions. As the barrier gets wider, the optimal pre-collision expansion time tends to increase.
At large barriers and long expansion times, however, momentum enhancement begins to rise as a function of barrier width. This surprising discontinuity in the slope of Gmax is a result of a change between the maximum G occurring at the front of the wavepacket, and being due mostly to the 1 ⁄ k tails acquired in the collision, to the maximum G occurring towards the back of the wavepacket, and being due to interference.
Figure 2.6↓ shows values of G extracted from clouds which have expanded before collision, and the site of barrier turn-off has been moved to one rms width before the centre of the cloud (meaning the barrier will have crossed less than half the cloud). The net effect is to reduce momentum enhancement due to tails since the phase discontinuity is placed in a region of lower amplitude, whereas momentum enhancement from interference has been enhanced, since the relevant interference occurs in a higher amplitude portion of the wavepacket. This shows that it is a qualitatively different effect. Figure 2.7↓ depicts a case where broadening the barrier increases the maximum value of G through the transitioning to this new effect.
figure Momentum Enhancement Model/BarrierWidths02.png
Figure 2.5 Momentum enhancement, Gmax, as a function of barrier width for different amounts of pre-collision expansion. Expansion is measured in time units of τ, which is the period of trap oscillation.
figure Momentum Enhancement Model/BarrierWidths03.png
Figure 2.6 Momentum enhancement, Gmax, as a function of barrier width for different amounts of pre-collision expansion. Expansion is measured in time units of τ, which is the period of trap oscillation. The collision is stopped when the barrier is within one rms width from the cloud centre. Moving the point of impact reduces momentum enhancement from the tails, but increases momentum enhancement due to interference. There is a net increase in momentum enhancement for broader barriers after longer expansion times.
figure Momentum Enhancement Model/BimodalFigure01.png
Figure 2.7 The transition between the maximum value of G being a product of the long tail, to being a product of the central dip. Barrier widths are measured in unexpanded cloud widths. The left-side panels show model predictions for the density profiles of clouds having undergone a collision (blue) and not having undergone a collision (green). The right-side panels plot the value of G as a function of integration limit. The point where maximal G is obtained is indicated by a green dot.

2.1.4 Discussion and Experiment

Momentum enhancement was initially considered to be a result of the 1 ⁄ k falloff in momentum space produced by the sharp phase step left in the condensate wavefunction. The analysis above predicts a second mechanism by which positive values of G are generated. This second mechanism is due strictly to matterwave interference and as such only appears in a cloud with a chirp.
It is assumed that application of this analysis to experiment will be done in a regime where mean field energy is low enough to be neglected at the time of collision. Parameters measured in cloud sizes will not reflect the size of the trapped cloud, but rather the effective size determined by the momentum distribution after the nonlinear expansion.
Figure 2.8↓ plots the maximum value of Gmax in a typical experimental setup. The cloud expands at a rate which corresponds to 30  nK, total time-of-flight expansion is limited to 30  ms, and measurements are done in position, not momentum space. Measuring in position space at a finite time after the collision further reduces the apparent violation, as enhanced momentum components must travel from the centre of the cloud.
figure Momentum Enhancement Model/ActualParams.png
Figure 2.8 Maximum observable momentum enhancement versus pre-collision expansion under typical experimental conditions. Solid lines indicate results for a π phase shift, and dotted lines indicate results for a 3π phase shift. The barrier used in the experiment will have a width of σB = 2.4  μm (Section 5.2.1↓), and a maximal expansion time of 30  ms is assumed. A second curve has been plotted for expected values using a σB = 0.76  μm barrier, as this may better reflect the dynamics of the measured system (see Section 5.3.1↓, for example). In either case the effect is around 1%, and will be a challenging measurement. The thin lines bracketing the curves give a sense of the tolerance of initial momentum uncertainty, and indicate the resulting measurement in the case where the scattered cloud has been shifted by ±0.1  μm.

2.2 Wavefunction Measurement

The phase recovery problem is well known in situations where physical measurements are made of wave phenomena. The Bose-Einstein condensate is an excellent tool for measuring coherent phenomena, but it continues to be a difficult and resource-consuming process to gain complete information about the condensate wavefunction. Existing methods 4,26,25,51 demonstrate or propose tomographic techniques which assemble multiple measurements to reconstruct both phase and amplitude information. Herein is proposed a technique that in some cases yields near-complete information about the phase and amplitude of a condensate in a single measurement.

2.2.1 Analytic Model

Momentum-space interferometry has been suggested in 3, using the collision-modified momentum distribution to locate the scattering potential. In this work I extend these ideas by considering a cloud with a non-uniform spatial phase (e.g. quadratic), and show that the resulting interferogram may be simply interpreted to recover the spatial phase profile of the cloud before the collision. The broader momentum distribution created by the collision can overlap and interfere in momentum space with any pre-existing momentum components that have travelled away from the collision site before the collision, and are assumed to be unaffected by the collision.
To predict the result of such an experiment, I follow the approach in 3, and assume the initial state to be a Gaussian cloud. I further assume that the cloud has a simple quadratic profile in phase, as would be acquired during ballistic expansion
(2.5) ψ(y) = exp( − y2)exp(iβy2)
where y is the spatial coordinate and β is a variable chosen to parametrize the quadratic phase of the cloud. The wavefunction in spatial coordinates just after the collision can be approximated as:
(2.6) ψ(y) = exp( − y2)exp(iβy2)exp(iφΘ(y))
where φ is the phase accumulated by the wavefunction during its transit across the barrier, and Θ is the Heaviside step function. To find the momentum distribution, I take the Fourier transform:
(2.7) ψ(k) = 12π − ∞ψ(y)exp( − iky)dy
Considering first the integral over positive y,
0exp( − y2 + iβy2 + iφ + iky)dy = 
(2.8) limy → ∞ − πσexp( − k2σ2)exp(iφ)[erf(ikσ − yσ2) +  erf(ikσ)]
where σ2 = 4 − 4iβ. This limit converges in the range where |(σ)| ≥ |(σ)|, which is true for any real value of β.
(2.9)  = πσexp( − k2σ2)exp(iφ)[1 −  erf(ikσ)]
and adding the integral over negative values
ψ(k) = πσexp( − k2σ2)[1 +  erf(ikσ) + exp(iφ) − exp(iφ)erf(ikσ)]
(2.10)  = 2πσexp( − k2σ2)[sinφ2 erf(ikσ) + cosφ2]

2.2.1.1 Interpretation

Figure 2.9↓ plots the |ψ(k)|2 distribution for various values of β and φ. A striking result of adding a quadratic phase profile to the cloud through free expansion is to add oscillations in the amplitude of the resulting momentum distribution. These oscillations may be interpreted as the interference within the wave packet of momentum components existing in the expanding cloud, with momentum components created during the collision with the barrier. This becomes intuitively clear if one considers that only a small subset of momenta in the cloud are affected by the collision, as the effect of the expansion is to spatially separate parts of the wavepacket with different momenta.
figure MugaPaper/WhyNotTryAgain/Fig2/analyticsample.png
Figure 2.9 Momentum distributions predicted by Equation 2.10↑. β and φ represent the chirp of the initial wavefunction and the phase shift applied by the barrier respectively.
To demonstrate this, I consider the case where φ = π, so Equation 2.10↑ becomes
(2.11) ψ(k) = 2πσexp( − k2σ2) erf(ikσ)
Clearly the oscillations are contained in the the term erf(ikσ), though their positions will be shifted by the envelope term. Experimentally, these corrections can be made quite simply by measuring a cloud that does not undergo collision, and dividing the data with oscillations. Not undergoing a collision is equivalent to setting φ = 0 in Equation 2.10↑, and dividing leaves only the oscillating term. The local oscillation frequency can be determined by taking the derivative with respect to k:
(2.12) ddkerf(ikσ) = iπ1 − iβexp(14k21 − iβ)
(2.13)  = iπ1 − iβexp((1411 − β2 + i14β1 − β2)k2)
and the oscillating term in the fringe pattern is thus
(2.14) exp(i14βk2)
in the case where β2≫1. For a scaling estimate, I estimate the curvature of the wavefunction phase in position space. Neglecting the finite size of the unexpanded cloud, the momentum, and thus the phase as a function of position can be written:
(2.15) x = vt = pmt = kmt
(2.16) k(x) = mtx
(2.17) φ(x) = 2k2(x)2mt = m2ℏtx2 = αx2
where α is defined as m2ℏt. The corresponding curvature in momentum space is
(2.18) φ(k) = t2mk2 = 14αk2
Comparing Equation 2.17↑ to Equation 2.5↑, it is evident that α = β and the oscillation in the interferogram correctly represents the spatial phase curvature at the time of collision. This validates the picture of created momenta interfering with otherwise unperturbed, existing momenta.

2.2.2 Single Shot Measurement

I have shown in Section 2.2.1.1↑ that in the case of a chirped Gaussian cloud, interference fringes produced through momentum enhancement during a collision provide an accurate phase measurement of the momentum spectrum of the colliding cloud. In this section I study a simple numeric method to extract this phase information from interferograms produced.
Figure 2.10↓ shows a measurement of the simple quadratic phase due to the ballistic expansion of a Gaussian cloud. The phase is extracted by locating the extrema in the interferogram and assigning each an increment of π. Panel (a) shows a simulated, one-dimensional profile of two clouds, one of which has undergone a collision, and one of which has not. Circles indicate the positions of the located extrema. Panel (b) plots these extrema, and fits their locations to a square root function with its sign flipped about a fitted central point. The square root function shows the correct scaling of phase as a function of position, but there is a phase shift between the two sides that is not a multiple of π. This phase shift is a result of the integral nature of the error function, and there is no analytical expression for its value which in general varies with the dispersion parameter β. It is also important to note that the central dip is not part of the interference pattern. The square root fit is thus used to locate the point of symmetry, and the phase values accorded to these peaks are flipped in sign on one side. The axes are then swapped and the result is the plot in panel (c). In this plot the phase shift between the two sides of the interferogram results in a vertical shift in the plot and does not affect the measured curvature value.
The resulting profile reflects the phase of the momentum-space wavefunction at the time of collision. To extract from this a single-shot measurement of the entire wavefunction, the amplitude of the momentum distribution must be estimated from the envelope of the interferogram. It is easier, however, to directly measure the amplitude of the momentum distribution by performing a time-of-flight measurement of a second cloud that does not undergo a collision, and this analysis will focus on this method. The full wavefunction can thus be assembled, and the position-space wavefunction of the cloud at the time of collision may be determined by an inverse Fourier transform.
In principle, this technique need not be limited to simple quadratic phase curvatures, and may be used to measure arbitrary wavefunctions. To demonstrate this I consider, as in 31, the focussing of an atom cloud with a parabolic magnetic field, where parasitic higher orders may be present. Figure 2.11↓ shows an interferogram created where the clouds have been subjected to cubic perturbations in their phase, and the recovery of this higher-order term using the peak-finding algorithm. This perturbation has been exaggerated for display reasons, and the algorithm generally performs better where the perturbation is small and the envelope is less distorted.
figure MugaPaper/WhyNotTryAgain/Fig34/Extrema_2.png
Figure 2.10 (a) The simulated momentum distribution of a BEC after 5  ms of free expansion, and a collision with a repulsive potential. Panel (b) shows the numbered extrema as a function of cloud position, and fitted to a square root function with a sign flip at a fitted central point. Panel (c) shows the same plot with each extremum assigned a phase of π radians, and flipped about the zero-point on one side.
figure MugaPaper/WhyNotTryAgain/Fig34/Extrema_cd.png
Figure 2.11 A cubic phase shift of 7.20 × 10 − 5 m has been added to the cloud just before the collision, and the algorithm recovers a phase shift of 7.18±0.56 × 10 − 5 m. Panel (a) shows the interferogram, and panel (b) shows the fitting of the peak positions to a cubic polynomial. The phase obtained in panel (b) and the envelope from the uncollided cloud in panel (a) have been combined to form the momentum-space wavefunction, which has been Fourier transformed to give the spatial wavefunction at the time of collision, shown in panel (c). The cloud in panel (a) is shown with an exaggerated shift for illustrative purposes.

2.2.2.1 Discussion

The above demonstrates a technique for the reconstruction of a condensate wavefunction using two different measurements of the momentum distribution - one measurement of an unperturbed cloud, and the other a momentum-enhanced interferogram. In this way the “envelope” term present in Equation 2.10↑ can be explicitly removed, and the oscillations, and thus the phase revealed. One question that remains unanswered is under which conditions the envelope must be measured separately, and when a full amplitude and phase measurement can be performed in “a single shot.” Without a separate amplitude measurement, the amplitude must be inferred from the interferogram, whose amplitude now represents the interference between the wavefunction to be measured and the enhanced momentum wavefunction. Furthermore, the phase of the interference fringes is itself shifted in a way that depends on the slope of the envelope, and correction relies on a good estimation of the local amplitude.
Both of these problems are diminished as the pre-collision expansion time is increased, and correspondingly, the size of the cloud compared to its original size. The greater this expansion, the better the information obtained in a single shot. As I have argued in Section 2.1.3↑, allowing greater expansion of the wavepacket before the collision means that a smaller fraction of the wavepacket is perturbed to produce enhanced momentum. This has the benefit of cutting a relatively smaller “notch” into the wavefunction, and also of producing a smaller amount of enhanced momentum to add to and interfere with the remainder of the packet. This means that an amplitude estimated by simply averaging over the fringes will approach the correct value, and it is in this limit that a wavefunction can be well-measured in a single shot. The drawback, of course, is the decreasing contrast of the increasingly rapid interference fringes, and this must be balanced with experimental noise limitations.
Finally, I note that this technique fits well into the framework of tomographic techniques such as FROG, where the “gate” is now supplied by the barrier, rather than a copy of the pulse, and the position and energy of the barrier can be varied in lieu of pulse delay. In cases where optimal experimental conditions are difficult to achieve, multiple measurements may be assembled to recover wavefunctions. Though not presented here, I have used these techniques to reconstruct wavefunctions using both sample data and the data presented in Chapter 5↓.

2.2.3 Position Measurement

A momentum-space interferogram with rapid fringes may be expected to perform well as a position-measuring tool, as positional shifts in the colliding potential (or equivalently the atom cloud) translate directly to phase shifts in the interferogram. In 3 the effect of a non-central phase shift was first considered, and I will extend this analysis to explicitly evaluate this as a measurement technique, and consider how dispersion affects this measurement. I consider a hypothetical measurement where a cloud is released from its trapping potential and collides with a potential at some time, after which the momentum distribution is measured. The potential is then displaced some amount and the experiment repeated. I consider a comparison made in the least squares sense:
(2.19) Φ = 1Nk[|ψa(k)|2 − |ψb(k)|2]2
where |ψa(k)|2 and |ψb(k)|2 are the interferograms produced by potentials at x = a and x = b respectively, and N is the number of datapoints in the trace. I consider two possibilities. It may be desired to know whether the displacement has taken place, in which case it is desirable to maximize Φ. It may also be desired to know the amount of displacement, in which case the quantity to be maximized is dΦ ⁄ dδ, where δ is the amount of movement of the potential.
figure Images/sample9ms.png
Figure 2.12 Momentum distributions of two clouds having interacted with the potential after 9  ms of free expansion. The potential was centered for one cloud and shifted 4.0  μm for the other, producing a maximal Φ for this drop time.
figure MugaPaper/WhyNotTryAgain/PositionMeasurement/chisquareds.png
figure MugaPaper/WhyNotTryAgain/PositionMeasurement/dchisquareds.png
Figure 2.13 Panel (a) shows Φ as a function of the position δ of the potential for different expansion times. The inset shows which expansion time optimizes this value for different δ. Panel (b) shows the same but for dΦdδ.
Figure 2.12↑ shows an example of a measurement of a 4  μm displacement. In this case the optimal Φ is generated by a 9  ms expansion of the cloud before the collision. It is clear why this expansion is optimal: the frequency of the interferogram is such that the central notch is replaced by the largest peak when the potential is shifted, providing minimal overlap. To generalize this result, the displacement of the potential should be matched to the wavefunction phase profile such that the phase difference between the two points is fixed.
Because the spatial phase of an expanding cloud is proportional to x2 ⁄ t, two points maintaining a constant phase separation should be expected to maintain a relationship such that (δx)2 ∝ t. Alternatively
(2.20) texpansion ∝ δx2Δφ
where Δφ is the phase difference to maintain between interferograms, and texpansion is the time the cloud is allowed to expand before the collision. Figure 2.13↑ shows plots of Φ (2.13↑a) and dΦdδ (2.13↑b) as a function of potential displacement for different expansion times. The optimal expansion times are summarized in the insets, and show this quadratic relationship. The inset in Figure 2.13↑(b) also shows a discontinuity at 4  μm, which corresponds to a change in the optimal phase shift Δφ to maintain. When the cloud has sufficient amplitude in the central few fringes, a simple π phase shift is optimal to maximize Φ. With more expansion however, the chirp in the fringes makes this less effective until it becomes preferable to use a larger phase shift and return to shorter expansions.
In measuring displacements which are comparable to the cloud size, the dynamics are different, and are treated to some extent in 3. For a transform-limited wavefunction, translating the potential does not move the central notch and Φ is better maximized by allowing some expansion before the collision, so the notch moves slightly with the potential. The maximization of |dΦ ⁄ dx| shows a similar result for very small displacements, but for displacements close to the cloud size, there is a region where short expansions are preferred. This corresponds to the case where the displaced potential sits on the part of the wavefunction with the highest derivative, and small displacements result in a large change in notch depth.

2.2.4 Experimental Limitations

The measurement as considered so far makes use of an infinitely narrow phase step, created by an infinitely thin barrier. Furthermore, the phase is extracted through study of the momentum distribution. In experiment, the width of the phase step will be limited by the numerical aperture of the optical system used to produce the barrier, and I propose, for simplicity, to measure the momentum distribution through a simple time-of-flight study, which only approximates a momentum measurement. The following subsections will deal with the effects of finite barrier thickness and corrections that must be made to account for a finite time of flight before the cloud is imaged.

2.2.4.1 Finite Barrier Thickness

In the laboratory, it is impossible to apply a step-function phase mask as is assumed in the above sections. In practice, the shape of the phase mask applied by a diffraction-limited optical barrier transiting across some portion of the cloud will be of the form
(2.21) φ(y)\varproptoφ2 erf(y2σB)
where σB is the rms radius of the focused spot, and will typically be the same order of magnitude of the cloud size, and φ is the magnitude of the phase written by the optical field. Figure 2.14↓ shows the effect of increasing the barrier size. Two effects are immediately clear. First, the interferogram loses its previously symmetrical form. With increasing spot size, the error function (erf) phase profile written on the wavefunction tends towards a linear gradient in its centre, effectively ‘‘pushing’’ the atoms to one side. Secondly, the visibility of the fringes is reduced, and eventually disappears. This makes it increasingly difficult to extract phase information as the barrier width increases. This does not in principle affect the value of phase curvature that is extracted, but will affect the extracted value by effectively changing the envelope function.
To improve the situation, the barrier may be turned off earlier in its transit across the cloud. This works because of the aforementioned asymmetry - the transit of the barrier reduces the visibility more on the side of the cloud that it has interacted with than on the side that it has not encountered. Figure 2.15↓ shows some resultant interferograms for different turn-off points, and counts the number of extrema located to make phase reconstructions. For an extremum to be considered useable, the contrast must exceed a level defined by the experimental signal to noise ratio. Collisions terminated early produce nearly double the number of extrema as those terminated later.
figure MugaPaper/WhyNotTryAgain/Fig5/figure5.png
Figure 2.14 The effect of increasing barrier width on the interferogram. A barrier of finite width has the effect of reducing fringe visibility, and also introducing asymmetry. Despite creating difficulty in the extraction, a broader barrier is still capable of extracting the correct phase curvature. Simulations plotted are for a cloud with 0.29  μm size in the trap, after 5  ms of free expansion, and circles represent peaks as located by the peak-finding algorithm.
figure MugaPaper/WhyNotTryAgain/Fig6/figure6.png
Figure 2.15 Useable extrema as a function of barrier thickness for impact points 10  μm right and left of the centre of the cloud, as well as in the centre. (a) shows the number of extrema above an arbitrary noise threshold in each interferogram for the three different impact points, as a function of barrier thickness. A collision at the front of the packet produces nearly double the number of fringes for barrier thicknesses up to nearly 1 μm. Panels (b), (c) and (d) show the interferograms for a barrier width of 0.3  μm, with located extrema indicated by circles.

2.2.4.2 The Far Field

So far all measurements have been considered in momentum space. The simplicity of this method, however, hinges on a simple, single shot measurement of the momentum wavefunction through a time of flight expansion. This is strictly a measurement of momentum only in the limit of infinite expansion time. My approach will be to begin from the approximation that a sufficiently expanded atom cloud will reflect a true momentum distribution, and then find a correction factor. Beginning from such a perfectly correlated cloud, the curvature measured using the technique of Section 2.2.2↑ will decrease in time as the cloud expands, according to
(2.22) βextracted ∝ 1t2
that is, as the cloud expands linearly, the curvature decreases quadratically. To extract the curvature at the time of collision, one therefore need only adjust the measured curvature by the square of the ratio of the time of collision to the time of measurement. Figure 2.16↓ shows the further correction that needs to be made due to the imperfect correlation of position and momentum. The correction is greater for clouds which are allowed to expand more before colliding, and the correction also declines at a slower rate. This behaviour can be intuitively understood as created momentum components asymptotically ‘‘catching up’’ with their pre-existing counterparts.
figure MugaPaper/WhyNotTryAgain/Fig8/Expansion_Correction.png
Figure 2.16 Inferred phase curvature as a function of post-collision expansion time, corrected for "actual" value input into simulation. Curves are shown for pre-collision expansion times of 3, 6, and 13  ms. Inferred values are obtained by multiplying measured spatial curvature by the square of the ratio of cloud sizes at the time of imaging and the time of collision.

3 Apparatus and Technique

In this chapter the experimental apparatus developed in the course of this work is discussed in detail, with particular emphasis on systems and components that have been redesigned from the previous iteration of the experiment. Background theory for certain aspects is also discussed.

3.1 Vacuum System

With transition temperatures in the hundreds of nanokelvin range, a Bose-Einstein condensation experiment using alkali vapours must enjoy extraordinary thermal isolation from its room-temperature environment. This is effected in the lab by ultra-high vacuum technology and strong magnetic fields which trap the atoms away from the chamber walls. The design of the vacuum system is detailed in 5 and 6, and an outline as well as a description of major modifications will be given here.
The rate of loss of atoms from a magnetic trap due to background gas collisions is a key determinant of evaporation efficiency, and ultimately the size of a condensate achievable with a given set of initial conditions. This is discussed in further detail in Section 3.3.6↓. Background gas collision rates are directly related to the pressure in the chamber, and as strong a vacuum as possible is therefore desired for evaporation to BEC. This is at odds, however, with the need to collect atoms from a thermal vapour into a magneto-optical trap. Typical solutions to this problem are the use of a dual MOT system where atoms are collected in a high-pressure region and then transferred to a low-pressure region, the use of dispensers where the vapour pressure can be dynamically adjusted by controlling the electrical current to the rubidium source, or the use of a Zeeman slower, which slows a thermal beam of atoms and uses this to load a trap in the low-pressure region. This experiment makes use of the dual MOT system, using two separate vacuum chambers coupled by a narrow tube and isolated with a pneumatically actuated gate valve.
A full schematic of the vacuum system is presented in Figure 3.1↓. The upper chamber, so named because it is farther from the table than the lower chamber, is the high-pressure chamber and contains a chunk of metallic rubidium, containing both the ^85Rb and ^87Rb isotopes in their natural ratio of 3:1. The pressure in this chamber is maintained at around 10 − 9 Torr by a Varian Turbo V-70LP pump whose nominal pumping speed is 70  l/s, and which is backed by a mechanical roughing pump. The metallic rubidium is heated to about 35  ^○C, close to its melting point, where the vapour pressure is approximately 10 − 6 Torr 8. The chamber has six quartz windows through which 1" MOT beams pass, a larger viewport to image the MOT, and an eighth window on the top of the chamber through which a “push” beam passes. The push beam drives atoms through a hole on the opposing side of the chamber which leads into the 5 inch long transfer tube mentioned above, through a Kurt J. Lesker SG0063-PCCF pneumatic gate valve and into the lower chamber. The SG0075-PCCF provides a UHV-quality seal and has an expected lifetime of 105 cycles.
The lower chamber is maintained at a pressure below 10 − 11 Torr, which is the lower limit of what may be reliably measured with the installed nude ionization gauge. This pressure is maintained by Varian VacIon Plus 150 Combination Pump, which combines a large ion pump with a titanium sublimation pump to achieve pumping speeds of over 500  l/s. In addition to the flange for the combination pump, there are connections on the vacuum chamber for a nude ionization gauge, the glass cuvette, a residual gas analyzer (RGA) and an unused (capped) 6” CFF flange.
With pressures below those that can be measured with standard ionization gauges, the quality of the vacuum is best assessed by measuring what is actually its primary figure of merit, the trap lifetime. Figure 3.2↓ shows a measurement of the trap lifetime, performed by measuring the number of atoms retained in a magnetic trap as a function of hold time, and fitting to an exponential decay. The lifetime is 153  s, which is consistent with values before the reconstruction, and suitable for evaporation to BEC.
figure Images/Vacuum System/VacSys5.png
Figure 3.1 Vacuum system diagram. (1) VacIon 150 combination ion / titanium sublimation pump (2) Added 11” Nipple Extension (3) Residual Gas Analyzer (RGA) (4) Pneumatic Gate Valve (5) Cuvette (6) Nude Ionization Gauge (not pictured, opposite RGA) (7) Upper Chamber: Includes 6 2.75” ports for MOT beams along 3 orthogonal axes, vertical 2.75” port for push beam, providing line of sight to lower chamber, and 4.5” camera viewport (8) 4-way cross opposite large viewport (not pictured); cross holds rubidium source, thermocouple gauges, turbo pump and roughing line.
figure Images/Vacuum System/LifetimePlot.png
Figure 3.2 Determination of magnetic trap lifetime. Atoms are held in a 70.5  G/cm quadrupole gradient and the atom number is plotted as a function of time, and fitted to an exponential function. The fit indicates a trap lifetime of 153±43  s. Error bars represent uncertainty in number measurement, which is high for large clouds, but still small compared to shot-to-shot fluctuation.

3.1.1 Modifications

After the failure of the gate valve in November 2006, the vacuum system was opened for its replacement. An excellent resource used in this rebuilding is 40. The major modification of the system was to add an 11” nipple extension between the vacuum chamber and the combination pump. This had the benefit of moving the large magnets of the ion pump much farther from the magnetic trap, as well as freeing space on the optical table for additional optics, and in particular the modular optical system which will be described in Section 3.2.3.1↓. Because the vacuum chamber had been mounted quite close to the table edge, a clamp was constructed so the pump could hang over the edge of the table. This construction is pictured in Figure 3.3↓, and also in Figure 3.1↑. A clamp was used to primarily to avoid sustained upward pulling forces on the threaded steel sheet of the optical table, as upward force has been known to spearate the sheet from the honeycomb layer below and cause “blistering” of the steel 53.
figure Images/clamp.png
Figure 3.3 Clamp to hang ion pump from side of table. Steel bars are welded into two “U” shapes and clamped to the table with the aid of four 1/4” threaded steel rods providing both tension (inner rods) and compression (outer rods). Holes for mounting are drilled into the steel bars and secure posts atop which sits a platform for the VacIon 150.

3.2 Optical System

The laser system in the first generation experiment consisted a chain of injection locked, home-built diodes, each with a maximum power of around 70  mW. Beams travelled nearly 2  m down the length of the table, and minor fluctuations in alignment were thus amplified to require frequent realignment of the lower MOT and also of the various injection locks. To improve on this design, the home-built diodes were replaced by a tapered amplifier, and long beam paths were replaced by optical fibre. This dramatically shortens the propagation lengths over which misalignments can occur, and alignment drift in the beam preparation appears only as a power fluctuation - a single parameter which can be measured and optimized. The fibre-bound transmission of the light has also made it possible to erect a large physical barrier on the table between elements generating and preparing the light, and the cuvette.

3.2.1 Background: Laser Cooling and Trapping

The first job of the optical system is to effect laser cooling. Of the nine orders of magnitude in temperature the entire cooling process needs to span, the first six are achieved by optical means. The magneto-optical trap has become a workhorse in ultracold matter, both cooling and trapping atoms from a thermal vapour. In the following sections I will outline some of the basic ideas of laser cooling and trapping. For a more comprehensive treatment, refer to 23.

3.2.1.1 The Doppler Process and the MOT

The Doppler effect is a well known phenomenon in which the apparent frequency of an emitter depends on the relative motion of the observer. As an observer moves towards the emitter, wave crests arrive more frequently, and the frequency appears higher. Conversely, an observer moving away observes a lower frequency. Because a radiation field can exert pressure on a neutral atom via scattering, and because this scattering occurs only over a narrow range of frequencies, the Doppler effect may be used to exert a velocity-dependent force which can then be used to cool a sample of atoms. To understand this process, one should first quantify the force applied by an optical field on an atom. Viewing a scattering event as an absorption event followed by a spontaneous emission event, one may deduce the average momentum shift incurred during such an event. During absorption, the atom simply absorbs the momentum of the photon, k, where k is the wavenumber of the atom. Spontaneous emission occurs in a random direction, and averaged over many events its contribution to the momentum of the atom is zero. The force on the atom can thus be expressed as 23:
(3.1) Fsp = ks0Γ ⁄ 21 + s0 + (2δ ⁄ Γ)2
where Γ is the natural linewidth of the atom, δ is the detuning of the optical field from resonance, and s0 = I ⁄ Isat is the saturation parameter which defines the intensity of the beam. This force is known as the radiation pressure force. The Doppler effect thus alters the force experienced by the atom by acting on δ, which in the laboratory frame will depend on the velocity of the atom. A laser beam can thus be used to produce a velocity-dependent force in one direction, and the addition of beams along different axes can be thus employed to slow fast-moving atoms and generate cooling. In particular, pairs of counter-propagating beams along three orthogonal axes can be used the produce three-dimensional cooling. The degree of cooling which can actually be achieved will ultimately be limited by spontaneous emission. Because every scattering event includes a recoil from this process in a random direction, it acts as a heating mechanism. The temperature at which this heating exactly balances the cooling is known as the Doppler limit. The Doppler temperature is TD = ℏΓ ⁄ 2kB where kB is the Boltzmann constant. The Doppler temperature for ^87Rb is approximately 140  μK.
Despite being able to achieve very low temperatures, simply counter-propagating laser beams provide no confinement, and cold atoms simply diffuse out of the cooling region. Condensation requires much lower temperatures, and much higher densities than can be achieved in this so-called Doppler molasses. The atoms can be confined and collected with the addition of a magnetic field gradient. The combination of the Doppler molasses and a magnetic field gradient is known as the magneto-optical trap (MOT), which has become ubiquitous in Bose-Einstein condensation experiments. The velocity dependence of the detuning δ in Equation 3.1↑ creates a velocity-dependent force to effect cooling, and similarly, the magnetic field gradient creates a spatially dependent force to effect confinement. To understand this, consider an atom with a F = 0 ground state and a F’ = 1 excited state in a magnetic field gradient. The energy of the excited state, and therefore the detuning of the optical field will be described by the Zeeman effect:
(3.2) E = mFgμBB
where mF is the magnetic quantum number of the atom, g is the Land g-factor, μB is the Bohr magneton and B is the magnitude of the magnetic field. The magnetic field employed in a MOT is typically generated by a pair of anti-helmholtz coils, producing a linear gradient in three dimensions:
(3.3) B = B(x^x + yŷ − 2z)
To understand how this field gradient produces a centering, restoring force, one must consider the internal structure of the atoms to be cooled. The hypothetical F’ = 1 atom has three magnetic sublevels (mF =  − 1, 0, 1), each of which experiences a different Zeeman shift according to Equation 3.2↑. A position-dependent force is generated if the counter-propagating cooling beams are polarized such that each couples the atom to a different one of these excited states, and thus has a different detuning and produces a different scattering force. This is effected by circularly polarized beams, which since counter-propagating, will be seen by the atoms to have opposite helicity. As an atom moves away from the centre of the trap, the local field value increases, bringing it closer to resonance with one of the beams. The detuning achieved by its motion assures that it scatters light preferentially from the beam towards which it is moving, and the polarization of this beam optically pumps the atom into the higher energy state, thus assuring a spatially dependent restoring force. It is important for the experimentalist to note that because of the negative sign in Equation 3.3↑, the beams along the ^x and ŷ axes have opposite polarizations to those along the axis.

3.2.2 Spectroscopy and Amplification

Because of its two ground states separated by nearly 7  GHz, the laser cooling of ^87Rb requires the use of two widely separated ranges of optical frequencies (See 8 for level diagrams, atomic properties and spectroscopic data). This BEC experiment makes use of the 52S1 ⁄ 2 → 52P3 ⁄ 2 transition, i.e. the D2 line. The cooling and imaging is achieved on the F = 2 → F’ = 3 transition, and optical pumping occurs on the F = 2 → F’ = 2 transition. The repumping transition, requiring much less light, removes atoms from the F = 1 ground state by driving the F = 1 → F’ = 2 transition. The linewidth of these transitions is approximately 6.06  MHz, and as such a frequency source with a sub-MHz bandwidth is desirable for Doppler cooling. The Vortex 6013 external cavity diode laser was chosen for this purpose, as its advertised linewidth is  < 300  kHz, determined for a 50  ms integration time. The frequencies of the Vortexes are controlled by two separate saturated absorption spectroscopy setups for the two frequency ranges. The signal from the spectroscopy controls the frequency of the Vortexes with the aid of a proportional-integral feedback system. The signal from the feedback circuit changes the current supplied the to laser, thus changing the frequency. A second feedback loop with a much longer time constant has been added, which integrates the error signal from the first. The output acts on the voltage supplied to the grating in the Vortex laser head, allowing more coarse-grained adjustment of the frequency and combatting drift in the system. This addition has significantly improved the stability of the experiment, as the lasers need only be relocked once or twice a day.
The 6013 has an output power of around 20  mW and must be amplified. The beam from the “trapping” Vortex (that is, the Vortex producing light to interact with F = 2 atoms) is used to injection lock two Sanyo DL7140 diodes contained in home-built housings providing temperature and current control. These diodes output approximately 70  mW. Light from one of these diodes is used to inject a tapered amplifier, and light from the second provides an imaging and optical pumping beam.
Access to the F = 1 → F’ = 2 transition was previously achieved with a home-built external cavity diode laser. This was found to be too unstable, and has been replaced with a second Vortex. The second Vortex is locked to the 1 → 2 crossover peak in the spectrum, approximately 32  MHz blue to the F = 1 → F’ = 2 transition. This light is used to injection lock a Sanyo diode, the light from which is then shifted in frequency by an acousto-optic modulator (AOM) and used, along with the trapping light, to inject the tapered amplifier. A full schematic of the beam layouts, as well as powers and detunings is provided in Figure 3.6↓.
The tapered amplifier is an Eagle Yard EYP-TPA-0780-01000, which is a GaAs semiconductor specified to be capable of delivering between 1.0 and 1.2  W of optical power, and 13  dB of amplification. Typically, 20  mW of input power is provided and an output of 1000 − 1100  mW is produced (approximately 17  dB gain), depending on laboratory climate. This value has been remarkably stable in the three years since installation. The mechanical and thermal housing for the tapered amplifier is similar to that described in 9, and was in fact manufactured in the Laboratoire Charles Fabry de l’Institut d’Optique. The mount is essentially a copper block sitting atop a thermoelectric element, allowing for efficient heating or cooling. The block also includes mounts for optics to focus the input beam onto the amplifier and collimate the output beam. The power supply for the both the amplifier current and the current for the cooling unit was constructed in house from a LDTC2/2 module from Wavelength Electronics, by our technologist Alan Stummer 10.
The amplifier is injected with light for both the trapping and repumping transitions. The two beams are combined on a polarizing beamsplitter and passed through a rotatable λ ⁄ 2 waveplate. This allows the ratio of powers in the amplified beam to be controlled as the amplifier ideally amplifies only vertically polarized light. The angle of the waveplate was selected empirically based on MOT performance, and the final waveplate angle was chosen to produce a ratio of about 8:1 trap light to repumper light polarized in the vertical direction. This is a much higher fraction of repumping light than is required for a MOT, but the highly nonlinear nature of the amplifier, which is operated in saturation, makes it difficult to determine the ratio of the frequencies in the final beam. Measurements of the amplified beams on a spectroscopic device show much more than an order of magnitude difference in their powers. There has been no attempt to determine the actual value.
It has also been found that the amplifier takes approximately two hours to “warm up” (Figure 3.5↓). More specifically, the optical power output of the amplifier equilibrates over this period, which may be due to thermal effects within the housing or possibly alignment drifts in the seed beams due to thermal equilibration of lasers or acousto-optics.
figure Images/MOPA/PowerCurrentPlot.png
Figure 3.4 Power output characteristic of tapered amplifier with a 20  mW seed. The amplifier is typically run just past saturation at 1.7  A, and outputs 1000 − 1100  mW of optical power. The current is selected such that the unit will operate at saturation and be insensitive to power fluctuations in the seed laser.
figure Images/MOPA/WarmupPlot.png
Figure 3.5 Curve showing the output power after turn-on of the amplifier. The output power takes nearly two hours to stabilize at maximum power.
figure Images/MOT Optics/BeamLayouts.png
Figure 3.6 Schematic of laser system. Trapping detunings are relative to the F = 2 → F’ = 3 transition on the D2 line, and repumping detunings are relative to F = 1 → F’ = 2. All waveplates are λ ⁄ 2, and are used for power division, except the waveplate before the tapered amplifier which controls the mixture of trapping and repumping light. The computer-controlled motorized waveplate is moved in and out of the beampath, not rotated, and takes 2  s to be inserted or removed. The AOM after the amplifier shifts light for the lower MOT, and the transmitted light (i.e. the zeroth order) is shifted by a different AOM and used for the upper MOT.

3.2.3 Fibres and Lower MOT Optics

To minimize day-to-day drift in beam alignment, optical fibres were installed to carry light most of the way to the cuvette. The advantage of this design is that the majority of the free-space beam manipulation for frequency done is before a single fibre coupler. The disadvantage is that optical fibre does not in general maintain the polarization of the transmitted light. The splitters chosen (Newport F-CPL-S12785) were indeed non-polarization-maintaining, and the remainder of the system could therefore not be constructed from polarization-maintaining fibre.

3.2.3.1 Modular Optics Design

To successfully laser cool atoms and contain them in a magneto-optical trap, the polarization of the trapping beams must be controlled. Since the fibres carrying light to the cuvette do not maintain polarization, correction needs to be made after the light’s exit from the fibre. Figure 3.7↓ provides a schematic of the solution employed. Even in the case of an arbitrary polarization exiting the fibre, the setup ideally allows total power transmission at the desired circular polarization. A series of optics is fastened in a rigid cage system, and can be mounted as a unit. The light diverging from the fibre first encounters a rotatable λ ⁄ 4 waveplate, which is actuated by a graduate student such that linearly polarized light is produced. The next element is a collimating lens. The collimating lens has been placed after the first waveplate in order to save space. Because the beam is uncollimated as it passes through the waveplate, its resultant polarization will depend on cross-sectional position, with up to 10% of the transmitted intensity being non-linearly polarized. The third element is a rotatable linear polarizer which transmits the linearly polarized light, and finally a second λ ⁄ 4 waveplate at an orientation fixed to the polarizer produces the desired circular polarization. A photodiode is mounted on the end of this battery. The photodiode measures the intensity of the light transmitted, and the graduate student can thus optimize the power throughput as needed. The azimuthally varying polarization produced by the configuration of the lens and waveplate however, is translated into a azimuthally varying intensity, and returning the unit to a previous power setting cannot guarantee an identical beam. It was attempted to correct this, but it could not be achieved without requiring significant reoptimization. This system has proven quite useful for maintaining relative beam strengths, as well as optimizing power throughput. It also provides an effective monitor for overall optical power when adjusting upstream elements that may affect the coupling into the fibre. Altogether, the implementation of the fibre and polarizing optics was a great success in reducing day-to-day problems in MOT alignment.
figure Images/MOT Optics/MOT_Optics_3_resize_labels_imp.png
Figure 3.7 Modular Optics for Lower MOT Beams are mounted in the ThorLabs Cage Mount system. The beam diverges from the fibre and first strikes a rotatable λ ⁄ 4 waveplate which converts its polarization to a linear polarization whose angle depends on the elliptical polarization emerging from the fibre. The beam is then collimated by a lens and travels further to a polarizer and second λ ⁄ 4 waveplate, also in a rotateable mount. The polarizer and waveplate have a fixed relative orientation, thus producing a fixed polarization, and the experimenter need only optimize the angle for power throughput.

3.2.4 Upper MOT Beams

The beams supplying light to the upper MOT have also been confined to fibre in the interest of stability of alignment. Power balance is guaranteed by retro-reflection and a slight convergence given to the beams to compensate for attenuation of the beam due to absorption by the atoms. Figure 3.8↓ shows a schematic representation of the upper MOT optical system.
figure Images/MOT Optics/UpperMOTLayout.png
Figure 3.8 Schematic of Upper MOT beam layout. 45  mW of power exits the fibre and is divided amongst the three beam paths. Beams are focused to converge slightly, so the retroreflected beam will be of equal intensity after attenuation by the atom cloud. The degree of focussing is exaggerated in the figure.

3.2.5 Push Beam

Atoms may be transferred from the upper chamber to the lower chamber simply by dropping them through the transfer tube. A far more efficient technique, however, is the use of a push beam, to drive the atoms downward. A beam is directed from above the vacuum system and passes through both MOTs before exiting the cuvette. The beam is focused at the centre of the upper MOT using an f = 10  cm lens above the chamber. The focusing is intended to provide a maximal push within the upper MOT where transverse cooling is still effective, thus creating an atomic beam with minimal divergence. The distance between the upper and lower MOTs is approximately 40  cm, and the temperature of the upper MOT is approximately 150  μK. The push beam is on continuously during loading, is approximately 13.5  mW, and is detuned 165  MHz to the red of the F = 2 → F’ = 3 transition. The large red detuning is favourable to the F = 2 → F’ = 2 transition, and light scattered outside the MOT, where the atoms cannot be repumped, will have an increased tendency to leave the atoms in the F = 1 ground state where they are invisible to the push beam. This in addition to the divergence of the beam ensures a beam of atoms with a low transverse temperature. Average transfer rates of around 100 million atoms per second are routinely achieved.

3.2.6 Imaging

Due to the very low temperature of a Bose-condensed system, all measurements of the atom cloud in this work are done though its interaction with optical fields. To learn about the cloud that has been produced, it is quite simply photographed. Several techniques exist for imaging ultracold atoms 21, two of which are employed in this study. Fluorescence imaging is the simple imaging of scattered light from the atom cloud, and is effected by pulsing the MOT beams on the atom cloud, usually for 1  ms. This technique is simple, but since light is scattered in all directions, to acquire sufficient signal for small, cold clouds causes too much heating of the cloud. A preferred technique is absorption imaging, whereby a resonant or near-resonant beam is passed through the atoms and imaged directly onto the camera. Through comparison with a reference beam where no atoms are present, the spatially dependent optical density of the cloud may be calculated. For a beam propagating along the x-direction, passing through a cloud with density n(x, y, z), the imaging beam will be modified according to the Beer-Lambert law:
(3.4) I(y, z) = I0(y, z)exp( − dx σn(x, y, z))
where I(y, z) and I0(y, z) are the imaging and reference beams respectively, and σ is the absorption cross-section of the atoms. A third, background image is obtained with no imaging beam and subtracted from both images, after which the images are divided to obtain a two-dimensional map of the column density. This technique integrates the density along the axis of beam propagation, and is thus limited to two dimensions unless a second imaging axis is used. A two-dimensional map, however, is sufficient for this experiment which seeks to study one-dimensional dynamics.
Because this technique relies on the direct comparison of two images, it is important that the source of illumination change as little as possible between the absorption and reference images. The camera is limited to a minimum of 7  ms between images, which leaves the imaging process vulnerable to vibrations of the components at typical acoustic frequencies. For this reason, the probe beam is coupled into an optical fibre which is brought as close as possible to the atoms. Much of the vibration, however, comes from the cuvette itself. Further improvement is made following a technique found in 20. Further improvement is made following a technique found in 20. Rather than simply dividing an absorption image by its subsequent reference image, many references from the same experimental configuration are pooled together, and a new reference image is composed from this pool for each absorption image. The generated references are the linear combinations of the pooled images that best match the absorption images outside the region of interest. Figure 3.9↓ shows a sample of the improvement obtained this way.
figure Images/Imaging/UnModPlusPlot.png figure Images/Imaging/ModPlusPlot.png
Figure 3.9 Image improvement through linear combination of reference images. The image in panel (a) is generated by dividing the absorption image by the reference image taken immediately after, whereas the image in panel (b) has been created with a reference generated from a pool of 25 images.
Images are recorded by a 12-bit Dalsa CA-D1-0256T-STDL CCD camera, which has a resolution of 255 × 256, and a pixel size of . Light from absorption or fluorescence imaging is, upon exit from the vacuum chamber, collected by a 1:1 transporting telescope comprising two f = 7.5  cm lenses 15  cm apart. The image is reformed 7.5  cm from the telescope, and is then imaged onto the CCD using a f = 5  cm lens, which is placed such that a magnification of 5.30±0.05 is achieved. To image larger clouds, mirrors are inserted to bypass the lens, and a Computar Macro10x compound lens is attached directly to the camera, providing a magnification of 0.267±0.005.

3.2.7 Optical Pumping

After the laser cooling phase, atoms will be in general unpolarized, and it can be expected that should the trap be turned on, only 15 of them would find themselves in the desired F = 2,  mF = 2 trapped state. To reduce this loss, atoms are optically pumped into this state using a circularly polarized laser beam. Atoms are pumped on the F = 2 → F = 2 transition and will enter a dark state (since there is no F’ = 2,  mF = 3). This is advantageous as an atom fully pumped will stop scattering light and no further force is applied to the atoms.
The optical pumping beam illuminates the cloud from below the cuvette. The light is prepared in frequency (Figure 3.6↑), and then brought to the chamber by an optical fibre. In the previous iteration of the experiment, the polarization of the light was prepared before coupling into the fibre. Because the fibre did not maintain polarization, this proved to be much too unstable, and the polarization was difficult to determine at the point of the atoms. The polarization is therefore prepared after the fibre. Upon exiting the fibre, the light is collimated and reflected towards the atoms, where the reflection is now implemented by a polarizing beamsplitter rather than a mirror. The polarized light is then passed through an adjustable λ ⁄ 4 waveplate where it becomes σ +  polarized. The illumination of the atoms is coordinated with the turn-on of the vertical compensation field, which provides a guiding field that acts as a quantization axis.

3.2.8 Dipole Barrier

3.2.8.1 Background: Optical Potentials

Magnetic potentials are well suited to trapping and cooling large clouds of atoms. To study quantum collisions, however, potentials with much smaller spatial features and potentials with much faster switching times are desired. Making use of the AC Stark shift in atomic transitions, ultracold atoms experiments often employ off-resonant laser beams to form these potentials. Because the potentials depend on the intensity of the light, they can be constructed with all the spatial and temporal flexibility of a laser beam.
An intuitive understanding of how a detuned laser field creates a potential for a neutral atom can be gained from a semiclassical picture. The oscillating electric field of the laser will induce an electric dipole in the atom, and the dipole will behave as a driven oscillator. If the dipole is driven above resonance, the dipole lags the driving force and oscillates anti-aligned with the electric field, whereas a dipole driven below resonance oscillates aligned with the field. The potential energy of the atom in an oscillating electric field is
(3.5) Udip =  − 12 p⋅E
where p is the induced dipole of the atom, and E is the electric field. Because the induced dipole p is proportional to the inducing field, the potential is proportional to |E|2 which is proportional to the intensity. An anti-aligned dipole will thus experience a repulsive potential, and a dipole driven below resonance will remain aligned with the field and experience an attractive potential. The force due to the gradient of this potential is known as the dipole force. A more sophisticated approach, like those found in 23,13 yields the following expression
(3.6) Udip(r) =  − 3πc22ω03(Γω0 − ω + Γω0 + ω)I(r)
where Γ is the inverse of the natural line width of the D2 line in ^87Rb, ω0 is the frequency of the transition and ω is the frequency of the driving field. In this experiment, as in most, the detuning is small compared to the laser frequency - about 1:1000 in this case. In such cases, it is common to make the rotating wave approximation, where the counter rotating term is ignored. The new relation is
(3.7) Udip(r) =  − 3πc22ω03ΓΔI(r)
where Δ = ω0 − ω, and is called the detuning. A second quantity of interest is the rate at which the optical potential will absorb and re-emit light quanta from the dipole field. In experiments with ultracold matter the energy deposited by a scattered photon is often large compared to the temperature of the cloud, and may lead to heating problems. Also in the rotating wave approximation, the scattering rate is given by
(3.8) Γsc(r) = 3πc22ℏω03(ΓΔ)2I(r)
An important aspect of this quantity is that the scattering rate declines as the inverse square of the detuning, whereas the dipole potential declines only as the inverse of the detuning. One can therefore obtain an identical potential with less scattering by using a more powerful laser at a larger detuning.
Optical potentials have found broad application in the field of ultracold matter. BECs have been created entirely in optical traps 22, and optical lattices arrange atoms for tests of condensed matter problems 47,48. Even Maxwell’s Demon has taken the form of a dipole barrier 46. Optical traps have the additional advantage that the potential can be essentially independent of the magnetic substate, thus opening an additional degree of freedom for the experiment.

3.2.8.2 Optics

The light for the barrier is produced by a Sanyo free-running diode laser, similar to those used for other beams in the experiment. The diode is selected for a frequency near the resonance wavelength at 780.1  nm. The frequency is monitored by a wavemeter, and maintained usually in the range of  + 100 to  + 300  GHz detuning through tuning of the temperature of the diode. The laser light passes through a mechanical shutter and is coupled into a fibre (ThorLabs P1-830A-FC-2) that is single mode at this wavelength. The fibre carries the beam closer to the atom cloud, both to filter the spatial mode of the beam and to reduce alignment drift. Figure 3.10↓ shows a schematic diagram of the barrier beam preparation.
Upon emerging from the fibre, the beam is collimated by a Thorlabs F230FC-B, f = 4.43 mm fibre collimation package, and the beam has a 1 ⁄ e2 diameter of 0.9  mm. The beam passes through a half-wave plate to (mostly) correct its polarization, and then through an AOM, which is used to control the power reaching the experiment. The beam next passes through a Galilean beam expander. The beam expander uses a ThorLabs LK1982L2-B f =  − 30  mm cylindrical lens, and a Thorlabs LJ1363L2-B f = 400.00  mm cylindrical lens, resulting in a beam size of 0.9 × 12  mm. Finally, the beam passes through a 1:1 telescope formed by two Thorlabs AC508-400-B achromatic doublets. This telescope does not change the shape of the beam, but serves to couple the barrier beam into the optical path of the absorption probe/camera. When focused by the ThorLabs AC127-050-B the beam converges to a near-diffraction-limited spot with a 1 ⁄ e radius of 2.5±0.1  μm along thin direction, and 20±1  μm along the broad direction (see Section 5.2↓ for measurements). The uncertainty refers to the variation in spot size along the one-dimensional “sheet” described in the next sub-section. The spot size is measured in situ using the camera, since the barrier is focused in the imaging plane. This number represents an upper limit since it is also near the resolution limit of the camera, though it is consistent with values measured using Ronchi rulings before the installation of the barrier in the experiment.
figure Images/Barrier/Schematic.png
Figure 3.10 Preparation of dipole barrier beam. Insets show schematic (i.e. not to scale) views of the beam cross-section at various points in the preparation. The z-axis is in the plane of the page, and the x-axis comes out of the page. The beam emerges from the fibre in a lowest order Gaussian mode, pictured in panel (a), at which point it is polarized. The beam scanned by the AOM and passed through a cylindrical telescope, producing a cross-section as in panel (b). The bright ellipse in the centre indicates the instantaneous intensity profile of the beam. The beam is then directed towards the atoms where it is refocused to the point where it will encounter the atoms.

3.2.8.3 Time-Averaged Potential

The AOM used to modulate the power of the barrier is also used to manipulate its position. The cylindrical telescope produces an asymmetric beam which is broad, but is still still not a one-dimensional sheet as it varies significantly in intensity over the lengthscale of a typical atom cloud. In order to “flatten” the intensity of the beam, the beam is rapidly scanned by the the AOM at a frequency of 100  kHz, which is much faster than atomic motional timescales and creates a time-averaged potential 43,55. The AOM is a NEOS 23080-1, which has an active aperture of 1  mm and has a scan range of approximately 50 − 100  MHz, corresponding to 20 diffraction-limited spots (the ratio of the scannable angle to the minimum divergence of a beam filling the aperture). In practice, the beam can be scanned over a range of 20 − 30  MHz about a central frequency if high efficiency into a single order is desired, and the barrier will be less than 10 spots across. The AOM can be frequency-modulated with a triangle wave to create a broad potential of around 100  μm across. In practice, however, the diffraction efficiency of the AOM is a function of diffraction angle (frequency), and a triangle wave produces a non-flat potential.
To flatten the potential, the waveform driving the AOM is frequency-modulated using a customized waveform. During the 10  μs sweep, the beam is held at less efficient frequencies for longer than the more efficient frequencies. To produce the customized waveform, a field-programmable gate array (FPGA) chip drives a digital-to-analog converter (DAC), whose output drives the frequency-modulation input of the RF source for the AOM (an Intraaction ME Series Modulator Driver). While it should be possible to measure the diffraction efficiency of the AOM as a function of frequency and calculate the optimal waveform, small changes in alignment change this efficiency curve and invalidate the measurement. A genetic algorithm was therefore designed, whereby the flatness of the barrier was measured directly by the CCD camera and fed back to the computer which could dynamically alter the customized waveform. A barrier of flatness  < 1% was achieved, usually within a few minutes. The base frequency of the FPGA is 100  MHz, allowing a resolution of 10  ns, or 1000 points in the customized waveform. Further work on the evaluation of the barrier is presented in Section 5.2↓.

3.3 Magnetic Fields

The creation of the BEC depends on the creation and manipulation of magnetic fields. From the first stage of cooling, magneto-optical trapping, the magnetic field must be carefully controlled and manipulated on the scale of milliseconds. The experiment as a whole is controlled by 8 sets of coils, each with different demands on strength and speed of fields that will be created.

3.3.1 Background

Optical cooling and trapping has limitations in terms of the densities and temperatures that can be simultaneously produced. In order to reach condensation, it becomes necessary to turn off the lights, and cool via different means. Neutral atoms may be confined in a magnetic potential by making use of the atom’s magnetic moment. According to Equation 3.2↑, an atom in a spatially dependent magnetic field has a spatially dependent energy shift, and can thus be trapped. An atom will seek the lowest energy position, and whether this is a field maximum or a field minimum depends on the sign of mFg. Rubidium-87 has two ground states: F = 1 whose Land g-factor is g =  − 1 ⁄ 2, and F = 2 whose Land g-factor is g = 1 ⁄ 2. Since it is not possible to create a local magnetic field maximum in free space, a minimum must be created, and trapped states will be F = 1,  mF =  − 1 and F = 2,  mF = 1, 2. The F = 2,  mF = 2 state is used in this experiment as it will be the most tightly confined for a given trap configuration.
A quadrupole field produced by anti-helmholtz coils and described by Equation 3.3↑ has been a successful magnetic trap for neutral atoms. A quadrupole trap provides a large-volume trap with strong confinement, and was the first to trap neutral atoms 32. This strong confinement makes it suitable for evaporative cooling (see Section 3.3.6↓), but achieving BEC in this trap is made impossible by a “hole” in its centre. The low field values in the centre of the trap, combined with the rapid change in field direction make it impossible for fast-moving atoms to remain properly aligned with the field, and such atoms may enter untrapped states and be ejected from the trap. This process of changing spin state is known as the Majorana spin-flip 33. The situation is actually worse for colder, denser atoms which spend more of their time close to the hole, and the hole thus generates heating as well as loss 15. Several solutions to this problem have been implemented. One involves optically “plugging” the hole with a blue-detuned laser beam 34, and another, more common solution is the Ioffe-Pritchard trap 35 which has a non-zero field minimum. The first trap to achieve BEC in ^87Rb, and the trap used in this experiment is the time-orbiting potential (TOP) trap 15.

3.3.1.1 TOP Trap

First developed by the Cornell group at JILA 15, the time orbiting potential (TOP) traps atoms in a magnetic potential with non-zero minimum by rapidly shifting the centre of a quadrupole trap. Effected by two pairs of helmholtz-oriented coils along orthogonal axes, the quadrupole field is offset by a constant but rotating bias field. The field rotates at a frequency that takes advantage of a difference in time scales. The magnetic field changes on a timescale that is slow compared to the atoms’ Larmor frequency, guaranteeing that the magnetic moments of the atoms maintain their alignment with the magnetic field, and enabling them to stay trapped. The minimum TOP fields used are a few gauss, giving Larmor frequencies of a few MHz. The other relevant timescale is the motional timescale of the atoms. This timescale is best described by the trapping frequencies, normally in the tens or hundreds of hertz. The effect of the added bias field is to shift the centre of the quadrupole gradient. If the force towards this centre changes directions, i.e. if the bias field changes directions before the atom can respond, the atom will experience a time-averaged potential. The zero-field centre is transported to the periphery of the trap where it rotates about the cloud and is known as the “circle of death”. Majorana spin-flips still occur, but only atoms with high enough energy to reach the instantaneous field zero will experience this. Not only does this prevent the ejection of atoms at the centre of the trap, but the bias field value may be varied to perform temperature-selective evaporation (see Section 3.3.6↓ for more detail).
In this experiment, the bias fields are applied along the two horizontal directions, i.e. in the plane of the optical table. The instantaneous field value is found by modifying the potential described in Equation 3.3↑, and is given by
(3.9) B(t) = (B0sin(ωt) + Bx)^x + Byŷ + (B0cos(ωt) + 2Bz)
where B0 is the bias field applied by the TOP coils, and ω is the frequency of rotation of the TOP fields, in this case 10  kHz. In this configuration the field zero actually traces out an ellipse in space, but the path is still referred to as a circle of death as the ellipse marks a path of constant potential energy. In general the bias field in the two quadratures need not be equal, nor does their relative phase need to be π ⁄ 2. Departure from these norms, however, would change the circle of death to an ellipse of death, energetically speaking, and evaporation due to both Majorana flips and radio-frequency radiation (see Section 3.3.6↓) would occur at a range of energies, and efficiency would suffer. Time-averaging this magnetic field, and keeping only terms up to second order in position gives the trap shape
(3.10) V = μBgmF[B0 + B24B0(x2 + 2y2 + 4z2)]
and corresponding trap frequencies of
(3.11) ωx, y, z = B(1, 2, 4)μBgmF2mB0
where m is the mass of a ^87Rb atom. The trap depth is another important parameter that is determined by the circle of death. The circle of death is a ring traced by the instantaneous field zero whose axes, according to Equation 3.9↑, can be defined by x’ = B0 ⁄ B and z’ = B0 ⁄ 2B. The corresponding trap depth is thus
(3.12) VCOD = μBgmFB04
which depends only on B0 and not B. For typical values where B0 is in the range of a few tens of gauss, 40  G in the case of this experiment, this trap will hold clouds of around 100  μK without significant rates of evaporation.

3.3.2 Upper Chamber Quadrupole Coils

In the upper chamber atoms are captured in a MOT from a rubidium vapour, and the quadrupole coils produce the necessary field. The coils comprise 200 winds each of 16AWG wire, and have a diameter of 2.75”. They produce 6  G/(cm⋅A), and are typically run at 3  A, producing 18  G/cm. The lower MOT is continuously loaded from the upper MOT, and as such the current is left on continuously, and no switching is necessary. An 8  Ω potentiometer is connected in parallel with one of the coils, and is used to centre the MOT over the transfer tube by shifting the magnetic field zero. Further detail and discussion on the design of these coils can be found in 5,6.

3.3.3 Lower Chamber Quadrupole Coils

The design requirements on the lower chamber quadrupole coils are far more demanding as they must provide the primary magnetic potential for trapping, and ultimate cooling of the cloud to BEC. As will be discussed in detail in Section 3.3.6↓, the tight confinement of an atom cloud is essential to its efficient cooling, and it is therefore desirable generate magnetic fields that are as large as possible. Large fields are produced by large electrical currents, which in turn dissipate large amounts of power. In addition to the production of large magnetic fields, a major design challenge becomes the dissipation of several kilowatts of power.
The design presented here replaces the previous design 5,6, which was the original implementation and which witnessed the experiment’s first BEC. This design, while sufficient to create BEC, lacked the mechanical stability and reliability (there were often leaks, and the coils often overheated) necessary to do more precise work. Furthermore, the new design improves on the previous design by offering nearly double the maximum field strength, a larger bore to improve optical access, and an generally more compact and efficient mechanical design. The coils were also designed with switchable sections allowing for the creation of strong, flat, DC fields with the aim of accessing the rubidium Feshbach resonance at 1007  G. Exceptional noise performace was desired as the width of the resonance has been measured to be approximately 200  mG 7, and this concern motivated the choice of power supply. Switchable sections have not yet been tested and are beyond the scope of this thesis.

3.3.3.1 Design Objectives

The single most significant difference in the new design was the move from a 300 turn coil contained in a plastic case through which water flowed, to an 80 turn coil made of hollow copper wire which allows water flow through the wire centre. The system of cooling used in the old design resulted in frequent leaks and overheating, and the poor mounting system resulted in visible motion of the coils when switching fields. The primary objective of redesign was to overcome these failings. Secondary objectives were increasing of field strengths, improving optical access and incorporating compensation coils into the design to improve their effectiveness and versatility.

3.3.3.2 Mechanical Design

A diagram of the coil mounting system is given in Figure 3.11↓, and more detailed engineering drawings of the components are provided in Appendix A↓. The mechanical mounts for the coils have been improved in several ways. The mounting system for the coil is designed such that all pieces are connected together, and the mount for the quadrupole coils, the TOP coils and three sets of compensation (bias) coils are combined in one rigid unit. The assembly is screwed into the table at four points, preventing rotation. Optical access is also improved, as the coil bore is now large enough to accommodate a ∅ = 1.2" lens tube, capable of mounting a 1" optic directly against the cuvette. The bias coils now match the axes of the quadrupole coils as well. One pair is oriented vertically, and the other two are oriented along, and perpendicular to the axis defined by the bore of the main coils. The various components of the mounts are held together with nylon screws where possible, and brass threaded rod where more strength is needed. Brass was chosen as it is non-magnetic.
Parts have been manufactured from Delrin due to its strength and its thermal properties. The gradient coils are secured by clamping along two axes. Each coil is held by two clamping pieces, one on top and one beneath. The clamps are machined to match the curvature of the coil along the top, and a lip fits along the back face of the coil to prevent the coils from moving apart. Clamping is effected vertically by nuts turned onto threaded brass rods, which run through both clamping pieces and are kept under tension. The coils are clamped along the bore axis by screws which go through the clamping pieces, and screw into threads in the TOP coil holders. Since the TOP coil holders receive screws from both sides, one from each coil, a fixed relative position of the two coils is assured. The x-axis TOP coils are fixed to these holders with epoxy, and the z-axis TOP coils are embedded in the quadrupole coils. The lower clamping pieces are screwed into legs which are in turn screwed into feet which can be fixed to the optical table. In addition, multiple threaded holes have been added to the legs and clamps to anticipate future needs.
figure Images/CC Drawings/ExplodeMount2.png
(a) Exploded view of coil mounting system.
figure coilsannotate.png
(b) View of the mounting system assembled with the gradient coils plus compensation and TOP coils attached.
Figure 3.11 Drawings 42 of the mechanical mounting system designed to hold the new gradient coils, as well as three sets of compensation coils and the y-axis TOP coils.

3.3.3.3 Current and Switching

The Agilent 6682A was chosen for the reconfigured experiment due to its superior noise characteristics, and is capable of delivering 5  kW of power at 240  A and 21  V. Figure 3.12↓ shows a schematic of the electrical connections used to drive the system in the standard, anti-helmholtz configuration. Electronics to create strong, flat fields for accessing the Feshbach resonance have not yet been assembled, and will not be discussed in this thesis.
To control the current output of the Agilent, the voltage limit was set to maximum, and it was run in constant current mode. The current was then controlled programmatically using an analog waveform from the LabVIEW control system. The current is not constant during the course of the experiment, however, and undergoes several ramps, typically on the order of 10  A/ms. The coils were measured with a DMM to have and inductance of 195±1  μH, and the maximum rate of change of current in the coils can thus be estimated as
(3.13) (dIdt)max = Vmax2L = 56  A/ms
where Vmax = 21  V, the power supply maximum, and the factor of two is to account for both coils (neglecting mutual inductance). This is adequate for trapping, compression and evaporative cooling of atoms, but the trap release should be much faster. The trap turn-off is ideally quasi-instantaneous, which means it must be fast with respect to the trapping frequency. Trap frequencies in this experiment will typically be 100 − 300  Hz, so the desired field turn-off must be ≪3  ms. For a fast turn-off, the extinguishing voltage should be as large as possible. To switch the current four power FETs are used in parallel. These FETs open the circuit and switch the current effectively instantaneously, but the inductive energy stored by the coils must be dealt with. Seven transient voltage suppressors (TVSs) are wired across the pair of coils. Similar to a Zener diode, the TVS will conduct electricity when its breakdown voltage is exceeded, and in contrast to the Zener diode, is designed to dissipate large amounts of power for a short time. The TVSs can be wired in series so the breakdown voltages will be additive, and the objective is to have as high a breakdown voltage as possible, yielding as fast a turn-off as possible (Eq. ), without damaging other circuit components. The FETs are thus chosen to have as high a breakdown voltage as possible, the tradeoff being that a FET with a higher breakdown voltage will be more resistive and dissipate more power. The APT30M17JFLL has a breakdown voltage of 300  V, and a drain-source on-state resistance of 17  mΩ. The TVSs employed are the 5KP15CA from American Microsemiconductor, and a breakdown voltage of 138±1  V is measured for seven in series, which is well clear of the 300  V breakdown voltage of the FETs.
Once the breakdown voltage and the rate of change of the current are known, the inductance of the coil assembly may be determined. Figure 3.14↓ shows the current in the coils as a function of time for the field turn-off for both a single coil and for the assembled coils using 6 TVSs, as well as the turn-off for the assembled coils with 7 TVSs, which is the standard configuration. The current as a function of time during turn-off is a linear curve, and the slope is given by Equation , where now Vmax is determined by the breakdown voltage of the TVSs. The inductances of the different configurations can thus be determined by solving for L. The inductance of a single coil is found to be 198±3  μH, consistent with the DMM measurement. The inductance of the installed coil assembly is measured to be 337±3 μH, and the mutual inductance of the installed coils is thus  − 59±3  μH.
figure Images/New Coils/ElectricalSchematic.png
Figure 3.12 Wiring schematic of the quadrupole coils. The Agilent 6682A drives up to 245  A through 8 strands of AWG10 wire (indicated by thick lines) to the coils. The current is determined by the analog current input of the driver, and is set by an analog waveform output by the computer. Four high-power FETs are used as switches for the current, which are actuated by a digital output from the computer. When the FETs are closed, a set of 7 TVSs reach their breakdown voltage and short the coils, thus quickly dissipating the energy stored in the magnetic field.
figure Images/New Coils/Field Measurements/Turnons.png
Figure 3.13 Turn-on of quadrupole field in constant current mode. The fastest possible turn-on is achieved by setting the power supply to output maximum current while the FETs are off, letting its capacitors charge, and then tolerating a large overshoot. Pictured are curves showing the turn-on for the current set to maximum while the FETs are off, and for the current set to the desired value while the FETs are off. Setting the current to maximum produces a faster turn-on, but also more overshoot.
figure Images/New Coils/Field Measurements/CoilTurnoffs.png
Figure 3.14 Turn-off of single coil and installed coil pair with 6 TVSs. The breakdown voltage of 6 TVSs is 117V, and the turn-off rates are 591  A/ms and 347  A/ms for the single coil and installed coils respectively. Inductances calculated from these rates are 198  μH and 337  μH. A third curve shows the switching of the current in the coil assembly using 7 TVSs.

3.3.3.4 Cloud Trajectories

Turn-off times much faster than the trap oscillation frequency are achieved in this way, but study of the cloud trajectories on turn-off reveals that the turn-off is not ideal. Figure 3.15↓ shows the trajectories of a clouds released from TOP traps with two different gradient strengths. The clouds are observed to move upwards during the turn-off, and to have corresponding upward velocities after the fields have been fully extinguished. A possible explanation for this behaviour is that superimposed on the magnetic trap is a DC field in the vertical direction, causing the magnetic field centre to move upwards during the turn-off of the gradient fields. The magnetic minimum of the TOP trap will sit atop the zero of the quadrupole field, and in the presence of a parasitic DC field the trap centre is
(3.14) xcentre = B’ ⁄ Bpar
where B is the gradient and Bpar is the hypothesized parasitic magnetic field. As the trap centre moves upward, the cloud experiences a restoring force upward. The resulting velocity is then proportional to the trap strength and the duration of the turn-off
(3.15) v’ = a × t = dt F(t)m
where a is the acceleration, m is the mass of an atom in the cloud, and F(t) is the restoring force due to the trap. For a linear turn-off, the acquired velocity is simply proportional to the trap strength times the turn-off duration. Velocities of released clouds are determined by fitting their trajectories to a second-order polynomial, and solving for the velocity at the time when field turn-off is complete. A summary of extracted velocities is given in Figure 3.16↓. The figure plots extracted velocity at turn-off versus the product of gradient strength before turn-off, and duration of turn-off, and shows a linear relationship. Possible sources of parasitic magnetic fields are the ion pump magnets, eddy currents or induced currents in coils not being switched.
figure Images/BEC/TurnoffFling/4TVS_Trajectories.png
Figure 3.15 Cloud trajectory during and after gradient field turn-off. Triangles indicate cloud position while the field is turning off, and circles indicate cloud positions after the turn-off is complete. Positions after turn-off are fitted to parabolic trajectories. Clouds have an initial upward velocity that is proportional to the gradient strength times the duration of the turn-off.
figure Images/BEC/TurnoffFling/DifferentTurnoffs.png
Figure 3.16 Turn-off velocities for different turn-off parameters. Velocities are determined from fits similar to those in Figure 3.15↑, and are found to be proportional to gradient strength times the duration of the turn-off.

3.3.3.5 Thermal Performance

A main criterion of the coil design was the ability to dissipate the waste heat produced during sustained operation at maximum currents. With the circuit set up as in Figure 3.12↑, the resistance at room temperature is around 73  mΩ, and equilibrates around 88  mΩ when 240  A are being driven. Since these data were taken, the 10  m, 4-strand, AWG 10 conducting cables have been replaced with 10  m, 8-strand, AWG 10 cables, which will reduce the total resistance by 2  mΩ at low currents, and even more at higher currents. It is now possible to run the system stably at the full current value of 245.22  A if water flow rates of 5.9  ml/s are maintained. At such a current the coils must each dissipate approximately 2044  W of power. The heating of the water is measured to be 63  ^○C, meaning that 1560  W is dissipated by the water, and about 500  W is dissipated into the coil environment.
In addition to increasing electrical resistance and possibly destroying materials, the waste heat has the potential distort the environment through thermal expansion of components and the air around them. Effects of air currents have not been studied formally, and are assumed to be negligible. To assess the distortion of trap geometry, the average temperature of the coils was slowly varied by heating the the input (i.e. cooling) water. The experiment was cycled continuously, producing and dropping condensates, and the cloud trajectories were studied. One set of results are shown in Figure 3.17↓, where the cloud position after 1.6  ms of expansion is observed to move 1.34±0.09  μm/K in the vertical direction, and 0.54±0.05  μm/K in the horizontal direction. Additional measurements are taken at 8.6  ms and 13.6  ms of expansion, and the results are summarized in Figure 3.18↓. With longer drop times, a stronger variation of position with respect to temperature is observed in the vertical direction, suggesting a temperature dependent velocity kick in addition to positional shift. By plotting the position/temperature correlation as a function of drop time, the slope and intercept give these parameters as 1.14±0.025  μm/K and 0.128±0.005  mm/(s⋅K) respectively, in the vertical direction. The horizontal cloud shift is 0.549±0.001  mm/K, which shows no significant change as a function of drop time.
The thermal expansion coefficient of copper is 16.5  μm/(m⋅K) 49, and the coil radius is 55  mm, so a trap at the centre might be expected to move 0.91  μm/K. It would seem the mounts are of some benefit in this respect, though these shifts are larger than hoped and potentially problematic. The coil temperature within a typical experimental cycle varies approximately 7  K. The mean temperature of the coils rises more slowly, and equilibrates during steady state running of the experiment. Measurements using thermistors embedded in the coils indicate that this equilibrium is reached after 5 to 10 minutes, however, measurements of the atom cloud indicate a somewhat slower process on the order of 45 minutes. For example, Figure 4.4↓ shows the average position of a dropped cloud drifting on this timescale. This temperature-dependent movement may be viewed as a design flaw, and in subsequent designs this positional shift may be mitigated by supporting the coils from the centre rather than from the bottom.
figure Images/New Coils/Temp Stuff/DropTempCorrelation.png
Figure 3.17 Vertical and horizontal cloud positions, as well as mean coil temperature for continuous running of the experiment. The mean coil temperature has been shifted by pre-heating the cooling water. The position of the cloud has been measured after 1.6  ms of free expansion and its correlation to the coil temperature extracted. A coefficient of 2.15±0.10  μm/K is found in the vertical direction, and 0.98±0.1  μm/K is found in the horizontal direction. Before extracting the correlation, the data were adjusted for the apparent lag between coil temperature change and cloud position change, and a shift of 5.9 minutes was found to maximize the correlation. This time is assumed to be related to thermal diffusion times in the apparatus.
figure Images/New Coils/Temp Stuff/SummaryMovement.png
Figure 3.18 Cloud position / coil temperature correlation at different drop times. Calculating the intercepts yields positional shifts of 1.14±0.025  μm/K in the vertical direction and 0.549±0.001  μm/K in the horizontal direction. The slopes yield values of 0.128±0.005  mm/(s⋅K) and  − 0.0035±0.0011  mm/(s⋅K) respectively.

3.3.3.6 Measuring the Gradient

Other magnetic fields in the experiment (i.e. DC compensation fields, TOP fields) have generally been measured by study of the absorption spectra of the rubidium atoms at various field strengths (Figure 3.21↓, for example). The gradient, however, is difficult to precisely determine in this fashion as the line is broadened by many different transitions and the finite spatial extent of the cloud, spanning a range of field strengths. Instead, the gradient field is measured in two ways. First, a MOT or magnetically trapped cloud is created in a strong magnetic gradient, and the position of the cloud is measured under the influence of different known DC fields produced by the compensation coils. The effect of the DC field is simply to translate the centre of the magnetic trap in a way inversely proportional to the gradient strength.
(3.16) dxdB0 = 1B
Second, the trap frequencies depend on the TOP fields and gradient field in a known way. The TOP fields are measured spectroscopically, and the gradient fields are thus extracted through the measurement of cloud oscillations in the trap. To find the gradient field in terms of the trap oscillation frequencies, one need rearrage Equations 3.10↑
(3.17) B’ = 2πfxmB02μB
(3.18) B’ = 2πfzmB0μB
where as before, B is the field gradient in the weak direction, B0 is the magnitude of the TOP field, assumed to be the same in both directions, m is the mass of rubidium and μB is the Bohr magneton. Figure 3.19↓ shows the results for the two measurements. The measurement of cloud movement due to bias fields gives a result of 1.76±0.02  G/(cm⋅A), and the measurements of trap frequencies in the horizontal and vertical directions give 1.75±0.05  G/(cm⋅A) and 1.77±0.05  G/(cm⋅A) respectively.
figure Images/New Coils/Field Measurements/GradientFromComp.png
figure Images/New Coils/Field Measurements/TrapOscillations.png
Figure 3.19 Determination of the gradient field. Panel (a) shows the motion of an atom cloud resulting from the application of a known, uniform, DC magnetic field. The applied field translates the trap centre as predicted by Equation 3.16↑, thus allowing the gradient to be deduced. Panels (b) and (c) show measurements of the trap oscillation frequencies, which are related to the gradient by Equations 3.17↑ and 3.18↑. An oscillation is initiated by abruptly displacing the trap centre, after which the cloud is held for some variable time (plotted on the x-axis) and dropped for 7.3  ms.

3.3.3.7 Summary of results

The new quadrupole coils produce a gradient of 1.76±0.02  G/(cm⋅A), and are capable of running 245  A sustainably, for a maximum gradient of 431±5  G/cm in the weak direction. This exceeds the previous limitation (235  G/cm) by nearly a factor of two. The new design does slightly worse in terms of power efficiency (5.9 vs 6.8  G/(cmW), which is mainly due to the larger bore (1.2" vs. 0.78"), but is mitigated by a more compact design. The more rigid design should ensure a more stable and reliable experiment, but fails in preventing trap movement due to thermal expansion.

3.3.4 TOP Coils

With several options available to “plug” the hole in the quadrupole trap, the TOP trap was chosen in this experiment for the potential flexibility of easily manipulating the shape of the trap using the rotating bias fields. This new design more fully realizes this potential by adding full independent control of the TOP axes, including separate control of amplitude and phase for the two field quadratures, the ability to ramp the fields up or down, as well as a USB interface to quickly and easily reprogram between runs. Furthermore, as will be shown in Section 3.3.6.4↓, the temperature of evaporation can be extremely sensitive to the value of the TOP field. In a typical approach to BEC, the evaporation temperature will be sensitive on the order of a few tens of nanokelvin per milligauss. As such a new feedback system has been employed to regulate the field which had previously plagued by instability due largely to heating of components. Another design goal was to increase the maximum TOP field from 40  G as reported in 6. The TOP field along the x-axis was doubled, and fields of up to 80  G were observed, whereas the maximum field in the z-direction, where the coils are nested in the quadrupole coils, could not be increased from 40  G.

3.3.4.1 Circuit and Driver

The design goal of the TOP field driver was to produce 20  A_p of current to each of two independent coil pairs at 10  kHz. Given the geometry of the coils, 20  A_p corresponds to approximately 80  G peak TOP field. Furthermore, the circuit must be able to ramp between different current values at a variable rate, and quickly turn off when desired.
As the TOP coils are a nearly purely inductive load, it is rather difficult to drive them directly. The solution to this problem, as in the previous design, is to build a resonant circuit in which the energy may be passed between capacitive and inductive (i.e. the coils) elements. Figure 3.20↓ shows a schematic of the TOP circuit. In series with the coils are placed capacitors which together with the inductors form a resonant circuit. Also in series with the coils is a 10  mΩ sense resistor, which provides feedback to the driving circuit. In this same loop is also a transformer. The transformer is placed in the circuit so a high-voltage, low-current amplifier may be used to drive a low-voltage, high-current circuit.
An FPGA is programmed by a computer through USB. Connected to the FPGA is a DAC which produces two sinusoids, one for each TOP coil, at the desired trap frequency. The two quadratures can have an arbitrary and dynamic relative phase. In the next stage, the sinusoids are passed to a pair of variable gain amplifiers which provide the power to the circuit. The gains of the amplifiers are set by proportional-integral (PI) feedback circuits which feedback on the current in the coils measured by the 10  mΩ sense resistors, and whose setpoints are set by DACs also programmed by the FPGA. These setpoints are manipulated to perform ramping of the field strengths during the experiment. The ramp is divided into segments, and ramp rates and durations are passed to the FPGA, which then executes the ramps in sequence, either immediately or waiting for a logic high from the trigger input. This approach allows for an arbitrary ramp shape, and a timestep resolution of 4  μs is used.
figure Images/TOP Trap/TOPSchematic.png
Figure 3.20 TOP circuit schematic. The FPGA outputs a sinusoidal voltage via an on-board, 10-bit DAC, which is then amplified by a variable gain amplifier (VGA). The signal then passes though a transformer, changing from high-voltage to high-current, through a 10  mΩ sense resistor and into a resonant circuit comprising the TOP coils and a tunable capacitor bank. The VGA gain signal is set by a PID feedback circuit, whose setpoint is input from the FPGA and and whose signal is obtained from the sense resistor, thus stabilizing the current. The TOP driver board has a USB input for programming ramp segments, as well is digital inputs to trigger ramping stages and to extinguish the signal. Also output from the board is the 250  kHz timing signal synchronizing the experiment.

3.3.4.2 Field Measurement

To quantify the fields generated, rubidium atoms are used as sensitive magnetometers. Due to the fine control and narrow linewidth of the optical frequency available in imaging, an absorption spectrum may be readily obtained by scanning the frequency of the absorption probe, and measuring the optical density of identical clouds. Two improvements in the electronics system were also necessary to make this possible (Section 3.4↓). First, the minimum pulse length of absorption light has been reduced from 100  μs to 8  μs, allowing imaging to take place in a nearly constant field (one period of the TOP oscillation is 100  μs). Secondly, the rotation of the TOP field is now synchronized with other components of the experiment, allowing repeated images to take place in the same nearly constant field. Figure 3.21↓ shows a calibration plot where the optical density is measured as a function of probe detuning for different values of TOP field. A weak TOP trap has been loaded in order to spin-polarize the atoms, and then the quadrupole gradient released while the rotating field is left on. The current is measured separately, and the field as a function of current can therefore be determined. Broadening due to the presence of mF = 1 atoms, as well as the variation of the magnetic field during the imaging pulse are accounted for by the fit.
figure Images/TOP Trap/Spectroscopy Calibration/Xaxis.png
Figure 3.21 Measurement of TOP fields using atomic spectroscopy. This plot yields a calibration of 3.92  G/A for the z-axis coils, and a similar plot (not shown) yields a calibration of 2.78  G/A for the x-axis coils. The width of the resonances grows partly due to the range of magnetic fields present during the pulse, but also due to a large fraction of atoms in the mF = 1 substate.

3.3.4.3 Turn-on and Turn-off

It is important that the TOP be turned on and off quickly to avoid unnecessary loss and heating. The turn-on occurs when the setpoint of the feedback circuit is set to some value, and the amplifier begins to drive power into the resonant circuit. Figure 3.22↓ shows measurements of the turn-on and turn-off of the z-axis TOP coil. The field turns on with an exponential time constant of τturnon = 360±20  μs and off with a time constant τturnoff = 300±20  μs. The turn-off of the TOP field is slightly faster as it is limited only by the quality of the oscillator, whereas the turn-on is limited by the bandwidth of the feedback circuit.
figure Images/TOP Trap/Turnon/TurnOnOffs.png
Figure 3.22 Turn-off (a) and turn-on (b) of the z-axis TOP Coils. The solid, oscillating lines are measurements of the coil current determined with a 10  mΩ sense resistor, and the dashed line is the rms value of the current. The rms current has been fit to an exponential function, and the time constant is determined to be τ = 360±20  μs (τ = 300±20  μs) in the case of the trap turn-on (turn-off). The turn-on of the TOP field is slightly slower as it is limited by the bandwidth of the feedback circuit used to control the field amplitude.

3.3.5 Compensation Coils

As the BEC experiment relies extensively on magnetic interactions, it is important to control background fields. Compensation coils create flat fields which serve to cancel background fields, and also to determine quantization axes when magnetically polarized atoms interact with laser fields. The vertical compensation field also serves to move the centre of the magnetic trap, compensating for the difference in position of the optical molasses and the magnetically trapped atoms. They are thus termed the compensation coils.
As described in Section 3.3.3.2↑, the mounting system for the quadrupole coils was designed to include mounts for three pairs of Helmholtz-oriented coils, intended to cancel spatially flat background fields. As part of the upgraded system, non-vertical compensation coils were made smaller, and mounted much closer to the atoms. In addition, the coils were built to be nearly Helmholtz in separation, and were oriented along the axes of the imaging beam and quadrupole coil bore. This orientation allows for more careful spectroscopy by allowing the coordinate systems defined by the magnetic fields and the laser beams to be accordant. Data for the three compensation coils is found in Table . The vertical compensation coils are driven by a BOP 36-12M bipolar power supply, and the other two coils are driven by home-built circuits. All coils can be switched on and off in approximately 1  ms, and are thus suitable for imaging spectroscopy.

3.3.6 The RF Coil and Evaporative Cooling

The previous section discussed magnetic trapping with the promise of further cooling via some other means. The other means is evaporative cooling. When atoms are loaded into the magnetic trap, they are typically a few microkelvin in temperature, and cannot be cooled further through contact with something colder. The final two or so orders of magnitude in cooling are achieved by selectively removing atoms from the cloud which carry more than the average kinetic energy of the sample, thus lowering the temperature. A summary of theoretical models for evaporative cooling has been presented in 14, and the most relevant aspects will be summarized here. The selective removal of energetic atoms is typically accomplished by lowering the trap depth such that atoms above a certain energy will leave the system. A more detailed look at the mechanisms for ejection in this system is given in Section 3.3.6.2↓ below.
The velocity distribution of atoms in a magnetic trap at temperature T is given by the Maxwell-Boltzmann distribution
(3.19) f(v) = 2π(mkBT)3v2exp( − mv22kBT)
where m is the mass of the atom. It should therefore be possible to decide the amount of cooling desired, and select a trap depth beyond which a single atom would carry a corresponding amount of energy. In practice, however, waiting for an atom to occupy such a high energy state could consume the tenures of one or even several graduate students. A more stringent limitation on the duration of the evaporation process is set by background processes such as background gas collisions and stray light absorption which limit the trap lifetime to something usually on the order of seconds or minutes. The experimentalist is forced to compensate by cutting deeper into the distribution, taking less energetic atoms and lowering the efficiency of the process. The great balancing act of evaporative cooling is choosing where to make this cut. To optimize cooling, both the evaporation rate and the efficiency must be high enough. By detailed balance, atoms in some section of the distribution must be introduced and removed at the same rate for the distribution to remain constant, and this process proceeds via elastic collisions in the cloud. For an evaporation process, a distribution truncated by evaporation will replace atoms in the truncated region at a rate approximately equal to the number of atoms in the tail times their collision rate, assuming the evaporative cut is not so deep as to greatly alter the distribution. The collision rate, in addition to background loss rate, is the other critical timescale in evaporative cooling. It is also important to note that in addition to transferring atoms to different energy states within the distribution, elastic collisions mediate the rethermalization of the cloud. That is, the removal of hot atoms results in a new Maxwell-Boltzmann distribution, but with a lower temperature. It has been shown that such a truncated cloud requires on average 2.7 elastic collisions per atom to fully rethermalize 37,38,39. The rate of elastic collisions at the centre of a cloud of trapped atoms is given by
(3.20) 1 ⁄ τel = nσv
where n is the density at the centre of the trap, σ is the elastic cross-section of the atoms and v is the velocity of an atom. The collision rate therefore increases with temperature and density. In most BEC experiments, both of these factors can be increased by adiabatic compression of the trap before evaporation. Adiabatic compression will be discussed further in Section 3.3.6.1↓.
To build a more detailed understanding of evaporative cooling and the important factors in its efficiency, I follow the work of 14, and consider a trap in which atoms of energy E > ηkBT are expelled from the trap, thus defining the factor η. For the distribution in Equation 3.19↑, and sufficiently high η, the fraction of atoms in the trap with energy larger than ηkBT can be approximated as 2e − ηη ⁄ π, and the corresponding evaporation rate is thus
(3.21)  =  − Nn0σvηe − η =  − Nτev
where N is the number of atoms in the trap, n0 is the peak density, σ is the elastic collision cross section, v = 8kBT ⁄ πm is the average thermal velocity and τev is defined as the time constant for evaporation. For evaporation to proceed quickly and efficiently, the collision rate should be maintained or increased.
Generally speaking, the goal of evaporative cooling is to arrive at Bose-Einstein condensation with the maximum number of atoms. Of primary importance is the amount of cooling that may be wrought per atom lost, and I therefore define
(3.22) α = d(lnT)d(lnN)
noting that evaporative cooling is essentially an exponential process. Table 3.1↓ lists how different quantities scale according to this parameter during evaporation. It is immediately clear that the tighter confinement of the quadrupole potential produces more rapid reduction in cloud size as cooling proceeds, thus raising the density and collision rates. A simple expression is found for α by assuming η to be large, thereby implying the distribution to be nearly unchanged from Equation 3.19↑. The average energy of an escaping atom is given by (η + κ)kBT, and α can thus be written as
(3.23) α = η + κδ + 3 ⁄ 2 − 1
where κ depends on η and the dimension of evaporation, and is usually between 0 and 1. Also, parametrizing trap loss as the “ratio of good collisions to bad collisions,” R = τloss ⁄ τev, and referring to Table 3.1↓, the collision rate can be shown to scale as
(3.24) d(nσv)dt ⁄ nσv = 1τel(α(δ − 1 ⁄ 2) − 1λ − 1R)
where the parameter δ has been introduced. In the case of a linear trap, δ = 3, and in the case of a harmonic potential, δ = 3 ⁄ 2. For the collision rate to increase its derivative must be positive and a minimum value of R can be defined to achieve so-called runaway evaporation.
(3.25) R\geqslantRmin = λα(δ − 1 ⁄ 2) − 1
If the experiment can be made to operate in the runaway regime, evaporation accelerates and becomes more efficient. Furthermore, the eventual success of evaporation will depend strongly and nonlinearly on the number and temperature of atoms present at the beginning of evaporation, as well as the trap configuration. Not all initial conditions can lead to condensation. It is also useful to define the quantity
(3.26) γ =  − d(lnD)d(lnN)
where D is the phase space density. The quantity γ is a generally accepted figure of merit for an evaporative process.
Quantity Exponent Exponent Exponent
(Harmonic Trap) (Linear Trap)
δ = 3 ⁄ 2 δ = 3
Atom Number: N 1 1 1
Temperature: T α α α
Volume: V δα 32α 3α
Density: n 1 − δα 1 − 32α 1 − 3α
Phase Space Density: D 1 − α(δ + 32) 1 − 3α 1 − 92α
Elastic Collision rate: 1 ⁄ τel 1 − α(δ − 12) 1 − α 1 − 52α
Table 3.1 Scaling laws for evaporative cooling in harmonic and linear (e.g. quadrupole) traps. Each quantity scales as Nx. Quantities scale considerably more favourably in a quadrupole trap.

3.3.6.1 Adiabatic Compression

Whether in a TOP or quadrupole trap, the efficiency of evaporation will depend ultimately depend on the number of elastic collisions that can be achieved per trap lifetime. In order to maximize this ratio, the trap is compressed immediately after loading to increase the collision rate. Table 3.2↓ summarizes the scaling of cloud parameters as a function of gradient strength under adiabatic compression, i.e., where phase space density is conserved. It is clearly advantageous to increase the collision rate with no penalty in phase space density, but there are limits. If the temperature becomes too high, p-wave collisions can lead to spin-changing collisions, leading to losses. This is mainly a problem in the quadrupole trap, as the rate of compression in the TOP trap is more likely to be limited by its trap depth and the resulting circle of death evaporation. Circle of death evaporation will begin around 693  μK for a 40  G TOP field. This effectively limits the cloud temperature to around 100  μK for efficient evaporation.
Quadrupole Trap TOP Trap
Temperature a2 ⁄ 3 a
Size a − 1 ⁄ 3 a − 1 ⁄ 2
Density a a3 ⁄ 2
Elastic Collision Rate a4 ⁄ 3 a2
Table 3.2 Scaling of cloud properties with gradient strength a for adiabatic compression in quadrupole and TOP traps. Scalings are derived through the application of the virial theorem while holding phase space density constant.

3.3.6.2 Evaporation Mechanism

In the case of the magnetic traps considered, evaporation is effected by producing a moveable region in space which changes the magnetic substate of the atoms to untrapped or anti-trapped states, thus expelling them from the trap. Because the region is moveable, and the trapping potential is spatially dependent, an effective “hole” can be cut into the trapping potential at an arbitrary potential energy corresponding to the energy of atoms to be evaporated.
To understand this ejection process, it is sufficient to consider a semiclassical picture, where an oscillating, perpendicular field exerts a torque on the magnetic dipole, changing the dipole’s orientation with respect to the guiding field. This radio-frequency field must be tuned to be resonant with the Larmor frequency of trapped atoms, which is
(3.27) ωLarmor = μBgB
where g is the Land g-factor and B is the magnitude of the guiding field. This applied field will have the effect of coupling atoms to different spin states, which is used to selectively eject atoms from the trap in RF evaporation.

3.3.6.3 Evaporation in the Quadrupole Trap

Because of its three-dimensional symmetry, the resonant surface of an RF field in a quadrupole trap is a spheroid with axes inversely proportional to gradient strength. Though this is in principle a three-dimensional evaporation surface, the polarization of the evaporating field means that evaporation will not occur near the top and bottom of the spheroid where the trapping and evaporating fields are aligned. Evaporation is effectively two-dimensional.

3.3.6.4 Evaporation in the TOP Trap

In a TOP trap, atoms may be released from the trap in a energy-selective way using two techniques. The first controllable loss mechanism is evaporation via the so-called circle of death. Atoms reaching the instantaneous field zero will undergo Majorana loss. The strength of the bias field will determine the location of the trap zero. The instantaneous field zero is located at
(3.28) x = B0Bsinωt
where ω is the frequency of TOP field rotation, B0 is the magnitude of the TOP field and B is the gradient field strength. The potential energy at the field zero, relative to the trap minimum, is found by substitution into Equation 3.10↑
(3.29) V = μBgmFB04
and the corresponding temperature of the cut is
(3.30) Tcut = μBgmFB02kB
The shape of this evaporation surface in space is essentially a toroid, as the field zero rotates. This creates a two-dimensional evaporation surface. Circle of death evaporation is typically employed while compressing the trap by reducing the TOP field, when lowering the trap depth is a necessary consequence. The concurrent processes of compression and evaporation has been dubbed “compraporation.” 17 It is slightly more efficient to add a radio-frequency (RF) field at the same energy to supplement the circle of death. The surface over which this evaporation occurs is an ellipsoid whose radius is so large that it appears nearly planar on the scale of the atom cloud. The rotation of this plane around the atoms gives a nearly cylindrical evaporation surface. A diagram of this is shown in Figure 3.23↓, and the temperature corresponding to this evaporation surface is given by
(3.31) Tcut = 2kBμBB0(hν − μBB02)2
figure Images/TOP Trap/TOPEvaporation.png
Figure 3.23 A schematic showing the evaporation surface of RF evaporation in a TOP trap. The plane of the page is the plane of TOP bias field rotation 14. The instantaneous field zero rotates about the cloud, and atoms are evaporated from the edge of the cloud furthest from the zero, where B = hνRF ⁄ μ. The time-averaged resonant (evaporation) surface is approximately barrel-shaped.

3.3.6.5 The RF Coil

Atoms trapped in the TOP trap must be spin polarized in the horizontal plane. A single loop of AWG27 wire is wrapped around the cuvette, approximately 2  cm from the atoms. This coil produces magnetic fields perpendicular to the magnetic moment of the trapped atoms, which are aligned to the rotating TOP bias field.

3.4 Electronics and Control

A further upgrade to the system was the installation of the National Instruments PCI-6229 as the master controller of the experiment. With this upgrade, the clock speed of the experiment was increased from 10  kHz to 250  kHz, reducing the shortest interval for experimental action from 100  μs to 4  μs. Furthermore, the experiment is now synchronized with the rotation of the TOP field, allowing for imaging at specific TOP field values (also measurement of the field), as well as repeatable and adjustable turn-offs of the trapping fields with respect to the phase of the TOP fields. Moreover the RF field frequency and quadrupole field amplitude are controlled directly by waveforms generated by the master control. This has eliminated much uncertainty in the timing of these fields.
The master clock is produced by the TOP electronics, and this clock is used to step through buffers of analog and digital outputs in the PCI-6229, as well as two PCI-6713 analog cards. Altogether, the system has 32 digital and 20 analog outputs.

4 Experimental Cycle and BEC

In this chapter I will describe the ramp to BEC using the upgraded vacuum system, laser system, TOP electronics and timing circuitry. The main result of this chapter is that the experimental cycle has been shortened from approximately 75  s to 45  s, and the atom number at the onset of condensation has been increased from 2 × 105 atoms to 6 × 105. Both are significant improvements, and results are compared to those reported in 6.
I will also discuss a key figure of merit for the proposed experiments, which is the predictability of the momentum at the time of release from the trap. This value is measured, and a calibration scheme is proposed and tested to compensate for large uncertainties.

4.1 Lower MOT Loading

The upper MOT has been left essentially unchanged in this iteration, and is measured to contain approximately 109 atoms at about 150  μK. This temperature is measured, as are others reported in this document, by determining via imaging the size of the cloud at different times during a time-of-flight expansion. The rms size of the cloud is related to its temperature by the equation
(4.1) xrms(t) = x02 + kBTmt2
where x0 is the rms size of the trapped cloud, kB is Boltzmann’s constant, T is the temperature, m is the mass of the atom and t is the time of free expansion. The lifetime in the trap is 1  s. Atoms collected in the upper MOT must be transferred from the high-pressure vapour cell to the lower chamber, which has a lower pressure and is thus suitable for evaporative cooling. When the loading is to begin, the gate valve is pneumatically opened by the control software, and a line-of-sight path is opened between the MOT in the upper chamber and the centre of the magnetic trap in the lower chamber. To transfer the atoms, a push beam is used to propel atoms between the chambers.
The loading phase typically takes 10 to 20 seconds, depending on the slight variations in laser powers and alignments from day to day. The software terminates the loading after either a fixed amount of time, or after the MOT fluorescence has achieved a certain level of brightness. To measure its brightness, the MOT is imaged by a lens onto a photodiode connected to the computer. Neither technique is found to be more effective than the other at stabilizing loading from shot to shot, but regulating the loading time is more effective at preventing drift. The reason for this is that the brightness measured depends much more strongly on laser power than the number of atoms collected in the MOT. Small variations in laser power may not strongly affect the loading rate, but strongly affect the photodiode signal.
It is estimated that the lower MOT contains up to 2 × 109 atoms, though high optical densities and mixed magnetic polarization make it difficult to accurately measure this number.

4.2 Magnetic Trap Loading and Evaporation

After the lower MOT has been loaded, its temperature is typically close to the Doppler temperature, and has been measured to be 150±15  μK. Sub-Doppler cooling is achieved by extinguishing the magnetic fields and pulsing on the MOT beams for 2  ms at  + 22  MHz detuning, then 0.1  ms at  + 26  MHz and finally 1.1  ms at  + 30  MHz. This “corkscrew molasses” phase is empirically optimized by adjusting the detuning and duration of these pulses: larger detunings result in less cooling, but also less heating due to light-assisted collisions 36. When beams are well-aligned smaller clouds can be cooled to around 10  μK, but more typically, temperatures of 50±20  μK are observed. Small variations in beam power balance and the density distribution of the MOT strongly affect the efficiency of this cooling process, and the time-consuming process of optimization is not deemed worthwhile when loading a poorly matched quadrupole trap. A 700  μs burst of optical pumping light aligns the atoms with a guiding field which acts as a quantization axis. Though only a weak guiding field is necessary, a field of 13.3±0.1  G is found to be optimal, and around 75% loading is achieved, typically leaving between 1.0 × 109 and 1.5 × 109 atoms in the magnetic trap. The magnetic trap is turned on quickly in order to prevent further expansion of the cloud, as a larger cloud acquires more potential energy in the trap. The atoms are loaded into a quadrupole potential rather than a TOP trap due to the more efficient evaporation possible in a quadrupole trap due to both its tighter confinement and its greater trap depth, which allows higher temperatures and greater collision rates.
The atom cloud should in principle be approximately centred on the magnetic field centre after the MOT cooling stage. The magnetic field centre is not, however, the equilibrium position for a trapped atom cloud, as the shape of the quadrupole trapping field is distorted by gravity. If the cloud is not loaded into the quadrupole trap at its equilibrium position, the extra potential energy will be converted to heat and hinder progress towards condensation. The vertical compensation field is therefore employed to shift the centre of the trap on loading. The cloud is loaded into a gradient of 70.5  G/cm, and a 2.0  G bias field is applied, which translates the trap approximately 0.28  mm. The value of the applied field is optimized empirically, as the optical pumping pulse pushes the cloud slightly before the trapping.
The most severe penalty associated with loading the quadrupole trap is the difficulty in mode-matching the trap. Large molasses clouds of around 50  μK and 1.5 × 109 atoms are loaded into the magnetic trap reach temperatures of approximately 400  μK. Clouds could be loaded into weaker TOP traps at temperatures of 100  μK, and slightly higher phase space density.
Once the atoms are loaded into the quadrupole trap, the trap is adiabatically compressed by ramping up the field strength. The trap is compressed in 1.2  s and the gradient is increased from 70.5  G/cm to 235  G/cm, ideally raising the temperature by a factor of 2.23, the density by a factor of 3.33 and most importantly, the elastic collision rate by a factor of 4.98 (refer to Table 3.2↑ for scaling relations).
Evaporation begins immediately after the adiabatic compression. The general procedure for designing the RF ramp in the quadrupole trap completely is empirical. The beginning frequency of the RF ramp was determined by setting η (defined in Section 3.3.6↑) according to the theory, such that the collision rate (determined using Equation 3.20↑) during evaporation would increase. For a linear potential with a ratio of good to bad collisions of about R = 200, optimal evaporation efficiency is achieved by setting η to approximately 6. The compressed cloud has a temperature of approximately 900  μK, and the beginning RF frequency is thus
(4.2) ν = ηkBT2h = 56  MHz
where T is the temperature of the cloud, kB is Boltzmann’s constant and h is Planck’s constant. The ramp is then divided into 5  MHz segments, and the rate of RF sweep through each segment is optimized for collision rate at the end of the segment. The resulting segments are then fitted to a polynomial and a smooth ramp, shown in Figure 4.1↓, is used.
figure Images/RF/Quad_Evap.png
Figure 4.1 Ramp for RF evaporation of atoms in quadrupole trap. The ramp has been optimized for collision rate in 5  MHz segments and the result has been fitted by a third-degree polynomial: f = 60 − 5.007t + 0.913t2 − 0.115t3
The TOP trap must eventually be loaded as Majorana losses cause loss and heating at rates that increase as the cloud temperature decreases. Majorana trap loss has been studied in 15, and an empirical model has been extracted in 16:
(4.3) Γm = 1.85m(μBBkBT)2
where Γm is the loss rate, B is the magnetic field gradient, μB is the Bohr magneton and m is the mass of the atom. It was found that cooling became inefficient around 200  μK, at which point the lifetime of the trap is expected to be around 10  s. It was at this point that atoms were transferred to the TOP trap.
A summary of the gain in phase space density per loss in atom number in this stage as well as the two following it is shown in Figure 4.3↓. The log-log slope for this stage is -2.3. The cooling efficiency during this stage could possibly be higher, but it is suspected that the trap is polluted by atoms in the also-trapped F = 2,  mF = 1 magnetic substate. Cloud sizes were consistently larger than expected from temperature and trap measurements, and spectroscopic measurements have estimated up to 1 ⁄ 3 of the atoms to be in this state. Atoms in the wrong trapped state can participate in spin-exchange collisions, leading to substantial loss and heating.

4.3 The TOP Trap, Compraporation and Evaporation

When evaporation in the quadrupole trap can no longer proceed efficiently due to losses, the atoms must be loaded into the TOP trap. When the TOP trap is first turned on, the TOP fields are 40  G and the quadrupole field is at its maximum value of 235  G/cm, yielding trap frequencies of νx = 39  Hz, νy = 55  Hz, and νz = 77  Hz. The transfer to the TOP trap is inherently lossy, as the transition from a zero-field “hole” in the centre to a circle of death sweeps the field zero through the cloud. Furthermore, the traps are rather poorly matched due to the much tighter confinement of the linear quadrupole potential. Improved matching should be possible by increasing the gradient (see Table 3.2↑ for scaling relations), but the circle of death cannot tolerate much higher temperatures. Attempts were made to achieve better matching in this way, and also by dynamically changing the gradient during loading, but no measureable improvement could be made over simply turning on the TOP.
To evaluate the success of the process, it is possible to compare to the ideal case. If the transfer is adiabatic, entropy is constant. The phase space density may be written as 50:
(4.4) nλdB3 = exp(52 + δ − SkBN)
where S is entropy, kB is the Boltzmann constant, N is the atom number and δ is the trap shape parameter introduced in Section 3.3.6↑. Under adiabatic transfer, the phase space density will therefore only depend on δ, and in a transfer from a quadrupole (δ = 3) to a harmonic trap (δ = 3 ⁄ 2), the phase space density is expected to decrease by a factor of e3 ⁄ 2 ≈ 4.5. A 170±10  μK cloud was loaded into a TOP with a B0 = 40  G bias field, and the resulting temperature was measured to be 95±10  μK, corresponding to a factor of 2.3 loss in cloud-averaged phase space density. A 17±4% loss of atoms makes the total loss in phase space density 2.8±0.3. The discontinuity evident in Figure 4.3↓ includes an additional factor of 22 as the ratio of peak to cloud-averaged phase space density is 22 in a harmonic trap and 8 in the quadrupole trap.

4.3.1 Compraporation

To move efficiently in the evaporation process, the trap may be further compressed by reducing the rotating bias field. The trap frequency is proportional to B0 − 1 ⁄ 2 (Equation 3.10↑). The circle of death, however, also provides an evaporation surface, so compression using the TOP field will also lower the trap depth, incurring evaporation. Circle of death evaporation is essentially one-dimensional, taking place on a ring (a toroid) in space. RF evaporation in the TOP trap, however, may be considered two-dimensional, as the evaporating surface is nearly cylindrical (barrel-shaped). For this reason, during the compraporation an RF field is added such that it provides an evaporation surface just inside the circle of death. The RF frequency corresponding to the circle of death can be found by noting that the instantaneous field value on the circle of death oscillates between B = 0 and B = 2B0:
(4.5) hνCOD = 12μB(2B0)
Pinning the RF frequency to the TOP field, the compraporation can proceed as a function of only one variable - the bias field strength. Similar to the previous phase, the optimization of the compraporation proceeds by dividing the scan range (from maximum to minimum bias field) into segments, and choosing the duration of each to maximize collision rate at the end of the segment. The resulting field ramp is a simple linear ramp lasting approximately 17  s 52.

4.3.2 Final RF Stage

After the TOP has been compressed, the cloud has reached 5 × 10 − 3 in phase space density, and a final RF sweep is required to reach condensation. The final ramp is a simple exponential lasting 2.5  s, and is determined by only two parameters. The starting frequency is set by the final frequency of the previous stage, and only the final frequency and the exponential time constant are varied. The optimized ramp is
(4.6) ν(t) = 2.5 + 2.5e − t ⁄ 3 MHz
where t is the time in seconds. This is optimized in much the same way as the other stages, searching for the parameters which yield the maximum efficiency. These values are often tuned from day to day to compensate for small changes in initial conditions and also, it is suspected, in the magnitude of the rotating bias field. The temperature of the final cut is extremely sensitive to the bias field value. Consider the end of this ramp, where ν = 3.59  MHz, and B0 = 4.5  G. The temperature of the evaporating surface is given by Equation 3.31↑ as approximately 3  μK, and the derivative is 10  nK/mG. Again, a summary of phase space density versus atom number for this and other stages is provided in Figure 4.3↓.

4.4 BEC

Using the new laser system and TOP driver, Bose-Einstein condensation transition has been observed at a maximum of 5.77 × 105 atoms, and 534  nK. This BEC is substantially larger than with the previous construction of the experiment 6, and the experimental cycle has also been shortened from approximately 75  s to approximately 45  s. Though it has not been quantified, a more significant improvement has been made in the day-to-day stability of the entire experiment. No longer does the lab lose days at a time to MOT realignment.
figure Images/BEC/PSDPlot.png
(a) Plot showing the final stage of RF evaporation. Condensation occurs at 534  nK with 5.77 × 103 atoms. Phase space density (D) has been calculated using the apparent temperature (i.e. the expansion rate), which gives inaccurate values when D\gtrapprox1. The figure should instead reflect the occupation of the ground state.
figure Images/BEC/BECTransition.png
(b) Profile plots. Clouds have expanded 10.8  ms. The nearly pure condensate in the final profile has 149±5 × 103 atoms. The frequency noted is the frequency of the RF field just before the release from the trap.
Figure 4.2 One example of Bose condensation in the redesigned system. Trap frequencies are νz = 203  Hz, νy = 124  Hz and νx = 94.6  Hz.
Figure a↑ shows a plot of the phase space density versus atom number for the largest BEC observed. The phase transition occurs at a transition temperature of Tc = 534  nK and Nc = 5.77 × 105. Figure b↑ shows the evolution of the one-dimensional density profile as the phase transition is crossed. The cloud is initially a thermal cloud and the profile is approximately Gaussian in shape. As the frequency of the RF field is lowered, atoms begin to accumulate in the lowest-energy state, and the profile takes on a bimodal appearance. Finally, the nearly pure condensate has a parabolic profile as described by the Thomas-Fermi approximation.
figure Images/BEC/Efficiency.png
Figure 4.3 Plot of atom number and phase space density on the path to BEC. The slopes of the best fit lines drawn give the value of γ = d(lnD) ⁄ d(lnN) (see Section 3.3.6↑), which show values of greater than 2. Points near the end of the compraporation and evaporation ramps have been excluded from the fit as further evaporation becomes inefficient. Error bars indicate run-to-run spread in atom number.

4.5 Momentum Stability

The momentum enhancement inequality (Eq. 2.1↑) offers the possibility of violation with minimal theoretical overhead. If a momentum measurement can be effectively made, the inequality may be violated without any detailed knowledge of the potential other than its sign, or any knowledge or assumptions about the dynamics of the collision or during the time after the collision. For this effective measurement of momentum to be made, it is essential that both the initial and final momenta of the clouds be well known. In this system, a single measurement is made, which is the position of the cloud at a known time. Considering a cloud undergoing no collision, this position measurement will be indicative of the momentum at the time of trap turn-off, provided the scatter in trap position is small.
Ideally the aim would be to produce clouds that from shot-to-shot have momenta with negligible variation, so that collided and uncollided clouds may be compared directly. In the absence of such stability, the initial momentum distribution of the colliding cloud may be post-dicted by the interference pattern if there is sufficient confidence in the theoretical model. As will be demonstrated in Chapter 5↓, the latter approach has failed, and the variation in cloud momentum must be studied. Figure 4.4↓ shows a measurement of the momentum scatter of Bose-condensed atom clouds, where clouds of identical preparation are dropped and their positions measured after some time of flight. In addition to random scatter in momentum, there is observed to be a slow drift in the mean momentum with a time constant of τ = 15.5  minutes, which may be related to thermal effects observed in Section 3.3.3.5↑. This drift accounts for nearly half the variance in the data, and is typical.
Though it is only possible to measure the position of the cloud, study of cloud scatter versus expansion time allows decoupling of the scatter due to momentum and due to positional uncertainty. Figure 4.5↓ shows the scatter measured in 95 datasets for different expansion times, comprising 859 individual images. Datasets for both the new and old coils are included, as the scatter was not measurably affected by this change. The scatter in velocity is determined to be 0.343±0.006  mm/s and the scatter in position is 0.63±0.07  μm. This uncertainty in velocity is significant compared to the velocity width of a typical condensed cloud (around 1.7  mm/s for the data in Chapter 5↓), and clearly dwarfs any observable momentum enhancement. As will be demonstrated in the following chapter, even this level of momentum uncertainty can bury the desired effect. The remainder of this section discusses a proposed solution to this problem, which is to make copies of the cloud before the collision which may share the perturbations causing the scatter, and which can be used as references so the pre-collision momentum is known.
figure Images/BEC/Scatter/CloudScatter.png
Figure 4.4 Sample histogram of atom cloud centre positions after 13.85  ms time of flight, and 44 repetitions. The mean rms width of the clouds in this trial was 1.1  mm/s, indicated by the error bars in (a). The measurement uncertainty of the cloud centres is approximately equal to the marker size. Panel (a) shows a plot of the cloud position as a function of time, with an exponential function fitted to the drift. Panel (b) is a histogram of the cloud positions in (a), and (c) is a histogram of the same data with the fit subtracted. This shows that the drift accounts for about half the variation in cloud position. The time constant of the fit to the drift is 15.5  min. The corrected histogram in panel (c) shows that the scatter in velocity is 0.4  mm/s.
figure Images/BEC/Scatter/PositionScatter.png
Figure 4.5 Collected scatter data for 95 datasets comprising 859 individual images. Datasets have been binned into 1  ms intervals for clarity of display. The plot shows that the scatter in velocity has a standard deviation of 0.343±0.006  mm/s and the scatter in position is 0.63±0.07  μm. Error bars are proportional to N − 1 ⁄ 2, where N is the number of points used in the dataset. Drift similar to that shown in panel (a) of Figure 4.4↑ has been subtracted from individual datasets where it was significant.

4.5.0.1 Stern-Gerlach Measurements

In order to make “copies” of a trapped condensate, one may take advantage of the magnetic structure of the atoms. In the previous chapter, the production of a monochromatic radio-frequency (RF) field to flip the magnetic substates of atoms in a particular region of space to untrapped states for evaporation was discussed. The same coil used to this end may also be used to flip all of the atoms simultaneously into a superposition of magnetic substates, which will then feel differential forces and spatially separate in the inhomogeneous magnetic field of the magnetic trap. The separation of different magnetic moments in an inhomogeneous magnetic field was most famously employed by Otto Stern and Walther Gerlach 41, and is known as the “Stern-Gerlach” effect. As it is desired to couple all of the atoms in the trap rather than a select few, the condition of monochromacity must be reversed, and the RF applied in a short, broadband burst. The magnetic moment of the atoms subjected to an RF field will oscillate at a Rabi frequency given by
(4.7) ωR = 12gμBBRF
where BRF is the amplitude of the applied RF field. The resulting populations of the magnetic substates measured in the basis of the guiding field is found by rotating the S = 2 spin vector using the rotation operator generated by Wigner’s formula 11:
dmm(j) = k( − 1)k − m + m(j + m)!(j − m)!(j + m)!(j − m)!(j + m − k)!k!(j − k − m)!(k − m + m)!
(4.8)  × (cosζ2)2j − 2k + m − m(sinζ2)2k − m + m
where ζ is the angle through which the vector is rotated, and the sum is taken over k where no factorial in the denominator has a negative argument. Figure 4.6↓ shows the predictions of this equation for an S = 2 system.
figure Images/RF/RabiSpin3.png
Figure 4.6 Rabi oscillations of magnetic spin state populations for an S = 2 system, produced by a pulse of varying area.
The maximum duration of the pulse is determined by the required bandwidth, as a broadband pulse is required to couple the entire cloud which resides in a spatially varying magnetic field. The cloud size is approximately 5  μm, and the field gradient 250  G/cm in the weak direction, meaning the atom cloud will experience field varying over roughly 100  mG. Referring to Equation 3.27↑ for the Larmor frequency, the frequency width of the pulse should thus be approximately 175  kHz, and the pulse duration should therefore be shorter than 36  μs. Figure 4.7↓ shows the result of a 6  μs pulse with a centre frequency of 5  MHz. After the spin states are created, the trap is left on so that the different spin states can separate. The mF = 2 spin state will remain in place, while the other states will begin to move downwards. The mF = 1 atoms have an equilibrium trapped position below that of the mF = 2 atoms, the mF = 0 atoms are untrapped, and the mF =  − 1,   − 2 atoms are anti-trapped. In addition to vertical forces, the atoms experience a force in the horizontal plane as well, giving the horizontal velocity gradient apparent in Figure a↓. This is due to asymmetric forces during the trap turn-off and the direction of the added velocity depends on the TOP phase at turn-off. Figure 4.8↓ shows the relative populations of these spin states as the pulse length is increased. As predicted, the pulse is no longer effective if the length exceeds a few tens of microseconds.
figure Images/BEC/SG/SG03_AnUnCrop.png
(a) Image of clouds outcoupled by an RF pulse. The horizontal velocity is due to forces during field turn-off, and depends on the TOP phase.
figure Images/BEC/SG/SGPlot.png
(b) Integrated profile
Figure 4.7 Outcoupled clouds. After condensation, the cloud is subjected to a 30-cycle (6  μs) pulse of 5  MHz radiation. The magnetic trap remains on for 1 additional ms to allow the different magnetic substates to separate. The clouds have then been released from the trap and allowed a further 11.1  ms of free expansion.
figure Images/BEC/SG/RabiFlopPlot.png
Figure 4.8 The oscillation of magnetic state populations under the influence of an RF pulse centered at 5  MHz. Populations are similar to theoretical predictions for the first several microseconds, at which point decoherence begins to degrade the signal. A period of oscillation is estimated to be τ = 30±2  μs, corresponding to a Rabi frequency of ωR = 2π × 33(2) kHz.
As the source of the scatter is unknown, it may be profitable to postdict the initial momentum of the collided cloud using one “copy,” or even two, three or four. I will restrict this study to two copies, with the aim of fitting all of the clouds in a single image, such that they have minimal overlap. To optimize the RF pulse, note that the maximal size for a cloud bracketed by two others is achieved when the population of mF = 1 or mF =  − 1 is maximized. Ignoring the mF =  − 1 maximum, this is achieved with a pulse area of π ⁄ 3 radians.

4.5.0.2 A Calibrated Position

To test this technique as a means to reduce scatter, a condensate was produced, and the RF field was pulsed as above. After 2  ms of further hold in the trap, and 22.3  ms of freefall, the clouds were imaged. The data consist of 25 images similar to that shown in Figure 4.9↓. To postdict the velocity of the mF = 1 cloud, the expected position is calculated by parameters analogous to the relative and centre-of-mass coordinates.
(4.9) m1p = m2 − (m2 − m0)m2 − m1m2 − m0
where m1p is the predicted position of the mF = 1 cloud, mn is the measured position of the mF = n atom cloud, and mm − mn denotes the mean separation between the mF = m and the mF = n clouds. The “centre-of-mass” in this case is set by the position of m2, and the relative coordinate is applied by fixing the ratio of the distance between the predicted position and the adjacent clouds. Figure 4.10↓ shows the fits to obtain correlations between different atom clouds, and Table 4.1↓ gives a summary of the scatter of, and the correlations amongst the different cloud positions. The scatter can be reduced by a factor of 2 by comparing to one adjacent cloud, and a total factor of 5 by comparing to two adjacent clouds.
These results provide some insight into the source of the scatter. Since the mF = 0 cloud experiences scatter similar to the other clouds, it is not likely that the scatter results from the details of the trap turn-off. There is also a large component of the scatter that is common-mode, suggesting a centre of mass velocity present before the RF pulse. Finally, Figure b↓ shows a correlation in the spacings between clouds which may be due to the position of the cloud at the time of the pulse, and thus the strength of field experienced during the cloud separation.
figure Images/BEC/SGFixit/3AtomClouds.jpg
figure Images/BEC/SGFixit/3AtomCloudProfile.png
Figure 4.9 Absorption image of mF = 2,  1,  0 clouds after 2  ms of Stern-Gerlach separation and 22.3  ms time of flight. The spacing has been optimized such that the clouds have maximum separation while still fitting in a single image with the desired 3  μm pixel size.
figure Images/BEC/SGFixit/mf1_vs_mf2_and_mf0_plot.png
(a) Correlation in scatter between mF = 1 and the two adjacent clouds
figure Images/BEC/SGFixit/residuals_plot.png
(b) There is correlation in the cloud spacings as well as positions, suggesting a breathing mode.
Figure 4.10 The positions of the three clouds, mF = 2,  1,  0 are studied for correlations.
Cloud Standard Deviation
in Velocity (mm/s)
mF = 2 0.389
mF = 1 0.340
mF = 0 0.378
(mF = 2) − (mF = 1) 0.195
(mF = 1) − (mF = 0) 0.163
m1p − (mF = 1) 0.069
Table 4.1 Correlations in position of atom cloud “copies.” The final row represents the standard deviation of the mF = 1 cloud position about its calibrated position, denoted by m1p and described by Equation 4.9↑. Calibrating with one adjacent cloud provides a factor of 2 reduction in scatter, while calibrating against two clouds provides a factor of 5 reduction.

5 Observation of Momentum Enhancement and Interference

As was discussed in Section 2.2↑, the momentum distribution of a colliding atom cloud is predicted to exhibit non-classical enhancement of momentum under certain conditions, and may be used to reconstruct the momentum wavefunction of the cloud at the time of collision. A series of experiments have been attempted in order to test these predictions, and wavefunctions have been extracted.

5.1 Experimental Setup and Procedure

A nearly pure condensate of around 60 × 103 atoms is created in a fully compressed trap, which has a frequency of 150  Hz along the dimension of the experiment, and 212  Hz and 106  Hz along the other directions. After the condensate has been produced, the trap is relaxed by reducing the quadrupole field from 235  G/cm to 179  G/cm, reducing the trap frequencies by the same factor. The main purpose of this adiabatic expansion is to improve the shot-to-shot stability of the cloud’s velocity as it leaves the trap, but has the added benefit of decreasing the barrier to cloud size ratio. The cloud is then released and allowed to propagate for some variable amount of time before encountering the dipole barrier. When some fraction of the cloud has traversed the barrier, the beam is quickly turned off and a sharp feature remains in the phase of the spatial wavefunction of the condensate. The cloud is then allowed further expansion so that the time of flight will produce a momentum measurement. A schematic of these experimental stages is given in Figure 5.1↓.
The dipole beam can be varied in its detuning about resonance for over 1  THz in either direction, by changing both current and temperature of the diode laser. In practice, the current is set near maximum, and the temperature is adjusted to tune the diode to one of several single-mode regions spaced a few tens of GHz apart. For the data presented in this thesis, the detuning of the barrier beam was close to  + 148  GHz from the D2 line which leads to a scattering rate of 1.27 × 10 − 4s − 1 at desired intensities. The beam is brought to a focus approximately 250  μm below the centre of the atom cloud at the time of its release. This corresponds to about 7.2  ms of freefall before a released atom cloud would encounter the barrier. The position of the barrier may be adjusted a small amount by vertically translating the final focussing lens, which is mounted on a translation stage. This is imprecise on the scale of microns, and quickly begins to distort the focus due to aberrations in the lens. A more precise technique is realized by applying a weak vertical field using the compensation coils, which can be used to translate the magnetic trap with a precision of  ~ 2  μm. The barrier position can then be measured by selecting a laser power high enough to reflect the atoms, and scanning its shutoff time while noting when falling atoms are blocked or transmitted. Further tuning of the barrier position at the time of collision may be done through study of the interference pattern itself. Explanation and data on this technique are presented in Section 5.2.3↓.
figure MugaPaper/WhyNotTryAgain/Fig1/Experiment.png
Figure 5.1 Schematic of experiment: The Bose-Einstein condensate is released from its trapping potential (a); and allowed to undergo ballistic expansion (b); the expanding atom cloud collides with a narrow optical potential (c); which is turned off when part of the cloud has traversed it (d); and the cloud undergoes further expansion (e); after which |ψ|2 is representative of momentum distribution at the moment of barrier turn-off.

5.2 The Dipole Barrier

This section contains work done to study the shape, location and intensity of the barrier after installation in the experiment.

5.2.1 Width

Although the barrier has been studied extensively before its installation, it is desirable to have an independent measurement of the focused spot as it is deployed in the experiment. Because the barrier beam propagates along the imaging axis, and because the barrier is focused in the imaging plane, it is possible to use the optical system described in Section 3.2.6↑ to image the focus of the barrier onto the camera CCD. Figure 5.2↓ shows an image of a barrier beam at its focus, as well as a Gaussian fit to determine the width. The cut is taken at the narrowest point in the beam - some points are as wide as σ = 2.6  μm.
figure Images/Barrier/Width/BarrierWidth.png
Figure 5.2 The cross-section of the barrier beam at its focus, imaged by the camera. Inset is the profile cut by the line in the figure. The camera pixel size is 3.02±0.05  μm. The beam is assumed to be Gaussian, and the fit indicates that the rms width is σ = 2.37  μm.

5.2.2 Intensity

In principle, the power required to write a π phase shift can be calculated with knowledge of the barrier detuning and focus size. In practice, the very small power required, coupled with losses in the system make it difficult to measure this value precisely with a calibrated photodiode or similar. A better method to determine the power reaching the atoms is to study the interaction between the atoms and the potential. A clear and easily measureable interaction with the atoms is their reflection from the barrier. The atoms should be reflected from the barrier when their kinetic energy is equal to or less than the peak of the optical potential. Using Equation 3.7↑ for the dipole potential, this can be expressed as
(5.1) \hboxKEatoms = 3πc22ω03(ΓΔ)2Imax
where Imax is the peak intensity of the laser beam. From this relation, if the shape of the barrier is known, the power can be reduced by the correct factor. The phase shift written on the wavefunction as a result of the interaction with the barrier is given by
(5.2) φ =  − ∞U(t)dt
and for a cloud moving over the barrier at velocity v,
(5.3) U(t) = exp( − (vt)22σ2)3πc22ω03(ΓΔ)2Imax
where σ is the rms width of the barrier. Substituting Equation 5.1↑ into Equation 5.3↑, and then inserting the result into Equation 5.2↑, the phase shift corresponding to the reflective barrier can be determined to be
(5.4) φreflection = π2mvσ
To achieve a π phase shift, the barrier intensity should therefore be reduced by a factor of mvσ ⁄ ℏ2π. Figure 5.3↓ shows the results of such a calibration. In this case the barrier width is σ = 2.45±0.1  μm and, the cloud velocity has been measured to be 56.8±0.5  mm/s, giving an adjustment factor of 73±2. Furthermore, by noting that the detuning is Δ = 148  GHz, and that the transverse width of the barrier is w = 300  μm, the optical power transmitted to the vacuum chamber may be estimated. This technique is used to bring the barrier power close to the desired value, but more fine tuning may be achieved by either comparing the measured momentum distribution with theory, or estimating the loss in kinetic energy of the cloud.

5.2.2.1 Central Dip

Figure 5.4↓ shows a series of images taken where the power of the barrier is increased from image to image. By matching the features in the clouds to the features in the theoretical curves, it is possible to make an estimate of the barrier power. Most notably, the descent of the large central dip to its lowest point should indicate a π phase shift.
figure Images/Barrier/reflected_cloud_withwords.png
figure Images/Barrier/Intensity Calibrati/BarrierIntensityCalibration.png
Figure 5.3 Calibration of barrier intensity using interactions with the atoms. Atoms are released from the trap and collide with the barrier after 5.7  ms. After a further 3.7  ms of expansion, an image is taken. Atoms with kinetic energy higher than the potential will be transmitted, and those with lower energy are reflected. Panel (a) shows a cloud that has been partially reflected, and panel (b) shows the result of measuring the reflected fraction as a function of barrier intensity. The red curve is a fit to the error function. The centre indicates the point where the barrier potential is equal to the average kinetic energy of the cloud, and the breadth is indicative of cloud temperature.
figure Images/Barrier/Intensity Calibrati/PowerSeries.png
Figure 5.4 Tuning of barrier intensity using numerical models. Each panel shows a cloud after collision plus expansion, with a barrier whose intensity is increased from frame to frame. By comparing the images with numerical models, the intensity calibration for the barrier is estimated.

5.2.2.2 Slowing of the Cloud

The barrier plus gravity form a conservative potential through which the atom travels, and atoms passing completely over the barrier should experience no net loss in kinetic energy. Atoms that sit upon the optical potential when it is removed however, have given up kinetic energy by climbing the potential, and can no longer regain it by sliding down the other side. The result is a decrease in the overall kinetic energy of the cloud. This loss of kinetic energy can be calculated for a Gaussian cloud and barrier as
(5.5) ΔKE =  − ∞|ψ(x)|2U(x)dx
where U(x) is the optical potential and |ψ(x)|2 = 12πσψexp( − x2 ⁄ 2σψ2) is the normalized wavefunction of the atom cloud. Also, the optical potential required to produce an Nπ phase shift in a cloud travelling at a velocity v may be written
(5.6) U(x) = v2πσbNπexp( − x2 ⁄ 2σb2)
where σb is the rms width of the dipole barrier. Completing the integral in Equation 5.5↑,
(5.7) ΔKE = π2vNσb2 + σψ2
and the corresponding shift in position can be calculated using experimental parameters. Assuming the shift in kinetic energy to be a small perturbation and keeping only the first order term,
(5.8) Δx = 1.53Nμm
Figure 5.5↓ shows a histogram of collided and uncollided clouds for a collision time of 7.5  ms and a subsequent expansion of 22.5  ms. The shift of 5.3±1.5  μm suggests a barrier applying a phase shift of 3.5±1.0π, according to Equation . A correction must also be made if the final barrier position is not at the centre of the cloud, but this will not be treated here.
figure Images/Momentum Enhancement/Collected Unmod/ColAndUncHist.png
Figure 5.5 Histograms showing the centre-of-mass positions of 148 (a) and 150 (b) clouds dropped 29  ms. The clouds in panel (b) have collided with an optical barrier after 7.8  ms of flight. The clouds which have collided have been shifted by 5.3  μm, corresponding to an approximately 3.5π phase shift. Some drift has been subtracted from the positions.

5.2.3 Location

The interference pattern also yields direct information about the position of the barrier at the time of collision. Due to the interference at the site of impact, a notch will appear at the site of the barrier when the collision was ended.
Final adjustments in the collision can be made choosing a phase shift large enough to provide a large central notch, and by noting that the position of the barrier during the collision determines the position of this notch in the final momentum distribution. Figure 5.6↓ shows the results of such a calibration. The time of collision for these clouds is nominally 7.8  ms, and has been varied by 100  μs from panel (a) to (b) to (c), thus shifting the impact site by 5.6  μm. The dark notch indicates the extinction of the momentum that was interacting with the barrier at the termination of the collision, and the impact site can thus be located. The clouds in Figure 5.6↓ have been allowed a further 22.2  ms of expansion after the collision, so the 5.6  μm shifts at the time of impact should correspond to shifts in notch position of
(5.9) 5.6  μm
in the final interferogram.
figure Images/Ripples/Time Series/BarrierPosition.png
Figure 5.6 Results of slight shifts in the impact position of the barrier. The images on the left show the absorption images of the clouds, and figures on the right show integrated density profiles. The insets show cartoons of the barrier position with respect to the cloud at the time of turn-off. Panel (a) shows the result of a collision that has been terminated when the barrier was 5.6  μm past the centre of mass. In panel (b) the collision has been terminated when the barrier was at the centre of mass, and in panel (c) the collision has been terminated when the barrier was 5.6  μm from reaching the centre of mass.

5.3 Momentum-Space Interferometry

The theoretical framework presented in Section 2.2↑ describes the emergence of “ripples” if the cloud is allowed to expand before colliding with the barrier. The frequency of these ripples is indicative of the phase of the momentum space wavefunction at the time of collision. Collisions were studied where the time of impact was varied from 5.15  ms to 8.15  ms. The subsequent expansion time was varied such that the total expansion time was 29.5±1  ms, allowing for some fluctuation in cloud velocity.

5.3.1 At a glance

Figure 5.7↓ shows an example of two clouds which have undergone this process. The data are clearly a product of the process described in Chapter 2↑, though there are some discrepancies. The large central peak appears in the position predicted by the model, as does the dark notch, though the contrast of the fringes is much higher than expected. From the model it is apparent that the contrast of the fringes could be increased in one of two ways. First, the contrast is increased with decreasing barrier width. Figure 5.8↓ shows the same data and model, but the barrier size in the model has been reduced by a factor of 3 from σ = 2.4  μm to σ = 0.80  μm. The contrast is improved, and looks similar to that of the interferogram. It also becomes apparent that the cloud in panel (b) shows oscillations that are nearly twice the expected frequency, whereas the cloud in panel (a) has fringes close to the correct frequency. While this yields attractive results, it is unlikely that the barrier has a sub-micron rms width. Figure 5.9↓ shows a more plausible method of increasing the fringe visibility, which is increasing the intensity of the barrier. Increasing the phase shift given by the barrier from π to 3π increases the contrast, but unlike decreasing the barrier width, the contrast decreases rapidly away from the collision site. The problem of fringe frequency also persists.
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_a_2pt4.png
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_b_2pt4.png
Figure 5.7 Two datapoints demonstrating momentum-space interference. The cloud in panel (a) has experienced a collision after 5.15  ms of freefall, and 23.72  ms of subsequent expansion. The cloud in panel (b) collided after 7.80  ms, and expanded a further 22.42  ms. Alongside the raw data is plotted a model generated by the approach in Section B.1.1↓. The collision was terminated at the centre of the cloud in panel (b), and 5  μm before the cloud centre in panel (a). The model assumes a barrier width of σ = 2.4  μm.
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_a.png
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_b.png
Figure 5.8 The same datapoints as in Figure 5.7↑, but with the barrier in the model reduced in width. The barrier has been reduced from the measured σ = 2.4  μm to σ = 0.76  μm, which has been arbitrarily chosen to match the contrast and asymmetry of the measured fringes.
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_a_4pi.png
figure Images/Ripples/150 Pts Curv Summary/TwoPoints_b_4pi.png
Figure 5.9 The same datapoints as in Figure 5.7↑, but with the barrier in the model increased in intensity (the width is again σ = 2.4  μm). The barrier has been increased in intensity to give a 3π phase shift rather than a π phase shift, a change which has been arbitrarily chosen to match the contrast of the measured fringes.

5.3.2 Measuring the Curvature

In Section 2.2↑ it was shown that peaks in the final interferogram of the collided atom cloud could be used to infer the phase of the colliding wave function. In this section a colliding wavefunction with quadratic phase is assumed, which should yield quadratically chirped interference fringes. Figure 5.10↓ shows the profile of the cloud from Figure 5.7↑(b), where the peaks have been located by a computerized peak-finding algorithm, and a curvature extracted through a fit of the peak locations as described in Section 2.2↑. Finally, Figure 5.11↓ shows a summary of 150 images where the curvature has been measured in this way. As suggested in the previous section, the clouds undergoing a collision at 5.15  ms nearly agree with the predicted value, and the agreement gets worse as the collision time increases.
figure Images/Ripples/150 Pts Curv Summary/FitErb001.png
Figure 5.10 A cloud after collision and free expansion. Panel (a) shows the smoothed, integrated profile of the cloud from panels (b) in Figs. 5.7↑-5.9↑, the peaks of which have been located. Error bars are estimated from the second derivative of the profile 56. The peak locations are each assigned a phase increment of π, and the resulting phases are plotted in panel (b), and fitted to a second-order polynomial to extract the curvature. The fit yields a curvature of 37.4±2.4 × 104 μm.
figure Images/Ripples/150 Pts Curv Summary/RawDataAndModel.png
Figure 5.11 Summary of curvature measured for fringes in data and FFT model (see Section B.1.1↓). The extracted curvatures are close to expected values for a pre-collision expansion time of 5.15  ms, but diverge from significantly at longer expansion times. The curvatures may trend slightly in the wrong direction, but are consistent with not changing at all.
figure Images/Ripples/150 Pts Curv Summary/KSpaceData.png
Figure 5.12 Data inverted to give k-space phase curvature. Error bars represent the scatter of values extracted from individual images, and points represent about 20 images each. The theoretical curve is the curvature 2mt due to dispersion. The model returns values that are systematically too low as the the far-field approximation has been made. The correction is known and is given in Section 2.2↑. The same correction should in principle be given to the data, which would push them even further from the theoretical value.

5.4 Momentum Enhancement

To study the predicted transient enhancement of momentum, it is necessary to compare the momentum distribution of a cloud undergoing a collision with the momentum distribution of a cloud undergoing no collision. Clouds similar to those illustrated in Figure 5.7↑ were immediately proceeded by clouds generated under identical experimental conditions, save that the barrier potential was not turned on. Comparison of one pair of collided and uncollided clouds is shown in Figure 5.13↓. As discussed in previous sections, any violation that is observed is strongly dependent on knowledge of the initial momenta of the clouds being compared. In experimental terms, the eventual positions of the clouds after the time of flight will have random shifts corresponding to the uncertainty in the initial momenta. If these shifts are known, the uncollided cloud can simply be “unshifted” for comparison. One possible corrective measure if this shift is not known a priori is to compare the centres of mass of the two clouds, which should have a fixed relationship as described by Equation . Also, if a large dataset is available, this correction can be determined empirically. Figure 5.13↓ shows a comparison of two sample datapoints where this correction been taken from the data in Figure 5.5↑ and applied, along with theoretical curves. The data curve shows a violation (where G > 0) that occurs just after the large central peak as predicted, and is somewhat larger than the modeled value. The traces are qualitatively similar, though this is not required for unambiguous violation of the inequality. As was discussed in Section 2.1.4↑, these results are very sensitive to the uncertainty in knowledge of the initial momentum state, and the exact degree of violation suffers from a correspondingly high uncertainty which is larger than the violation itself. The correction used for the centres of mass also depends on the barrier position at the end of the collision, and thus the initial momentum. With improved momentum stability in the experiment, a stronger statement of violation could be made, and qualitative features leave little doubt that the desired effect has been observed.
figure Images/Momentum Enhancement/OnePtViolation.png
Figure 5.13 Comparison of collided and uncollided clouds to determine violation of the inequality in Equation 2.1↑. Panel (a) shows the integrated profile of a cloud that has collided at 7.80  ms with the potential barrier, as well as the integrated profile of a cloud that has undergone no collision. Due to the large scatter in the data, the profiles of the clouds have been adjusted to have their centres of mass separated by 5.3  μm, the average separation for centres of mass as shown in Figure 5.5↑. Panel (b) shows a plot of G, the difference between the integrals from  − ∞ to x of the two momentum distributions. The value of Gmax is noted for comparison, but suffers from a very high statistical uncertainty, which is discussed in the text. Panels (c) and (d) show the modelled results for the experimental conditions producing panels (a) and (b).

6 Conclusions

The main results of this thesis may be grouped into three parts. First, my theoretical work extends an idea by Muga et al.1, and considers the transient enhancement of momentum in a realistic Bose-Einstein condensation apparatus, leading to the proposal of a technique for wavefunction reconstruction. Second, a description is given of work to upgrade all major systems of a Bose-Einstein condensation apparatus. Third and finally, I have conducted an experiment where a Bose-Einstein condensate is made to collide with an optical potential, and have thereby demonstrated the predicted momentum enhancement.

6.0.0.1 Theory

Muga et al. 1 first proposed that quantum particles, during a collision with a repulsive potential, could exhibit a transient enhancement in momentum forbidden to classical particles. I have considered in detail how to demonstrate this prediction using the macroscopic wavefunction of a Bose-Einstein condensate. I have shown that the internal interaction energy of the condensate is comparable in energy to the enhanced momentum components, and would thus lead to an ambiguous measurement of enhancement. I have also shown that a second obstacle to observation of enhanced momentum is the minimum width of the optical potential that can be created. As a solution to both the ambiguity of the interaction energy and the finite barrier width, I consider for the first time a collision with a wavepacket of non-uniform phase. Allowing freefall of the condensate before the collision lowers the density, and thus the interaction energy, and accrues a quadratic phase profile across the cloud. I have shown that such a cloud permits the observation of enhanced momentum with wider barriers, and indeed I have uncovered a regime in which momentum enhancement is increased by increasing barrier width. I have demonstrated that momentum enhancement should be observable in a parameter regime attainable by the apparatus described in later chapters.
Consideration of a cloud with a spatially non-uniform phase also leads to a novel technique for wavefunction measurement. I show that the appearance of amplitude modulations, or “ripples” in the momentum-space wavefunction during the collision may be interpreted as the interference between “enhanced” momentum components and the previously existing wavefuntion. I demonstrate a technique whereby the amplitude and phase of a wavefunction immediately before the collision can be reconstructed, and detail limits in which the entire wavefunction can be determined in a single measurement. Outside these limits, the technique is also suitable for tomographic reconstruction.

6.0.0.2 Apparatus

The second major result of this thesis is my successful upgrade of nearly all components of the apparatus. Through modification of the vacuum system, I have generated greater optical access to the atom cloud, and also reduced parasitic magnetic fields experienced by it. I have overhauled and simplified the entire laser system of the experiment, providing a more stable and powerful arrangement. A chain of injection-locked diodes has been replaced with a single, high-powered amplifier, the light from which is divided and distributed by optical fibre, vastly improving alignment stability. I have redesigned the coils providing magnetic fields to the lower chamber, incorporating them into a highly stable mechanical mount. The new assembly provides nearly double the magnetic field gradient, and is cooled more efficiently and reliably. I have aligned compensation field coils with principal axes of the experiment and synchronized the oscillation of the TOP fields with other experimental timing. In addition to this synchronization, the TOP driver has also been upgraded to supply more current to the coils, and to be more flexible in the ramping of fields. I have rebuilt the entire software and hardware system governing the timing of the experiment, improving the temporal resolution from 100  μs to 4  μs, and improving integration among experimental components. The summary result of these improvements has been a shortening of the experimental cycle from approximately 75  s to 45  s, and an increase of the atom number at the onset of condensation from 2 × 105 atoms to 6 × 105. Perhaps most significant, however, has been an unquantified improvement in day-to-day stability of the experiment.

6.0.0.3 Data

I have conducted an experiment as proposed in the theory section, where a bose-condensed cloud was made to collide with an optical potential. I have detailed the addition to the apparatus of a one-dimensional “sheet of light” which acts as a repulsive potential to a falling atom cloud. Through imaging the barrier directly and studying its interaction with the cloud, I have characterized this barrier to be approximately Gaussian in shape, with an rms width of σ = 2.45±0.1  μm. I have shown how the intensity may be tuned from the strong limit where it acts as a mirror, down to the weak limit where it acts as a phase mask. In the experiments presented, I have cooled the condensate to a point where there is no significant thermal fraction, and then released it from the trap. The cloud undergoes a free expansion in the range of 5.15 to 8.15  ms, at which point it collides with the optical potential whose height and position are varied from shot to shot. The potential is removed effectively instantaneously during the collision, and the momentum distribution at that moment is then studied through a time-of-flight measurement.
By analysis of the resulting interferograms, I have shown that the transient enhancement of high momentum components has indeed occurred, and the inequality in Equation 2.1↑ has been violated. The exact extent of the violation cannot be measured, however, due to problematic experimental uncertainties, mainly related to the unpredictability of the momentum state of condensed atoms before the collision. I have measured the momentum uncertainty of the cloud upon release from the trap to be 0.343±0.006  mm/s, and I have demonstrated a novel technique which uses near-identical copies of the cloud to postdict this momentum after a collision experiment is completed. This technique reduces the momentum uncertainty by a factor of 5, and may be employed in future work to increase confidence in the momentum enhancement result. It will likely be preferable, however, to reduce this uncertainty by other means. Evidence I have presented in this thesis suggests oscillations of the trapped cloud, which could possibly be suppressed through better regulation of the magnetic and acoustic environments, and perhaps even cooling the cloud in a purely optical potential.
As predicted, I have observed asymmetric modulations in the density of the expanding cloud, consistent with the model of momentum-space interfence. The fringes however, despite showing quadratically varying spacing, have a higher contrast and a lower frequency than expected, and the fringe frequency does not scale as expected with increasing pre-collision expansion times. This is certainly an exciting subject for future work. Despite these inconsistencies between experimental and model parameters, it is clear that the predicted, novel effect has been observed through the interaction of a Bose-Einstein condensate with an optical potential.

A Coil Mounts: Engineering Drawings

Components for the coil mounts were manufactured by the machine shop in the Department of Physics, University of Toronto. The mechanical drawings below were submitted to the shop, and prepared by Rockson Chang.
figure Images/New Coils/Coil Mount CAD/Drawings/Drawing2pp_Page_1.png
figure Images/New Coils/Coil Mount CAD/Drawings/Drawing2pp_Page_2.png
figure Images/New Coils/Coil Mount CAD/Drawings/Drawing2pp_Page_3.png
figure Images/New Coils/Coil Mount CAD/Drawings/Drawing2pp_Page_4.png

B Calculations

B.1 Simulation Methods

Discussion of momentum enhancement under different experimental conditions quickly leaves the realm of what can be modeled analytically. Most of the profiles and results in this section, including Figure 2.2↑, are produced using one-dimensional numerical modelling. This subsection briefly outlines the approaches employed to model wavepacket behaviour.

B.1.1 Fast Fourier Transform Method

This most simplistic method produces profies by propagating the experiment in phases. First, consider a Gaussian wavepacket in one dimension
(B.1) ψ(x, t0) = exp( − x2 ⁄ 2σ2)
Its Fourier transform is calculated and propagated in time by adding dispersion
(B.2) ψ(k, t0) =  F[ψ(x, t0)]
(B.3) ψ(k, t1) = ψ(k, t0)exp(ik22mt1)
and after an inverse Fourier transform, a phase shift is applied to the spatial wavefunction that corresponds to the phase written by the barrier. The barrier is assumed to be Gaussian in shape, and the written phase therefore takes the form of the error function
(B.4) ψ(x, t1) =  F − 1[ψ(k, t1)]
(B.5) ψ(x, t1) = ψ(x, t1)exp[iφ2erf((x-x_0) ⁄ 2σ)
and finally the final propagation is done in a manner similar to the first
(B.6) ψ(k, t1) =  F[ψ(x, t1)]
(B.7) ψ(k, t2) = ψ(k, t1)exp(ik22m(t2 − t1))
and
(B.8) ψ(x, t2) =  F − 1[ψ(k, t2)]

B.1.2 Split-Operator Technique

The split-operator technique allows the explicit definition of the potential, and propagates the initial state forward in time. This technique is useful in that it can incorporate the actual dynamics of the optical and magnetic potentials, as well as the nonlinear dynamics. In predicting interference patterns, it was not found that this technique produced significantly different results, and was abandoned in favour of the technique described in the previous section. For predicting momentum enhancement, the correction is sometimes important.
The well-known split-operator technique is a finite-difference method wherein the kinetic and potential energies of the Hamiltonian are separated and propagated alternately. If the Hamiltonian can be so written
(B.9) H = ^T + Û
Where ^T contains the kinetic energy portion of the Hamiltonian, and Û the potential energy portion. The propagation operator is then
(B.10) eit(^T + Û)
which is approximated as
(B.11) eit12^TeitÛeit12^T
Propagation is achieved in a similar fashion to the previous section. Propagation is cut into timesteps smaller than any other relevant timescales, and in every timestep the wavefunction is Fourier transformed to alternately apply the kinetic and potential propagators.

B.1.3 Variational Model

Finally, a variational method is employed as in 2 to model the nonlinear expansion of the condensate, and is used to estimate rates of expansion which are used to set initial conditions for other modeling techniques. This technique accounts for the nonlinear interactions during early expansion, and is used to describe condensates in the Thomas-Fermi regime. Neglecting the kinetic energy term and solving for the ground state of the GPE yields
(B.12) ψ(r)ψTF(r) = μ − U(r, 0)Ng
inside the region where where μ − U(r, 0), and zero outside this region. μ is the chemical potential, U is the trapping potential, N is the number of atoms and g is the interaction parameter. In the case of a harmonic potential, the ground state is a paraboloid. The chemical potential can be found by normalizing the wavefunction
(B.13) μ = 12ω(15Namω)2 ⁄ 5
where ω is the geometric mean of the trap frequencies. An interesting result of 2 is that the expanding condensate retains its shape, and the cloud can be characterized by the three semi-axes
(B.14) Rj(t) = λj(t)Rj(0)
where Rj(t) is the distance from the cloud centre to the edge of the parabola (i.e. the semi-axis) along dimensions j = 1, 2, 3, and λ is a scaling parameter. The evolution of the scaling parameters is given by the relation
(B.15) ¨λj = ωj2(0)λjλ1λ2λ3 − ωj2(t)λj
and ωj is the trap frequency along axis j. This work is generally only concerned with dynamics after the trap turn-off, when the second term is zero.

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