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

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 374; no. 2069; pp. 1 - 15
• Main Authors: Zangara, Pablo R., Bendersky, Denise, Levstein, Patricia R., Pastawski, Horacio M.
• Format: Journal Article
• Language: English
• Published: THE ROYAL SOCIETY 13-06-2016
• Subjects: Autocorrelation; Chaos theory; Energy; Entropy; Mathematical moments; Nuclear magnetic resonance; Quantum decoherence; Quantum mechanics; Thermodynamics; Time dependence
• ISSN: 1364-503X; 1471-2962

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 M1,1, 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 ∼ (M1,1)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 (T2) and that of the perturbation (τΣ). Our analysis suggests that the short-time decay, characterized by the time scale τΣ, is greatly enhanced by the complex processes that occur beyond T2. This would ultimately lead to the experimentally observed T3, which was found to be roughly independent of τΣ but closely tied to T2.