The interplay between interactions and disorder in closed quantum many-body systems is relevant for thermalization phenomenon. In this paper, we address this competition in an infinite temperature spin system by means of the Loschmidt echo (LE), which is based on a time reversal procedure. This quantity has been formerly employed to connect quantum and classical chaos, and in the present many-body scenario we use it as a dynamical witness. We assess the LE time scales as a function of disorder and interaction strengths. The strategy enables a qualitative phase diagram that shows the regions of ergodic and nonergodic behavior of the polarization under the echo dynamics.
The interplay between interactions and disorder in closed quantum many-body systems is relevant for thermalization phenomenon. In this paper, we address this competition in an infinite temperature spin system by means of the Loschmidt echo (LE), which is based on a time reversal procedure. This quantity has been formerly employed to connect quantum and classical chaos, and in the present many-body scenario we use it as a dynamical witness. We assess the LE time scales as a function of disorder and interaction strengths. The strategy enables a qualitative phase diagram that shows the regions of ergodic and nonergodic behavior of the polarization under the echo dynamics.
A local excitation in a quantum many-particle system evolves deterministically. A time-reversal procedure, involving the invertion of the signs of every energy and interaction, should produce an excitation revival: the Loschmidt echo (LE). If somewhat imperfect, only a fraction of the excitation will refocus. We use such a procedure to show how non-inverted weak disorder and interactions, when assisted by the natural reversible dynamics, fully degrade the LE. These perturbations enhance diffusion and evenly distribute the excitation throughout the system. Such a dynamical paradigm, called ergodicity, breaks down when either the disorder or the interactions are too strong. These extreme regimes give rise to the well known Anderson localization and Mott insulating phases, where quantum diffusion becomes restricted. Accordingly, regardless of the kinetic energy terms, the excitation remains mainly localized and out-of-equilibrium, and the system behaves non-ergodically. The LE constitutes a fair dynamical witness for the whole phase diagram since it evidences a surprising topography in which ergodic and non-ergodic phases interpenetrate each other. Furthermore, we provide an estimation for the critical lines separating the ergodic and non-ergodic phases around the Mott and Anderson transitions. The energy uncertainties introduced by disorder and interaction shift these thresholds towards stronger perturbations. Remarkably, the estimations of the critical lines are in good agreement with the phase diagram derived from the LE dynamics.Received: 20 Novembre 2014, Accepted: 29 June 2015; Edited by: C. A. Condat, G. J. Sibona; Reviewed by: A. De Luca, Laboratoire de Physique Theorique, ENS & Institut Philippe Meyer, Paris, France; DOI: http://dx.doi.org/10.4279/PIP.070012Cite as: P R Zangara, P R Levstein, H M Pastawski, Papers in Physics 7, 070012 (2015)This paper, by P R Zangara, P R Levstein, H M Pastawski, is licensed under the Creative Commons Attribution License 3.0.
A local excitation in a quantum many-spin system evolves deterministically. A time-reversal procedure, involving the inversion of the signs of every energy and interaction, should produce the excitation revival. This idea, experimentally coined in nuclear magnetic resonance, embodies the concept of the Loschmidt echo (LE). While such an implementation involves a single spin autocorrelation M(1,1), i.e. a local LE, theoretical efforts have focused on the study of the recovery probability of a complete many-body state, referred to here as global or many-body LE MMB Here, we analyse the relation between these magnitudes, with regard to their characteristic time scales and their dependence on the number of spins N We show that the global LE can be understood, to some extent, as the simultaneous occurrence of N independent local LEs, i.e. MMB∼(M(1,1))(N/4) This extensive hypothesis is exact for very short times and confirmed numerically beyond such a regime. Furthermore, we discuss a general picture of the decay of M1,1 as a consequence of the interplay between the time scale that characterizes the reversible interactions (T(2)) and that of the perturbation (τ(Σ)). Our analysis suggests that the short-time decay, characterized by the time scale τ(Σ), is greatly enhanced by the complex processes that occur beyond T(2) This would ultimately lead to the experimentally observed T(3), which was found to be roughly independent of τ(Σ) but closely tied to T(2).
Abstract If a magnetic polarization excess is locally injected in a crystal of interacting spins in thermal equilibrium, this ‘excitation’ would spread as consequence of spin–spin interactions. Such an apparently 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 initial condition. Then, if at a certain time, say t / 2 , an experimental protocol reverses the many-body dynamics by changing the sign of the effective Hamiltonian, it would drive the system back to the initial non-equilibrium state at time t . As a matter of fact, the reversal is always perturbed by small experimental imperfections and/or uncontrolled internal or environmental degrees of freedom. This limits the amount of signal M ( t ) recovered locally at time t . The degradation of M ( t ) accounts for these perturbations, which can also be seen as the sources of decoherence. This general idea defines the Loschmidt echo (LE), which embodies the various time-reversal procedures implemented in nuclear magnetic resonance. Here, we present an invitation to the study of the LE following the pathway induced by the experiments. With such a purpose, we provide a historical and conceptual overview that briefly revisits selected phenomena that underlie the LE dynamics including chaos, decoherence, localization and equilibration. This guiding thread ultimately leads us to the discussion of decoherence and irreversibility as an emergent phenomenon. In addition, we introduce the LE formalism by means of spin–spin correlation functions in a manner suitable for presentation in a broad scope physics journal. Last, but not least, we present new results that could trigger new experiments and theoretical ideas. In particular, we propose to transform an initially localized excitation into a more complex initial state, enabling a dynamically prepared LE. This induces a global definition of the LE in terms of the raw overlap between many-body wave functions. Our results show that as the complexity of the prepared state increases, it becomes more fragile towards small perturbations.
If a magnetic polarization excess is locally injected in a crystal of interacting spins in thermal equilibrium, this 'excitation' would spread as consequence of spin-spin interactions. Such an apparently 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 initial condition. Then, if at a certain time, say , an experimental protocol reverses the many-body dynamics by changing the sign of the effective Hamiltonian, it would drive the system back to the initial non-equilibrium state at time t. As a matter of fact, the reversal is always perturbed by small experimental imperfections and/or uncontrolled internal or environmental degrees of freedom. This limits the amount of signal M(t) recovered locally at time t. The degradation of M(t) accounts for these perturbations, which can also be seen as the sources of decoherence. This general idea defines the Loschmidt echo (LE), which embodies the various time-reversal procedures implemented in nuclear magnetic resonance. Here, we present an invitation to the study of the LE following the pathway induced by the experiments. With such a purpose, we provide a historical and conceptual overview that briefly revisits selected phenomena that underlie the LE dynamics including chaos, decoherence, localization and equilibration. This guiding thread ultimately leads us to the discussion of decoherence and irreversibility as an emergent phenomenon. In addition, we introduce the LE formalism by means of spin-spin correlation functions in a manner suitable for presentation in a broad scope physics journal. Last, but not least, we present new results that could trigger new experiments and theoretical ideas. In particular, we propose to transform an initially localized excitation into a more complex initial state, enabling a dynamically prepared LE. This induces a global definition of the LE in terms of the raw overlap between many-body wave functions. Our results show that as the complexity of the prepared state increases, it becomes more fragile towards small perturbations.
In the long quest to identify and compensate the sources of decoherence in many-body systems far from the ground state, the varied family of Loschmidt echoes (LEs) became an invaluable tool in several experimental techniques. A LE involves a time-reversal procedure to assess the effect of perturbations in a quantum excitation dynamics. However, when addressing macroscopic systems one is repeatedly confronted with limitations that seem insurmountable. This led to the formulation of the central hypothesis of irreversibility stating that the timescale of decoherence, T3, is proportional to the timescale of the many-body interactions we reversed, T2. We test this by implementing two experimental schemes based on Floquet Hamiltonians where the effective strength of the dipolar spin-spin coupling, i.e., 1/T2, is reduced by a variable scale factor k. This extends the perturbation timescale, ${T}_{\mathrm{\ensuremath{\Sigma}}}$, in relation to T2. Strikingly, we observe the superposition of the normalized Loschmidt echoes for the bigger values of k. This manifests the dominance of the intrinsic dynamics over the perturbation factors, even when the Loschmidt echo is devised to reverse that intrinsic dynamics. Thus, in the limit where the reversible interactions dominate over perturbations, the LE decays within a timescale, ${T}_{3}\ensuremath{\approx}{T}_{2}/R$ with $R=(0.15\ifmmode\pm\else\textpm\fi{}0.01)$, confirming the emergence of a perturbation independent regime. These results support the central hypothesis of irreversibility.
The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many-body localized phase, which is characterized by emergent local integrals of motion and provides a generic example of nonergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a noninteracting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence these probes allow one to get insights into the relation between physical operators and local integrals of motion and access the operator spreading in the many-body localized phase.