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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, Australia 2JILA, 440 UCB, University of Colorado, Boulder, Colorado 80309, USA 3Kirchhoff-Institut fu¨r Physik, Universita¨t 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, 6 Newcastle upon Tyne NE1 7RU, United Kingdom 1 0 The problem of understanding how a coherent, macroscopic Bose–Einstein condensate (BEC) 2 emergesfromthecoolingofathermalBosegashasattractedsignificanttheoreticalandexperimental n interest over several decades. The pioneering achievement of BEC in weakly-interacting dilute a atomic gases in 1995 was followed by a number of experimental studies examining the growth J of the BEC number, as well as the development of its coherence. More recently there has been 2 interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, 2 in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topologicalstructuressuchasvorticesandsolitonsinquenchedcold-atomexperimentshasenabled ] the verification of the Kibble–Zurek mechanism predicting the density of topological defects in s continuous phase transitions, first proposed in the context of the evolution of the early universe. a g ThischapterreviewsprogressintheunderstandingofBECformation,anddiscussesopenquestions - and future research directions in the dynamics of phase transitions in quantum gases. t n a u I. INTRODUCTION approachestothedescriptionofnonequilibriumandnon- q zero-temperature quantum gases [1]. t. The equilibrium phase diagram of the dilute Bose gas We note that the past decade has seen the observa- a exhibitsacontinuousphasetransitionbetweencondensed tionofBECinanumberofdiverseexperimentalsystems m and noncondensed phases. The order parameter charac- beyond ultracold atoms, including exciton-polaritons, - teristicofthecondensedphasevanishesabovesomecriti- magnons, andphonons, whicharecoveredinotherchap- d n caltemperatureTc andgrowscontinuouslywithdecreas- ters of this volume. Many of the universal aspects of o ing temperature below this critical point. However, the condensate formation also apply to these systems. c dynamical process of condensate formation has proved [ to be a challenging phenomenon to address both theo- 1 retically and experimentally. This formation process is II. THE PHYSICS OF BEC FORMATION v a crucial aspect of Bose systems and of direct relevance 7 to all condensates discussed in this book, despite their 9 evident system-specific properties. Important questions The essential character of the excitations and collec- 1 tiveresponseofacondensedBosegasiswelldescribedby leading to intense discussions in the early literature in- 6 perturbativeapproachesthattakeastheirstartingpoint clude the timescale for condensate formation, and the 0 the breaking of the U(1) gauge symmetry of the Bose role of inhomogeneities and finite-size effects in “closed” . 1 systems. Theseissuesarerelatedtotheconceptofspon- quantum field. This approach can be extended further 0 to provide a kinetic description of excitations in a con- taneous symmetry breaking, its causes, and implications 6 densed gas weakly perturbed away from equilibrium [2]. for physical systems (see, for example, the chapter by 1 The description of the process of formation of a Bose- : Snoke and Daley in this volume). v Einstein condensate in a closed system begins, however, In this chapter we give an overview of the dynam- i intheoppositeregimeofkineticsofanon-condensedgas. X ics of condensate formation and describe the present Over the past decades, there have been many studies us- understanding provided by increasingly well controlled r ingmethodsofkinetictheorytoinvestigatetheinitiation a cold-atomexperimentsandcorrespondingtheoreticalad- ofBose–Einsteincondensation. Itisnowwellestablished vances over the past twenty years. We focus on the thatthesedescriptionsbreakdownnearthecriticalpoint, growth of BECs in cooled Bose gases, which, from a and in particular in any situation in which the forma- theoretical standpoint, requires a suitable nonequilib- tion process is far from adiabatic. A number of different rium formalism. A recent book provides a more com- theoretical methodologies have been applied to the issue plete introduction to a number of different theoretical of condensate formation, but most have converged to a similar description of the essential physics. The prevail- ing view is that a classical non-linear wave description ∗ [email protected] — a form of Gross–Pitaevskii equation — can describe 2 the nonequilibrium dynamics of the condensation pro- Experimental attempts in the 1980s to achieve Bose cess, which involves in general aspects of weak-wave tur- condensation of spin-polarized hydrogen (see the chap- bulenceand, inmoreaggressivecoolingscenarios, strong ter by Greytak and Kleppner for an overview and re- turbulence. The classical field describes the highly occu- centdevelopments),andexcitonsinsemiconductorssuch pied modes of the gas at finite temperature and out of as Cu O, inspired renewed theoretical interest in Bose- 2 equilibrium. gas kinetics. Eckern developed a kinetic theory [12] for A summary of the consensus picture of condensate Hartree–Fock–Bogoliubov quasiparticles appropriate to formation in a Bose gas cooled from above the critical the relaxation of the system on the condensed side of temperature is as follows. Well above the critical point thetransition. SnokeandWolferevisitedthequestionof the coherences between particles in distinct eigenstates the kinetics of approach to the condensation transition of the appropriate single-particle Hamiltonian are neg- by undertaking numerical calculations of the quantum ligible and the system is well described by a quantum Boltzmann equation [13]. They found in particular that Boltzmann kinetic equation for the occupation numbers the bosonic enhancement of scattering rates in the de- of these single-particle modes. As cooling of the gas pro- generate regime offset the increased number of scatter- ceedsduetointer-particlecollisionsandinteractionswith ing events required for rethermalisation in this regime, an external bath, if one is present, the occupation num- such that re-equilibration of a shock-cooled thermal dis- bersoflower-energymodesincrease. Oncephasecorrela- tributiontakesplaceontheorderofthreetofourkinetic tionsbetweenthesemodesbecomesignificant,thesystem collision times, τkin = (ρσvT)−1, where ρ is the particle is best described in terms of an emergent quasiclassical density, σ is the collisional cross section, and the mean field, which may in general exhibit large phase fluctua- thermal velocity vT = (3kBT/m)1/2. These results im- tions,topologicalstructuresandturbulentdynamics,the ply that a Boltzmann-equation description of this early nature of which may vary over time and depend on the kinetic regime is valid even for short-lived particles such specific details of the system — including its dimension- as excitons, as the particle lifetime is long compared to ality, density, and strength of interactions. This regime this equilibration timescale. is sometimes referred to as a nonequilibrium quasicon- Over time a comprehensive picture of the process of densate, in analogy to the phase-fluctuating equilibrium condensation of a quench-cooled gas has emerged, and regimesoflow-dimensionalBosesystems[3,4]. Theeven- comprisesthreedistinctstagesofnonequilibriumdynam- tual relaxation of this quasicondensate establishes phase ics: a kinetic redistribution of population towards lower coherenceacrossthesample,producingthestatethatwe energy modes in the non-condensed phase, development routinely call a Bose–Einstein condensate. of an instability that leads to nucleation of the condensate and a subsequent build-up of coherence, and finally condensate growth and phase ordering. In the midst of A. The pre-condensation kinetic regime increasingly intensive efforts to achieve Bose condensation in dilute atomic gases, by then including the new system of alkali-metal vapours, these stages were anal- Early investigations of the kinetics of condensation of ysed in more detail in the early 1990s, beginning with agasofmassivebosonsbeganwithstudiesofsuchasys- a series of papers by Stoof [14–18], and by Svistunov, temcoupledtoathermalbathwithinfiniteheatcapacity, Kagan, and Shlyapnikov [19–22]. consisting of phonons [5] or fermions [6–8]. These works inherited ideas from earlier studies of condensation of InRef.[19], Svistunovdiscussedcondensateformation photons in cosmological scenarios [9]. In a homogeneous in a weakly interacting, dilute Bose gas, with so-called system, condensation is signified by a delta-function sin- gas parameter ζ = ρ1/3a (cid:28) 1, where a is the scattering gularity of the momentum distribution at zero momen- length. In a closed system, a cooling quench generically tum (see, e.g., Ref. [10]). Levich and Yakhot found [6] leadstoaparticledistributionwhich,belowsomeenergy that an initially non-degenerate equilibrium ideal Bose scaleε ,exceedstheequilibriumoccupationnumbercor- 0 gas brought in contact with a bath with a temperature responding to the total energy and particle content. En- belowT woulddevelopsuchasingularityatzeromomen- ergy and momentum conservation then imply that a few c tum only in the limit of an infinite evolution time (see particlesscatteredtohigh-momentummodescarryaway also Ref. [11]). These same authors subsequently found a large fraction of the excess energy associated with this that the introduction of collisions between the bosons over-occupation, allowing the momentum of a majority lead to the “explosive” development of a singular peak of the particles to decrease. Should the characteristic at zero momentum after a finite evolution time [7, 8]. energy scale of the overpopulated regime be sufficiently They were careful to point out, however, the approxima- small, ε (cid:28) (cid:126)2ρ2/3/m ∼ k T , mode-occupation num- 0 B c tions involved in their treatment of interactions, and in- bers in this regime will be much larger than unity, and deed that the development of such coherence invalidates thesubsequentparticletransportinmomentumspaceto- the assumptions underlying the quantum Boltzmann de- wards lower energies is described by the quantum Boltz- scription, conjecturing that “the system in the course of mann equation in the classical-wave limit [19–21]. This phase transition passes through a stage which may be is valid for modes with energies above the scale set by identified as a period of strong turbulence” [7]. the chemical potential µ = gρ ∼ ζk T of the ultimate B c 3 equilibrium state, where g = 4π(cid:126)2a/m is the interac- significant many-body corrections once the interaction (cid:82) tion constant for particles of mass m. At lower ener- energy g dkn of particles with momenta below a k(cid:46)p k gies, the phase correlations between momentum modes givenscalepexceedsthekineticenergyatthatscale,and become significant, and a description beyond the quan- theseareindeedtheprevailingconditionswhenphaseco- tum Boltzmann equation is required. herence emerges and the condensate begins to grow [19]. We note that for open systems such as exciton- Inaseriesofpapers[14,15,17,18],Stooftookaccount polariton condensates, the quasi-coherent dynamics of ofthesemany-bodycorrectionsanddevelopedatheoryof such low-energy modes will in general be sensitive to the condensatenucleationrestingonkineticequationsincor- driving and dissipation corresponding to the continual poratingaladder-resummedmany-bodyT-matrixdeter- decay and replenishment of the bosons. Such external mined from a one-particle-irreducible (1PI) effective ac- coupling candramatically alter thebehaviour of thesys- tion or free-energy functional. In the 1PI formalism the tem, and its effects on condensate formation dynamics propagators appearing in the effective action are taken are a subject of current research — see, e.g., Refs. [23– as fixed, determined in this case by the initial thermal 25] and the chapter by Altman et al.. Hereafter, unless Bose number distribution and the spectral properties of otherwise specified, the theoretical developments we dis- a free gas. cuss pertain to closed systems in which the bosons un- Constructed within the Schwinger–Keldysh closed- dergoing condensation are conserved in number during time-path framework, the method allows the determina- the formation process. tionofthetimeevolutionoftheself-energyandthusofan By assuming the scattering matrix elements in the effectivechemicalpotentialforthezero-momentummode waveBoltzmannequationtobeindependentofthemode through the phase transition. During the kinetic stage, energies, Svistunov discussed several different transport once the system has reached temperatures below the scenarioswithintheframeworkofweak-waveturbulence, interaction-renormalized critical temperature, the self- in analogy to similar processes underlying Langmuir- energy renders the vacuum state of the zero-momentum wave turbulence in plasmas [26]. He concluded that mode metastable. Stoof found that this modification of the initial kinetic transport stage of the condensation theself-energyoccursonatimescale∼(cid:126)/k T andgives B c process evolves as a weakly non-local particle wave in rise to a small seed population in the zero mode, n ∼ 0 momentum space. Specifically, he proposed that the ζ2ρ, within the kinetic time scale τ ∼ (cid:126)/(ζ2k T ). kin B c particle-flux wave followed the self-similar form n(ε,t)∼ He argued that, following this seeding, the system un- ε1(t)−7/6f(ε/ε1(t)), with ε1(t) ∼ (t−t∗)3, and scaling dergoes an unstable semi-classical evolution of the low- function f falling off as f(x) ∝ x−α for x (cid:29) 1, with energy modes. Taking interactions between quasiparti- α = 7/6. Following the arrival of this wave at time cles into account he found that the squared dispersion t∗ (cid:39)t0+(cid:126)ε0/µ2, a quasi-stationary wave-turbulent cas- ω(p)2 becomes negative for p (cid:46) (cid:126)(cid:112)an0(t), i.e., below cade forms in which particles are transported locally, a momentum scale of the order of the inverse healing frommomentumshelltomomentumshell,fromthescale length associated with the density n (t) of the existing 0 ε0 of the energy concentration in the initial state to the condensed fraction. As a result, the condensate grows low-energy regime ε(cid:46)µ where coherence formation sets linearly in time over the kinetic time scale τ . The kin in. growthprocesseventuallyceasesduetotheconservation The wave-kinetic (or weak-wave turbulence) stage of oftotalparticlenumber,whereafterthefinalkineticequi- condensate formation following a cooling quench was libration of quasi-particles takes place over a time scale investigated in more detail by Semikoz and Tkachev ∼(cid:126)/(ζ3k T )asdiscussedpreviouslybyEckern[12],and B c [27, 28], who solved the wave Boltzmann equation nu- by Semikoz and Tkachev [28]. merically and found results consistent with the above scenario, albeit with a slightly shifted power-law exponent α (cid:39) 1.24 for the wave-turbulence spectrum. Later C. Turbulent condensation dynamical classical-field simulations of the condensation formation process by Berloff and Svistunov [29] further The semi-classical scenario of Stoof is built on the as- corroborated the above picture. sumptions that the cooling quench has driven the sys- temtothecriticalpointinaquasi-adiabaticfashion,and that the neglect of thermal fluctuations and nonequilib- B. The formation of coherence rium over-occupations in the self-energy is justified [18]. However, as previously pointed out in Ref. [7], a more It has been known for some time that a kinetic Boltz- vigorous quench may drive the system into an interme- mann equation model is unable to describe the develop- diate stage of strong turbulence, where the coherences mentofamacroscopiczero-momentumoccupationinthe betweenwavefrequenciesleadtotheformationofcoher- absence of seeding or other modifications [6, 13, 19]. In ent structures, such as vortices, that have a significant any event, the quantum Boltzmann equation ceases to influence on the subsequent dynamics. The main pro- be valid in the high-density, low-energy regime in which cesses and scales governing this stage were discussed in condensation occurs. The two-body scattering receives detail by Kagan and Svistunov [21, 22]. As a result of 4 excessparticlesbeingtransportedkineticallyintotheco- band using standard techniques [42], yielding equations herentregime(wavenumbersbelowthefinalinverseheal- of motion for the occupations of the condensate mode √ inglength,k (cid:46)ξ−1 ∼ aρ),thedensityandphaseofthe andthelow-lyingexcitedstatescontainedintheconden- Bosefieldfluctuatestronglyonlengthscalesshorterthan sate band. A simple BEC growth equation derived from ξ. Thegrowingpopulationatevensmallerwavenumbers this approach provided a reasonable first estimate of the then implies, according to Refs. [19–21], the formation time of formation for the 87Rb and 23Na BECs of the of a quasicondensate over the respective length scales, JILA [36] and MIT [37] groups, respectively. as the coherent evolution of the field according to the The first experiment to explicitly study the forma- Gross–Pitaevskiiequationcausesthedensityfluctuations tion dynamics of a BEC in a dilute weakly interacting tostronglydecreaseattheexpenseofphasefluctuations. gas was performed by the Ketterle group at MIT, us- This short-range phase-ordering occurs on a time scale ing their newly developed technique of non-destructive τc ∼ (cid:126)/µ ∼ (cid:126)/(ζkBTc). Depending on the flux of ex- imaging [43]. Beginning with an equilibrium gas just cess particles entering the coherent regime, this leads to above the critical temperature, they performed a sud- quasicondensate formation over a minimum length scale den evaporative cooling “quench” by removing all atoms lv > ξ (see Sects. IVA and IVB) [30]. The phase, how- above a certain energy. The subsequent evolution led to ever, remains strongly fluctuating on larger length scales the formation of a condensate, with a characteristic S- duetotheformationoftopologicaldefects—vortexlines shapedcurveforthegrowthincondensatenumber. This andrings. Thesevorticesappearintheformofclumpsof wasinterpretedasevidenceofbosonicstimulationinthe strongly tangled filaments [31] with an average distance growth process, and they fitted the simple BEC growth betweenfilamentsoforderlv. Ifthecoolingquenchissuf- equation of Ref. [44] to their experimental observations. ficiently strong to drive the system near a non-thermal However, the measured growth rates did not fit the the- fixedpoint,cf.Sect.IVB,thisquasicondensateischarac- ory all that well. terised by new universal scaling laws in space and time. Gardiner and co-workers subsequently developed an The work of Kagan and Svistunov laid the founda- expanded rate-equation approach incorporating the dy- tionsforstudyingtheroleofsuperfluidturbulenceinthe namics of a number of quasiparticle levels [45, 46]. This process of Bose–Einstein condensation. Kozik and Svis- formalism predicted faster growth rates, mostly due to tunovhavesubsequentlyelucidatedthedecayofthevor- the enhancement of collision rates by bosonic stimula- tex tangle via the transport of Kelvin waves created on tion, but still failed to agree with the experimental data. the vortex filaments through their reconnections, which Onelimitationofthisapproachwasthatitneglectedthe can itself assume a wave-turbulent structure [32–35]. evaporative cooling dynamics of the thermal cloud, in- stead treating it as being in a supersaturated thermal equilibrium. III. CONDENSATE FORMATION The details of the evaporative cooling were simulated EXPERIMENTS in two closely related works by Davis et al. [47] and Bi- jlsma et al. [48]. The former was based on the quantum A. Growth of condensate number kinetic theory of Gardiner and Zoller, while the latter emerged as a limit of the field-theoretical approach of We now provide a historical overview of both experi- Stoof [17, 18] and the “ZNG” formalism previously de- ments and theory related to condensate formation in ul- veloped for nonequilibrium trapped Bose gases [49] by tracold atomic gases. The first experiments to achieve Zaremba, Nikuni, and Griffin. The latter authors used a Bose–Einstein condensation in 1995 [36, 37] reached the broken-symmetry approach to derive a quantum Boltz- phase space density necessary for quantum degeneracy mann equation for non-condensed atoms coupled to a using the technique of evaporative cooling [38] — the Gross–Pitaevskii equation for the condensate [49, 50], steady removal of the most energetic atoms, followed by thereby extending their two-fluid model for trapped rethermalisation to a lower temperature via atomic col- BECs [50], which was based on the pioneering work of lisions. These experiments, which concentrated on the Kirkpatrick and Dorfman [51–54]. The ZNG methodol- BECatomnumberastheconceptionallysimplestobserv- ogy has since been used successfully and extensively to able,providedanindicationofthetimescaleforconden- studyavarietyofnonequilibriumphenomenainpartially sation in trapped atomic gases, in the range of millisec- condensed Bose gases, such as the temperature depen- onds to seconds. This gave the impetus for the develop- denceofcollectiveexcitations,asreviewedinRef.[2]. As mentofaquantumkinetictheorybyGardinerandZoller thismethodologyisexplicitlybasedonsymmetrybreak- using the techniques of open quantum systems. They ing, it cannot address the initial seeding of a BEC, or first considered the homogeneous Bose gas [39], before any critical physics arising from fluctuations. However, extending the formalism to trapped gases [40, 41]. Their it can model continued growth once a BEC is present. methodology split the system into a “condensate band”, The works of Davis et al. [47] and Bijlsma et al. [48] containingmodessignificantlyaffectedbythepresenceof both introduced approximations to the formalisms they aBEC,anda“non-condensateband”containingallother were built on, assuming that the condensate grew adi- levels. Amasterequationwasderivedforthecondensate abatically in its ground state, and treating all non- 5 condensed atoms in a Boltzmann-like approach. Both ing of an atomic Bose cloud brought in close proximity papers boiled down to simulating the quantum Boltz- to a dielectric surface, due to the selective adsorption of mann equation in the ergodic approximation, in which high-energy atoms. More recently, similar experiments the phase-space distribution depends on the phase-space havebeenundertakenbytheDurham[64]andTu¨bingen variablesonlythroughtheenergy[55]. Despitethediffer- groups [65], with the observed rates of loss in the latter ent approaches, the calculations were in excellent agree- case explained accurately by non-ergodic ZNG-method ment with one another — yet still quantitatively dis- calculations of the evaporative cooling dynamics. Exam- agreed with the MIT experimental data [43]. This dis- ple results are shown in Fig. 1(b). agreement has remained unexplained. A second study of evaporative cooling to BEC in a dilute gas was performed by the group of Esslinger and Hänsch in Munich [56]. In this experiment the Bose B. Other theories for condensate formation cloud, which was again initially prepared in an equilibrium state slightly above Tc, was subjected to a contin- For completeness, here we briefly outline other theo- uous rf field inducing the ejection of high-energy atoms reticalmethodsthatcanbeappliedtocondensateforma- from the sample. By adjusting the frequency of the ap- tion. Ageneralisedkineticequationforthermallyexcited plied field and thus the energies of the removed atoms, Bogoliubov quasiparticles was obtained by Imamovic- these authors were able to investigate the growth of Tomasovic and Griffin [66] based on the application of the condensate for varying rates of evaporative cooling. theKadanoff–BaymnonequilibriumGreen’sfunctionap- Davis and Gardiner extended their earlier approach [47] proach [67] to a trapped Bose gas. This kinetic equation toincludetheeffectsofthree-bodylossandgravitational reduces to that of Eckern [12] in the homogenous limit sag on the cooling of the 87Rb cloud in this experi- and to that of ZNG [49] when the quasiparticle charac- ment[57]. Theircalculationsyieldedexcellentagreement ter of the excitation spectrum is neglected. Walser et with the experimental data of Ref. [56] within its statis- al. [68, 69] derived a kinetic theory for a weakly inter- tical uncertainty for all but the slowest cooling scenarios acting condensed Bose gas in terms of a coarse grain- considered. An example is shown in Fig. 1(a). ing of the N-particle density operator over configura- In1997Pinkseet al.[58]experimentallydemonstrated tional variables. Neglecting short-lived correlations be- thatadiabaticallychangingthetrapshapecouldincrease tween colliding atoms in a Markov approximation, they the phase-space density of an atomic gas by up to a fac- obtained kinetic equations for the condensate and non- tor of two and conjectured that this effect could be ex- condensate mean fields which were subsequently shown ploited to cross the BEC transition in a thermodynami- to be microscopically equivalent [70] to the nonequilib- cally reversible fashion. This scenario was subsequently rium Green’s function approach of Ref. [66]. Exactly the realised in the MIT group by Stamper-Kurn et al. [59] same kinetic equations were derived by Proukakis [71], by slowly ramping on a tight “dimple” trap formed from within the formalism of his earlier quantum kinetic for- an optical dipole potential on top of a weaker harmonic mulation [72, 73], based on the adiabatic elimination of magnetic trap. This experiment was the setting for the rapidly-evolving averages of non-condensate operators, firstapplicationofastochasticGross–Pitaevskiimethod- ideas which fed into the development of the ZNG ki- ology [60], previously developed from a nonequilibrium netic model [74]. Although elegant, these formalisms formalism for Bose gases by Stoof [18]. This is based have not provided a tractable computational methodol- onthemany-bodyT-matrixapproximation,andusesthe ogy for modelling condensate formation away from the Schwinger–Keldyshpathintegralformulationofnonequi- quasistatic limit. librium quantum field theory to derive a Fokker-Planck A non-perturbative method for the many-body dy- equation for both the coherent and incoherent dynamics namics of the Bose gas far from equilibrium has been of a Bose gas. The classical modes of the gas were rep- developed by Berges, Gasenzer, and co-workers [75–78]. resented by a Gross–Pitaevskii equation, with additional This two-particle irreducible (2PI) effective-action ap- dissipative and noise terms resulting from a collisional proach provides a systematic way to derive approximate coupling to a thermal bath with a temperature T and Kadanoff–Baym equations consistent with conservation chemical potential µ. laws such as those for energy and particle number. In Proukakis et al. [61] subsequently used this methodol- contrast to 1PI methods, single-particle correlators are ogy tostudy the formation ofquasicondensates ina one- determined self-consistently by these equations. This dimensional dimple trap. A much later experiment [62] approach allows the description of strongly correlated investigated the dynamics of condensate formation fol- systems, and has been exploited in the context of tur- lowing the sudden introduction of a dimple trap, and bulent condensation [79–81] where it provides a self- included quantum-kinetic simulations that were in good consistently determined many-body T matrix. This 2PI agreement with the data. effective-action approach is useful for studying strongly A novel method of cooling a bosonic cloud to con- interacting systems such as 1D gases with large coupling densation was introduced in 2003 by the Cornell group constant [82], or relaxation and (pre-)thermalization of at JILA [63], who demonstrated the evaporative cool- strongly correlated spinor gases [83]. 6 EVAPORATIVECOOLINGOFCOLDATOMSATSURFACES PHYSICALREVIEWA90,023614(2014) possible collision partners. Because the density of the atom 1.5 cloud can vary considerably, we use an adaptive Cartesian 1.0 gridinrealspaceasoutlinedin[77],whilekeepingaglobal N timestep. 500.5 Our initial state is a thermal cloud in equilibrium with a 1 1.0 temperature T. This state is calculated using self-consistent Hartree-Fock as outlined in [78]. In addition to the thermal N 0.5 1.5 cloud, the initial state requires a small condensate “seed” 5 t(s) 0 to allow for C collisions, and hence condensate growth; 1 12 the number of atoms in the seed is obtained using the 0.5 Bose-Einsteindistribution,assumingµ 0[38]. (a) c = (b) Interactionsbetweenthesurfaceandtheatomsaremodeled bycalculatingthesingle-correctionfunction[Eq.(7)]forthe generalizedGross-Pitaevskiiequation[Eq.(1)]andcombining 0 it with a linear imaginary potential to remove condensate 0 1 2 3 atoms,effectivefromthepositionwherethetrapopens[79]. t (s) In addition, we annihilate test particles that are beyond this openingpoint,resultinginanatomlossforthethermalcloud. FIG.2.(Coloronline) TotalatomnumberN againsttimet for FIG. 1. (a) Growth of an atomic Bose–Einstein condensate modelled with the quantum Boltzmann equation. The experiment These two processes lead to a reduction in the total atom three different trap-surface separations: x 68µm (gold solid beganwitha87RbBosegasinanelongatedharmonictrapwithN =(4.2±0.2)×106atomsatatempse=ratureofT =(640±30) numberinthesystem. icurve),30µm(reddashedcurve),and15µm(blackdotteidcurve). nK,beforeturningonrfevaporativecoolingwithatruncationenerPgoyinotsf1co.4rrkeBspTon.dTtoheexspoelriidmaenntdalddoattateadndlitnheescsuhrvoews ctohreretsphoenodretical calculationswithastartingnumberofNi =4.2×106 andNi =4.4to×si1m0u6laatitoonms.sThreesdpoet-cdtaisvheeldy.goTladkceunrvferoshmowRseaf.si[m56ul]a.ti(obn)foSrurface evaporation leading to the fIoVr.mRaEtSioUnLToSf a BEC, showing the totxal at6o8mµmnwumithboeutrcofollrisitohnrs,eie.e.d,CifferenCt clo0u.dT-hseugrrfaaycveerdtiicsatlances. s = 12= 22= The linesHaraevitnhgesreetsuupltosuorfcZoNmGputsaitmiounlaaltmioondse,l,thweepnooiwntesmaprelofyromdeaxspheedrilmineemnta.rkTsthheepdoointt-wdahesnhthgeoaldtomlincleouidsrfeoarchaesZitNsfiGnaslihmoludlation neglectiintgtocsotluldisyiosnusrfaincetheveapthoreartmivaelccolooluindg,.dWeemsohnoswtrathteinrgestuhltast mopdoseiltlioinngattthe 0fu.lTlhdeygnraaymhaicshsedofartehaeshtohwesrmthealshcifltooufdthiescnurevceessary = foraquoafnstiimtautlaivtieonusnfdoerrtswtaonddifinfegreonfttgheeomexetprieersi.mInenSte.c.TIhVeAin,sweetshowwhsetnhtehetsoutrafalcneupmosbitieorn,itshvearrmiedalbyclou2.d5µnumm.Tbheer,inasentdshcoownsdaensate ± numberd,irfreoctmlytcoopmptaorebtohtetoormywrietshpeexcpteivriemlye,ntatsoaexfaumnicnteiotnheoefxtteimnte. TbarkeaekndofrwonmofRtheef.cl[o6u5d].atomnumbersagainsttimeforxs =68µm fromthepointwhenthecloudreachesitsholdingposition.Thesolid towhichthemodelcapturestheimportantphysicalprocesses. curveshowsthetotalatomnumber,thedashedcurvecorrespondsto WethengoontoconsiderasimplermodelsysteminSec.IVB, thermalatoms,andthedottedcurvecorrespondstothecondensate C.withOathvieerwptoioonpetiemriizninggcpoanradmeentesrastteo-cforeramteathteiopnurestor In 2004, the Sengstock group observed the formation atomnumber. largestcondensatees.xperiments of a BEC at constant temperature [87]. Working with a The experiments were performed using the apparatus spin-1 system, they prepared a partially condensed gas described in [17]. Clouds of 87Rb atoms were loaded into temperatures are slightly above the critical temperature for consisting of m =±1 states. Spin collisions within the thaTnheevriaenanptahaoretroeamatxaiivacelnhiducpiomreotrbclatieipnorgnwotaifotnhdeinxfωrcpeyrqeeuraeismωneczeitenhste2aωπplxhm=a8se52etπr-has×odpsd1ass6c−er1oadtdinhesnest−hr-e1 BcoEnWCdeencspoaemtrifoopnr,omTneced,nfFotthsreapnsoiimdpeuuallalatgitoaesnds[8ut0hs]i.engmthFese=ex0pesritmateen,talwhich = = × then quickly thermalised. When the population of the sityofraadqiualanditruecmtiogna.sTahnedclfoourdmwaascoinnitdiaelnlysaptree.paWredebwritihefltyhe parameters[81].Weplotthetheoreticalandexperimentalatom mentiotnratphceemntehreatreadfoisrtacnocmexpslete1n3e5sµs.mfromasiliconsurface, mnuFmb=ers0agcaoinmstptoimneenintFriega.c2h(ead“ttimheescerriietsic”a).lWneucmonbseidrear new ≈ definedasthex 0plane.Atthispoint,therewasnegligible BthEeCtimeme etrge0d.toTbheetheexppeorinimt wenhtenwtahsemcloodudellreedacwheisthitsasim- An experimental t=echnique that has proved to be ex- = overlapbetweenthecloudandthesurface.Thecloudwasthen pfilnealrahoteldepqousiatitoino,ni.ndicatedbythegrayverticaldashedline. tremelyusefulformulti-componentquantumgasesisthe transportedalongthex axisatavariablespeedtoavariable Sincetheabsolutesurfacepositionmayvaryduetodrifts,we method of sympathetic cooling, in which an atomic gas distance,xs,fromthesurfaceandheldforavariableholdtime. peIrfnortmheedsaarmanegeyoefasri,mKuleattitoenrslew’sithMvaIrTyinggroxustpoopbetrafionrtmheed an is cooleIndorbdyervtoirmtueeasuorfetihtserceomlaliisniionngaaltomintneurmabcetiroNn,twheitchlouad ebxepstefiritm.Inentthisinsewnshei,cthhethsiemyuldaitisotnilslesdervaedBaEsCacfarloibmratoionne trap secondwgaassswoifftalytobrmousg,hdtibsatcikngtouiitsshineidtiaflrpoomsititohne,afiftresrtwehiitchhewre mtoionl:imfourmthetoxsano1t4h,er29[,88an].dA72nµonm-zceurrov-etse,mthpeebreasttufirtesBEC isotopicpaerlfloyrmoerdbtiymei-notfe-flrnigahltmqueaasnutreummenntsuamndbCerCsD, iwmhaigcihng.is wwaesrefoobrtmaineeddiwn≈ithana osipmtuiclaatelddcilpooulde-sutrrfaapce, sbeepfaorarteionaosfecond itself subject to, e.g., evaporative cooling. This tech- 15,30,and68µm,respectively,wellwithintheexperimental trap with a greater potential depth was brought nearby. nique was first demonstrated by Myatt et al. [84] in a uncertainties.Figure2showstheevolutionofthetotalnumber A. Losscurves Atomsofsufficientthermalenergywereabletocrossthe gas comprising two distinct spin states of 87Rb, and was ofatoms,N,remaininginthecloudduringthecourseofthe 1. Timeseries bsiamrruileartiobne,twwitehecnutrvheestcwororepspootnednitnigatlomnuinmiemriaca,lpreospuultslaatnidngthe subsequentlyemployedtocoolasingle-componentFermi Webeginbyconsideringatomlosscurvesasafunctionof speocionntsdcotrrraespp.onEdvinegnttouaexllpyertimheenfitarlstdactao.ndensate evaporated, gas to degeneracy by Schreck et al. [85]. timewhenthecloudisbroughtintooverlapwiththesurface. andThaesgeocldonsodlidcocnudrveenasnadtegoflodrsmtaerdpoiinntsthaerenfoerwthgelxobal trap s In aIsnimtheilaerxpsepriimrietn,tsin, th2e00cl9outdhweaIsntgruansscpioortgedrotuopthiensuFrlfoac-e m68inµimmuhmold. point,thereddashedcurveandredopencircle=s rence uinse1dseanndtrhoepldysteaxticohnaarnygaetabefitnwaleheonldcpoominptofonreunptstoo2f.5as. areforthex 30µmholdpoint,andtheblackdottedcurve s two-speTchireeseh87oRldbp-o4i1nKtswBeoresecognasisdemreidx:txure t1o4,i2n9d,uancde7B2EµCm. anFdibnlaaclklyc,rwos=esemsaernetfioornthaexrecen1t5eµxmpehroilmdpeonitntb.yTotghievegroup s s ≈ = in oneTohefsethweereceosmtimpaotnedenfrtosm[t8h6e]p.oinTtwhheeretwthoetrgaapsceosmpwleetreely oafnSidceharoefchkowatthIensnusrbfarcuecpko,siwtiohnoadffeecmtsotnhestrreamtaeidnintgheatofimrst ex- broughotpcenloesdeantdoadlleogfethneeraatocmysbwyerecoloosltintogt,heasftuerfracwe.hRicehfertehnece pneurmimbeer,ntwaelvparryodthuectsiuornfacoefpaosiBtioEnCbysole2l.5yµbmyfloarsethrecool- ± strengtmheoasfutrhemee4n1tKs retvreaaplepdinthgatptoetmepnetriaatlurwe-areslaatdediadbraifttisccaolluyld i6n8gµ[m89c]u.rvTe,hshisowfenaatstwheasgramyahdaeshepdoasrseiablineFbigy.2la.ser cooling increassehdif,tbthyeipnotsriotiodnucoifntgheasnurofapcteicbayludpiptool1e0pµomte,nhteinaclettohe onFaornaalrlrvoawlu-elsinoefwxisd,twhetorabsnesrivteioannoonftr8i4vSiarl,lorsessuclutrivneg; in a givenvaluesforx areapproximate;thisisthedominantsource thelossratesincreasetoamaximumasthecloudisbrought which the 87Rb compsonent was largely insensitive. In a low Doppler-limit temperature of just 350 nK. A “light- oferror.Theinitialcloudtemperatureswere130nKforx to the surface. Once the cloud reaches its final position, the s single-component system this would lead to an increase≈ shift”laserbeamwasintroducedatthecentreofthetrap 14µmandx 29µm,and140nKforx 72µm.These losses swiftly reduce. The transfer between these regimes is s s inthetemperature≈andleavethephase-spac≈edensityun- so that the atoms in that region no longer responded to affected. However, in the dual-species setup the 87Rb023614t-h3e laser cooling, after which an additional dimple trap cloud acted as a thermal reservoir, suppressing the tem- wasintroducedtoconfinetheatoms. Repeatedlycycling perature increase of the 41K component and causing it the dimple trap on and off resulted in the formation of to cross the BEC threshold. several condensates [89]. 7 D. Low-dimensional Bose systems and phase of (quasi-) one-dimensional systems can be realized in fluctuations elongated 3D traps [105]. In such a regime, the Bose gas behaves much as a conventional three-dimensional Bose condensate,exceptthatthecoherencelengthofthegasis The experiments described above were in the three- shorterthanthesystemextentalongthelongaxisofthe dimensional (3D) realm, in which long-wavelength phase trap. Astudyofcondensateformationinthisregimewas fluctuationsarestronglysuppressedawayfromthevicin- performedbytheAmsterdamgroupofWalraven[106]in ityofthephasetransition. Inlowerdimensionalsystems 2002 in an elongated 23Na cloud. Similarly to the MIT suchfluctuationsareenhanced,leadingtodramaticmod- experiment [43], they performed rapid quench cooling of ifications to the physics of the degenerate regime. In a their sample from just above the critical temperature. two-dimensional(2D)system,thermalfluctuationsofthe However, the system was in the hydrodynamic regime in phase erode the long-range order associated with true theweaklytrappeddimension,i.e.,themeandistancebe- condensation, leaving only so-called quasi-long-range or- tweencollisionswasmuchshorterthanthesystemlength. der characterised by correlation functions that decay al- gebraically with spatial separation [3]. A more complete It was argued that the system rapidly came to a lo- analysisrevealstheimportanceofvortex-antivortexpairs cal thermal equilibrium in the radial direction, resulting in this phase-fluctuating “quasi-condensed” regime [90]. in cooling of the cloud below the local degeneracy tem- Such pairs undergo a so-called Berezinskii–Kosterlitz– perature over a large spatial region and generating an Thouless (BKT) deconfinement transition at some finite elongated quasicondensate. However, the extent of this temperature, above which even quasi-long-range order is quasicondensatealongthelongaxisofthetrapwaslarger lost and superfluidity is extinguished. Two-dimensional than that expected at equilibrium, leading to large am- Bose systems are of particular interest due to their nat- plitude oscillations. The momentum distribution of the ural realisation in systems such as liquid helium films cloud was imaged via “condensate focussing”, with the andthefactthatthedegenerateBosequasiparticlessuch breadth of the focal point giving an indication of the as excitons and polaritons in semiconductor systems are magnitude of the phase fluctuations present in the sam- typically confined in a planar geometry. An insightful ple. This interesting experiment was somewhat ahead of overview of BKT physics can be found in the chapter by its time, with theoretical techniques unable to address Kim, Nitsche and Yamamoto in this volume. many of the nonequilibrium aspects of the problem. In 2007 the group of Aspect from Institut d’Optique There have been numerous experimental realisations also studied the formation of a quasicondensate in an of (quasi-)2D Bose gases in cold-atom experiments [91– elongated three-dimensional trap via continuous evapo- 96],withmostnotabletheobservationsofthermallyacti- rativecooling[107]inasimilarfashiontotheearlierwork vated vortices via interferometric measurements [91] and by Köhl et al. [56]. As well as measuring the condensate the direct probing of the equation of state and scale in- number, they also performed Bragg spectroscopy dur- variance of the 2D system [96] (see the chapter by Chin ing the growth to determine the momentum width and andRefs.[97–101]forrelatedtheoreticalconsiderations). hence the coherence length of the system. They found Further details and a lengthy discussion of the interplay that the momentum width they measured rapidly de- between BKT and BEC in homogeneous and trapped creased with time to the width expected in equilibrium systems can be found in Ref. [102]. Although theoreti- for the instantaneous value of the condensate number. cal works on the dynamics of such systems have existed Modelling of the growth of the condensate population for some time, little experimental work on the formation using the methodology of Ref. [57] produced results in dynamics of condensates in these systems has been un- good agreement with the experimental data, apart from dertaken (aside from the quasi-2D Kibble-Zurek works anunexplaineddelayof10–50ms,dependingontherate discussed in the following section). Considerable discus- of evaporation. sion is currently taking place regarding the emergence and nature of the BKT transition in driven-dissipative polariton condensates: experimentalists have observed evidence for quasi-long-range order [103, 104] (see Kim IV. CRITICALITY AND NONEQUILIBRIUM et al.’s chapter), but the nature of the transition and its DYNAMICS “nonequilibrium” features are topics of current debate [24, 25] (see also the chapter by Keeling et al.). As Bose–Einstein condensation is a continuous phase In one dimension, the effects of phase fluctuations are transition,thetheoryofcriticalphenomena[108]predicts even more pronounced, leading to the complete destruc- thatinthevicinityofthecriticalpointthecorrelationsof tion of long-range order and superfluidity at any finite the Bose field obey universal scaling relations. In partic- temperature. Manyexperimentswithcoldatomsinelon- ular,thescalingofcorrelationsatandnearequilibriumis gated “cigar-shaped” traps have investigated the physics governedbyasetofuniversalcriticalexponentsandscal- of such (quasi-) one-dimensional systems, though again, ingfunctions,independentofthemicroscopicparameters little work has been done on the formation dynamics of the gas. For a homogeneous system close to critical- of these degenerate samples. We note, however, that ity, standard theory predicts that the correlation length quasicondensate regimes somewhat analogous to those ξ, relaxation time τ, and first-order correlation function 8 G(x)=(cid:104)ψ†(x)ψ(0)(cid:105) obey scaling laws Although an important topic in its own right, the greatest significance of the equilibrium theory of critical ξ τ ξ = 0 , τ = 0 , G(x)=(cid:15)ν(d−2+η)F((cid:15)νx), (1) fluctuations to studies of condensate formation is that it |(cid:15)|ν |(cid:15)|νz provides a basis for generalisations of concepts such as critical scaling laws and universality classes to the do- with (cid:15) = T/T − 1 the reduced temperature, ν and z c mainofnonequilibriumphysics. Intheremainderofthis the correlation length and dynamical critical exponents, sectionwediscusstwosuchextensions: theKibble–Zurek η the scaling dimension of the Bose field, and F a uni- mechanism (KZM), and the theory of non-thermal fixed versal scaling function. The static Bose gas belongs to points. theXY (orO(2))universalityclass,andisthusexpected tohavethesamecriticalexponentsassuperfluidhelium, i.e., in 3D, ν (cid:39)0.67 and η =0.038(4) [109]. The critical A. The Kibble–Zurek mechanism dynamicsofthesystemareexpectedtoconformtothose of the diffusive model denoted by F in the classification of Ref. [110], implying a value z =3/2 for the dynamical The theory of the Kibble–Zurek mechanism leverages critical exponent. the well-established results of the equilibrium theory of Theinfluenceofcriticalphysicsissignificantlyreduced criticality to make immediate predictions for universal in the conditions of harmonic confinement typical of ex- scalingbehaviourinthenonequilibriumdynamicsofpas- perimental Bose-gas systems, as compared to homoge- sage through a second-order phase transition. The un- neous systems. Within a local-density approximation, derlying idea — that causally disconnected regions of the inhomogeneous thermodynamic parameters of the space break symmetry independently, leading to the for- system imply that only a small fraction of atoms in the mation of topological defects — was first discussed by gas enter the critical regime, and so global observables Kibble [121], who predicted that the distribution of de- arerelativelyinsensitivetotheeffectsofcriticality. Nev- fects following the transition would be determined by ertheless, a few experiments have attempted to observe the instantaneous correlation length of the system as it aspects of the critical physics of trapped Bose gases. passes through the Ginzburg temperature [122]. Zurek In a homogeneous gas the introduction of interparti- later emphasised [123] the importance of dynamic crit- cle interactions has no effect on the critical temperature ical phenomena [110] in such a scenario. In particular, at the mean-field level, but the magnitude and even the the scaling relations (1) imply that both the correlation sign of the shift due to critical fluctuations was debated length and the characteristic relaxation time of the sys- forseveraldecades(seeRef.[111]andreferencestherein) tem diverge as the critical point is approached ((cid:15) → 0), before being settled by classical-field Monte-Carlo calcu- imposing a limit to the size of spatial regions over which lations [112, 113]. An experiment by the Aspect group ordercanbeestablishedduringthetransition. Topologi- carefully measured a shift in critical temperature of the caldefectswillthusbeseeded,withadensitydetermined trapped gas, but was unable to unambiguously infer any bythecorrelationlengthatthetimethesystem“freezes” beyond-mean-fieldcontributiontothisshift[114,115]. A during the transition, and will subsequently decay in the laterexperimentbythegroupofHadzibabicmadeuseof symmetry-broken phase. The more rapidly the system a Feshbach resonance to control the interaction strength passes through the critical point, the shorter the corre- in 41K, and found clear evidence of a positive beyond- lation length that is frozen in, and therefore more topo- mean-field shift [116] (see also the chapter by Smith in logicaldefectswillform. Adimensionalanalysispredicts this volume). that a linear ramp (cid:15)(t) = −t/τQ of the reduced temper- In 2007 the ETH Zu¨rich group of Esslinger revisited ature through the critical point on a characteristic time their experiments on condensate formation and the co- scaleτQ resultsinadistributionofspontaneouslyformed herenceofathree-dimensionalBECwithanewtool: the defects with a density nd that scales as [124] ability to count single atoms passing through an optical cavity below their ultra-cold gas [117]. They out- n ∝τ(p−d)ν/(1+νz), (2) d Q coupled atoms from two different vertical locations from their sample as it was cooled, realising interference in where d is the dimensionality of the sample and p is the the falling matter waves. By monitoring the visibility of intrinsic dimensionality of the defects. the fringes, they were able to measure the growth of the Zurek initially described the KZM in the context of coherence length as a function of time. Using the same vortices in the λ-transition of superfluid 4He [123]. Al- optical cavity setup, the Esslinger group subsequently though vortices are observed in the wake of this transi- measured the coherence length of their Bose gas as it tion, it is difficult to identify them as having formed due was driven through the critical temperature by a small to the KZM rather than being induced by, e.g., inadver- backgroundheatingrate,anddeterminedthecorrelation- tent stirring [124] (see also the chapter by Pickett in this lengthcriticalexponenttobeν =0.67±0.13[118]. Their volume). The prospect of generating vorticity in atomic resultsareshowninFig.2(a). Classical-fieldsimulations BECsbymeansoftheKZMwasfirstdiscussedbyAnglin of their experiment were in reasonable agreement, deter- and Zurek in 1999 [125]. However, it was not until the mining ν =0.80±0.12 [119]. 2008experimentoftheAndersongroupattheUniversity 9 8 10 (a) (b) 7 6 2.0 m) 1 µ h ξ ( 5 m L n lengt 34 0.1 10-3 10-2 m{ H o ati el 2 1.0 orr C 1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.1 0.4 1.6 Reduced temperature (T-Tc)/Tc tQ s FIG. 2. Critical phenomena in BECs. (a) Divergence of the equilibrium correlation length ξ as a function of the reduced H L temperature, and the fitting of the critical exponent, giving the result ν =0.67±0.13. Inset: Double logarithmic plot of the same data. Taken from Ref. [118]. (b) Log-log plot of the dependence of the correlation length, here labelled (cid:96), as a function of the characteristic time τ of the quench through the BEC phase transition. The solid line corresponds to a Kibble–Zurek Q power-law scaling (cid:96)∝τb with b=0.35±0.04, in agreement with the beyond-mean-field prediction b=1/3 of the so-called F Q model [110] and inconsistent with the mean-field value b=1/4. This in turn implies a value z =1.4±0.2 for the dynamical critical exponent. Taken from Ref. [120]. ofArizona[126]thatspontaneouslyformedvorticeswere BECs is the inhomogeneity of the system in the exper- first observed in such a system (see also Ref. [127]). imental trapping potential, which is typically harmonic. From the point of view of a local-density approxima- The observations of spontaneous vortices in Ref. [126] tion, this inhomogeneity implies that the instantaneous were supported by numerical simulations using the coherence length and relaxation timescale are spatially stochastic projected Gross-Pitaevskii equation descrip- varying quantities, and that the transition occurs at dif- tion of Gardiner and Davis [128]. Their results are ferent times in different regions of space as the system shown in Fig. 3. This formalism is essentially a vari- is cooled. Following preliminary reports of the exper- ant of the Gardiner-Zoller quantum kinetic theory, ob- imental observation of dark solitons following the for- tained by making a high-temperature approximation to mation of a quasi-one-dimensional BEC by the group of the condensate-band master equation and then exploit- Engels at the University of Washington [134], Zurek ap- ing the quantum-classical correspondence of the Wigner plied the framework of the KZM to a quasi-1D BEC in representation to obtain a stochastic classical-field de- a cigar-shaped trap to estimate the scaling of the num- scriptionofthecondensateband[128,129]. Althoughde- ber of spontaneously generated solitons as a function of rived using different theoretical techniques, the resulting the quench time [135, 136]. Witkowska et al. [137] nu- description is similar to the stochastic Gross-Pitaevskii merically studied cooling leading to solitons in a com- equation of Stoof [18, 60], both in terms of its phys- parableone-dimensionalgeometry. Zurek’smethodology ical content and its computational implementation — for inhomogeneous systems was applied by del Campo see, e.g., discussion in Refs. [130, 131]. A related phase- et al. [138] to strongly oblate geometries in which vortex space method originating in quantum optics known as filaments behave approximately as point vortices in the thepositive-Prepresentationhasalsobeenappliedtoul- plane, an idealisation of the geometry of the experiment tracold gases [132]. This has been used to investigate of Weiler et al. [126]. cooling of a small system towards BEC by Drummond and Corney [133], who observed features consistent with Lamporesi et al. [139] recently reported the sponta- spontaneously formed vortices. Despite formally being neous creation of Kibble–Zurek dark solitons in the for- a statistically exact method, for interacting systems it mationofaBECinanelongatedtrap,andfoundthescal- tends to suffer from numerical divergences after a rela- ing of the number of observed defects with cooling rate tively short evolution time. ingoodagreementwiththepredictionsofZurek[135]. It Itseemslikelythatspontaneouslyformedvorticesand waslaterrealisedthattheapparentsolitonswereactually other defects were present in earlier BEC-formation ex- solitonic vortices [140]. The effects of inhomogeneity in periments, but not observed due to the practical diffi- such experiments can be mitigated by the realisation of culties inherent in resolving these defects in experimen- “box-like” flat-bottomed trapping geometries. The Dal- tal imaging — and indeed the fact that these experi- ibard groupin Paris has observed the formation of spon- ments were not attempting to investigate whether such taneous vortices in a quasi-2D box-like geometry, and structures were present. Another difficulty in identify- found scaling of the vortex number with quench rate in ing quantitative signatures of the KZM in experimental good agreement with the predictions of the KZM [141]. 10 b (b) 6 1 (aa) 5 0.8 y oms) 4 0.6 abilit (cc) 5N (10 at0 23 0.4 ortex prob V 1 0.2 d (d) 0 0 2 3 4 5 6 Time (s) FIG.3. SpontaneousvorticesintheformationofaBose–Einsteincondensate. (a)Squares: Experimentallymeasuredcondensate population as a function of time. Solid line: Condensate number from stochastic Gross–Pitaevskii simulations. Dashed line: probabilityoffindingoneormorevorticesinthesimulationsasafunctionoftime,averagedover298trajectories. Theshaded area indicates the statistical uncertainty in the experimentally measured vortex probability at t = 6.0 s. It was observed in experimentthattherewasnodiscerniblevortexdecaybetween3.5sand6.0s(b)Experimentalabsorptionimagestakenafter 59 ms time-of-flight showing the presence of vortices. (c) Simulated in-trap column densities at t=3.5 s [indicated by the left vertical dotted line in (a).] (d) Phase images through the z = 0 plane, with plusses (open circles) representing vortices with positive (negative) circulation. Adapted from C. N. Weiler et al. [126]. We also note further work by the Dalibard group [142] 87Rb-133Cs [147] Bose-Bose mixtures has also been ob- verifyingtheproductionofquench-inducedsupercurrents served. The competing growth dynamics of the two im- in a toroidal or “ring-trap” geometry [143] analogous to miscible components in the formation of such a binary the annular sample of superfluid helium considered in condensate have recently been investigated theoretically Zurek’s original proposal [123]. in the limit of a sudden temperature quench [148] (see Experimental investigations of the KZM in dilute also Refs. [149–151] for related critical scaling in other atomic gases have largely focused on the imaging of de- Hamiltonian quenches). These investigations indicate fects in the wake of the phase transition — either fol- the rich nonequilibrium dynamics possible in these sys- lowing time-of-flight expansion [126, 139, 141] or in situ tems, including strong memory effects on the coarsening [140]. However, the accurate extraction of critical scal- of spontaneously formed defects and the potential “mi- ingbehaviourfromsuchobservationsishamperedbythe crotrapping” of one component in spontaneous defects large background excitation of the field near the transi- formed in the other. tion, and the relaxation (or “coarsening”) dynamics of defects in the symmetry-broken phase. An alternative approachistomakequantitativemeasurementsofglobal B. Non-thermal fixed points properties of the system following the quench. Perform- ing quench experiments in a three-dimensional box-like A general characterisation of the relaxation dynamics geometry,theHadzibabicgroupinCambridge[120]made ofquantummany-bodysystemsquenchedfaroutofequi- careful measurements of the scaling of the correlation librium remains a largely open problem. In particular, length with quench time. From the measured scaling it is interesting to ask to what extent analogues of the law,theseauthorswereabletoinferabeyond-mean-field universal descriptions arising from the equilibrium the- valuez =1.4±0.2forthedynamicalcriticalexponentfor ory of critical fluctuations may exist for nonequilibrium this universality class. Some of the results of Ref. [120] systems. Arecentadvancetowardsansweringsuchques- are displayed in Fig. 2(b). tions has been made in the development of the theory of Thepossibilitiesforthetrappingandcoolingofmulti- non-thermal fixed points: universal nonequilibrium con- component systems in atomic physics experiments have figurationsshowingscalinginspaceand(evolution)time, naturally lead to investigations of the spontaneous for- characterised by a small number of fundamental prop- mation of more complicated topological defects during erties. The theory of such fixed points transposes the a phase transition. Although such experiments have so- concepts of equilibrium and diffusive near-equilibrium far largely focused on the formation of defects following renormalisation-group theory to the real-time evolution a quench of Hamiltonian parameters [144, 145], the for- ofnonequilibriumsystems. Thesedevelopmentsprovide, mationofnontrivialdomainstructuresfollowinggradual forexample,aframeworkwithinwhichtounderstandthe sympathetic cooling in immiscible 85Rb-87Rb [146] and turbulent,coarsening,andrelaxationdynamicsfollowing