Bose einstein condensation and atom cavity interactions

These include the all-optical production of aBose-Einstein condensate in a 1 µm wavelength crossed beam dipole trapand direct mechanical translation of cold atoms into a high finesse cav

INTERACTION OF ATOMS AND LIGHT IN A HIGH FINESSE CAVITY

KYLE JOSEPH ARNOLD

B.S Eng Physics, University of Illinois Urbana-Champaign B.S Mathematics, University of Illinois Urbana-Champaign

A THESIS SUBMITTED FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

CENTRE FOR QUANTUM TECHNOLOGIES

NATIONAL UNIVERSITY OF SINGAPORE

2012

I herewith declare that the thesis is my original work and

it has been written by me in its entirety I have duly acknowledged all the sources of information which have

been used in the thesis.

The thesis has also not been submitted for any degree in

any university previously.

Kyle Joseph Arnold

1 December 2012

First and foremost, I would like to thank my supervisor, Dr Murray rett We have worked closely over the years and without the wealth ofatomic physics, optics, and electronics knowledge I have received from him,the work in the thesis would not have been possible.

Bar-Next I would like to thank Markus Baden, my partner on the cavityexperiments In particular, for many fruitful physics discussions and takingthe time to proof read my thesis Also, for introducing me to python, mygo-to tool for scientific computing and source of many quality plots in thisthesis

Many thanks to my other fellow PhD students, Arpan Roy, Chuah Leng, and Nick Lewty, who, though not directly involved in my experiments,have all contributed to our common efforts in developing the lab Thanksalso to the many RAs who have helped out in the lab, in particular AndrewBah who produced the 3D-rendered experiment schematics for my thesis.I’m grateful for work of our CQT support staff, especially our procurementofficer, Chin Pei Pei, our electronics support staff, Joven Kwek and Gan EngSwee, and our machinists, Bob and Teo, who have made numerous partsfor me on short order

Boon-Finally, I would like to thank my wife, Vicky, for her continuous loveand support during these years, and my parents who having always beensupportive of my chosen path even though it has taken me to distant landsfar from home

List of Tables ix

1.1 Outline of the Thesis 3

2 Dipole Trapping and All-Optical Bose-Einstein Condensation 5 2.1 Introduction 5

2.2 Background 7

2.2.1 Dipole traps 7

2.2.2 Laser cooling 11

2.2.3 Evaporative cooling 12

2.2.4 Scaling laws 12

2.2.5 Atom losses due to inelastic collisions 15

2.3 Discussion of crossed beam traps 16

2.3.1 General thermal distribution of a trapped gas 17

2.3.2 Crossed beam distribution: numeric solution 18

2.3.3 Crossed beam distribution: approximate analytic solution 18

2.3.4 Thermalization in crossed beam traps 21

2.3.5 Analysis of a recent cross-beam result 23

2.3.6 Elliptical beams 24

2.3.7 Basics of Bose-Einstein condensates 25

2.4 Experimental setup 27

2.4.1 Cooling lasers 27

2.4.2 Imaging diagnostics 30

2.4.3 MOT 32

2.4.4 Dipole trap loading 33

2.4.5 Trap lifetime 34

2.4.6 Hyperfine changing collisions 35

2.4.7 Measuring trap frequencies 36

2.4.8 Thermal lensing 37

2.5 Bose-Einstein condensation experiment 38

2.5.1 Primary beam geometry 38

2.5.2 Primary beam free evaporation 39

2.5.3 Primary beam forced evaporation 39

2.5.4 Secondary beam geometry 41

2.5.5 Cross-beam compression 41

2.5.6 Observation of a condensate 43

2.5.7 Comments of observing a bi-modal distribution 46

2.6 Summary 46

3 Collective Cavity Quantum Electrodynamics with Multiple Atom Lev-els 49 3.1 Introduction 49

3.2 Cavity quantum electrodynamics 50

3.2.1 Jaynes-Cummings model 50

3.2.2 Real systems: dissipation 55

3.2.3 Cavity QED for N multi-level atoms 58

3.2.4 Semi-classical model for multi-level atoms 63

3.3 Experimental setup 66

3.3.1 High finesse cavity 66

3.3.2 Cavity laser system 69

3.3.3 Detection 71

3.3.4 Optical lattice transport 71

3.3.5 808 nm intra-cavity FORT 72

3.3.6 Optical pumping 74

3.4 Experimental results: cavity transmission spectra 74

3.4.1 Experiment procedure 75

3.4.2 Two-level atoms: the cycling transition 76

3.4.3 Multi-level atoms: π-probing 77

3.4.4 Driving both cavity modes 78

3.4.5 Optical pumping by the cavity field 79

3.5 Summary 81

4 Self-Organization of Thermal Atoms Coupled to a Cavity 83 4.1 Introduction 83

4.2 Derivation of the threshold equations 87

4.2.1 Lattice geometry 90

4.2.2 Traveling wave geometry 94

4.3 Experimental set-up and methods 95

4.3.1 Dual-wavelength high finesse cavity 95

4.3.2 Cavity laser system 97

4.3.3 Detection 100

4.3.4 Atom transport: translation of the dipole trap 101

4.3.5 1560 nm intra-cavity FORT 103

4.4 Experimental results: self-organization threshold scaling 106

4.4.1 Experimental procedure 107

4.4.2 Comparison to the threshold equations 107

4.4.3 Lattice geometry threshold results 109

4.4.4 Traveling wave geometry threshold results 110

4.4.5 Discussion of threshold scaling 111

4.5 Experimental results: dynamics of self-organization 113

4.5.1 Lattice geometry 113

4.5.2 Traveling wave geometry 120

4.6 Summary 122

5 Bragg Scattering, Cavity Cooling, and Future Directions 123 5.1 Introduction 123

5.2 Bragg scattering 124

5.2.1 Future Bragg scattering related experiments 127

5.3 Cavity cooling of atomic ensembles 128

5.3.1 Cavity cooling via the collective mode 129

5.3.2 Cavity cooling via self-organization 134

5.3.3 Conclusions and future experimental directions for cavity cooling 138 5.4 Future directions 139

A High finesse cavities: technical details 141 A.1 ATF mirrors 141

A.1.1 Brief History of low-loss mirrors 141

A.1.2 Mirrors from ATF: 2008-2011 142

A.1.3 Mirror handling and cleaning 144

A.2 Contamination of mirrors by Rb 145

A.3 Cavity construction 146

B Self-organization threshold equations 149 B.1 Lattice geometry 150

B.2 Traveling wave geometry 151

C Self-organization: temperature, entropy and phase space density 153

This thesis details experimental investigations into the interaction of anultracold atomic ensemble with a single mode high finesse optical cavity.

To this end, simple and efficient experimental methods are developed tocool and transport atoms These include the all-optical production of aBose-Einstein condensate in a 1 µm wavelength crossed beam dipole trapand direct mechanical translation of cold atoms into a high finesse cavityover ∼ 1 cm

First, we study the cavity transmission spectra for weak driving of a singlemode cavity coupled to a cold ensemble of rubidium atoms The multi-levelstructure of the atoms together with the collective coupling to the cavitymode leads to complex spectra which depend on atom number and probepolarization We model the linear response of the system as collective spinwith multiple levels coupled to a single mode of the cavity The observedspectra are in good agreement with this reduced model

Second, we study transverse pumping of a thermal ensemble of atoms pled to a cavity which results in self-organization The differences betweenprobing with a traveling wave and a retro-reflected lattice are investigated

cou-We derive threshold conditions for self-organization in both scenarios andverify a threshold scaling consistent with the mean field prediction over arange of atom numbers and cavity detunings

Most recently, a 2D lattice potential is used to organize the atoms into aBragg crystal, and coherent scattering into the cavity is observed withoutthreshold This configuration is ideal for future investigations into eithercavity sideband cooling of the collective motion or simulation of Dicke modelvia the collective spin

2.1 Scaling for forced evaporation in optical traps (η = 10) 14

2.2 Example initial conditions for a 1.06 µm cross beam trap (η = 8) 22

3.1 High finesse cavity parameters 67

4.1 Dual-wavelength high finesse cavity parameters 96

A.1 First coating run 144

A.2 Second coating run 144

2.1 Schematic and photograph of the experimental apparatus 7

2.2 Density distribution in cross beam trap 19

2.3 Cross beam potential profile 19

2.4 Fraction of atoms in wings vs η 20

2.5 Migration of atoms to wings during evaporation 22

2.6 Trap volume for elliptical beams 25

2.7 Schematic of master laser optics 28

2.8 Schematic of slave laser optics 28

2.9 Frequency tuning scheme for cooling lasers 29

2.10 Schematic of imaging system 30

2.11 Loading sequence for dipole traps 33

2.12 Dipole trap loading vs repump intensity The peak corresponds to a repump intensity of ≈ 5 µW/cm2 33

2.13 Atoms loaded vs FORT power 34

2.14 Atoms loaded into FORT vs MOT number 34

2.15 Trap lifetime for several Rb source currents 35

2.16 Hyperfine changing collisional losses 36

2.17 Measuring trap frequencies by parametric excitation 37

2.18 Measured radial trap frequencies of the primary beam 38

2.19 Free evaporation in the primary trap 40

2.20 Forced evaporation in the primary trap only 42

2.21 Composite trap evaporation cycle 43

2.22 BEC Images 44

2.23 Condensate fraction below the critical temperature 45

3.1 Cavity experiment schematic and photo 50

3.2 Jaynes-Cummings ‘ladder’ of dressed states 52

3.3 Vacuum-Rabi splitting 52

3.4 Eigenenergies for N-atoms 54

3.5 Transmission spectrum including dissipation 56

3.6 Schematic of experiment configuration 58

3.7 Collective enhancement example 61

3.8 Collective coupling to multiple hyperfine transitions 63

3.9 Analytic eigenspectrum for N -alkali atoms 64

3.10 Transmission spectrum for dispersion theory 65

3.11 Vibration isolation stack 68

3.12 Experiment cavity 68

3.13 PZT circuit 68

3.14 Schematic of experiment laser system 70

3.15 Intra-cavity FORT lifetimes 73

3.16 Optical pumping scheme 75

3.17 Lifetime after optical pumping 75

3.18 Experiment configuration for an effective two-level system 76

3.19 Transmission spectrum on cycling transition 76

3.20 Transmission spectra probing π-transitions 77

3.21 Transmission spectra probing both cavity modes 78

3.22 Optical pumping by cavity probe field 80

4.1 Schematic and picture of experiment apparatus 84

4.2 Schematic of self-organization 85

4.3 Self-organization potentials 92

4.4 Picture of dual-wavelength cavity 97

4.5 Cavity birefringence 98

4.6 780/1560 nm laser setup 99

4.7 Beatnote between our 1560 nm laser and a frequency comb phase-locked to a narrow reference laser 101

4.8 Cavity coupling and detection set-up 102

4.9 Lowest electronic states of 87Rb 104

4.10 Cavity mode overlap for 780/1560 nm lattices 105

4.11 Measured cavity coupling at nodes and anti-nodes 106

4.12 Heating rate due the scattering of probe beam 109

4.13 Heating due to adiabatic compression by the probe beam 109

4.14 Results of threshold measurements in the lattice geometry 110

4.15 Results of threshold measurements in the traveling wave geometry 111

4.16 Self-organization traces: lattice probe, very large dispersive shift, ˜∆c< −κ115 4.17 Self-organization traces: lattice probe, very large dispersive shift, ˜∆c= −κ116 4.18 Self-organization traces: lattice probe, large dispersive shift 117

4.19 Self-organization traces: lattice probe, µ → 1 118

4.20 Heating due to non-adiabatic dynamics 120

4.21 Self-organization traces: traveling wave probe, unstable configuration 121

4.22 Self-organization traces: traveling wave probe, stable configuration 121

5.1 Bragg scattering 125

5.2 Suppression of Bragg scattering 126

5.3 Entropy phase diagram for self-organization 136

5.4 Entropy and phase space density gain 136

5.5 Realization of Dicke model 139

A.1 Various cavities designs 145

A.2 Cavity construction set-up inside laminar flow hood 146

C.1 Entropy and phase space density across threshold 160

Since the discovery of laser cooling over two decades ago [1, 2, 3, 4, 5], the use of laserlight to precisely manipulate both internal and external atomic degrees of freedom hasbeen an essential tool for atom physics research Near to an atomic resonance, scat-tering of laser light via spontaneous emission is the dominant process Laser coolingharnesses this scattering to cool the motion of atoms down to micro-Kelvin temper-atures Additionally, optical pumping [6] methods utilize spontaneous emission toprepare atoms in a specific internal state For laser light far detuned from an atomicresonance, spontaneous emission is suppressed and the dominant processes are coherentscattering and the dispersive interaction with the light field In this regime, even moreprecise control is possible; by using Raman processes [7] to coherently manipulate theinternal electronic state or the motional state of the atom, and by using the dipoleforce which arises from the dispersive interaction of the induced dipole with the lightfield For sufficiently large detuning, the optical potential resulting from the dipoleforce is effectively conservative [8] Far-off resonance optical traps (FORT) are a ver-satile tool for the trapping and manipulation of cold atoms with which a variety oftrapping geometries can be realized, such as optical lattice potentials [9] and crossedbeam traps [10] Furthermore, with evaporative cooling, the FORT offers an all-opticalroute to producing Bose-Einstein condensate (BEC) [11] as a practical alternative tothe standard magnetic trapping methods [5, 12]

Inside a high finesse optical resonator, the normally weak dipole interaction is hanced by multiple passes of the light such that a single photon may strongly couple

en-to a single aen-tom Although the physics of cavity quantum electrodynamics (QED) was

first described nearly 50 years ago by the Jaynes-Cummings model (JCM) [13], only inthe last 20 years has mirror technology advanced to the point where the strong couplingregime has been experimentally accessible This has opened the way for experimentaldemonstration of phenomena which predicted by the JCM, such as the photon blockadeeffect [14] and the vacuum-Rabi splitting due to a single atom [15, 16] For a sufficientlygood cavity, an atom can even be trapped by the dispersive backaction from the field of

a single photon [17] Unlike a free-space FORT, the intra-cavity optical potential in adriven cavity depends on the position of the atom [18] and thus strongly couples to theatomic motion This coupling can result in dissipative forces [19] which have been used

to cool a single atom [20] Cavity cooling via coherent scattering is of particular interestbecause it can, in principle, be applied to any polarizable particles [21, 22] Extend-ing cavity cooling schemes to ensembles of particles in a cavity remains a tantalizingpossibility and area of active theoretical [23, 24] and experimental research [25, 26].For an ensemble of atoms trapped inside a cavity driven by a laser field, the atom-field coupling increases due to the interaction of N atoms with the single cavity mode.This system is described quantum mechanically by the Tavis-Cummings model [27] inwhich the atoms couple collectively to the field with an effective rate√N greater thanthe single atom rate The √N enhancement of the coupling is readily observed, forexample, in the normal mode splitting [28, 29] The dynamical effects of backaction onthe atomic motion due to the dispersive interaction are complicated by the fact thatthe total dispersive coupling depends on the positions of all of the atoms As a result,the motion of one atom couples to all the others via long-range cavity-mediated lightforces [30] Theoretical work has shown that consequently the cooling rates for somecavity cooling mechanisms do not scale favorably with N [31, 32] Recently however,cavity cooling of a single collective phonon mode at a collectively enhanced rate has beenexperimentally demonstrated [25] Whether this can be used to efficiently cool all modes

of an ensemble [25, 26], and whether this result holds for transverse pumping [33, 34, 35]are still under investigation

When driving an atomic ensemble with a laser field transverse to the cavity, thebehavior of the system is significantly different and interesting phenomena emerge Co-herent scattering into the cavity mode is enhanced by constructive interference only ifthe atoms are spatially ordered to match the phase of the driving and cavity fields.For uniformly distributed atoms, scattering is suppressed by destructive interference

In a standing wave cavity, the backaction on the atoms from the scattered field givesrise to a phase transition to a spatially organized array for sufficiently strong pump-ing [36] The first experiments to observe self-organization [37, 38] transversely pumped

a thermal ensemble falling through a cavity In addition to collectively enhanced herent scattering, these experiments reported cooling and a deceleration of the center

co-of mass motion resulting from resonator induced light forces Later experiments with

a BEC in a standing wave cavity explored the transition from a superfluid to the organized phase [39, 40], which was mapped to the Dicke Hamiltonian and associatedquantum phase transitions [41, 42] However, there are still open questions related toself-organization, specifically concerning the effective threshold scaling [34, 43] and theextent to which self-organization can be used for cavity cooling of a thermal ensemble[36, 44]

In the following chapters of this thesis, I describe work spanning five years and threegenerations of experimental apparatus touching on many of these topics The focus

of our research has evolved over time, together with the capability of the apparatus.The structure of this thesis follows the evolution of the experimental apparatus with achapter devoted to each The common thread throughout is the dispersive interactionbetween light and atomic ensembles First, in free space, where optical potentialsare used for the trapping and manipulation of atoms Second, in a cavity, where thecollective dispersive interaction with the cavity gives rise to the dispersive shift anddynamic optical potentials

Chapter 2 In our first experiment, we all-optically produce a BEC in crossed beamdipole trap of 1 µm wavelength Our experimental setup is relatively simplecompared to earlier 1 µm all-optical BEC experiments and thus could be easilyintegrated into more complex setups, such as a cavity QED experiment Wediscuss the effects of beam geometry on the thermal distribution and highlight theobstacles to condensing all-optically in short wavelength traps The techniquesfor cooling, trapping, and manipulating atoms described in this chapter form thefoundation for experiments described in subsequent chapters

Chapter 3 In our second experiment, we integrate trapped ultracold rubidium atomswith a high finesse optical cavity in the strong coupling regime For a weaklydriven cavity, we study the transmission spectra of a single mode cavity stronglycoupled to an ensemble of multi-level atoms The linear response of this system

is that of a collective spin coupled to multiple levels for which the ing effective couplings to the cavity mode are enhanced by √N We develop areduced quantum mechanical model for the system which is in good agreementwith the experimental results We also interpret the results from a semi-classicalperspective using linear dispersion theory

correspond-Chapter 4 Presently, in our third experiment, we employ a dual wavelength highfinesse cavity which allows for trapping of the atoms at every alternate anti-node

of the cavity mode This configuration is ideal for studying coherent scatteringinto the cavity from a transverse pumping field In this chapter, we detail oursystematic study of self-organization of thermal atoms for two probing geometries:

a retro-reflected lattice [37, 38, 43] and a traveling wave Self-organization in alinear cavity with a traveling wave probe has not been previously reported Wederive threshold conditions for both probe configurations in the mean field limit

We experimentally measure the scaling of the self-organization threshold over awide range of parameters and characterize the behavior of the induced dynamicalpotentials

Chapter 5 By using an additional transverse trapping lattice with our third iment apparatus, we are are able to organize the atom into a Bragg crystal bystatic FORT potentials alone We report our observation of threshold-free coher-ent scattering into the cavity and discuss potential future research directions withsuch a system Demonstrating cavity cooling of atomic ensembles has been one

exper-of our long standing research objectives, but has thus far produced null results

We end on a discussion of the practical and fundamental difficulties of coolingtrapped atomic ensembles coupled to a single mode cavity

Dipole Trapping and All-Optical Bose-Einstein Condensation

This chapter covers the first generation experimental apparatus, pictured in Fig 2.1, and is largely based on ’All-optical Bose-Einstein condensation in a 1.06 µm dipole trap’ K J Arnold and M D Barrett Optics Communications 284, 3288 (2011).

Since the first observation of a Bose-Einstein condensate (BEC) in 1995 [45, 46], therehas been a tremendous volume of experimental and theoretical work which continues tothis day So many labs around the world produce BEC that it can now be consideredroutine, though still by no means trivial The most common method is to use radiofrequency (rf) induced forced evaporation of atoms in a magnetic trap An alternativemethod, first demonstrated in 2001 [11], uses only optical trapping Since then, severalatomic species have been successfully condensed via this method For instance, using

a 10 µm wavelength CO2 laser, 87Rb [11, 47], 133Cs [48], and 23Na [49] have all beencondensed Using a near 1 µm wavelength fiber laser,87Rb [50, 51, 52], 133Cs [53], and

52Cr [54] have been condensed

The advantages of the all-optical approach over magnetic trapping include a tively simple experimental setup, comparatively high repetition rates, and the ability

rela-to trap arbitrary spin states The primary difficulty is associated with relaxing thetrapping potential to induce evaporation Relaxation of the trap leads to a continual

decrease in the collision rate as the evaporation progresses The decreasing collisionrate can stagnate the evaporation process before degeneracy is achieved In the earliestall-optical BEC experiments [11, 47], researchers were able to achieve sufficiently highinitial phase space and spatial densities by directly loading a 10.6 µm-wavelength trapthat BEC was reached before the evaporation process stagnated.

The original goal for the experiment described in this chapter was to produce BEC

in exactly the same simple crossed-beam optical trap geometry used in [11], exceptwith a 1.06 µm wavelength trap rather than a 10.6 µm wavelength trap The 1.06 µmwavelength is practically advantageous over the 10.6 µm wavelength because specialoptics and vacuum windows are not required This is important for integration intomore complex experiments which have limited optical access, such as our subsequentcavity QED experiments Although previous experiments had produced BEC in near

1 µm wavelength traps, they generally started from worse initial conditions and insteadused methods of varying complexity to circumvent a stagnating evaporation rate InRef [50] a mobile lens system dynamically compressed the atoms to offset the decreasingdensity In other experiments evaporation was forced without relaxing the optical trapusing either a strong magnetic field gradient [53], or a large displaced auxiliary beam[52]

Our original goal proved elusive, however, as the low beam divergence due to the

1 µm wavelength was, unexpectedly, a significant complicating factor for the crossedbeam trap geometry Later in this chapter, we will discuss the nature of the thermaldistribution in crossed beam traps in order to understand the source of our early diffi-culties Although focused primarily on cross beam traps, the considerations discussedwould equally apply to any trapping geometry composed of disparate volumes

Ultimately, we were able to produce a BEC all-optically in a 1 µm cross-beam trap

We used a method similar in spirit to the dynamic compression experiment [50], butwith the same minimal complexity of the crossed beam geometry Fig 2.1 shows aschematic representation of our experiment next to a photograph of the experimentchamber A single tightly focused dipole trap is loaded directly from a magneto-opticaltrap (MOT) and only later in the evaporation is a second transverse beam used to thecompress the atoms Using this approach, we are able to a achieve a gain in phasespace density of 105, higher than is typically possible in all-optical traps We reach

Figure 2.1: Schematic representation and photograph of the experiment Not all optics used in the experiment are present in the photograph.

degeneracy after 3 seconds of evaporation resulting in a mostly pure condensate of3.5 × 104 atoms

where α(ω) is the frequency dependent polarizability The interaction potential is thengiven by

Udip= −1

2hpEi = −

1

where the field intensity is I = 20c| ˜E|2, and the factor of 1/2 is because the dipole

is an induced, not a permanent one The dipole force arises from the gradient of theinteraction potential and is thus a conservative force proportional to the gradient ofthe laser intensity

While the real part of the polarizability governs to dispersive interaction which givesrise to the dipole force, the imaginary part governs the absorption Power absorbed bythe oscillating dipole is re-radiated via spontaneous emission processes at a scatteringrate

Γ

From these equations two key aspects of far-off resonant traps (FORT) are readilyobserved First, for ∆ < 0 (known as ”red detuning”) the potential is negative andthus atoms are attracted to the light field Second, while the potential scales as I/∆,the scattering rate scales as I/∆2 Therefore by going to large detuning and highintensity, the scattering rate can be made negligibly small while maintaining the samepotential depth.

2.2.1.2 Multi-level alkali atoms

Here we consider the results specifically for the ground state of alkali atoms, wherespin-orbit coupling leads to the D-line doublet of S1/2→ (P1/2, P3/2) Coupling to thenuclear spin then produces the hyperfine structure of the ground and excited states Inthis case, the potential and spontaneous scattering rate for an atom in a ground statewith total angular momentum F and magnetic quantum number mF are given by [8],

Udip(r) = πc

2Γ2ω03

2

∆2 2,F

∆2 1,F

so that the potential is mF-state independent

Note that the RWA has been made in the above equations For the experiments insubsequent chapters, 87Rb will be trapped in both 1064 nm and 1560 nm wavelengthFORTs where the counter-rotating terms should not be neglected Keeping the counterrotating terms results in a correction factor to the potential depth (Eq 2.10) of 1.15for a 1064 nm FORT, and 1.33 for a 1560 nm FORT

2.2.1.3 Rayleigh vs Raman scattering

The total scattering rate given by Eq 2.11 is composed of two parts: coherent Rayleighscattering does not change the internal state, and incoherent Raman scattering for

which the atom changes hyperfine or Zeeman level in the S1/2 ground state manifold.The ratio of these two rate is given by

2.2.1.4 Single focused beam FORT

The simplest possible trapping geometry is a single Gaussian laser beam of wavelength

λ focused to a waist w0, and thus with Rayleigh range zR = πw20/λ The FORTpotential is determined from the intensity profile of the Gaussian beam, i.e

U (r, z) = − U0

1 + (zz

R)2

!exp

and

ωax=

s2U0

mz2R =

s2U0λ2

To get a general idea of typical FORT parameters, 10 W from a 1064 nm wavelengthlaser focused to a 40 µm waist has a depth of U0/kb = 600 µK, radial trapping fre-quencies 2π × 1.9 kHz, and an axial tapping frequency 2π × 12 Hz The scatteringrate is only Γsc = 2π × 0.35 Hz Since recoil temperature for an elastically scattered

1064 nm photon is Trec ≈ 190 nK, the heating rate due to off-resonant scattering is

˙

T ≈ TrecΓsc ≈ 63 nK/s, which is very low relative to the trap depth

to natural linewidth Γ of the atomic transition and results in a minimum temperature

of the atoms Tdoppler = ~Γ/2kb [57] For 87Rb, this Doppler limit is 146 µK [58].Even lower temperatures can be reached by sub-Doppler cooling methods [3, 59] Sub-Doppler cooling methods are still reliant on photon scattering processes and so arelimited by the recoil energy of the spontaneously emitted photons In practice, sub-Doppler cooling is limited to ∼ 10Trec For the 780 nm cooling transition of 87Rb,

Trec ≈ 360 nK, and thus temperatures on the order of a few µK can be reached [60].For creating a Bose-Einstein condensate, the quantity of interest is the phase spacedensity ρ = nΛ3, where Λ =p2π~2/mkbT is the thermal de-Broglie wavelength and n

is the particle density Thus we want to not only cool the atoms via laser cooling, butalso achieve high densities In a MOT, the typical particle density is ∼ 1010cm−3which

is limited by radiation pressure from reabsorption of light scattered by the trappedatoms and collisions with excited state atoms [56] By increasing the magnetic fieldgradients as well as increasing the detuning of the cooling lasers, a MOT can be com-pressed to higher densities as a result of the reduced rate of excitation Densities ashigh as 5 × 1011 cm−3 are reported by this method [61] Alternatively by creating a

“dark spot” in the center of MOT where there is no repump intensity and thus a muchlower rate of excitation, densities close to 1012 cm−3 have been reported [62] Usingthese laser-cooling methods, the highest reported phase-space densities in a MOT are

in range 10−5 to 10−4 [63, 64] In order to cool atoms to higher phase space densities,methods which do not require excitation of the atom must be employed The onlysuccessful methods have been forced evaporation in either a magnetic trap or FORT

2.2.3 Evaporative cooling

Evaporation is the natural process by which the highest energy atoms escape the fining potential, thereby lowering the average thermal energy of the remaining trappedatoms Collisions continually redistribute energy between atoms to bring the totaldistribution towards the Maxwell-Boltzmann distribution By this thermalization pro-cess, atoms in the high energy tail of the distribution, which have sufficient energy

con-to escape the trap, are replenished The process eventually stagnates for decreasingtemperature as the tail of distribution above of the trap depth becomes exponentiallysmall The evaporation must then be forced by lowering the trap depth, truncating theMaxwell-Boltzmann distribution at ever lower energies

In magnetic traps, this is accomplished by driving rf-transitions to remove the higherenergy atoms Because atoms higher in the trapping potential have a larger Zeemanshift, they can be selectively addressed, flipping their spin, and thus ejecting themfrom the trap as they experience an anti-trapping potential Ramping down the rffrequency, the effective trap depth decreases as lower energy atoms are removed Sincethe trapping frequencies remain the same, both the density and collision rate increase

as the evaporation progresses This is known as ‘runaway’ evaporation because therate at which the trap depth can be efficiently lowered accelerates By this method thefirst BEC was created, which required nearly a 107 gain in phase space density fromforced evaporation [45] A general review of evaporative cooling methods, includingrf-induced evaporation, can be found in the report of Ketterle and Druten [65]

In optical traps, the evaporation is usually forced by lowering the laser beam tensity Although this method has the advantage of simplicity, a major limitation

in-is that the thermalization rate slows as evaporation progress Because the trappingfrequencies decrease along with the trap depth, the density and collision rate simulta-neously decrease even while the phase space density increases We investigate this morequantitatively by considering the scalings laws for evaporation in optical potentials

2.2.4 Scaling laws

In order to get a rough idea of how the atom number N , phase space density ρ, andelastic collision rate γ, vary as the potential depth U is lowered, it is useful to considerthe simple scaling laws derived in [66] These scaling laws provide simple analytic

expressions which are in good agreement with a Boltzmann equation model We brieflyoutline the derivation in [66] here.

An important parameter for evaporation is the ratio of the trap depth to ture, η ≡ U0/kbT As an initial condition, we assume free evaporation processes havealready stagnated such that η is large Furthermore, we assume that the evaporationforced by lowering the optical potential proceeds sufficiently slowly that η remains con-stant For large η, the atoms are well confined near the bottom of the potential which

tempera-is, to good approximation, a harmonic oscillator potential Thus the energy is given by

E = 3N kbT , from which one finds the rate equation

˙

From this equation, we see there are two contributions to the energy loss: the first

is associated with the evaporative loss of atoms, and the second with a decreasingtemperature due to adiabatic lowering of the potential The average energy taken bysingle atom which escapes from the trap is (η + α)kbT , where 0 ≤ α ≤ 1 [67] Thefactor α depends on the trap geometry, but for any potential which is harmonic nearthe bottom, α = (η − 5)/(η − 4) [66] The energy loss rate due to evaporation is thus

˙

Eevap = η0N k˙ bT , where η0 = η + α To find the rate of energy loss due to changingpotential, we note that the phase space density, given by ρ = N (hν)3/(kbT )3 for aharmonic oscillator in the classical regime, is adiabatically invariant at fixed N Thusthe quantity ν/T is constant, where ν is the geometric mean trap frequency Given

E ∝ T , at fixed N , and ν ∝ √U , we have dtd(√U /E) = 0 from which we find therate due to the changing potential ˙Epot= (3N kbT /2)( ˙U /U ) Equating ˙Eevap+ ˙Epot to

Eq 2.16 we have

η0N k˙ bT +3N kbT

2

˙U

U = 3 ˙N kbT + 3N kbT ˙ (2.17)Substituting U = ηkbT , by definition, and ˙U = ηkbT , assuming constant η, we find˙

˙N

32

1

η0− 3

˙U

We are particularly interested in the scaling of phase space density, for which we see

Table 2.1: Scaling for forced evaporation in optical traps (η = 10).

phase space density atom number trap depth collision rate

Let us consider the numbers listed in Table 2.1, which are calculated for a typical

η = 10 We can understand the importance of the initial conditions because gainingsignificantly more than 103in phase space density is impractical This is for two reasons.First, while controllably lowering the trap depth to 1/100 of the maximum depth isstraight forward, going to 1/1000 and beyond becomes experimentally problematic.Second, the collision rate sets the time scale for how fast the potential depth can belowered while maintaining thermalization Eventually the loss of phase space densityfrom non-evaporative atom losses, which were neglected in the scaling laws, overtakesgain of phase space density from evaporation

Experiments which have produced BEC by directly loading crossed beam trapsand simply lowering the potential mostly report starting phase spaced densities >

10−3 [11, 47, 49] In the experiment reported in Ref [68], they start at a phase spacedensity 2 × 10−4 and are unable to achieve condensation, stagnating at ρf ≈ 0.2 after

decreasing the potential by a factor of 100 These results are all consistent with thescaling laws and demonstrate the necessity of ρi∼ 10−3 as a starting condition in theabsence of more sophisticated methods.

One outlying example is Ref [47] which reports production of a BEC in a singlebeam trap starting from ρi = 1.2 × 10−4 while only decreasing the beam power by afactor of 140 In this case, it is most likely that gravity is playing a significant roll

in truncating the trap near the end of the evaporation The ratio of their reportedinitial temperature and critical temperature suggests Ui/Uf ≈ 800, assuming constant

η, which is much closer to the expected scaling

2.2.5 Atom losses due to inelastic collisions

The scaling laws in the previous section neglect the effects of atom losses throughinelastic collisions Different collisional processes contribute to one-, two-, and three-body loss rates, labeled α, β, and L respectively As a result, the density of the atomiccloud decreases at the rate

of α/nbg ∼ 5 × 10−9 cm−3/sec, where nbg is the particle density of the backgroundgas Experiments which load a MOT from Rb vapor pressure, such as ours, typicallyoperate at a vacuum in the range of 10−10 to 10−9 Torr, which corresponds to a traplifetime on the order of 10 seconds We report the lifetime due to background collisionsfor our experiment in Sec 2.4.5 Other more complex experiments which employ coldatom sources such a Zeeman slower [70] or 2D MOT [71] are able to reach a vacuum

of ∼ 10−12Torr and thus background limited lifetimes exceeding 10 minutes

Two-body losses

There are two principle mechanisms for two-body losses [72]: light assisted collisionsand hyper-fine ground state changing collisions Light-assisted collisions occur when

near-resonant light excites a pair of nearby atoms to a molecular potential The atomsare accelerated together and gain sufficient energy before relaxing to the ground state,via radiative emission, such that both atoms are ejected from the trap [73, 74] Thistype of collision can be avoided by ensuring all near-resonant light, such as from theMOT and repump lasers, is completely blocked (e.g by mechanical shutters) afterloading the FORT.

Hyperfine ground state changing collisions can occur when at least one of the liding atoms is in the upper hyperfine ground state, |F = 2i for 87Rb During thecollision, the atom undergoes a spin-flip from the |F = 2i to the |F = 1i ground state,resulting in a release of energy corresponding to the ground state splitting of 6.8 GHz.This is over 100 times the typical depth of a dipole trap, and so both atoms are lostafter the collision These collision can be avoided by pumping all the atoms into the

col-|F = 1i ground state manifold For a measurement of the two-body loss rate β due tohyperfine changing collisions in our experiment, see Sec 2.4.6

Three-body losses

Three-body recombination results in two atoms forming a molecule with the third atomcarrying away resulting energy, and thus all three atoms are lost from the trap Therate has been measured for 87Rb to be L ∼ 10−29cm6s−1 [75, 76] Thus three-bodylosses only become an issue for peak densities > 1014 cm−3, which can be the case incrossed beam dipole traps [11] For the experiments in this thesis, however, three-bodylosses do not play a significant role

Due to the unfavorable scaling discussed in Sec 2.2.4, the key to reaching degeneracy indipole traps is good initial conditions The crossed beam configuration is particularlyadvantageous for facilitating high initial densities and phase space densities A simplepicture is that the comparatively large volume of the individual beams allows effectiveloading of a large number of atoms from the MOT During free-evaporation immediatelyafter loading, the atoms redistribute towards a Boltzmann distribution with most ofremaining atoms settling in the much smaller volume of the crossed beam region Inthis way densities ni > 1014 cm−3 and phase space densities ρi > 10−3, roughly two

orders magnitude higher than in the compressed MOT, are quickly obtained [11, 47].From these starting conditions, evaporating to BEC is straightforward.

But this simple picture is not everything In our initial experiments, we expected

to load a 1.06 µm wavelength crossed beam trap with similar results to those earlier periments using the 10.6 µm wavelength CO2 laser Much like the experiment reported

ex-in Ref [68], which was virtually identical to ours, we had difficultly gettex-ing such able starting conditions In this section, we present a detailed analysis of the thermaldistribution of crossed beam traps to understand how the difference in wavelength hassuch a significant impact

favor-2.3.1 General thermal distribution of a trapped gas

For a thermal gas in a infinitely deep potential, the phase-space distribution is given

by [67]

f0(r, p) = n0Λ3exp[−(U (r) + p2/2m)/kbT ], (2.23)where n0 is the peak density located at the minimum of the potential, Λ is the thermalde-Broglie wavelength By integrating over momentum states we find the familiarthermal density distribution

N = n0

Z

V

exp(−U (r)/kbT )P (3/2, t− U (r)/kbT ) d3r (2.27)

2.3.2 Crossed beam distribution: numeric solution

Here we consider the potential formed by two orthogonal dipole traps both focused to

a waist w0 and each with potential depth U0 Following methods described in Ref [77],the integral of Eq 2.27 can be simplified by introducing the function V (U ), which isthe volume of space enclosed by the equipotential surface U This reduces Eq 2.27 tothe one dimension form

exp(−ηu)P (3/2, η(β − u))dV

where η = U0/kbT , β = t/kbT , and u = U/U0 Note that u ∈ [0, 2] for the crossedbeam potential We do not have an analytic expression for dVdu, but can calculate itnumerically As discussed in Ref [77], it is necessary to take β < 2 truncating thetrap depth due to the divergence in the density of states at the top of the trap Forrelevant values of η, the divergence only becomes numerically significant for β & 1.95and, in practice, the trap depth is typically less than this due to external effects such

as gravity For 1.85 < β < 1.95 the integrals only vary by 10% and smaller values of βare only relevant when the potential is significantly modified by an external influence

In Fig 2.2, we compare the numerically calculated density profiles of 10.6 µm and1.06 µm wavelength crossed beam traps for w0 = 40 µm and η = 8 The difference indensity profiles for the two wavelengths is easily interpreted by considering the geometry

of the two potentials, which are plotted in Fig 2.3 Although the potentials are identical

in the cross region (µ < 1), the shallow divergence of the 1.06 µm wavelength trapresults in much weaker confinement outside the cross region (µ > 1) The comparativelylarge increase in the density of states outside the cross region in the 1.06 µm wavelengthtrap results in a significant fraction of the atoms remaining outside the center regioneven for a thermalized distribution This is after free evaporation has already stagnated,given that η = 8, and the truncation of the distribution is negligible

2.3.3 Crossed beam distribution: approximate analytic solution

We can find approximate analytic expressions for the number of atoms in the centerregion, Nc, and number of atoms in the wings, Nw, by considering the integral in

Figure 2.2: Density profiles found by

nu-meric integration for w 0 = 40 µm, η = 8,

and β = 1.9.

Figure 2.3: Optical potential depth along the axis of one beam in a crossed beam trap For both wavelengths the beam waists are w 0 = 40 µm.

Eq 2.29 separately for the two regions µ < 1 and µ > 1 In the center region (µ < 1),the potential is, to a good approximation, independent of the divergence of the beams

as can be seen in Fig 2.3 Since the density is weighted towards lower values of

u, we can make a harmonic approximation to the potential and neglect the effects

of truncation The volume of the center region is given by V (u) = π3w3

0u3/2 in theharmonic approximation Thus we have

Nc = n0

Z 1 0

exp (−ηu) P (3/2, η(β − u))dV

≈ πn0w

3 0

2

Z ∞ 0

exp (−ηu) u1/2d u = n0w

3 0

4

 πη

exp (−ηu) P (3/2, η(β − u))dV

= n0e−η

Z β−1 0

exp −ηu0
P 3/2, η(β − 1 − u0)
dV

du0d u0 (2.33)where we have made the substitution u0 = u−1 Note in Eq 2.33 the factor e−η out thefront can be interpreted as the reduction factor of the density in the wings relative to the

Figure 2.4: Comparison of the fraction of atoms in the wings outside the dimple (blue lines) and the peak density (red lines) as a function of η for 1.06 µm and 10.6 µm traps The red and blue dashed lines are the analytic approximations from equations (2.35) and (2.36) respectively The solid lines result from numerical integration of the exact equations (2.30) and (2.32) with β = 1.9 All calculations assume an atom number N = 2 × 10 6 and beam waists w 0 = 40 µm.

trap center For u > 1 the function V (u) is, to a good approximation, independent ofthe beam intersection and can be approximated by the sum of two independent singlefocus beams As before, they can be approximated by two infinitely deep harmonicoscillator potentials, each with a volume V (u) = 2π3 w02zRu3/2 Integration of Eq 2.33yields

Nw= n0w02zR

 πη

ηπ

between a CO2 laser with λ = 10.6 µm and a fiber laser with λ = 1.06 µm We seethat agreement of the analytic expressions (dashed lines) to the numeric result (solidlines) is reasonably good considering the crude approximations made Under typicalexperimental conditions, the initial free evaporation proceeds quickly, due to the verylarge densities and corresponding collision rates, until stagnating near η ∼ 8 For bothwavelengths there is a relatively small fraction of atom in the wings for this value of η.However, during forced evaporation, the collision rate drops which can result in a slightdecrease in η Although this drop in η is inconsequential for a CO2 laser trap, it canhave a substantial effect for a 1.06 µm laser trap with large number of atoms migrating

to the wings While this migration of atoms to the wings does result in a furtherdecrease in the density, the main impact is on the thermalization and evaporation rates

of the overall distribution

2.3.4 Thermalization in crossed beam traps

The collision rate sets the time scale for how fast the evaporation can be forced without

a significant increase in η The problem with the atoms in the wings is their vastlyreduced collision rate as compared to atoms in the center region To illustrate theproblem we consider the experiment of two 8 W 1.06 µm beams with waist w0 = 40 µmand the initial conditions shown in Table 2.2 These would normally be ideal conditions

to start evaporation as discussed in Sec 2.2.4 The collision rate in the center region

is sufficiently high to maintain continuous thermalization as evaporation is forced Foratoms in the wings however, the density is suppressed the factor e−η, and typicallythe collision rate will be less than the single beam axial trapping frequency Thus it isreasonable to assume that the atoms move independently in the wings until they reachthe center of the trap where they have a high probability of undergoing a collision Thethermalization rate will therefore be roughly determined by the axial frequency of theindividual beams

When we attempting forced evaporation with the same geometry and initial ditions of this example, we observe a migration of the atoms into the wings, which isshown in Fig 2.5 When the power is not ramped down, nearly all the atoms are in thecross region of the trap after 2 seconds, as expected from Eq 2.35 for large η As thepower is lowered, atoms ‘evaporate’ from the center of the trap into the wings wherethere is insufficient density to thermalize with other atoms in the wings and evaporate

con-Table 2.2: Example initial conditions for a 1.06 µm cross beam trap (η = 8).

mean trap frequency (dimple) ω¯ 1.9 2π kHz

Figure 2.5: Fraction of atoms in the wings after linearly ramping the optical power from

an initial power of 12 W (6 W per beam) to a final power over 2 seconds.

from the trap entirely, at least on a reasonable timescale The wings thus act as areservoir for atoms of higher energy than those in the dimple These atoms primarilythermalize only with the colder atoms in the dimple at a rate determined by the axialfrequency This rate is too slow to maintain thermaliztion during the 2 second rampand, as a result, η increases and essentially all the atoms end up in the wings

Although we abandoned starting from a crossed beam trap because of this cation, there are several ways one might circumvent atoms accumulating in the wings.The issue is that we want to take advantage of the trapping volume of the individualbeams to load a large number of atoms, but do not want atoms to be trapped in thewings during evaporation To this end, one could turn on an additional potential gra-dient, from either a magnetic or optical field, after loading to truncate the trap depth

compli-so that atoms are not trapped in the wings This would function like the ‘RF knife’but in an optical trap Both magnetic [53] and optical [52] methods have already beenemployed in this way to achieve ‘run-away’ evaporation in optical traps A simplermethod would be to put one of the crossed beams at an angle so that gravity truncatesthe trap depth in the wings However, this is disadvantageous because fewer atoms will

be loaded into the dimple during the initial free evaporation Finally, one could sider decreasing the waist in order to increase the axial frequency which scales as λ/w3

con-0.However, this is also disadvantageous because the peak density would increase by thesame order of magnitude As illustrated in Fig 2.4, peak densities at w0 = 40 µm al-ready exceed 1014cm−3where three-body losses become significant [75] Increasing thedensity further would simply result in a substantial loss of atoms by this mechanism.Rather than any of these methods, which either complicates the experiment orhave other disadvantages, we used a cross beam trap with a modified geometry andevaporation cycle to reach BEC Briefly, a single tightly focused beam is loaded fromthe MOT and only later in the evaporation is a secondary cross beam used to compressthe atoms The experimental setup is the same as for loading directly into a crossbeam trap, and thus quite simple However, this method allows us to (1) load a largenumber of atom from the MOT given the relatively large trapping volume of the singlebeam, (2) avoid accumulation of atoms in the wings which disrupt thermalization, (3)circumvent the unfavorable scaling of optical evaporation by boosting the collision ratethrough recompression, and (4) avoid high densities and thus three-body losses Theexperiment is described in further detail in Sec 2.5

2.3.5 Analysis of a recent cross-beam result

Some time after our BEC results, Lauber et al were able to produce a BEC in across-beam 1070-nm wavelength trap loaded directly from a MOT [78] It is interesting

to consider their results in the context of the discussion thus far Their crossed beamsare focused to an average waist of 43 µm with a total beam power of 12 W They statethat atoms initially in the ‘wings’ are rapidly spilled at the beginning of evaporationcycle This could be due to several factors such as astigmatism in beam (see Sec 2.4.8)

or gravity if either beam is at a modest angle Regardless of the reason, if the atoms inthe wings are quickly spilled then they are not being efficiently loaded into the centerregion This is consistent with their unfavorable starting conditions of Ni = 3 × 105,

ni= 1.1 × 1013, and ρi= 2 × 10−5 These conditions are roughly what one would expect

by loading the center region only from a compressed MOT and complete loss of atoms

in the wings As discussed in Sec 2.2.4, it is very difficult to reach BEC from theseconditions due to the adverse scaling laws Indeed, Lauber et al employ logarithmicphoto diodes to actively stabilize their beam power over three order of magnitude andrequire a long 12 second evaporation cycle Their MOT is loaded from a Zeeman slowerwhich results in a low background pressure and allows for long evaporation times Inall of our experiments the dipole trap in loaded from a vapor pressure MOT, and theone-body loss rate precludes such a slow evaporation (see Sec 2.4.5) Their end result

of a BEC with 15 × 103 atoms after evaporation over Ti/Tcrit = 3700 is consistent withthe scaling laws

In summary, while Ref [78] has shown it is clearly possible to evaporate to BECfrom directly loading of ∼ 1 µm crossed dipole trap, this is not inconsistent with ourarguments that the ∼ 1 µm wavelength is geometrically unfavorable, in particular ascompared to the ∼ 10 µm wavelength Although the problems with thermalization wereavoided by dumping the atoms in the wings, the bi-partite nature of the cross beam trapwas not utilized to produce the high initial phase space densities from which a largercondensate can be easily produced in the much shorter time of a few seconds [11, 47]

2.3.6 Elliptical beams

We briefly discuss how focusing a dipole trap to an elliptical waist can be a usefulmethod for engineering better trap geometries with increased harmonic confinementbut still large capture volume Let us consider the potential due to a single Gaussianbeam focused to confocal waists wx and wy,

U (r, z) = U0





1q

1 + (zz

Rx)2

1q

We define an ellipticity parameter ξ ≡ wx/wy, and let wx = √ξw0 and wy = w0/√ξ

so that the trap depth is constant as we vary ξ In the harmonic approximation, thetrapping frequencies are then

ωx = ξ

s4U0

mw2 0

ωy = 1ξ

s4U0

mw2 0

ωz =

s

U0

mz2 R

ξ2(1 + 1

ξ4) (2.37)