Loschmidt echo in many-spin systems: contrasting time-scales of local and global measurements

TL;DR

This paper investigates the relationship between local and global Loschmidt echoes in many-spin quantum systems, revealing how their characteristic time scales depend on system size and interactions, with implications for understanding quantum reversibility.

Contribution

It establishes a quantitative link between local and global Loschmidt echoes, showing the global echo as a product of local echoes, and analyzes the decay mechanisms of the local echo in relation to system interactions.

Findings

Global LE approximates the N/4 power of local LE for short times

Decay of local LE is influenced by reversible interaction time scale T2

Experimentally observed decay time T3 is independent of perturbation strength

Abstract

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 $T_{2}$.