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Register a new userLoschmidt echo in many-spin systems: a quest for intrinsic decoherence and emergent irreversibility
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Added by
Pablo Ren\\'e Zangara
Publication date
2017
fields
Physics
and research's language is
English
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If a magnetic polarization excess is locally injected in a crystal of interacting spins, 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 certain time, say $t/2$, an experimental protocol reverses the many-body dynamics, 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 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, ultimately leading to the discussion of irreversibility as an emergent phenomenon. In addition, we introduce the LE formalism by means of spin-spin correlation functions and 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.
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The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.
Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $varsigma={rm Tr}rho^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $bar M(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $varsigma(t)$ and $bar M(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of $varsigma$ and $bar M$ has a regime where it is dominated by the classical Lyapunov exponents
Evaluating the role of perturbations versus the intrinsic coherent dynamics in driving to equilibrium is of fundamental interest to understand quantum many-body thermalization, in the quest to build ever complex quantum devices. Here we introduce a protocol that scales down the coupling strength in a quantum simulator based on a solid-state nuclear spin system, leading to a longer decay time T2, while keeping perturbations associated to control error constant. We can monitor quantum information scrambling by measuring two powerful metrics, out-of-time-ordered correlators (OTOCs) and Loschmidt Echoes (LEs). While OTOCs reveal quantum information scrambling involving hundreds of spins, the LE decay quantifies, via the time scale T3, how well the scrambled information can be recovered through time reversal. We find that when the interactions dominate the perturbation, the LE decay rate only depends on the interactions themselves, T3 ~ T2, and not on the perturbation. Then, in an unbounded many-spin system, decoherence can achieve a perturbation-independent regime, with a rate only related to the local second moment of the Hamiltonian.
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 NMR, 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 here as global or many-body LE $M_{MB}$. Here, we analyze the relation between these magnitudes, in what concerns 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. $M_{MB}sim left( M_{1,1}right) ^{N/4}$. 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 $M_{1,1}$ as a consequence of the interplay between the time scale that characterizes the reversible interactions ($T_{2}$) and that of the perturbation ($tau _{Sigma }$). Our analysis suggests that the short time decay, characterized by the time scale $tau _{Sigma }$, 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 $tau _{Sigma }$ but closely tied to $T_{2}$.
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 non-ergodic 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 non-interacting 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 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.
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