Formation of Bose–Einstein condensates

Matthew J. Davis,1, 2, ⇤ Tod M. Wright,1 Thomas Gasenzer,3 Simon A. Gardiner,4 and Nick P. Proukakis5

1School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia2JILA, 440 UCB, University of Colorado, Boulder, Colorado 80309, USA

3Kirchho↵-Institut fur Physik, Universitat Heidelberg,Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

4Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom

5Joint Quantum Centre (JQC) Durham-Newcastle,School of Mathematics and Statistics, Newcastle University,

Newcastle upon Tyne NE1 7RU, United Kingdom

The problem of understanding how a coherent, macroscopic Bose–Einstein condensate (BEC)emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimentalinterest over several decades. The pioneering achievement of BEC in weakly-interacting diluteatomic gases in 1995 was followed by a number of experimental studies examining the growthof the BEC number, as well as the development of its coherence. More recently there has beeninterest in connecting such experiments to universal aspects of nonequilibrium phase transitions,in terms of both static and dynamical critical exponents. Here, the spontaneous formation oftopological structures such as vortices and solitons in quenched cold-atom experiments has enabledthe verification of the Kibble–Zurek mechanism predicting the density of topological defects incontinuous phase transitions, first proposed in the context of the evolution of the early universe.This chapter reviews progress in the understanding of BEC formation, and discusses open questionsand future research directions in the dynamics of phase transitions in quantum gases.

I. INTRODUCTION

The equilibrium phase diagram of the dilute Bose gasexhibits a continuous phase transition between condensedand noncondensed phases. The order parameter charac-teristic of the condensed phase vanishes above some criti-cal temperature Tc and grows continuously with decreas-ing temperature below this critical point. However, thedynamical process of condensate formation has provedto be a challenging phenomenon to address both theo-retically and experimentally. This formation process isa crucial aspect of Bose systems and of direct relevanceto all condensates discussed in this book, despite theirevident system-specific properties. Important questionsleading to intense discussions in the early literature in-clude the timescale for condensate formation, and therole of inhomogeneities and finite-size e↵ects in “closed”systems. These issues are related to the concept of spon-taneous symmetry breaking, its causes, and implicationsfor physical systems (see, for example, the chapter bySnoke and Daley in this volume).

In this chapter we give an overview of the dynam-ics of condensate formation and describe the presentunderstanding provided by increasingly well controlledcold-atom experiments and corresponding theoretical ad-vances over the past twenty years. We focus on thegrowth of BECs in cooled Bose gases, which, from atheoretical standpoint, requires a suitable nonequilib-rium formalism. A recent book provides a more com-plete introduction to a number of di↵erent theoretical

⇤ mdavis@physics.uq.edu.au

approaches to the description of nonequilibrium and non-zero-temperature quantum gases [1].

We note that the past decade has seen the observa-tion of BEC in a number of diverse experimental systemsbeyond ultracold atoms, including exciton-polaritons,magnons, and phonons, which are covered in other chap-ters of this volume. Many of the universal aspects ofcondensate formation also apply to these systems.

II. THE PHYSICS OF BEC FORMATION

The essential character of the excitations and collec-tive response of a condensed Bose gas is well described byperturbative approaches that take as their starting pointthe breaking of the U(1) gauge symmetry of the Bosequantum field. This approach can be extended furtherto provide a kinetic description of excitations in a con-densed gas weakly perturbed away from equilibrium [2].The description of the process of formation of a Bose-Einstein condensate in a closed system begins, however,in the opposite regime of kinetics of a non-condensed gas.Over the past decades, there have been many studies us-ing methods of kinetic theory to investigate the initiationof Bose–Einstein condensation. It is now well establishedthat these descriptions break down near the critical point,and in particular in any situation in which the forma-tion process is far from adiabatic. A number of di↵erenttheoretical methodologies have been applied to the issueof condensate formation, but most have converged to asimilar description of the essential physics. The prevail-ing view is that a classical non-linear wave description— a form of Gross–Pitaevskii equation — can describe

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the nonequilibrium dynamics of the condensation pro-cess, which involves in general aspects of weak-wave tur-bulence and, in more aggressive cooling scenarios, strongturbulence. The classical field describes the highly occu-pied modes of the gas at finite temperature and out ofequilibrium.

A summary of the consensus picture of condensateformation in a Bose gas cooled from above the criticaltemperature is as follows. Well above the critical pointthe coherences between particles in distinct eigenstatesof the appropriate single-particle Hamiltonian are neg-ligible and the system is well described by a quantumBoltzmann kinetic equation for the occupation numbersof these single-particle modes. As cooling of the gas pro-ceeds due to inter-particle collisions and interactions withan external bath, if one is present, the occupation num-bers of lower-energy modes increase. Once phase correla-tions between these modes become significant, the systemis best described in terms of an emergent quasiclassicalfield, which may in general exhibit large phase fluctua-tions, topological structures and turbulent dynamics, thenature of which may vary over time and depend on thespecific details of the system — including its dimension-ality, density, and strength of interactions. This regimeis sometimes referred to as a nonequilibrium quasicon-

densate, in analogy to the phase-fluctuating equilibriumregimes of low-dimensional Bose systems [3, 4]. The even-tual relaxation of this quasicondensate establishes phasecoherence across the sample, producing the state that weroutinely call a Bose–Einstein condensate.

A. The pre-condensation kinetic regime

Early investigations of the kinetics of condensation ofa gas of massive bosons began with studies of such a sys-tem coupled to a thermal bath with infinite heat capacity,consisting of phonons [5] or fermions [6–8]. These worksinherited ideas from earlier studies of condensation ofphotons in cosmological scenarios [9]. In a homogeneoussystem, condensation is signified by a delta-function sin-gularity of the momentum distribution at zero momen-tum (see, e.g., Ref. [10]). Levich and Yakhot found [6]that an initially non-degenerate equilibrium ideal Bosegas brought in contact with a bath with a temperaturebelow Tc would develop such a singularity at zero momen-tum only in the limit of an infinite evolution time (seealso Ref. [11]). These same authors subsequently foundthat the introduction of collisions between the bosonslead to the “explosive” development of a singular peakat zero momentum after a finite evolution time [7, 8].They were careful to point out, however, the approxima-tions involved in their treatment of interactions, and in-deed that the development of such coherence invalidatesthe assumptions underlying the quantum Boltzmann de-scription, conjecturing that “the system in the course ofphase transition passes through a stage which may beidentified as a period of strong turbulence” [7].

Experimental attempts in the 1980s to achieve Bosecondensation of spin-polarized hydrogen (see the chap-ter by Greytak and Kleppner for an overview and re-cent developments), and excitons in semiconductors suchas Cu2O, inspired renewed theoretical interest in Bose-gas kinetics. Eckern developed a kinetic theory [12] forHartree–Fock–Bogoliubov quasiparticles appropriate tothe relaxation of the system on the condensed side ofthe transition. Snoke and Wolfe revisited the question ofthe kinetics of approach to the condensation transitionby undertaking numerical calculations of the quantumBoltzmann equation [13]. They found in particular thatthe bosonic enhancement of scattering rates in the de-generate regime o↵set the increased number of scatter-ing events required for rethermalisation in this regime,such that re-equilibration of a shock-cooled thermal dis-tribution takes place on the order of three to four kineticcollision times, ⌧kin = (⇢�vT )�1, where ⇢ is the particledensity, � is the collisional cross section, and the meanthermal velocity vT = (3kBT/m)1/2. These results im-ply that a Boltzmann-equation description of this earlykinetic regime is valid even for short-lived particles suchas excitons, as the particle lifetime is long compared tothis equilibration timescale.

Over time a comprehensive picture of the process ofcondensation of a quench-cooled gas has emerged, andcomprises three distinct stages of nonequilibrium dynam-ics: a kinetic redistribution of population towards lowerenergy modes in the non-condensed phase, developmentof an instability that leads to nucleation of the conden-sate and a subsequent build-up of coherence, and finallycondensate growth and phase ordering. In the midst ofincreasingly intensive e↵orts to achieve Bose condensa-tion in dilute atomic gases, by then including the newsystem of alkali-metal vapours, these stages were anal-ysed in more detail in the early 1990s, beginning witha series of papers by Stoof [14–18], and by Svistunov,Kagan, and Shlyapnikov [19–22].

In Ref. [19], Svistunov discussed condensate formationin a weakly interacting, dilute Bose gas, with so-calledgas parameter ⇣ = ⇢1/3a ⌧ 1, where a is the scatteringlength. In a closed system, a cooling quench genericallyleads to a particle distribution which, below some energyscale "0, exceeds the equilibrium occupation number cor-responding to the total energy and particle content. En-ergy and momentum conservation then imply that a fewparticles scattered to high-momentum modes carry awaya large fraction of the excess energy associated with thisover-occupation, allowing the momentum of a majorityof the particles to decrease. Should the characteristicenergy scale of the overpopulated regime be su�cientlysmall, "0 ⌧ ~2⇢2/3/m ⇠ kBTc, mode-occupation num-bers in this regime will be much larger than unity, andthe subsequent particle transport in momentum space to-wards lower energies is described by the quantum Boltz-mann equation in the classical-wave limit [19–21]. Thisis valid for modes with energies above the scale set bythe chemical potential µ = g⇢ ⇠ ⇣kBTc of the ultimate

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equilibrium state, where g = 4⇡~2a/m is the interac-tion constant for particles of mass m. At lower ener-gies, the phase correlations between momentum modesbecome significant, and a description beyond the quan-tum Boltzmann equation is required.

We note that for open systems such as exciton-polariton condensates, the quasi-coherent dynamics ofsuch low-energy modes will in general be sensitive to thedriving and dissipation corresponding to the continualdecay and replenishment of the bosons. Such externalcoupling can dramatically alter the behaviour of the sys-tem, and its e↵ects on condensate formation dynamicsare a subject of current research — see, e.g., Refs. [23–25] and the chapter by Altman et al.. Hereafter, unlessotherwise specified, the theoretical developments we dis-cuss pertain to closed systems in which the bosons un-dergoing condensation are conserved in number duringthe formation process.

By assuming the scattering matrix elements in thewave Boltzmann equation to be independent of the modeenergies, Svistunov discussed several di↵erent transportscenarios within the framework of weak-wave turbulence,in analogy to similar processes underlying Langmuir-wave turbulence in plasmas [26]. He concluded thatthe initial kinetic transport stage of the condensationprocess evolves as a weakly non-local particle wave inmomentum space. Specifically, he proposed that theparticle-flux wave followed the self-similar form n(", t) ⇠"1(t)�7/6f("/"1(t)), with "1(t) ⇠ (t � t⇤)3, and scalingfunction f falling o↵ as f(x) / x�↵ for x � 1, with↵ = 7/6. Following the arrival of this wave at timet⇤ ' t0 + ~"0/µ2, a quasi-stationary wave-turbulent cas-cade forms in which particles are transported locally,from momentum shell to momentum shell, from the scale"0 of the energy concentration in the initial state to thelow-energy regime " . µ where coherence formation setsin.

The wave-kinetic (or weak-wave turbulence) stage ofcondensate formation following a cooling quench wasinvestigated in more detail by Semikoz and Tkachev[27, 28], who solved the wave Boltzmann equation nu-merically and found results consistent with the abovescenario, albeit with a slightly shifted power-law expo-nent ↵ ' 1.24 for the wave-turbulence spectrum. Laterdynamical classical-field simulations of the condensationformation process by Berlo↵ and Svistunov [29] furthercorroborated the above picture.

B. The formation of coherence

It has been known for some time that a kinetic Boltz-mann equation model is unable to describe the develop-ment of a macroscopic zero-momentum occupation in theabsence of seeding or other modifications [6, 13, 19]. Inany event, the quantum Boltzmann equation ceases tobe valid in the high-density, low-energy regime in whichcondensation occurs. The two-body scattering receives

significant many-body corrections once the interactionenergy g

Rk.p dknk of particles with momenta below a

given scale p exceeds the kinetic energy at that scale, andthese are indeed the prevailing conditions when phase co-herence emerges and the condensate begins to grow [19].

In a series of papers [14, 15, 17, 18], Stoof took accountof these many-body corrections and developed a theory ofcondensate nucleation resting on kinetic equations incor-porating a ladder-resummed many-body T -matrix deter-mined from a one-particle-irreducible (1PI) e↵ective ac-tion or free-energy functional. In the 1PI formalism thepropagators appearing in the e↵ective action are takenas fixed, determined in this case by the initial thermalBose number distribution and the spectral properties ofa free gas.

Constructed within the Schwinger–Keldysh closed-time-path framework, the method allows the determina-tion of the time evolution of the self-energy and thus of ane↵ective chemical potential for the zero-momentum modethrough the phase transition. During the kinetic stage,once the system has reached temperatures below theinteraction-renormalized critical temperature, the self-energy renders the vacuum state of the zero-momentummode metastable. Stoof found that this modification ofthe self-energy occurs on a time scale ⇠ ~/kBTc and givesrise to a small seed population in the zero mode, n0 ⇠⇣2⇢, within the kinetic time scale ⌧kin ⇠ ~/(⇣2kBTc).He argued that, following this seeding, the system un-dergoes an unstable semi-classical evolution of the low-energy modes. Taking interactions between quasiparti-cles into account he found that the squared dispersion!(p)2 becomes negative for p . ~

pan0(t), i.e., below

a momentum scale of the order of the inverse healinglength associated with the density n0(t) of the existingcondensed fraction. As a result, the condensate growslinearly in time over the kinetic time scale ⌧kin. Thegrowth process eventually ceases due to the conservationof total particle number, whereafter the final kinetic equi-libration of quasi-particles takes place over a time scale⇠ ~/(⇣3kBTc) as discussed previously by Eckern [12], andby Semikoz and Tkachev [28].

C. Turbulent condensation

The semi-classical scenario of Stoof is built on the as-sumptions that the cooling quench has driven the sys-tem to the critical point in a quasi-adiabatic fashion, andthat the neglect of thermal fluctuations and nonequilib-rium over-occupations in the self-energy is justified [18].However, as previously pointed out in Ref. [7], a morevigorous quench may drive the system into an interme-diate stage of strong turbulence, where the coherencesbetween wave frequencies lead to the formation of coher-ent structures, such as vortices, that have a significantinfluence on the subsequent dynamics. The main pro-cesses and scales governing this stage were discussed indetail by Kagan and Svistunov [21, 22]. As a result of

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excess particles being transported kinetically into the co-herent regime (wave numbers below the final inverse heal-ing length, k . ⇠�1 ⇠ p

a⇢), the density and phase of theBose field fluctuate strongly on length scales shorter than⇠. The growing population at even smaller wave numbersthen implies, according to Refs. [19–21], the formationof a quasicondensate over the respective length scales,as the coherent evolution of the field according to theGross–Pitaevskii equation causes the density fluctuationsto strongly decrease at the expense of phase fluctuations.This short-range phase-ordering occurs on a time scale⌧c ⇠ ~/µ ⇠ ~/(⇣kBTc). Depending on the flux of ex-cess particles entering the coherent regime, this leads toquasicondensate formation over a minimum length scalelv > ⇠ (see Sects. IVA and IV B) [30]. The phase, how-ever, remains strongly fluctuating on larger length scalesdue to the formation of topological defects — vortex linesand rings. These vortices appear in the form of clumps ofstrongly tangled filaments [31] with an average distancebetween filaments of order lv. If the cooling quench is suf-ficiently strong to drive the system near a non-thermalfixed point, cf. Sect. IVB, this quasicondensate is charac-terised by new universal scaling laws in space and time.

The work of Kagan and Svistunov laid the founda-tions for studying the role of superfluid turbulence in theprocess of Bose–Einstein condensation. Kozik and Svis-tunov have subsequently elucidated the decay of the vor-tex tangle via the transport of Kelvin waves created onthe vortex filaments through their reconnections, whichcan itself assume a wave-turbulent structure [32–35].

III. CONDENSATE FORMATIONEXPERIMENTS

A. Growth of condensate number

We now provide a historical overview of both experi-ments and theory related to condensate formation in ul-tracold atomic gases. The first experiments to achieveBose–Einstein condensation in 1995 [36, 37] reached thephase space density necessary for quantum degeneracyusing the technique of evaporative cooling [38] — thesteady removal of the most energetic atoms, followed byrethermalisation to a lower temperature via atomic col-lisions. These experiments, which concentrated on theBEC atom number as the conceptionally simplest observ-able, provided an indication of the time scale for conden-sation in trapped atomic gases, in the range of millisec-onds to seconds. This gave the impetus for the develop-ment of a quantum kinetic theory by Gardiner and Zollerusing the techniques of open quantum systems. Theyfirst considered the homogeneous Bose gas [39], beforeextending the formalism to trapped gases [40, 41]. Theirmethodology split the system into a “condensate band”,containing modes significantly a↵ected by the presence ofa BEC, and a “non-condensate band” containing all otherlevels. A master equation was derived for the condensate

band using standard techniques [42], yielding equationsof motion for the occupations of the condensate modeand the low-lying excited states contained in the conden-sate band. A simple BEC growth equation derived fromthis approach provided a reasonable first estimate of thetime of formation for the 87Rb and 23Na BECs of theJILA [36] and MIT [37] groups, respectively.

The first experiment to explicitly study the forma-tion dynamics of a BEC in a dilute weakly interactinggas was performed by the Ketterle group at MIT, us-ing their newly developed technique of non-destructiveimaging [43]. Beginning with an equilibrium gas justabove the critical temperature, they performed a sud-den evaporative cooling “quench” by removing all atomsabove a certain energy. The subsequent evolution led tothe formation of a condensate, with a characteristic S-shaped curve for the growth in condensate number. Thiswas interpreted as evidence of bosonic stimulation in thegrowth process, and they fitted the simple BEC growthequation of Ref. [44] to their experimental observations.However, the measured growth rates did not fit the the-ory all that well.

Gardiner and co-workers subsequently developed anexpanded rate-equation approach incorporating the dy-namics of a number of quasiparticle levels [45, 46]. Thisformalism predicted faster growth rates, mostly due tothe enhancement of collision rates by bosonic stimula-tion, but still failed to agree with the experimental data.One limitation of this approach was that it neglected theevaporative cooling dynamics of the thermal cloud, in-stead treating it as being in a supersaturated thermalequilibrium.

The details of the evaporative cooling were simulatedin two closely related works by Davis et al. [47] and Bi-jlsma et al. [48]. The former was based on the quantumkinetic theory of Gardiner and Zoller, while the latteremerged as a limit of the field-theoretical approach ofStoof [17, 18] and the “ZNG” formalism previously de-veloped for nonequilibrium trapped Bose gases [49] byZaremba, Nikuni, and Gri�n. The latter authors used abroken-symmetry approach to derive a quantum Boltz-mann equation for non-condensed atoms coupled to aGross–Pitaevskii equation for the condensate [49, 50],thereby extending their two-fluid model for trappedBECs [50], which was based on the pioneering work ofKirkpatrick and Dorfman [51–54]. The ZNG methodol-ogy has since been used successfully and extensively tostudy a variety of nonequilibrium phenomena in partiallycondensed Bose gases, such as the temperature depen-dence of collective excitations, as reviewed in Ref. [2]. Asthis methodology is explicitly based on symmetry break-ing, it cannot address the initial seeding of a BEC, orany critical physics arising from fluctuations. However,it can model continued growth once a BEC is present.

The works of Davis et al. [47] and Bijlsma et al. [48]both introduced approximations to the formalisms theywere built on, assuming that the condensate grew adi-abatically in its ground state, and treating all non-

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condensed atoms in a Boltzmann-like approach. Bothpapers boiled down to simulating the quantum Boltz-mann equation in the ergodic approximation, in whichthe phase-space distribution depends on the phase-spacevariables only through the energy [55]. Despite the di↵er-ent approaches, the calculations were in excellent agree-ment with one another — yet still quantitatively dis-agreed with the MIT experimental data [43]. This dis-agreement has remained unexplained.

A second study of evaporative cooling to BEC in a di-lute gas was performed by the group of Esslinger andHansch in Munich [56]. In this experiment the Bosecloud, which was again initially prepared in an equilib-rium state slightly above Tc, was subjected to a contin-uous rf field inducing the ejection of high-energy atomsfrom the sample. By adjusting the frequency of the ap-plied field and thus the energies of the removed atoms,these authors were able to investigate the growth ofthe condensate for varying rates of evaporative cooling.Davis and Gardiner extended their earlier approach [47]to include the e↵ects of three-body loss and gravitationalsag on the cooling of the 87Rb cloud in this experi-ment [57]. Their calculations yielded excellent agreementwith the experimental data of Ref. [56] within its statis-tical uncertainty for all but the slowest cooling scenariosconsidered. An example is shown in Fig. 1(a).

In 1997 Pinkse et al. [58] experimentally demonstratedthat adiabatically changing the trap shape could increasethe phase-space density of an atomic gas by up to a fac-tor of two and conjectured that this e↵ect could be ex-ploited to cross the BEC transition in a thermodynami-cally reversible fashion. This scenario was subsequentlyrealised in the MIT group by Stamper-Kurn et al. [59]by slowly ramping on a tight “dimple” trap formed froman optical dipole potential on top of a weaker harmonicmagnetic trap. This experiment was the setting for thefirst application of a stochastic Gross–Pitaevskii method-ology [60], previously developed from a nonequilibriumformalism for Bose gases by Stoof [18]. This is basedon the many-body T-matrix approximation, and uses theSchwinger–Keldysh path integral formulation of nonequi-librium quantum field theory to derive a Fokker-Planckequation for both the coherent and incoherent dynamicsof a Bose gas. The classical modes of the gas were rep-resented by a Gross–Pitaevskii equation, with additionaldissipative and noise terms resulting from a collisionalcoupling to a thermal bath with a temperature T andchemical potential µ.

Proukakis et al. [61] subsequently used this methodol-ogy to study the formation of quasicondensates in a one-dimensional dimple trap. A much later experiment [62]investigated the dynamics of condensate formation fol-lowing the sudden introduction of a dimple trap, andincluded quantum-kinetic simulations that were in goodagreement with the data.

A novel method of cooling a bosonic cloud to con-densation was introduced in 2003 by the Cornell groupat JILA [63], who demonstrated the evaporative cool-

ing of an atomic Bose cloud brought in close proximityto a dielectric surface, due to the selective adsorption ofhigh-energy atoms. More recently, similar experimentshave been undertaken by the Durham [64] and Tubingengroups [65], with the observed rates of loss in the lattercase explained accurately by non-ergodic ZNG-methodcalculations of the evaporative cooling dynamics. Exam-ple results are shown in Fig. 1(b).

B. Other theories for condensate formation

For completeness, here we briefly outline other theo-retical methods that can be applied to condensate forma-tion. A generalised kinetic equation for thermally excitedBogoliubov quasiparticles was obtained by Imamovic-Tomasovic and Gri�n [66] based on the application ofthe Kadano↵–Baym nonequilibrium Green’s function ap-proach [67] to a trapped Bose gas. This kinetic equationreduces to that of Eckern [12] in the homogenous limitand to that of ZNG [49] when the quasiparticle charac-ter of the excitation spectrum is neglected. Walser et

al. [68, 69] derived a kinetic theory for a weakly inter-acting condensed Bose gas in terms of a coarse grain-ing of the N -particle density operator over configura-tional variables. Neglecting short-lived correlations be-tween colliding atoms in a Markov approximation, theyobtained kinetic equations for the condensate and non-condensate mean fields which were subsequently shownto be microscopically equivalent [70] to the nonequilib-rium Green’s function approach of Ref. [66]. Exactly thesame kinetic equations were derived by Proukakis [71],within the formalism of his earlier quantum kinetic for-mulation [72, 73], based on the adiabatic elimination ofrapidly-evolving averages of non-condensate operators,ideas which fed into the development of the ZNG ki-netic model [74]. Although elegant, these formalismshave not provided a tractable computational methodol-ogy for modelling condensate formation away from thequasistatic limit.

A non-perturbative method for the many-body dy-namics of the Bose gas far from equilibrium has beendeveloped by Berges, Gasenzer, and co-workers [75–78].This two-particle irreducible (2PI) e↵ective-action ap-proach provides a systematic way to derive approximateKadano↵–Baym equations consistent with conservationlaws such as those for energy and particle number. Incontrast to 1PI methods, single-particle correlators aredetermined self-consistently by these equations. Thisapproach allows the description of strongly correlatedsystems, and has been exploited in the context of tur-bulent condensation [79–81] where it provides a self-consistently determined many-body T matrix. This 2PIe↵ective-action approach is useful for studying stronglyinteracting systems such as 1D gases with large couplingconstant [82], or relaxation and (pre-)thermalization ofstrongly correlated spinor gases [83].

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(a)

EVAPORATIVE COOLING OF COLD ATOMS AT SURFACES PHYSICAL REVIEW A 90, 023614 (2014)

possible collision partners. Because the density of the atomcloud can vary considerably, we use an adaptive Cartesiangrid in real space as outlined in [77], while keeping a globaltime step.

Our initial state is a thermal cloud in equilibrium with atemperature T . This state is calculated using self-consistentHartree-Fock as outlined in [78]. In addition to the thermalcloud, the initial state requires a small condensate “seed”to allow for C12 collisions, and hence condensate growth;the number of atoms in the seed is obtained using theBose-Einstein distribution, assuming µc = 0 [38].

Interactions between the surface and the atoms are modeledby calculating the single-correction function [Eq. (7)] for thegeneralized Gross-Pitaevskii equation [Eq. (1)] and combiningit with a linear imaginary potential to remove condensateatoms, effective from the position where the trap opens [79].In addition, we annihilate test particles that are beyond thisopening point, resulting in an atom loss for the thermal cloud.These two processes lead to a reduction in the total atomnumber in the system.

IV. RESULTS

Having set up our computational model, we now employit to study surface evaporative cooling. We show the resultsof simulations for two different geometries. In Sec. IV A, wedirectly compare theory with experiment to examine the extentto which the model captures the important physical processes.We then go on to consider a simpler model system in Sec. IV B,with a view to optimizing parameters to create the purest orlargest condensates.

The experiments were performed using the apparatusdescribed in [17]. Clouds of 87Rb atoms were loaded intoan atom chip trap with frequencies !x = 2"!16 rads s"1

in the axial direction and !y = !z = 2"!85 rads s"1 in theradial direction. The cloud was initially prepared with thetrap center at a distance xs # 135 µm from a silicon surface,defined as the x = 0 plane. At this point, there was negligibleoverlap between the cloud and the surface. The cloud was thentransported along the x axis at a variable speed to a variabledistance, xs , from the surface and held for a variable hold time.In order to measure the remaining atom number N , the cloudwas swiftly brought back to its initial position, after which weperformed time-of-flight measurements and CCD imaging.

A. Loss curves

1. Time series

We begin by considering atom loss curves as a function oftime when the cloud is brought into overlap with the surface.In the experiments, the cloud was transported to the surfacein 1 s and held stationary at a final hold point for up to 2.5 s.Three hold points were considered: xs # 14, 29, and 72 µm.These were estimated from the point where the trap completelyopened and all of the atoms were lost to the surface. Referencemeasurements revealed that temperature-related drifts couldshift the position of the surface by up to 10 µm, hence thegiven values for xs are approximate; this is the dominant sourceof error. The initial cloud temperatures were 130 nK for xs #14 µm and xs # 29 µm, and 140 nK for xs # 72 µm. These

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FIG. 2. (Color online) Total atom number N against time t forthree different trap-surface separations: xs = 68 µm (gold solidcurve), 30 µm (red dashed curve), and 15 µm (black dotted curve).Points correspond to experimental data and the curves correspondto simulations. The dot-dashed gold curve shows a simulation forxs = 68 µm without collisions, i.e., C12 = C22 = 0. The gray verticaldashed line marks the point when the atom cloud reaches its final holdposition at t = 0. The gray hashed area shows the shift of the curvewhen the surface position is varied by ±2.5 µm. The inset shows abreakdown of the cloud atom numbers against time for xs = 68 µmfrom the point when the cloud reaches its holding position. The solidcurve shows the total atom number, the dashed curve corresponds tothermal atoms, and the dotted curve corresponds to the condensateatom number.

temperatures are slightly above the critical temperature forcondensation, Tc, for an ideal gas [80].

We performed the simulations using these experimentalparameters [81]. We plot the theoretical and experimental atomnumbers against time in Fig. 2 (a “time series”). We considerthe time t = 0 to be the point when the cloud reaches itsfinal hold position, indicated by the gray vertical dashed line.Since the absolute surface position may vary due to drifts, weperformed a range of simulations with varying xs to obtain thebest fit. In this sense, the simulations served as a calibrationtool: for the xs # 14, 29, and 72 µm curves, the best fitswere obtained with a simulated cloud-surface separation of15, 30, and 68 µm, respectively, well within the experimentaluncertainties. Figure 2 shows the evolution of the total numberof atoms, N , remaining in the cloud during the course of thesimulation, with curves corresponding to numerical results andpoints corresponding to experimental data.

The gold solid curve and gold star points are for the xs =68 µm hold point, the red dashed curve and red open circlesare for the xs = 30 µm hold point, and the black dotted curveand black crosses are for the xs = 15 µm hold point. To givean idea of how the surface position affects the remaining atomnumber, we vary the surface position by ±2.5 µm for the68 µm curve, shown as the gray hashed area in Fig. 2.

For all values of xs , we observe a nontrivial loss curve;the loss rates increase to a maximum as the cloud is broughtto the surface. Once the cloud reaches its final position, thelosses swiftly reduce. The transfer between these regimes is

023614-3

(b)

FIG. 1. (a) Growth of an atomic Bose–Einstein condensate modelled with the quantum Boltzmann equation. The experimentbegan with a 87Rb Bose gas in an elongated harmonic trap with Ni = (4.2±0.2)⇥106 atoms at a temperature of Ti = (640±30)nK, before turning on rf evaporative cooling with a truncation energy of 1.4kBT . The solid and dotted lines show the theoreticalcalculations with a starting number of Ni = 4.2⇥ 106 and Ni = 4.4⇥ 106 atoms respectively. Taken from Ref. [56]. (b) Surfaceevaporation leading to the formation of a BEC, showing the total atom number for three di↵erent cloud-surface distances.The lines are the results of ZNG simulations, the points are from experiment. The dot-dash gold line is for a ZNG simulationneglecting collisions in the thermal cloud, demonstrating that modelling the full dynamics of the thermal cloud is necessaryfor a quantitative understanding of the experiment. The inset shows the total number, thermal cloud number, and condensatenumber, from top to bottom respectively, as a function of time. Taken from Ref. [65].

C. Other pioneering condensate-formationexperiments

There are a number of experimental methods otherthan evaporative cooling to increase the phase-space den-sity of a quantum gas and form a condensate. We brieflymention them here for completeness.

An experimental technique that has proved to be ex-tremely useful for multi-component quantum gases is themethod of sympathetic cooling, in which an atomic gasis cooled by virtue of its collisional interaction with asecond gas of atoms, distinguished from the first eitherisotopically or by internal quantum numbers, which isitself subject to, e.g., evaporative cooling. This tech-nique was first demonstrated by Myatt et al. [84] in agas comprising two distinct spin states of 87Rb, and wassubsequently employed to cool a single-component Fermigas to degeneracy by Schreck et al. [85].

In a similar spirit, in 2009 the Inguscio group in Flo-rence used entropy exchange between components of atwo-species 87Rb-41K Bose gas mixture to induce BECin one of the components [86]. The two gases werebrought close to degeneracy by cooling, after which thestrength of the 41K trapping potential was adiabaticallyincreased, by introducing an optical dipole potential towhich the 87Rb component was largely insensitive. In asingle-component system this would lead to an increasein the temperature and leave the phase-space density un-a↵ected. However, in the dual-species setup the 87Rbcloud acted as a thermal reservoir, suppressing the tem-perature increase of the 41K component and causing itto cross the BEC threshold.

In 2004, the Sengstock group observed the formationof a BEC at constant temperature [87]. Working with aspin-1 system, they prepared a partially condensed gasconsisting of mF = ±1 states. Spin collisions within theBEC components populated the mF = 0 state, whichthen quickly thermalised. When the population of themF = 0 component reached the critical number a newBEC emerged. The experiment was modelled with a sim-ple rate equation.

In the same year, Ketterle’s MIT group performed anexperiment in which they distilled a BEC from one trapminimum to another [88]. A non-zero-temperature BECwas formed in an optical dipole trap, before a secondtrap with a greater potential depth was brought nearby.Atoms of su�cient thermal energy were able to cross thebarrier between the two potential minima, populating thesecond trap. Eventually the first condensate evaporated,and a second condensate formed in the new global trapminimum.

Finally, we mention a recent experiment by the groupof Schreck at Innsbruck, who demonstrated the first ex-perimental production of a BEC solely by laser cool-ing [89]. This feat was made possible by laser coolingon a narrow-linewidth transition of 84Sr, resulting in alow Doppler-limit temperature of just 350 nK. A “light-shift” laser beam was introduced at the centre of the trapso that the atoms in that region no longer responded tothe laser cooling, after which an additional dimple trapwas introduced to confine the atoms. Repeatedly cyclingthe dimple trap on and o↵ resulted in the formation ofseveral condensates [89].

7

D. Low-dimensional Bose systems and phasefluctuations

The experiments described above were in the three-dimensional (3D) realm, in which long-wavelength phasefluctuations are strongly suppressed away from the vicin-ity of the phase transition. In lower dimensional systemssuch fluctuations are enhanced, leading to dramatic mod-ifications to the physics of the degenerate regime. In atwo-dimensional (2D) system, thermal fluctuations of thephase erode the long-range order associated with truecondensation, leaving only so-called quasi-long-range or-der characterised by correlation functions that decay al-gebraically with spatial separation [3]. A more completeanalysis reveals the importance of vortex-antivortex pairsin this phase-fluctuating “quasi-condensed” regime [90].Such pairs undergo a so-called Berezinskii–Kosterlitz–Thouless (BKT) deconfinement transition at some finitetemperature, above which even quasi-long-range order islost and superfluidity is extinguished. Two-dimensionalBose systems are of particular interest due to their nat-ural realisation in systems such as liquid helium filmsand the fact that the degenerate Bose quasiparticles suchas excitons and polaritons in semiconductor systems aretypically confined in a planar geometry. An insightfuloverview of BKT physics can be found in the chapter byKim, Nitsche and Yamamoto in this volume.

There have been numerous experimental realisationsof (quasi-)2D Bose gases in cold-atom experiments [91–96], with most notable the observations of thermally acti-vated vortices via interferometric measurements [91] andthe direct probing of the equation of state and scale in-variance of the 2D system [96] (see the chapter by Chinand Refs. [97–101] for related theoretical considerations).Further details and a lengthy discussion of the interplaybetween BKT and BEC in homogeneous and trappedsystems can be found in Ref. [102]. Although theoreti-cal works on the dynamics of such systems have existedfor some time, little experimental work on the formationdynamics of condensates in these systems has been un-dertaken (aside from the quasi-2D Kibble-Zurek worksdiscussed in the following section). Considerable discus-sion is currently taking place regarding the emergenceand nature of the BKT transition in driven-dissipativepolariton condensates: experimentalists have observedevidence for quasi-long-range order [103, 104] (see Kimet al.’s chapter), but the nature of the transition and its“nonequilibrium” features are topics of current debate[24, 25] (see also the chapter by Keeling et al.).

In one dimension, the e↵ects of phase fluctuations areeven more pronounced, leading to the complete destruc-tion of long-range order and superfluidity at any finitetemperature. Many experiments with cold atoms in elon-gated “cigar-shaped” traps have investigated the physicsof such (quasi-) one-dimensional systems, though again,little work has been done on the formation dynamicsof these degenerate samples. We note, however, thatquasicondensate regimes somewhat analogous to those

of (quasi-) one-dimensional systems can be realized inelongated 3D traps [105]. In such a regime, the Bose gasbehaves much as a conventional three-dimensional Bosecondensate, except that the coherence length of the gas isshorter than the system extent along the long axis of thetrap. A study of condensate formation in this regime wasperformed by the Amsterdam group of Walraven [106] in2002 in an elongated 23Na cloud. Similarly to the MITexperiment [43], they performed rapid quench cooling oftheir sample from just above the critical temperature.However, the system was in the hydrodynamic regime inthe weakly trapped dimension, i.e., the mean distance be-tween collisions was much shorter than the system length.

It was argued that the system rapidly came to a lo-cal thermal equilibrium in the radial direction, resultingin cooling of the cloud below the local degeneracy tem-perature over a large spatial region and generating anelongated quasicondensate. However, the extent of thisquasicondensate along the long axis of the trap was largerthan that expected at equilibrium, leading to large am-plitude oscillations. The momentum distribution of thecloud was imaged via “condensate focussing”, with thebreadth of the focal point giving an indication of themagnitude of the phase fluctuations present in the sam-ple. This interesting experiment was somewhat ahead ofits time, with theoretical techniques unable to addressmany of the nonequilibrium aspects of the problem.

In 2007 the group of Aspect from Institut d’Optiquealso studied the formation of a quasicondensate in anelongated three-dimensional trap via continuous evapo-rative cooling [107] in a similar fashion to the earlier workby Kohl et al. [56]. As well as measuring the condensatenumber, they also performed Bragg spectroscopy dur-ing the growth to determine the momentum width andhence the coherence length of the system. They foundthat the momentum width they measured rapidly de-creased with time to the width expected in equilibriumfor the instantaneous value of the condensate number.Modelling of the growth of the condensate populationusing the methodology of Ref. [57] produced results ingood agreement with the experimental data, apart froman unexplained delay of 10–50 ms, depending on the rateof evaporation.

IV. CRITICALITY AND NONEQUILIBRIUMDYNAMICS

As Bose–Einstein condensation is a continuous phasetransition, the theory of critical phenomena [108] predictsthat in the vicinity of the critical point the correlations ofthe Bose field obey universal scaling relations. In partic-ular, the scaling of correlations at and near equilibrium isgoverned by a set of universal critical exponents and scal-ing functions, independent of the microscopic parametersof the gas. For a homogeneous system close to critical-ity, standard theory predicts that the correlation length⇠, relaxation time ⌧ , and first-order correlation function

8

G(x) = h †(x) (0)i obey scaling laws

⇠ =⇠0|✏|⌫ , ⌧ =

⌧0|✏|⌫z , G(x) = ✏⌫(d�2+⌘)F(✏⌫x), (1)

with ✏ = T/Tc � 1 the reduced temperature, ⌫ and zthe correlation length and dynamical critical exponents,⌘ the scaling dimension of the Bose field, and F a uni-versal scaling function. The static Bose gas belongs tothe XY (or O(2)) universality class, and is thus expectedto have the same critical exponents as superfluid helium,i.e., in 3D, ⌫ ' 0.67 and ⌘ = 0.038(4) [109]. The criticaldynamics of the system are expected to conform to thoseof the di↵usive model denoted by F in the classificationof Ref. [110], implying a value z = 3/2 for the dynamicalcritical exponent.

The influence of critical physics is significantly reducedin the conditions of harmonic confinement typical of ex-perimental Bose-gas systems, as compared to homoge-neous systems. Within a local-density approximation,the inhomogeneous thermodynamic parameters of thesystem imply that only a small fraction of atoms in thegas enter the critical regime, and so global observablesare relatively insensitive to the e↵ects of criticality. Nev-ertheless, a few experiments have attempted to observeaspects of the critical physics of trapped Bose gases.

In a homogeneous gas the introduction of interparti-cle interactions has no e↵ect on the critical temperatureat the mean-field level, but the magnitude and even thesign of the shift due to critical fluctuations was debatedfor several decades (see Ref. [111] and references therein)before being settled by classical-field Monte-Carlo calcu-lations [112, 113]. An experiment by the Aspect groupcarefully measured a shift in critical temperature of thetrapped gas, but was unable to unambiguously infer anybeyond-mean-field contribution to this shift [114, 115]. Alater experiment by the group of Hadzibabic made use ofa Feshbach resonance to control the interaction strengthin 41K, and found clear evidence of a positive beyond-mean-field shift [116] (see also the chapter by Smith inthis volume).

In 2007 the ETH Zurich group of Esslinger revisitedtheir experiments on condensate formation and the co-herence of a three-dimensional BEC with a new tool: theability to count single atoms passing through an opti-cal cavity below their ultra-cold gas [117]. They out-coupled atoms from two di↵erent vertical locations fromtheir sample as it was cooled, realising interference inthe falling matter waves. By monitoring the visibility ofthe fringes, they were able to measure the growth of thecoherence length as a function of time. Using the sameoptical cavity setup, the Esslinger group subsequentlymeasured the coherence length of their Bose gas as itwas driven through the critical temperature by a smallbackground heating rate, and determined the correlation-length critical exponent to be ⌫ = 0.67±0.13 [118]. Theirresults are shown in Fig. 2(a). Classical-field simulationsof their experiment were in reasonable agreement, deter-mining ⌫ = 0.80 ± 0.12 [119].

Although an important topic in its own right, thegreatest significance of the equilibrium theory of criticalfluctuations to studies of condensate formation is that itprovides a basis for generalisations of concepts such ascritical scaling laws and universality classes to the do-main of nonequilibrium physics. In the remainder of thissection we discuss two such extensions: the Kibble–Zurekmechanism (KZM), and the theory of non-thermal fixedpoints.

A. The Kibble–Zurek mechanism

The theory of the Kibble–Zurek mechanism leveragesthe well-established results of the equilibrium theory ofcriticality to make immediate predictions for universalscaling behaviour in the nonequilibrium dynamics of pas-sage through a second-order phase transition. The un-derlying idea — that causally disconnected regions ofspace break symmetry independently, leading to the for-mation of topological defects — was first discussed byKibble [121], who predicted that the distribution of de-fects following the transition would be determined bythe instantaneous correlation length of the system as itpasses through the Ginzburg temperature [122]. Zureklater emphasised [123] the importance of dynamic crit-ical phenomena [110] in such a scenario. In particular,the scaling relations (1) imply that both the correlationlength and the characteristic relaxation time of the sys-tem diverge as the critical point is approached (✏ ! 0),imposing a limit to the size of spatial regions over whichorder can be established during the transition. Topologi-cal defects will thus be seeded, with a density determinedby the correlation length at the time the system “freezes”during the transition, and will subsequently decay in thesymmetry-broken phase. The more rapidly the systempasses through the critical point, the shorter the corre-lation length that is frozen in, and therefore more topo-logical defects will form. A dimensional analysis predictsthat a linear ramp ✏(t) = �t/⌧Q of the reduced temper-ature through the critical point on a characteristic timescale ⌧Q results in a distribution of spontaneously formeddefects with a density nd that scales as [124]

nd / ⌧(p�d)⌫/(1+⌫z)Q , (2)

where d is the dimensionality of the sample and p is theintrinsic dimensionality of the defects.

Zurek initially described the KZM in the context ofvortices in the �-transition of superfluid 4He [123]. Al-though vortices are observed in the wake of this transi-tion, it is di�cult to identify them as having formed dueto the KZM rather than being induced by, e.g., inadver-tent stirring [124] (see also the chapter by Pickett in thisvolume). The prospect of generating vorticity in atomicBECs by means of the KZM was first discussed by Anglinand Zurek in 1999 [125]. However, it was not until the2008 experiment of the Anderson group at the University

9

10-3 10-2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.1

1

10

0

1

2

3

4

5

6

7

8

Cor

rela

tion

leng

th !

(µm

)

Reduced temperature (T-Tc)/Tc

(a)

0.1 0.4 1.6

2.0

1.0

tQ HsL

{HmmL

(b)

FIG. 2. Critical phenomena in BECs. (a) Divergence of the equilibrium correlation length ⇠ as a function of the reducedtemperature, and the fitting of the critical exponent, giving the result ⌫ = 0.67 ± 0.13. Inset: Double logarithmic plot of thesame data. Taken from Ref. [118]. (b) Log-log plot of the dependence of the correlation length, here labelled `, as a functionof the characteristic time ⌧Q of the quench through the BEC phase transition. The solid line corresponds to a Kibble–Zurekpower-law scaling ` / ⌧ b

Q with b = 0.35 ± 0.04, in agreement with the beyond-mean-field prediction b = 1/3 of the so-called Fmodel [110] and inconsistent with the mean-field value b = 1/4. This in turn implies a value z = 1.4 ± 0.2 for the dynamicalcritical exponent. Taken from Ref. [120].

of Arizona [126] that spontaneously formed vortices werefirst observed in such a system (see also Ref. [127]).

The observations of spontaneous vortices in Ref. [126]were supported by numerical simulations using thestochastic projected Gross-Pitaevskii equation descrip-tion of Gardiner and Davis [128]. Their results areshown in Fig. 3. This formalism is essentially a vari-ant of the Gardiner-Zoller quantum kinetic theory, ob-tained by making a high-temperature approximation tothe condensate-band master equation and then exploit-ing the quantum-classical correspondence of the Wignerrepresentation to obtain a stochastic classical-field de-scription of the condensate band [128, 129]. Although de-rived using di↵erent theoretical techniques, the resultingdescription is similar to the stochastic Gross-Pitaevskiiequation of Stoof [18, 60], both in terms of its phys-ical content and its computational implementation —see, e.g., discussion in Refs. [130, 131]. A related phase-space method originating in quantum optics known asthe positive-P representation has also been applied to ul-tracold gases [132]. This has been used to investigatecooling of a small system towards BEC by Drummondand Corney [133], who observed features consistent withspontaneously formed vortices. Despite formally beinga statistically exact method, for interacting systems ittends to su↵er from numerical divergences after a rela-tively short evolution time.

It seems likely that spontaneously formed vortices andother defects were present in earlier BEC-formation ex-periments, but not observed due to the practical di�-culties inherent in resolving these defects in experimen-tal imaging — and indeed the fact that these experi-ments were not attempting to investigate whether suchstructures were present. Another di�culty in identify-ing quantitative signatures of the KZM in experimental

BECs is the inhomogeneity of the system in the exper-imental trapping potential, which is typically harmonic.From the point of view of a local-density approxima-tion, this inhomogeneity implies that the instantaneouscoherence length and relaxation timescale are spatiallyvarying quantities, and that the transition occurs at dif-ferent times in di↵erent regions of space as the systemis cooled. Following preliminary reports of the exper-imental observation of dark solitons following the for-mation of a quasi-one-dimensional BEC by the group ofEngels at the University of Washington [134], Zurek ap-plied the framework of the KZM to a quasi-1D BEC ina cigar-shaped trap to estimate the scaling of the num-ber of spontaneously generated solitons as a function ofthe quench time [135, 136]. Witkowska et al. [137] nu-merically studied cooling leading to solitons in a com-parable one-dimensional geometry. Zurek’s methodologyfor inhomogeneous systems was applied by del Campoet al. [138] to strongly oblate geometries in which vortexfilaments behave approximately as point vortices in theplane, an idealisation of the geometry of the experimentof Weiler et al. [126].

Lamporesi et al. [139] recently reported the sponta-neous creation of Kibble–Zurek dark solitons in the for-mation of a BEC in an elongated trap, and found the scal-ing of the number of observed defects with cooling ratein good agreement with the predictions of Zurek [135]. Itwas later realised that the apparent solitons were actuallysolitonic vortices [140]. The e↵ects of inhomogeneity insuch experiments can be mitigated by the realisation of“box-like” flat-bottomed trapping geometries. The Dal-ibard group in Paris has observed the formation of spon-taneous vortices in a quasi-2D box-like geometry, andfound scaling of the vortex number with quench rate ingood agreement with the predictions of the KZM [141].

10

2 3 4 5 60

0.2

0.4

0.6

0.8

1

Time (s)

Vort

ex

pro

babili

ty

0

1

2

3

4

5

6

N0 (

10

5 a

tom

s)

a(a)b

c

d

(b)

(c)

(d)

FIG. 3. Spontaneous vortices in the formation of a Bose–Einstein condensate. (a) Squares: Experimentally measured condensatepopulation as a function of time. Solid line: Condensate number from stochastic Gross–Pitaevskii simulations. Dashed line:probability of finding one or more vortices in the simulations as a function of time, averaged over 298 trajectories. The shadedarea indicates the statistical uncertainty in the experimentally measured vortex probability at t = 6.0 s. It was observed inexperiment that there was no discernible vortex decay between 3.5 s and 6.0 s (b) Experimental absorption images taken after59 ms time-of-flight showing the presence of vortices. (c) Simulated in-trap column densities at t = 3.5 s [indicated by the leftvertical dotted line in (a).] (d) Phase images through the z = 0 plane, with plusses (open circles) representing vortices withpositive (negative) circulation. Adapted from C. N. Weiler et al. [126].

We also note further work by the Dalibard group [142]verifying the production of quench-induced supercurrentsin a toroidal or “ring-trap” geometry [143] analogous tothe annular sample of superfluid helium considered inZurek’s original proposal [123].

Experimental investigations of the KZM in diluteatomic gases have largely focused on the imaging of de-fects in the wake of the phase transition — either fol-lowing time-of-flight expansion [126, 139, 141] or in situ

[140]. However, the accurate extraction of critical scal-ing behaviour from such observations is hampered by thelarge background excitation of the field near the transi-tion, and the relaxation (or “coarsening”) dynamics ofdefects in the symmetry-broken phase. An alternativeapproach is to make quantitative measurements of globalproperties of the system following the quench. Perform-ing quench experiments in a three-dimensional box-likegeometry, the Hadzibabic group in Cambridge [120] madecareful measurements of the scaling of the correlationlength with quench time. From the measured scalinglaw, these authors were able to infer a beyond-mean-fieldvalue z = 1.4±0.2 for the dynamical critical exponent forthis universality class. Some of the results of Ref. [120]are displayed in Fig. 2(b).

The possibilities for the trapping and cooling of multi-component systems in atomic physics experiments havenaturally lead to investigations of the spontaneous for-mation of more complicated topological defects duringa phase transition. Although such experiments have so-far largely focused on the formation of defects followinga quench of Hamiltonian parameters [144, 145], the for-mation of nontrivial domain structures following gradualsympathetic cooling in immiscible 85Rb-87Rb [146] and

87Rb-133Cs [147] Bose-Bose mixtures has also been ob-served. The competing growth dynamics of the two im-miscible components in the formation of such a binarycondensate have recently been investigated theoreticallyin the limit of a sudden temperature quench [148] (seealso Refs. [149–151] for related critical scaling in otherHamiltonian quenches). These investigations indicatethe rich nonequilibrium dynamics possible in these sys-tems, including strong memory e↵ects on the coarseningof spontaneously formed defects and the potential “mi-crotrapping” of one component in spontaneous defectsformed in the other.

B. Non-thermal fixed points

A general characterisation of the relaxation dynamicsof quantum many-body systems quenched far out of equi-librium remains a largely open problem. In particular,it is interesting to ask to what extent analogues of theuniversal descriptions arising from the equilibrium the-ory of critical fluctuations may exist for nonequilibriumsystems. A recent advance towards answering such ques-tions has been made in the development of the theory ofnon-thermal fixed points: universal nonequilibrium con-figurations showing scaling in space and (evolution) time,characterised by a small number of fundamental prop-erties. The theory of such fixed points transposes theconcepts of equilibrium and di↵usive near-equilibriumrenormalisation-group theory to the real-time evolutionof nonequilibrium systems. These developments provide,for example, a framework within which to understand theturbulent, coarsening, and relaxation dynamics following

11

the creation of various kinds of defects and nonlinear pat-terns in a Kibble-Zurek quench.

The existence and significance of non-thermal scal-ing solutions in space and time was discussed by Bergesand collaborators in the context of reheating after early-universe inflation [79, 80] and then generalised by Berges,Gasenzer, and coworkers to scenarios of strong matter-wave turbulence [81, 152]. For the condensation dynam-ics of the dilute Bose gas discussed here, the presence of anon-thermal fixed point can exert a significant influencein the case of a strong cooling quench [30, 153, 154].

As an illustration, we consider a particle distribu-tion that drops abruptly above the healing-length scalek⇠ =

p8⇡a⇢, as depicted on a double-logarithmic scale

in Fig. 4 (dashed line). In order for the influence of thenon-thermal fixed point to be observed, the decay of n(k)above Q ' k⇠ is assumed to be much steeper than thequasi-thermal scaling that develops in the kinetic stageof condensation following a weak quench [27, 28], as dis-cussed in Sect. II. Such a distribution would, e.g., re-sult from a severe cooling quench of a thermal Bosegas initially just above the critical temperature whereT > |µ|/kB such that the Bose-Einstein distribution hasdeveloped a Rayleigh–Jeans scaling regime where n(k) ⇠2mkBT/(~k)2. The modulus of the chemical potential ofthis state determines the momentum scale Q where theflat infrared scaling of the distribution goes over to theRayleigh–Jeans scaling at larger k. If this chemical po-tential is of the order of the ground-state energy of thepost-quench fully condensed gas, (~Q)2/2m ' |µ| ' g⇢,with g = 4⇡~2a/m, then the energy of the entire gasis concentrated at the scale Q ' k⇠ after the quench.This is a key feature of the extreme nonequilibrium ini-tial state from which a non-thermal fixed point can beapproached. We note that, if in this state there is nosignificant zero-mode occupation n0, the respective oc-cupation number at Q is on the order of the inverse ofthe diluteness parameter, nQ ⇠ ⇣�3/2.

In analogy to the weak-wave-turbulence scenario [19,20, 27, 28] discussed in Sect. II A, the initial overpop-ulation of modes with energies ⇠ (~Q)2/2m leads toinverse particle transport while energy is transportedto higher wavenumbers, as indicated by the arrows inFig. 4 [30, 153, 154]. However, the inverse transportinvolves non-local scattering and thus does not repre-sent a cascade. Furthermore, in contrast to the case ofa weak quench [14, 15, 18–20, 28, 29], in which weak-wave turbulence produces a quasi-thermal momentumdistribution that relaxes quickly to a thermal equilib-rium distribution, here the inverse transport is charac-terised by a strongly non-thermal power-law scaling inthe infrared. Specifically, the momentum distributionn(k) ⇠ k�d�2 ⇠ k�5 in d = 3 dimensions [81] providesthe “smoking-gun” of the influence of the non-thermalfixed point. Semiclassical simulations by Nowak, Gasen-zer and collaborators [30, 155–157] showed that this scal-ing is associated with the creation, dilution, coarseningand relaxation of a complex vortex tangle, as predicted on

Synthetic · Zagreb · 30 Sep 2015

Now: Strong cooling quench!

log

nk

log k

particles removed by cooling quench

initial distribution after quench

kξ

nQ ≈ (ρa 3)-1/2

Q

FIG. 4. Sketch of the evolution of the single-particle momen-tum distribution nk(t) of a Bose gas close to a non-thermalfixed point (after Ref. [153]). Starting from the extreme initialdistribution (dashed line, see main text for details) produced,e.g., by a strong cooling quench, a bidirectional redistributionof particles in momentum space (arrows) builds up a quasi-condensate in the infrared while refilling the thermal tail atlarge momenta. The particle transport towards zero momen-tum is characterised by a self-similar scaling evolution in spaceand time, n(k, t) = (t/t0)

↵n([t/t0]�k, t0), with characteristic

scaling exponents ↵, �. Note the double-logarithmic scale.

phenomenological grounds in Ref. [21], and other types of(quasi-)topological excitations in low-dimensional, spinorand gauge systems [158–162].

The dynamics in the vicinity of the fixed point arecharacterised by an anomalously slow relaxation of thetotal vortex line length, which exhibits an algebraic de-cay ⇠ t�0.88 (see Ref. [158] for analogous results in the2D case). At the same time, the condensate populationgrows as n0(t) ⇠ t2 [30, 153], a significant slowing com-pared to the ⇠ t3 behaviour observed for weakly nonequi-librium condensate formation [30, 163].

In the vicinity of the fixed point the momentum dis-tribution is expected to follow a self-similar scaling be-haviour in space and time in the infrared, n(k, t) =(t/t0)↵n([t/t0]�k, t0). For a 3D Bose gas, these scalingexponents have recently been numerically determined tobe ↵ = 1.66(12), � = 0.55(3), in agreement with theanalytically predicted values ↵ = �d, � = 1/2 [153].This behaviour, here corresponding to the dilution andrelaxation of vortices leading to a build-up of the con-densate population, represents the generalisation of crit-ical slowing-down to real-time evolution far away fromthermal equilibrium. At very late times, the systemleaves the vicinity of the non-thermal fixed point, typ-ically when the last topological patterns decay, and fi-nally approaches thermal equilibrium [32–35, 158, 164].This equilibrium state corresponds to a fully establishedcondensate superimposed with weak sound excitations.

In summary, non-thermal fixed points are nonequilib-rium field configurations, exhibiting universal scaling intime and space, to which the system is attracted if suit-ably forced — e.g., in the case of Bose condensation, fol-

12

lowing a su�ciently strong cooling quench. In the vicin-ity of such fixed points, the relaxation of the field is crit-ically slowed down and the dynamics exhibit self-similartime evolution, governed by new critical exponents andscaling functions. The possibility of categorising systemsinto generalised “universality classes” associated with thenew critical exponents is a fascinating prospect and thesubject of current research [23, 24].

Finally in this section we note that prethermalisationor “pre-Gibbsianisation”, i.e., the approach of a statecharacterised by a Generalised Gibbs ensemble, usuallyin near-integrable systems (see the chapter by Schmied-mayer), represents a special case of a Gaussian non-thermal fixed point, meaning that the e↵ective couplingof the prethermalised modes vanishes. It is expected thatthe exponents ↵ and � in such a situation can becomevery small compared to unity. Only at very late timesthe remaining e↵ects of interactions may eventually drivethe system away from the fixed point towards a thermalstate.

V. CONCLUSIONS AND OUTLOOK

In this chapter we have provided a brief introductionto the scenario of the formation of a Bose–Einstein con-

densate, and physics related to the dynamics of the BECphase transition. We have given a fairly comprehensivereview of the experiments studying the formation of sim-ple, single-component BECs in three-dimensional atomicgases, with brief mentions of how such features are af-fected by reduced e↵ective dimensionality or in caseswhere more than one condensate may co-exist. How-ever, the underlying physics described here is relevant toseveral other systems, most notably exciton-polaritonsconfined in strictly two-dimensional geometries featuringpumping and decay, where experiments on condensateformation have also been performed [165–167].

An interesting question is what are the similarities anddi↵erences between these systems, and others such asBECs of photons [168] and magnons [169]. Furthermore,what can phase transitions in quantum gases teach usabout phase transitions that cannot be accessed experi-mentally, such as inflationary scenarios of early-universeevolution? This was one of the motivating questions inthe formulation of the Kibble–Zurek mechanism, as wellas in the development of the theory of non-thermal fixedpoints. It remains to be seen what we can learn aboutsuch matters as the formation of cosmological topologi-cal defects and (possibly) baryon asymmetry by studyingnanokelvin gases here on earth.

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