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Loschmidt echo in many-spin systems: a quest for intrinsic decoherence

and emergent irreversibility

Pablo R. Zangara1 and Horacio M. Pastawski1

1Instituto de Fısica Enrique Gaviola (CONICET-UNC) and Facultad de Matematica,Astronomıa, Fısica y Computacion (FaMAF),

Universidad Nacional de Cordoba, 5000, Cordoba, Argentina

If a magnetic polarization excess is locally injected in a crystal of interacting spins in thermalequilibrium, this “excitation” would spread as consequence of spin-spin interactions. Such an appar-ently irreversible process is known as spin diffusion and it can lead the system back to “equilibrium”.Even so, a unitary quantum dynamics would ensure a precise memory of the non-equilibrium ini-tial condition. Then, if at certain time, say t/2, an experimental protocol reverses the many-bodydynamics by changing the sign of the effective Hamiltonian, it would drive the system back to theinitial non-equilibrium state at time t. As a matter of fact, the reversal is always perturbed by smallexperimental imperfections and/or uncontrolled internal or environmental degrees of freedom. Thislimits the amount of signal M(t) recovered locally at time t. The degradation of M(t) accountsfor these perturbations, which can also be seen as the sources of decoherence. This general ideadefines the Loschmidt echo (LE), which embodies the various time-reversal procedures implementedin nuclear magnetic resonance. Here, we present an invitation to the study of the LE followingthe pathway induced by the experiments. With such a purpose, we provide a historical and con-ceptual overview that briefly revisits selected phenomena that underlie the LE dynamics includingchaos, decoherence, localization and equilibration. This guiding thread ultimately leads us to thediscussion of decoherence and irreversibility as an emergent phenomenon. In addition, we introducethe LE formalism by means of spin-spin correlation functions in a manner suitable for presentationin a broad scope physics journal. Last, but not least, we present new results that could triggernew experiments and theoretical ideas. In particular, we propose to transform an initially localizedexcitation into a more complex initial state, enabling a dynamically prepared LE. This induces aglobal definition of the LE in terms of the raw overlap between many-body wave functions. Ourresults show that as the complexity of the prepared state increases, it becomes more fragile towardssmall perturbations.

I. IRREVERSIBILITY AND

DECOHERENCE: A CONCEPTUAL AND

HISTORICAL OVERVIEW

A. On the emergent hierarchical structure

of Nature

Our knowledge about the Universe surround-ing us is far from being harmoniously unified.Instead, we have a large catalogue of almost in-dependent disciplines that constitute, with dif-ferent degree of success, our comprehension onhow Nature works. Quantum Physics, Chem-istry, Biology, and their subfields belong to sucha hierarchical catalogue that could end withPsychology and Sociology. Each of these fieldsof knowledge provides partial descriptions that

correspond to different levels of reality [1]. Therelation between the basic laws of a disciplinein terms of the ones ruling the previous level inthe hierarchical tree (usually called reduction) ishighly nontrivial as the passage from one levelto another may involve conceptual and math-ematical discontinuities or “phase transitions”[2–4]. Among these, life and consciousness haveproved to be the most elusive, with only a fewhints as to when and why they emerge [5, 6].Within the realm of Physics, these transitions

occur between different domains characterizedby specific energy, time and length scales. Here,we have reasons to be more optimistic since wehave well-developed experimental and mathe-matical tools. To begin with, let us consider atextbook example which is close to our centralproblem of irreversibility, namely the derivation

2

of equilibrium Thermodynamics from classicalStatistical Mechanics (SM). The former, one ofthe driving forces behind the Industrial Revolu-tion, describes the properties of matter in termsof pressure, volume, temperature, among othermacroscopic variables. The latter, mainly de-veloped by James C. Maxwell, Ludwig Boltz-mann and Josiah W. Gibbs, provides a proba-bilistic description of a system composed by N“atomic” constituents. These elementary enti-ties, and the way in which they interact, are de-scribed by means of Classical Mechanics (CM).In fact, Boltzmann [7, 8] considered “the worldas a mechanical system of an enormously largenumber of constituents, and of an immenselylong period of time” (t),i.e. what we currentlycall the thermodynamic limit of N → ∞ andthen t → ∞. Then, simple macroscopic rela-tions (thermodynamic equations of state) canbe derived from SM by focusing on certain mi-croscopic observables and taking such a limit.

While the previous reduction frameworkworks in principle, and to a large extent inpractice, it fails when dealing with specific andhighly nontrivial physical situations: the crit-ical points. Indeed, a critical point indicatesthe onset of a phase transition, where the ther-modynamic variables, expressed as functions ofcertain control variable, become non-analyticor even diverge. Here, the reduction paradigmhas to be replaced by the emergence of qualita-tively new phenomena. As stated by Sir MichaelBerry [3],

“Thermodynamics is a continuum theory, soreduction has to show that density fluctuationsarising from interatomic forces have a finite(and microscopic) range. This is true every-where except at the critical point, where thereare fluctuations on all scales up to the samplesize. Thus, at criticality the continuum limitdoes not exist, corresponding to a new state ofmatter. In terms of our general picture, thecritical state is a singularity of thermodynam-ics, at which its smooth reduction to statisticalmechanics breaks down; nevertheless, out of thissingularity, emerges a large class of new ‘crit-ical phenomena’, which can be understood by acareful study of the large-N asymptotes.”

Our everyday experience corresponds to amacroscopically large classical domain, whereour falling glass shatters, our cup of coffee coolsdown, and the ice cube left out of the refrig-erator melts. All of these irreversible phenom-ena are manifestations of the well known Sec-ond Law of Thermodynamics, i.e. things evolvetowards an unavoidable increase of entropy. In-deed, the physical description of such a macro-scopic domain involves time-asymmetric equa-tions of motion, such as hydrodynamic and dif-fusive ones. As already stated, the elemen-tary or microscopic constituents of each macro-scopic entity belong to a “more fundamental”level, whose physical description involves time-symmetric equations of motion, such as Newto-nian dynamics or, even better, the Schrodingerequation. In other words, while the microscopicworld is reversible, the macroscopic one is not.This paradoxical contrast has fed a recurrentcontroversy since the end of the nineteenth cen-tury to the present day. Our position on thisissue relies on the emergence of irreversibilityfrom a microscopically reversible quantum dy-namics provided that the thermodynamic limit(TL) N → ∞ is appropriately considered.This article constitutes an invitation to the

study of irreversibility as en emergent phe-nomenon following a pathway much entangledwith a series of NMR experiments led by Patri-cia Levstein and Horacio Pastawski at Cordoba,Argentina [9–11]. In what follows, we discussand summarize the key concepts on this regardin a manner suitable for a broad readership.Novel results and new insights of interest forspecialists are discussed in Sec. III.

B. The irreversibility paradox in the

classical world

In his attempt to reconcile the irreversible na-ture of the Second Law with the reversible New-tonian laws of motion underlying SM, Boltz-mann considered the evolution of a gas com-posed by colliding particles which is preparedout of equilibrium. Such a complex systemwas then described according to a kinetic equa-

3

tion that irreversibly leads the system to equi-librium. Here, the time-reversal symmetry isremoved through the assumption of “molecu-lar chaos” or stosszahl-ansatz. This hypoth-esis implies that after each particle collision(characterized by a typical time) the memoryof the previous state is lost. Boltzmann’s ap-proach finally led him to the celebrated Htheorem, which is the first formal justificationof the Second Law. At that time, JosephLoschmidt raised the objection that for ev-ery possible trajectory that leads to equilib-rium there will be another trajectory, equallypossible, that would lead to the initial out-of-equilibrium state. Therefore, it would be pos-sible to revert the velocities of every particleto get again a low entropy state. As an answer,Boltzmann emphasized on the extreme practicaldifficulty of achieving the time reversed evolu-tion proposed by Loschmidt, allegedly by saying“it is you who would invert the velocities!” [12].

Boltzmann himself improved his theory trans-forming it into a probabilistic description. Moreprecisely, the separation between microscopicand macroscopic scales, is exactly what enablesus to predict the typical evolution of a partic-ular macroscopic system. In fact, this consti-tutes the modern wisdom that explains the socalled “irreversibility paradox”. As stated byJoel Lebowitz [13] (see also [14, 15]),

“... several interrelated ingredients which to-gether provide the sharp distinction between mi-croscopic and macroscopic variables required forthe emergence of definite time-asymmetric be-havior in the evolution of the latter despite thetotal absence of such asymmetry in the dynam-ics of individual atoms. They are: (a) the greatdisparity between microscopic and macroscopicscales, (b) the fact that events are, as put byBoltzmann, determined not only by differentialequations, but also by initial conditions, and (c)the use of probabilistic reasoning: it is not ev-ery microscopic state of a macroscopic systemthat will evolve in accordance with the secondlaw, but only the ‘majority’ of cases—a majoritywhich however becomes so overwhelming whenthe number of atoms in the system becomes verylarge that irreversible behavior becomes a near

certainty.”

The key point here is that, already in theclassical Boltzmann’s approach, the notion ofthe emergent irreversibility depends somehowon the conditions under which the passage fromthe microscopic to the macroscopic domain isperformed. More precisely, the main qualita-tive features of this paradigm have only a slightdependence on the specific details of the under-lying microscopic dynamics. However, there ex-ist microscopic properties that do determine thequantitative description of the macroscopic evo-lution [13]. In other words, the derivation ofhydrodynamic and diffusive macroscopic equa-tions ultimately depends on the behavior of mi-croscopic trajectories. These microscopic de-tails can include two mutually complementaryproperties: an extreme sensitivity towards ini-tial conditions, i.e. chaoticity, and a tendencytowards a uniform distribution of the state inthe available phase space, i.e. mixing.

Let us explore the two ingredients just men-tioned in the context of few-body CM. Accord-ing to CM, the state of a system, composed byN particles in d dimensions, is described as apoint X in a (2dN)-dimensional phase space. Ifthe system is conservative, the energy is the pri-mary conserved quantity, and the phase spaceis restricted to a hypersurface Ω of 2dN − 1 di-mensions usually called energy shell. In this sce-nario, chaos implies that two trajectories in Ω,starting from points separated by an arbitrar-ily small distance δ0, will separate exponentiallyas a function of time, δ(t) ∼ δ0 exp[λt]. Here,the typical inverse time λ is a Lyapunov charac-teristic exponent. The mixing property impliesthat the system evolves over time so that anygiven region contained in Ω eventually overlapswith any other given region in Ω. This can bethought as an “intertwined” picture of the phasespace. None of these properties are satisfied inthe case of fully integrable systems, since theirsolutions are regular and non-dense orbits in Ω.But if integrability is completely broken, chaosand mixing imply that the orbits become irregu-lar and cover Ω densely. This means that an ac-tual trajectory X(t) will be arbitrarily close toevery possible configuration within Ω, provided

4

that enough time has elapsed. This last obser-vation embodies the concept of ergodicity: anobservable can be equivalently evaluated by av-eraging it for different configurations in Ω or byits time-average along a single trajectory X(t).Such a property is the cornerstone of -classical-SM since it sets the equivalence between theGibbs’ description in terms of ensembles andBoltzmann kinetic approach. Furthermore, weremark that this result does not depend on thelimit N → ∞.

By the early 1950’s, Enrico Fermi, JohnPasta and Stanislaw Ulam (FPU) [16] tried tostudy when and how the integrability break-down could lead to an ergodic behavior withina deterministic evolution. They considered astring of harmonic oscillators perturbed by an-harmonic forces in order to verify that thesenonlinearities can lead to energy equi-partitionas a manifestation of ergodicity. Even thoughUlam himself stated “The motivation then wasto observe the rates of mixing and thermaliza-tion...” [17], the results were not those expected:“thermalization” dynamics did not show up atall. Instead, the dynamics of the FPU problem,at least for the small number of particles consid-ered, evidenced remarkable recurrences much asthose invoked by Henri Poincare in his famousrecurrence theorem [8, 18, 19].

Nowadays, the striking FPU results are un-derstood in terms of the theory of chaos [20].More precisely, the microscopic instabilities aresystematically described by the theory of dy-namical chaos developed by Boris Chirikov [21,22]. In fact, the onset of dynamical chaos can beidentified with the transition from non-ergodicto ergodic behavior [23]. A more general cri-terion for the onset of ergodicity is given bythe Kolmogorov-Arnold-Moser (KAM) theorem[24]. It predicts that a weak nonlinear pertur-bation of an integrable system destroys the con-stants of motion only locally in the regions ofresonances. In other regions of the phase space,islands of quasi-periodic motion persist. Al-though a direct application of KAM theoremto the FPU model suffers from technical diffi-culties, it constitutes the main indication thatone should not naively expect that weak non-

linear perturbations ensure ergodicity. In addi-tion, it is crucial to remember that real physicalsystems are neither closed nor finite in a strictsense. Already Boltzmann used this argumentto conjure up the peril that cyclic or recurrentmotion posed on SM [7, 8]:“In practice, however, the walls are continu-

ously undergoing perturbations, which will de-stroy the periodicity resulting from the finitenumber of molecules”.Furthermore, the final way out should be

found outside the realm of CM, in a descrip-tion already foreseen by Boltzmann, an emer-gent from a many-body (quantum) descriptionin the TL N → ∞ [7, 8]:“Since today it is popular to look forward to

the time when our view of Nature will have beencompletely changed, I will mention the possibil-ity that the fundamental equations for the mo-tion of individual molecules will turn out to beonly approximate formulas which give averagevalues, resulting from the probability calculusfrom the interactions of many independent mov-ing entities forming the surrounding medium–asfor example metheorology laws are valid only onaverage values obtained by long series of obser-vations using the probability calculus. These en-tities must be of course so numerous and mustact so rapidly that the correct average values areattained in millionths of a second.”

C. A coherent quantum world

Is it possible to formulate an straightfor-ward extension of the previous physical picturewithin Quantum Mechanics (QM)? Any closedand finite quantum system involves a discreteenergy spectrum and evolves quasi-periodicallyin the Hilbert space, which becomes the quan-tum analogue to the classical phase space. Asin CM, a closed and finite quantum system isintrinsically reversible due to the unitarity ofthe evolution operator. However, there exists acrucial difference: while integrability, ergodicityand chaos are well-established concepts withinCM, their extensions in QM are much less clear.The notion of integrability in the QM liter-

5

ature may refer to different criteria. Contraryto CM, the existence of “N independent (lo-cal) conserved mutually commuting linearly in-dependent operators” does not necessarily im-ply that the system is “exactly solvable” in thequantum domain [25, 26]. In addition, the con-cept of ergodicity in QM has also generated in-tense debate, even after the recent rediscoveryof the John von Neumann’s Quantum ErgodicTheorem [27, 28]. Last, but not least, the classi-cal definition of chaos as the sensitivity to initialconditions does not apply to a wave equationas Schrodinger’s. Indeed, we will discuss belowthat the quantum signature of dynamical chaoshad to be found as an instability of a quantumevolution towards perturbations in the Hamil-tonian [29].

1. Decoherence and the theory of open quantumsystems

A phenomenological description of irre-versible dynamics within QM can be performedby postulating that the system of interest is incontact with an environment. This implies theremoval of the assumption that the system is“closed”. As stated by Goran Lindblad [30],“It seems that the only possibility of introduc-

ing an irreversible behavior in a finite system isto avoid the unitary time development altogetherby considering non-Hamiltonian systems. Oneway of doing this is by postulating an interac-tion of the considered systems S with an exter-nal system R like a heat bath or a measuringinstrument.”The first consequence of the coupling to in-

finitely many environmental degrees of freedomseems to be the destruction of quantum weird-ness within the system. More precisely, thismeans the attenuation of the interferences ev-idenced in specific observables and, ultimately,the appearance of classicality [31]. Since thisdegradation is originated in the loss of phase-coherence between the components of specificquantum superpositions, such a process is calleddecoherence. This automatically implies thatthe open system dynamics is irreversible.

A theoretical framework, developed in the1960’s by Leo Kadanoff and Gordon Baym [32]and by Leonid Keldysh [33], describes the non-equilibrium SM and it intrinsically deals withopen-system dynamics. Such a framework usesthe tools of Quantum Field Theory and it ul-timately involves the TL. Precisely, this limitmay shadow the nature of the approximationsinvolved. A more “controlled” formalism to de-scribe open-system dynamics had to wait un-til 1976, when Lindblad’s work [30] was in-dependently and simultaneously complementedby Vittorio Gorini, Andrzej Kossakowski, andGeorge Sudarshan [34]. They provide the math-ematical structure of the quantum master equa-tions (QME), which in turn can be understoodas generalized Liouville-von Neumann differen-tial equations for the reduced density matrix.In this approach, the system under study is fi-nite and specific assumptions are made to de-scribe its environment. These physical hypoth-esis, which are basically known as the Marko-vian approximation, were already known anddiscussed in the 1950’s by Felix Bloch [35] andUgo Fano [36]. As summarized by Karl Blum[37],

“It is assumed that R has so many degrees offreedom that the effects of the interaction withS dissipate away quickly and will not react backonto S to any significant extent so that R re-mains described by a thermal equilibrium distri-bution at constant temperature, irrespective ofthe amount of energy and polarization diffusinginto it from the system S. In other words, it isassumed that the reaction of S on R is neglectedand the correlations between S and R, inducedby the interaction, are neglected.”

The QME approach has proved to be opera-tionally successful in describing the dynamics ofopen systems, as in the case of nuclear magneticresonance (NMR) [38, 39] and quantum optics[40, 41]. The more prominent model within thetheory of open quantum systems is the two-level system in the presence of a structured en-vironment [42]. In particular, the Kadanoff-Baym-Keldysh approach was recently appliedto such a model with the specific purpose ofdescribing quantum dynamical phase transitions

6

[43]. This novel phenomenon, firstly observed inNMR [44], corresponds to a functional change ornon-analyticities in the dynamics of specific ob-servables. More recently, these transitions werealso described by means of the QME approach[45]. In any case, the environment is required tobe already in the TL, a requisite automaticallysatisfied in the experimental realm.

2. Closed quantum systems: to equilibrate or tolocalize?

As in the original Boltzmann’s gas of collidingmolecules, let us consider now a closed quan-tum system composed by a large number N ofinteracting particles, e.g. fermions or spins in alattice. We assume that the initial state of thesystem (t = 0) is given by ρ0, which does notcommute with the time-independent Hamilto-nian H. In other words, ρ0 is a non-equilibriumstate. Its time evolution is given by

ρt = exp

[

− i

~tH

]

ρ0 exp

[

i

~tH

]

,

which, being dependent on time throughquasiperiodic functions, is nearly recurrent and,strictly speaking, it cannot describe the ap-proach to any “equilibrium” state. This ismerely the result of an unitary dynamics.Given a specific observable, say O, the time-

evolution of its expectation value is:

〈O(t)〉 = tr(

ρtO)

.

Surprisingly, provided that O fulfils certainconditions, equilibration can occur for 〈O(t)〉.Here, equilibration means that after some tran-sient behavior, 〈O(t)〉 reaches some “station-ary” value and remains close to it for most ofthe time [27, 28]. There exists several theo-retical and experimental questions around suchidea [46, 47]. For instance, one may naturally

ask on the conditions required for O, H , and ρ0in order to observe the equilibration of 〈O(t)〉.Indeed, it seems that a key requirement for equi-libration is the locality of O. Loosely speaking,

this means that O involves only a small num-ber of sites or particles. The idea of localityprovides for a natural argument in favor of anapparently irreversible behavior of 〈O(t)〉. In-deed, if we “observe” a local subsystem thatinvolves only a small fraction of the degrees offreedom of the entire system, the coupling tothe rest of the system mimics a coupling to anenvironment. Therefore, as in the above discus-sion on open system dynamics, the relaxation of〈O(t)〉 occurs due to the presence of a large en-vironment “observing” the subsystem. In sucha case, the complete system is said to act as itsown environment.

Given an interacting many-body system, isit natural to expect that local observables doequilibrate? The answer is no, since there existstriking phenomena in which the equilibrationmechanisms are completely inhibited. In fact, aparadigmatic example can be drawn from a veryactive area in physics: the quantum localization.The concept was originally developed by PhilipW. Anderson [48], who described the absence ofdiffusion of spin excitations in a disordered sys-tem. This phenomenon was observed by GeorgeFeher in electron-nuclei double resonance ex-periments performed in doped semiconductors[49]. While the problem had indeed a many-body nature, Anderson succeed in simplifyingit as a system of non-interacting electrons ina d-dimensional disordered lattice. He realizedthat its dynamics described by Bloch states canchange dramatically when a perturbing disor-dered potential exceeds a critical value. Thisis known as the extended-to-localized transitionor Anderson’s localization (AL) [49]. Indeed, ifthe disorder is small, single-particle states areessentially described in terms of the scatteringof Bloch states with some finite lifetime. Pre-cisely, this physical picture was employed byRobert Laughlin to re-examine quantum trans-port in a random potential as a problem ofQuantum Chaos [52]. He proved that the Lya-punov characteristic exponent of the classicalelectron motion in such a potential can be iden-tified with the collision rate 1/τ appearing inOhm’s law. This constitutes a conceptual linkbetween chaos and diffusive transport. Then,

7

for small disorder, one can generally say thata local excitation is described as a wave packetthat ultimately diffuses throughout the system.But AL involves a paradigm shift: when the dis-order is large enough, the eigenstates of the sys-tem become localized in the real space. Thus,in contrast to the diffusive picture, localizationimplies that the excitation remains close to itsinitial location and transport phenomena is nolonger possible [50, 51]. In some sense, the sameimpurities that produce chaos and Lyapunovexponents, which make possible diffusion, dissi-pative transport and equilibration [52], end upconspiring against them. For dimension d ≤ 2,even a weak disorder ensures the onset of AL.

A step beyond the standard AL problem cor-responds to the original problem of localizationin interacting systems. As a matter of fact,adding interactions increases the effective di-mensionality of the problem, and hence signif-icantly magnifies its complexity. A clean (or-dered) system can already evidence a transitioninto an insulating phase if the interactions arestrong enough. This is the case of the Mott-Hubbard transition, whose most paradigmaticcase occurs in a crystal with one electron peratomic orbital. John Hubbard showed by meansof a Hartree-Fock calculation that for stronglocal electron-electron repulsion, an otherwisehalf-filled band of electronic states would splitinto an occupied band and an unoccupied one[53]. This drives the system from a metal-lic phase to a insulator one [54, 55], a situ-ation that would persist at fillings where thestrong electron-electron Coulomb repulsion for-bids the identification of single-particle bands.If the system is disordered, the naive expecta-tion would be that interactions prevent the ex-istence of AL essentially for two reasons. First,collisional dephasing would destroy the specificinterferences needed to localize. Second, includ-ing interactions automatically implies an expo-nential increase in the fraction of the Hilbertspace required to describe the excitation dy-namics. This, in turn, constitutes a more favor-able scenario for diffusion, consistent with thefact that a large d prevents localization. Nev-ertheless, a different kind of transition does oc-

cur between extended and localized many-bodyphases, which is called Many-Body Localization(MBL) [56, 57]. The MBL is a dynamical tran-sition that results when both perturbations toBloch states, i.e. interactions and Anderson dis-order, are present. As in the AL case, an inter-acting many-body system is said to be localizedif the diffusion or transport of excitations be-comes frozen, and therefore a memory of theinitial conditions is preserved in local observ-ables for long times [58]. However, there is ahierarchical difference between AL and MBL:while the former deals with the eigenstates in asingle particle Hilbert space, the latter relies onthe properties of the many-body eigenstates inthe much bigger Fock space.

The study of localization phenomenon, andparticularly the MBL, is extremely attractivefrom fundamental grounds. Its importancetraces back to the FPU problem and the (classi-cal) theoretical framework developed to under-stand it. As in the attempt by Fermi and collab-orators, the current aim is to study simple quan-tum models that could go parametrically froman ergodic to a non-ergodic quantum dynamics.Moreover, a fundamental question is whethersuch a transition occurs as a smooth crossoveror it has a critical value, a sort of generalizedquantum KAM threshold [59]. As a matter offact, the MBL transition is the promising can-didate in the quantum realm. If the many-bodystates are extended, then one may expect thatthe system is ergodic enough to behave as itsown environment, and, as stated before, equi-libration is enabled. Quite on the contrary, ifthe many-body states are localized, any initialexcitation would remain out-of-equilibrium. Inthis case, equilibration is precluded. Thereforethe MBL would evidence the sought thresholdbetween ergodic and non-ergodic behaviors.

During the last years, the literature dealingwith the dynamics of equilibration and ther-malization has grown overwhelmingly (see Refs.[47] and [58]). On the experimental side, anextreme degree of isolation and control hasbeen achieved, such as in the case of ultracoldquantum gases [60], trapped ions [61] and alsoNMR [62]. On the theoretical side, the use

8

of the Hubbard model [53] has proven usefulto address the dynamics of strongly-interactingmany-body systems [63–65]. In addition, forspin systems, numerical studies have providedevidence indicating that the competence be-tween disorder and interactions may lead tohighly non-trivial dynamical phase diagrams[115, 116].

D. Echoes from the future: NMR comes

back

We are now ready to focus on our major ques-tion. Namely, to what extent would equilibra-tion, as introduced above, could turn the excita-tion spreading into an irreversible phenomenon?In order to tackle such a question, we considerspecifically a system composed by N quantumspins which is perturbed from a high temper-ature equilibrium by injecting a local polariza-tion excess. In addition, we assume that thesystem can indeed equilibrate, i.e. dynamicscan lead to a homogeneous distribution of thepolarization. Such an equilibration of the po-larization is a consequence of a mechanism usu-ally called spin diffusion [66, 67]. However,even though a particular observable seems tohave reached “equilibrium”, a unitary quantumdynamics would ensure a precise memory ofthe non-equilibrium initial condition. In otherwords, the initial state is completely encodedinto correlations (eventually non-local) presentin the evolved state.If some experimental protocol could manage

to achieve the inverse evolution operator, i.e.to reverse the many-body dynamics, then itwould drive the system back to the initial non-equilibrium state [68]. This “echo idea” has re-mained at the heart of NMR. In fact, the firstNMR time-reversal experiments were performedby Erwin Hahn in the 1950’s [69]. There, thetotal polarization, that sums up contributionsfrom individual spins, when lying perpendicu-lar to the external magnetic field, will ideallyprecess indefinitely. However, each independentspin contributing to the polarization precessesat different velocity due to the local inhomo-

geneities of the magnetic field. This dephasingproduces the decay of the observed polarization.Hahn’s procedure involves the inversion of thesign of the Zeeman energy in order to reversethese precessions. If such an inversion is per-formed at time t/2, it will produce the refocus-ing of the total polarization at time t, whichis the well known “spin echo”. Nevertheless,the sign of the energy scale associated to thespin-spin interactions is not inverted and, ac-cordingly, the echo signal M(t) becomes pro-gressively degraded with t. This decay has acharacteristic time scale called T2, which, in acrystal, characterizes these many-body interac-tions.

By the early 1970’s, Horst Kessemeier, Won-Kyu Rhim, Alex Pines, and John Waugh im-plemented the reversal of the dynamics inducedby the spin-spin dipolar interaction which wasmissed by the Hahn’s procedure [70, 71]. Thisresults in the “magic echo” signal, which indi-cates the recovery of a global polarization state.Two decades later, Richard Ernst and collabo-rators introduced the “polarization echo” [72].There, the polarization along the external fieldis injected locally at some labeled spins. As theglobal polarization is a conserved quantity inthis experiment, the local polarization diffusesaway for a time t/2 due to the spin-spin dipo-lar interaction. Since it originates in many-spininteractions, this spreading occurs in time scaleT2. By the implementation of a time-reversalprocedure involving a change in the sign of thespin-spin energy at time t/2, some polarizationM(t) is recovered at time t in the same initialspots where it was injected.

While the success of these time reversalechoes unambiguously evidenced the determin-istic and quantum nature of spin-dynamics inNMR, it is unavoidable that the reversal re-sults degraded by uncontrolled internal or en-vironmental degrees of freedom, or by imper-fections in the pulse sequences. In terms ofthe theory of open quantum systems, these per-turbations would constitute sources of deco-herence. Quite often, their precise assessmentcould only be evidenced through the time rever-sal experiment [9, 73]. In addition, it was no-

9

ticed that, since in the “polarization echo” ex-periments the total polarization is a conservedquantity, an imperfectly recovered M(t) wouldbe roughly proportional to inverse of the num-ber of spins among which the polarization is dis-tributed. This gives a clear entropic meaning for− ln[M(t)][11]. Most importantly, the degrada-tion of M(t) in such experiments, where themany-spin interactions are globally and quiteperfectly time-reversed, occurs in a new timescale T3. Remarkably, the experiments indi-cate that T3 seems to be much shorter than anynaive estimation of the characteristic time scaleof the perturbations, say τΣ. Thus, the questionis whether the complexity inherent to a largenumber of correlated spins would enhance thefragility of the procedure under perturbations[9].

E. The Loschmidt echo: irreversibility as

an emergent phenomenon

The time-reversal procedures just describedcan be thought as a quantum realization of thegedanken experiment suggested by Loschmidt.As mentioned above, he proposed the rever-sal of the particles’ velocities as a mechanismfor undoing the increase of entropy. Actually,the aforementioned NMR time-reversal experi-ments are specific procedures for backtrace thetime evolution [74]. This led Patricia Levsteinand Horacio Pastawski to define the “Loschmidtecho” (LE) as an idealization which embodiesthe various, eventually imperfect, time-reversalprocedures implemented in NMR [75].A next generation of experiments in organic

crystals [9–11], suggested that the experimentaltime scale for irreversibility T3 never exceedsmore than a few times T2, and indeed, it remainstied to it [10, 11]. Thus, it seems that one is leftwith a sort of unbeatable limit,

T2 . T3 ≪ τΣ.

The most immediate conclusion would be thatthere is still an uncontrolled source of decoher-ence not described by τΣ, an explanation thatboth theoreticians and experimentalists would

subscribe to [76]. However, the experimentshinted that this “perturbation independent de-cay” (PID) has a much deeper origin, with T3acting as a sort of inverse Lyapunov exponent[10]. Since this is in fact an intrinsic propertyof the system (in the absence of external per-turbations), the PID can also be understood asan intrinsic decoherence rate. We should stressthat until now, this observation holds for LEexperiments where polarization is a conservedmagnitude. Those of the magic echo type failto fulfill this requirement [77].

When the decay rate 1/T3 (of a LE with po-larization conserved) is plotted against the in-tensity of the most relevant non-inverted term,namely the residual non-secular interaction, itsaturates to a finite value: T3 ∼ 4T2. This is ex-plicitly shown in Fig. 1. This plot resembles astandard resistivity vs. temperature plot in animpure metal, where the finite resistivity offsetat the zero temperature limit is determined bythe impurity scattering [78], i.e. by chaos [52].Analogously, T3 in the zero perturbation limit,keeps tied to the time scale that characterizesthe reversible many-body interaction. This ob-servation led to postulate the Central Hypoth-esis of Irreversibility: in an infinite many-spinsystem far away from its ground state, the com-plex dynamics amplifies the action of any smallnon-inverted interaction to the degree that suchcomplex (reversible) dynamics provides for thedominant time-scale. Thus, reversible inter-actions responsible for spin diffusion turn outto be the determinant contribution to the irre-versibility rate.

The previous hypothesis triggered the theo-retical study of simpler problems in which theLE can be evaluated systematically. In partic-ular, the role of a “testing bench” was playedby single particle systems whose classical coun-terpart is chaotic. There, the LE is defined as[29]

M(t) =

∣

∣

∣

∣

〈ψ| exp[ i~(H0 + Σ)t] exp[− i

~H0t] |ψ〉

∣

∣

∣

∣

2

,

(1)i.e. the square of the overlap between twoone-body wave functions, one of them being

10

FIG. 1. The LE decay rate 1/T3 as a function of the perturbation’s characteristic rate 1/τΣ. Both quantitiesare rescaled by the T2 time scale, which corresponds to the reversible or controlled Hamiltonian. Thus, thevertical axis corresponds to the irreversibility time-scale, while the horizontal one is associated with thestrength of the perturbation. In the limit of vanishing perturbation, the irreversibility time-scale saturatesat a fraction ∼ 1

4of the reversible time-scale. This drastically differs from what expected for a single

molecule: a perfectly reversible dynamics as the perturbation goes to zero. Adapted from [79].

evolved by an unperturbed Hamiltonian H0 andthe other by a perturbed one H0 + Σ. A pio-neering prediction by Asher Peres [80] pointedthat, if the unperturbed evolution is classicallychaotic, the long time limit M(∞) would yieldan evenly spread excitation (not expected forintegrable systems). Intermediate times probeda more subtle behavior. Semiclassical calcula-tions showed that at a finite energy ε and smallperturbation strengths (but exceeding the spec-tral discreteness), the LE decay rate 1/T3 equalsΓ(Σ, ε), i.e. the broadening estimated from aFermi golden rule (FGR) calculation [29, 81].This lifetime is proportional to the square ofthe perturbation strength. Exponential decayswere soon found for different observables andmethods [82, 83]. However, the real surprisewas the existence of a regime in which the de-cay rate of the LE corresponds to the classicalLyapunov exponent [29, 84]. More specifically

1/T3 = min[Γ/~, λ].

The LE Lyapunov regime is a particular PIDthat holds for a semiclassical initial state built

from a dense spectrum with a perturbationabove certain critical threshold Σc. Notably,Σc falls inversely proportional to the energy εof the state. Since ε → ∞ is equivalent to~ → 0, we may say that Σc vanishes in theclassical limit, providing the elusive quantum-classical limit [85].

One should point out that the LE Lyapunovregime may not be fully equivalent with themore natural Lyapunov growth of correlationfunctions at short times which is, by now, muchused to address the so called “information para-dox” of black holes in the context of AdS-CFT[86–91]. In fact, the search for diverging correla-tions was inspired by the expected growth of theuncertainties of an electron wave packet prop-agating in a dirty metal [92]. Such an evolu-tion of the correlation functions holds up to theEhrenfest or “scrambling” time, when differentportions of a spread wave packet are scatteredby different impurities. At this point we shouldremember that the LE Lyapunov regime holdsuntil the much longer Thouless time, when thealready scrambled excitation had fully spread

11

through the whole system [84, 93].

The discovery of the PID or Lyapunovregimes in specific LE evaluations was a verybig leap that triggered the interest of the Quan-tum Chaos and Quantum Information commu-nities [94, 95]. Indeed, the mere existence ofa PID regime would pose a great challenge onthe controllability of quantum devices as it evi-dences an intrinsic fragility of quantum dynam-ics towards minuscule perturbations. The sen-sitivity to perturbations or fragility of quantumsystems [96–98] has become a major problemthat transversely affects several fields, e.g. pos-sible chaoticity of quantum computers [99, 100],NMR quantum information processing [101–104], quantum criticality [105, 106] and, morerecently, quantum control theory [107]. Nev-ertheless, the ultimate numerical or analyticalproof that would support the above hypothesis,and thus explain the experimental observations,is still lacking. The reason for such a theoret-ical bottleneck relies on the inherent difficultyin addressing many-body dynamics. Precisely,the association of many-body complexity witha form of chaos [108, 109], could provide a ra-tional for the experimentally observed PID.

An unavoidable starting point for the analysisof PID is the identification of the conditions forthe “perturbation dependent decay” that mani-fests at the relatively short times when exponen-tial decay described by FGR is manifested [110].As mentioned above, in single-particle systemswith a semiclassical excitation, the FGR holdsfor weak perturbations up to threshold givenby the classical Lyapunov exponent that indi-cates the onset of the PID. So, could a many-body system have a vanishing threshold for thePID? If that is the case, it could show up as aquantum dynamical phase transition in the TL(N → ∞). As a matter of fact, a standard ex-periment involves a crystalline sample with aninfinitely large number of spins. In other words,experiments are “already in the TL”. In con-trast, any numerical approach to assess many-spin dynamics can only cope with a strictly fi-nite N . This is the key point where the originaldiscussion on an emergent irreversibility comesback into scene: an appropriate finite size scal-

ing is required in order to grasp the emergentmechanism that rules irreversibility in the TL.Indeed, one should increase progressively N go-ing from small systems to larger ones with acontrolled perturbation. There, the emergentbehavior would follow only in a precise order ofthe limits: first N → ∞ and then τΣ → ∞.

Reference [111] constitutes a first attempt topursue such an ambitious program. The nu-merical detection of the PID was not achievedand, in fact, it may stay beyond the state-of-the-art numerical techniques [112]. However, aneffective FGR regime was reported being consis-tent with an emergent picture of irreversibility.Keeping in mind the crucial order of the lim-its stated above, in an infinitely large system,an infinitesimal perturbation is associated to afinite τΣ. If a unitary evolution keeps the polar-ization equilibrated during a time t > τΣ, suchan equilibration becomes irreversible, and itstime-reversal would be completely ineffectual.

In what follows, inspired by the LE exper-iments in NMR, we provide a framework fortheoretical evaluations of LE of the polarizationtype. We do not consider here LE of the magicecho type, as in this case neither the numericalresolution of small systems [113] nor the actualexperiments on scaled Hamiltonians [77] seemto evidence a PID regime. We start by sum-marizing the LE formulation in spin systems asa local autocorrelation function. Moreover, asintroduced in Ref. [112], we discuss the localcorrelation in terms of two global ones. One ofthese global correlations is defined as the many-body LE and resembles the standard one-bodyLE definition embodied by Eq. (1). The otheris an initially fast growing multi-spin cross cor-relation that later decays. This formalizationleads us to address a fundamental question un-derlying the LE literature [114]: what is the re-lation between the LE as defined in one-bodysystems, i.e. the overlap of wave functions, andthe LE as defined in many-body systems, i.e. aspin correlation function? Precisely, we proposeand discuss a dynamical preparation protocol totransform the local LE into a global one. Such adynamically prepared LE (DPLE) can, in turn,lead to a new series of NMR experiments to sys-

12

tematically address the fragility of strongly cor-related many-body systems in the presence ofsmall perturbations.

II. THE LOSCHMIDT ECHO

FORMULATION

A. The local autocorrelation function

Following the early LE experiments [9–11], letus discuss and formalize here the spin systemand the time reversal protocol. We consider Nspins 1/2 whose initial condition is described byan infinite temperature state, i.e. a completelydepolarized mixture, plus a locally injected po-larization,

ρ0 =1

2N(I+ 2Sz

1 ). (2)

Here, the spin 1 is polarized while the othersare not, i.e. tr[Sz

i ρ0] =12δi,1.

The polarization, which is initially placed ina single spin, diffuses all around due to the spin-spin interactions. More precisely, a many-spinHamiltonian H0 rules such a forward evolutionof the system up to a certain time tR. At thatmoment, an inversion of the sign of H0 is per-formed, leading to a symmetric backward evo-lution. Typically, H0 stands for a truncateddipolar Hamiltonian. Nevertheless, there areunavoidable perturbations, denoted by Σ, thatcould arise from the incomplete control of thespin Hamiltonian, acting on both periods. Theprocedure ends up with a local measurement inthe same spin that was originally polarized. SeeFig. 2. The evolution operators for each tR-periods are U+(tR) = exp[− i

~(H0 + Σ)tR] and

U−(tR) = exp[− i~(−H0 + Σ)tR] respectively.

Thus, it is natural to define the LE operator

as:

ULE(2tR) = U−(tR)U+(tR), (3)

which produces an imperfect refocusing at time2tR. The local measurement of the polarization,performed at site 1, defines the local LE:

M1,1(t) = 2tr[Sz1 ULE(t)ρ0U

†LE(t)] = 2tr[Sz

1 ρt].(4)

Here, we choose as free variable t = 2tR, thetotal elapsed time in the presence of the per-turbation. The time dependence of ρt in theSchrodinger picture is,

ρt = ULE(t)ρ0U†LE(t). (5)

Using Eq. (2), and after some algebraic ma-nipulation, the LE can be explicitly written asa correlation function:

M1,1(t) =1

2N−2tr[U †

LE(t)Sz1 (0)ULE(t)S

z1 (0)]

=tr[Sz

1 (t)Sz1 (0)]

tr[Sz1 (0)S

z1 (0)]

. (6)

Here, the time dependence is written accordingto the Heisenberg picture,

Sz1 (t) = U †

LE(t)Sz1 (0)ULE(t). (7)

Notice that Eq. (6) is an explicit correlationfunction at the same site but different times,i.e. an autocorrelation. This correlation hasbeen recently employed to address many-bodylocalization in spin systems [115, 116].

If we use the identity Sz1 = S+

1 S−1 − 1

2 I inEq. (6), the invariance of the trace under cyclic

permutations ensures that tr[Sz1 (t)S

z1 (0)] =

tr[S−1 (0)Sz

1 (t)S+1 (0)] − 1

2 tr[Sz1 (t)]. Since

tr[Sz1 (t)] = tr[Sz

1 (0)] = 0, then:

M1,1(t) = 2∑

i

1

2N−1〈i| S−

1 (0)U †LE(t)S

z1 (0)ULE(t)S

+1 (0) |i〉

= 2∑

i∈A

1

2N−1〈i| U †

LE(t)Sz1 ULE(t) |i〉 . (8)

13

exp[-i t / ]R h(H + )0 S exp[-i(-H / ]0 R h+ )tS

Time0 tR 2tR

a) b) c)

FIG. 2. The pictorial scheme of the dynamics involved in M1,1(t). (a) A local excitation in injected in ahigh temperature spin system. The corresponding state is given by Eq. (2). The system evolves ruled by

the Hamiltonian H0 + Σ. Spin diffusion leads to the spreading of the excitation until a time t = tR (b). At

that time, the sign of H0 is inverted. A backward evolution takes place ruled by −H0+Σ. At time t = 2tR(c), at the same initial spot, a local measurement is performed.

Here, the states |i〉 correspond to the compu-tational Ising basis for N spins. In the secondline, we restrict the sum to the set A of numbersi that label basis states which have the 1st spinpointing up, i.e. i ∈ A⇔ Sz

1 |i〉 = + 12 |i〉. Equa-

tion (8) is indeed an explicit way to rewrite Eq.(4) in the form of an ensemble average. Fromthe computational point of view, we are leftwith two possible ways to evaluate, without anytruncation, the correlation function M1,1(t).

The first (and naive) alternative would be thestorage and manipulation of the complete den-sity matrix (whose size scales as ∼ 2N × 2N ).Additionally, the time-dependence representedin Eq. (5) would eventually require the diag-onalization of the Hamiltonian matrix. Thisstrategy has strong limitations due to the mem-ory constraints of any hardware and thus onewould hardly achieve systems larger than N ∼12. The second alternative would be the inde-pendent (trivially parallel) computation of eachof the 2N−1 expectation values in Eq. (8). Insuch a case, one would handle single vectors(whose size scales as ∼ 2N ). The evolution op-

erators, in turn, can be implemented accordingto the Trotter-Suzuki formula up to a desiredaccuracy [117].

Remarkably, there is a parallel way to suc-cessfully approximate the previous calculationusing quantum superpositions. Since Sz

1 is a lo-cal (“one-body”) operator, its evaluation in Eq.(8) can be replaced by the expectation value ina single superposition state [118],

M1,1(t) = 2 〈Ψneq| U †LE(t)S

z1 ULE(t) |Ψneq〉 ,

(9)where:

|Ψneq〉 =∑

k∈A

exp[−iφk]√2N−1

|k〉 . (10)

Here, φk are random phases uniformly dis-tributed in [0, 2π). As a matter of fact, thestate defined in Eq. (10) is a random superposi-tion that can successfully mimic the dynamics ofensemble calculations and provides a quadraticspeedup of computational efforts [118] (for sim-ilar implementations, see also [119, 120]).

14

B. From local to global LE

According to the basis states introducedabove, the initial state in Eq. (2), can be writ-ten as

ρ0 =∑

j∈A

2−(N−1) |j〉 〈j| . (11)

Using Eqs. (2), (8) and (11), we can rewriteM1,1(t) as introduced in Ref. [121],

M1,1(t) = 2

∑

i∈A

∑

j∈A

1

2N−1

∣

∣

∣〈j| ULE(t) |i〉∣

∣

∣

2

− 1

2

.

(12)After some manipulation,

M1,1(t) = 2

∑

i∈A

∑

j∈A

1

2N−1

∣

∣

∣〈j| ULE(t) |i〉∣

∣

∣

2

− 1

2

=∑

i∈A

1

2N−1

∣

∣

∣〈i| ULE(t) |i〉∣

∣

∣

2

+∑

j∈A (j 6=i)

∣

∣

∣〈j| ULE(t) |i〉∣

∣

∣

2

−∑

j∈B

∣

∣

∣〈j| ULE(t) |i〉∣

∣

∣

2

.

(13)

Here, B stands for the complement of A, i.e.j ∈ B ⇔ Sz

1 |j〉 = − 12 |j〉. We follow Ref. [112]

to split the contributions in M1,1(t). Then, thefirst sum in Eq. (13) is defined as the many-body or global LE, denoted by MMB(t),

MMB(t) =∑

i∈A

1

2N−1

∣

∣

∣〈i| ULE(t) |i〉∣

∣

∣

2

, (14)

and stands for the average probability of revivalof the many-body states. As a matter of fact,this magnitude resembles the original LE defini-tion stated in Eq. (1). Moreover, a widely em-ployed extension of the LE in many-body sys-tems corresponds to single overlaps of specificmany-body wave functions, i.e. a specific single

term in the sum of Eq. (14). Such an approachhas been performed in many scenarios, such ascriticality [105], non-Markovianity in open sys-tems [122, 123], orthogonality catastrophe [124],equilibration dynamics that follows a quantumquench [125–128], many-body localization [129],among others.

The second sum in Eq. (13) represents theaverage probability of changing the configura-tion of any spin except the 1st. The third sumstands for the average probability that the 1st

spin has actually flipped, i.e. of all those pro-cesses that do not contribute to M1,1(t). Thesum of these terms defines a correlation functionMX(t),

MX(t) =∑

i∈A

1

2N−1

∑

j∈A (j 6=i)

∣

∣

∣〈j| ULE(t) |i〉

∣

∣

∣

2

−∑

j∈B

∣

∣

∣〈j| ULE(t) |i〉

∣

∣

∣

2

. (15)

This balance of probabilities leads to the appro-priate asymptotic behavior ofM1,1(t) accordingto the symmetries that constrain the evolution.

The decomposition

M1,1(t) =MMB(t) +MX(t) (16)

15

has been recently studied in Ref. [112]. Thevery short-time perturbative behavior of eachof these quantities has been specifically quanti-fied: MMB(t) decreases as 1−(N/4) (t/τΣ)

2 and

MX(t) increases initially as (N/4− 1) (t/τΣ)2.

In this specific context, τΣ is defined as thecharacteristic local time-scale of the perturba-tion. The precise balance between MMB(t) andMX(t) provide for the initial decay of M1,1(t),

which at very short times is given by 1−(t/τΣ)2.

These expansions indicate an extensivity rela-tion between MMB(t) and M1,1(t) based on thefact that the former decays N times faster thanthe latter. This has been interpreted [112] asa consequence of statistical independence (atleast valid for short-times): the probability ofrefocusing a complete many-spin state is essen-tially N times the probability of refocusing asingle spin configuration (up or down).

For intermediate times, the experimental evi-dence indicates that many-body interactions be-come crucial to provide for decay of M1,1(t).

More precisely, reversible interactions in H0

are responsible for the observed M1,1(t) de-cay rates. This experimental observation mo-tivates an analisys of the perturbation seriesbeyond the short-time regime. As alreadyhinted in [112], a general term in these expan-

sions would be proportional to (t/τΣ)2 (t/T2)

2,with coefficients that increase rapidly as a func-tion of N . These terms account for the ap-pearce of high order many-body correlations.In the case of MX(t), it would indicate an in-crease even faster than the one indicated above.This is fully consistent with the idea of chaos-induced scrambling of quantum information asrecently discussed in Ref. [130]. Neverthe-less, this growth could not persist indefinitelysince MX(t) should ultimately decay, much asin a Multiple Quantum Coherence experimentwhere two-spin coherences (i.e. correlations)should give place to four-spin coherences andso on [38, 131].

III. DYNAMICALLY PREPARED

LOSCHMIDT ECHO

A. Operational ideas

Let us now discuss how to explicitly trans-form the local LE, i.e. M1,1(t), into a globalmeasure of the reversibility of the many-bodydynamics. Here, the strategy is based on dy-namical preparation, which provides for an in-creasing complexity of the initial state in a con-trollable way. Basically, the procedure consistsof a sequence, schematized in Fig. 3, which isgiven by: preparation, standard LE, perfect re-versal of the preparation and, finally, a localmeasurement. The preparation results from anevolution, controlled by a Hamiltonian Hp, thatoccurs during a time tp. Then, a standard LEprocedure is performed as discussed above, i.e.an imperfect forward-backward evolution. Af-ter that, a symmetric tp backward evolution

ruled by −Hp unravels the preparation, lead-ing to a local observation. Such a local LE isnow denoted byM1,1(t, tp). The main idea hereis that the local measurement, i.e. Fig. 3(e),would be equivalent to the overlap between two“equilibrated” many-body wave functions thatcorrespond to the states of the system afterpreparation, i.e. the overlap of the states inFig. 3(b) and Fig. 3(d). In this sense, the localautocorrelation M1,1(t, tp) would take the formof the standard LE definition as in Eq. (1), butnow, in terms of many-body wave functions.The preparation scheme encodes the local ex-

citation into an extended state. Since the LEprocedure is performed after such preparation,the sensitivity under perturbations is evaluatedin an initially correlated state. In order to quan-tify such an observation, let us first consider asingle basis state |j〉, j ∈ A, that is prepared

according to Up(tp) = exp[− i~Hptp]:

Up(tp) |j〉 =∑

k

cjk(tp) |k〉 =∣

∣

∣Ψ(j)⟩

. (17)

If Hp does not exhibit any particular symme-try, dynamics would not be restricted and oneshould assume that the summation in k effec-

16

exp[-iH t / ]P P h exp[-i(-H / ]0 R h+ )tS

Time0 tP

exp[iH t / ]P P h

t +tP Rt + tP R2 2 2t + tP R

a) b) c) d) e)

exp[-i(H / ]0 R h+ )tS

FIG. 3. The preparation scheme is included in the protocol shown in Fig. 2. (a) The initial state given byEq. (2) evolves leading to a dynamically correlated state represented in (b). The time reversal procedure isperformed afterwards (c)-(d). A “perfect” reversal of the preparation finally leads to a local measurementin (e).

tively runs over the complete Hilbert space.Moreover, the quantum superposition in Eq.(17) is often assumed to be a sort of “chaotic”superposition of the basis states |k〉 [109, 132].In our case, this observation is particularly trueif tp is larger than the time needed to drive thepolarization into an equilibrated value (say, τeq)[111]. In such an extreme case, the coefficientscik(tp) fluctuate so randomly as function of theindices j and k that we end up replacing

∣

∣

∣Ψ(i)⟩

=∑

k

cjk(tp) |k〉 −→

−→∑

k

exp[−iφk]√2N

|k〉 = |Φ〉 ,(18)

where, as before, φk are random phases

uniformly distributed in the interval [0, 2π).Two different realizations of these randomphases, say φk and

φk

, lead to two dif-

ferent states,

|Φ〉 =∑

k

exp[−iφk]√2N

|k〉 ,

∣

∣

∣Φ⟩

=∑

k

exp[−iφk]√2N

|k〉 ,

which are typically almost orthogonal:

∣

∣

∣

⟨

Φ | Φ⟩∣

∣

∣

2

=

∣

∣

∣

∣

∣

∑

k

exp[−i(φk − φk)]

2N

∣

∣

∣

∣

∣

2

∼ O(2−N ).

(19)

If we replace ULE(t) by U †p (tp)ULE(t)Up(tp) in

Eq. (15),

MX(t, tp) =∑

i∈A

1

2N−1

∑

j∈A (j 6=i)

∣

∣

∣〈j| U †p(tp)ULE(t)Up(tp) |i〉

∣

∣

∣

2

−∑

j∈B

∣

∣

∣〈j| U †p (tp)ULE(t)Up(tp) |i〉

∣

∣

∣

2

=∑

i∈A

1

2N−1

∑

j∈A (j 6=i)

∣

∣

∣

⟨

Ψ(j)∣

∣

∣ ULE(t)∣

∣

∣Ψ(i)⟩∣

∣

∣

2

−∑

j∈B

∣

∣

∣

⟨

Ψ(j)∣

∣

∣ ULE(t)∣

∣

∣Ψ(i)⟩∣

∣

∣

2

. (20)

Here comes our first specific assumption. Let us replace each of the coherent superpositions

17

states, namely∣

∣Ψ(i)⟩

and∣

∣Ψ(j)⟩

, by incoherentsuperpositions as in Eq. (18). Then, the twosummations in Eq. (20) would essentially yieldthe same outcomes and this leads us to expectthat MX(t, tp) ∼ O(2−N ). In addition,

MMB(t, tp) =

=∑

i∈A

1

2N−1

∣

∣

∣〈i| U †p(tp)ULE(t)Up(tp) |i〉

∣

∣

∣

2

=∑

i∈A

1

2N−1

∣

∣

∣

⟨

Ψ(i)∣

∣

∣ ULE(t)∣

∣

∣Ψ(i)⟩∣

∣

∣

2

.

Here, the replacement proposed in Eq. (18) hasa practical relevance. Indeed, when replacing∣

∣Ψ(i)⟩

−→ |Φ〉, Eq. (21) consists in the average

of 2N−1 overlaps, each of them being mathemat-ically equivalent. Then, it is enough to keep justone of these overlaps,

MMB(t, tp) ∼∣

∣

∣〈Φ| ULE(t) |Φ〉∣

∣

∣

2

. (21)

If our previous observation about MX(t, tp) be-ing exponentially small is indeed verified, Eq.(16) automatically implies that M1,1(t, tp) ∼

MMB(t, tp). Then,

M1,1(t, tp) ∼∣

∣

∣〈Φ| ULE(t) |Φ〉∣

∣

∣

2

. (22)

We stress here that the previous expectations,represented by Eqns. (21) and (22), would be-come valid in the limit of tp ≫ τeq . Addi-tionally, we do not expect that these equalitieshold for any choice of Hp. In particular, weneed that such a Hamiltonian can create corre-lations involving a large extent of the Hilbertspace H. In practice, this justifies the specificmodel adopted and described in the followingSection. Furthermore, the choice is not a purelyacademic issue, since it is also motivated by apossible experimental realization.Let us briefly comment on the case in which

Hp does exhibit a particular symmetry, forinstance, the conservation of spin projection

[Hp,∑N

n=1 Szi ] = 0. Such a case naturally in-

duces the decomposition of H in terms of sub-spaces Smz

with definite spin projection quan-tum number mz =

∑N

n=1 Szn. Then, the su-

perposition in Eq. (17),∣

∣

∣Ψ(j)[Smz

]

⟩

would be re-

stricted to the specific projection subspace Smz

according to the initial basis state |j〉. As a con-sequence, one should replace in Eq. (18) the co-

herent superposition∣

∣

∣Ψ(j)[Smz

]

⟩

by a random one

also defined in such a subspace∣

∣Φ[Smz]

⟩

. Thereasoning is analogous as before but one has toaverage each of the subspaces,

M1,1(t, tp) ∼ (1−M∞)∑

mz

Dmz

∣

∣

∣

⟨

Φ[Smz

]

∣

∣

∣ULE(t)

∣

∣Φ[Smz]

⟩

∣

∣

∣

2

+M∞. (23)

Here, Dmzstands for the statistical weight of

the subspace Smzand M∞ is the corresponding

asymptotic value of the LE. If the total spin pro-jection in the z direction is conserved (i.e. mz

is good quantum number), then one should ex-pect thatM∞ ∼ N−1. This last asymptotic be-havior occurs when the dynamics is sufficiently

complex to distribute the polarization homoge-neously among the spins in the system, as re-ported in [110].

18

B. The many-spin model with non-secular

interactions

FIG. 4. Square lattice with periodic boundary con-ditions, N = 16. Solid lines correspond to the spinpairs 〈i, j〉

in Eq. (24), and dashed lines to the

pairs 〈i, j〉♦in Eq. (25).

In order to try out the previous expectations,we address the evaluation of the LE in a specificspin system. As in the experiments [9–11], weconsider a truncated dipolar Hamiltonian,

Hdip =

N∑

〈i,j〉

J0

[

2Szi S

zj −

(

Sxi S

xj + Sy

i Syj

)]

,

(24)

where the superscript in Hdip corresponds to

the summation 〈i, j〉 of first nearest neighborsin a square lattice with periodic boundary con-ditions, as depicted in Fig. 4 with solid lines.Additionally, J0 stands for the appropriate en-ergy units. In analogy, the dipolar coupling be-tween next nearest neighbors in the square lat-tice is given by

H♦dip =

N∑

〈i,j〉♦

J0

[

2Szi S

zj −

(

Sxi S

xj + Sy

i Syj

)]

,

(25)where, accordingly, 〈i, j〉♦ stands for the pairsof spins connected by dashed lines in Fig. 4.Then, our LE procedure is defined according to

the following choice:

H0 = Hdip,

Σ = λH♦dip, (26)

where we fix the coefficient λ = 0.1.In addition, we consider also the double-

quantum (DQ) Hamiltonian [133, 134],

Hdq =

N∑

〈i,j〉

J0

(

Sxi S

xj − Sy

i Syj

)

, (27)

H♦

dq =

N∑

〈i,j〉♦

J0

(

Sxi S

xj − Sy

i Syj

)

, (28)

with the same convention as above. This inter-action, being proportional to S+

i S+j + S−

i S−j ,

does not conserve spin projection in the z di-rection since it mixes subspaces with δmz = 2.Given the remarkable degree of control that

can be achieved in NMR quantum simulators,in the last years the DQ Hamiltonian has beenintensively employed to study the interplay be-tween decoherence and correlations in large spinarrays [101–104, 131]. In addition, it has alsobeen employed to address localization phenom-ena [135, 136]. Indeed, the DQ Hamiltonian canbe used to create, in a controllable way, clus-ters of correlated spins that can serve as initialstates for more sophisticated protocols. Thismotivates the choice

Hp =H

dq + H♦

dq√2

,

which provides for the preparation dynamics.

C. LE numerical evaluation

We show in Fig. 5 the evaluation of the cor-relation functions M1,1(t), MMB(t) andMX(t),according to Eq. (26). One may expect thatM1,1(t) and MMB(t) differ radically since thelatter should decay, as stated above, ∼ N timesfaster than the former. Strictly speaking, as dis-cussed in Ref. [112], Mη

1,1 ≃ MMB, with η ≃N/4, which for the case considered (N = 16)

19

0 10 20

0.0

0.2

0.4

0.6

0.8

1.0

M1,1(t)

MX(t)

MMB(t)

t [ /J0]

M1,

1(t) =

MM

B(t)

+ M

X(t)

FIG. 5. The local autocorrelation function or localLE M1,1(t) (black triangles), and its two non-localcontributions: MMB(t) (blue squares) and MX(t)(green circles). On the one hand, MMB(t) exhibitsa fast decay (roughly N times faster than that ofM1,1(t)) and it asymptotically goes to zero. Onthe other hand, MX(t) exhibits a rapid growth un-til it reaches a maximum, and afterwards it decays.Precisely, this decay of MX (t) determines the in-termediate and long-time decay of M1,1(t) and itsasymptotic value. As discussed in the text, the caseconsidered (N = 16) is still too small to empha-size these observations. The features in the dynam-ics of these correlation functions will become moreprominent as N increases, which may be beyond thestate-of-the-art numerical techniques.

corresponds to η ≃ 4. Such a small exponentis basically the reason why the two correlationfunctions do not separate each other consider-ably. An ideal finite size scaling would involvea progression of systems satisfying η ≫ 1. Inshort, our N = 16 is still too small. This ob-servation indicates that, within the state-of-the-art numerical techniques, finding numerical evi-dence of an emerging PID remains a major chal-lenge [111, 112].We address the evaluation of the DPLE in

Fig. 6. In particular, Fig. 6-(a) shows a for-ward autocorrelation function that correspondsto the dynamical preparation, i.e.

P1,1(t) = 2 〈Ψneq| U †p(tp)S

z1 Up(tp) |Ψneq〉 .

(29)

The time-evolution of the local polarizationat site 1 is monitored and it is observed thatit stabilizes near zero within the characteristictime of equilibration τeq ∼ 5~/J0. This leadsus to choose specific preparation times tp . τeqand tp ≫ τeq. Accordingly, Fig. 6-(b) showsM1,1(t, tp) for such choices of preparation times.If tp . τeq , the larger the tp, the faster the decayofM1,1(t, tp). When tp exceeds τeq , a saturationregime is observed, in whichM1,1(t, tp) becomesindependent of tp. In fact, Fig. 6-(b) showsthatM1,1(t, tp = 3~/J0) ≃M1,1(t, tp = 15~/J0)(purple stars and open circles, respectively). Ingeneral, all the curves representing M1,1(t, tp),for tp > τeq, collapse into a single one.The fact thatM1,1(t, tp) no longer depends on

the precise value of tp (provided that it exceedsτeq) indicates that the specific phases in the

state Up(tp) |Ψneq〉 have become non-relevantfor the dynamics of the polarization. This ideacan be generalized to say that outcome of a localmeasurement is independent of many non-localcorrelations present in an evolved many-bodystate. These “irrelevant” correlations are, inturn, responsible for encoding the precise mem-ory of the initial state, i.e. the evolution is stillunitary.In order to address the saturation aforemen-

tioned (tp > τeq), we compare M1,1(t, tp) andMMB(t, tp) for tp = 6~/J0 in Fig. 7. Inaddition, as stated in Eqns. (21) and (22),

we include the overlap | 〈Φ| ULE(t) |Φ〉 |2, being|Φ〉 the random superposition state defined inEq. (18). The coincidence between these threecurves is remarkable. It is also worthwhile tonotice the clear exponential nature of the de-cay.The previous observation essentially states

the equivalence M1,1(t, tp) ≃MMB(t, tp) in theregime where tp > τeq . Moreover, it identifiessuch saturation as the overlap between two ran-dom superposition states that evolve ruled byperturbed and unperturbed Hamiltonians. Thisis essentially the many-body extension of thesemiclassical LE definition [29]. Then, whenthe dynamical preparation creates a sufficientlycomplex state, a given perturbation yields a lo-cal LE that is same as a global one for the same

20

0 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20

0.0

0.2

0.4

0.6

0.8

1.0 (b)

t [ /J0]

P 1,1(t)

(a)

t [ /J0]

M1,1(t

,t p)

FIG. 6. (a) Forward (local) autocorrelation function P1,1(t) defined as in Eq. (29). Vertical dotted linesindicate the evolution times chosen to play the role of tp (preparation time in the DPLE protocol). Thecorresponding DPLE, M1,1(t, tp), are shown in (b). In particular, tp = 0~/J0 (black squares); tp = 0.3~/J0

(green upper triangles); tp = 0.6~/J0 (red circles); tp = 0.9~/J0 (blue down triangles); tp = 1.5~/J0 (greydiamonds); tp = 3~/J0 (purple stars); tp = 15~/J0 (open circles).

0 5 10 15 20 25

0.01

0.1

1

M1,

1 & M

MB

t [ /J0]

FIG. 7. The DPLE. M1,1(t, tp = 6~/J0), solidblack line, and MMB(t, tp = 6~/J0), blue dottedline. The overlap of random superposition states| 〈Φ| ULE(t) |Φ〉 |

2 is plotted with a green dashedline.

perturbation.

The mentioned saturation is relevant for thepicture of equilibration discussed in SectionIC 2. When the preparation dynamics equili-brates the polarization, the state Up(tp) |Ψneq〉can be replaced by the “equilibrated” state |Φ〉,as far as the evaluation of polarization is con-

cerned. The random phases in |Φ〉 correspondto the fact that equilibration of a local observ-able is reached when the global correlations areirrelevant or at least redundant for such observ-able. A similar argument has already been dis-cussed precisely in the context of the LE [111],and it provides a hint for theoretical investi-gations on the onset of equilibration for localobservables in closed many-body systems.

Our observations here may also provide newstrategies to understand the experimental re-sults discussed in Sec. I E. Since the LE decayaccelerates as a function of tp, the fragility ofthe reversal procedure in the presence of per-turbations can be systematically quantified asa function of tp. More precisely, this indicatesthat the preparation dynamics contributes tothe time-scale of the LE decay. According tothe experimental hints, this contribution wouldultimately be the dominant term in the time-scale of LE decay. Of course, this last scenariocorresponds to the TL, and thus an appropriatefinite size scaling would be needed to confirm it.

21

IV. CONCLUSIONS

Starting from a conceptual and historical dis-cussion on the microscopic foundations of theSecond Law of thermodynamics as an emergentof Classical Mechanics, we have introduced andmotivated the study of the Loschmidt echo (LE)as tool that could help us to reveal the origin ofirreversibility in a Quantum Mechanical frame-work. In particular, we discussed the LE asdefined for spin systems, which corresponds tothe original NMR polarization echo experimentswhere the global polarization is a conservedobservable. Such experimental LE evaluationshave supported an emergent, and quite para-doxical, picture for irreversibility, here embod-ied by our Central Hypothesis of Irreversibil-ity: in an infinite many-spin system in a highlymixed state, any arbitrarily small perturbationis amplified by the progressively increased com-plexity resulting from the many-body dynamics,which then becomes the dominant time-scale.Thus, the reversal procedure would ultimatelybe degraded within a time scale determined bysuch complex, but reversible, many-body inter-actions.

By formulating the LE in spin systems as alocal autocorrelation function, we were able todefine the non-local or global LE, and we pro-posed a protocol to transform the local LE intoa global one. This means that a local LE can beemployed to measure a global overlap betweenmany-body states. This modified LE procedureintroduces a dynamical preparation of the ini-tial state, which creates correlations by meansof an (ideally reversible) evolution. In practice,our numerical results confirm that the morecorrelated the state is, the more fragile underperturbations it becomes, which is manifestedin much shorter, perturbation dependent, timescales. Moreover, the decay saturates at a spe-cific time-scale that corresponds to the globalLE of random superposition states. This occurs

precisely when the preparation time tp exceedsthe equilibration time τeq. of the polarization.Such a saturation indicates that the local LEno longer depends on the precise value of tp. Inother words, a local observation of the polar-ization does not depend on the specific phasesencoded in the many-body evolved state.Our observations provide for a possible way to

understand the experimentally observed PID.Indeed, we showed that reversible dynamicstransforms the original local excitation stateinto a more complex and sensitive one, which inturn is shown to be much more likely to be af-fected by any small residual perturbation. Thetime scale at which complexity is being gener-ated could then appear as the dominant time-scale. This dynamical regime, however, has notyet been reached in numerical simulations, asthe number of involved spins does not seem tobe large enough. Thus, while we are not yet ina position to present a definite numerical testfor the Central Hypothesis of Irreversibility, itis worthwhile to mention that the state prepara-tion scheme discussed here can be implementedin a variety of actual NMR experiments. Itsapplication to finite and infinite systems in theform of Loschmidt echo of the polarization- andmagic echo types, could shed further light ontheir strikingly different behaviors, and more-over, into the elusive reversibility paradox.

V. ACKNOWLEDGEMENTS

We thank Fernando Pastawski for his carefuland critical reading of the manuscript. HMPacknowledges Alexei Kitaev for his hospitalityand fruitful discussions at Caltech, and Ar-turo Lopez Davalos and Francisco de la Cruzfor early inspiring lectures on irreversibility anddissipation. We acknowledge financial supportfrom CONICET, ANPCyT, SeCyT-UNC andMinCyT-Cor. This work used Mendieta Clusterfrom CCAD at UNC, that is part of SNCAD-MinCyT, Argentina.

22

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