Nonequilibrium dynamics of closed interacting quantum systems.

Anatoli Polkovnikov1, Krishnendu Sengupta2, Alessandro Silva3, Mukund Vengalattore4

1Department of Physics, Boston University, Boston, MA 02215, USA2 Theoretical Physics Department, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India3 The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy4 Department of Physics, Cornell University, Ithaca, NY, 14853,USA

This colloquium gives an overview of recent theoretical and experimental progress in the areaof nonequilibrium dynamics of isolated quantum systems. We particularly focus on quantumquenches: the temporal evolution following a sudden or slow change of the coupling constants ofthe system Hamiltonian. We discuss several aspects of the slow dynamics in driven systems andemphasize the universality of such dynamics in gapless systems with specific focus on dynamicsnear continuous quantum phase transitions. We also review recent progress on understandingthermalization in closed systems through the eigenstate thermalization hypothesis and discussrelaxation in integrable systems. Finally we overview key experiments probing quantum dynamicsin cold atom systems and put them in the context of our current theoretical understanding.

Contents

I. Introduction 1

II. Nearly adiabatic dynamics in quantum systems 3A. Universality in a nutshell 3B. Universal dynamics near quantum critical points 3C. Slow dynamics in gapped and gapless systems. 7D. Effects of finite temperature. 8E. Open problems 9

III. Effects of Integrability and its breaking: ergodicityand thermalization. 9A. Quantum and classical ergodicity. 10B. Nonergodic behavior in integrable systems: the

generalized Gibbs ensemble 11C. Breaking integrability: eigenstate thermalization. 14D. Outlook and open problems: quantum KAM threshold

as a many-body delocalization transition ? 15

IV. Experimental progress in quantum dynamics incold atoms and other systems 16

V. Outlook 20

VI. acknowledgements 20

References 20

I. INTRODUCTION

In the past two decades the outlook of condensed mat-ter physics has been deeply and unexpectedly revolu-tionized by a few experimental breakthroughs in atomicphysics, quantum optics and nanoscience. In synthesis,crucial advances in these fields have made it possibleto realize artificial systems (e.g. optical lattices, quan-tum dots, nanowires) that are described to a very highdegree of accuracy by models (e.g. Hubbard, Kondo,and Luttinger models) whose physics has been a subjectof intense investigation in various contexts ranging fromhigh temperature superconductivity to low temperaturetransport in metals. It is fair to say that this experi-mental progress has changed the way theory and exper-

iment look at each other. In the past, effective modelswere largely devised to explain the low energy physicsof highly complex systems. The situation has now beenreversed so that one can experimentally realize and simu-late the physics of such models. On one hand, the designand realization of interacting many-body systems couldin principle be used to perform practical tasks, such asquantum information processing (Farhi et al., 2001). Onthe other hand, direct simulations of simple models couldhelp resolving important problems in condensed matterphysics. But most importantly, the availability of ex-perimental controllable systems whose properties can beaccurately described by simple models provides unprece-dented opportunity to explore several new frontiers ofcondensed matter physics including the nonequilibriumdynamics in closed interacting quantum systems.

Equilibrium systems can often be understood usinga combination of a mean field theory, renormalizationgroup, and universality. This allows us to understandlow temperature experimental data obtained in complexsystems, such as interacting electrons in solids, in termsof simple effective models containing a few relevant pa-rameters. Away from equilibrium the situation is muchless clear. While some progress was made in the past forclassical systems (Schmittmann and Zia, 1995), there areno rigorously justified generalizations of any of these ap-proaches to generic quantum nonequilibrium systems. Itis thus not obvious that the theoretical study of the dy-namics of simplified models would accurately describe ex-periments of more complex systems. In addition there arefewer available tools for analyzing dynamics of even sim-ple interacting models. In this respect cold atomic gasesand nanostructures make possible what would be arduousotherwise: a fruitful comparison between nonequilibriumtheories based on simple models and carefully designedexperiments with tunable system parameters.

Finding systematic ways to understand the nonequilib-rium physics of interacting systems is not only of funda-mental importance, but could also be crucial for futuretechnologies. A quantum computer, for example, will

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definitely require the capability of performing real timemanipulations of interacting quantum systems. Thoughlarge scale quantum computers are yet to be on the hori-zon, it is evident that quantum coherent dynamics will beone of the focus points of various experimental systemsand of future technologies.

Nonequilibrium dynamics is a potentially a vast field:there are many ways to take a system out of equilibrium,such as applying a driving field or pumping energy orparticles in the system through external reservoirs as intransport problems. It is thus of utter importance tofocus on simple, yet fundamental protocols. In this Col-loquium we will concentrate on the simplest paradigm:the study of the nonequilibrium dynamics of closed inter-acting quantum systems following a change in one of thesystem parameters (quantum quench). Such a change,which could be either fast or slow, is particularly in-teresting when it takes the system through a quantumphase transition involving macroscopic changes in thestate of the many-body system at the initial and the finalpoint. Seminal work in this direction includes ground-breaking experiments (Greiner et al., 2002a,b) showingboth the feasibility of observing a quantum phase transi-tion in cold atoms and the possibility of observing quan-tum coherent dynamics. Following this work, a numberof different experiments explored the dynamics of coldatom systems driven across BCS-BEC crossover (Re-gal, 2006), polar and ferromagnetic phases of spinor con-densates (Sadler et al., 2006), insulating and superfluidphases of ultracold bosons (Tuchman et al., 2006) andmany others (see Ref. (Bloch et al., 2008) for a review).

These experiments stimulated an active theoretical re-search in the relatively unexplored area of quantum dy-namics in closed interacting systems. An interestingcharacteristic common to these systems is that despiteof the absence of energy exchange with an environmentand of the consequent global relaxation, it neverthe-less frequently possible to look at the long time dynam-ics and characterize it in terms of an asymptotic stateattained by physical (measurable) observables (Crameret al., 2008; Flesch et al., 2008; Gogolin et al., 2011; Lin-den et al., 2009; Reimann, 2008; Rigol et al., 2008). Inconnection to this, it is possible to categorize recent re-search on the subject of this Colloquium in two mainquestions:

• What is universal in the dynamics of a system fol-lowing a quantum quench ?

• What are the characteristics of the asymptotic,steady state reached after a quench ? When is itthermal ?

In this Colloquium we will discuss both of these ques-tions extensively. We shall outline our current level ofunderstanding of these issues and chart out the outstand-ing open questions in the field. In Sec. II we will focuson the first question and describe, from various points ofview, the universal aspects of nearly adiabatic dynamics

near quantum critical points as well as in generic gappedand gapless systems. We will argue that the proximityto the adiabatic limit allows us to make specific univer-sal predictions of scaling of various quantities such as thedefect density and heat with the quench rate.

In Sec. III, we will discuss recent progress in under-standing thermalization of a quantum system following aquench. In classical systems active interest in this topicwas stimulated by the celebrated work of Fermi Pastaand Ulam on the dynamics of a one-dimensional (1D)anharmonic chain (Campbell et al., 2005; Fermi et al.,May 1955; Porter et al., 2009) which demonstrated theabsence of such thermalization. It was realized muchlater that the nonlinearity of the interaction is not suf-ficient for thermalization which occurs, in this system,only if the initial amplitude of interaction exceeds a cer-tain threshold (Izrailev and Chirikov, 1966). Below thisthreshold, the solution splits into solitons and retainsits quasi-periodic nature (Zabusky and Kruskal, 1965)which is a consequence of the Kolmogorov-Arnold-Moser(KAM) theorem (Tabor, 1989). In quantum systems,the question of sufficient criteria for thermalization hasremained largely unaddressed so far. Some experimentalprogress in this direction has been made by a recent ex-periment from Kinoshita et. al. (Kinoshita et al., 2006)on non-thermalizing dynamics of 1D bosons with shortrange interactions. This experiment constitutes the firstclear demonstration of the fact that a nearly integrablequantum interacting many-particle system does not ther-malize for a very long time. Currently, the question ofextension of KAM theorem to quantum systems is a sub-ject of active theoretical debate (see e.g. Ref. (Olshaniiand Yurovsky, 2009)).

Finally, we note that many important topics concern-ing the physics of closed interacting systems did notfind space in this Colloquium. Most important amongthese are the tools that are being developed to de-scribe theoretically the physics of interacting systemsout of equilibrium. Among such methods we men-tion density-matrix renormalization group (DMRG) andtime-evolving block decimation (TEBD) for analyzingequilibrium and nonequilibrium 1D systems (Schollwock,2005; Schollwock and White, 2006; Vidal, 2003, 2004;White, 1992) and higher dimensional ones (Verstraeteet al., 2008), the Keldysh technique (Kamenev andLevchenko, 2009) which is particularly helpful for deriv-ing quantum kinetic equations, and closely related func-tional integral methods (Gasenzer, 2009; Plimak et al.,2001; Rey et al., 2005). Cold atom experiments alsoprompted rapid developments in phase space methods,where quantum dynamics is represented as an evolu-tion in the classical phase space (Blakie et al., 2008;Polkovnikov, 2010). These methods were originally de-veloped and applied to various problems in single-particledynamics (Hillery et al., 1984; Zurek, 2003) and inde-pendently in quantum optics in the context of coherentstates (Gardiner and Zoller, 2004; Walls and Milburn,1994). There are other reviews available in literature

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(see the references above) which specifically target theseareas.

II. NEARLY ADIABATIC DYNAMICS IN QUANTUMSYSTEMS

A. Universality in a nutshell

Universality (or insensitivity to microscopic details) isone of the crucial concepts of modern condensed matterphysics. It naturally emerged from one of the milestonesof modern physics: the renormalization group. In con-densed matter physics universality is a very powerful toolfor the understanding of continuous (second order) phasetransitions, both classical (Chaikin and Lubensky, 1995;Landau and Lifshitz, 1980) and quantum (Sachdev, 1999;Sondhi et al., 1997; Vojta, 2003). As a consequence of thedivergence of the correlation length, a system undergo-ing such a continuous phase transition is typically scaleinvariant in the vicinity of the critical point and can becharacterized by relatively simple massless field theories,which permit a classification of perturbations driving thesystem away from the critical point. Consequently, uni-versality manifests itself in the scaling behavior of var-ious quantities such as the order parameter, (free) en-ergy, susceptibilities and correlations functions near thecritical point. In this review we will focus mostly onquantum phase transitions occurring at zero tempera-ture upon the variation of a control parameter λ througha critical point λc. A standard example of universalityis the fact that the exponent ν characterizing the diver-gence of the correlation length ξ ∼ 1/|λ − λc|ν near thequantum critical point (QCP) is insensitive to the mi-croscopic details of the system and depends only on theuniversality class of the transition, determined by the di-mensionality, overall symmetries and range of the inter-actions. For classical (thermal) phase transitions similaruniversality manifests in the divergence of the relaxationtime τrel ∼ 1/|λ − λc|zν , where z is the dynamical crit-ical exponent. For quantum phase transitions the expo-nent z can be associated with a vanishing energy scale∆ ∼ |λ− λc|zν , which can be either a gap or a crossoverscale where the spectrum changes qualitatively. By theuncertainty principle this energy scale corresponds toa divergent time scale, which typically describes thecrossover in the scaling behavior of unequal time cor-relation functions. Phase transitions can be also char-acterized by singular susceptibilities, which are in turnconnected through the fluctuation-dissipation theorem tothe correlation functions of conjugate variables (e.g., themagnetic susceptibility is related to the correlation func-tion of the magnetization). At critical points these corre-lation functions have often power law scaling behavior atlong distances, e.g. 〈m(x)m(x′)〉 ≈ 1/|x− x′|2α. The ex-ponent α sets the scaling dimension of the correspondingoperator m(x): dim[m(x)] = α. Because similar correla-tion functions can enter different susceptibilities not all

the scaling exponents are independent but must satisfyscaling relations (Chaikin and Lubensky, 1995; Vojta,2003).

As mentioned in the introduction, the idea of univer-sality makes it possible to interpret experimental data ob-tained in real systems in terms of effective models witha few parameters. Universality can be ultimately un-derstood using the renormalization group, which showsthat as a system is coarse grained to lower energies andlonger length scales, more and more parameters of itsoriginal, ab initio description become unimportant (ir-relevant), while the remaining few (relevant) parametersdefine an effective low energy model. A standard exampleof universality in this context is the scaling relation be-tween energy and momentum of elementary excitations,ε ∝ kz, controlled by the dynamical exponent z whichdepends on the symmetries of the system. In particular,z = 1 in most phases characterized by a continuous bro-ken symmetry (crystals, superfluids, anti-ferromagnets),z = 2 in ferromagnets, where there is an additional con-servation law of the order parameter.

Universality is well established and understood in equi-librium. It is, however, crucial for many experimentallyrelevant situations to understand the extent to which thisconcept can be extended to out of equilibrium physics.Can irrelevant interactions turn out to be important awayfrom equilibrium? Since there are many ways to take asystem out of equilibrium, for which specific protocolswill universality emerge and which details of the proto-col are potentially important? Below we will focus on re-cent studies addressing these important issues in closedinteracting quantum systems, and in particular on thedynamics of a system whose parameters are dynamicallytuned either through a quantum critical point, or in gen-eral within a gapless/gapped phase.

B. Universal dynamics near quantum critical points

Let us start by considering the simplest nonequilibriumprotocol (Dziarmaga, 2005; Polkovnikov, 2005; Zureket al., 2005): the system is prepared in its ground stateand is then driven through a QCP by changing an exter-nal parameter λ in time. As long as the rate change of thegap in the spectrum ∆ 1 caused by changing λ is smallerthan the square of the gap one can expect the system toapproximately follow the ground state adiabatically (wewill revisit this statement in the next section). However,the vanishing of the gap at λ = λc implies that the sys-tem will always violate adiabaticity close to the quantumcritical point, no matter how slowly the parameters arechanged. It is then natural to ask how many excitations

1 We use the word “gap” for brevity. However, the system can begapless on one or both sides of the transition (e.g. superfluid-insulator transition). Then ∆ would denote a crossover energyscale vanishing at the QCP.

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will be generated while passing though the critical pointand how their density as well as generated entropy andenergy will depend on the rate of change of λ.

A similar question known under the name of theKibble-Zurek mechanism (KZ) (Kibble, 2007, 1976;Zurek, 1985, 1996) has been addressed in the past decadesfor classical phase transitions. In that case the excita-tion density of defects is described by a simple scalingargument (Zurek, 1985, 1996). Suppose that the tuningparameter, for example external temperature T , slowlydecreases in time across the critical value Tc: T = Tc−υt.The system will respond adiabatically (quasi-statically ifthe system is not thermally insulated) up to some closevicinity of the critical point, where adiabaticity will beviolated as a result of the divergence of the relaxationtime (τrel ∼ 1/|T −Tc|zν) and the dynamics will becomediabatic (sudden). The adiabatic response is once againresumed after the system moves out of the vicinity ofthe critical point. Zurek suggested a very simple crite-rion for separating such adiabatic and diabatic (impulse)regimes: the time to reach the critical point t = |T−Tc|/υshould be equal to the relaxation time. This immedi-ately introduces the time and length scales characterizingthe adiabatic to diabatic crossover: t? ∼ 1/|υ|zν/(zν+1),ξ? ∼ 1/|υ|ν/(zν+1). The violation of adiabaticity impliesthat order can not form on distances larger than ξ? lead-ing to the formation of a domain structure with a char-acteristic distance ξ? between the domain boundaries.In two and three dimensional systems, when the orderparameter is characterized by a continuous broken sym-metry, the points where several domains meet correspondto vortices or vortex rings. These are robust topologicalexcitations with a very long life time (see Fig. 1). Sinceξ? determines the average distance between the defectstheir density is given by a simple universal expression

nex ∼ (ξ?)d ∼ |υ|dν/(zν+1). (1)

The universality of the KZ prediction above is manifest inthe appearance of the universal critical exponents z and νin the scaling law. This scaling was confirmed in experi-ments in liquid crystals (Ducci et al., 1999). Experimentsin other systems (superconductors (Maniv et al., 2003),arrays of Josephson junctions (Monaco et al., 2006)) ob-served the production of topological defects with a powerlaw scaling on the quench rate but gave a different ex-ponent. The KZ scaling was also confirmed theoreti-cally using stochastic dynamics (Ginzburg-Landau dy-namics with Langevin noise or Glauber dynamics) wheretemperature changes in time (Krapivsky, 2010; Lagunaand Zurek, 1997; Rivers, 2001; Yates and Zurek, 1998),though there are also works suggesting various modi-fications (Biroli et al., 2010; Hindmarsh and Rajantie,2000). One can also interpret Eq.(1) as a measure ofnon-adiabaticity near the critical point. It is naturallyexpected that other measures like non-adiabatic energyproduction and entropy generation will display similaruniversality. These measures might be preferable overnex in situations where it is difficult to identify defects.

30 µm

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A

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Time after quench

FIG. 1 Defect generation after a quench in a spinorcondensate - These images show the transverse magnetiza-tion density of spinor condensates for variable evolution timesafter a quench to a ferromagnetic state, revealing a spatiallyinhomogeneous formation of ferromagnetic domains. The ori-entation φ and amplitude A are depicted by the color andbrightness according to the color wheel shown. Inset: Aninstance of a spin vortex spontaneously created during thequench. For reference, the length scale corresponding to thecharacteristic healing length ξ is also shown. (Adapted fromRef. (Sadler et al., 2006). See Sec. IV for more details.)

Finding the scaling of these quantities remains an openquestion.

The arguments above were recently generalized to thecrossing of quantum phase transitions (Dziarmaga, 2005;Polkovnikov, 2005; Zurek et al., 2005) (for recent reviewson this subject see Refs. (Dziarmaga, 2010; Gritsev andPolkovnikov, 2010). As discussed before in the quantumcase the parameter to be varied is not temperature Tbut rather the coupling λ tuning the system through thequantum critical point. In order to obtain the scalingfor the number of excitations produced in the quantum

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case let us first recall the Landau-Zener analysis of thecrossover between adiabatic and nonadiabatic dynamicsin a simple driven two-level system:

Hlz = g(t)σz + ∆σx., (2)

where g(t) = υt. If the system was initially prepared inthe ground state at t → −∞, the probability of transi-tion to the excited state at t → +∞ is (Landau, 1932;Majorana, 1932; Stuckelberg, 1932; Zener, 1932)

pex = exp[−πγ], (3)

where we introduced the Landau-Zener parameter γ =∆2/υ. Notice that the limit γ � 1 corresponds to theadiabatic limit with an exponentially suppressed tran-sition probability while γ � 1 corresponds to the dia-batic limit where the transition happens with probabil-ity close to unity. Hence when the rate of change ofthe energy splitting between two levels becomes larger orcomparable to the energy splitting squared one observesa crossover from adiabatic to diabatic dynamics. An al-ternative qualitative explanation of this result has beerecently formulated (Damski and Zurek, 2006).

The Landau-Zener argument can be straightforwardlyextended to the crossing of a QCP. The characteristicenergy scale which changes in time is now the gap ∆.As we discussed earlier this gap universally depends onthe tuning parameter λ: ∆(λ) ∼ |λ − λc|zν ∼ |υt|zν ,where we assumed that the dependence λ(t) can be lin-earized near the QCP: λ(t) ≈ λc + υt. Comparing therate of change of the gap with its square, i.e. solvingthe equation d∆/dt ≈ ∆2, we find the energy scale atwhich adiabaticity breaks down is ∆? ∼ |υ|zν/(zν+1). Atthis point the system is characterized by the length scaleξ? ∼ |υ|−ν/(zν+1), which can be interpreted as the typ-ical length scale of fluctuations of the order parameter.Beyond this point the adiabatic approximation breaksdown and fluctuations at longer scales cannot adiabati-cally follow the ground state. This results in the creationof defects with typical distance ξ between them and den-sity nex ∼ |ξ?|d ∼ |υ|dν/(zν+1). This scaling is identicalto the classical one, Eq. (1) with λ → T , and was pro-posed independently in Refs. (Polkovnikov, 2005; Zureket al., 2005). There is a simple quasi-particle interpreta-tion for this scaling: assuming that the excitations in thesystem are characterized by isolated quasi-particles then

their density can be found from nex ≈∫ ∆?

0dερ(ε), where

ρ(ε) is the single-particle density of states near the QCP.In uniform d-dimensional systems ρ(ε) ∼ εd/z−1, whichagain reproduces Eq. (1).

The scaling in Eq.(1) was verified in a series of ex-act solutions of the dynamics across the QCP in in-tegrable systems whose dynamics can be mapped intoa series of Landau-Zener transitions of a few quasi-particle modes. In particular, it has been verified forvarious spin models in one and two dimensions whichcan be mapped to noninteracting fermions (Mukherjeeet al., 2007), for models where low energy excitations

near phase transitions can be described by bosonic Gold-stone modes (Dziarmaga et al., 2008; Lamacraft, 2007;Polkovnikov, 2005), the sine-Gordon model, where el-ementary excitations are solitons and breathers withfractional statistics (De Grandi et al., 2008, 2010b),graphene (Dora et al., 2010; Dora and Moessner, 2010).This scaling was also extended to disordered systems,like a disordered Ising spin chain, where it was foundthat nex ∼ 1/ log2(υ) (Caneva et al., 2007; Dziarmaga,2006), as expected from Eq. (1) due to the divergence ofthe exponent z near the critical point (Fisher, 1995).

The scaling in Eq.(1) can be generalized to the case ofnonlinear dependence of the tuning parameter on time,λ(t) ∼ λc ± υ|t|r, where considerations similar to thoseleading to Eq. (1) give (Barankov and Polkovnikov, 2008;De Grandi et al., 2010a; Sen et al., 2008):

nex ∼ |υ|dν/(zνr+1). (4)

In all cases υ in Eq. (4) plays the role of the adiabaticparameter: the limit υ → 0 corresponds to the adiabaticlimit (this interpretation is valid even for instantaneousquenches r = 0, where υ plays the role of the quenchamplitude). This suggests the dynamics can be sys-tematically analyzed using adiabatic perturbation theory(De Grandi and Polkovnikov, 2010; Polkovnikov, 2005;Rigolin et al., 2008), i.e. expanding the transition am-plitudes to the instantaneous eigenstates of the systemin powers of υ. Using such analysis in Refs. (De Grandiet al., 2010a,b) it was shown that the scaling (4) can bederived from the scaling of the adiabatic fidelity, definedas the overlap of the time-dependent wave function withthe instantaneous ground state:

F (t) = |〈ψ(t)|ψgs(t)〉|. (5)

In particular, for λ(t) = λc + υtr/r!θ(t), where θ(t) is astep function,

Pex(υ) = 1− F (t)2 ≈ Ld|υ|2χ2r+2(λc), (6)

where

χ2r+2(λ) =1

Ld

∑n 6=0

|〈n|V |0〉|2

(En − E0)2r+2, (7)

is the adiabatic (fidelity) susceptibility of the order 2r+2(χ2 is the conventional fidelity susceptibility (Gu and Lin,2009)). Here En are the eigenenergies and V is the op-erator coupled to the parameter λ: V = ∂λH(λ)|λ=λc .If the perturbation is local and spatially uniform , i.e.V =

∫ddxu(x), then the scaling dimension of the adia-

batic fidelity susceptibility is obtained from a straight-forward generalization of the result of Refs. (Gu andLin, 2009; Venuti and Zanardi, 2007), i.e. dim[χ2r+2] =2∆u − 2z(r + 1) − d, where ∆u ≡ dim[u] is the scalingdimension of u(x).

Let us now discuss from this general perspective thearguments leading to the generalized scaling relation

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Eq. (4). If the scaling dimension of the susceptibility isnegative, this implies that χ2r+2 diverges at the criticalpoint. In this case from Eq. (6) we find that asymptoti-cally at υ → 0

Pex(υ) ∼ |υ|2L2d+2z(r+1)−2∆u . (8)

From Eq. (8) we see that the probability of excit-ing the system becomes of order one when L ∼1/|υ|1/(d+z(r+1)−∆u). This length scale can be inter-preted as the typical distance between elementary exci-tations (defects) and thus we find that instead of Eq. (8)we get

nex ∼ |υ|d/(d+z(r+1)−∆u). (9)

This expression reduces to Eq. (4) if u(x) is a relevantoperator driving the system to the new phase. Indeed,in this case λ

∫ddxu(x) should have the same scaling di-

mension as the gap, i.e. z, which immediately impliesthat the scaling dimension of u(x) is ∆u = d + z − 1/νand that dim[χ2r+2] = d− 2zr − 2/ν (De Grandi et al.,2010b; Schwandt et al., 2009). Notice finally that if thescaling dimension of χ2r+2 is positive then the asymp-totics in Eq. (8) gives a subleading correction to the reg-ular analytic part, Pex ≈ const Ld υ2, coming from thehigh energy (ultra-violet) contribution to the susceptibil-ity. We will discuss its importance in the next section.

Other possible generalizations of the scaling law Eq.(1)involve studies of defect production in systems where thedynamics describes the passage through quantum crit-ical lines. A concrete example of such a situation oc-curs in the transverse-field XY model (Divakaran et al.,2008; Mukherjee et al., 2008). Here the quench takes onethrough a gapless line where the critical point occurs at

the same momenta (~k = 0 for the present case) at eachpoint on the line. A detailed analysis shows that in suchcases, for critical lines with z = ν = 1, the defect den-sity still obeys a universal scaling law albeit with a dif-ferent power: n ∼ υ1/3 (Divakaran et al., 2008; Mondalet al., 2009; Mukherjee et al., 2008). The second situationinvolves the 2D Kitaev model (Sengupta et al., 2008),where a quench once again involves the passage througha gapless line with an energy gap vanishing for differentmomenta at different points on the line. It can be shownthat in such a case, the defect density scales as

√υ for

2D Kitaev model instead of the expected n ∼ υ behaviorfor 2D systems with z = ν = 1 (Sengupta et al., 2008). Ageneralization of these results for linear quenches throughcritical lines with arbitrary z and ν has also been workedout (Mondal et al., 2008, 2009; Mukherjee et al., 2008;Sengupta et al., 2008). Many other situations involv-ing anisotropic phase transitions and quenching throughmulti-critical points were analyzed in literature leading tovarious deviations from the scaling (4) (Bermudez et al.,2009; Deng et al., 2008; Sen and Vishveshwara, 2010).

In order to detect experimentally the density of ex-citations generated by passing through a QCP, it isevident that one should distinguish between situations

where such excitations are long-lived quasi-particles (asfor nearly integrable systems) or decay after being cre-ated (as for non-integrable systems). In the first case,the presence of excitations above the ground state couldfor example be detected by measuring correlation func-tions long after the quench. This has been shown for aQuantum Ising chain linearly tuned through its quan-tum critical point (Cherng and Levitov, 2006). Thepresence of defects with respect to the ferromagneticground state lead to exponentially decaying correlationsof the order parameter superimposed, for slow enoughquenches, to characteristic oscillations with period scal-ing with the quench velocity. This second feature isobserved for abrupt quenches as well (Sengupta et al.,2004) and is a consequence of the integrability of themodel (Rossini et al., 2010). If in turn we consider ageneric non-integrable system, it is necessary to expressdeviations from adiabaticity in terms of quantities, suchas the excess energy or the entropy generated by passingthrough the QCP, which are not sensitive to the decay ofquasi-particles, but can still be related to the density ofexcitations created close to the quantum critical point.Energy can be unambiguously determined both exper-imentally and numerically for both integrable and non-integrable systems and its scaling with the rate of quenchcan be used to differentiate between different nonadia-batic regimes (Eckstein and Kollar, 2010; Moeckel andKehrein, 2010; Polkovnikov and Gritsev, 2008). The ex-cess energy or equivalently heat (Q) (Polkovnikov, 2008b)generated during the quench process is in general univer-sal if the process ends near the critical point. The scalingofQ is associated with the singularity of the susceptibilityχ2r+1 at the critical point (De Grandi and Polkovnikov,2010), which implies that for relevant perturbations inthe thermodynamic limit

Q ∼ |υ|(d+z)ν/(zνr+1). (10)

Unless one considers cyclic processes, the drawback ofusing heat as a measure of non-adiabaticity is that it ishard to separate it from the adiabatic part of the energychange, corresponding to the limit υ → 0. Moreover,if the position of the QCP is not exactly known, theheat becomes sensitive to the nonuniversal details of thespectrum at the final point of the evolution. A way outcould be to measure the higher moments of the energy orthe whole distribution function of the energy, connectedto the statistics of the work (Silva, 2008; Talkner et al.,2007) in finite size systems. In particular, in the case ofabrupt quenches close to critical points the statics of thework is characterized by sharp edge singularities (Paraanand Silva, 2009; Silva, 2008). A related natural measureof non-adiabaticity is obtained by focusing on the entropysince entropy is conserved only for slow (adiabatic) pro-cesses, while is expected to increase as the system passesthrough the QCP. Moreover entropy production can bedetected experimentally in certain systems, e.g. in coldatoms by driving the system to the weakly interactingregime, where the relation between entropy and energy

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is known (Luo et al., 2007). Theoretically, the quantifi-cation of entropy production in a closed quantum systemis rather subtle. Indeed, the von Neumann’s entropy ofthe entire system, being conserved throughout unitaryevolution (Landau and Lifshitz, 1980), cannot be a goodcharacterization of deviations from adiabatic dynamics.However, the concept of diagonal-entropy (Polkovnikov,2008a), defined as Sd = −

∑n ρnn ln(ρnn), where ρnn are

the diagonal matrix elements of the density matrix inthe instantaneous basis, avoids this difficulty. In station-ary systems, the diagonal entropy is nothing but the vonNeumann’s entropy of the time averaged density matrix,also called diagonal ensemble. It is clear that the diago-nal entropy is generated only due to nonadiabatic transi-tions and thus satisfies the key requirement of the ther-modynamic entropy: it is conserved for adiabatic pro-cesses, and can only increase or stay constant in closedsystems if the initial state is stationary (Polkovnikov,2008a). For initial equilibrium states the diagonal en-tropy also satisfies fundamental thermodynamic relation:dE = TdS − Fdλ, where F = −〈∂λH〉 is the generalizedforce. For particular noninteracting models, the scalingof the diagonal entropy was found to be the same asthat of the density of quasi-particles (4) (De Grandiet al., 2010a,b; Mukherjee et al., 2008). It is also possi-ble to analyze the entanglement entropy (Calabrese andCardy, 2004, 2005; Refael and Moore, 2004; Vidal et al.,2003), i.e. the von Neumann’s entropy of the reduceddensity matrix of a part of the system, and in particularat its time evolution following a quench (Cincio et al.,2007; Pollmann et al., 2010; Sengupta and Sen, 2009).For specific 1D spin systems it was found that the entan-glement entropy scales logarithmically with the quenchtime (Cincio et al., 2007; Pollmann et al., 2010). Noticehowever that, at the moment, it is unclear how one canmeasure entanglement in many body systems and the en-tanglement entropy in particular (see some suggestions inRefs. (Klich and Levitov, 2009; Klich et al., 2006)) andwhat its relation with the thermodynamic entropy is.

Finally, another interesting question that has receivedattention is the connection between microscopic dynam-ics and thermodynamics in the semiclassical limit. Ingeneral, in classical systems there is no simple analogue tothe instantaneous energy levels, the key concept in anal-ysis of quantum systems. Such analogue, however, doesexist in the case of periodic motion. Then in the semiclas-sical limit the stationary levels are found from the Bohrquantization (or more accurately from the WKB approx-imation (Landau and Lifshitz, 1981)), which states thatthe reduced action in the stationary orbit should be quan-tized. In classical mechanics it is known that the reducedaction is an adiabatic invariant, i.e. it is conserved for theadiabatic evolution (Landau and Lifshitz, 1982). Fromthe previous discussion applied to quantum systems wecan deduce that near singularities like second order phasetransitions, conservation of adiabatic invariants shouldbe violated and this is indeed the case (Landau and Lif-shitz, 1982). In Refs. (Altland et al., 2009; Itin and

Torma, 2009a,b) slow dynamics was analyzed for a par-ticular many-body generalization of the Landau-Zenermodel (closely related to the Dicke model) in the semi-classical limit. It was found that the nontrivial powerlaw scaling of the number of excitations in this system(similar to Eq. (1)) follows from the changes of adiabaticinvariants near the singularity, which in turn correspondsto a quantum critical point in the thermodynamic limit.It is interesting that quantum fluctuations in this prob-lem entered only through the initial distribution of theadiabatic invariants but not through the equations of mo-tion. The corresponding truncated Wigner approxima-tion turned out to be very accurate in all regimes of thedynamics (Altland et al., 2009; Kiegel, 2009). It wouldbe very important to understand precise connections be-tween transitions among microscopic energy levels in thequantum case and changes of suitable generalizations ofadiabatic invariants in the classical limit.

C. Slow dynamics in gapped and gapless systems.

Up to now we have discussed the universal dynam-ics resulting from the variation of a control parameterλ through a quantum critical point. However the dy-namics of interacting quantum systems has interestingregimes even when the system is fully gapped or gaplessfor the entire duration of the protocol. The classifica-tion of these regimes is important in order to understanddissipation and to develop optimal protocols minimizingnon-adiabatic effects. Many of these questions are cur-rently a subject of intense theoretical research in differentcontexts, from quantum computation to transport.

The general formulas Eq.(6)-(7), which describe thedensity of excitations Pex(υ) generated by a variationof the control parameter, tell us that if the system re-mains fully gapped throughout the evolution, then Pex

and nex will have a quadratic scaling with υ wheneverthe susceptibility χ2r+2 evaluated at the initial and finalcouplings is finite (De Grandi and Polkovnikov, 2010;Rigolin et al., 2008). A similar argument shows that theheat Q is also quadratic in υ if χ2r+1 is finite. Thisquadratic scaling is characteristic of any quantum sys-tem. Let us point that in the standard Landau-Zenerproblem in the slow limit ∆2 � υ if we start in theground state at t → −∞ and let the system evolve upt → +∞, the transitions to excited states are exponen-tially suppressed as a result of destructive interferencebetween multiple transitions (Vitanov, 1999; Vitanov andGarraway, 1996). At the same time in the intermediatestages of the evolution the transition probability reachesmuch higher values which scale only quadratically withthe rate υ. For example, if one considers a process whichstarts at t0 → −∞ the transition probability to the in-stantaneous excited state at the moment t in the slowlimit υ/∆2 � 1 can be approximated by (Vitanov, 1999):

pex ≈υ2∆2

16(g(t)2 + ∆2)3. (11)

8

If t > 0 there is an additional exponential term whichleads to Eq. (3) at t → +∞. If the process starts att0 = 0 exactly in the symmetric point, where g(0) = 0the quadratic asymptotics (11) is also recovered (Cuc-chietti et al., 2007). This scaling occurs as a result ofthe discontinuity of the first derivative of g(t) at themoment where the process starts or ends or followinga discontinuity in any other point of the evolution (seee.g. Refs. (Damski and Zurek, 2006; Divakaran et al.,2010) for particular cases). Likewise if there is a discon-tinuity in the second order derivative of g(t) asymptoti-cally the transition probability in the LZ problem scalesquadratically with acceleration. More generally for theprotocol g(t) = g0 + υ(t − t0)r/r! θ(t − t0), where θ(t)is the step function, one can show that (De Grandi andPolkovnikov, 2010)

pex(t→∞) ≈ υ2∆2

16(g20 + ∆2)2r+1

. (12)

As we discussed in the previous section this formula ap-plies even to sudden transitions (r = 0) where it reducesto the result of the ordinary perturbation theory. Thesame expression applies to the reverse process. If boththe initial and final couplings are finite then the result-ing transition probability is asymptotically determinedby the sum of probabilities associated with discontinu-ities of derivatives of g(t) at the initial and final timesof the evolution plus additional interference terms whichhighly oscillate in the slow limit. Let us point out thatin the LZ problem (and in general in gapped systems)one can suppress power law asymptotics of the transitionprobability by starting and ending the protocol infinitelysmoothly, e.g. g(t) ∼ g0 + g1 exp[−τ/(t − t0)]θ(t − t0).In this case only the non-analytic term in the transitionprobability survives and we are back to Eq. (3) where υis the time derivative of g(t) near the symmetric pointwhere g(t) = 0.

In gapless systems the situation becomes qualitativelydifferent. In this case the adiabatic susceptibilities candiverge leading to non-analytic dependence of the corre-sponding quantities on υ, as in the case of the crossing ofa quantum critical point. For example, it is straightfor-ward to see that for marginal perturbations in a genericgapless phase the scaling dimension of the adiabatic sus-ceptibility dim[χ2r+2] = d− 2zr. It becomes negative inlow dimensions d < 2zr leading to a non-analytic scal-ing of the density of excitations with υ. Thus dependingon dimensionality in gapless systems one expects at leasttwo different regimes of the response of the system toa slow external perturbation: analytic and non-analytic.These regimes were first suggested in Ref. (Polkovnikovand Gritsev, 2008) together with a third regime whereadiabaticity is violated in the thermodynamic limit andQ or nex become proportional to a power of the systemsize or some other large length scale associated with someirrelevant operator. In this regime, which can be realizedin low-dimensional bosonic systems, the scaling Eq. (1)is violated. At the moment it is unclear how generic it is

and what sets the scaling of various quantities.

A close inspection of the adiabatic susceptibility showsthat in general the analytic (quadratic) part of the heatand energy of excitations on υ comes from the high en-ergy (or ultra-violet) part of the spectrum, while thenon-analytic part comes from low energies. This was in-deed shown to be the case in several situations, fromthe sine-Gordon model (De Grandi et al., 2010b), to theFalicov-Kimball model (Eckstein and Kollar, 2010) andthe turning on interactions in a Fermi liquid (Moeckeland Kehrein, 2010). As we pointed above the ultra-violettransitions can be suppressed by avoiding discontinuitiesin λ(t) and its derivatives. However, this is not neces-sarily the case for the low energy non-analytic contribu-tion. To see this we need to reexpress the excess energy(or density of excitations) in terms of the total time ofthe process τ . Doing this it was found that in an in-sulating, gapped phase, the details of the protocol areimportant and smoother protocols lead to a suppressionof non-adiabatic effects, while in a gapless phase makingλ(t) smoother does not affect the heating (Eckstein andKollar, 2010). This result can be again understood by an-alyzing the scaling dimension of the susceptibility χ2r+1.According to our discussion for generic gapless systemsits scaling dimension is negative when d + z < 2zr.Then Q ∼ υ(d+z)/zr (this result immediately follows fromEq. (10) by taking the limit ν →∞). On the other hand,υ is related to the total quench time as υ ∼ 1/τ r. Thuswe see that in this case Q ∼ 1/τ (d+z)/z, i.e. indeed in-dependent on r. On the other hand for positive scalingdimension of χ2r+1 which is the case for d + z > 2zrand which is always true in gapped systems we haveQ ∼ υ2 ∼ 1/(τ)2r. Since in the adiabatic limit τ is largewe see that indeed the heat can be suppressed by increas-ing r and making λ(t) smoother. An interesting openquestion is finding an optimal protocol for minimizing thenon-adiabatic effects within given time τ . It is plausiblethat the optimal power is determined by a vanishing scal-ing dimension of the corresponding adiabatic susceptibil-ity χ2r+1. The questions of finding protocols minimizingnon-adiabatic effects for gapped systems (with possibil-ity of crossing isolated quantum critical points) were alsoaddressed by approximately minimizing the transitionprobability and identifying the Riemannian metric ten-sor underlying the adiabatic evolution (Rezakhani et al.,2010, 2009). Studying the optimization of the protocoltaking a system through a QCP it was found the optimalexponent of λ(t) ∼ |t|rsign(t) near the QCP scales log-arithmically with the quench time, r ∝ ln(τ) (Barankovand Polkovnikov, 2008). This result was also extendedto systems with external confining potential (Collura andKarevski, 2010) .

D. Effects of finite temperature.

In the discussion above we always implicitly assumedthat the system is initially prepared in the ground state.

9

An interesting and genuine question is how finite tem-perature effects modify the picture. In isolated systemstemperature enters through initial conditions: the sys-tem is prepared in the initial finite temperature equilib-rium state and is then dynamically driven out of equi-librium. How is the response of the system affectedby the initial thermal fluctuations ? One naturally ex-pects that while the transitions to high energy states(quadratic in υ) will not be affected by small temper-atures in the system, the transitions to the low energystates, which determine the non-analytic contribution toheat and density of excitations, will be very sensitive totemperature. In Refs. (De Grandi et al., 2010a,b) (seealso Ref. (Gritsev and Polkovnikov, 2010)) studying aparticular Sine-Gordon model in the two limits where itcould be mapped to free bosons and free fermions, it wasshown that the statistics of quasi-particles enters the scal-ing of both Q and nex making dynamics more adiabatic(compared to the zero temperature case) for fermions dueto Pauli blocking and less adiabatic for bosons due toBose enhancement. These results were not yet extendedto generic interacting systems.

Another aspect of thermalization, the influence of thecoupling to an environment setting the temperature onthe slow dynamics near quantum critical points, has beenstudied in Refs. (Fubini et al., 2007; Mostame et al., 2007;Patane et al., 2009a; Patane et al., 2008; Patane et al.,2009b). This setup allows one to analyze the effects ofthermal smearing and of dephasing/dissipation on thedynamics of a quantum critical system. Using a combi-nation of kinetic equations and scaling arguments it wasfound that in this situation the excess energy has twouniversal contributions, one still given by Eq. (10), whilethe second involving a universal power of temperaturereplacing the universal power of υ (Patane et al., 2008).

E. Open problems

While the physics described above is definitely an im-portant example of the emergence of universality in thedynamics of interacting quantum systems, it is evidentlya piece, albeit important, of the puzzle that has to becomposed in order to understand to which extent thestandard concepts of statistical physics can be applied tononequilibrium problems. Understanding the meaningof relevance or irrelevance of a perturbation in genericnonequilibrium processes, extending the notion of uni-versality to nonequilibrium systems, as well as the con-cept of renormalization group, is a task (or dream) thatcertainly requires the solution of many specific problems,and a close comparison between experiments and theory.

So far most of the theoretical research focused on ana-lyzing slow dynamics for global quenches, where the ex-ternal perturbation couples to the whole system. Howthese results can be extended to local or spatially nonuni-form perturbations is an open question. At one extremelimit, one can imagine performing a quench only locally.

Then the rest of the system could be seen as a thermalbath. The analysis of a special case of dynamics of atransverse field Ising model where the tuning parame-ter linearly depends both on time and space has shownthat excitations are generally suppressed by nonunifor-mity of the tuning parameter (Dziarmaga and Rams,2010). This suggests that quantitative and qualitativedifferences may emerge when some of the symmetries ofthe system, e.g. translational, are broken in the quenchprocess.

Another important issue concerns the connections be-tween adiabaticity in thermodynamics and microscopicdynamics. One of the consequences of the thermody-namic adiabatic theorem is that no heat can be gen-erated in an isolated system during an infinitesimallyslow process. More generally according to the secondlaw of thermodynamics in the Thompson’s (Kelvin’s)form for any cyclic process the system can only in-crease its energy, i.e. the heat should be always non-negative as long as one starts in equilibrium. Thisstatement, which is obvious if the system is initially inthe ground state, has been proven microscopically for aclass of passive initial states (Allahverdyan and Nieuwen-huizen, 2002, 2005; Boksenbojm et al., 2010; Thirring,2002), whose initial density matrix is stationary (diag-onal) and monotonically decreasing function of energy:(ρn − ρm)(εm − εn) ≥ 0. This statement also directlyfollows from analyzing transitions between microscopicenergy levels (Polkovnikov, 2008b). Likewise many state-ments of thermodynamics related to behavior of entropyincluding the second law and fundamental thermody-namic relations are recovered using the conecpt of diago-nal entropy (Polkovnikov, 2008a). At the same time thereare many open questions remaining: what are the timescales involved in the definition of adiabaticity ? Howone can microscopically define adiabatic time scales ininteracting systems and why these time scales are muchshorter than inverse distance between many-body levels(see e.g. discussion in Ref. (Balian, 1991)) ? And finally,what is the role of integrability in nonequilibrium ther-modynamics? These questions are closely connected tothe microscopic origin of conventional dissipation, whichin turn is also very likely related to the combination ofnonadiabatic creation of the elementary excitations andtheir following relaxation or dephasing. From the discus-sion above, we can anticipate anomalous dissipation nearcritical points and in gapless low-dimensional systems.

III. EFFECTS OF INTEGRABILITY AND ITSBREAKING: ERGODICITY AND THERMALIZATION.

Let us now turn to one of the most natural questionsto be addressed when studying the dynamics of a closedmany-body quantum system: are interactions within thesystem sufficient to make the system behave ergodically ?If we focus on local degrees of freedom, e.g. a few spins ina spin chain, can the rest of the system be always thought

10

as an effective thermal bath? And if this is not possible,are there some observable effects on the system dynam-ics? While these questions are definitely connected toquantum ergodicity (Goldstein et al., 2010), a topic witha long history dating back to the early days of quantummechanics (Deutsch, 1991; Mazur, 1968; von Neumann,1929; Pauli and Fierz, 1937; Peres, 1984; Srednicki, 1994;Suzuki, 1971), the past few years have brought a greatdeal of progress in the context of closed many-body sys-tems. The main motivation came from recent experi-ments on low dimensional cold atomic gases describedin some detail in Sec. IV in this Colloquium (Greineret al., 2002b; Kinoshita et al., 2006). The experimentalavailability of essentially closed (on the time scales of ex-periments) strongly correlated systems together with theawareness of the conceptual importance of these issuesin a number of areas (e.g. transport problems, many-body localization, integrable and non-integrable dynam-ics) have stimulated a lot of interest on quantum ther-malization. Below we will give a synthetic view on anumber of recent important developments on this sub-ject, starting with the discussions of the general conceptsof ergodicity and thermalization, and then moving to thediscussion of many-body systems and integrability.

A. Quantum and classical ergodicity.

While the idea of ergodicity is well defined in classicalmechanics, the concept of quantum ergodicity is some-what less precise and intuitive. Classically, an interactingsystem of N particles in d dimensions is described by apoint X in a (2 d N)-dimensional phase space. The intu-itive content of the word ”ergodic”, i.e. the equivalence ofphase space and time averages, can be then formalized byrequiring that if we select an initial condition X0 havinginitial energy H(X0) = E, where H is the Hamiltonianof the system, then

δ(X −X(t)) ≡ limT→∞

1

T

∫ T

0

dt δ(X −X(t)) = ρmc(E),

(13)where ρmc(E) is the microcanonical density of the systemon the hyper-surface of the phase space of constant en-ergy E, andX(t) is the phase space trajectory with initialcondition X0. Of course if this condition is satisfied by alltrajectories, then it is also true for every observable. Weimmediately see that in order to have ergodicity, the dy-namics cannot be arbitrary: the trajectories X(t) haveto cover uniformly the energy hyper-surface for almostevery initial condition X0.

The most obvious quantum generalization of this no-tion of ergodicity is arduous (von Neumann, 1929). Letus first of all define a quantum microcanonical densitymatrix: given a Hamiltonian with eigenstates | Ψα〉 ofenergies Eα, a viable definition of the microcanonicalensemble is obtained by coarse graining the spectrumon energy shells of width δE, sufficiently big to con-tain many states but small on macroscopic scales. De-

noting by H(E) the set of eigenstates of H having en-ergies between E and E + δE, we define ρmc(E) =∑α∈H(E) 1/N | Ψα〉〈Ψα |, where N is the total num-

ber of states in the micro-canonical shell. Let us nowask the most obvious question: given a generic initialcondition made out of states in a microcanonical shell,| Ψ0〉 =

∑α∈H(E) cα | Ψα〉, is the long time average

of the density matrix of the system given by the micro-canonical density matrix? The answer to this questionfor a quantum system is, unlike in the classical case, al-most always no, as J. von Neumann realized already in1929 (von Neumann, 1929). More precisely, if we assumethe eigenstates of the system not to be degenerate, thetime average is

| Ψ(t)〉〈Ψ(t) | =∑α

| cα |2| Ψα〉〈Ψα |= ρdiag, (14)

where | Ψ(t)〉 is the time evolved of | Ψ0〉. This object isknown in the modern literature as the diagonal ensem-ble (Rigol, 2009; Rigol et al., 2008, 2007). Notice nowthat the most obvious definition of ergodicity, i.e. therequirement ρmc = ρdiag, implies that | cα |2= 1/N forevery α, a condition that can be satisfied only for a veryspecial class of states. Quantum ergodicity in the strictsense above is therefore almost never realizable (Gold-stein et al., 2010; von Neumann, 1929).

Our common sense and expectations, which very fre-quently fail miserably in the quantum realm, make usnevertheless believe that, in contrast with the argumentsabove, macroscopic many-body systems should behaveergodically almost always, unless some very special con-ditions are met (e.g. integrability). The key to under-stand ergodicity in therefore to look at quantum systemsin a different way, shifting the focus on observables ratherthan on the states themselves (Mazur, 1968; von Neu-mann, 1929; Peres, 1984). Given a set of macroscopicobservables {Mβ} a natural expectation from an ergodicsystem would be for every | Ψ0〉 on a microcanonical shellH(E)

〈Ψ(t) |Mβ(t) | Ψ(t)〉 →t→+∞ Tr[Mβ ρmc] ≡ 〈Mβ〉mc,(15)

i.e. that looking at macroscopic observables long after thetime evolution started makes the system appear ergodicfor every initial condition we may choose in H(E). Oneneeds a certain care in defining the infinite time limithere, since literally speaking it does not exist in finitesystems because of quantum revivals. A proper way tounderstand this limit is to require that Eq. (15) holds inthe long time limit at almost all times. Mathematicallythis means that the mean square difference between theLHS and RHS of Eq. (15) averaged over long times isvanishingly small for large systems (Reimann, 2008). Toavoid dealing with these issues ergodicity can be definedusing the time average, i.e. requiring that

〈Ψ(t) |Mβ(t) | Ψ(t)〉 = Tr[Mβ ρdiag] = 〈Mβ〉mc. (16)

Notice that if the expectation value of Mβ relaxes to awell defined state in the sense described above, this state

11

will coincide with the time averaged state and the twodefinitions of ergodicity Eq.(15)-(16) will be equivalent.If the conditions above are satisfied then in loose termsρmc can be considered as equivalent to ρdiag . J. vonNeumann proved that if the system satisfies some verynatural requirements (e.g. absence of resonances), andthe set {Mβ} is constructed in such a way as to definemacrostates of the system, which obviously requires theobservables to be coarse grained on the various micro-canonical shells H(E) and mutually commuting, then aform of ergodicity is observed (sometimes referred to anormal typicality). In particular, for every | Ψ0〉 andalmost every set {Mβ} the diagonal and microcanoni-cal ensembles are equivalent (Goldstein et al., 2010; vonNeumann, 1929). More recently it was proven that thewhole density matrix of a small subsystem of a biggersystem which is placed initially in a typical eigenstateis described by the canonical ensemble (Popescu et al.,2006). In Ref. (Gogolin et al., 2011) these results werefurther extended to the problem of measurement and de-coherence. Particular care is nevertheless needed in relat-ing these statements to the dynamics and thermalizationof actual many body systems, since physical initial con-ditions in quenched system almost never correspond toeigenstates of a new Hamiltonian.

B. Nonergodic behavior in integrable systems: thegeneralized Gibbs ensemble

While the statements above are very general, their ap-plication to specific systems is not at all straightforward.Looking at a concrete many-body system, it is of primaryinterest not just to find out whether in principle a set ofmacroscopic observables that behave ergodically exists,but whether specific and natural observables, such as themagnetization for spin chains, density for cold atomicgases, or various correlation functions behave ergodicallyor not. In this respect, experiments tell us that ergodic-ity is not at all guaranteed (Kinoshita et al., 2006) if theclosed system is integrable or nearly integrable. Whilethis fact was expected (Barouch et al., 1970; Girardeau,1969, 1970; Mazur, 1968; Suzuki, 1971) recent research onthe dynamics of integrable systems has focused on find-ing ways to predict the asymptotical states taking intoaccount integrability, i.e. the presence of many constantsof motion.

Let us discuss how this can be done qualitatively us-ing the simplest example of integrable system, a periodicharmonic chain of finite length described by the Hamil-tonian

H =

M−1∑j=1

[p2j

2m+mν2

2(xj − xj+1)2

], (17)

where xj are deviations of particles from the equilibriumpositions and pj are their momenta; we use the identi-fication xM ≡ x0. Let us imagine that initially we de-form the system in a particular way and ask how this

deformation evolves with time. We note that since thisis a harmonic system described by linear equations ofmotion the following analysis also applies to quantumsystems. From elementary physics we know that the ini-tial deformation splits into normal modes characterizedby the quasi-momenta qn = 2πn/M , where n is integern ∈ [0,M − 1], and the dispersion ωq = 2ν sin q/2. Thissystem obviously does not thermalize even at long timesbecause there is no energy exchange between modes.This does not imply though that it can not reach a welldefined asymptotic state in the long time limit (Bartheland Schollwock, 2008; Cramer and Eisert, 2010). To il-lustrate this point consider for example the displacementof the j-th atom at time t after some initial displacement:

xj(t) =1√M

M−1∑n=0

xqn(t)eiqnj . (18)

xq(t) = Aq cos[ωqt], where Aq is a complex amplitude de-termined by initial conditions (for simplicity we assumedan initial stationary state). Let us now analyze quali-tatively the dynamics of this system. At short times,provided that the initial modulation is smooth and onlymodes with small momenta (q � π) are excited we canlinearize the spectrum ωq ≈ νq. Clearly in this case werecover periodic motion of the wave-packet with a periodequal to the ratio of the system size and sound velocity:T = M/ν. This persistent motion is characteristic of theabsence of any relaxation. However, as time gets longerdeviations of the dispersion from linear become more im-portant. In particular, when t∗(ωn+1 +ωn−1− 2ωn) ∼ 1,where ωn is the central frequency of the wavepacket, cor-relations between phases among the different modes arelost and they can be treated as essentially random num-bers. For our model t∗ ∼ M2/ωn. At long times t � t∗

the different momentum modes become uncorrelated andthe system reaches the asymptotic stationary state in asense we defined earlier (it can be found close to thatstate at almost all times). For this asymptotic state theonly relevant information about the initial conditions isencoded in the M mode amplitudes |Aq| or equivalentlyin their squares |Aq|2 proportional to the occupancies ofthe modes of energy Eq, which are the integrals of mo-tion. Note though that there are special modes corre-sponding to momenta q and −q which are exactly degen-erate. The correlations between Aq and A−q = A?q thusnever disappear and in general one needs to fix M addi-tional constraints representing the relative phases of thecomplex amplitudes. For example, if the initial configu-ration is symmetric xj = x−j then Aq is real, meaningthat the phases of all modes are identical. Then it iseasy to check that with this constraint 〈x2

j (t)〉 acquiresspatial dependence on j even in the long time limit. Thisdependence can not be recovered by fixing only modeoccupancies. Only when these phases are unimportante.g. they average to zero or if there are no degenera-cies between the normal modes the asymptotic state isfully fixed by the integrals of motion. Thus in contrast

12

with ergodic systems where only the energy needs to befixed, the long time behavior of our integrable model canbe reproduced by fixing the M integrals of motion andpossibly ∼ M other constraints if there are degenera-cies. While the number of commuting (local) integrals ofmotion is large, equal to the system size M , it is vastlysmaller than the total number of states which scales withM exponentially.

Let us now see how these considerations are transposedin many-body systems, focusing on another simple inte-grable model, the Quantum Ising chain (Sachdev, 1999)described by the Hamiltonian: H0 = −

∑i σ

xi σ

xi+1 + gσzi .

Here σx,zi are the spin operators at site i and g is thestrength of the transverse field. This model gives one ofthe simplest examples of quantum phase transition, witha quantum critical point at gc = 1 separating two mutu-ally dual gapped phases, a quantum paramagnet (g > gc)and a ferromagnet (g < gc).

In the Quantum Ising chain the local transverse magne-tization, Mx =

∑i σ

xi is a non-ergodic operator (Barouch

et al., 1970; Girardeau, 1969, 1970; Mazur, 1968). Tosee this, it is useful to employ a Jordan-Wigner trans-formation that reduces the problem to a free fermionmodel (Sachdev, 1999). In terms of the fermionic op-erators ck relative to modes of momentum k = 2πn/Lthe Hamiltonian takes the form

H = 2∑k>0

(g − cos(k))(c†kck − c−kc†−k)

+ i sin(k)(c†kc†−k − c−kck), (19)

Under this mapping the transverse magnetization be-

comes Mx = −2∑k>0(c†kck − c−kc

†−k). The eigenmodes

γk of energy Ek = 2√

(g − cos(k))2 + sin(k)2 diagonal-izing the Hamiltonian are related to the fermionic op-erators ck by a Bogoliubov rotation, ck = cos(θk)γk −i sin(θk)γ†−k, with tan(2θk) = sin(k)/(g − cos(k)). InHeisenberg representation the operators γk acquire sim-ple time dependence: γk(t) = γk(0) exp[−iEkt]. As inthe previous problem of a harmonic chain if the energiesEk are incommensurate, at sufficiently long times differ-ent momentum modes become statistically independentfrom each other. This statement does not apply to modeswith opposite momenta k and −k which have identicalenergies. However, if these correlations are not importantthen in the long time limit (see below) each mode can be

characterized by the conserved quantity nk = 〈γ†kγk〉. Letus now consider unitary dynamics of the transverse mag-netization starting with a generic initial condition | Ψ0〉.The time evolution of the operator Mx(t) expressed interms of the eigenmodes of the Hamiltonian is

Mx(t) = −2∑k>0

cos(2θk)(γ†kγk − γ−kγ†−k)

+ i sin(2θk)(γ−kγke−2iEkt − γ†kγ

†−ke

2iEkt). (20)

In the long time limit only the diagonal terms in the sumsurvive, while the off diagonal ones, describing creation or

destruction of two fermions average to zero. Therefore forany initial condition | Ψ0〉 the asymptotic value attainedby the transverse magnetization is

〈Mx(t)〉 = −2∑k>0

cos(2θk)(〈γ†kγk〉 − 〈γ−kγ†−k〉). (21)

This asymptotic value is therefore perfectly described bythe set of the occupation numbers nk.

The result above leads one to conjecture that theasymptotic state is described by a Gibbs-like statisticalensemble of the type (Rigol et al., 2007)

ρG =e−

∑k λkγ

†kγk

Z, (22)

where the Lagrange multipliers λk are fixed by requiring

that nk ≡ 〈Ψ0 | γ†kγk | Ψ0〉 = Tr[ρGγ†kγk] = 〈γ†kγk〉G.

The ensemble defined above in Eq. (22) can be seen as aparticular case of the ensemble

ρG =e−

∑α λαIα

Z, (23)

known as the generalized Gibbs ensemble (GGE) orthe maximum entropy ensemble and introduced byJaynes (Jaynes, 1957) to describe the equilibrium state ofa system possessing N constant of motions Ik. A recentconjecture (Rigol et al., 2007) proposed to use the GGEto describe the asymptotic state of a generic quantumintegrable model. This proposal had however to face twoobvious subtleties. It is first of all to be specified howto choose the Ik in Eq.(23). Indeed, if all constants ofmotion would be admissible, including non-local ones,then one would obviously and tautologically describe theasymptotic state (both for integrable and non-integrablesystems), as one can easily see by choosing as Ik the pro-jectors onto the eigenstates of the Hamiltonian. The wayout comes however by observing that in standard ther-modynamics the Gibbs ensemble emerges for small sub-systems from the assumption of statistical independencebetween sufficiently big subsystems. In this derivationthe additivity of a conserved quantity - energy - plays acrucial role. This is the reason why the probability ofa given configuration is exponential in energy and note.g. in energy squared (Kardar, 2007). Similar argu-ments apply to any additive integrals of motion so thatstatistical independence and invariance of the ensembleto the choice of a subsystem of an integrable system putsstrong constraints on the choice of the integrals of mo-tion in GGE when the latter is applied to subsystemsof an integrable system. In this respect the average oc-cupation numbers of different momentum modes used inEq. (22) become approximately additive for small subsys-tems. This approximate additivity of integrals of motionwas recently discovered in Ref. (Cassidy et al., 2011) foranother integrable system of one-dimensional hard-corebosons. In particular, it was noticed that the integrals ofmotion Iα and the lagrange multipliers λα in that case

13

can be written as a smooth functions of α/N implyingthat in large systems the argument of the exponent en-tering Eq. (23) can be written in the extensive (additive)

form:∑α λαIα ≈ L

∫ 1

0dξ, I(ξ)λ(ξ), where ξ = α/L.

This suggests that GGE can be defined through a smoothfunction λ(ξ), which replaces the temperature in the er-godic systems.

There is a second subtlety in applying the GGE toquantum systems. Here the most natural definition ofintegrability is based on requirement that the systemhas well defined quasi-particles that maintain their iden-tity upon scattering (see Ref. (Caux and Mossel, 2011)for a more detailed discussion), i.e. scattering is purelyelastic and there is no production of particles or dis-sipation associated to it (Mussardo, 2009; Sutherland,2004). This notion can be made precise in continuumintegrable models, such as the Luttinger liquid or theSinh-Gordon model, which can emerge as low energy de-scriptions of other integrable models, such as the criticalXXZ chain and the Lieb-Liniger gas. In these systemsit is natural to associate the Iα to the occupation num-bers of the quasi-particle states. More specifically, con-sidering a generic one dimensional relativistically invari-ant integrable system with say a single species of quasi-particles of mass m, energy E = m cosh(θ) and momen-tum p = m sinh(θ) (θ is the rapidity), the quasi-particles

can be described by annihilation operators A(θ) satis-

fying the algebra A(θi)A(θj) = S(θi − θj)A(θj)A(θi),where S is the S-matrix of the two particle scattering.Similar relations are valid for the products of creation,and creation-annihilation operators (Mussardo, 2009).

Since the Hamiltonian is by definition diagonal in A(θ),H =

∫dθE(θ)A†(θ)A(θ), and every eigenstate can be

written as | θ1, . . . , θn〉 = A†(θ1) · · · · · A†(θn) | 0〉, withθ1 > · · · > θn, in this case it is rather natural to postulatethe form

ρG =e−

∫dθλ(θ)A†(θ)A(θ)

Z, (24)

for the generalized Gibbs ensemble (Fioretto and Mus-sardo, 2010). This ensemble is a direct generalization ofthe GGE for the Quantum Ising Model, where S = −1.For this general class of integrable systems and a spe-cific class of translationally invariant initial states it wasindeed shown the long-time limit of the average of localoperators is well described by this ensemble (Fioretto andMussardo, 2010). Such initial states can be written as

| Ψ0〉 = N e−∫dθ K(θ) A†(θ)A†(−θ), (25)

which in turn are similar to the so-called integrableboundary states in statistical field theory (Ghoshal andZamolodchikov, 1994). Such states naturally emergein experimentally relevant systems, for example whenstudying dephasing in split quasi-1d condensates (Grit-sev et al., 2007) or in the Quantum Ising model, whenstudying a quantum quench from a transverse field γi toa transverse field γf (Silva, 2008).

A very interesting idea related to the GGE was sug-gested by Gurarie (Gurarie, 1995) to explain the steadystate of a driven nearly integrable system. It was shownthat the steady state distribution of the wave amplitudescorresponding to different momenta (see Ref. (Zakharovet al., 1992) for details) can be obtained by taking theprobability density ρ ∝ exp[−F ], where F is a (complex)combination of the approximate integrals of motion foundperturbatively. In terms of this ensemble one recovers thecorrect power law distribution of the amplitudes of waveswith the momentum and other observables.

Another view towards elucidating the validity of thegeneralized Gibbs ensemble has been pursued for specialquenches in a 1D Bose-Hubbard model (Cramer et al.,2008) and in integrable systems with free quasiparti-cles (Barthel and Schollwock, 2008). It was shown that,upon tracing all degrees of freedom of the system outsidea small region of space and under specific conditions, thelocal density matrix tends asymptotically to ρG. Morerecently a series of recent theoretical (Flesch et al., 2008)and experimental (Trotzky et al., 2011) works on thedynamics of Bose-Hubbard models has proven the re-laxation of local observables in these system to a max-imum entropy ensemble consistent with the constraintsof the dynamics. A hint towards the generalization ofρG for Bethe Ansatz integrable systems was proposedin Ref. (Barthel and Schollwock, 2008). The GGE wasalso tested in a number of models, from Luttinger liq-uids (Cazalilla, 2006; Iucci and Cazalilla, 2009) and freebosonic theories (Calabrese and Cardy, 2007), to inte-grable hard-core boson models (Rigol et al., 2007) andHubbard-like models (Eckstein and Kollar, 2008; Kol-lar and Eckstein, 2008). In all cases, it was shown tocorrectly predict the asymptotic momentum distributionfunctions for a variety of systems and quantum quenches.

At this point is should be stressed that as we discussedbefore the GGE does not always give complete descrip-tion of the asymptotic state of the system. In the simpleexample of the harmonic chain we saw that for genericinitial conditions it is necessary to specify 2N real con-stants or N complex amplitudes in order to correctly de-scribe the asymptotic state even if we focus exclusivelyon local observables. For a quantum Ising chain, more-over, ρG can be interpreted as a grand-canonical dis-tribution with an energy dependent chemical potentialµk = Ek − λk. It is evident now that if we consider

the correlations of δnk = γ†kγk − 〈γ†kγk〉, the occupation

numbers of different eigenmodes, the GGE necessarilypredicts 〈δnkδnk′〉 = 0. Likewise the GGE predicts the

correlators of the type 〈γ†k(t)γ−k(t)〉 are always equal tozero. For a generic initial state | Ψ0〉 both statements arenot necessarily true: by e.g. breaking translational in-variance in the initial state one could have 〈Ψ0 | δnkδnk′ |Ψ0〉 6= 0 and 〈Ψ0 | γ†kγ−k | Ψ0〉 6= 0. Notice that the meresurvival of off-diagonal correlations of this type when theevolution starts with a non-translationally invariant statesignals in a sense the integrability of the model, i.e. theexistence of well defined quasiparticles γk. Indeed, follow-

14

ing the argument of Gangardt and Pustilnik (Gangardtand Pustilnik, 2008), if the Hamiltonian of the system istranslationally invariant but integrability is broken, theoff-diagonal correlators are expected to decay to zero forany initial condition, thereby restoring the translationalinvariance in the asymptotic state. Finally, notice thatoff-diagonal correlations might influence the asymptoticsof physically relevant observables: a simple example isthe asymptotic value of 〈(Mx(t))2〉, which for a genericnon-translationally invariant condition | Ψ0〉 cannot bepredicted using the GGE.

A very important open question is to understand un-der which general circumstances the GGE can be ap-plied. For free fermionic and bosonic systems the GGEwas argued to hold for local observables (Barthel andSchollwock, 2008; Cramer et al., 2008). For more generalintegrable systems this is not evident at all. For examplein the case of the Quantum Ising chain the two signif-icant observables, the transverse magnetization σxi andthe order parameter σzj , are local in the spin representa-tion. However, this locality does not translate directly totheir representation in terms of the quasiparticles of themodel: while σxi retains a local character in terms of γi,σzi does not. Will the asymptotic dynamics of any localoperator be represented by the GGE, or just that of localoperators in the quasi-particle fields ? Do the symmetriesof the initial state play any role in this ? Answering thesequestions appears to be crucial to understand the role ofintegrability in the dynamics of many-body systems.

Another very important question is whether all natu-ral observables of an integrable system behave necessar-ily nonthermally. The answer to it appears to be no, aspointed out recently (Rossini et al., 2009). The key to un-derstand this issue seems to be again locality with respectto the quasi-particles diagonalizing the model. Thus inthe Quantum Ising Model it was shown that while thetransverse magentization is non-ergodic, the correlatorsof the order parameter σz following a quench of the trans-verse field relax as in a thermal state with an effectivetemperature Teff set by the initial energy of the systemE = 〈Ψ0(gi) | H(gf ) | Ψ0(gi)〉. At low Teff this relaxationappears to be universal, i.e. determined only by the lowenergy scattering properties of quasi-particles (Rossiniet al., 2009). Analogous studies for an XXZ chain hinttowards a different behavior of local and non-local opera-tors with respect to quasi-particles (Canovi et al., 2011).The situation is much less clear for quenches with higheffective temperatures, where the universal character ofthe low energy theory is lost (Barmettler et al., 2009).

C. Breaking integrability: eigenstate thermalization.

When integrability is explicitly broken with a strongenough perturbation one naturally expects ergodic be-havior to emerge for all observables (Kollath et al., 2007;Manmana et al., 2007; Rigol, 2009; Rigol et al., 2008;Roux, 2009, 2010). The quest for the necessary con-

ditions for thermalization to occur (i.e. how stronglyshould integrability be broken, which spectral propertiesshould the system display) is an important problem inmany different fields, from mathematical and statisticalphysics to quantum chaos (Deutsch, 1991; Peres, 1984;Rigol et al., 2008; Srednicki, 1994, 1999). In classical sys-tems the intense research on this subject was stimulatedby the study of dynamics of a nonlinear chain of cou-pled oscillators by Fermi Pasta and Ulam (FPU) (Fermiet al., May 1955), where instead of thermalization regularquasi-periodic oscillations were observed. Later it was re-alized that the FPU problem is nearly integrable and thatthere is a finite threshold for the chaotic behavior (Camp-bell et al., 2005) . In quantum systems the situation wasfar less clear: while different views on this issue emergedfrom time to time, the key towards a clear understandingof quantum thermalization appears to be linked to theemergence of quantum chaotic behavior (Peres, 1984).In particular, it has been proposed that the emergenceof thermal behavior is linked to the pseudo-random formof natural observables once represented in the eigenbasisof the Hamiltonian (Peres, 1984). This observation hasbeen made more precise by conjecturing that thermal-ization in quantum chaotic systems occurs eigenstate-by-eigenstate, i.e. the expectation value of a natural observ-able 〈Ψα | A | Ψα〉 on an eigenstate | Ψα〉 is a smoothfunction of its energy Eα being essentially constant oneach microcanonical energy shell (Deutsch, 1991; Sred-nicki, 1994, 1999). If this happens, then ergodicity andthermalization in the asymptotic state follow for everyinitial condition sufficiently narrow in energy (e.g. local-ized in a microcanonical shell), as one can easily under-stand using the diagonal ensemble. This hypothesis isknown as eigenstate thermalization (ETH).

In order to understand how eigenstate thermalizationcan emerge, let us consider a quantum gas of N par-ticles of mass m with hard-core interactions (Srednicki,1994). Srednicki pointed out that in the time evolution ofthis system starting with an initial condition | Ψ0〉 suf-ficiently narrow in energy, the momentum distributionwill always relax to the Maxwell-Boltzmann distributionfMB(p) as long as the eigenstates of the system | Ψα〉 canbe considered as pseudo-random superpositions of planewaves, i.e. have a diffusive nature in phase space. This re-quirement should be satisfied as a result of the chaoticityof the system, the so-called Berry’s conjecture. CallingX = (x1,x2, , . . . ,xN ) the coordinates of the particlesand P = (p1,p2, . . .pN ) their momenta, Berry’s conjec-ture states that the eigenstates have the form

Ψα(X) = N∫dPAα(P)δ(P2 − 2mEα)eiP·X, (26)

with Aα(P) being pseudo-random variables with gaus-sian statistics, 〈Aα(P)Aβ(P′)〉 = δβαδ

(3N)(P+P′)/δ(P2−P′

2). Notice that we are disregarding the symmetrization

of the wave function, see (Srednicki, 1994) for a discus-sion of this aspect. If the properties above are assumedit is easy to prove that on average in the thermodynamic

15

limit the momentum distribution function is

〈f(p)〉 =

∫dp2dp3 . . . 〈| Ψα(p,p2, . . . ) |2〉

=e−

p2

2mkT

(2πmkT )3/2= fMB(p), (27)

where the temperature is set by the equipartition law asEα = 3/2NkT . Notice that this is expected to happenfor every eigenstate of energy close to Eα, as required bythe ETH. Hence thermal behavior will follow for everyinitial condition sufficiently narrow in energy.

For generic many-body systems, such as Hubbard-likemodels and spin chains, the close relation between break-ing of integrability and quantum chaotic behavior is aknown fact (Poilblanc et al., 1993). In particular, finitesize many-body integrable systems are characterized bythe Poisson spectral statistics while the gradual breakingof integrability by a perturbation leads to a crossover tothe Wigner-Dyson statistics. The latter is typically asso-ciated, in mesoscopic systems or billiards, with diffusivebehavior and can be taken as a signature of quantumchaos (Imry, 1997). In many-body disordered systemsthe emergence of the Wigner-Dyson statistics was ar-gued to be an indicator of the transition between metallic(ergodic) and insulating (non-ergodic) phases (Mukerjeeet al., 2006; Oganesyan and Huse, 2007). Inspired bythese close analogies, recent studies gave a boost to ourunderstanding of the crossover from non-ergodic to ther-mal behavior as integrability is gradually broken and ofthe origin of ergodicity/thermalization in systems suffi-ciently far from integrability (Biroli et al., 2010; Kollathet al., 2007; Manmana et al., 2007; Rigol, 2009; Rigolet al., 2008). In particular, a careful study of the asymp-totics of density-density correlators and momentum dis-tribution function for hard-core bosons in 1d showed thatthe transition from non-thermal to thermal behavior infinite size systems takes the form of a crossover con-trolled by the strength of the integrability breaking per-turbation and the system size (Rigol, 2009). Moreoverthere is a universality in state to state fluctuations ofsimple observables in this crossover regime (Neuenhahnand Marquardt, 2010), which goes hand-by-hand withan analogous transition from Poisson to Wigner-Dysonlevel statistics (Rigol and Santos, 2010; Santos and Rigol,2010a). When integrability is broken by sufficientlystrong perturbation ergodic behavior emerges (Neuen-hahn and Marquardt, 2010; Rigol, 2009; Rigol and San-tos, 2010), which in turn appears to be related to thevalidity of the ETH (Rigol et al., 2008). In this con-text, the anomalous, non-ergodic behavior of integrablemodels has been reinterpreted as originating from widefluctuations of the expectation value of natural observ-ables around the microcanonical average (Biroli et al.,2010).

All these statements apply to the asymptotic (or timeaveraged) state. So far the relaxation in time, in par-ticular in the thermodynamic limit, has received much

less attention. In a series of studies of relaxation infermionic Hubbard models subject to quenches in theinteractions strength it has been argued that for suffi-ciently rapid quenches relaxation towards thermal equi-librium occurs through a pre-thermalized phase (Moeckeland Kehrein, 2008, 2010). Similar two step dynamics oc-curs in quenches of coupled superfluids where initial fast“light cone” dynamics leads to a pre-thermalized steadystate, which then slowly decays to the thermal equilib-rium through the vortex-antivortex unbinding (Matheyand Polkovnikov, 2010). In Ref. (Burkov et al., 2007) avery unusual sub-exponential in time decay of correlationfunctions was predicted and later observed experimen-tally (Hofferberth et al., 2007) for relaxational dynamicsof decoupled 1d bosonic systems.

D. Outlook and open problems: quantum KAM thresholdas a many-body delocalization transition ?

The arguments above clearly pointed to the connectionbetween thermalization in strongly correlated systemsand in chaotic billiards. This analogy however , ratherthan being the end of a quest, opens an entire new kindof questions, which are a current focus of both theoreti-cal and experimental research. In particular, we do knowthat in a number of models of strongly correlated parti-cles eigenstate thermalization is at the root of thermalbehavior (Rigol et al., 2008). What is the cause of eigen-state thermalization in a generic many-body system, i.e.the analogue of the diffusive eigenstates in phase spaceof Berry’s conjecture ? And most importantly, while in afinite size system the transition from non-ergodic to er-godic behavior takes the form of a crossover, what hap-pens in the thermodynamic limit ? Is the transition fromergodic to non-ergodic behavior still a crossover or it issharp (a quantum KAM threshold ) ?

At present, research on these questions has juststarted. An interesting idea that has recently emerged isthat the study of the transition from integrability to non-integrability in quantum many-body systems is deeplyconnected to another important problem at the frontierof condensed matter physics: the concept of many-bodylocalization (Altshuler et al., 1997; Basko et al., 2006),which extends the original work of Anderson on single-particle localization (Anderson, 1958). We note that re-lated ideas were put forward in studying energy transferin interacting harmonic systems in the context of large or-ganic molecules (Leitner and Wolynes, 1996; Logan andWolynes, 1990). More specifically, it has been noticedthat a transition from localized to delocalized states ei-ther in real space (Pal and Huse, 2010) or more gener-ally in quasi-particle space (Canovi et al., 2011) is closelyconnected to a corresponding transition from thermal tonon-thermal behavior in the asymptotics of significantobservables. For weakly perturbed integrable models,the main characteristic of the observables to display suchtransition appears to be again their locality with respect

16

to the quasiparticles (Canovi et al., 2011). This con-nection with many-body localization becomes more clearon the basis of a recently proposed way to quantify thetransition from non-ergodic to ergodic behavior in manybody systems (Olshanii and Yurovsky, 2009). The au-thors consider an integrable Hamiltonian H0 with a weaknon-integrable perturbation λV . Formulating essentiallya generalization of the Berry’s conjecture and makingsome additional assumptions they showed that the devi-ations from thermal behavior in the expectation value ofobservables can be quantified according to the formula

〈Ψ(t) |A |Ψ(t)〉 − 〈A〉mc ≈ η(〈Ψ0(t) |A |Ψ0(t)〉0 − 〈A〉mc),

where | Ψ0(t)〉 = exp[−iH0t] | Ψ0〉 is the time evolvedstate with respect to the integrable Hamiltonian, while| Ψ(t)〉 = exp[−i(H0 + λV)t] | Ψ0〉 evolves with thenon-integrable one. The key ingredient in this formulais the parameter η, defined as the average over themicrocanonical shell of the inverse participation ratioηα =

∑n | 〈ϕn | Ψα〉 |4, where | ϕn〉 are the eigen-

states of the integrable Hamiltonian H0 and | Ψα〉 arethe eigenstates of H0 +λV. Notice that when the systemis close to integrability η ' 1 but as the strength of Vincreases, η is roughly proportional to the inverse of thenumber of states N hybridized by the perturbation.

Using this formula it is now possible to understandhow many-body localization enters the scenario (Canoviet al., 2011; Pal and Huse, 2010): an abrupt transitionat a certain λc from localized to delocalized states inquasi-particle space corresponds to a sharp decrease ofthe inverse participation ratio η from a value O(1) to avalue negligibly small and tending to zero in the ther-modynamic limit, essentially O(1/N (λ)), where N (λ) isthe total number of states in an energy window of widthof the order of the matrix elements of the perturbation.This would lead to an abrupt transition from non-thermalto thermal behavior at λc, a transition qualitatively cor-responding to the physics of a quantum KAM thresh-old. Notice that on the delocalized side of this transi-tion the eigenstates are expected to be of diffusive na-ture (in quasi-particle space), i.e. a natural generaliza-tion of the form postulated by Berry’s conjecture. Suchtransition have been studied extensively in confined elec-tronic system, following a seminal paper by Althsuler,Kamenev, Levitov and Gefen (Altshuler et al., 1997)and on interacting electron systems with localized sin-gle particle states (Basko et al., 2006). While the de-pendence η(λ) was analyzed numerically in certain smallsystems (Canovi et al., 2011; Neuenhahn and Marquardt,2010; Santos and Rigol, 2010a,b), eventual emergence ofa sharp KAM-like threshold in the thermodynamic limitremains an open question.

The ETH also suggested a new way on looking at quan-tum relaxational dynamics as dephasing in the many-body basis. In particular, the information about theasymptotic state is fully contained in the diagonal el-ements of the density matrix, which do not change intime if the Hamiltonian is constant. So the process of

time evolution in this picture is equivalent to averagingof oscillating off-diagonal elements of the density matrixto zero. In a way this picture is different from conven-tional thinking based on kinetic theory of thermaliza-tion through collisions between quasi-particles and thetime evolution of their distribution function. This ap-parent difference is hidden in the complicated structureof the many-body eigenstates. Our intuition is based onthinking about dynamics in the approximate basis, e.g.a basis of independent quasi-particles. The precise rela-tion between the many-body and kinetic approaches tothermalization is still an open question. Another poten-tially intriguing possibility is understanding thermaliza-tion as a renormalization group process, where time evo-lution results in averaging over high-energy degrees offreedom. If one deals with approximate noninteractingbasis then because of interactions the process of elimi-nating high energy states affects dynamics of low energymodes and hence in renormalization of the low energy dy-namics. In Ref. (Mathey and Polkovnikov, 2010) it wasshown that such renormalization process indeed can ex-plain real time dynamics in a two-dimensional sine Gor-don model and the emerging nonequilibrium Kosterlitz-Thouless transition. In Ref. (Moeckel and Kehrein, 2009)similar ideas were put forward to analyze dynamics of in-teracting fermions using the flow equation method. Atthe moment it is unclear whether using such real timerenormalization group one can analyze relaxational longtime dynamics in generic interacting systems.

IV. EXPERIMENTAL PROGRESS IN QUANTUMDYNAMICS IN COLD ATOMS AND OTHER SYSTEMS

As we mentioned before, the study of nonequilibriumdynamics of quantum many-body systems has been in-creasingly motivated by a series of advances in the field ofultracold atomic and molecular gases. Due to the conflu-ence of various features, these mesoscopic quantum sys-tems are in many ways near-ideal systems for the studyof nonequilibrium quantum phenomena.

Firstly, quantum gases can exhibit a remarkably highdegree of isolation from environmental sources of deco-herence and dissipation. Thus, to an excellent approxi-mation, during duration of experiments they can be re-garded as closed quantum systems. Further, the dilutenature of these gases and exceptionally low temperaturesresult in long timescales of dynamical effects (typicallyon the order of milliseconds or longer) allowing for time-resolved studies of nonequilibrium processes resultingfrom phase-coherent many-body dynamics. Such stud-ies are hardly possible in conventional condensed mattersystems.

Secondly, an array of techniques have been developedto dynamically tune various parameters of the Hamil-tonian governing these quantum gases. This has madeit possible to realize various prototypical nonequilibriumprocesses such as quantum quenches discussed above.

17

Quenches across phase transitions have been realized toinvestigate the onset and formation of long range orderand the mechanism underlying the spontaneous forma-tion of topological defects. The latter is closely related tothe KZ mechanism described earlier in the text. A quan-titative experimental study of the defect, entropy and en-ergy production resulting from such quantum quenchesshould allow for an accurate comparison with the theo-retical predictions.

Lastly, the ability to engineer and experimentally real-ize model Hamiltonians of archetypal correlated systemscoupled with a detailed knowledge of the microscopic in-teractions make ultracold atomic gases a tantalizing sys-tem for applications ranging from the quantum simula-tion of strongly correlated systems to the adiabatic quan-tum computation. In addition to the form of the modelHamiltonian, experimental control can also be achievedover the effective dimensionality of the ultracold gas mak-ing it possible to investigate the nontrivial interplay be-tween fluctuations, interactions and dimensionality.

From a technological perspective, there is an increasingthrust towards engineering ultracold atomic many-bodysystems for applications in quantum metrology (Appelet al., 2009; Esteve et al., 2008; Leroux et al., 2010; Meiseret al., 2008; Riedel et al., 2010; Vengalattore et al., 2007).A deeper understanding of the dynamics of interactingmany-body systems and the mechanisms of decoherenceand dissipation in these systems is of crucial importancein this context.

Motivated by these factors, a number of experimentshave been performed in recent years using ultracold quan-tum gases to investigate topics including quantum coher-ent dynamics in optical lattices, quenches across quan-tum phase transitions and thermalization in low dimen-sional systems. For the purposes of this colloquium,we distinguish between classes of nonequilibrium exper-iments both in terms of the general protocol as wellas the questions being addressed by these experiments.(i) Nonequilibrium states of many body atomic systemswherein the high degree of isolation of the atomic sys-tem from the environment allows for the creation ofmetastable or highly excited many-body states with longlifetimes, (ii) Quantum quench experiments in whichone or more parameters of the Hamiltonian are changedrapidly to create an out-of-equilibrium state of the many-body system and (iii) Dynamical tuning of the Hamilto-nian in order to study quantum coherent dynamics of aninteracting many-body system.

These experimental advances have stimulated a veryactive theoretical research in the area of nonequilibriumquantum dynamics in interacting many-body systems.Among the issues most debated in recent literature isthe relation between thermalization in isolated quantumsystems and quantum integrability. In this regard, a re-cent pioneering study on thermalization in 1D Bose gaseswas performed in Ref. (Kinoshita et al., 2006). In thisexperiment, a blue detuned 2D optical lattice was used tocreate arrays of tightly confined tubes of ultracold 87Rb

atoms. The depth of the lattice potential far exceeded theenergy of the ultracold gas ensuring negligible tunnelingamong the tubes. The array of tubes was then placed in asuperposition of states of momentum ±2p0 by the appli-cation of a transient optical phase grating. The impartedkinetic energy was small compared to the energy requiredto excite the atoms to the higher transverse states and thegases remained one-dimensional. This out-of-equilibriumsystem was then allowed to evolve for variable durationsbefore the momentum distribution was probed by absorp-tion imaging of the gas (see Fig. 2).

It was found that, while the initial momentum distri-butions exhibit some dephasing on account of trap an-harmonicities, the dephased distribution remains non-gaussian even after thousands of collisions. This is indistinct contrast to the gaussian distributions observedwhen the 2D optical lattice is adiabatically imposed onan equilibrium 3D Bose gas. This remarkable observationthat the nonequilibrium Bose gases do not equilibrate onthe timescales of the experiment appears consistent withthe fact that this system is a very close experimental re-alization of a Lieb-Liniger gas with point-like collisionalinteractions - an integrable quantum system in whichonly elastic pairwise interactions can occur. Apparentlythe experimental technicalities such as anharmonicitiesor the axial potential are insufficient to sufficiently liftintegrability in this system.

In addition to unambiguously showing the absenceof thermalization within experimental timescales in thismodel realization of the Lieb-Liniger gas, this study alsopoints the way towards addressing more general ques-tions on integrability and ergodicity. Starting from anintegrable system, modifications such as the addition offinite range interactions, tunneling between the 1D tubesor the imposition of axial potentials one can tunably liftintegrability and analyze emergence of irreversability andthermalization. This experiment largely motivated muchof the theoretical work discussed in the previous section.

Another issue that has attracted a lot of attention isthe search for universal effects either in the nonequilib-rium dynamics following a quantum quench or in the adi-abatic dynamics near a quantum critical point which wedescribed earlier in the text. In particular, the issue ofnon-adiabatic dynamics near quantum phase transitionshas been the focus of recent experimental studies on con-densate formation in a dilute, weakly interacting Bosegas that is rapidly cooled past the BEC phase transition(Weiler et al., 2008). This process was found to be ac-companied by the spontaneous formation of topologicaldefects, i.e. vortices, in the nascent superfluid. This canbe phenomenologically understood as being due to theformation of isolated superfluid regions of a characteristicsize ξ, each with a random relative phase. These isolatedregions then gradually merge to give rise to global phasecoherence. In this process, regions which enclose phaseloops of 2π are constrained by the nature of the super-fluid, i.e. the continuity of the wavefunction, to have avanishing superfluid density at the core. Thus, the KZ

18

FIG. 2 Time-of-flight absorption images of an ensem-ble of 1D Bose gases - Ultracold atoms are confined inarrays of 1D optical traps. Optical pulses are used to placethe atoms in a superposition of ±2~k momentum states. Thegas is then allowed to evolve for variable durations before be-ing released from the trap and photographed to reveal themomentum distribution. The false color in each image isrescaled to show detail. The non-gaussian nature of the mo-mentum distribution clearly indicates an absence of thermal-ization.(Adapted from (Kinoshita et al., 2006))

mechanism predicts a density of vortices that scales as1/ξ2.

In this experiment, a magnetically trapped thermal gasof 87Rb atoms was cooled by radiofrequency (rf) evapo-ration to temperatures below the BEC transition tem-perature. The quench rate, i.e. the rate of cooling, wascontrolled by varying the rate at which the rf frequencywas ramped down. Following a brief duration of equili-bration, vortices are detected by absorption imaging ofthe gas after ballistic expansion. Allowing for some un-certainty in the ability to discern a vortex due to line-of-sight integration in these images, it was found that abouta quarter of the images showed at least one vortex core.

The rate of cooling during the quench was limited bythe collision rate between atoms in the trapped gas dur-ing evaporative cooling. This resulted in a limited dy-namic range for the quench rate. Also, the rapid de-crease of the thermal fraction following the formation ofthe condensate led to a low damping rate for the vortices.A faster quench rate, realized through a trap with tighterconfinement or increased density or via sympathetic cool-

ing with another species, could result in the observationof an increased number of vortices during the quench. Inturn, this would potentially allow for quantitative tests ofthe predicted scaling of vortex number with the quenchrate and the extraction of dynamic critical exponents.

While the formation of a superfluid by quenching thetemperature is seeded by thermal fluctuations, ultracoldatomic gases also potentially allow for the realization ofphase transitions initiated purely by quantum fluctua-tions (Greiner et al., 2002a; Sadler et al., 2006).

A particularly intriguing study of a quench past suchquantum phase transitions was carried out in a degener-ate F = 1 spinor Bose gas of 87Rb (Sadler et al., 2006).These gases, with a spin degree of freedom arising from anon-zero hyperfine spin F , are quantum fluids that maysimultaneously exhibit the phenomena of magnetism andsuperfluidity, both of which result from symmetry break-ing and long range order. Owing to rotational symmetry,the contact interactions between two atoms can be char-acterized by the total spin of the colliding pair. In thecase of a F = 1 spinor gas, these interactions give riseto a mean field energy given by n(c0 + c2〈F〉2) where thecoupling strengths c0,2 are related to the s-wave scatter-ing lengths in the total spin f = 0, 2 channels (Ho, 1998;Ohmi and Machida, 1998). In addition to the mean fieldinteractions, a finite external magnetic field B imposesa quadratic Zeeman energy (QZE) that scales as q〈F 2

z 〉with q = (µBB)2/4∆hf where µB is the Bohr magnetonand ∆hf is the energy splitting between the ground statehyperfine manifolds.

For a F = 1 condensate of 87Rb, the competing influ-ences of the spin-dependent interaction and the QZE giverise to a continuous quantum phase transition between a‘polar’ and a ferromagnetic phase. Rapidly tuning theexternal magnetic field from large values (q � |c2n|) tosmall values (q � |c2n|) quenches the spinor gas fromthe polar phase to the ferromagnetic phase. The ensu-ing growth of ferromagnetic domains was directly de-tected by in situ imaging (see Fig. 1). It was foundthat the resulting texture of ferromagnetic domains wasspatially inhomogeneous can characterized by a typicallength scale that was related to the spin healing lengthξ = ~/

√2m|c2n|. Concurrent with the appearance of

these domains, the spin textures revealed the sponta-neous formation of polar-core spin vortices. These topo-logical defects are characterized by a non-zero spin cur-rent but no mass current. The origin of these spin vor-tices is also rooted in the KZ mechanism. It was shownthat, for slow quenches, the number of such vortices is ex-pected to scale as τ−1/6 where τ is the time over whichthe spinor gas is swept into the ferromagnetic state (Saitoet al., 2007).

The weak spin-dependent interactions inherent to thisspinor gas also allow for nondestructive detection of thevortices and studies of their dynamics. In addition, theweak coupling between the spin and mass degrees of free-dom make it straightforward to realize extremely lowspin temperatures to examine the role of quantum fluc-

19

tuations in seeding this phase transition (Klempt et al.,2010). These features make spinor quantum fluids a richsystem to investigate the quench dynamics and KZ mech-anism past quantum phase transitions between differentmagnetically ordered phases. In addition, corrections tothe KZ scaling imposed by long range interactions (Ven-galattore et al., 2008), conservation laws and finite tem-perature effects can also be studied.

Yet another range of experimental studies is made pos-sible by the tunability of atomic interactions using a Fes-hbach resonance. This technique allows the rapid dy-namic control of the s-wave scattering length by meansof a time-varying external magnetic field. This abilitywas utilized in a recent study of a strongly interactingtwo-component Fermi mixture (Jo et al., 2009). Start-ing from an initially weak, repulsive interaction betweenthe two Fermionic species, the interactions were rapidlyincreased by tuning the magnetic field to the vicinity ofthe Feshbach resonance. The subsequent decrease in theatomic loss rate, the increase in the size of the trappedgas and the increase in kinetic energy as measured intime-of-flight images were interpreted as an indicationof the Stoner transition to a ferromagnetic state. How-ever, in a later theoretical work this interpretation wasquestioned and an alternative explanation based on rapidmolecule formation was suggested (Ref. (Babadi et al.,2009)). Thus, a direct in-situ measurement of local mag-netization is necessary to understand whether ferromag-netism plays a role in this experiment.

In addition to the thermalization dynamics acrossphase transitions, the long coherence times inherent toultracold gases also makes it possible to study the quan-tum coherent dynamics of many-particle systems. A par-ticularly dramatic instance of such coherent many-bodydynamics was illustrated in the collapse and revival of thematter wave field of a Bose condensate (Greiner et al.,2002b). Here, the interaction-induced dynamical evolu-tion of a matter wave field was clearly revealed in themultiple matter wave interference patterns obtained af-ter releasing the gas from the lattice. This work hasalso been extended to the time-resolved observation ofsuperexchange processes in optical ‘superlattice’ poten-tials (Trotzky et al., 2008). Similar demonstrations ofcollisional coherence have also been shown in spinor Bosegases (Chang et al., 2005; Kronjager et al., 2005). Dueto the internal degrees of freedom in a spinor gas, thedynamics in this fluid is due to coherent spin-mixing col-lisions. In a trapped gas that is well described by thesingle mode approximation (SMA), these coherent colli-sions can lead to the periodic and reversible formation ofcondensates in initially unpopulated spin states.

Further, in certain situations, coherent interactions canalso lead to quantum correlations (Sorensen et al., 2001).Schemes that might realize such entangled many-particlestates have received attention due to potential applica-tions in quantum information processing and metrology.The dynamical evolution of such entangled states in thepresence of quantum or thermal fluctuations is obviously

of great interest. A recent experiment investigated thisevolution in low dimensional two-component Bose gaseswith adjustable interactions (Widera et al., 2008), findingthat quantum fluctuations play a crucial role in the phasediffusion dynamics of low dimensional systems. Morerecently, the dynamical control of a Bose-Einstein con-densate confined in a strongly driven optical lattice wasdemonstrated (Lignier et al., 2007). By periodically mod-ulating the lattice potential, the tunneling parameter Jwas shown to be suppressed in a phase coherent manneropening the possibility of driving quantum phase transi-tions using this technique.

The isolation of ultracold atomic gases from externalsources of dissipation also makes it possible to study re-laxation dynamics driven purely by intrinsic mechanisms.Such mechanisms should set the timescales for adiabaticquantum computing or the simulation of strongly cor-related lattice models. A recent experiment along theselines investigated the evolution of excited states of the re-pulsive Fermi-Hubbard system (Strohmaier et al., 2010).Here, doubly occupied lattice sites (doublons) were cre-ated by modulating the lattice and the subsequent decayof the system to thermal equilibrium was monitored overtime. It was shown that the lifetime of these doublonsscales exponentially with the ratio of the interaction en-ergy to kinetic energy, in fair agreement with theoreticalpredictions. It was argued that the dominant mechanismdriving this relaxation was a high-order scattering pro-cess involving several fermions (Sensarma et al., 2010).

While this colloquium places an emphasis on experi-ments involving ultracold atomic gases, there is a range ofother mesoscopic quantum systems which also lend them-selves to studies on quantum nonequilibrium dynamics.For the sake of completeness, we briefly review a few ofthese systems here. Defect formation following a quenchwas first studied in the context of vortices in liquid crys-tals (Chuang et al., 1991). This has since been followedby similar studies in various mesoscopic systems includ-ing isolated superconducting loops where the defects as-sume the form of spontaneous fluxoids (Monaco et al.,2009), superconducting thin films (Maniv et al., 2003)and multi-Josephson junction loops (Monaco et al., 2006,2002). A cumulative view of these studies would indicatethat the influence of finite size effects, thermal fluctua-tions and dimensionality on the production of topologicaldefects by the KZ mechanism is as yet unclear and a topicthat warrants further study.

Another potential system for the study of nonequilib-rium dynamics of many-particle states arises from rapidadvances in the field of photonics. There have been sev-eral proposals (Chang et al., 2008; Greentree et al., 2006)for the dynamical creation of strongly correlated pho-tonic states using photon-photon interactions mediatedby a nonlinear optical medium. The realization of statessuch as a Tonks gas of photons have been proposed usinghollow-core optical fibers, tapered optical fibers, photonsin coupled cavities and surface plasmons on conductingnanowires. Such strongly correlated photon states should

20

have applications in metrology, sub-shot noise interfer-ometry and the quantum emulation of exotic spin mod-els.

V. OUTLOOK

One of the ultimate goals of the new field of quantumdynamics is to develop a systematic understanding ofnonequilibrium phenomena in strongly interacting quan-tum many-body systems. A few of the most significantopen questions along this avenue are readily identified :How can we classify nonequilibrium behavior in closedmany-body systems? What is the general relation be-tween integrability and dynamics? What is the dynam-ical effect of a gradual breaking of integrability? Whatare the effects of dissipation on these nonequilibrium pro-cesses? Can we understand time evolution of interactingsystems through the renormalization group? Answeringthese and other questions allied with systematic, quanti-tative studies of possible nonequilibrium quantum phasetransitions and the extraction of dynamical critical ex-ponents, are just a few of the many tantalizing programsto be pursued. The rapidly developing sophisticationand precision of ultracold atomic experiments and otherexperimental systems should allow for close and directcomparison between theoretical predictions and ad hocexperiments.

The realization of robust techniques for the experimen-tal study of such systems and the development of theo-retical tools to describe nonequilibrium many-body pro-cesses should bode for tantalizing opportunities in thisnascent field, potentially leading to a deeper understand-ing of the principles governing nonequilibrium many-body phenomena and establishing robust connections be-tween microscopic dynamics and statistical physics.

VI. ACKNOWLEDGEMENTS

A.P. was supported by NSF (DMR-0907039), AFOSRFA9550-10-1-0110 and the Sloan Foundation. K.S.thanks DST, India for financial support under ProjectNo. SR/S2/CMP-001/2009. M. V. was supported bythe Sloan Foundation.

References

Allahverdyan, A. E., and T. M. Nieuwenhuizen, 2002, PhysicaA: Statistical Mechanics and its Applications 305, 542.

Allahverdyan, A. E., and T. M. Nieuwenhuizen, 2005, Phys.Rev. E 71, 046107.

Altland, A., V. Gurarie, T. Kriecherbauer, andA. Polkovnikov, 2009, Phys. Rev. A 79, 042703.

Altshuler, B. L., Y. Gefen, A. Kamenev, and L. S. Levitov,1997, Phys. Rev. Lett. 78, 2803.

Anderson, P. W., 1958, Phys. Rev. 109, 1492.Appel, J., P. J. Windpassinger, D. Oblak, U. B. Hoff,

N. Kjaergaard, and E. S. Polzik, 2009, PNAS 106, 10960.

Babadi, M., D. Pekker, R. Sensarma, A. Georges, andE. Demler, 2009, arXiv:0908.3483 .

Balian, R., 1991, From Microphysics to Macrophysics(Springer, Berlin, Heidelberg).

Barankov, R., and A. Polkovnikov, 2008, Phys. Rev. Lett.101, 076801.

Barmettler, P., M. Punk, V. Gritsev, E. Demler, and E. Alt-man, 2009, Phys. Rev. Lett. 102, 130603.

Barouch, E., B. M. McCoy, and M. Dresden, 1970, Phys. Rev.A 2, 1075.

Barthel, T., and U. Schollwock, 2008, Phys. Rev. Lett. 100,100601.

Basko, D. M., I. L. Aleiner, and B. L. Altshuler, 2006, Ann.Phys. (NY) 321, 1126.

Bermudez, A., D. Patane, L. Amico, and M. A. Martin-Delgado, 2009, Phys. Rev. Lett. 102, 135702.

Biroli, G., L. F. Cugliandolo, and A. Sicilia, 2010, Phys. Rev.E 81, 050101.

Blakie, P. B., A. S. Bradley, M. J. Davis, R. J. Ballagh, andC. W. Gardiner, 2008, Advances in Physics 57, 363.

Bloch, I., J. Dalibard, and W. Zwerger, 2008, Reviews of Mod-ern Physics 80, 885.

Boksenbojm, E., B. Wynants, and C. Jarzynski, 2010, Phys-ica A: Statistical Mechanics and its Applications 389, 4406.

Burkov, A. A., M. D. Lukin, and E. Demler, 2007, Phys. Rev.Lett. 98, 200404.

Calabrese, P., and J. Cardy, 2004, J. Stat. Mech. , P06002.Calabrese, P., and J. Cardy, 2005, J. Stat. Mech. , P04010.Calabrese, P., and J. Cardy, 2007, J. Stat. Mech: Th. and

Exp. , P06008.Campbell, D. K., P. Rosenau, and G. Zaslavsky, 2005, Chaos

15, 015101.Caneva, T., R. Fazio, and G. E. Santoro, 2007, Phys. Rev. B

76, 144427.Canovi, E., D. Rossini, R. Fazio, G. Santoro, and A. Silva,

2011, Phys. Rev. B 83, 094431.Cassidy, A. C., C. W. Clark, and M. Rigol, 2011, Phys. Rev.

Lett. 106, 140405.Caux, J.-S., and J. Mossel, 2011, J. Stat. Mech. , P02023.Cazalilla, M. A., 2006, Phys. Rev. Lett. 97, 156403.Chaikin, P. M., and T. C. Lubensky, 1995, Principles of con-

densed matter physics (Cambridge Univesrity Press, Cam-bridge).

Chang, D. E., V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin,and E. A. Demler, 2008, Nature Phys. 4, 884.

Chang, M. S., Q. S. Qin, W. X. Zhang, L. You, and M. S.Chapman, 2005, Nature Phys. 1, 111.

Cherng, R. W., and L. S. Levitov, 2006, Phys. Rev. A 73,043614.

Chuang, I., R. Durrer, N. Turok, and B. Yurke, 1991, Science251, 1336.

Cincio, L., J. Dziarmaga, M. M. Rams, and W. H. Zurek,2007, Phys. Rev. A 75, 052321.

Collura, M., and D. Karevski, 2010, Phys. Rev. Lett. 104,200601.

Cramer, M., C. M. Dawson, J. Eisert, and T. J. Osborne,2008, Phys. Rev. Lett. 100, 030602.

Cramer, M., and J. Eisert, 2010, New J. Phys. 12, 055020.Cucchietti, F. M., B. Damski, J. Dziarmaga, and W. H. Zurek,

2007, Phys. Rev. A 75, 023603.Damski, B., and W. H. Zurek, 2006, Phys. Rev. A 73(6),

063405.De Grandi, C., R. A. Barankov, and A. Polkovnikov, 2008,

Phys. Rev. Lett. 101(23), 230402.

21

De Grandi, C., V. Gritsev, and A. Polkovnikov, 2010a, Phys.Rev. B 81, 012303.

De Grandi, C., V. Gritsev, and A. Polkovnikov, 2010b, Phys.Rev. B 81, 224301.

De Grandi, C., and A. Polkovnikov, 2010, Lect. Notes inPhys. 802, 75.

Deng, S., G. Ortiz, and L. Viola, 2008, Europhys. Lett. 84,67008.

Deutsch, J., 1991, Phys. Rev. A 43, 2046.Divakaran, U., A. Dutta, and D. Sen, 2008, Phys. Rev. B 78,

144301.Divakaran, U., A. Dutta, and D. Sen, 2010, Phys. Rev. B

81(5), 054306.Dora, B., E. V. Castro, and R. Moessner, 2010, Phys. Rev. B

82, 125441.Dora, B., and R. Moessner, 2010, Phys. Rev. B 81, 165431.Ducci, S., P. L. Ramazza, W. Gonzalez-Vinas, and F. T. Arec-

chi, 1999, Phys. Rev. Lett. 83, 5210.Dziarmaga, J., 2005, Phys. Rev. Lett. 95, 245701.Dziarmaga, J., 2006, Phys. Rev. B 74, 064416.Dziarmaga, J., 2010, Adv. in Phys. 59, 1063.Dziarmaga, J., J. Meisner, and W. Zurek, 2008, Phys. Rev.

Lett. 101, 115701.Dziarmaga, J., and M. M. Rams, 2010, New J. of Phys. 12,

055007.Eckstein, M., and M. Kollar, 2008, Phys. Rev. Lett. 100,

120404.Eckstein, M., and M. Kollar, 2010, New J. of Phys. 12,

055012.Esteve, J., C. Gross, A. Weller, S. Giovanazzi, and M. K.

Oberthaler, 2008, Nature 455, 1216.Farhi, E., J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren,

and D. Preda, 2001, Science 292, 472.Fermi, E., J. Pasta, and S. Ulam, May 1955, Document LA-

1940 .Fioretto, D., and G. Mussardo, 2010, New J. of Phys. 12,

055015.Fisher, D. S., 1995, Phys. Rev. B 51, 6411.Flesch, A., M. Cramer, I. McCulloch, U. Schollwoeck, and

J. Eisert, 2008, Phys. Rev. A 78, 033608.Fubini, A., G. Falci, and A. Osterloh, 2007, New J. Phys. 9,

134.Gangardt, D. M., and M. Pustilnik, 2008, Phys. Rev. A 77,

041604(R).Gardiner, C., and P. Zoller, 2004, Quantum Noise (Springer-

Verlag, Berlin Heidelberg), third edition.Gasenzer, T., 2009, European Physical Journal - Special Top-

ics 168, 89.Ghoshal, S., and A. B. Zamolodchikov, 1994, Int. J. Mod.

Phys. A 9, 3841.Girardeau, M. D., 1969, Phys. Lett. 30A, 442.Girardeau, M. D., 1970, Phys. Lett. 32A, 67.Gogolin, C., M. P. Mller, and J. Eisert, 2011, Phys. Rev. Lett.

106, 040401.Goldstein, S., J. L. Lebowitz, R. Tumulka, and N. Zanghi,

2010, European Phys. J. H 35, 173.Greentree, A. D., C. Tahan, J. H. Cole, and L. C. L. Hollen-

berg, 2006, Nature Phys. 2, 856.Greiner, M., O. Mandel, T. Esslinger, T. W. Hansch, and

I. Bloch, 2002a, Nature 415, 39.Greiner, M., O. Mandel, T. W. Hansch, and I. Bloch, 2002b,

Nature 419, 51.Gritsev, V., E. Demler, E. Lukin, and A. Polkovnikov, 2007,

Phys. Rev. Lett. 99, 200404.

Gritsev, V., and A. Polkovnikov, 2010, in UnderstandingQuantum Phase Transitions, edited by L. Carr (Taylor &Francis, Boca Raton).

Gu, S.-J., and H.-Q. Lin, 2009, Europhys. Lett. 87, 10003.Gurarie, V., 1995, Nucl. Phys. B 441, 569.Hillery, M. A., R. F. O’Connell, M. O. Scully, and E. P.

Wigner, 1984, Phys. Rep. 106, 121.Hindmarsh, M., and A. Rajantie, 2000, Phys. Rev. Lett. 85,

4660.Ho, T. L., 1998, Phys. Rev. Lett. 81, 742.Hofferberth, S., I. Lesanovsky, B. Fischer, T. Schumm, and

J. Schmiedmayer, 2007, Nature 449, 324.Imry, Y., 1997, Introduction to mesoscopic physics (Oxford

University Press).Itin, A., and P. Torma, 2009a, arXiv:0901.4778 .Itin, A. P., and P. Torma, 2009b, Phys. Rev. A 79, 055602.Iucci, A., and M. A. Cazalilla, 2009, Phys. Rev. A 80, 063619.Izrailev, F. M., and B. V. Chirikov, 1966, Soviet Physics Dok-

lady 11, 32.Jaynes, E., 1957, Phys. Rev. 106, 620.Jo, G.-B., Y. R. Lee, J.-H. Choi, C. A. Christensen, T. H.

Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle,2009, Science 325, 1521.

Kamenev, A., and A. Levchenko, 2009, Advances in Physics58, 197.

Kardar, M., 2007, Statistical Physics of Particles (CambridgeUniversity Press).

Kibble, T., 2007, Physics Today 60, 47.Kibble, T. W. B., 1976, J. Phys. A 9, 1387.Kiegel, C., 2009, Diploma Thesis: Semiclassical approach

to driven many-particle systems (Institute for TheoreticalPhysics, University of Cologne, Germany).

Kinoshita, T., T. Wenger, and D. S. Weiss, 2006, Nature 440,900.

Klempt, C., O. Topic, G. Gebreyesus, M. Scherer, T. Hen-ninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt,2010, Phys. Rev. Lett. 104, 195303.

Klich, I., and L. Levitov, 2009, Phys. Rev. Lett. 102, 100502.Klich, I., G. Refael, and A. Silva, 2006, Phys. Rev. A 74,

032306.Kollar, M., and M. Eckstein, 2008, Phys. Rev. A 78, 013626.Kollath, C., A. M. Lauchli, and E. Altman, 2007, Phys. Rev.

Lett. 98, 180601.Krapivsky, P. L., 2010, J. of Stat. Mech.: Theor. and Exper-

iment , P02014.Kronjager, J., C. Becker, M. Brinkmann, R. Walser, P. Navez,

K. Bongs, and K. Sengstock, 2005, Phys. Rev. A 72,063619.

Laguna, P., and W. H. Zurek, 1997, Phys. Rev. Lett. 78,2519.

Lamacraft, A., 2007, Phys. Rev. Lett. 98, 160404.Landau, L., 1932, Phys. Z. Sowjetunion 1, 88.Landau, L., and E. Lifshitz, 1981, Quantum Mechanics: Non-

Relativistic Theory, Vol. 3 (Butterworth-Heinemann, Ox-ford, UK).

Landau, L., and E. Lifshitz, 1982, Mechanics: Course of The-oretical Physics Vol. 1 (Butterworth-Heinemann, Oxford,UK).

Landau, L. D., and E. M. Lifshitz, 1980, StatisticalPhysics, Third Edition, Part 1: Volume 5 (Butterworth-Heinemann, Oxford, UK).

Leitner, D. M., and P. G. Wolynes, 1996, Phys. Rev. Lett.76, 216.

Leroux, I. D., M. H. Schleier-Smith, and V. Vuletic, 2010,

22

Phys. Rev. Lett. 104, 073602.Lignier, H., C. Sias, D. Ciampini, Y. Singh, A. Zenesini,

O. Morsch, and E. Arimondo, 2007, Phys. Rev. Lett. 99,220403.

Linden, N., S. Popescu, A. Short, and A. Winter, 2009, Phys.Rev. E 79, 061103.

Logan, D. E., and P. G. Wolynes, 1990, J. Chem. Phys. 93,4994.

Luo, L., B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas,2007, Phys. Rev. Lett. 98, 080402.

Majorana, E., 1932, Nuovo Cimento 9, 43.Maniv, A., E. Polturak, and G. Koren, 2003, Phys. Rev. Lett.

91, 197001.Manmana, S. R., S. Wessel, R. M. Noack, and A. Muramatsu,

2007, Phys. Rev. Lett. 98, 210405.Mathey, L., and A. Polkovnikov, 2010, Phys. Rev. A 81,

033605.Mazur, P., 1968, Physica 43, 533.Meiser, D., J. Ye, and M. J. Holland, 2008, New J. of Phys.

10, 073014.Moeckel, M., and S. Kehrein, 2008, Phys. Rev. Lett. 100,

175702.Moeckel, M., and S. Kehrein, 2009, Ann. Phys. 324, 2146.Moeckel, M., and S. Kehrein, 2010, New J. of Phys. 12,

055016.Monaco, R., M. Aaroe, J. Mygind, R. J. Rivers, and V. P.

Koshelets, 2006, Phys. Rev. B 74, 144513.Monaco, R., J. Mygind, and R. J. Rivers, 2002, Phys. Rev.

Lett. 89, 080603.Monaco, R., J. Mygrind, R. J. Rivers, and V. P. Koshelets,

2009, Phys. Rev. B 80, 180501.Mondal, S., D. Sen, and K. Sengupta, 2008, Phys. Rev. B 78,

045101.Mondal, S., K. Sengupta, and D. Sen, 2009, Phys. Rev. B 79,

045128.Mostame, S., G. Schaller, and R. Schutzhold, 2007, Phys.

Rev. A 76, 030304(R).Mukerjee, S., V. Oganesyan, and D. Huse, 2006, Phys. Rev.

B 73, 035113.Mukherjee, V., U. Divakaran, A. Dutta, and D. Sen, 2007,

Phys. Rev. B 76, 174303.Mukherjee, V., A. Dutta, and D. Sen, 2008, Phys. Rev. B 77,

214427.Mussardo, G., 2009, Statistical Field Theory. An Introduc-

tion to Exactly Solved Models in Statistical Physics (OxfordUniversity Press).

Neuenhahn, C., and F. Marquardt, 2010, arXiv:1007.5306 .von Neumann, J., 1929, Z. Phys. 57, 30.Oganesyan, V., and D. A. Huse, 2007, Phys. Rev. B 75,

155111.Ohmi, T., and K. Machida, 1998, J. Phys. Soc. Jpn. 67, 1822.Olshanii, M., and V. Yurovsky, 2009, arXiv:0911.5587 .Pal, A., and D. A. Huse, 2010, Phys. Rev. B 82, 174411.Paraan, F. N. C., and A. Silva, 2009, Phys. Rev. E 80, 061130.Patane, D., L. Amico, A. Silva, R. Fazio, and G. E. Santoro,

2009a, Phys. Rev. B 80, 024302.Patane, D., A. Silva, L. Amico, R. Fazio, and G. E. Santoro,

2008, Phys. Rev. Lett. 101, 175701.Patane, D., A. Silva, F. Sols, and L. Amico, 2009b, Phys.

Rev. Lett. 102, 245701.Pauli, W., and M. Fierz, 1937, Z. Phys. 106, 572.Peres, A., 1984, Phys. Rev. A 30, 504.Plimak, L. I., M. Fleischhauer, M. K. Olsen, and M. J. Collett,

2001, arXiv:cond-mat/0102483 .

Poilblanc, D., T. Ziman, J. Bellissard, F. Mila, and G. Mon-tambaux, 1993, Europhys. Lett. 22 (7), 537.

Polkovnikov, A., 2005, Phys. Rev. B 72, 161201(R).Polkovnikov, A., 2008a, Ann. of Phys. 326, 486.Polkovnikov, A., 2008b, Phys. Rev. Lett. 101, 220402.Polkovnikov, A., 2010, Annals of Phys. 325, 1790.Polkovnikov, A., and V. Gritsev, 2008, Nat. Phys. 4, 477.Pollmann, F., S. Mukerjee, A. G. Green, and J. E. Moore,

2010, Phys. Rev. E 81, 020101.Popescu, S., A. J. Short, and A. Winter, 2006, Nature Physics

2, 745.Porter, M., N. Zabusky, B. Hu, and D. Campbell, 2009, Amer-

ican Scientist 97, 214.Refael, G., and J. E. Moore, 2004, Phys. Rev. Lett. 93,

260602.Regal, S., 2006, experimental realization of BCS-BEC

crossover physics with a Fermi gas of atoms, PhD. The-sis, University of Colorado, Boulder, 2006; C. Regal, D. S.Jin, arXiv:cond-mat/0601054.

Reimann, P., 2008, Phys. Rev. Lett. 101, 190403.Rey, A. M., B. L. Hu, E. Calzetta, and C. W. Clark, 2005,

Phys. Rev. A 72, 023604.Rezakhani, A. T., D. F. Abasto, D. A. Lidar, and P. Zanardi,

2010, Phys. Rev. A 82, 012321.Rezakhani, A. T., W.-J. Kuo, A. Hamma, D. A. Lidar, and

P. Zanardi, 2009, Phys. Rev. Lett. 103, 080502.Riedel, M. F., P. Bohi, Y. Li, T. W. Hansch, A. Sinatra, and

P. Treutlein, 2010, Nature 464, 1170.Rigol, M., 2009, Phys. Rev. Lett. 103, 100403.Rigol, M., V. Dunjko, and M. Olshanii, 2008, Nature 452,

854.Rigol, M., V. Dunjko, V. Yurovsky, and M. Olshanii, 2007,

Phys. Rev. Lett. 98, 050405.Rigol, M., and L. F. Santos, 2010, Phys. Rev. A 82,

011604(R).Rigolin, G., G. Ortiz, and V. H. Ponce, 2008, Phys. Rev. A

78, 052508.Rivers, R., 2001, J. of Low Temp. Phys. 124, 41.Rossini, D., A. Silva, G. Mussardo, and G. Santoro, 2010,

Phys. Rev. B 82, 144302.Rossini, D., A. Silva, G. Mussardo, and G. E. Santoro, 2009,

Phys. Rev. Lett. 102, 127204.Roux, G., 2009, Phys. Rev. A 79, 021608.Roux, G., 2010, Phys. Rev. A 81(5), 053604.Sachdev, S., 1999, Quantum Phase Transitions (Cambridge

University Press).Sadler, L. E., J. M. Higbie, S. R. Leslie, M. Vengalattore, and

D. M. Stamper-Kurn, 2006, Nature 443, 312.Saito, H., Y. Kawaguchi, and M. Ueda, 2007, Phys. Rev. A

76, 043613.Santos, L. F., and M. Rigol, 2010a, Phys. Rev. E 81, 036206.Santos, L. F., and M. Rigol, 2010b, Phys. Rev. E 82, 031130.Schmittmann, B., and R. Zia, 1995, Phase transitions

and critical phenomena vol. 17 (editors C. Domb and J.Lebowitz) (Academic Press, London), chapter Statisticalmechanics of driven diffusive systems.

Schollwock, U., 2005, Rev. Mod. Phys. 77, 259.Schollwock, U., and S. White, 2006, in Effective models

for low-dimensional strongly correlated systems, edited byG. G. Batrouni and D. Poilblanc (AIP, Melville, New York),chapter Methods for Time Dependence in DMRG, p. 155.

Schwandt, D., F. Alet, and S. Capponi, 2009, Phys. Rev. Lett.103, 170501.

Sen, D., K. Sengupta, and S. Mondal, 2008, Phys. Rev. Lett.

23

101, 016806.Sen, D., and S. Vishveshwara, 2010, Europhys. Lett. 91,

66009.Sengupta, K., S. Powell, and S. Sachdev, 2004, Phys. Rev. A

69, 053616.Sengupta, K., and D. Sen, 2009, Phys. Rev. A 80, 032304.Sengupta, K., D. Sen, and S. Mondal, 2008, Phys. Rev. Lett.

100, 077204.Sensarma, R., D. Pekker, E.Altman, E. Demler,

N. Strohmaier, D. Greif, R. Jordens, L. Tarruell,H. Moritz, and T. Esslinger, 2010, Phys. Rev. B 82,224302.

Silva, A., 2008, Phys. Rev. Lett. 101, 120603.Sondhi, S. L., S. M. Girvin, J. P. Carini, and D. Shahar, 1997,

Rev. Mod. Phys. 69, 315.Sorensen, A., L. M. Duan, J. I. Cirac, and P. Zoller, 2001,

Nature 409, 63.Srednicki, M., 1994, Phys. Rev. E 50, 888.Srednicki, M., 1999, J. Phys. A: Math. Gen. 32, 1163.Strohmaier, N., D. Greif, R. Jo”rdens, L. Tarruell, H. Moritz,

and T. Esslinger, 2010, Phys. Rev. Lett. 104, 080401.Stuckelberg, E. C. G., 1932, Helv. Phys. Acta 5, 369.Sutherland, B., 2004, Beautiful Models (Wold Scientific).Suzuki, M., 1971, Physica 51, 277.Tabor, M., 1989, Chaos and Integrability in Nonlinear Dy-

namics: An Introduction (Wiley, New York).Talkner, P., E. Lutz, and P. Hanggi, 2007, Phys. Rev. E 75,

050102.Thirring, W., 2002, Quantum Mathematical Physics

(Springer, New York).Trotzky, S., P. Cheinet, S. Folling, M. Feld, U. Schnorrberger,

A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, andI. Bloch, 2008, Science 319, 295.

Trotzky, S., Y.-A. Chen, A. Flesch, I. P. McCul-loch, U. Schollwock, J. Eisert, and I. Bloch, 2011,arXiv:1101.2659 .

Tuchman, A. K., C. Orzel, A. Polkovnikov, and M. Kasevich,

2006, Phys. Rev. A 74, 051601.Vengalattore, M., J. M. Higbie, S. R. Leslie, J. Guzman, L. E.

Sadler, and D. M. Stamper-Kurn, 2007, Phys. Rev. Lett.98, 200801.

Vengalattore, M., S. R. Leslie, J. Guzman, and D. M.Stamper-Kurn, 2008, Phys. Rev. Lett. 100, 170403.

Venuti, L. C., and P. Zanardi, 2007, Phys. Rev. Lett. 99,095701.

Verstraete, F., V. Murg, and J. Cirac, 2008, Advances inPhysics 57, 143.

Vidal, G., 2003, Phys. Rev. Lett. 91, 147902.Vidal, G., 2004, Phys. Rev. Lett. 93, 040502.Vidal, G., J. I. Latorre, E. Rico, and A. Kitaev, 2003, Phys.

Rev. Lett. 90, 227902.Vitanov, N. V., 1999, Phys. Rev. A 59, 988.Vitanov, N. V., and B. M. Garraway, 1996, Phys. Rev. A 53,

4288.Vojta, M., 2003, Rep. Prog. Phys. 66, 2069.Walls, D., and G. Milburn, 1994, Quantum Optics (Springer-

Verlag, Berlin).Weiler, C. N., T. W. Neely, D. R. Sherer, A. S. Bradley, M. J.

Davis, and B. P. Anderson, 2008, Nature 455, 948.White, S., 1992, Phys. Rev. Lett. 69, 2863.Widera, A., S. Trotzky, P. Cheinet, S. Folling, F. Gerbier,

and I. Bloch, 2008, Phys. Rev. Lett. 100, 140401.Yates, A., and W. H. Zurek, 1998, Phys. Rev. Lett. 80, 5477.Zabusky, N. J., and M. D. Kruskal, 1965, Phys. Rev. Lett.

15, 240.Zakharov, V., V. L’vov, and G. Falkovich, 1992, Kolmogorov

spectra of turbulence, Vol. 1 (Springer-Verlag, Berlin).Zener, C., 1932, Proc. R. Soc. London, Ser. A 137, 696.Zurek, W. H., 1985, Nature 317, 505.Zurek, W. H., 1996, Phys. Rep. 276, 177.Zurek, W. H., 2003, Rev. Mod. Phys. 75, 715.Zurek, W. H., U. Dorner, and P. Zoller, 2005, Phys. Rev. Lett.

95, 105701.