Echoes are ubiquitous phenomena in several branches of physics, ranging from acoustics, optics, condensed matter and cold atoms to geophysics. They are at the base of a number of very useful experimental techniques, such as nuclear magnetic resonance, photon echo and time-reversal mirrors. Particularly interesting physical effects are obtained when the echo studies are performed on complex systems, either classically chaotic, disordered or many-body. Consequently, the term Loschmidt echo has been coined to designate and quantify the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system, or equivalently to characterize the stability of quantum evolution in the presence of perturbations. Here, we present the articles which discuss the work that has shaped the field in the past few years.
Echoes are ubiquitous phenomena in several branches of physics, ranging from acoustics, optics, condensed matter and cold atoms to geophysics. They are at the base of a number of very useful experimental techniques, such as nuclear magnetic resonance, photon echo and time-reversal mirrors. Particularly interesting physical effects are obtained when the echo studies are performed on complex systems, either classically chaotic, disordered or many-body. Consequently, the term Loschmidt echo has been coined to designate and quantify the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system, or equivalently to characterize the stability of quantum evolution in the presence of perturbations. Here, we present the articles which discuss the work that has shaped the field in the past few years.
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 concept of irreversibility is necessary to justify the statistical description of many-body systems. Here, the authors study the dynamics of observable echoes in quantum many-body systems occurring when the time-evolution is effectively inverted. For nonintegrable systems they find that small imperfections in the time-reversal procedure that alter the state of the system lead to exponentially decaying echoes. As the corresponding decay rate is largely independent of the strength of the perturbation, an intrinsic sensitivity to the perturbation is revealed. This implies irreversibility of the dynamics for all practical purposes, similar to the situation in chaotic classical systems.