Prethermalization and universal dynamics in near-integrable quantum systems
PPrethermalization and universal dynamics innear-integrable quantum systems
Tim Langen
JILA, University of Colorado and NIST, Boulder, CO 80309, USAE-mail: [email protected]
Thomas Gasenzer
Kirchhoff-Institut f¨ur Physik, Ruprecht-Karls-Universit¨at Heidelberg,Im Neuenheimer Feld 227, 69120 Heidelberg, GermanyExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur SchwerionenforschungGmbH, Planckstraße 1, 64291 Darmstadt, GermanyE-mail: [email protected]
J¨org Schmiedmayer
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien,Stadionallee 2, 1020 Wien, AustriaE-mail: [email protected]
12 May 2016
Abstract.
We review the recent progress in the understanding of the relaxation ofisolated near-integrable quantum many-body systems. Focusing on prethermalizationand universal dynamics following a quench, we describe the experiments with ultracoldatomic gases that illustrate these phenomena and summarize the essential theoreticalconcepts employed to interpret them. Our discussion highlights the key topics that linkthe different approaches to this interdisciplinary field, including the generalized Gibbsensemble, non-thermal fixed points, critical slowing and universal scaling. Finally,we point to new experimental challenges demonstrating these fundamental features ofmany-body quantum systems out of equilibrium. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l rethermalization and universal dynamics in near-integrable quantum systems Contents1 Introduction 22 Prethermalization, non-thermal fixed points and universal dynamics 6
The relaxation of isolated quantum many-body systems is a fundamental unsolvedproblem connecting many different fields of physics. Examples range from the dynamicsof the early universe and quark-gluon plasmas to coherence and transport in condensedmatter physics and quantum information. Consequently relaxation and non-equilibriumdynamics encompass a wide range of phenomena on vastly different energy, length, andtime scales.In the last years, rapid progress in the field was triggered by experiments withultracold quantum gases [1], which allow probing of the fundamental processes and timescales of these phenomena [2–22]. These experiments opened up a new path to testingtheoretical concepts, revealing their implications for realistic systems.In the experiments, an initially prepared, trapped gas is suddenly ‘quenched’ out ofequilibrium. This is typically achieved by a parameter change in the trapping potentialor the interaction properties.
Relaxation , i.e., the evolution to a new (quasi-) stationarystate is subsequently probed through temporally and spatially resolved measurementsof suitably selected observables. Central questions that have been studied range from rethermalization and universal dynamics in near-integrable quantum systems non-thermal steady states and the microscopic dynamics that establishthese states to the emergence of aspects of macroscopic classical statistical mechanicsfrom the microscopic unitary quantum evolution. A fundamental problem thereby ishow unitary evolution, which conserves the von Neumann entropy S = − Tr[ ˆ ρ ln ˆ ρ ] ofthe full quantum state ˆ ρ of an isolated system can lead to relaxation and thermalization (the approach of a steady state well described by a Gibbs ensemble).In the establishment of (quasi-)steady states conservation laws play a key role.Consequently, isolated many-body systems (approximately) described by quantumintegrable models with a large number of conserved quantities are of particular interest.Ultracold atomic gases are ideally suited to realize and probe such systems. In particular,they allow studying different mechanisms that break the integrability. This opens theunique possibility to investigate the intricate relationship between thermalization andintegrability in quantum systems. Emergence of macroscopic physics: The generalized Gibbs ensemble.—
Informationtheory provides a particularly powerful approach [23] to studying such steady states.As pointed out by Jaynes in the 1950s, once one realizes that thermodynamic entropyand information entropy are the same concept, one may take entropy as the startingpoint and consider statistical mechanics as a form of statistical inference [24, 25].From this point of view, the maximum-entropy state takes the central role, being theleast biased estimate possible on given information, or the maximally non-committalconfiguration with regard to missing information [24]. The maximum-entropy principleleads directly to the standard thermodynamical ensembles, which are constrained onlyby quantities like energy or particle number [26]. However, as highlighted by thework of Rigol and collaborators [27], it has recently become clear that the late-timebehavior of isolated integrable quantum many-body systems is fundamentally differentfrom that of non-integrable ones [28–32]. Further conservation laws constrain thedynamics of integrable systems and thus, appropriately generalized Gibbs ensembles (GGE) are expected on the grounds of the maximum-entropy argument [24, 25]. TheseGGEs were indeed observed in experiment, in the relaxation of a 1D quantum gas[9]. When deriving statistical ensembles from the microscopic physical propertiesone usually has to complement the basic equations of motion with further physicalassumptions, in particular ergodicity. Jaynes’ statistical, information-based approachis, however, independent of such assumptions [24] and, hence, it is applicable not onlyto the special subclass of equilibrium states but also to dynamical evolution processes[25, 27, 28, 33, 34]. The questions arising in the above context have been quickly takenup by experimenters [22], as will be discussed in detail in Sect. 3.
Microscopic relaxation processes: Prethermalization.—
Given the new types ofstationary states discussed above, it is also important to better understand the relevantmicroscopic processes leading to these states. However, a general framework for therelaxation dynamics of isolated quantum many-body systems quenched far out ofequilibrium largely remains an open problem. rethermalization and universal dynamics in near-integrable quantum systems thermalfixedpointprethermalization,drift to equilibrium directthermalization differentinitialconditions Figure 1.
Schematics of prethermalization [35, 36] (left panel, adapted from [37])and a non-thermal fixed point [38] (NTFP, right panel, adapted from [39]), basedon the ideas of a renormalization group flow. The left panel depicts possible timeevolutions in a two-dimensional projection or sub-manifold of the space of many-bodystates. The arbitrarily chosen axes represent running ‘coupling’ parameters. Theseare related, e.g., to the various Lagrange parameters of a generalized Gibbs ensemble(GGE, cf. Sects. 2.1.2 and 3.3.4), of which all but the Gibbs-ensemble parametersvanish in the thermal state (green ‘thermal fixed point’). Hence, the space of suchtrajectories is in general not restricted to the two dimensions chosen in the sketch.A near-integrable system quickly approaches a prethermalized state (red line), which,owing to conservation laws, retains memory of the initial conditions and then slowlydrifts to the thermal fixed point. Depending on the particular choice of initial condition(e.g. quench), different, non-universal early-time evolutions (blue trajectories) occurbefore the system starts to show universal behavior. In contrast, in non-integrablesystems, direct thermalization on a single time scale as in kinetic damping is thegeneric pathway. The red drift line can be regarded as a partial fixed point which isapproached quickly from generic initial conditions and where the system experiencescritical slowing down. Along the red line, correlation functions C ( p, t ) can show scalingbehavior in space and time, e.g., C ( p, t ) = t α f ( t β p ), with a universal scaling function f (cf. also Fig. 3). In the case of prethermalization to a GGE in a (near-)integrablesystem, one expects α = β = 0 ( (cid:39)
0) such that the system gets (almost) stuck whenreaching the red line. Instead of ending at the stable thermal fixed point, the red linecould also lead to a partially stable NTFP. In this case, the blue ‘direct thermalization’lines are replaced by trajectories leading away from the NTFP, e.g., towards a differentfinal thermal fixed point (see also sketch in the right panel). In this situation, moregeneral scaling with α , β (cid:54) = 0 is expected on the red line, where the algebraic timeevolution nonetheless means slowing down of the evolution (cf. arrows in right panel). The key question is which types of non-trivial ‘pathways’ such systems can, ingeneral, take under the influence of conservation laws. In particular for integrable andnear-integrable quantum systems, one expects special relaxation characteristics. It isthen especially interesting to find those degrees of freedom which need to be taken intoaccount when performing a maximum-entropy analysis of the relaxed states. Generically,relaxation within a simple kinetic framework is expected to be described by exponentiallaws in time, with a rate defining the time scale. In contrast, (near-) integrable systems rethermalization and universal dynamics in near-integrable quantum systems
Universal (scaling) dynamics and non-thermal fixed points.—
Prethermalizationhas first been proposed on the basis of ideas of a renormalization-group flow [35, 36,40, 41], see Fig. 1 for an illustration. In the wider context of relaxation dynamics,‘prethermalization’ means the approach of any (partially) universal intermediate statewhich is still out of equilibrium with respect to asymptotically long time evolution.The state being universal means that it can be described mathematically in terms of alimited set of parameters and/or functions which only depend on a corresponding set ofsymmetries obeyed in the dynamical evolution of the state. These characteristics are,however, independent of the particular physical realization and of the specific initialconfiguration resulting from the quench. If the state is only partially universal thedynamics is dominated by the universal characteristics while non-universal propertiesremain depending, e.g., on the particular initial state (also the final thermal state beingpartially universal in this respect).As an example, during the prethermalization stage, measurable correlations couldbe well reproduced by assuming the system to be described by a GGE containing alimited set of conserved operators. These operators are directly related to intrinsicsymmetry properties of the system’s Hamiltonian, while, in contrast, the values of theLagrange multipliers pertaining to the conserved quantities are non-universal as theydepend on values of these quantities in the initial state.Another example is prethermalization to an algebraically-slowly evolving state. Forinstance, phase-ordering kinetics arising after a quench across a symmetry-breakingphase transition could be described in terms of universal distribution functions showingself-similar coarsening evolution in space and time [42]. The same is true for wave-turbulent kinetics where the self-similar evolution appears in the form of cascadingtransport of e.g. energy between different scales [43], similar to classical fluid turbulence.Besides slowing-down, the property common to these evolutions is scaling behavior,with evolution time as scaling parameter. This scaling is reminiscent of equilibrium rethermalization and universal dynamics in near-integrable quantum systems
2. Prethermalization, non-thermal fixed points and universal dynamics
We start by reviewing the theoretical basis for discussing relaxation of quantum systemsdescribed by (near-)integrable models. Important concepts are the GGE resultingfrom a pure statistical approach, as well as, on the microscopic dynamical level,prethermalization, the approach of non-thermal fixed points, and universal (scaling)dynamics. See also other contributions to this volume [53–60] for more detaileddiscussions of the theoretical background.
Integrability and its consequences have beencrucial for understanding the process that inhibit or allow thermalization in classicalmechanics. In one of the first numerical experiments, Fermi, Pasta, and Ulam studiedthe evolution of a chain of harmonic oscillators with non-linear couplings therebyobserving quasi-periodic instead of ergodic behavior. This surprising result can beascribed to the integrability of the model, which allows for quasi-periodic motion insteadof thermalization [61]. Integrability of classical model systems results from the existenceof a full set of conserved quantities, which restrict the evolving system to a toroidalsub-region of the total phase space. The above findings have ultimately led to thedevelopment of chaos theory, which became the basis for the understanding of classicalthermalization [62].In quantum systems the meaning of integrability is less clear [63]. Differentdefinitions that have been used include the existence of a complete set of algebraicallyindependent mutually commuting conserved operators [64] (with the flaw that it is,according to the spectral theorem, fulfilled by any hermitian Hamiltonian as one could rethermalization and universal dynamics in near-integrable quantum systems N point-interacting bosons ona line [68, 69], the one-dimensional Hubbard model [70], and variants of the Heisenbergmodel [66]. Note, however, that the above criteria are not one-to-one identical implying,e.g., that non-integrable models can nonetheless be analytically solvable [71]. As willbe discussed below, in the context of prethermalization, it is the number of spatially(quasi-)local conserved charges which appears to matter.The question whether and how quantum systems thermalize has regained muchimpetus through ultracold atomic gases described by (near-)integrable mathematicalmodels. A first explicit demonstration of a strongly interacting one-dimensional Bosegas not showing any thermalization over a period corresponding to many collision timeswas delivered in the experiment by Kinoshita et al. [2]. Their system is in principledescribed by the integrable Lieb-Liniger (LL) model [68, 69]. In the specific experimentalenvironment, it is, however, expected to become weakly non-integrable due to thepresence of the transverse degrees of freedom and the additional longitudinal harmonictrapping potential. Details of this experiment are discussed in Sect. 3. Previous studieshad demonstrated the applicability of a description in terms of the hard-core or Tonks-Girardeau limit [72] of the LL model [73, 74]. In this limit, the model maps to a system ofnon-interacting spin-less fermions, which is analytically solvable and due to the absenceof scattering does not thermalize. These experiments sparked a new research direction,asking whether the time evolution under the constraint of conserved quantities could atall lead to a thermalized final state [27–29]. Despite following a strictly unitary evolution,generic isolated, non-integrable quantum systems are now expected to exhibitthermalization in accordance with classical experience. If only the total energy isconserved, one observes chaotic behavior and thermalization, i.e., local observablesrelax to a canonical or Gibbs ensemble with an effective temperature [27–32, 75–93]. Acommon picture thereby is that a finite region of a system is thermalized by interactionwith the rest acting as a bath. The question, however, why a non-integrable quantumsystem, in contrast to integrable ones, can be successfully described by such a smallnumber of conserved quantities is a still a matter of intense research.Despite the lack of a unique definition of integrability, it is now generally acceptedthat in the quantum case, as in the classical, additional conserved quantities mayslow down if not even inhibit thermalization. However, even in the absence of strictthermalization, relaxation and the emergence of thermal-like properties are still possible.Reviving the statistical arguments by Jaynes [24, 25], Rigol et al. demonstrated, for the rethermalization and universal dynamics in near-integrable quantum systems | Ψ(0) (cid:105) of atranslationally invariant system described by an integrable model with Hamiltonianˆ H . Then, stationary n -point correlation functions of local operators ˆ O a ( x ) in thethermodynamic limit are given by a generalized Gibbs ensemble (GGE),lim t →∞ (cid:104) Ψ( t ) | n (cid:89) a =1 ˆ O a ( x a ) | Ψ( t ) (cid:105) = Tr (cid:20) ˆ ρ GGE n (cid:89) a =1 ˆ O a ( x a ) (cid:21) , (1)with | Ψ( t ) (cid:105) = exp( − i ˆ Ht/ (cid:126) ) | Ψ(0) (cid:105) andˆ ρ GGE = 1 Z exp (cid:16) − (cid:88) m λ m ˆ I m (cid:17) . (2)Here ˆ I m denotes a full set of conserved quantities, Z = Tr exp( − (cid:80) m λ m ˆ I m ) is thepartition function and m is a positive integer. A separate Lagrange multiplier λ m is associated with each of the conserved quantities. These numbers are obtained bymaximisation of the von Neumann entropy S = − Tr[ ˆ ρ GGE ln ˆ ρ GGE ], under the conditionthat the expectation values of the conserved quantities are fixed to their initial valuesin the thermodynamic limit,Tr[ ˆ I m ˆ ρ GGE ] = (cid:104) ˆ I m (cid:105) ( t = 0) . (3)The emergence of such a maximum-entropy state is not in contradiction to a unitaryevolution according to quantum mechanics [25]. It rather reflects that the true quantumstate is indistinguishable from the maximum-entropy ensemble with respect to a set ofobservables [30]. It was pointed out in [94, 95] that for the GGE conjecture to hold itis essential to restrict the observables to finite subsystems. Hence, the GGE describesthe long-time limit of sufficiently local observables.The GGE is a direct generalization of the well-known thermodynamical ensembles.In cases where only the total energy and the particle number are conserved, it reducesto the grand-canonical ensemble, where temperature ( λ = β = 1 /k B T ) and chemicalpotential ( λ = − βµ ) play the role of the respective Lagrange multipliers [26]. Weemphasize that in the cases described here, in most of the basic examples, there isonly one species of undistinguishable particles such that there is no natural numberconservation arising from intrinsic particle properties. The conserved quantities can,e.g., measure functions of the fractions of particles in different collective motionalstates (e.g. quasiparticle momentum modes). More generally, they can also be given bycorrelations of higher order in some fundamental field describing the system’s properties.A question not yet fully resolved is precisely which conserved quantities ˆ I m needto be included in the definition of the GGE. The current belief is that they should beconstructed out of local or quasi-local conserved operators [53, 57], i.e. ˆ I m = (cid:80) j i m ( j ), rethermalization and universal dynamics in near-integrable quantum systems i m ( j ) acts non-trivially onlyaround the site j if the charge is local, while it has exponential tails if it is quasi-local.Consider the case of a free theory, in which Fock mode operators ˆ a p , ˆ a † p diagonalizethe Hamiltonian, ˆ H = (cid:80) p ω p ˆ a † p ˆ a p , with mode frequencies ω p . Conserved quantities maythen be written in the additive form ˆ I m = (cid:80) p f ( m ) p ˆ a † p ˆ a p with a set of scalar functions f ( m ) p . (Note that in free models, the above local charges are linearly related to the modeoccupation operators ˆ a † p ˆ a p such that either of them can be used to define the GGE, withEq. (3) applying to the densities.) While total energy and particle number are obtainedwith f p = ω p and f p = 1, respectively, further conserved quantities of the free theoryare, e.g., simply the occupation numbers themselves, f ( m ) p = δ pm . In this case, timeevolution is identical to dephasing, leading to stationary values of operators on finitesubsystems.In the case of interacting theories, however, the question of relevant conservedquantities is not yet fully settled [96]. The question which are the relevant conservedquantities has been discussed intensely [97–102], in particular the question whetheror not quasi-local conserved operators are required in the construction of GGEs[53, 57, 103–105].The GGE is now believed to be the final state of relaxation for generic quantumintegrable systems [27, 30–34, 53–57, 60, 87, 93–95, 97–133]. These studies focusedon many different models and theories, including the Luttinger model [55, 106–108], hard-core bosons in one dimension [27, 29, 109–112], conformal field theories[54, 87, 93], the infinite-dimensional Falicov-Kimball model [113], the Lieb-Liniger model[56, 98, 102, 114–120], the quantum Ising chain in a transverse field [57, 87, 103, 108, 121–125], and Heisenberg spin chains [53, 95, 97, 99–102, 104, 108, 126–128]. We note thatwithin the conditions set by integrability, various effective ‘temperatures’ have beenproposed on the basis of static [86, 87, 134, 135] and dynamic [136, 137] properties.These definitions, however, in general depend on the details of the initial conditions.We note that ergodicity may also be broken due to many-body localization [138–140]. Hence, generalising the concept and taking into account the influence of theexternal potential, the GGE has also been suggested as a description for such many-body localized states [58, 140].As will be discussed in detail in Sect. 3, dynamics, relaxation and prethermalizationof near-integrable quantum systems have been studied extensively in experiment [2, 4,5, 8, 9, 17, 141, 142]. These experiments immediately lead to the important questionwhat happened if certain quantities were only approximately conserved. It has beenconjectured that in this case an isolated system will first relax to a metastable statedescribed by a GGE [143–158]. In the context of near-integrable systems, this behaviorhas been related to prethermalization (cf. Sect. 2.2 below; it can also be regarded asan alternative definition of prethermalization). Eventual thermalization happens, incontrast, on a much longer time scale [159]. It thereby remains unclear in which wayand how far integrability has to be perturbed to allow for actual thermalization [157]. rethermalization and universal dynamics in near-integrable quantum systems (a) (c) O cc upa t i on nu m be r pe r m ode Time [m -1 ] 500 100 50p=0.32 m ABp=0.78 m ABp=1.38 m AB
2p [m]0 A Initial fermion distribution B t [s] -6 -5 -4 -3 -2 -1 n ( t ; p i ) / n L p i i = 0i = 32 (b) (d) -1 ] E qua t i on o f s t a t e w = p / (cid:161)
500 100 50 -1 ] w = p / (cid:161) h = 1.0h = 0.5 p [1/ L ] (cid:79) [ n K ] t t t (cid:105) = (cid:105) x t t t [s]0200400 e n e r g y / n L [ s - h / (cid:47) ] total en.interaction energykinetic en. Figure 2.
Prethermalization. (a) Time evolution of Fermion occupation number n ( f ) ( t ; p ) in a low-energy quark-meson model, for three different momentum modes asa function of time. The evolution is shown for two different initial conditions with the same energy density. (b) The ratio of pressure p over energy density w as a functionof time. The inset shows the early stages for two different couplings and demonstratesthat the prethermalization time is independent of the interaction details. Adapted from[36]. Note the logarithmic scale for times t ≥ m − . (c) Time evolving momentum( p ) spectrum of a uniform dilute 1D Bose gas after an interaction quench heatingthe gas, shown for the lowest 32 modes. A fast short-time kinetic prethermalizationperiod is followed by a long quasi-stationary drift to the final equilibrium distribution.(d) Momentum and time dependent temperature variable Θ( t ; p ) obtained by fittingthe distribution (c) to n ( t ; p ) = [exp { ( (cid:126) ω ( p ) − µ ) /k B Θ( t ; p ) } − − , at differenttimes t ν , with chemical potential µ ensuring ω ( p ) to be gapless. As long as Θ( t ; p )strongly depends on p , a generalized Gibbs ensemble applies. Only at very large times,Θ represents the temperature of the sample. Note that the chosen approximationintroduced an integrability breaking effect. The inset shows the energy contributionsbecoming stationary after prethermalization. Adapted from [85]. This question is of strong conceptual interest, in particular as the related problem hasbeen very well studied in classical mechanics. In that case, the famous Kolmogorov-Arnold-Moser (KAM) theorem quantifies the effect of a weak non-integrability on thedynamics [30]. No corresponding theorem is known in the quantum case [154].
The statistical maximum-entropy approach does not answer the fundamental question,how the respective many-body states are actually reached given a certain microscopic rethermalization and universal dynamics in near-integrable quantum systems
Kinetic prethermalization sets in very rapidly as a consequence of the loss ofphase information, i.e., of coherence between modes with different eigenfrequencies,see Figs. 2a, c. This dephasing is independent of the non-linear interactions betweenquasiparticles. Defining a kinetic temperature in terms of the total mean kinetic energy,it is found that this temperature can take on its near-thermal value already after kineticprethermalization, see inset of Fig. 2d. As the total energy is conserved, this is also truefor the potential energy stored in the interactions between particles (Figs. 2b, d).(II)
Loss of details of the initial conditions occurs on a somewhat larger time scale,which, however, can still be much smaller than the final equilibration time. Duringthis period, different initial conditions lead to transient states which show similarcorrelations, provided conserved quantities such as energy and particle number are thesame. This applies, e.g., to mode occupation numbers of quantum gases, cf. Fig. 2a.These need to be redistributed, which takes longer than dephasing, but not yet the longtime needed to assume their final thermal values. rethermalization and universal dynamics in near-integrable quantum systems ω ( p ) is known, one may define momentum-dependent mode ‘temperatures’ Θ( t ; p ).This is done by fitting the transient correlations such as the time-dependent single-particle momentum distribution to their expected final canonical form. For example, n ( t ; p ) = { exp[( (cid:126) ω ( p ) − µ ) /k B Θ( t ; p )] − } − for an ideal Bose system, see [35, 36] andFig. 2d, also for parameter definitions. Then, a momentum dependence of Θ( t ; p ) signalsthat final thermalization has not been reached yet. Such a non-thermal distribution isequivalent to the system being described by a GGE.(III) Thermalization or long-time equilibration leading eventually to detailedbalance [26, 160, 161]. This requires a suitable collisional redistribution within theparticle spectra and can take much longer than the first two steps. It can also bepartially or completely inhibited in integrable quantum systems due to the large numberof conserved quantities, and only allowed through weak integrability breaking effects.If the system consists of more than one species of particles, chemical equilibration candefine another time scale in the overall process [36, 162]. In the example shown inFig. 2c, stages II and III coincide.As compared to the near-integrable systems discussed in Sect. 2.1.1, prethermaliza-tion in the above high-energy-physics examples seemingly does not rely on integrabilityof the underlying model. However, the same principles apply there. The near-integrabledynamics is given by the early-time mean-field dephasing evolution of the interactingquantum fields, cf. also [143–158]. The fast time scale of dephasing is set by the slowestin the spectrum, i.e., by the inverse rest mass. As long as dephasing dominates theweak non-linear interactions between quasiparticles, or, e.g., the thermalizing processeshave not yet reached resonant amplification [36], the dynamics is nearly Gaussian andthus nearly integrable. This implies that the occupation number of any relevant fieldmode which is set to its value by the initial quench is an approximately conserved quan-tity. Note that in more than one spatial dimension, isotropization may play anotherimportant role within the early evolution starting from anisotropic initial conditions[163].Studies of the non-linear Schr¨odinger model of one-dimensional dilute Bose andFermi gases by means of non-perturbative functional-integral techniques have shown asimilar separation of time scales [83, 85, 164]. They clearly showed kinetic pretherma-lization as an early-time dephasing period. After this dephasing, the total kineticand potential energies were found to be of approximately equal size and close to theirfinal values. This behavior is reminiscent of the virial theorem for harmonic-oscillatortype potentials and thus provides an additional signature that the system during theprethermalization stage essentially represents a series of decoupled harmonic oscillators.First experimental studies of prethermalization were performed by Gring et al. [5],see also [6, 8, 9, 22, 141, 142], splitting a one-dimensional condensate into two copies andmeasuring differences in the subsequent dynamics of the two halves. Details of these rethermalization and universal dynamics in near-integrable quantum systems l , become stationary at atime on the order of t ∼ l/ c when the region fully falls into the light cone with apexcentered at t = 0. This ‘horizon effect’ reflects the fundamental Lieb-Robinson bound[167] prescribing that information can spread only at a finite group velocity c . The effectwas studied numerically [168, 169] and in experiment [7, 15].In a different context, prethermalization has been potentially realized in dynamicswhere magnetisation domains and defects are being created following a quench ina low-dimensional spinor Bose-Einstein condensate [10–12, 170, 171]. This type ofprethermalization [172] will be further discussed in the context of non-thermal fixedpoints and universal dynamics in Sects. 2.3 and 3.5. The observation and description of prethermalization and GGEs leads to the questionto what extent these phenomena are universal. It can be argued that they can bedescribed within the more general framework of non-thermal critical states arising withina renormalization-group approach to time evolution, see Fig. 1 for an illustration.These ideas draw from the concept of universal critical scaling of correlationfunctions in equilibrium, which has been very successful in classifying and characterisingmatter near continuous phase transitions [44–47]. Within the renormalization-groupapproach one basically looks at a physical system through a microscope and comparesthe pictures seen at different resolutions with each other, i.e., at different magnificationsof the lens. When looking for critical scaling, one takes the correlations in, e.g., thespatial patterns seen at a particular resolution and continuously changes the magnifica-tion of the lens, watching how the correlations change. Near the phase transition, rethermalization and universal dynamics in near-integrable quantum systems s ,to different spatial scales causes the correlation function C ( x ; s ) to be rescaled accordingto C ( x ; s ) = s ζ f ( x/s ). Hence, whatever the resolution s is, the correlations can bewritten solely in terms of the universal exponent ζ and the universal scaling function f .If the above ’renormalization-group flow’ of correlations under a change of the scaleparameter s does not change C by any means, the scaling function f must have reacheda pure power-law behavior f ( x ) ∼ x ζ as is seen from the above scaling relation. Thissituation is called a fixed point of the flow.These ideas have been extended to the time evolution from some non-equilibriuminitial state, where the time t takes the role of the scaling parameter s . Thereby,increasing time t can mean both, reducing the microscope resolution (resolvingincreasingly larger spatial scales) or increasing it. A non-thermal fixed point is afixed point as defined above, in a time-evolution scenario where t takes the role of s . Functional renormalisation-group approaches for describing this real-time scalingevolution were presented in [173–175].In realistic physical situations, a fixed point is reached only approximately suchthat C ( x ; s ) = s ζ f ( x/s ) holds but f is not a pure power-law, i.e., it retains informationof scales such as a correlation length ξ . Near a non-thermal fixed point, one has C ( x, t ) = t α f ( t − β x ), with two universal exponents α and β . Hence, the correlationlength would change as a power of time, ξ ( t ) ∼ t β . The time evolution taking power-lawcharacteristics is equivalent to critical slowing down.Berges, Rothkopf, and Schmidt proposed that in the reheating of the post-inflationary universe non-thermal fixed points could arise, which excessively delayedthermalization [38, 174]. They manifest themselves in the above universal scalingbehavior in time [176], see also [51, 52] for more recent reviews.The theory of non-thermal fixed points extends the concepts of equilibrium anddiffusive near-equilibrium renormalization-group theory (see [47] for a seminal review)to the real-time evolution of far-from-equilibrium systems. Critical scaling phenomena inspace and time are strongly reminiscent of turbulence in classical fluids [43, 177] as wellas superfluids [178, 179]. For example, according to the seminal theory of Kolmogorov,eddies created in a fluid break down into successively smaller eddies until they becomeof the size set by dissipation of kinetic energy into heat. This energy cascade to smallerand smaller scales builds up a non-equilibrium steady state.The concept of non-thermal fixed points also naturally includes relaxation dynamicswhich exhibits coarsening and phase-ordering kinetics [42] following the creation ofdefects and nonlinear patterns, e.g., in a quench across a phase transition. In thefollowing, this is illustrated by an example of a non-thermal fixed point approachedafter a quench in an ultracold Bose system. The example shows in particular thatthere are more general prethermalization phenomena expected beyond the realm ofnear-integrable quantum systems in one spatial dimension. rethermalization and universal dynamics in near-integrable quantum systems Synthetic · Zagreb · 30 Sep 2015
Now:
Strong cooling quench! l og n k log k particles removed by cooling quench n ( k , t ) ~ t α f ( t β k ) n ( k , t ) ~ t α ' f ' ( t β ' k ) initial distribution after quench k ξ n Q ≈ ( ρa ) -1/2 Q n ( k , t ) ~ t α f ( t β k ) Figure 3.
Self-similar scaling in time and space close to a non-thermal fixed point.The sketch indicates the evolution of the single-particle radial number distribution n ( k, t ) as function of momentum k of a Bose gas for two different times t (solidlines). Sketch after Ref. [176]. Starting from the extreme initial distribution n ( k, t )(dashed line) produced, e.g., by a strong cooling quench, a bidirectional redistributionof particles in momentum space (arrows) occurs. This builds up a quasicondensatein the infrared while refilling the thermal tail at large momenta. The particletransports towards zero as well as large momenta is characterized by self-similarscaling evolution in space and time, n ( k, t ) = ( t/t ) α n ([ t/t ] β k, t ), with characteristicscaling exponents α , β , in general different for the two directions. The infraredtransport (blue arrow) conserves particle number quasi-locally in momentum spacewhile energy is conserved in the redistribution of short-wavelength fluctuations (redarrow). Note the double-logarithmic scale. In a 3D dilute Bose gas with density ρ ,particle mass m , and s-wave scattering length a , the condition for an initial statewhich allows approaching the non-thermal fixed point is ( (cid:126) Q ) / m (cid:39) | µ | (cid:39) gρ . Here, g = 4 π (cid:126) a/m , i.e. Q (cid:39) k ξ = (8 π (cid:126) aρ ) / is on the order of the inverse healing length.Hence, if no significant zero-mode occupation n ( k = 0 , t ) is present, the respectiveoccupation number at Q is on the order of the inverse of the diluteness parameter, n ( Q, t ) ∼ ( ρa ) − / . Consider the dynamics following a strong cooling quench in an ultracold Bosegas, leading ultimately to the formation of a Bose condensate, see Fig. 3 as well as[39, 176, 180, 181]. Qualitatively this proceeds as follows. In a closed system, a coolingquench, removing particles with the highest energies, quite generically leads to a non-equilibrated particle distribution n ( k, t ) (dashed line in Fig. 3). This distribution, belowsome energy scale ε , exceeds the thermal equilibrium occupation number determinedby the mean energy per mode [182–185]. Energy and momentum conservation thenimply a bi-directional redistribution of particles: while a few particles are scattered tohigh-momentum modes and carry away a large fraction of the excess energy associatedwith this over-occupation, the majority of the particles is scattered to lower momenta[39, 176, 180]. The redistribution is indicated by the arrows in Fig. 3.Semiclassical simulations of the Gross-Pitaevskii model in three dimensions showedthat this behavior is associated with the creation, dilution, coarsening and relaxation of a rethermalization and universal dynamics in near-integrable quantum systems n ( k, t ) to larger spatial scales(i.e. smaller momenta) reflects the decay in the number of defects [189–192] and thusthe increase in mean inter-vortex distances. This diluting ensemble of vortices can beconsidered as a type of superfluid (quantum) turbulence.The resulting inverse transport is described by characteristic non-thermal scalingfunctions, separately in the low- and high-energy regions, as sketched in Fig. 3. Theseshift self-similarly as time proceeds (note the double-log scale). For the momentum-space occupation number the scaling relations in time and space translate to n ( k, t ) =( t/t ) α n ([ t/t ] β k, t ), with universal exponents α and β . For a dilute Bose gas in d = 3dimensions, possible scaling exponents have recently been numerically determined to be α = 1 . β = 0 . α = βd , β = 1 / (2 − η ), when assuming η (cid:39) z = 1 /β . Different such exponents, notcontained in the classes considered usually [42] are found at later times of the vortex-dilution process [189], indicating that the concept of non-thermal fixed points can leadbeyond usual phase-ordering kinetics.Non-thermal fixed points [38, 174] were discussed in various contexts, includingstrong wave turbulence in low-energy Bose gases [193], in relativistic scalar models[194, 195], as well as abelian [196] and non-abelian gauge theory [197] They werefurthermore related to classical Burgers turbulence [175]. Early proposals of non-thermalfixed points can be found in [40, 41].Besides vortices in 3D gases, other types of (quasi-) topological and non-topologicalbut strongly non-linear excitations can play a role in the manifestation of the scaling. Forexample, solitons can form ensembles of nonlinear quasiparticles in a one-dimensionalsystem [198]. In two spatial dimensions, Onsager-type [199] ensembles of logarithmicallyinteracting vortices and antivortices determine the behavior near the non-thermal fixedpoint [189]. The latter type of systems has recently been explored extensively inexperiment [200–202] and theory [203–213] investigating the role of quantum vorticesin 2D dynamics and studying connections with aspects of two-dimensional classicalturbulence. Further examples of non-thermal fixed point scaling have been discussedfor pseudo-spin and spinor gases [214–217], holographic superfluids [218], and gaugesystems [196, 219]. The latter indicate that such excitations may also be present in the rethermalization and universal dynamics in near-integrable quantum systems α and β are close to zero such that there is no further time evolution of the scale. When thesystem finally thermalizes, it departs from the fixed point, and the time evolution canbecome non-universal and non-scaling.Dynamics near non-thermal fixed points goes beyond the mere dephasing of thefundamental quasiparticle modes as discussed in the previous sections. Non-thermalfixed points address a wider spectrum of prethermalization phenomena: Broadlyspeaking, in the cases where non-linear excitations are involved, the degrees of freedomwhich become independent and decorrelate in the prethermalization stage, first haveto emerge in the early non-linear time evolution, cf. also [226, 227, 230–233] andSect. 3.5 in the context of quenches near an equilibrium critical point. As a consequence, prethermalization may be seen as the concept overarching all these phenomena.We however also note that wave-turbulent cascades could be captured within theconcept of a GGE, with conserved operators defined through universal scaling functions[229]. In this case, GGEs have been defined which are applicable for driven stationarywave-turbulent states and thus do not take into account the self-similar time evolutiongoverned by the exponents α and β .Scrutinizing these concepts in experiment and deepening their mathematicalfoundations is an exciting task for future research. Non-thermal fixed points so far havenot been identified in experiment. However, theoretical work suggests that they mayhave played an important role in a number of situations where the approach of a thermalstate has seemingly been suppressed, e.g., in the spinor-gas experiments discussed inSect. 3.5. rethermalization and universal dynamics in near-integrable quantum systems
3. Experiments: Relaxation dynamics in near-integrable systems
The experimental study of relaxation dynamics requires well-controlled and trulyisolated quantum systems. In this context, ultracold neutral atoms provide uniqueopportunities because of the large set of available experimental methods to isolate,manipulate and probe these systems [1, 22, 234].In the following, we will highlight a series of experiments in which bosonic gaseswere strongly confined in two spatial dimensions, effectively restricting their motionto the single remaining dimension. This realizes a system that is very close to theintegrable Lieb-Liniger gas [68, 69, 235]. The systems are not perfectly one-dimensionalbecause the two tightly confining directions are technically still present and are felt bythe atoms [236, 237]. Controlling the influence of the remaining transverse degrees offreedom and of the longitudinal trapping potential provides an ideal setting to studythe relaxation in near-integrable quantum systems.We note that this restricts our discussion to a particular aspect of the large numberof non-equilibrium phenomena that can and have recently been explored using coldatoms. For a broader overview of the rapid experimental progress in this field we referthe reader to more general reviews [1, 22, 30–32, 63, 238].
The experimental realization of a one-dimensional (1D) Bose gas follows the standardprocedure used for the production of Bose-Einstein condensates, employing laser cooling,trapping, and evaporative cooling [234]. In typical experiments, the cold quantumdegenerate gas of atoms is harmonically confined in all three spatial dimensions. Tocreate a situation where the gas is effectively 1D, a very strong anisotropy needs tobe realized in a way that the transverse confinement dominates all other energy scales.In the following, we assume the transverse confinement to be cylindrically symmetricand characterized by a trapping energy (cid:126) ω ⊥ . The size of the transverse ground state is l = (cid:112) (cid:126) /mω ⊥ , where m denotes the atomic mass. To realize a one-dimensional system,the temperature T and the interaction-determined chemical potential µ have to fulfillthe conditions k B T < (cid:126) ω ⊥ and µ < (cid:126) ω ⊥ [237, 239]. This is equivalent to stating thatthe lowest transverse excited states in the trap have a negligible occupation, i.e., thatthe trapped quantum gas is transversely in the ground state. It is interesting to notethat, for a weakly interacting Bose gas, the essential criterion to achieve, µ < (cid:126) ω ⊥ ,is in fact independent of the transverse confinement scale ω ⊥ . It is fulfilled whenever2 a s n <
1, where a s is the atomic s -wave scattering length and n is the linear density.For example, in typical experiments with Rb ( a s = 5 . n < /µ m.Such strongly anisotropic geometries can be realized using either magneticmicrotraps on atom chips [240, 241], or optical dipole traps [242], or with optical lattices[243]. The first two allow implementations of single 1D systems, the latter realize many,nearly identical 1D systems in parallel. rethermalization and universal dynamics in near-integrable quantum systems time z ( µ m ) a b Figure 4.
Quantum Newton’s cradle realized with a 1D Bose gas [2]. (a) Long-livedoscillations in momentum space demonstrate the absence of thermalization in thisnear-integrable system. A period of τ = 13 ms is shown. (b) Examples of expandedmomentum distributions for γ = 4 and three different evolution times t = τ = 34 ms(green curve), t = 15 τ (blue), t = 30 τ (red), which clearly reveal a non-thermal nature.Figure adapted from Ref. [2]. In a 1D system, all transverse degrees of freedom are effectively frozen out andexcitations can only propagate along the longitudinal, weakly confining direction. Forbosons, this leads to markedly different behavior than in three-dimensional (3D) Bose-Einstein condensates. In a 3D BEC, only the lowest momentum mode is macroscopicallyoccupied as the gas is cooled to lower and lower temperatures [234]. In a 1D confinement,the scaling of the density of states ρ ( E ) ∝ √ E leads to a large occupation of manymomentum modes. This is the origin of strong density and phase fluctuations, whichprevent the creation of long-range order [244, 245] and lead to a more complex set ofpossible equilibrium quantum states [239]. The key parameters determining the stateof the system are the temperature T and the interaction parameter γ = mg / (cid:126) n where g = 2 (cid:126) a s /ml is the 1D interaction strength. For typical temperatures T reached in experiments and γ (cid:29) γ (cid:28) The first result visualizing how integrability influences the relaxation in such a systemwas the experiment by Kinoshita et al. [2]. Atoms were trapped in an optical latticeproviding strong confinement in two transverse directions, realizing a 2D array of 1DBose gases. Changing the strength of the radial confinement allowed for a tuning of γ all the way from weak to strong interactions [74]. By applying an optical phasegrating [248], a superposition of two longitudinal momentum states with opposite signwas imposed on the trapped gas. Given these initial conditions, the atoms startedto oscillate in the trap, much like a Newton’s cradle. These oscillations were directlyimaged, and revealed a persistent non-thermal distribution. rethermalization and universal dynamics in near-integrable quantum systems The intricate microscopic dynamics that result from the interplay of relaxation,thermalization and integrability were observed in a series of experiments in Vienna [5–7, 9, 252]. In these experiments, a 1D Bose gas was created using a single magneticmicrotrap on an atom chip [240, 241, 253].Note that experiments with single systems allow for measurements that areconceptually different from the ones performed with ensembles of 1D systems in anoptical lattice. In the latter, individual 1D gases slightly differ, e.g., in atom number,and because they are all realized and probed in parallel only ensemble averages areaccessible. Due to the central-limit theorem one obtains Gaussian distributions. Incontrast, only single systems allow exploring the full probability distribution functionsof a quantum observable [141, 254] (for theoretical descriptions see [165, 166, 255, 256]).Moreover, correlation functions and cumulants which are of higher order than meanvalues and variances become accessible [257]. These methods give significantly deeperinsight into the underlying quantum states and can thus provide comprehensive detailsabout the many-body dynamics and the resulting relaxed states.
The atom chip microtrap used in theexperiments enabled a precise dynamical control over the trap. In this way, the initialharmonic transverse confinement could be transformed into a fully tunable double-wellpotential by applying strong radio-frequency (RF) dressing of the magnetic sub-statesof the atoms [258, 259]. This split the gas into two almost identical halves, realizinga quench and creating a non-equilibrium state characterized by the quantum noiseintroduced by the splitting.This situation is best illustrated by analyzing a splitting process that is performedfast compared to the longitudinal dynamics so that t split < ξ h /c = (cid:126) /µ . Here, ξ h = (cid:126) /mc is the healing length, and c = (cid:112) µ/m is the speed of sound. In this limit, no correlations rethermalization and universal dynamics in near-integrable quantum systems Figure 5.
Outline of the Vienna experiments described in [5, 8, 9, 22, 142]. (a,b)A phase fluctuating 1D quasi-condensate is coherently split, creating two 1D gaseswith almost identical phase profiles ϕ L ( z ) and ϕ R ( z ) (represented by the solid blacklines). The gases are then allowed to evolve in the double-well potential for sometime t evol , which leads to strong fluctuations in the local phase difference ∆ ϕ ( z ) anda decrease of the phase correlation length λ ϕ . The question the experiment aims atanswering is whether and how this dynamical state reaches the thermal equilibriumstate of two independent quasi-condensates. In these, the phase difference between the1D gases fluctuates strongly, and the correlation length λ thermal is determined by theirtemperature and density. (c) The phase difference ∆ ϕ ( z ) between the two 1D gasesis probed through time-of-flight matter-wave interference of the two gases. The localrelative phase is directly transformed to a local phase shift of the interference pattern.This relative phase shift or the contrast C ( L ) of the axially integrated interferencepattern can then be used as a direct measure of the strength of the relative phasefluctuations. Figure adapted from Ref. [5]. can build up along the axial direction, and the splitting happens independently ateach point in the gas. The process can be intuitively pictured as a local beam splitterwhere each atom is independently put into the left or right half of the new systemwith probabilities p L and p R = 1 − p L , respectively. The corresponding probabilitydistribution for the local number of particles N on either side is therefore binomial.These splitting fluctuations cause a locally fluctuating interaction energy and hence setthe system out of equilibrium.The outline of the experimental scheme is shown in Fig. 5. After the splittingquench the system was let to evolve for a variable time. Subsequently, all trappingpotentials were switched off, and the gases rapidly expanded transversally. This reducedthe internal interaction energy to zero on a timescale ∼ /ω ⊥ such that the gasexpanded mostly ballistically. This stopped the many-body evolution and froze thestate. The matter waves of the two 1D gases in the double-well trap overlapped andformed a matter-wave interference pattern that could be measured by standard imaging rethermalization and universal dynamics in near-integrable quantum systems slow further decay ? to thermal equilibrium? M ean s qua r ed c on t r a s t < C > light cone like evolu6on prethermalized state generlized Gibbs ensemble Figure 6.
Relaxation behavior after splitting a single 1D quantum gas into two asrevealed by the square of the interference contrast integrated over the central 110 µ mof the 1D interference pattern. The graph shows the time evolving mean squaredcontrast (cid:104)C (cid:105) , integrated over the full length of the 1D system. A decreasing (cid:104)C (cid:105) reveals the growing fluctuations in the interference pattern as it gets more ‘wiggly’.Initially, the contrast decays quickly due to dephasing of the approximate eigenmodesof the near-integrable Hamiltonian. This dephasing spreads through the system in alight-cone-like fashion [7], cf. Fig. 8. The system then relaxes towards a quasi-steady,prethermal state [5] which is characterized by a generalised Gibbs ensemble (GGE) [9].The fast initial relaxation reflects the build-up of a prethermal coherence-length scale,which can be related to the fast approach of the red line on one of the blue trajectoriesin Fig. 1. On longer timescales, the system shows further relaxation, and it is akey future challenge to separate and distinguish any further relaxation from processescaused by outside influences like heating due to trap instability or atom loss. Figureadapted from Ref. [5]. techniques [260]. Because of the fast reduction of the interaction energy, interactionsduring expansion could be neglected and the position of the fringes after time of flightapproximately reflected the difference ∆ ϕ ( z ) = ϕ L ( z ) − ϕ R ( z ) between the phases ofthe two 1D gases. The phase fluctuations can be characterized by means of the phasecorrelation function, C ( z, z (cid:48) ) ∼ (cid:104) Ψ ( z )Ψ † ( z )Ψ † ( z (cid:48) )Ψ ( z (cid:48) ) (cid:105) ∼ (cid:104) e i ∆ ϕ ( z,t ) − i ∆ ϕ ( z (cid:48) ,t ) (cid:105) . (4)Here, Ψ and Ψ denote bosonic field operators describing the two halves of thesystem [7, 9]. This function provides a measure for the correlations of the phase betweentwo different points z and z (cid:48) along the length of the system. It contains only theexperimentally measured relative phases ∆ ϕ ( z ) and ∆ ϕ ( z (cid:48) ), and could thus be directlycalculated from the data. Besides this local phase ∆ ϕ ( z ) extracted from the relativeshift of the interference pattern, a second important observable in the experiments wasthe mean squared contrast (cid:104)C (cid:105) of the interference patterns, integrated over a specificlength scale [165, 166, 255, 256].Repeating the experiment many times with identical initial conditions allowedstudying the fluctuation dynamics of this relative phase and its relation to thermal rethermalization and universal dynamics in near-integrable quantum systems Figure 7. (a) Dependence of the effective prethermalization temperature T eff on thelinear atom density ρ , and (b) independence of T eff of the temperature T of the systembefore splitting, corrected for the scaling of T eff with density. The (black) solid linecorresponds to the theoretical prediction k B T eff = gn D /
2. Figure adapted from [5]. states. The observed dynamics as seen through the mean squared contrast (cid:104)C (cid:105) of theinterference pattern is summarized in Fig. 6 (where the contrast C is not to be confusedwith the correlation function C ). As the splitting was performed rapidly, the two halvesof the system were fully coherent immediately after the splitting. The relative phasefield was close to zero along the whole length of the system, resulting in straight fringesand thus in a large integrated mean squared contrast. Over time, the dynamics ledto a randomization of the relative phase ∆ ϕ ( z ), to ‘ wiggly ’ interference patterns and adecrease of the mean square contrast (cid:104)C (cid:105) . After asufficiently long evolution time of the pair of 1D Bose gases after the splitting, a steadystate was observed. This state was found to be characterized by thermal full distributionfunctions of (cid:104)C (cid:105) and exponentially decaying correlations, very much like the thermalequilibrium state for a quasi 1D Bose gas [5, 6]. However, a much lower temperaturewas measured than the temperature of the initial gas before the splitting. Moreover,the longitudinal coherence length was also significantly larger than the expected thermaldecay length. The relative degrees of freedom of the system were thus looking thermal,but with an effective temperature T eff that was significantly lower than the initialtemperature T in .Concomitant theoretical studies [165, 166] revealed the observed extremely lowtemperature to be a result of the particular quench that was performed. Splitting thesystem creates new degrees of freedom the canonical coordinates of which are given bythe local phase difference and the relative atom number between the two systems. Whilethe individual halves of the system still fluctuate strongly with the initial temperature T in , only a very small amount of energy is introduced into the relative degrees of freedomvia the quantum shot noise of the splitting process. The fast quench, in particular,leads to equipartition of this energy ε split = gn D / T eff = ε split /k B = gn D / k B . The theoretical model thus predicts that the effectivetemperature should be independent of the initial temperature, and exhibit a linear rethermalization and universal dynamics in near-integrable quantum systems evolutionftimef(ms) c r o ss o v e r f d i s t an c e f z c ( µ m ) ab ff... z ( z ) C ffffffffff pha s e f c o rr e l a t i on ff un c t i on f C ( z ) relativefdistancefz=z − z´f(µm) evolutionftimef(ms) c r o ss o v e r f d i s t an c e f z c ( µ m ) ab ff... z ( z ) C ffffffffff pha s e f c o rr e l a t i on ff un c t i on f C ( z ) relativefdistancefz=z − z´f(µm) Figure 8.
Temporal spreading of the prethermalized fraction. (a) Experimental phasecorrelation functions C (¯ z, t ) (filled circles) compared to theoretical calculations (solidlines), as a function of the relative distance ¯ z = z − z (cid:48) between two longitudinalpositions. The evolution time t increases from top to bottom. The final (green) lineis the theoretical prediction for the relaxed, fully prethermalized state. At each time t , correlation functions follow this prediction up to a crossover distance ¯ z c ( t ) beyondwhich the system remembers the initial long-range phase coherence. (b) Position ofthe crossover distance ¯ z c as a function of t , revealing the light-cone-like emergenceof the thermal correlations of the prethermalized state. The slope of the solid linecorresponds to twice the phonon velocity of the system. Figure adapted from Ref. [7]. scaling with the 1D density. This behavior was indeed observed in the measurementsand is shown in Fig. 7. It represents precisely the kinetic prethermalization [35, 36] thatis discussed in Sect. 2.2.Note that in the experiments, the quench introduced white-noise fluctuations intothe relative degrees of freedom. This leads, within good approximation, to a GGE withonly one Lagrange multiplier corresponding to the inverse of the effective temperature, β eff = ( k B T eff ) − . This GGE thus takes the form of a genuine Gibbs ensemble, with ε split (cid:39) ω k n k (0) (cid:39) const., independent of wave number k , where n k (0) is the modeoccupation number right after the quench. An important question is howthe above prethermalized quasi-steady state is established after the quench. In theexperiments, the two-point phase correlation function C ( z, z (cid:48) ) (see Fig. 8) allowed for adetailed study of these dynamics [7, 8, 142]. Directly after the splitting the correlationswere close to C ( z, z (cid:48) ) ≡
1, which reflected the full coherence of the relative phase. Forany given evolution time t after the splitting, C ( z, z (cid:48) ) revealed that the system hadalready established the prethermalized correlations up to a distance z − z (cid:48) = 2 c t , withthe speed of sound c . At larger separations of z and z (cid:48) , the system still retained theinitial long-range order imposed by the quench.This demonstrates how the thermal correlations of the prethermalized state wereestablished locally and then spread through the system in a light-cone-like fashion.Hence, the phononic excitations of the system could be interpreted as informationcarriers that propagate correlations through the system. This reflects the basic Lieb- rethermalization and universal dynamics in near-integrable quantum systems a b e x pe r i m en tt heo r y A z (µm)4-point 6-point 10-pointz (µm) −
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20 0 20 − l o c a l χ l o c a l χ l o c a l χ z ( µ m ) z ( µ m ) z (µm) Figure 9.
Observation of a generalized Gibbs ensemble (GGE). (a) Two-pointphase correlations visualizing the emergence of the GGE. The correlation functionsshow a characteristic maximum on the diagonal and a decay of correlationsaway from the diagonal (cf. Fig. 8). Additional correlations on the anti-diagonal are the result of the enhanced occupation of even (with respect tothe longitudinal trap) quasiparticle modes, in good agreement with a theoreticalmodel assuming multiple conserved quantities. (b) Examples of cuts throughexperimental 4-, 6-, and 10-point correlation functions. The GGE describes wellalso phase correlations up to 10 th order. From left to right, we plot the cuts C ( z , , z , C ( z , − , z , C ( z , , , z , − , C ( z , − , , z , − , − C ( z , , , z , − , z , − , − , , −
4) and C ( z , − , − , z , − , − , − , z , − , − µ m and were chosen as representative cases for ourhigh-dimensional data. Adapted from Ref. [9]. Robinson bound limiting the spreading of information to a finite group velocity, asoriginally introduced for lattice spin models [167] and studied both numerically [168, 169]and experimentally [15]. The limits observed here extend these ideas to continuousmodels, as previously put forward for quenches in conformal field theories [86, 87].
The experimentally observedprethermalized state can be described by a single effective temperature T eff . Althoughthis state was observed in a system that is very well described by an integrable model,this state is practically indistinguishable from a thermal state. As mentioned above, thekey to this surprising result lies in the particular quench protocol that was employed. rethermalization and universal dynamics in near-integrable quantum systems mode m m ode o cc upa t i on s m / µ phonon modes are number squeezed m o d e o cc up a o n mode Figure 10.
The generalized Gibbs ensemble (GGE) observed in [9]. Shown are themean occupation numbers n m (in units of ε m /µ ) of the quasiparticle modes with index m that define the GGE shown in Fig. 9. The distribution reveals that the occupationof the lowest even (odd) modes are increased (decreased) as compared to the quantumnoise occupation found in a single-temperature prethermalized state observed afteran instantaneous quench (dashed line). Modes with occupation number below this‘quantum-noise’ line are number squeezed. Figure adapted from Ref. [9]. According to this protocol, all conserved quasiparticle modes of the relative degrees offreedom were prepared with the same energy ε split (cid:39) k B T eff .In a subsequent experiment, Langen et al. [9] modified this quench protocol bychanging the speed with which the double well was established during the splittingprocess. They observed a relaxed state, which showed enhanced correlations outside thelight cone, strikingly visible as a cross in the non-translation-invariant phase correlations C ( z, z (cid:48) ) reproduced in Fig. 9a. A detailed analysis showed that, in fact, not less than tenLagrange multipliers, describing the occupation of the ten lowest-energy quasiparticlemodes of the relative degrees of freedom, were necessary for a precise description of theobserved correlations (Fig. 10). It is interesting to note that a series of these modes,especially n = 1 and 3 and, to a lesser extent, n >
5, had occupations that weresignificantly below the quantum-noise limit of a fast quench. This could be due to thefact that the employed quench created a non-trivial many-body quantum state withnumber-squeezed excitations. Studying these will be part of future research.Theoretically, the GGE has been predicted to be applicable to describe localcorrelation functions of prethermalized states defined in a finite-size region, see thediscussion in Sect. 2.1.2. Moreover, since the GGE represents the optimum ensemble onthe basis of available information, deviations of relaxed states from the GGE descriptionare generally expected to become manifest first in higher-order correlation functions.Basic phase correlation functions up to 10 th order have been measured [9], and arbitrarilychosen two-dimensional cuts through these are shown in Fig. 9b. Similarly to the two-point correlation functions, they were found to be in very good agreement with thetheoretical model. As demonstrated in [9], the specifically varying mode occupations andthus Lagrange multipliers are necessary to correctly describe the measured correlations, rethermalization and universal dynamics in near-integrable quantum systems An outstanding question concerns the furtherevolution after the establishment of the prethermalized state described by a GGE.Will the system further relax and eventually reach thermal equilibrium? The slowdecay of the mean squared contrast (cid:104)C (cid:105) over long limes seen in Fig. 6 hints at such afurther relaxation. It has been predicted that higher-order scattering processes [249–251] will induce this relaxation, and numerical calculations [261] gave further evidence.Experimentally, this is a very difficult problem, because the smallest heating due to,e.g., a shaking of the trap can mimic the respective processes. Further experimentalinvestigations are currently under way.An interesting further aspect concerns the effect of the finite size of the studiedsystem, in particular the possibility of coherent revivals at finite times. For quenches inconformal field theories, partial as well as full revivals were predicted to occur at integermultiples of the system size divided by twice the speed of sound, and relations to theformation of black holes existing through the AdS-CFT correspondence were pointedout [93, 132]. A detailed study of the light-cone-like propagation of excitations andthe build-up of a dephased, prethermalized state after the initial splitting quench gaveconcrete predictions that revivals should be observable in box-like potentials, but aresuppressed by the harmonic confinement of the current experiments [8]. Both of theseproblems will be at the center of future developments and experiments. The relaxation to a prethermalized instead of a thermal state also has importantimplications for the preparation of the initial pre-quench gas in such experiments. Thestandard technique to prepare these gases at ultracold temperatures is evaporativecooling, which relies on the selective removal of the most energetic particles from atrapped gas and the subsequent rethermalization of the gas to a lower temperaturethrough elastic collisions. For efficient cooling this cycle is being repeated continuously,increasing the phase-space density of a gas at the cost of reducing the total number ofatoms. However, in a pure 1D setting, two-body collisions do not lead to a redistributionof energy and momentum and thus, intuitively, such systems should not be cooledthrough particle dissipation.In principle, the known conditions that break integrability in these systems couldbe an explanation for the observed cooling efficiency deep into the 1D regime. For rethermalization and universal dynamics in near-integrable quantum systems Figure 11.
Using prethermalization for the preparation of a 1D Bose gas. Shown isthe universal scaling relation between temperature and atom number observed in thedissipative cooling of a near-integrable 1D Bose gas. Figure adapted from Ref. [262]. example, for realistic confinement strengths, the 1D condition and thus the integrabilityof the system can be broken in collisions where there is enough energy availableto access transverse excited states. However, these thermalizing two-body collisionsare suppressed by at least a factor of exp( − (cid:126) ω ⊥ /k B T ) in a non-degenerate bosonicgas [249]. Consequently, these collisions freeze out as soon as the gas enters the 1Dregime. Other higher-order processes that can lead to thermalization are three-bodycollisions [249, 251] or phonon-phonon scattering [250, 261]. However, their expectedthermalization time scales are much larger than the cooling time in typical experiments.As in the case of prethermalization the key to understanding the cooling process ofthe gas is dephasing. In an experiment by Rauer et al. [262], the outcoupling processwas revealed to be nearly homogenous and independent of the mode energy. While thiswould, as discussed above, not lead to cooling, a significant decrease in temperaturedown to k B T ∼ . (cid:126) ω ⊥ was observed. In particular, all correlation functions remainedclose to their thermal form for all times despite the constant removal of atoms.The dynamics resulting from such a homogeneous particle dissipation can beintuitively understood within a Luttinger-liquid picture, describing the low-energydynamics of the underlying Lieb-Lininger model [263]. Each of the phonon modes inthis model contributes to the fluctuations in the gas through a density and a phasequadrature, in analogy to the position and momentum quadratures of a harmonicoscillator. The free evolution of such a mode k with energy (cid:126) ω k can be visualized as arotation of the corresponding Wigner function with the frequency ω k . In this picture, asudden homogeneous outcoupling of atoms leads to a decrease in average density withthe density fluctuations around this average being scaled down correspondingly. Thisinstantaneous density reduction therefore extracts energy from the density quadratureof the phonon modes while leaving the phase quadrature unchanged. The systemreacts to this reduction by dephasing, redistributing the remaining energy between the rethermalization and universal dynamics in near-integrable quantum systems The experiments discussed in the previous subsection employed an entirely 1D coolingprocess that suppressed excitations of the gas. The opposite case of a strong coolingquench into a single 1D trap can also be realized and is expected to show similarities toquenches to or across a critical point. In a different context, quenches across a criticalpoint were realized in the experiments of the Stamper-Kurn [10, 11, 170, 171], Sengstock[12], and Oberthaler [13, 14] groups with spinor gases, see [238] for a recent review.Quenches leading to solitary excitations, studying Kibble-Zurek-type physics [181, 264–267], were performed in experiment by Lamporesi, Ferrari, et al. [16], Schneider, Blochand collaborators [18], the Dalibard group [19, 20], and the Hadzibabic group [21]. Allof these contributed towards studying dynamical critical scaling and universality.Employing semiclassical Truncated-Wigner-type simulations, Barnett et al. [172]analyzed the spinor gas quench experiments [10, 11, 170, 171], showing that the systemdid not thermalize at appreciable time scales but rather reached a quasi-steady regimethat evolved anomalously slowly in time. The long-time evolution after the short-time instabilities and growth of magnetization [268–276] was characterized by a quasi-steady state with exponentially decaying spatial correlations. The time scale for Landaudamping was estimated to be beyond the reach of the experiment. It was concludedthat the system did not thermalize at appreciable time scales but rather reached aquasi-steady regime that evolves anomalously slowly in time and which they calledprethermalized.A slow long-time residual growth of the spin domains created in the quenchesacross the ferromagnetic phase transition was reported in [171] and conjectured to signalcoarsening dynamics. Coarsening as self-similar rescaling in time and space has beendiscussed for many different systems within the theory of phase-ordering kinetics [42],including spinor condensates [217, 268, 277, 278]. These analyses typically assumedissipative dynamics, with, in some cases, additional ‘reversible’ (non-dissipative)coupling between the modes [278].In the experiments of Nicklas et al. [13, 14], performing quenches to either side of aquantum critical point in a quasi 1D pseudo-spin 1/2 gas, scaling has been revealed ofboth, characteristic time and spatial scales with respect to the parametric distance of thefinal Hamiltonian to criticality, see Fig. 12. Theoretical studies of these experiments, likeof similar quench protocols in transverse-field Ising chains [121–123], indicated that theintermediate-time critical scaling properties of correlation lengths can be understoodin terms of a GGE for a prethermalized state [279]. For previous discussions ofprethermalization and/or the approach of a GGE in quenches near criticality see also[226, 227, 230–233] for isolated systems, or [280] for open systems. Starting with a rethermalization and universal dynamics in near-integrable quantum systems (a) space rescaling (b) time rescaling immiscible miscible h J z ( y ) J z ( y ) i y y ( µ m) y y " / ( µ m) 0 10 0 2 4 6 y y ( µ m) y y " / ( µ m)00.51 h J z ( y ) J z ( y ) i ⇠ L d d o m a i n s i z e L d ( µ m )
10 1 " " c o rr . l e n g t h ⇠ ( µ m ) " c h a r . t i m e ⌧ ( m s ) c o rr . l e n g t h ⇠ ( µ m ) ⌧ Figure 12.
Spatial and temporal scaling of the spin-spin correlations measured in aquasi 1D pseudo-spin 1 / c wherea transition from a paramagnetic to a ferromagnetic phase is triggered. (a) Spatialcorrelations after quenches to different relative proximities ε = Ω / Ω c − y → yε ν of the distance dependence with mean-field exponent ν = 1 /
2, fall on a universal curve. Bottom row: ε -dependence of the characteristiclength scales deduced from the correlation functions. The straight lines reveal valuesfor the critical exponent ν on either side of the transition. (b) Temporal scaling ofthe spin-spin correlations. After the quench to the miscible side of the transition, thecorrelation length ξ , measuring the exponential decay seen in the top right panel in(a), for any ε ≥
0, shows first a linear rise and later settles in an oscillatory mannerto a finite saturated value, see [14] for more details. The characteristic time τ fordifferent ε is obtained as the intersection point of the linear fits to the initial rise of ξ ( t ) (grey symbols for ε = 0 .
1) and to the behaviour after ξ deviates from this rise.The procedure of determining the intersection is exemplarily shown in the upper panelfor ε = 0 .
23. In the lower panel we compare the extracted τ ( ε ) to a mean-field scalingwith νz = 1 / τ ∼ / ∆, with gap∆( ε ) = Ω c (cid:112) ε ( ε + 1), also applicable at larger ε (cid:38) ground state far away from the critical point, the quench close to criticality maps thisstate onto new, nearly number-conserved quasiparticle degrees of freedom. The resultingoccupation numbers of these quasiparticles define the Lagrange parameters of the GGE.It is nevertheless found that, the closer the quench of the pseudo-spin gas is tuned to thecritical point, the better the correlations are described by a single effective temperatureparameter [279]. See [54, 86, 87] for earlier studies in conformal field theories.In the light of fast experimental progress, the verification of non-thermal fixed pointsas well as universal scaling dynamics as a more general kind of prethermalization in near-integrable quantum systems is at the horizon. This will include demonstrations of theuniversal coarsening of defects, strong non-linear excitations and turbulent dynamics. rethermalization and universal dynamics in near-integrable quantum systems Integrability does not only affect the thermalization of cold atomic systems, but also hasimportant effects on their transport dynamics. This was observed in an experiment byRonzheimer et al. [17] in which the expansion of initially localized atoms in homogeneous1D and 2D optical lattices was studied. K atoms were prepared in the combinedpotential of a 3D optical lattice and an additional harmonic confinement. Tunable inter-actions allowed the realization of both the hard-core boson limit and bosons with finiteinteractions. The harmonic confinement was then decreased in one or two directions sothat the atoms could expand in a 1D or 2D optical lattice potential. It was observedthat the fastest ballistic expansion of hard-core bosons proceeds in all integrable limits ofthe system, where the many constants of motion inhibit diffusive scattering. Deviationsfrom these limits significantly suppressed the expansion and led to signatures of diffusivedynamics. Interestingly, for finite interactions, deviations from the ballistic expansionof hard-core bosons only occurred because the gas was subject to an interaction quench,implying the higher energies the weaker the interactions are [281].
4. Conclusions and outlook
Advances in preparing and controlling low-dimensional ultracold gases have lead toan enormous, renewed interest in integrable and near-integrable model systems asprototypes for studying dynamics, relaxation, and thermalization in quantum physics.Insight from experiments has lead to rapid progress in linking the microscopic quantumdynamics of atoms and molecules with the macroscopic properties of matter providinga unique connection between quantum many-body physics and statistical physics. Inparticular they provide a unique window onto the emergence of (classical) statisticalensembles in the evolution of isolated many-body quantum systems and thus thetransition from quantum physics at the micro-scale to our classical world.The experimental progress in this field continues at a remarkable speed and willallow new insights into previously unattainable phenomena. Examples include theability to control the interaction strength [282], to observe and control quantum many-body systems locally at the single-atom level [283–287], the use of optimal controlschemes to manipulate their (global) external degrees of freedom [288–291]. Moreover,the extension to gases with fermionic statistics [292], gauge fields [293], spin-orbitcoupling [294] or long-range interactions [295–297] should bear many new interestingaspects.With these capabilites, ultracold atomic gases offer themselves as a ‘quantumsimulator’ for universal dynamics of systems which are difficult to access directly [298].Beyond the immediate implications for simple low-energy degenerate quantum gases,phenomena such as topological configurations in solids, in soft matter, the dynamicsof the quark-gluon plasma created in heavy-ion collisions, dynamics of the big bang[299, 300], or the reheating of the post-inflationary universe come into sight.
EFERENCES
Acknowledgments
The authors thank J. Berges, F. Brock, H. Cakir, I. Chantesana, S. Czischek, E. G. DallaTorre, M. Davis, E. Demler, J. Eisert, S. Erne, F. Essler, C. Ewerz, R. Geiger, M. Gring,A. Johnson, M. Karl, V. Kasper, M. Kastner, K. Kheruntsyan, M. Kronenwett,M. Kuhnert, P. Kunkel, D. Linnemann, S. Mathey, I. Mazets, L. McLerran, W. Muessel,E. Nicklas, B. Nowak, M. Oberthaler, J. Pawlowski, A. Pi˜neiro Orioli, M. Pr¨ufer,B. Rauer, H. Salman, A. Samberg, C. Scheppach, C. Schmied, J. Schole, T. Schweigler,D. Sexty, H. Strobel, T. Wright, and J. Zill for discussions and collaboration on thetopics described here. They acknowledge funding by the EU (SIQS, ERC advancedgrant QuantumRelax, FET-Proactive grant AQuS, Project No. 640800), by the AustrianScience Fund (FWF), and by the National Science Foundation under Grant No. NSFPHY11-25915. T.L. acknowledges support by the Alexander von Humboldt Foundationthrough a Feodor Lynen Fellowship.
References [1] I. Bloch, J. Dalibard and W. Zwerger,
Rev. Mod. Phys. (2008) 885.[2] T. Kinoshita, T. Wenger and D. S. Weiss, Nature (2006) 900.[3] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm and J. Schmiedmayer,
Nature (2007) 324.[4] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ock, J. Eisertet al.,
Nature Phys. (2012) 325.[5] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl et al., Science (2012) 1318.[6] M. Kuhnert, R. Geiger, T. Langen, M. Gring, B. Rauer, T. Kitagawa et al.,
Phys. Rev. Lett. (2013) 090405.[7] T. Langen, R. Geiger, M. Kuhnert, B. Rauer and J. Schmiedmayer,
NaturePhys. (2013) 640.[8] R. Geiger, T. Langen, I. E. Mazets and J. Schmiedmayer, New J. Phys. (2014) 053034.[9] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert et al., Science (2015) 207.[10] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore and D. M.Stamper-Kurn,
Nature (2006) 312, [ cond-mat/0605351 ]. EFERENCES
Phys. Rev.Lett. (2008) 170403.[12] J. Kronj¨ager, C. Becker, P. Soltan-Panahi, K. Bongs and K. Sengstock,
Phys.Rev. Lett. (2010) 090402.[13] E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, P. G. Kevrekidiset al.,
Phys. Rev. Lett. (2011) 193001.[14] E. Nicklas, M. Karl, M. H¨ofer, A. Johnson, W. Muessel, H. Strobel et al.,
Phys.Rev. Lett. (2015) 245301, [ ].[15] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß T. Fukuharaet al.,
Nature (2012) 484.[16] G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo and G. Ferrari,
Nat. Phys. (2013) 656.[17] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P.McCulloch et al., Phys. Rev. Lett. (2013) 205301.[18] S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Rieraet al.,
Proc. Nat. Acad. Sci. (2015) 3641.[19] L. Corman, L. Chomaz, T. Bienaim´e, R. Desbuquois, C. Weitenberg,S. Nascimb`ene et al.,
Phys. Rev. Lett. (2014) 135302.[20] L. Chomaz, L. Corman, T. Bienaim´e, R. Desbuquois, C. Weitenberg,S. Nascimb`ene, J. Beugnon, and J. Dalibard,
Nat. Comm. (2015) 6162.[21] N. Navon, A. L. Gaunt, R. P. Smith and Z. Hadzibabic, Science (2015) 167,[ ].[22] T. Langen, R. Geiger and J. Schmiedmayer,
Ann. Rev. Cond. Mat. Phys. (2015) 201.[23] C. E. Shannon and W. Weaver, The mathematical theory of communication . TheUniversity of Illinois Press, Urbana, IL, 1949.[24] E. T. Jaynes,
Phys. Rev. (1957) 620.[25] E. T. Jaynes,
Phys. Rev. (1957) 171.[26] K. Huang,
Statistical Mechanics . Wiley, New York, 1987.[27] M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii,
Phys. Rev. Lett. (2007)050405.[28] M. Rigol, V. Dunjko and M. Olshanii, Nature (2008) 854.[29] M. Rigol,
Phys. Rev. Lett. (2009) 100403.[30] A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore,
Rev. Mod. Phys. (2011) 863.[31] J. Eisert, M. Friesdorf and C. Gogolin, Nature Phys. (2015) 124.[32] C. Gogolin and J. Eisert, Rep. Prog. Phys. , 056001 (2016), [ ].[33] M. Girardeau, Phys. Lett. A (1969) 442. EFERENCES
Phys. Lett. A (1970) 67.[35] G. Aarts, G. F. Bonini and C. Wetterich, Phys. Rev. D (2000) 025012.[36] J. Berges, S. Borsanyi and C. Wetterich, Phys. Rev. Lett. (2004) 142002,[ hep-ph/0403234 ].[37] C. Wetterich, ‘Prethermalization’, talk given at RETUNE, Heidelberg, 2012 .[38] J. Berges, A. Rothkopf and J. Schmidt, Phys. Rev. Lett. (2008) 041603,[ ].[39] B. Nowak, J. Schole and T. Gasenzer,
New J. Phys. (2014) 093052,[ ].[40] L. M. A. Bettencourt and C. Wetterich, Phys. Lett.
B430 (1998) 140,[ hep-ph/9712429 ].[41] G. F. Bonini and C. Wetterich,
Phys. Rev. D (1999) 105026.[42] A. J. Bray, Adv. Phys. (1994) 357.[43] V. E. Zakharov, V. S. L’vov and G. Falkovich, Kolmogorov Spectra of TurbulenceI: Wave Turbulence . Springer, Berlin, 1992.[44] N. Goldenfeld,
Lectures on phase transitions and the renormalization group .Frontiers in physics. Addison-Wesley, 1992.[45] J. Zinn-Justin,
Quantum Field Theory and Critical Phenomena . Internationalseries of monographs on physics. Clarendon Press, 2004.[46] S. Ma,
Modern Theory of Critical Phenomena . Advanced book classics. Perseus,2000.[47] P. C. Hohenberg and B. I. Halperin,
Rev. Mod. Phys. (1977) 435.[48] S. Sachdev, Quantum Phase Transitions . Cambridge University Press, 2000.[49] M. Henkel, H. Hinrichsen and S. L¨ubeck,
Non-Equilibrium Phase Transitions:Volume 1: Absorbing Phase Transitions . Theoretical and Mathematical Physics.Springer Netherlands, 2008.[50] U. C. T¨auber,
Critical Dynamics. A Field Theory Approach to Equilibrium andNon-Equilibrium Scaling Behaviour . Cambridge University Press, 2014.[51] J. Berges and D. Mesterh´azy,
Nucl. Phys. Proc. Suppl. (2012) 37,[ ].[52] B. Nowak, S. Erne, M. Karl, J. Schole, D. Sexty and T. Gasenzer, in
Proc. Int.School on Strongly Interacting Quantum Systems Out of Equilibrium, LesHouches, 2012 (to appear) , ArXiv e-prints (2013), [ ].[53] E. Ilievski, M. Medenjak, T. Prosen and L. Zadnik,
ArXiv e-prints (2016),[ ].[54] P. Calabrese and J. Cardy,
J. Stat. Mech. ].[55] M. A. Cazalilla and M.-C. Chung,
J. Stat. Mech. ].[56] J.-S. Caux,
J. Stat. Mech. ]. EFERENCES
J. Stat. Mech. ].[58] R. Vasseur and J. E. Moore,
J. Stat. Mech. ].[59] A. De Luca and G. Mussardo,
J. Stat. Mech. ].[60] L. Vidmar and M. Rigol,
J. Stat. Mech. ].[61] G. P. Berman and F. M. Izrailev,
Chaos (2005) .[62] M. Rigol, in Quantum Gases: Finite Temperature and Non-EquilibriumDynamics (Vol. 1 Cold Atoms Series) (N. P. Proukakis, S. A. Gardiner,M. J. Davis, and M. Szymanska, eds.), Imperial College Press, London, 2013.[ ].[63] J.-S. Caux and J. Mossel,
J. Stat. Mech.: Theor. Exp. (2011) P02023.[64] J. Von Neumann and R. T. Beyer,
Mathematical foundations of quantummechanics; 1996 ed.
Landmarks in Mathematics and Physics. Princeton Univ.Press, Princeton, NJ, 1955.[65] B. Sutherland.
Beautiful Models: 70 years of exactly solved quantum many-bodyproblems.
World Scientific, Singapore, 2004.[66] V. E. Korepin, N. M. Bogoliubov and A. G. Izergin,
Quantum Inverse ScatteringMethod and Correlation Functions . CUP, Cambridge, UK, 1997.[67] G. Mussardo,
Statistical Field Theory, An Introduction to Exactly Solved Modelsin Statistical Physics . Oxford University Press, 2009.[68] E. H. Lieb and W. Liniger,
Phys. Rev. (1963) 1605.[69] E. H. Lieb,
Phys. Rev. (1963) 1616.[70] B. S. Shastry,
Phys. Rev. Lett. (1986) 2453.[71] D. Braak, Phys. Rev. Lett. (2011) 100401.[72] M. Girardeau,
J. Math. Phys. (1960) 516.[73] B. Paredes, A. Widera, V. Murg, O. Mandel, S. F¨olling, I. Cirac et al., Nature (2004) 277.[74] T. Kinoshita, T. Wenger and D. S. Weiss,
Science (2004) 1125.[75] J. M. Deutsch,
Phys. Rev. A (1991) 2046.[76] M. Srednicki, Phys. Rev. E (1994) 888.[77] H. Tasaki, Phys. Rev. Lett. (1998) 1373.[78] J. Berges and J. Cox, Phys. Lett.
B517 (2001) 369, [ hep-ph/0006160 ].[79] J. Berges,
Nucl. Phys.
A699 (2002) 847, [ hep-ph/0105311 ].[80] J. Berges, S. Borsanyi and J. Serreau,
Nucl. Phys.
B660 (2003) 51,[ hep-ph/0212404 ].[81] S. Popescu, A. J. Short and A. Winter,
Nature Phys. (2006) 754,[ quant-ph/0511225 ].[82] P. Calabrese and J. Cardy, J. Stat. Mech.: Theor. Exp. (2005) P04010.
EFERENCES
Phys. Rev. A (2005)063604, [ cond-mat/0507480 ].[84] J. Eisert and T. J. Osborne, Phys. Rev. Lett. (2006) 150404.[85] J. Berges and T. Gasenzer, Phys. Rev. A (2007) 033604, [ cond-mat/0703163 ].[86] P. Calabrese and J. Cardy, Phys. Rev. Lett. (2006) 136801.[87] P. Calabrese and J. Cardy, J. Stat. Mech.
P06008 (2007) .[88] M. Rigol and L. F. Santos,
Phys. Rev. A (2010) 011604.[89] G. Biroli, C. Kollath and A. M. L¨auchli, Phys. Rev. Lett. (2010) 250401.[90] M. C. Ba˜nuls, J. I. Cirac and M. B. Hastings,
Phys. Rev. Lett. (2011)050405.[91] M. Eckstein, M. Kollar and P. Werner,
Phys. Rev. Lett. (2009) 056403.[92] M. Rigol,
Phys. Rev. Lett. (2014) 170601.[93] J. Cardy,
J. Stat. Mech. (2016) 023103, [ ].[94] T. Barthel and U. Schollw¨ock,
Phys. Rev. Lett. (2008) 100601.[95] C. Gogolin, M. P. M¨uller and J. Eisert,
Phys. Rev. Lett. (2011) 040401.[96] B. Bertini, D. Schuricht and F. H. L. Essler,
J. Stat. Mech.: Theor. Exp. (2014) P10035.[97] M. Fagotti and F. H. L. Essler,
J. Stat. Mech.: Theor. Exp. (2013) P07012.[98] J. De Nardis, B. Wouters, M. Brockmann and J.-S. Caux,
Phys. Rev. A (2014) 033601.[99] M. Fagotti, M. Collura, F. H. L. Essler and P. Calabrese, Phys. Rev. B (2014) 125101.[100] B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol and J.-S. Caux, Phys. Rev. Lett. (2014) 117202.[101] B. Pozsgay, M. Mesty´an, M. A. Werner, M. Kormos, G. Zar´and and G. Tak´acs,
Phys. Rev. Lett. (2014) 117203.[102] G. Goldstein and N. Andrei,
Phys. Rev. A (2014) 043625.[103] M. Fagotti and F. H. L. Essler, Phys. Rev. B (2013) 245107.[104] E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler and T. Prosen, Phys. Rev. Lett. (2015) 157201.[105] F. H. L. Essler, G. Mussardo and M. Panfil,
Phys. Rev. A (2015) 051602.[106] M. Cazalilla, Phys. Rev. Lett. (2006) 156403.[107] A. Iucci and M. A. Cazalilla, Phys. Rev. A (2009) 063619.[108] M. A. Cazalilla, A. Iucci and M.-C. Chung, Phys. Rev. E (2012) 011133.[109] M. Rigol, A. Muramatsu and M. Olshanii, Phys. Rev. A (2006) 053616.[110] D. M. Gangardt and M. Pustilnik, Phys. Rev. A (2008) 041604(R).[111] C. Gramsch and M. Rigol, Phys. Rev. A (2012) 053615. EFERENCES
Phys. Rev. A (2014) 013609.[113] M. Eckstein and M. Kollar, Phys. Rev. Lett. (2008) 120404.[114] J. Mossel and J.-S. Caux,
New J. Phys. (2012) 075006.[115] J.-S. Caux and R. M. Konik, Phys. Rev. Lett. (2012) 175301.[116] J.-S. Caux and F. H. L. Essler,
Phys. Rev. Lett. (2013) 257203.[117] M. Collura, S. Sotiriadis and P. Calabrese,
Phys. Rev. Lett. (2013) 245301.[118] S. Sotiriadis and P. Calabrese,
J. Stat. Mech.: Theor. Exp. (2014) P07024.[119] J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer and M. J. Davis,
Phys.Rev. A (2015) 023611.[120] J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer and M. J. Davis, NewJ. Phys. , 045010 (2016), [ ].[121] P. Calabrese, F. H. L. Essler and M. Fagotti, Phys. Rev. Lett. (2011) 227203.[122] P. Calabrese, F. H. L. Essler and M. Fagotti,
J. Stat. Mech. Theor. Exp. (2012)P07016, [ ].[123] P. Calabrese, F. H. L. Essler and M. Fagotti,
J. Stat. Mech. Theor. Exp. (2012)P07022, [ ].[124] F. H. L. Essler, S. Evangelisti and M. Fagotti,
Phys. Rev. Lett. (2012)247206.[125] M. Kastner and M. van den Worm,
Physica Scripta (2015) 014039.[126] J. Mossel and J.-S. Caux,
New J. Phys. (2010) 055028.[127] B. Pozsgay, J. Stat. Mech.: Theor. Exp. (2013) P07003.[128] M. Mierzejewski, P. Prelovˇsek and T. Prosen,
Phys. Rev. Lett. (2014)020602.[129] D. Fioretto and G. Mussardo,
New J. Phys. (2010) 055015.[130] B. Pozsgay, J. Stat. Mech.: Theor. Exp. (2011) P01011.[131] G. Mussardo,
Phys. Rev. Lett. (2013) 100401.[132] J. Cardy,
Phys. Rev. Lett. (2014) 220401.[133] M. Perarnau-Llobet, A. Riera, R. Gallego, H. Wilming and J. Eisert,
ArXive-prints (2015), [ ].[134] D. Rossini, A. Silva, G. Mussardo and G. E. Santoro,
Phys. Rev. Lett. (2009) 127204.[135] A. Mitra and T. Giamarchi,
Phys. Rev. Lett. (2011) 150602.[136] L. Foini, L. F. Cugliandolo and A. Gambassi,
Phys. Rev. B (2011) 212404.[137] L. Foini, L. F. Cugliandolo and A. Gambassi, J. Stat. Mech.: Theor. Exp. (2012) P09011.[138] A. Pal and D. A. Huse,
Phys. Rev. B (2010) 174411.[139] M. Serbyn, Z. Papi´c and D. A. Abanin, Phys. Rev. Lett. (2013) 127201.
EFERENCES
Phys. Rev. Lett. (2013) 067204.[141] D. A. Smith, M. Gring, T. Langen, M. Kuhnert, B. Rauer, R. Geiger et al.,
NewJ. Phys. (2013) 075011.[142] T. Langen, M. Gring, M. Kuhnert, B. Rauer, R. Geiger, D. A. Smith et al., Eur.Phys. J. ST (2013) 43, [ ].[143] M. Moeckel and S. Kehrein,
Phys. Rev. Lett. (2008) 175702.[144] M. Moeckel and S. Kehrein,
Ann. Phys. (NY) (2009) 2146, [ ].[145] M. Moeckel and S. Kehrein,
New J. Phys. (2010) 055016, [ ].[146] A. Rosch, D. Rasch, B. Binz and M. Vojta, Phys. Rev. Lett. (2008) 265301.[147] M. Kollar, F. A. Wolf and M. Eckstein,
Phys. Rev. B (2011) 054304.[148] J. Marino and A. Silva, Phys. Rev. B (2012) 060408.[149] M. van den Worm, B. C. Sawyer, J. J. Bollinger and M. Kastner, New J. Phys. (2013) 083007.[150] M. Marcuzzi, J. Marino, A. Gambassi and A. Silva, Phys. Rev. Lett. (2013)197203.[151] F. H. L. Essler, S. Kehrein, S. R. Manmana and N. J. Robinson,
Phys. Rev. B (2014) 165104.[152] N. Nessi, A. Iucci and M. A. Cazalilla, Phys. Rev. Lett. (2014) 210402.[153] M. Fagotti,
J. Stat. Mech.: Theor. Exp. (2014) P03016.[154] G. P. Brandino, J.-S. Caux and R. M. Konik,
Phys. Rev. X (2015) 041043.[155] B. Bertini and M. Fagotti, J. Stat. Mech.: Theor. Exp. (2015) P07012.[156] M. Babadi, E. Demler and M. Knap,
Phys. Rev. X (2015) 041005.[157] B. Bertini, F. H. L. Essler, S. Groha and N. J. Robinson, Phys. Rev. Lett. (2015) 180601.[158] M. Buchhold, M. Heyl and S. Diehl,
ArXiv e-prints (2015), [ ].[159] M. Stark and M. Kollar,
ArXiv e-prints (2013), [ ].[160] G. S. Agarwal,
Z. Phys. (1973) 409.[161] L. M. Sieberer, A. Chiocchetta, A. Gambassi, U. C. T¨auber and S. Diehl,
Phys.Rev. B (2015) 134307.[162] J. Berges, AIP Conf. Proc. (2005) 3, [ hep-ph/0409233 ].[163] J. Berges, S. Borsanyi and C. Wetterich,
Nucl. Phys.
B727 (2005) 244,[ hep-ph/0505182 ].[164] M. Kronenwett and T. Gasenzer,
Appl. Phys. B (2011) 469, [ ].[165] T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, V. Gritsev andE. Demler,
Phys. Rev. Lett. (2010) 255302.[166] T. Kitagawa, A. Imambekov, J. Schmiedmayer and E. Demler,
New J. Phys. (2011) 073018. EFERENCES
Commun. Math. Phys. (1972) 251.[168] M. Cramer, C. M. Dawson, J. Eisert and T. J. Osborne, Phys. Rev. Lett. (2008) 30602.[169] J. Eisert, M. Cramer and M. B. Plenio,
Rev. Mod. Phys. (2010) 277.[170] M. Vengalattore, J. Guzman, S. R. Leslie, F. Serwane and D. M. Stamper-Kurn, Phys. Rev. A (2010) 053612.[171] J. Guzman, G.-B. Jo, A. N. Wenz, K. W. Murch, C. K. Thomas and D. M.Stamper-Kurn, Phys. Rev. A (2011) 063625.[172] R. Barnett, A. Polkovnikov and M. Vengalattore, Phys. Rev. A (2011) 023606.[173] T. Gasenzer and J. M. Pawlowski, Phys. Lett.
B670 (2008) 135, [ ].[174] J. Berges and G. Hoffmeister,
Nucl. Phys.
B813 (2009) 383, [ ].[175] S. Mathey, T. Gasenzer and J. M. Pawlowski,
Phys. Rev. A (2015) 023635.[176] A. P. Orioli, K. Boguslavski and J. Berges, Phys. Rev. D (2015) 025041,[ ].[177] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov . CUP, Cambridge, UK,1995.[178] W. Vinen,
J. Low Temp. Phys. (2006) 7.[179] M. Tsubota,
J. Phys. Soc. Jpn. (2008) 111006, [ ].[180] J. Berges and D. Sexty, Phys. Rev. Lett. (2012) 161601, [ ].[181] M. J. Davis, T. M. Wright, T. Gasenzer, S. A. Gardiner and N. P. Proukakis, in
Universal Themes of Bose-Einstein Condensation , edited by D. W. Snoke, N. P.Proukakis and P. B. Littlewood (Cambridge University Press) to appear 2016,
ArXiv e-prints (2016), [ ].[182] B. Svistunov,
J. Mosc. Phys. Soc. (1991) 373.[183] Y. Kagan, B. V. Svistunov and G. V. Shlyapnikov, [Zh. Eksp. Teor. Fiz. 101,528 (1992)] Sov. Phys. JETP (1992) 279.[184] D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. (1995) 3093.[185] D. Semikoz and I. Tkachev, Phys. Rev. D (1997) 489, [ hep-ph/9507306 ].[186] B. Nowak, D. Sexty and T. Gasenzer, Phys. Rev. B (2011) 020506(R),[ ].[187] B. Nowak, J. Schole, D. Sexty and T. Gasenzer, Phys. Rev. A (2012) 043627,[ ].[188] Y. Kagan and B. V. Svistunov, [Zh. Eksp. Teor. Fiz. 105, 353 (1994)] Sov.Phys. JETP (1994) 187.[189] J. Schole, B. Nowak and T. Gasenzer, Phys. Rev. A (2012) 013624,[ ].[190] E. Kozik and B. Svistunov, Phys. Rev. Lett. (2004) 035301. EFERENCES
J. Low Temp. Phys. (2009) 215,[ ].[192] C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau and S. Rica,
Phys. Rev.Lett. (2005) 263901.[193] C. Scheppach, J. Berges and T. Gasenzer, Phys. Rev. A (2010) 033611,[ ].[194] J. Berges and D. Sexty, Phys. Rev. D (2011) 085004, [ ].[195] T. Gasenzer, B. Nowak and D. Sexty, Phys. Lett.
B710 (2012) 500, [ ].[196] T. Gasenzer, L. McLerran, J. M. Pawlowski and D. Sexty,
Nucl. Phys. A (2014) 163, [ ].[197] J. Berges, S. Scheffler and D. Sexty,
Phys. Lett.
B681 (2009) 362, [ ].[198] M. Schmidt, S. Erne, B. Nowak, D. Sexty and T. Gasenzer,
New J. Phys. (2012) 075005, [ ].[199] L. Onsager, Nuovo Cim. Suppl. (1949) 279.[200] C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis and B. P.Anderson, Nature (2008) 948, [ ].[201] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson,
Phys. Rev. Lett. (2010) 160401, [ ].[202] T. W. Neely, A. S. Bradley, E. C. Samson, S. J. Rooney, E. M. Wright, K. J. H.Law et al.,
Phys. Rev. Lett. (2013) 235301.[203] A. S. Bradley and B. P. Anderson,
Phys. Rev. X (2012) 041001.[204] M. T. Reeves, B. P. Anderson and A. S. Bradley, Phys. Rev. A (2012) 053621.[205] M. T. Reeves, T. P. Billam, B. P. Anderson and A. S. Bradley, Phys. Rev. Lett. (2013) 104501.[206] M. T. Reeves, T. P. Billam, B. P. Anderson and A. S. Bradley,
Phys. Rev. A (2014) 053631.[207] T. P. Billam, M. T. Reeves, B. P. Anderson and A. S. Bradley, Phys. Rev. Lett. (2014) 145301.[208] T. Simula, M. J. Davis and K. Helmerson,
Phys. Rev. Lett. (2014) 165302.[209] M. T. Reeves, T. P. Billam, B. P. Anderson and A. S. Bradley,
Phys. Rev. Lett. (2015) 155302.[210] T. P. Billam, M. T. Reeves and A. S. Bradley,
Phys. Rev. A (2015) 023615.[211] A. J. Groszek, T. P. Simula, D. M. Paganin and K. Helmerson, Phys. Rev. A ,043614 (2015), [ ].[212] X. Yu, T. P. Billam, J. Nian, M. T. Reeves and A. S. Bradley, ArXiv e-prints (2015), [ ].[213] H. Salman and D. Maestrini,
ArXiv e-prints (2016), [ ].[214] M. Karl, B. Nowak and T. Gasenzer,
Sci. Rep. (2013), [ ]. EFERENCES
Phys. Rev. A (2013) 063615,[ ].[216] S. Heupts, Master thesis (unpublished), Universit¨at Heidelberg, 2014.[217] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. (2016) 025301.[218] C. Ewerz, T. Gasenzer, M. Karl and A. Samberg,
JHEP (2015) 070,[ ].[219] M. Mace, S. Schlichting and R. Venugopalan, Phys. Rev. D (2016) 074036,[ ].[220] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, Phys. Rev. D (2014) 114007, [ ].[221] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, JHEP (2014) 054, [ ].[222] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan,
Phys. Rev. Lett. (2015) 061601, [ ].[223] J. Berges, B. Schenke, S. Schlichting and R. Venugopalan,
Nucl. Phys.
A931 (2014) 348, [ ].[224] L. P. Kadanoff and G. Baym,
Quantum Statistical Mechanics . Addison-Wesley,second ed., 1995.[225] A. Bransch¨adel and T. Gasenzer,
J. Phys. B (2008) 135302, [ ].[226] A. Chiocchetta, M. Tavora, A. Gambassi and A. Mitra, Phys. Rev. B (2015)220302.[227] A. Maraga, A. Chiocchetta, A. Mitra and A. Gambassi, Phys. Rev. E (2015)042151.[228] M. Marcuzzi and A. Gambassi, Phys. Rev. B (2014) 134307.[229] V. Gurarie, Nucl. Phys. B (1995) 569.[230] B. Sciolla and G. Biroli,
Phys. Rev. B (2013) 201110.[231] A. Chandran, A. Nanduri, S. S. Gubser and S. L. Sondhi, Phys. Rev. B (2013) 024306.[232] P. Smacchia, M. Knap, E. Demler and A. Silva, Phys. Rev. B (2015) 205136.[233] A. Maraga, P. Smacchia and A. Silva, ArXiv e-prints (2016), [ ].[234] W. Ketterle, D. S. Durfee and D. M. Stamper-Kurn, in
Proceedings of theInternational School of Physics – Enrico Fermi (M. Inguscio, S. Stringari andC. E. Wieman, eds.), p. 67, IOS Press, Amsterdam, 1999.[235] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac and M. Rigol,
Rev. Mod.Phys. (2011) 1405.[236] F. Gerbier, Europhys. Lett. (2004) 771.[237] P. Kr¨uger, S. Hofferberth, I. E. Mazets, I. Lesanovsky and J. Schmiedmayer, Phys. Rev. Lett. (2010) 265302.
EFERENCES
Rev. Mod. Phys. (2013) 1191.[239] D. S. Petrov, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. Lett. (2000) 3745.[240] R. Folman, P. Kr¨uger, J. Schmiedmayer, J. H. Denschlag and C. Henkel, Adv.At. Mol. Opt. Phys. (2002) 253.[241] J. Reichel and V. Vuletic, Atom chips . John Wiley & Sons, 2011.[242] R. Grimm, M. Weidem¨uller, and Yu. B. Ovchinnikov,
Adv. At. Mol. Opt. Phys. (2000) 95.[243] I. Bloch, Nature Phys. (2005) 23.[244] N. D. Mermin and H. Wagner, Phys. Rev. Lett. (1966) 1133.[245] P. C. Hohenberg, Phys. Rev. (1967) 383.[246] M. Olshanii,
Phys. Rev. Lett. (1998) 938.[247] V. Dunjko, V. Lorent and M. Olshanii, Phys. Rev. Lett. (2001) 5413.[248] A. Cronin, J. Schmiedmayer and D. Pritchard, Rev. Mod. Phys. (2009) 1051.[249] I. E. Mazets, T. Schumm and J. Schmiedmayer, Phys. Rev. Lett. (2008)210403.[250] A. Andreev,
Sov. Phys. JETP (1980) 1038.[251] S. Tan, M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. (2010) 090404.[252] T. Langen,
Non-equilibrium Dynamics of One-Dimensional Bose Gases .Springer International Publishing, Cham, 2015.[253] R. Folman, P. Kr¨uger, D. Cassettari, B. Hessmo, T. Maier and J. Schmiedmayer,
Phys. Rev. Lett. (2000) 4749.[254] S. Hofferberth, I. Lesanovsky, T. Schumm, A. Imambekov, V. Gritsev,E. Demler et al., Nature Phys. (2008) 489, [ ].[255] V. Gritsev, E. Altman, E. Demler and A. Polkovnikov, Nature Phys. (2006)705.[256] A. Imambekov, V. Gritsev and E. Demler, Ultracold Fermi gases, Proc. Internat.School Phys. Enrico Fermi, 2006 , ch. Fundamental noise in matterinterferometers. IOS Press, Amsterdam, The Netherlands, 2007.[257] T. Schweigler, V. Kasper, S. Erne, B. Rauer, T. Langen, T. Gasenzer et al.,
ArXiv e-prints (2015), [ ].[258] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth,I. Bar-Joseph et al.,
Nature Phys. (2005) 57.[259] S. Hofferberth, I. Lesanovsky, B. Fischer, J. Verdu and J. Schmiedmayer, NaturePhys. (2006) 710.[260] D. Adu Smith, S. Aigner, S. Hofferberth, M. Gring, M. Andersson,S. Wildermuth et al., Optics Express (2011) 8471.
EFERENCES
Phys. Rev. A (2011) 023618.[262] B. Rauer, P. Griˇsins, I. E. Mazets, T. Schweigler, W. Rohringer, R. Geiger et al., Phys. Rev. Lett. (2016) 030402.[263] P. Griˇsins, B. Rauer, T. Langen, J. Schmiedmayer and I. E. Mazets,
Phys. Rev.A (2016) 033634.[264] T. W. B. Kibble, J. Phys. A: Math. Gen. (1976) 1387.[265] W. H. Zurek, Nature (1985) 505.[266] P. Laguna and W. H. Zurek,
Phys. Rev. Lett. (1997) 2519–2522.[267] B. Damski and W. H. Zurek, Phys. Rev. Lett. (2010) 160404.[268] A. Lamacraft,
Phys. Rev. Lett. (2007) 160404.[269] H. Saito, Y. Kawaguchi and M. Ueda, Phys. Rev. A (2007) 013621.[270] H. Saito, Y. Kawaguchi and M. Ueda, Phys. Rev. A (2007) 043613.[271] B. Damski and W. H. Zurek, Phys. Rev. Lett. (2007) 130402.[272] M. Uhlmann, R. Sch¨utzhold and U. R. Fischer, Phys. Rev. Lett. (2007)120407.[273] R. W. Cherng, V. Gritsev, D. M. Stamper-Kurn and E. Demler, Phys. Rev. Lett. (2008) 180404.[274] M. Baraban, H. F. Song, S. M. Girvin and L. I. Glazman,
Phys. Rev. A (2008) 033609.[275] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus et al., Phys. Rev. Lett. (2009) 195302.[276] J. D. Sau, S. R. Leslie, D. M. Stamper-Kurn and M. L. Cohen,
Phys. Rev. A (2009) 023622.[277] S. Mukerjee, C. Xu and J. E. Moore, Phys. Rev. B (2007) 104519.[278] K. Nam, B. Kim and S. J. Lee, J. Stat. Mech.: Theor. Exp. (2011) P03013.[279] M. Karl. PhD thesis, Ruprecht-Karls-Universit¨at Heidelberg, 2016.[280] P. Gagel, P. P. Orth and J. Schmalian,
Phys. Rev. Lett. (2014) 220401.[281] S. Sorg, L. Vidmar, L. Pollet and F. Heidrich-Meisner,
Phys. Rev. A (2014)033606.[282] E. Timmermans, P. Tommasini, M. Hussein and A. Kerman, Phys. Rep. (1999) 199.[283] R. B¨ucker, A. Perrin, S. Manz, T. Betz, C. Koller, T. Plisson et al.,
New J.Phys. (2009) 103039.[284] W. S. Bakr, J. I. Gillen, A. Peng, S. F¨olling and M. Greiner, Nature (2009)74.[285] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch and S. Kuhr,
Nature (2010) 68, [ ]. EFERENCES
Science (2011) 336, [ ].[287] E. Kaminishi, T. Mori, T. N. Ikeda and M. Ueda,
Nature Phys. (2014) 1050,[ ].[288] R. B¨ucker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller et al., NaturePhys. (2011) 608.[289] R. B¨ucker, T. Berrada, S. V. Frank, J. Schaff, T. Schumm, J. Schmiedmayeret al., J. Phys. B: At. Mol. Opt. Phys. (2013) 104012.[290] S. van Frank, A. Negretti, T. Berrada, R. B¨ucker, S. Montangero, J.-F. Schaffet al., Nature Commun. (2014).[291] S. van Frank, M. Bonneau, J. Schmiedmayer, S. Hild, C. Gross, M. Cheneauet al., Arxiv e-prints (2015), [ ].[292] S. Nascimb`ene, N. Navon, K. Jiang, F. Chevy and C. Salomon,
Nature (2010) 1057.[293] N. Goldman, I. Spielman et al.,
Rept. Progr. Phys. (2014) 126401.[294] V. Galitski and I. B. Spielman, Nature (2013) 49.[295] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig et al.,
Nature (2014) 198.[296] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. Hazzard, A. M. Rey et al.,
Nature (2013) 521.[297] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut et al.,
Nature (2016) 194.[298] I. Bloch, J. Dalibard and S. Nascimbene,
Nature Phys. (2012) 267.[299] C.-L. Hung, V. Gurarie and C. Chin, Science (2013) 1213.[300] J. Schmiedmayer and J. Berges,
Science341