One contribution of 12 to a theme issue 'Loschmidt echo and time reversal in complex systems' . 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 M MB . 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. M MB ∼ (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 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 (τ Σ ). Our analysis suggests that the short-time decay, characterized by the time scale τ Σ

We address the question o f how w eak perturbations, w hich are quite ineffective in sm all m any-body system s, can lead to decoherence and hence to irreversibility w hen they proliferate as the system size increases. This question is at the heart o f solid-state N M R . T here, an initially local polarization spreads all over due to spin-spin interactions that conserve the total spin projection, leading to an equilibration o f the polarization. In principle, this quantum dynam ics can be reversed by changing the sign o f the H am iltonian. How ever, the reversal is usually perturbed by nonreversible interactions that act as a decoherence source. T he fraction o f the local excitation recovered defines the L oschm idt echo (LE), here evaluated in a series o f closed N spin system s w ith ali-to-all interactions. T he m ost rem arkable regim e o f the LE decay occurs w hen the perturbation induces proliferated effective interactions. We show that if this perturbation exceeds som e low er bound, the decay is ruled by an effective F erm i golden rule (FG R). Such a low er bound shrinks as N increases, becom ing the leading m echanism for L E decay in the therm odynam ic lim it. O nce the polarization stayed equilibrated longer than the FG R tim e, it rem ains equilibrated in spite o f the reversal procedure. D O l: 10.1103/PhysR evA .91.042112 PACS num ber(s): 03.65.Y z, 05.70.L n, 75.10.Jm , 7 4 .2 5 .nj

We consider the degradation of the dynamics of a Gaussian wave packet in a harmonic oscillator under the presence of an environment. This last is given by a single non-degenerate two level system. We analyze how the binary degree of freedom perturbs the free evolution of the wave packet producing decoherence, which is quantified by the Loschmidt echo. This magnitude measures the reversibility of a perturbed quantum evolution. In particular, we use it here to study the relative "fragility" of coherent superpositions (cat states) with respect to incoherent ones. This fragility or sensitivity turns out to increase exponentially with the energy separation of the two components of the superposition.