Isolated systems of many particles often appear to behave irreversibly.though they are in principle dynamically reversible.In some cases this reversibility can be made evident in 'echo • phenomena.in which the system is literally restored to a dynamical statc which existed in the past.Whether such echoes are in principle possible for all isolated systems is not known.However.they do not require the absence of interactions among the particles.Thus the inhomogeneaus and 'magic sandwich' echoes are aspects of the same phcnomenon.
Isolated systems of many particles often appear to behave irreversibly.though they are in principle dynamically reversible.In some cases this reversibility can be made evident in 'echo • phenomena.in which the system is literally restored to a dynamical statc which existed in the past.Whether such echoes are in principle possible for all isolated systems is not known.However.they do not require the absence of interactions among the particles.Thus the inhomogeneaus and 'magic sandwich' echoes are aspects of the same phcnomenon.
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.
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.
In spin systems, the decay of the Loschmidt echo in the time-reversal experiment (evolution-perturbation-time-reversed evolution) is linked to the generation of multiple-quantum (MQ) coherences. Unlimited growth of the MQ coherences leads to irreversibility of dynamics. In some cases, one can expect that the deviation of the Loschmidt echo and the second moment of the MQ intensities distribution are linear in time. The criteria of such behavior, called weak irreversibility, are formulated. The proposed approach can be extended beyond spin systems, in order to analyze some general aspects of reversibility of many-body quantum dynamics.
The problem of reconciling the irreversibility of thermodynamics with the completely reversible mechanics of the ultimate constituents of the thermodynamical system is examined from an operational viewpoint. The informational nature of entropy is demonstrated, and the famous paradoxes of statistical mechanics, due to Loschmidt and Zermelo, are resolved with its aid. Spin echo experiments are shown to realize the conditions of Loschmidt's reflection paradox, and used to illustrate how reversibility occurs only with perfect “memory” or information storage, while “forgetting” or loss of information implies irreversibility.
We employ the Loschmidt echo, i.e., the signal recovered after the reversal of an evolution, to identify and quantify the processes contributing to decoherence. This procedure, which has been extensively used in single-particle physics, is employed here in a spin ladder. The isolated chains have 1/2 spins with XY interaction and their excitations would sustain a one-body-like propagation. One of them constitutes the controlled system S whose reversible dynamics is degraded by the weak coupling with the uncontrolled second chain, i.e., the environment E. The perturbative SE coupling is swept through arbitrary combinations of XY and Ising-like interactions, that contain the standard Heisenberg and dipolar ones. Different time regimes are identified for the Loschmidt echo dynamics in this perturbative configuration. In particular, the exponential decay scales as a Fermi golden rule, where the contributions of the different SE terms are individually evaluated and analyzed. Comparisons with previous analytical and numerical evaluations of decoherence based on the attenuation of specific interferences show that the Loschmidt echo is an advantageous decoherence quantifier at any time, regardless of the S internal dynamics.
Loschmidt's paradox asks why macroscopic irreversibility is universal despite the time-reversal symmetry of microscopic dynamics. We argue that irreversibility is not a property of the dynamics but of accessibility: chaotic evolution drives phase-space structure below the quantum resolution scale ℓℏ, at a critical time tc=λ−1ln(δ0/ℓℏ), after which the time-reversed microstate exists as a valid solution of Hamilton's equations but cannot be selected by any physically admissible operation. The mechanism operates entirely within the semiclassical regime tc≤tE, where classical geometry is exact. This provides a dynamical resolution of the Loschmidt paradox.
The quantum foundation is established using a Krylov-complexity framework: we prove that for any H(t)=H(−t), the quantum Lyapunov exponent satisfies λLforward=λLbackward. The arrow of time is not in the dynamics. The mechanism predicts sigmoid fidelity decay, logarithmic scaling of tc with λ−1, and ensemble-size independence of the inaccessibility threshold -- all consistent with three decades of Loschmidt echo experiments and confirmed in a stadium-billiard simulation reported here. Underlying everything: quantum mechanics conserves information exactly. Entropy, defined as the logarithm of the multiplicity Ω -- the number of possibilities consistent with the available information -- can only increase when information becomes operationally inaccessible. The second law reflects not a breakdown of microscopic reversibility, but the dynamical inaccessibility of the information required to reverse it.