Evaporative Cooling in a Strongly

Confining Microchip Trap

Submitted in partial fulfillment of the requirements of theDegree of Master of Science in Physics

by

Lindsay J. LeBlancUniversity of Toronto

August 24, 2005

Contents

1 Introduction 1

2 Theory of evaporative cooling 22.1 Basic considerations . . . . . . . . . . . . . . . . . . . . . . . 22.2 Models used for calculations of evaporative cooling . . . . . . 42.3 The Amsterdam model . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Distribution of energies in a trapped gas . . . . . . . . 52.3.2 Thermodynamics in a trapped gas . . . . . . . . . . . 72.3.3 Background loss effects . . . . . . . . . . . . . . . . . 92.3.4 Three-body loss . . . . . . . . . . . . . . . . . . . . . . 102.3.5 Effective energy removal: α . . . . . . . . . . . . . . . 112.3.6 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Runaway evaporation and efficiency . . . . . . . . . . . . . . 122.5 Removal of atoms from a magnetic trap using RF radiation . 13

3 Calculating the evaporative cooling process 153.1 Implementation in MATLAB . . . . . . . . . . . . . . . . . . 153.2 Applying the model . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Optimization of evaporation . . . . . . . . . . . . . . . 16

i

3.2.2 Required lifetime . . . . . . . . . . . . . . . . . . . . . 183.2.3 The µEM advantage . . . . . . . . . . . . . . . . . . . 19

4 Experimental implementation of evaporative cooling 204.1 Experimental overview . . . . . . . . . . . . . . . . . . . . . . 204.2 Methods for collecting data . . . . . . . . . . . . . . . . . . . 21

4.2.1 Measuring the extent and temperature of the atom cloud 224.3 Trap characterisics . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.1 Trap frequency measurements . . . . . . . . . . . . . . 244.3.2 Trap profile measurements . . . . . . . . . . . . . . . . 244.3.3 Trap lifetime . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Generating radio frequency radiation for evaporation . . . . . 27

5 Comparison of theoretical and experimental results 275.1 Connections between the model and the experiment . . . . . 27

5.1.1 Error Analysis . . . . . . . . . . . . . . . . . . . . . . 285.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 Experiment 1: Fast evaporation . . . . . . . . . . . . . 295.2.2 Experiment 2: Slow evaporation . . . . . . . . . . . . 295.2.3 Comparisons between experiment and calculations . . 31

6 Extensions to the project 34

7 Conclusions 35

A Appendix 36A.1 Calculation of temperature-number relationship, α . . . . . . 36A.2 Calculation of time evolution of number . . . . . . . . . . . . 36

ii

1 Introduction

I learned when I was in high school that matter, like light, could be con-sidered both a particle and a wave. It was several years before I understoodwhat that meant, and seven years later, I find this seeming paradox thesubject of my research. This quantum mechanical phenomenon manifestsitself through the statistics particles obey - Bose-Einstein or Fermi-Dirac- and these statistical effects become evident at low energies. Such quan-tum degenerate regimes have been experimentally accessible for just overa decade now, being first demonstrated in bosons with the observation ofBose-Einstein Condensation (BEC) in 1995 [1], followed by Fermi degen-eracy in 1999 [2]. The excitement surrounding the achievement of thesesystems is founded in their diverse application to the study of physical phe-nomena. Bose systems have been used to study such things as matter waveinterferometry [3] and the Mott-insulator phase [4]. Degenerate Fermi gaseshas been proposed as models for studying condensed matter systems includ-ing the the BEC-BCS crossover [5] and the Hubbard model [6]. Connectionshave also been made to studies in high energy physics [7].

The last experimental stage needed to create these low energy, low tem-perature systems is known as evaporative cooling, and that is the subjectof this report. Evaporative cooling is a familiar concept to most people. Itexplains how a cup of tea cools down and how a human body maintains itstemperature through perspiration. These effects are a result of the mostenergetic water molecules, those that can overcome the heat of vaporiza-tion, leaving the system with more than the average energy. Consequently,the average energy of that which is left behind is reduced along with thetemperature.

In the context of atomic physics, evaporative cooling is somewhat differ-ent process. Where water molecules need to overcome the heat of vaporiza-tion to escape the system, trapped atoms undergo no such energy dependentfirst order phase transition. The hotter atoms in a trap will experience theouter reaches of the potential while the cooler atoms tend towards the centre.Evaporative cooling relies on the ability to remove only atoms whose energyorbits extend the furthest. Selecting the point at which this removal oc-curs, the trap depth, determines how much energy escapes with each atomfrom the trap. The remaining atoms, through collisions with each other,will rethermalize to a temperature lower than that which they had with thehotter atoms around.

As an extreme example, imagine a trap with N atoms whose averageenergy is 〈E〉, where only those atoms with energy N〈E〉 could escape the

1

trap. Since the energy in a trap is exchanged through random collisionsbetween particles, one partical should eventually gain this energy and leavethe trap. When it escapes, it will have taken all of the energy from theother particles, leaving those remaining with zero energy. This example isimpractical, as will be discussed later, but is essential in understanding whyevaporative cooling will work in an atomic system.

This report will discuss a basic and fairly general theory of evapora-tive cooling, applicable to many experimental set-ups. It will focus on theexperiment at the Toronto Ultracold Atoms Laboratory, where a stronglyconfining microelectromagnetic (µEM) “chip” trap is used to trap bosonic87Rb and fermionic 40K. These traps allow fast [8] and efficient evaporationbecause the tight confinement causes atoms to collide more frequently, whichincreases the rate of rethermalization,

The report is set up as follows: §2 establishes the theory of evaporativecooling, which is based on the Amsterdam model; §3 demonstrates how thecalculations are implemented; §4 discusses the the experimental impleme-nation of evaporative cooling; §5 makes comparisons between the calculatedand experimental results; §6 gives suggestions for extensions to this work;and §7 concludes the report.

2 Theory of evaporative cooling

2.1 Basic considerations

A precursor to studying the interesting phenomena in quantum gases isentering degeneracy, a regime in which the quantum statistical nature ofa collection of particles becomes evident. Alkali atomic systems are ex-cellent candidates for such studies, since these atoms can be trapped inrelatively weak magnetic potentials, they are easily cooled using laser cool-ing techniques, and they can be evaporatively cooled using radio frequencyradiation. As stated in the Introduction, there are two types of statisticsobeyed at low temperatures, Bose-Einstein for bosons, and Fermi-Dirac forfermions. The characteristics of degeneracy are very different in these twosystems. The first demonstrates a second order phase transition to BECat low temperatures, while the second evolves into a Fermi Degenerate Gas(FDG) without a phase transition.

Evaporative cooling is based on the supposition that the particles in-volved will rethermalize to lower temperatures after the hot ones are re-moved. Elastic collisions between particles provide the necessary energyredistribution and randomization that will lead to an equilibrium-like dis-

2

tribution of energies after a certain number of collisions. Primarily, evap-orative cooling is happening in the classical regime, in which the Maxwell-Boltzmann distribution of energies characterizes the temperature of the sys-tem.

The necessity for collisions involves a consideration of the statistics ofthe atoms before degeneracy is reached. At low temperatures, only s-wavescattering is allowed, which for fermions, is prohibitive. The wave functionof a fermion is, by definition, anti-symmetric under exchange of particles,and is inconsistent with the symmetric s-wave process. The Pauli Exclusionprinciple dictates that fermions cannot collide at low temperatures; neithercan they rethermalize to lower temperatures and evaporative cooling is inef-fective. This problem can be overcome by implementing sympathetic cooling[9], in which collsions between bosonic and fermionic particles are used torethermalize the ensemble while evaporating one or both species. This in-direct method of cooling fermions is beyond the scope of this report andthough it is discussed briefly in §6, bosonic systems will be the focus.

Bosons are ideal candiates for evaporative cooling, as their s-wave scat-tering cross-sections at low temperatures are described by σ = 8πa2. Thetransition to BEC occurs when there is a macroscopic number of particles inthe ground state of the system, which happens when the phase space densityof the gas exceeds g3/2(1) = 2.612 [10].

Ultimately, it is the phase space density that needs to be optimized.One definition for this quantity is ‘the number of atoms per unit volumein 6-dimensional phase space.’ One can imagine that as this density in-creases, the individual atoms begin to “see” each other and fundamentalissues of quantum indeterminancy emerge. When the number density ofatoms is high enough and the thermal de Broglie wavelengths of the atomslong enough, the interparticle spacing is less than the wavelength and distin-guishing between particles becomes impossible. This view is demonstratedin a quantitative definition of phase space density,

ρ = n0Λ3 (1)

where n0 = N/V is the peak number density and Λ is the thermal de Brogliewavelength,

Λ =(

2π~2

mkT

)1/2

, (2)

where m is the mass of the particle, k is the usual Boltzmann constant, andT is the temperature of the cloud. Examination of Eqs. (1) and (2) reveals

3

that there are two processes by which the phase space density is increased:increasing the density of the cloud and decreasing its temperature.

The fundamental ideas behind evaporative cooling were first proposedin 1989 [11]. The crucial concept is that the removal of the most energeticatoms in a trap can, under the right conditions, result in increasing densityand decreasing temperature.

2.2 Models used for calculations of evaporative cooling

Upon the successful implementation of evaporative cooling in the mid-1990’s, several models detailing the microscopic mechanisms behind the pro-cess were developed. Most of these models used classical physics to describethe interaction of the atoms as the evaporative cooling process progressed,which is justified considering that the statistics of particles only significantlydiverge from the classical Boltzmann distribution when quantum statisticaleffects are evident, at which point the evaporation ends.

A review by Ketterle and van Druten in 1996 [12] gives an overview ofevaporative cooling and presents a simple model used to develop a qualitativeunderstanding of the process. An earlier analytic model of the evaporationprocess was developed in the same group [13] where a number of discretetruncations to the thermal distribution are made. Though this model encap-sulates a starting point for understanding evaporation, it does not accuratelyrepresent the continuous truncation used in experiment. A more detailedversion was presented by Walraven in his 1995 review [14], hereafter referredto as the “Amsterdam” model. Classical kinetic theory is used to determinethe evolution of the energy distribution for a trapped gas, by integratingthe Boltzmann collision integral. Analytic expressions are established foran experimentally important class of potentials, including power law po-tentials and the Ioffe-Pritchard trap [15]. This model, however, relies onthe assumption of “sufficient ergodicity,” that is, that the mixing of energybetween the degrees of freedom in a trap is ergodic, which may not be acorrect assumption under all experimental conditions.

Abandonning the realm of analytic expressions, many models have sim-ulated the evaporation process numerically. A similar approach to the Am-sterdam model was undertaken by Sackett et. al. [16], where they integratethe same expression numerically and optimize the trajectory step-by-step.Wu and Foot [17] developed a model based on “Bird’s method,” a MonteCarlo simulation in which the position and velocity of individual atoms istracked. Their claim was that this model made fewer assumptions than someof the previous models, and could be used in a wider range of situations.

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Models were also developed using the quantum statistics that becomerelevant towards the end of the evaporation trajectory. Reference [18] is anexample of such a model.

In choosing a model on which to base the following calculations, I consid-ered the usefulness of each in an experimental group aiming to understandits experiment. The need to use quantum statistics to explain this mostly-classical phenomenon seemed superfluous, but the simplest models were tooimprecise to compare quantitatively with experiment. The analyticity of theAmsterdam model was appealing in that it would give a better understand-ing of the actual processes happening. The exactness developed for powerlaw potentials, ones that would be used in real experiments, also made itattractive.

2.3 The Amsterdam model

The Amsterdam model’s use of microscopic kinetic theory to develop an-alytic expressions for the evolution of the trapped gas has been shown to bean accurate representation of the process when compared to direct numer-ical integration [15]. Because it is an analytical model, there are a numberof assumptions that must be considered before it is implemented.

2.3.1 Distribution of energies in a trapped gas

The Amsterdam model is based on the assumption that the energy ofthe atoms in the trap is the only relevant parameter. This assumption ofsufficient ergodicity is important because it allows us to treat the ensembleof atoms in the language of densities of states rather than considering thetrajectories of individual particles as they navigate the trapping potential.The energy density of states

g(ε) =2π(2m)3/2

(2π~)3

∫U(r)≤ε

d3r√

ε− U(r) (3)

can be written such that it contains all the necessary information about thetrapping potential. More specifically, for “power-law potentials”, that is,those with polynomial trapping potentials of the form

Utrap(x,y,z) = x1/δx + y1/δy + z1/δz (4)

where the sumδ = δ1 + δ2 + δ3 (5)

5

is the scaling parameter that provides the relationship between the volumeand temperature of the trapped particles, the density of states takes a par-ticularly simple form,

g(ε) = APLεδ+1/2 (6)

where APL is a coefficient determined by performing the integration (3). Formost regimes of interest in this report, a harmonic trapping potential of theform

U(r) =12mω2r2 (7)

is used, where ω is the trap averaged oscillation frequency and r =√

x2 + y2 + z2

is the radial position of the particles. In such a trap, δ = 32 and APL =

12(1/~ω)3.

A second major assumption is that this distribution of energies is de-scribed by the Boltzmann distribution even as the ensemble is disturbedfrom equillibrium. Together with an expression for the density of states (6),the distribution function f(ε) allows us to describe the number of particlesthat can be found with an energy between ε and ε + dε as g(ε)f(ε)dε. Ref-erence [15] describes in detail how one may use the Boltzmann equation ofclassical kinetic theory [19] to derive an expression for the evolution of thedistribution of energy in the a trap:

g(ε4)f(ε4) =mσ

π2~3

∫dε1dε2dε3dε4δ(ε1ε2ε3ε4)g(min[ε1, ε2, ε3, ε4])

{f(ε1)f(ε2)− f(ε3)f(ε4)} (8)

where σ is the scattering cross section (which for 87Rb is 6.89 × 10−16 m2

[20]), the subscripts 1 and 2 refer to two colliding particles before collision,subscripts 3 and 4 refer to those same particles after collision, the deltafunction ensures conservation of energy during the collision, the min func-tion ensures we are working with the density of states of the lowest energyparticle in the collision, and the bracketed term with products of distribu-tion functions describes the probability that the two particles will collide.This differential equation can be solved to give an expression for the evolu-tion of the distribution function, f(ε4), as a function of time. This is notsuch a useful quantity from the point of view of the experimentalist; instead,Eq. (8) is used to derive analytic expressions for the relevant thermodynamicquantities.

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2.3.2 Thermodynamics in a trapped gas

To understand the evaporative cooling process in an ensemble of manyatoms, we must rely on thermodynamics to provide the the sorts of numbersthat we can measure in the laboratory, like atom number, temperature, ordensity. It turns out that it is enough to calculate only two of these quantitiesand use scaling laws to determine the rest (see §2.3.6); here, these will beatom number and ensemble energy.

First, we must determine how the atom number of a trapped gas changeswhen atoms are escaping from the cloud as a result of a finite trap depth.The basic assumption of these calculations is that any atom with an energyabove that of the trap depth, εt = ηkT , will leave the trap. The otherassumption, as stated in §2.3.1, is that the Maxwell-Boltzmann distributionis accurate in this non-equilibrium situation.

The derviations of these quantities can be found in Ref. [14]. For thespecial case of the power-law potential, these quantities are described interms of incomplete gamma functions.

The rate of change in atom number due to evaporation is found to be

Nev = −n20σe−ηv

Λ3

kTVev, (9)

where v =√

8kTπm is the average relative speeds of colliding particles, n0 is

the peak density, and

Vev =Λ3

kT

∫ εt

0dεg(ε)

[(εt − ε− kT ) e−ε/kT + kTe−η

](10)

represents a characteristic volume of evaporation.A quantity that is more easily understood is the rate of of evaporation,

1/τev where τev is a characteristic evaporation time. If we define the collisionrate for these trapped atoms as

1τcoll

= n0vσ, (11)

then we can also look at the ratio between these two rates, called λ, where

λ =τev

τcoll=

Ve

Veve−η. (12)

From this expression, it is obvious how the trap depth affects the evaporation- as η increases, there is an exponential increase in the characteristic timefor evaporation.

7

To improve the evaporative cooling process, one can imagine that contin-uously lowering the depth of the trap as the atoms become colder would behelpful. Indeed, if temperature is decreasing, η is actually increasing for con-stant εt throughout the evaporation process, and as Eq. (12) demonstrates,this increases the time required for evaporation. Lowering the trap depthis known as forced evaporative cooling and is essential to reaching quantumdegeneracy in atomic systems. This procedure gives rise to additional atomloss unrelated to the pure evaporative losses and this must be taken intoaccount when considering overall atom loss,

N = Nev + Nt, (13)

where Nt is the loss due to truncation. In the notation of the Amsterdammodel,

Nt

N=

ξεt

εt. (14)

The second thermodynamic quanitity of interest is the internal energyof the collection of atoms. The total energy is

E = (3/2 + ζ) NkT, (15)

where ζ replaces γ of the Amsterdam notation. The energy lost due toevaporation is

Eev = Nev

[εt +

(1− Xev

Vev

)kT

], (16)

where

Xev =Λ3

kT

∫ εt

0dεg(ε)

[kTe−ε/kT + (ε− εt − kT ) e−η

], (17)

such that the term(1− Xev

Vev

)kT expresses the amount of “extra” energy a

particle that escapes the trap has, on average. The energy loss due to thecontinuous lowering of the trap is

Et = εtNt. (18)

With the evolution of number and energy determined from kinetic theory,the evaporative cooling process can be evalutated.

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2.3.3 Background loss effects

In the development of the model to this point, no consideration has beenmade for the loss mechanisms inherent in a real experiment. The first ofthese, and the most important to consider in our experiment, is that of col-lisions with the background, or untrapped, atoms. These untrapped atomswill be thermalized with the walls of the chamber, which are at room tem-perature. The energy of these atoms far exceeds the trap depth (which istypically 1 - 100µK) such that upon collision with a trapped atom, bothhave a great probability of exiting the collision untrapped.

To collect atoms to trap, small amounts of atomic vapour are dispensedinto the vacuum cell. This atomic vapour is collected by the MOT beams andsubsequently trapped in the magnetic trap (see Ref. [21]). Though there aremany benefits to the single-chamber set-up we implement, one of its faultsis that it results in a high vacuum pressure in the main chamber. Thereare many untrapped atoms in the same volume as those which are trapped,increasing the probability of a bad collision and losing an atom without thebenefit of its having more than the average energy. This loss mechanism canbe measured by measuring the lifetime of atoms in the magnetic trap (see§4.3.3), a characteristic decay time in the number of atoms.

To take into account this loss of atoms, we consider a total loss factor,N . With evaporative and background loss effects taken into account,

N = Nev + Nbkgd + Nt. (19)

The interesting quantities are loss rates, so we divide through by N . SinceNev/N = 1/τev is given by (9) and Nbkgd/N = 1/τloss is an experimentallydetermined quantity, we can determine the overall loss rate for the process,

N

N=

Nev

N+

Nbkgd

N+

Nt

N= Nev

(1 +

λ

R

)+

Nt

N, (20)

where λ is defined in Eq. (12) and R = τbkgd/τcoll is similarly defined as theratio of good to bad collisions. For a large R, that is, many good (leadingto evaporation) collisions and few bad (leading to loss) collisions, the termτev/τbkgd = λ/R is small and the effect of background collisions is negligible.To establish this criterion, the collision rate of the atomic sample can beincreased and/or the number of background atoms to cause lossy collisionscan be decreased.

To determine the effect on the energy of the atoms being lost, we assumethat for each atom lost from the trap due to background collisions, an energy

9

equal to the average energy per atom E = kT is lost. The total energy lossis expressed in a manner similar to (19),

E = Eev + Ebkgd + Et, (21)

and the overall energy lost can be expressed as

E = Nev

[η +

(1− Xev

Vev

)+

λ

R(3/2 + ζ)

]kT + ηkTNt. (22)

The energy loss rate is simply this equation divided by the internal energy,E = (3/2 + ζ)NkT .

2.3.4 Three-body loss

In addition to background collisions, three-body processes can contributeto losses from the trap. When three particles are sufficiently close together,two of them can form a dimer or molecular state, usually in a highly excitedvibrational state, and the third atom carries off this excess energy. Sincethe energy involved is typically much greater than the trap depth, all threeatoms will be lost from the trap [22]. This process involving three particles isproportional to the square of the local density, n2(r), and only becomes im-portant at high densities. Quantitatively, the loss due three-body processesis given by

N3−body =∫

d3rG2n3(r), (23)

where G2 is an experimentally determined constant, which for 87Rb has beenmeasured as 1.1(±0.3)×10−28 cm6/s [22]. This means that this loss processis only important as the density approaches 1014cm−3, which happens onlynear the end of the experiment. At this point, the assumption of classicalstatistics is beginning to break down and the simulation of the evaporationtrajectory is already questionable. As a result, I have included the effectof three-body losses in the calculation of losses from the trap, but do notinclude the effect in calculating, for example, the parameter α, in the spiritof making a first order approximation. In a harmonic trap, the rate at whichatoms are lost due to three body collisions is

N3−body

N=

1τ3−body

=G2n

20

33/2. (24)

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2.3.5 Effective energy removal: α

Determining the reduction in temperature for each evaporatively cooledatom is important for determining whether the scheme is feasible. The suc-cess of evaporative cooling depends on whether the atoms will be sufficientlycooled before all of the atoms are lost from the trap. A quantitative measuregenerally known as “α” is defined as the ratio of rate of change in temperatreto rate of change of number,

α =T /T

N/N=

d lnT

d lnN. (25)

To calculate this is a matter of relating energy to temperature using

E

E=

N

N+

T

T. (26)

and Eqs. (20) and (22). The details are algebraically intensive and includedin Appendix A.1 with the final result:

α =η +

(1− Xev

Vev

)+ (3/2 + ζ)

(3/2 + ζ)(1 + λ

R(1 + ξ))− λ

Rξη + ξ(1− Xev

Vev.) . (27)

2.3.6 Scaling Laws

To determine the evolution of the various measurable quantities in anensemble of trapped atoms scaling laws are implemented. The time evolutionof only one quantity is determined and the rest are found through theirrelationship to this one quantity using α and the characteristics of the trap.

The quantities we can measure and are interested in are atom number,N , temperature, T , atom density, n, collision rate, nσv, and phase spacedensity, ρ. In my calculations, I have chosen to use atom number as theprimary quantity on which to base my calculations, as it is the most naturalin terms of thinking about loss from the trap. Therefore, all quantities areexpressed in terms of their proportionality to atom number.

Temperature is the most obvious quantity; this was just calculated as α.Therefore, we can say

T ∝ Nα. (28)

The second “easy” relationship of note is that the volume of the trap, V ,scales as

V ∝ T δ, (29)

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Quantity, Q Scaling Exponent, xNumber of atoms, N 1Temperature, T αVolume, V αδDensity, n 1− αδCollision rate, nσv 1− α(δ − 1/2)Phase space density, ρ 1− α(δ + 3/2)

Table 1. Scaling relationships for some thermodynamic quantities:, Q ∝ Nx.

a characteristic of the power-law potential. These two basic relationshipscomplete that which is needed to construct the scaling relationship of anyof the quantities listed above, as each is a function only of number, tem-perature and volume (through density). The scaling relations useful in theevaporative cooling process are summarised in Table 1, after [12].

2.4 Runaway evaporation and efficiency

As has been described earlier, it is important to know whether it is possibleto remove enough energy to provide cooling before all of the atoms are lost.In the real-life situation where background losses exist, it is imporant tomaintain an ever-increasing collision rate to sustain the evaporation process.The regime where this is the case is known as “runaway evaporation” andcan be qualitatively evaluated by considering the scaling exponents. If theratio of

d ln(nσv)d lnN

= 1− α(η, R, δ)[δ − 1/2] < 0, (30)

then there is an increase in collision rate, nσv, as number, N , decreases. Inthe case of a harmonic trap, the runaway condition is statisfied while α > 1.

For a measure of the efficiency of evaporation, we take our cue fromexperimental considerations. The ultimate goal is to improve the phasespace density of the atomic cloud while losing as few atoms as possible. Thisallows us to maintain a high ratio of signal-to-noise. Stated quantitatively,the ratio between the change in phase space density to the change in atomnumber should be considered,

γ = −d ln(ρ)d lnN

= α(η, R, δ)[δ + 3/2]− 1, (31)

where the dependence of α on η, R, and δ is important for establishingconditions under which efficient evaporation will occur.

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4 5 6 7 8 9 10

1

1.5

2

2.5

3

3.5

4

η

ef

fi

ci

en

cy

Figure 1. Efficiency of evaporation, γ, at a collision rate of n0σv = 500s−1

and various lifetimes. The top curve represents a lifetime of 9s, with 7s, 5s,and 3s falling in order beneath.

Figure 1 demonstrates the efficiency, γ as a function of trap depth, ηunder a variety of experimental conditions. As the ratio of good-to-badcollisions, R, is increased, the efficiency improves and the optimal value ofη increases. This behaviour would approach that of the extreme exampleof the Introduction, that is, as mechanisms of loss are eliminated, the mostefficient trap depth, η, increases towards the total energy of the trap.

2.5 Removal of atoms from a magnetic trap using RF radi-ation

Discussion of evaporation to this point has assumed that the atoms arein a trap whose depth can be changed without affecting the shape of thepotential below the cut-off. The ability to control the trap in this manneris important to the success of the evaporative cooling process and can berealised using a magnetic trap and RF radiation.

Magnetic traps are able to confine atoms that have a non-zero dipolemoment. This dipole moment, µ, is the result of the intrinsic and extrinsicangular momenta of the charged constituents of the atom interacting withthe magnetic field. In alkali atoms in the ground state, there is no orbitalangular momentum of the valence electron, and the effect of the nuclear

13

spin is negligible. The magnetic moment is primarily due to the spin of thevalence electron, which leads to a magnetic dipole moment,

µ = −2µB

~S (32)

where µB is the Bohr magneton, and S is the spin operator for the orbitalelectron. If this quantity is projected onto the total angular momentum ofthe atom, F = I + S, where I is the angular momentum associated withthe spin of the nucleus, we find [23]

µ = −gF µB

~F (33)

wheregF =

F (F + 1)− I(I + 1) + S(S + 1)F (F + 1)

. (34)

The energy associated with the interaction of the dipole moment and amagnetic field is

U = −µ · B = mF gF µB|B| (35)

where mF represents the projection of the angular momentum along thedirection of the magnetic field. This is the potential energy of an atom in amagnetic trap.

If the atom has a positve gF mF , then a magnetic field with at minimumwill trap the atoms; if the atom has a negative gF mF , it requires a magneticfield with a maximum to do the trapping. Since magnetic field maxima areforbidden [24], only those atoms with gF mF > 0 are trappable.

The next question is one of changing the trap depth. Radio frequency(RF) radiation can be used to couple the hyperfine levels of an atom, chang-ing the value of mF from one that is trappable to one that is not. Selectionrules dictate that changes in the hyperfine levels can occur only in steps ofmF = ±1, so the change in energy, equal to the energy of the RF photon is

~ωRF = gF µB|B|. (36)

The fact that the Zeeman splitting is dependent on the magnetic field meansthat one frequency will couple atoms with energies associated with one valueof the magnetic field. In this way, higher frequencies will eject hotter atoms,and as the frequency is reduced, colder atoms can be removed from the trap.

The atom used in the experiments detailed in this report is 87Rb, whichis trapped in the |F = 2,mF = 2〉 hyperfine state. For atoms in the F = 2state, there are five mF levels, all of which are coupled to their neighbours in

14

the presence of resonant RF radiation. Examining this in the dressed atompicture, one finds that states of the atoms are adiabatically transformedfrom mF = 2 to mF = −2 via a multiple-photon process [12].

The combination of a magnetic trap and controllable RF radiation allowsfor the implementation of forced evaporation in an atomic system.

3 Calculating the evaporative cooling process

3.1 Implementation in MATLAB

Determining the evolution of an evaporative cooling process involves mak-ing calculations after many discrete time steps. I chose to use MATLAB forits abilities to handle numeric calculations and its built-in functions, includ-ing the incomplete gamma functions.

Numerical implementation involves calculating the analytic equationsdeveloped in §2 and evaluating them at discrete time steps. The naturalquantity to calculate at each time step is the number of atoms remaining,after which the scaling laws are used to calculate the remaining quantities.The primary task is to determine the initial conditions and the evolution ofatom number as a function of time.

In order to establish a starting point, a number of initial conditionsare required, including atom number, temperature, and the trap charac-teristics. Knowledge of these three allows the calculation of the remainingvalues, including collision rate, density, and phase space density, throughthe relationship

N =∫

all spacen(r)d3r =

∫all space

n0e−U(r)/kT d3r (37)

with the definitions of collision rate (Eq. 11) and phase space density (Eq. 1).The time evolution of the atom number is found by integrating the dif-

ferential equation for N/N , the details of which are found in Appendix A.2,yielding the result

N(t) = N(0) exp[−t

1/τev + 1/τloss

1− ξα

], (38)

where 1/τloss = 1/τbkgd + 1/τ3−body. This calculation is discretized in timesteps (dt) that are less than the inverse collision rate by a factor of 10(dt = τcoll/10), and each value is calculated at each time step. For example,

15

the calculation of number takes the form:

Ni+1 = Ni exp[−dti

1/τev,i + 1/τloss

1− ξiαi

]. (39)

where i denotes the values calculated from the previous time step (or initialconditions, if appropriate). Values of α and ξ are dependent on i if weconsider time dependent values of η.

Calculating the evolution of a general evaporation process is straight-forward using these methods. The challenge is to find a useful and efficientmeans of acheiving degeneracy with these calculations. An appropriate valuefor the trap depth, η must be selected at each time step in the calculationsuch that effective evaporation can happen without loss processes interfer-ing.

Two considerations are made when choosing a value of η in these calcula-tions. The first is that we wish to remain in a regime of runaway evaporation.Evaporation can occur without entering this regime [16], but it is preferred.The restriction imposed by Eq. (30) provides a lower bound for η.

The second consideration is efficiency. Using the definition describedin §2.4, Eq. (31) is maximised at each time step to give the best η to beused in the following time step. Together with the lower bound for runawayevaporation, an optimal evaporation trajectory can be calculated, duringwhich the phase space density is improved as much as possible while losing asfew atoms as possible. This infomation is easily converted into a trajectoryof RF frequencies that can be implemented in an experiment.

3.2 Applying the model

Running simulations of the evaporation process under realistic experi-mental conditions is useful for understanding the process and determiningexperimental parameters with which to begin. Using efficiency as the figureof merit, evaporation is simulated with or without optimization algorithms,in various trapping configurations, with different trap lifetimes.

As an example, consider a cylindrically symmetric harmonic trap withtrapping frequencies of 2π × 30 Hz in the axial direction and 2π × 800 Hzin the radial direction. Assume the lifetime of the trap is 5 s and initialconditions are N = 1× 107 atoms at a phase space density of ρ = 1× 10−6.These conditions are used in the examples which follow, unless otherwisestated.

3.2.1 Optimization of evaporation

16

Figure 2. Comparison of evaporation efficiencies with constant and optimizedvalues of η. Efficiency, γ, is the negative slope of the trajectories on this log-logplot. Trajectories end when ρ = 2.612. Inset is the trajectory taken by η as afunction of time, associated with the “top”, highlighted curve. All other valuesof η remain constant throughout the trajectory.

Consider the effect of optimizing of the trap depth as the evaporationprogresses. Figure 2 compares the results for a trajectory that has beenoptimized to those with constant-η trajectories. The optimized (highlighted)trajectory has the highest efficiency, though values of η held at the initialoptimal value and one between initial and final values also achieve nearlythe same efficiency. This independence of the value of η over this rangeindicates the robustness of the evaporative cooling scheme. Degeneracy isnot achieved when the relative trap depth is always equal to the final optimalvalue. This high value for η restricts the onset of runaway evaporation andthe loss processes dominate from the beginning of the experiment.

In this optimized trajectory, as with any with reasonably good startingconditions, increasing the value of η as the evaporation progresses increasesefficiency. In the runaway regime, the collision rate is increasing and with itthe ratio of good-to-bad collisions, R. Improving R is equivalent to movingfrom a lower to a higher curve in Fig. 1, where the peak efficiency moves tohigher values of η as R increases. Were there no bad collisions, the optimal

17

Figure 3. Evaporation efficiency as a function of trap lifetime. a) Results ofsimulations at various lifetimes. Top to bottom curves: τlife = 100s, 50s, 20s,10s, 5s, 2s, 0.5s. Efficiency, γ, is the negative slope of the curve on this log-logplot. b) Required initial atom number for a given efficiency as a function oftrap lifetime. N ∝ τ−1.5

trap depth would the infinite. This is where the extreme example of theIntroduction is found; it cannot be realised because there will always be amechanism of loss.

3.2.2 Required lifetime

The magnetic trap lifetime also has an effect on the success of the exper-iment. Figure 3(a) demonstrates the efficiency of evaporation for severaltrap lifetimes. The efficiency continues to increase as lifetime increases andthe ratio of good-to-bad collisions increases. Again, consideration of the ex-treme example of the Introduction establishes this relationship. Longer traplifetimes means that waiting for the super-energetic atoms to leave is feasi-ble, while short lifetimes means losing more less-energetic atoms to acheivethe necessary reduction in ensemble energy. Extremely short lifetimes resultin failure of the experiment such that reaching degeneracy is impossible.

A single chamber setup results in higher vacuum pressures in the mag-netic trap, which means lower trap lifetimes. In addition, in an experimentthat uses a µEM trap, the lifetime of the atoms in the magnetic trap is com-promised as compared to more traditional traps - noise in the magnetic-field

18

providing current caused by the roughness of the wire surface can spin-flipatoms to an untrapped state [25]. As a result, efficiency as a function oftrap lifetime is an important consideration in our laboratory. Figure 3(b)demonstrates the required number of atoms with which one must begin inorder to attain a specified efficiency, γ, as a function of lifetime. If lifetimeis reduced, the number of atoms loaded into the trap must be improved tomaintain an efficient evaporation. The linear fits in Fig. 3(b) give a scalingrelationship between N , and trap lifetime, τlife, that is preserved for effi-ciencies between γ = 2 and γ = 4. In this harmonic trap, the relationshipbetween the trap lifetime and the number of atoms required to acheive agiven efficiency is

N ∝ τ−1.5life . (40)

3.2.3 The µEM advantage

As a final insight, the trapping frequencies can be changed in the modelto see the effects of different degrees of confinement on the efficiency ofevaporation. Since in a harmonic trap it is the trap-averaged frequency (i.e.ω = 2πf) which affects the results, this is used as the metric for comparison.Figure 4(a) demonstrates behaviour as might be expected; traps with greatertrap frequencies have better efficiencies than those with less confinement, asthe degree of confinement determines the collision rate of the ensemble. Atonly half the nominal frequency, the evaporation fails to bring the cloud todegeneracy.

Figure 4(b) shows how attaining a given efficiency means loading moreatoms as trap frequencies are decreased. As above, a scaling relationship isestablished from these results, such that over γ = 2 to 4, N ∝ f−3.

These simulations show that one might expect evaporative cooling towork with the experimental conditions we have available in the laboratory.In particular, the high trapping frequencies available with the µEM trapwill increase the efficiency of evaporation while relaxing the requirements forthe lifetime. Figure 5 demonstrates the relationship between trap lifetimeand trap frequencies for a consistent set of starting conditions. Attaining agreater trap lifetime allows the experimentalist to “get away” with lower trapfrequencies and vice versa. The scaling relationship between trap lifetimeand trapping frequencies is τ ∝ f−2.

19

Figure 4. Comparison of evaporation efficiencies with various lifetimes. a.)Results of simulations at various lifetimes. Top to bottom curves: f = 5f0,f = 2f0, f = f0, f = 0.5f0, and f = 0.1f0, where f0 = (800 × 800 × 30)1/3.Efficiency, γ, is given by the negative slope of each curve. b.) Required atomnumber as a function of trap averaged frequency, f , at given efficiencies. N ∝f−3.

4 Experimental implementation of evaporative cool-ing

4.1 Experimental overview

The means by which we implement evaporative cooling require that theatoms be relatively cold at the beginning of the process. The techniquesof laser cooling and trapping are implemented prior to evaporative cool-ing in order to attain these temperatures. A detailed explanation of theexperimental apparatus and the process by which atoms are prepared forevaporative cooling is given in Ref. [21], and a short summary of these stepsis outlined below.

The first stage of the experiment is the magneto-optical trap (MOT). Sixcircularly polarized laser beams whose frequencies are precisely controlledand tuned below resonance intersect at the magnetic field minimum of aquadropole trap. This scheme selectively removes momentum from fasteratoms and creates a cloud of slowly moving (cold) atoms at the intersection

20

Figure 5. Required trapping frequency as a function of lifetime at givenefficiencies. τlife ∼ f−2. Starting conditions are N = 1 × 107 atoms andρ = 1× 10−6.

of the beams [26]. The temperature in the MOT is limited by the Dopplertemperature, characteristic of the transition linewidth. In 87Rb, this Dopplerlimit is 144µK.

Before moving to a magnetic trap, the atoms undergo optical molasses,a short duration during which the atoms are cooled to sub-Doppler tem-peratures using only optical fields. Optical pumping is also done here, aprocess that moves all of the atoms into an appropriate hyperfine state toavoid spin-exchange losses in the magnetic trap. Here, phase space densitiesof up to 10−5 are reached.

The atoms are transferred into a magnetic trap, in which they are con-fined by the force associated with the magnetic dipole moment of the theatoms. The minimum of this macroscopic magnetic trap is moved throughspace until it transfers the atoms to a more tightly confining µEM trap,where a magnetic field minimum is formed slightly below the surface of asilicon substrate. This magnetic trap is the result of the combination mag-netic fields produced by current running through gold wires fabricated on thesubstrate and weak external magnetic fields [27]. It is in this tight magnetictrap the evaporation occurs.

4.2 Methods for collecting data

In all experiments we conduct, our data takes the form of a two dimen-sional image. We use either absorption or fluorescence imaging. Absorption

21

position, x ( µm)

po

sitio

n,

y (

µm)

−400 −200 0 200 400

−200

−100

0

100

200

−400 −200 0 200 4000

0.05

0.1

0.15

0.2

position, x ( µm)

Op

tica

l De

pth

Figure 6. Left: Absorption image of 87Rb with 2.1 × 106 atoms after 3 mstime of flight. Right: Gaussian fit to column averaged data of the absorptionimage. The fit gives a value of σx = (151 ± 4)µm which corresponds to atemperature of (26.5± 1.5)µK.

imaging relies on the information we can obtain from the shadow of an atomcloud. A beam of resonant light is shone through the cloud of atoms directlyinto the camera and a second, background, image is taken when no atomsare present. The ratio of intensity between these two images at each pointdepends on the amount light absorbed, which gives direct information aboutthree dimensional density integrated along the imaging direction. The twodimensional shadow of the atoms completes the information necessary fordetermining the number of atoms present. Fluorescence imaging relies onthe re-radiation by the atoms after they have absorbed light. The camerais placed at right angles to the incident radiation, collecting the maximumamount of scattered light proportional to the atom number. Fluorescenceimaging is used as a qualitative tool only while absorbtion imaging is usedto determine the atom numbers and densities.

4.2.1 Measuring the extent and temperature of the atom cloud

In addition to being able to determine the atom numbers and densities,these two dimensional images allow us to measure the dimensions of thecloud. By fitting the density distribution of the atoms to a specified function,often a Gaussian, the lengths and widths of the clouds can be determinedand compared. Figure 6 shows the absorption image of a 87Rb cloud andthe corresponding Gaussian fit in the horizontal direction.

Temperature is an important parameter when conducting experiments

22

leading to and including quantum degeneracy. When considering evapo-rative cooling, it is a necessary parameter to determine the phase spacedensity.

The standard method of determining the temperature in atomic gasesis to use a time-of-flight (TOF) measurement. When a trapped gas is al-lowed to expand, its spatial extent is proportional to the initial velocitiesof the atoms, and therefore, the temperature. The size of the cloud and itstemperature are related using

σi =√

σ2i,0 + v2

rmst2, (41)

where σi is the e−√

2 radius of cloud in the ith dimension, σi,0 is the intialradius of the cloud, vrms is the root-mean-square one-dimensional velocityof the atoms and t is the time for which they have been allowed to expand(the “time of flight”). By taking measurements of the size of the cloud andusing Gaussian fits to determine σi at different times of flight, we can useof a fit of these data to extract vrms. From this, the relationship

12mv2

rms =12kT (42)

allows us to calculate the temperature.A simpler (and quicker) way of measuring the temperature is to do a “one

shot” measurement in which the temperature is determined by measuring σi

and t as above, but calculating σi,0 using the characteristics of the harmonictrap. The initial cloud size in a harmonic trap is

σi,0 =

√kT

mω2i

. (43)

Substituting this result, Eq. (41) can be solved for temperature. This can isfeasible in a harmonic trap because the potential is separable into potentialand kinetic energy terms. In the case where the trap is not harmonic, aone-shot measurement of temperature can be performed if images are takenat long times of flight. The cloud enters the asymmtotic region of Eq. (41),that is t is large enough that σ ≈ vrmst. Measuring σi and controlling tallows the calculation of temperature.

4.3 Trap characterisics

The characteristics of the trap are important for determining processesthat happen in it. Most importantly, we need to know the shape of the

23

trap; we must determine if it falls into the class of power-law potentials andif so, which power it obeys. Calculations of the potential created by themagnetic fields involved indicate that the trap is harmonic for temperatures

kT . mF gF µB|BIoffe |, (44)

where BIoffe is an external field applied to the chip trap to pull the trapaway from a magnetic field zero and avoid Majorana spin-flip losses. Thisfield can be determined by measuring the trap bottom using RF cuts, andusing the atomic properties of 87Rb to determine the relationship betweenthe RF energy and the magnetic field. In the “slow evaporation” experimentto be described, the trap bottom was found to be 3.515 MHz and the trapis approximately harmonic for T < 340µK.

4.3.1 Trap frequency measurements

The shape of the potential surface of a harmonic trap, and its confine-ment, is described by the frequency, ωi, where i can stand for any of thethree dimensions. Our trap has cylindrical symmetry with two characteristicfrequencies, one in the radial direction and one in the longitudinal direction.Re-expressing the potential (Eq. 7) to elucidate this symmetry leads to

U(r,z) =12mω2

rr2 +

12mω2

zz2. (45)

An atom trapped in a harmonic trap will behave like any classical particlein a harmonic oscillator potential, that is, it will oscillate with the trapfrequency. In order to measure this oscillation, we disturb the potentialby changing one magnetic field, quickly returning to the regular potential,and measuring the centre position of the atoms as a function of time afterthe disturbance. We find that the trap centre moves periodically; by fittinga damped sinusoid to these measurements, we are able to attain a trapfrequency. The measurement is made in both the radial and longitudinaldirections, and Fig. 7 shows an example of such a measurement.

4.3.2 Trap profile measurements

As a means of ensuring that our trap is harmonic and that the trapfrequency measurements are valid, we employ an alternate way of lookingat the trap geometry. The images taken by the camera provide informationabout the density distribution of the atoms in the trap. In the classical

24

Figure 7. Measurement of trap frequency for 40K, “Intermediate trap”. Datais fit to a decaying exponential; the fit parameter for frequency yields a radialoscillation frequency of frad = (47± 1) Hz.

regime, the density distribution is directly related to the potential throughthe relationship

n(r) = n0e−U(r)/kT . (46)

Given that we can measure both the density and temperature of the atoms,the potential remains the only unknown quantity. Upon rearrangement ofEq. (46), we find that

U(r) = −kT ln(

n(r)n0

), (47)

where we can ignore the n0 by setting it to unity, for it produces only anoffset, which we can ignore.

Using this method in the “compressed” trap, we find that the densityprofile method gives a potential energy surface that looks very much likethe one calculated for these conditions (see Fig. (8)). Of note, however,are the finer structures that become evident at lower temperatures. It ap-pears that the large harmonic well fragments into smaller wells, which arealso accurately approximated by harmonic potentials. We believe that thisfragmentation is a result of defects in the wire through which current runsto provide the magnetic field. At low temperatures, the atoms will clusterinto one of these wells and behave as though trapped in a potential withcharacteristics of this more tightly confining well. It turns out that the trapfrequencies for these smaller, fragmented wells are greater than that of the

25

Figure 8. Calculation of trap profile from density measurements at ∼ 7µK;dashed line represents a theoretical calculation of the potential. Absorptionimage is below. The higher density regions of the trap correspond to a lowerpotential energy. Of interest in this trap is the “dimple,” a region of tighterconfinement that dominates at low temperatures. This has the effect of im-proving the collision rate and, therefore, efficiency of evaporation.

large, main well, and as such, the collision rate and density of the ensembleis improved, increasing the efficiency of evaporation.

4.3.3 Trap lifetime

Measuring the trap lifetime is important for determining the losses inher-ent to the experiment. We assume that the only losses at the intial densitieswill be due to background collisions and not three-body loss, the next mostinfluential loss mechanism. This is a good approximation considering ourinitial densities are orders of magnitude lower than the critical density of1014 cm−3 at which three-body loss becomes a dominant mechanism.

To measure the trap lifetime, we hold the atoms in the magnetic trap,and measure the number of atoms left after a variable hold time. These dataare fitted to a decaying exponential, and we extract from them a lifetime,

26

τlife, as defined through the fitting function:

N = N0e−t/τlife . (48)

4.4 Generating radio frequency radiation for evaporation

The RF radiation used in the experiment is generated using Direct DigitalSynthesis (DDS), which provides stable and accurate frequency control. Asa chip trap is being used, the antenna for this RF radiation is a wire on thesubstrate located near the trapping wire.

To implement forced evaporation, the frequency of the radiation is con-tinuously lowered. Piecewise-linear sections are used to approximate thenearly-exponential behaviour predicted by the simulations. Optimization ofthe ramp is done with the ultimate goal of acheiving an efficient evapora-tion, that is, by increasing the phase space density while retaining as manyatoms as possible.

5 Comparison of theoretical and experimental re-sults

5.1 Connections between the model and the experiment

The ultimate goal of this project was to create a model that would ac-curately predict the viablity of evaporative cooling under a given set ofexperimental conditions. In order to evaluate its success, the results of theexperiment need to be compared with those of the model.

The method by which the comparison is made is to take the starting num-ber, phase space density and trap lifetime, and use them with an additionalpiece of information from the experiment, the RF ramp. The informationgiven in the RF ramp can be converted into information about the trapdepth as a function of time. Using the relationship

η =1|gF |

h(fRF − fbottom)kT

(49)

where fbottom is the trap bottom, an experimentally determined frequencyat which all of the atoms are removed from the trap corresponding to theminimum in the magnetic potential. Information of the temperature at eachpoint allows for a calculation of η, and with this, the evolution of the entireevaporation process can be calculated.

27

To fully connect the experimental results to those which are calculated,we must be able to convert the values we measure into meaningful param-eters to be compared with the model. Temperature and number are twoparameters easily measured and compared, and values for density can befound from the characteristics of a harmonic trap (Eq. (37)) or measur-ing the spatial extent of the cloud to determine its volume and calculatingdensity with information about the number of atoms.

5.1.1 Error Analysis

Given that the calculated values use as inputs values which contain uncer-tainties, this error must be taken into account in order to develop a basis forcomparison. The calculated model takes as inputs the number, phase spacedensity, trap frequencies and RF trajectory. We can assume that the errorin the RF trajectory is negligible due to the high precision of the digitalsource. Uncertainties in the remaining values are all considered.

To deduce the measure of uncertainty inherent in the calculated valuesalong the trajectory due to these various sources of error, I have used a“Monte Carlo” type approach, where random errors are assigned to eachof these values and the evaporation trajectory is computed many times.The error assigned to each of these quantities is weighted with a normaldistribution around the nominal value with a standard deviation equal tothe value of the bounds of the error bar.

The error in each calculated value is determined by looking at the rangeof values at one time. An error bar can be calculated from this distribution,which is normal, by obtaining the standard deviation of the data. Errorbars are calculated at many points along the evaporation path and includedas dashed lines in the plots to follow.

5.2 Experimental results

Two sets of data are presented in comparison to this model. A third setof data specifically for this report was attempted, but due to some experi-mental difficulties, the data collected was flawed. In the two sets that havebeen collected, the trapping potentials and the goals of the experimentswere different. In the first, the evaporation trajectory was experimentallyoptimized to reach degeneracy in a Bose system, while the second trajectorywas optimized for sympathetic cooling of fermions, though the data is withbosons present only. The resulting efficiency is, as expected, lower than thatof the first set of data.

28

0 1000 2000 3000 4000 5000 6000 7000

0

5

10

15

Time (ms)

RF Frequency (MHz)

14.923 MHz @ t = 0

9.923 MHz

5.923 MHz

3.923 MHz

1.923 MHz

0.923 MHz 0.223 MHz

0.123 MHz

0.043 MHz

↑24.371 MHz @ t = 0

5.371 MHz

1.371 MHz

0.401 MHz

0.171 MHz

0.046 MHz

Figure 9. RF ramps. Dashed line represents fast evaporation experiment;solid line represents slow evaporation experiment. RF frequencies are labelledin reference to the experimentally determined trap bottoms.

5.2.1 Experiment 1: Fast evaporation

In the first experiment, the evaporation ramp was optimized experimen-tally for final phase space density by changing the durations and end pointsof the short linear ramps that comprise the overall trajectory. Figure 9(dashed line) demonstrates this experimental RF ramp. Some initial evap-orative cooling was done in a more tightly confining trap, but as densitiesbegan to approach those where three-body losses are dominant, the trap wasrelaxed (trap frequencies reduced) to allow for evaporation to occur withoutthree-body loss. Results in this report will focus only on this second stage ofthe experiment. A graphical comparison of the measured results and thosegiven by the model is shown in Fig. 10.

5.2.2 Experiment 2: Slow evaporation

In the second experiment, the evaporation ramp was optimized for thesuccess of the sympathetic cooling procedure when both fermions and bosons

29

Figure 10. A selection of parameters from experiment and model for fastevaporation. Uncertainties in (a)-(c) are indicated by the dashed line for themodel and error bars for the model. (a) Atom number, N , as a function oftime; (b) Phase space density, ρ, as a function of time; (c) Trap depth, η,as a function of time; (d) Efficiency plot for experiment (open symbols) andmodel (closed symbols) as calculated at fixed time points. Dashed line connectscalculated points as a guide to the eye.

30

were present in the trap. As a result, the evaporation is much longer andin a less tightly confining trap from the beginning. Data was taken for thisramp when bosons were alone in the trap and is presented here. The RFramp used here is shown in Fig. 9 (solid line). Figure 11 demonstrates agraphical comparison of experimental and modelled results.

5.2.3 Comparisons between experiment and calculations

In Figs. 10 and 11, plots (a) - (c) show the temporal evolution of the evap-orative cooling process. Figures (a) show the atom number as a function oftime, with good agreement between the model and measurements. Compar-ison of phase space density in Figs. (b) show good agreement, though thereare points which lie outside the error bound for both experiments. Fig-ures (c) show the evolution of trap depth and the poor agreement betweencalculated and measured results.

In the second experiment, the third experimental point lies away fromthe modelled data and the overall trend in all plots, indicating there maybe a problem with the measurement of this point. As well, the final pointsin the trajectories tend to fall away from the modelled results. This effectis likely due to the fact that it is a classical process that is being calculatedwhile the last points are in a regime where the quantum statistical effectsare important.

More fundamental, however, is the systematic disagreement in the quani-ties indirectly related to temperature. In particular, the agreement of phasespace density with respect to time is slightly off, with measured points fallingbelow the calculated ones. This same disagreement arises in the comparisonof trap depth, η. Both of these values rely on a measurement and calculationof temperature, and these discrepancies would indicate that temperature isbeing over-estimated in the measurements.

The atom number calibration for this experiment has yet to be per-formed, and as such, there is a systematic error not represented by the errorbars in these plots. A discrepancy of up to a factor of 2 in atom numberis possible, and this may explain much of the inconsistency between the ex-perimental and modelled data. In particular, this would explain the betteragreement in the Fig. 10(a) with atom number, N as compared to Figs.10(b) and (c), both of which rely on calculation of temperature. This sys-tematic error is absorbed by the model in the calculation of atom numberwith the input of atom number as an initial condition, while for phase spacedensity and especially η, the indirect measure of temperature would exposethis systematic error.

31

Figure 11. A selection of parameters from experiment and model for slowevaporation. Uncertainties in (a)-(c) are indicated by the dashed line for themodel and error bars for the model. (a) Atom number, N , as a function oftime; (b) Phase space density, ρ, as a function of time; (c) Trap depth, η,as a function of time; (d) Efficiency plot for experiment (open symbols) andmodel (closed symbols) as calculated at fixed time points. Dashed line connectsmodelled points as a guide to the eye.

32

Other issues that may lead to inconsistencies between the measured andcalculated results include the possible anharmonicity of the trapping poten-tial. The defects in the wire that create potential dimples in the trap profile(see Fig. 8) mean that trap frequencies may change during the evaporation,which is not considered in the model. If the trap is initially less compressedthan it is towards the end of the experiment, this could explain part of theover-estimation of temperature.

The suggestion that the atoms enter the hydrodynamic regime along thelong axis of the anisotropic trap is also a potential source of inconsistency inthese data. Since collision rates are calculated to be on the order of hundredsper second, while the trap frequency in the long direction is only tens of Hz,an atom will collide many times before traversing the axial dimension. Theassumption of sufficient ergodicity may break down here. It remains unclearwhether one must adjust the effective collision rate in the calculation toaccount for these ineffective collisions (that “hot” atoms will remain in thetrap and redistribute their energies before reaching the point where theycan be ejected), or whether the mixing of the atomic motions into the otherdimensions eliminates these effects. At present, these effects are ignored inthe modelling, though they will be investigated in the future.

Figures 10(d) and 11(d) show the efficiency plots for the experimentaland modelled results. Agreement between experimental and calculated re-sults is good, though agreement in number is better than that in phasespace density. Regardless of the agreement between individual points, theagreement between the efficiency (negative slope of the trend) between theexperiment and model is excellent, validating the conclusions reached in §3.2about the conditions necessary for efficient evaporation.

In the first, fast experiment, fits to the experimental data and to the mod-elled data give efficiencies of γ = 4.00±0.05 and γ = 4.10±0.06, respectively.This efficiency γ ≈ 4 is exceptionally good among quantum gas experimentsand comes as a result of a reasonable trap lifetime (τlife ∼ 6s) and a tightlyconfining harmonic trap (with f = ((741± 2)× (741± 2)× (25.6± 0.3))1/3

Hz).In the second, slower experiment, fits to the data give efficiencies of

γ = 3.0± 0.2 and γ = 3.4± 0.2 for the experimental and modelled efficien-cies, respectively, lending more credibility to the results of §3.2. The overallefficiency in this slow evaporation is less than that of first experiment, asmight be expected. Here, the lifetime is similar (τlife ∼ 5s), but the trap fre-quencies have been relaxed to f = ((560± 5)× (560± 5)× (31.4± 0.1))1/3

Hz due to complications that arise in the presence of the fermionic species

33

[28].The agreement between the experimental and modelled results is rea-

sonable and justifies the use of the model in describing real experimentalconditions. The agreement between the efficiencies of experiment and cal-culation is especially promising, lending credibility to the results of §3.2and validating its conclusions. This model is a good tool for predicting thesuccess of an evaporative cooling process.

6 Extensions to the project

As a continuation of this project, there remain a number of questions thatcould be answered. In one vein, a more detailed approach to the calculationsdone in this report could be taken, while in another, these results couldbe extend to include a fermionic species and the prospect of sympatheticcooling.

To better understand the entire evaporative cooling trajectory, the modelmust be extended beyond the harmonic approximation. The beginning ofthe fast evaporation that was not analyzed in this report includes times whenthe temperature exceeds the limit of Eq. (44) and the trap takes on non-harmonic characteristics. The analytics of this trap have been calculated byother group members, and the best approach for analyzing the evaporativepath in this regime and its extension into low temperatures would be totake the approach of Ref. [16] where numerical integrations of Eq. (8) for anarbitrary trapping potential can be done.

The goal in our laboratory at present is to use sympathetic cooling toreach quantum degeneracy in a system of fermions. In this process, a “bath”of bosons is evaporatively cooled in the presence of fermions. Collisions be-tween bosons and fermions rethermalize the fermions to the boson tempera-ture, which is being continuously reduced via evaporation. One prospect forfuture work would be to calculated the evaporative cooling process for thisdual species set-up. To do this, one would have to take into account the factthat though the bosonic species is being evaporated as before, there are nowcollisions with a second species, a species that is not being evaporated. Thisshould reduce the efficiency of evaporation, especially when the number ofboth species is nearly equal.

Models for this process have been developed with varying degrees ofdetail. Delannoy et. al. developed a simple model for the cooling of twospecies of equal mass when only one is being evaporated [29]. This modelwas extended to species with unequal masses by Mosk et. al. [9], a step

34

which would be necessary in our dual species (40K and 87Rb) experiment.A model based on the same Boltzmann equations as the Amsterdam model(c.f. Eq. (8)) was explored by F. Schreck [30] by calculating the distributionsof bosons and fermions separately and using only boson-boson and boson-fermion collisions to redistribute the energy. Assuming the same successwith this model as with that used in this report, developing an extensionto these calculations to quantify sympathetic cooling should lead to a gooddescription of the processes seen in experiment.

7 Conclusions

Evaporative cooling is an effective means by which to achieve quantumdegeneracy in atomic systems, playing the crucial last role in the experi-ment. The removal of the most energetic atoms of the atomic ensembleallows those remaining to rethermalize to lower temperatures and increaseddensities. This process has been explored using classical kinetic theory, andseveral models have established to calculate the results of the process. Usingthese calculations, predictions can be made as to the success of evaporativecooling under various experimental conditions establishing the relationshipsbetween requirements for trap lifetime, confinement and starting conditions.In particular, there is a strong relationship between the confinement of theand the lifetime whereby increasing one relaxes the restriction on the other.In a µEM trap, especially strong confinement allows for meagre trap life-times while maintaining efficient cooling.

A comparison of the results of based on the Amsterdam model withthose measured in experiment show reasonable agreement, demonstratingthe usefulness of calculations of the process. Good efficiencies of evaporationup to γ = 4 are found in the µEM trap. The future holds the prospect ofusing the cold bosonic system as a way of cooling a fermionic ensemble andachieving degeneracy there.

35

A Appendix

A.1 Calculation of temperature-number relationship, α

Using the definition of α, Eq. (25), the first task is to find expressions ford(lnT ) and d(lnN). As has been demonstrated earlier, the expressions forthe energies are expressible as function of the numbers, and it is the goal tofind an expression for d(lnT ) in terms of d(lnN). Three-body losses are notconsidered in this calculation.

The first step is to write out E/E using Eq. (22),

E

E=

Nev

[η +

(1− Xev

Vev

)+ λ

R (3/2 + ζ)]

+ ηNt

N(3/2 + ζ=

N

N+

T

T. (50)

Next, a rearrangement of Eq. (20) for the overall loss from the trap leads to

Nev =N − Nt(1 + λ

R

) (51)

which can be substituted into (50). A key approximation in this calculationis that of quasi-static ramping - the relative trap depth, η changes veryslowly compared to either temperature or number, so for Nt = ηkT ,

Nt

N= ξ

T

T(52)

is a valid approximation. Under this approximation,

T

T

1 + ξη +

(1− Xev

Vev

)+ λ

R (3/2 + ζ)(1 + λ

R

)(3/2 + ζ)

− ξη

3/2 + ζ)

=

N

N

η +(1− Xev

Vev

)+ λ

R (3/2 + ζ)(1 + λ

R

)(3/2 + ζ)

− 1

, (53)

which can be rearranged and simplified to give Eq. (27).

A.2 Calculation of time evolution of number

The calculation for the time evolution of the atom number begins withEq, (20), rewritten in the form

−N

N=

1τev

+1

τloss+

1τt

, (54)

36

representing losses in atom number due to evaporation, loss processes, andtruncation. We must consider the temperature loss rate and its relationshipto number loss rate through α, Eq. (25) to properly account for the last termof this equation. As above, we assume η is approximately constant and allowthat the rate due to truncation is directly proportional to temperature lossrate through ξ. Then,

− T

T= α

(1

τev+

1τloss

)+ ξα

T

T(55)

which can be rearranged and divided through by α to yield an expression

N

N= −

(1

τev+ 1

τloss

1− ξα

)(56)

which is simply integrated to give the final result, Eq. (38).

37

References

[1] H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E. A.Cornell, Science 269 198 (1995); K.B. Davis, M.-O. Mewes, M.R. An-drews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle,Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, C.A. Sackett, J.J.Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); see alsoC.A. Sackett, C.C. Bradley, M. Welling, and R.G. Hulet, Appl. Phys.B B65, 433 (1997).

[2] B. DeMarco and D.S. Jin, Science 285, 1703 (1999); A.G. Truscott,K.E. Strecker, W.I. McAlexander, G.B. Partridge, and R.G. Hulet,Science 291, 2570 (2001); F. Schreck, L. Khaykovich, K.L. Corwin,G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett.87, 080403 (2001); S.R. Granade, M.E. Gehm, K.M. O’Hara, and J.E.Thomas, Phys. Rev. Lett. 88, 120405 (2002); G. Roati, F. Riboli, G.Modugno, and M. Inguscio, Phys. Rev. Lett. 89, 150403 (2002).

[3] A. Peters, K.Y. Chung and S. Chu, Nature, 400, 849 (1999).

[4] M. Greiner, O. Mandel, T. Esslinger, T.W. Hansch and I. Bloch, Nature,415, 39 (2002).

[5] For examples, see: C.A. Regal, M. Greiner, and D.S. Jin, Phys. Rev.Lett, 92, 040403 (2004); M.W. Zweirlein, J.R. Abo-Shaeer, A. Schi-rotzek, C.H. Schunck, W. Ketterle, Nature, 435, 1047 (2005).

[6] W. Hofstetter, J.I. Cirac, P. Zoller, E. Demler, and M.D. Lukin, Phys.Rev. Lett., 89, 220407 (2002).

[7] E. Shuryak, Nuc. Phys. A, 750, 64 (2005).

[8] W. Hansel, P. Hommelhoff, T.W. Hansch, and J. Reichel, Nature, 413,498 (2001); H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossman, and C.Zimmermann, Phys. Rev. Lett. 87, 230401 (2001).

[9] A. Mosk, S. Kraft, M. Mudrich, K. Singer, W. Wohlleben, R. Grimm,and M. Weidemuller, Appl. Phys. B, 73, 791 (2001).

[10] R.K. Pathria, Statistical Mechanics, 2nd ed. (Butterworth-Heinemann,Oxford, 1996).

[11] H. Hess. Phys. Rev. B. 34, R3479 (1986).

38

[12] W. Ketterle and N.J. van Druten, Adv. At. Mol. Opt. Phys. 37, 181,(1996).

[13] K.B. Davis, M.-O. Mewes, and W. Ketterle, Appl. Phys. B. 60, 155(1995).

[14] J.T.M. Walraven, in Quantum Dynamics of Simple Systems, 44th Scot-tish Universities Summer School in Physics., editted by G.-L. Oppo,S.M. Barnett, E. Riis, and M. Wilkinson (SUSSP Publications, Edin-burgh, and Institute of Physics Publishing, London, 1996), p 315.

[15] O.J. Luiten, M.W. Reynolds, and J.T.M. Walraven, Phys. Rev. A 53,381 (1996).

[16] C.A. Sackett, C.C. Bradley, and R.G. Hulet, Phys. Rev. A, 55, 3797(1997).

[17] H. Wu and C.J. Foot, J. Phys. B.: At. Mol. Opt. Phys. 29, L321 (1996);H. Wu, E. Arimondo, and C.J. Foot, Phys. Rev. A 56, 560 (1997).

[18] M. Holland, J. Williams, and J. Cooper, Phys. Rev. A 55, 3670 (1997).

[19] K. Huang. Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).

[20] E.G.M. van Kempen, S.J.J.M.F. Kokkelmans, D.J. Heinzen, and B.J.Verhaar, Phys. Rev. Lett. 88, 093201 (2002).

[21] S. Aubin, M.H.T. Extavour, S. Myrskog, L.J. LeBlanc, J. Esteve, S.Singh, P. Scrutton, D. McKay, R. McKenzie, I. Leroux, A. Stummer,and J.H. Thywissen, J. Low Temp. Phys. 140, 377 (2005)

[22] J. Soding, D. Guery-Odelin, P. Desbiolles, F. Chevy, H. Inamori, andJ. Dalibard, Appl. Phys. B. 69 257, (1999).

[23] I.D. Leroux, B.A.Sc. Thesis, University of Toronto, 2005 (unpublished).

[24] W.H. Wing, Prog. Quant. Electr. 8, 181 (1984).

[25] J. Fortagh, H. Ott, S. Kraft, A. Gunther, and C. Zimmermann, Phys.Rev. A 66, 041604(R) (2002).

[26] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping,(Springer-Verlag, New York, 1999).

[27] M.H.T. Extavour, M.Sc. Thesis, University of Toronto, 2004 (unpub-lished).

39

[28] At temperatures near those with which we begin evaporation, the scat-tering cross-section between 40K and 87Rb is reduced by a factor of ∼100, and a slow evaporation to ensure rethermalization is necessary. Inaddition, the trap frequencies must be reduced to avoid collapse be-tween the BEC and fermions at low temperatures (collapse reference).

[29] G. Delannoy, S.G. Murdoch, V. Boyer, V. Josse, P. Bouyey, and A.Aspect, Phys. Rev. Lett., 63, 051602(R) (2001).

[30] F. Schreck, Ph.D. Thesis, Ecole Normale Superieure (2002).

40