Loschmidt echo as a robust decoherence quantifier for many-body systems

We employ the Loschmidt echo, i.e., the signal recovered after the reversal of an evolution, to identify and quantify the processes contributing to decoherence. This procedure, which has been extensively used in single-particle physics, is employed here in a spin ladder. The isolated chains have $1/2$ spins with $XY$ interaction and their excitations would sustain a one-body-like propagation. One of them constitutes the controlled system $\mathcal{S}$ whose reversible dynamics is degraded by the weak coupling with the uncontrolled second chain, i.e., the environment $\mathcal{E}$. The perturbative $\mathcal{SE}$ coupling is swept through arbitrary combinations of $XY$ and Ising-like interactions, that contain the standard Heisenberg and dipolar ones. Different time regimes are identified for the Loschmidt echo dynamics in this perturbative configuration. In particular, the exponential decay scales as a Fermi golden rule, where the contributions of the different $\mathcal{SE}$ terms are individually evaluated and analyzed. Comparisons with previous analytical and numerical evaluations of decoherence based on the attenuation of specific interferences show that the Loschmidt echo is an advantageous decoherence quantifier at any time, regardless of the $\mathcal{S}$ internal dynamics.

Figure 1

The spin system. (a) Open boundary conditions. (b) Closed boundary conditions (ring-like). Continuous (green) connections represent interactions that can be inverted to obtain the Loschmidt echo. Dash (blue) lines represent non-controllable interactions. The first spin (black circle) is initially polarized and the rest of the spins are in one of the high temperature configurations.

Figure 2

Time regimes of the local Loschmidt echo. Numerical results for a ring of five spins weakly coupled to another identical ring by an $XY$ interchain interaction ($\alpha =0$). The $M_{\mathrm{LE}}$ is plotted as a function of the total evolution time $t=2t_R$. Note that the stronger the interchain coupling $J_{\mathcal{SE}}^{}$, the faster the saturation regime is reached (dashed line). From top to bottom, the different curves correspond to the coupling parameters $J_{\mathcal{SE}}^{}$—$0.001,$ $0.01,$ $0.025,$ $0.05,$ $0.1,$$0.25,$ and $0.5$—in units of $J_{\mathcal{E}}^{}$.

Figure 3

(a) Short-time behavior of the Loschmidt echo. The dotted line is the quadratic decay [Eq. (24)]. (b) Short-time behavior of the survival probability for an XY spin chain described by ${\widehat{H}}_{\mathrm{total}}=J_{\mathcal{SE}}^{}({\widehat{S}}_1^{+}{\widehat{S}}_2^{-}+{\widehat{S}}_2^{+}{\widehat{S}}_1^{-})+J_{\mathcal{E}}^{}{\sum }_{n=2}^{\infty }({\widehat{S}}_n^{+}{\widehat{S}}_{n+1}^{-}+{\widehat{S}}_{n+1}^{+}{\widehat{S}}_n^{-})$, where the first spin is the system and the others constitute the environment. This system is known to satisfy the FGR [60]. Note the similarity between (a) and (b) in departing from the quadratic decay, which evinces the spreading time $t_s$. The dashed line shows a plot of Eq. (22) for $J_{\mathcal{SE}}^{}=0.5J_{\mathcal{E}}^{}$.

Figure 4

Onset of the LE exponential decay. On the log scale, the dashed lines are the ideal exponentials determined by the fitting, shown here for representative values. The delay in the entrance into the exponential regime is indicated by arrows pointing to the first data used in the fitting. Solid lines represent the interpolative Eq. (22), used with the corresponding ${\sigma }^2$ and $\Gamma$.

Figure 5

(a) Decay rates for NMR-relevant interchain couplings, in units of $J_{\mathcal{E}}^{}/\hslash$. The slopes and offsets depend on the value of $\alpha$, i.e., on the relative weight between the Ising (dephasing) and the $XY$ (polarization transfer) contributions to the interchain coupling. (b) Scaled rates showing the additivity of the XYand Ising contributions to the FGR rate as a function of ${\alpha }^2$. For comparison, the rates obtained from mesoscopic echo (ME) degradation data obtained in Ref. [31] are also plotted.

Figure 6

Spin propagation in the presence of an Ising interchain interaction, represented as a single fermion moving through a binary alloy landscape. A filled circle is a spin up or a fermion, and an open circle represents a spin down or a hole. All the processes are considered on the short time scale given by a single Trotter time step. (a) Forward dynamics within the upper chain $\mathcal{S}$, in the presence of the “random site energies” produced by the Ising interaction with the lower chain $\mathcal{E}$. (b) If $\mathcal{E}$ remains frozen, the backward evolution in $\mathcal{S}$ is imperfect because the signs of the “site energies” are not inverted. (c) A particular evolution of $\mathcal{E}$ provides a perfectly reversed dynamics for $\mathcal{S}$. In this case, the local portion of the potential landscape, which determined the excitation's forward evolution, is reversed.