Nonmonotonic short-time decay of the Loschmidt echo in quasi-one-dimensional systems

We study the short-time stability of quantum dynamics in quasi-one-dimensional systems with respect to small localized perturbations of the potential. To this end, we analytically and numerically address the decay of the Loschmidt echo (LE) during times that are short compared to the Ehrenfest time. We find that the LE is generally a nonmonotonic function of time and exhibits strongly pronounced minima and maxima at the instants when the corresponding classical particle traverses the perturbation region. We also show that, under general conditions, the envelope decay of the LE is well approximated by a Gaussian, and we derive explicit analytical formulas for the corresponding decay time. Finally, we demonstrate that the observed nonmonotonic nature of the LE decay is only pertinent to one-dimensional (and, more generally, quasi-one-dimensional) systems, and that the short-time decay of the LE can be monotonic in a higher number of dimensions.

Figure 1

One-dimensional quantum particle in the presence of a localized potential barrier. See text for discussion.

Figure 2

Top figure: LE $M(t)$ for a free particle with the perturbation potential $V(q)$ given by Eq. (18). The horizontal dashed line represents the LE saturation value $M(\infty )$ given by Eq. (28). Bottom figure: Perturbation potential as seen by the unperturbed classical particle [i.e., $V(q_1(t))$]. The system is characterized by $p_0=50$, $\sigma =0.1$, $a-q_0=1$, $\gamma =0.3$, and $V_0=150$.

Figure 3

Top figure: LE $M(t)$ for a particle on a ring with the perturbation potential $V(q)$ given by Eq. (18). The dashed line represents the envelope decay $\stackrel{̲}{M}(t)$ calculated in accordance with Eqs. (29) and (31). Bottom figure: Perturbation potential as seen by the unperturbed particle [i.e. $V(q_1(t))$]. The system is characterized by $L=1$, $p_0=200$, $\sigma =0.1$, $a-q_0=0.3$, $\gamma =0.1$, and $V_0=1500$.

Figure 4

A quantum particle in a one-dimensional potential well. The unperturbed potential is given by $U(q)$, and the perturbed potential by $U(q)+V(q)$. The total energy $E$ of the system is such that both unperturbed and perturbed classical motions have the same turning points $q_{-}$ and $q_{+}$.

Figure 5

Top figure: LE $M(t)$ for a particle in a parabolic well with the perturbation potential $V(q)$ given by Eq. (18). The dashed line represents the envelope decay $\stackrel{̲}{M}(t)$ calculated in accordance with Eqs. (29) and (41). Bottom figure: Perturbation potential as seen by the unperturbed particle [i.e. $V(q_1(t))$]. The system is characterized by $\omega =60$, $q_0=-1$, $p_0=10$, $\sigma =0.1$, $a=0$, $\gamma =0.1$, and $V_0=200$. Note that the LE decay obtained by the average-potential TGA (red curve) is essentially indistinguishable from the exact result (blue thick curve) for most of the time range.

Figure 6

Top figure: LE $M(t)$ for a particle in an anharmonic well with the perturbation potential $V(q)$ given by Eq. (18). The dashed line represents the envelope decay $\stackrel{̲}{M}(t)$ calculated in accordance with Eqs. (29) and (41). Bottom figure: Perturbation potential as seen by the unperturbed particle, [i.e. $V(q_1(t))$]. The system is characterized by $\omega =60$, $\epsilon =-400$, $q_0=-1$, $p_0=10$, $\sigma =0.1$, $a=0$, $\gamma =0.1$, and $V_0=200$.

Figure 7

LE decay for a free particle in two dimensions under the action of the perturbation potential given by Eq. (46). The system is characterized by $p_0=50$, $\sigma =0.1$, $a-q_0=1$, $V_0=150$, and $\gamma =0.3$. The $M(t)$ curves correspond to seven different values of the $y$ extent of the perturbation region: from bottom to top $\delta =0.1$, $0.2$, $0.3$, $0.35$, $0.4$, $0.5$, and $\infty$.

Figure 8

Time evolution of the unperturbed (left column) and perturbed (right column) wave function. The top snapshot shows the initial state. The white ellipse in the snapshots of the right column is centered at the position of the perturbation maximum and its major and minor semiaxes are equal to $\gamma =0.3$ and $\delta =0.1$, respectively. The system is characterized by $p_0=50$, $\sigma =0.1$, $a-q_0=1$, $V_0=150$, $\gamma =0.3$, and $\delta =0.1$ and exhibits a LE decay represented by the red (lowest) curve in Fig. 7.