Unitary equilibrations: Probability distribution of the Loschmidt echo

Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly solvable example of a quasifree system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double-peaked, universal form.

Figure 1

Typical behavior of the Loschmidt echo for the Ising model in transverse field. All curves refer to a size of $L=100$. In the upper left panel the relaxation time $T_R$ is indicated. Using arguments as in Sec. 3d (see also notes [23, 24]) one is able to show that, in the quantum Ising model at criticality, the first revival time is exactly $T_{\mathrm{rev}}=L$ (upper left panel).

Figure 2

Relaxation time for the Loschmidt echo in the Ising model in transverse field. The critical points are at $h_c=\pm 1$. Clearly we observe divergences at critical points when $\delta h$ is small. On the line $h^{(1)}=h^{(2)}$ $\mathcal{L}(t)=1$ and so there is no relaxation or even dynamics.

Figure 3

Rescaled Loschmidt echo ${\mathcal{L}}_L(t/\sqrt{L})$ for the Ising model in transverse field. The state which defines the average is critical: $h^{(1)}=1$ while $H$ is at $h^{(2)}=2.5$. The same data collapse feature is observed when choosing different $h^{(i)}\mathrm{s}$, although the variance of this Gaussian is sensitive to that.

Figure 4

From top to bottom, mean, and variance of Loschmidt echo at size $L=100$. The region where the variance is large shrinks when increasing the system size $L$. The height of the peak however remains constant.

Figure 5

Approximate exponential behavior. Parameters are $L=18,h^{(1)}=0.3,h^{(2)}=1.4$. When $\delta h$ is large this behavior is observed even for moderate sizes (here $L=18$), (upper panel). The thick line reproduces $\vartheta (x)e^{-x/\overline{\mathcal{L}}}/\overline{\mathcal{L}}$ with $\overline{\mathcal{L}}$ given by Eq. (16). In the lower panel we plot the Fourier series ${\overline{\mathcal{L}}}_{\mathrm{disc}}(\omega )$ given by Eq. (20).

Figure 6

Gaussian behavior for $\delta h$ small but away from criticality. Parameters are $L=20,h^{(1)}=0.1,h^{(2)}=0.11$. Note that the distribution function is an extremely peaked Gaussian (upper panel). The thick line is a Gaussian with mean and variance given by Eqs. (16) and (17). In the bottom panel one can notice that many frequencies contribute to the LE. The one-particle contribution $c(\omega )$, Eq. (22), is given by the black dots while the red curve gives the true spectral decomposition ${\overline{\mathcal{L}}}_{\mathrm{disc}}(\omega )$ given by Eq. (20).

Figure 7

Typical double-peak structure behavior. Parameters are $L=40,h^{(1)}=0.99,h^{(2)}=1.01$. Upper panel: probability distribution (histogram) together with the result of the approximation given in Eq. (18) (thick line). The mean $\overline{\mathcal{L}}$ is taken from Eq. (9) while the parameters $A$ and $B$ are obtained by computing the two largest spectral weights [see Eq. (21)]. Lower panel: the light curve (red online) is the Fourier series ${\overline{\mathcal{L}}}_{\mathrm{disc}}(\omega )$ (projected spectral density), black curve shows the highest coefficient $c(\omega )$ given by Eq. (22), together with allowed frequencies (black dots).

Figure 8

Double-peak distribution approaching an exponential one when increasing system size $L$ at fixed $h^{(i)}$. Parameters are $h^{(1)}=0.9,h^{(2)}=1.2$ and chain length are $L=10,20,30,40$.

Figure 9

Gaussian distribution approaching an exponential one when increasing system size $L$ at fixed $h^{(i)}$. Parameters are $h^{(1)}=0.2,h^{(2)}=0.6$ and chain length are $L=20,30,40,80,120$.

Figure 10

Variance of the magnetization as given by Eq. (10) for $L=80$. A signature of criticality are the cusps at $h^{(2)}=\pm 1$.

Figure 11

The probability distribution for the magnetization has only Gaussian behavior. Here parameters are $L=40$, $h^{(1)}=0.9$, $h^{(2)}=1.01$. The continuous line is a Gaussian with mean and variance given by Eqs. (24) and (25).

Figure 12

Typical behavior of the function $s(t)$ (black line) in the thermodynamic limit. The red line is the approximation given by Eq. (A1). Parameters are $h^{(1)}=1.3$, $h^{(2)}=2$.