Quasiclassical approach to quantum quench dynamics in the presence of an excited-state quantum phase transition

The dynamics of a quantum system following a sudden, highly nonadiabatic change of its control parameter (quantum quench) is studied with quasiclassical techniques. Recent works have shown, using exact quantum mechanical approach, that equilibration after quantum quench exhibits specific features in the presence of excited-state quantum phase transitions. In this paper, we demonstrate that these features can be understood from the classical evolution of the Wigner function in phase space.

Figure 1

Comparison of the critical subspace $M=2j$ (upper row) and a noncritical one $M=3j/2$ (lower row). Parameters used are $j=20,\omega =2{\omega }_0$. (a) Energy spectra $\varepsilon \equiv E/{\omega }_{0}j$ as functions of the control parameter $\lambda$. (b) Scaled mean value of the quasispin $z$ projection $J_z$ as a function of energy for fixed $\lambda /{\lambda }_c=5$ as computed from the respective spectrum in panel (a). The red dashed line in the upper panel represents the the critical energy ${\varepsilon }_c$.

Figure 2

Comparison of the critical subspace $M=2j$ (upper row) and a noncritical one $M=3j/2$ (lower row). Parameters used are $j=20,\omega =2{\omega }_0$. Classical Hamiltonian $h_{cl}\equiv H_{\mathrm{cl}}/{\omega }_{0}j$ in the phase space for fixed $\lambda =5{\lambda }_c$ is shown together with the cut $v\equiv h_{cl}(x,p=0)$.

Figure 3

Sketch of the critical quench protocols used in the paper. Upper row: Forward quench critical protocol. Lower row: Backward quench critical protocol. The red line indicates the equal energy of the stationary point and the value $v\left(x\right)$ at the coordinate of the center of mass of the initial Wigner function.

Figure 4

Forward quench in a the noncritical subspace $M=3j/2$. The parameters were chosen as $j=200,\omega =2{\omega }_0,{\lambda }_{\mathrm{i}}=0,{\lambda }_{\mathrm{f}}=5{\lambda }_c$. (a) Comparison of quantum and classical survival probability. The green line indicates the power-law decay $\propto 1/\tau$ of the revivals. (b) Same as in (a) with the linear scale. (c) Snapshots of the classically evolved Wigner function in the phase space for multiple values of $\tau$. The pink lines represent energy equivalue lines of the final classical Hamiltonian. The respective time points are marked by crosses in panel (a). The initial state $|{\psi }_{\mathrm{i}}\rangle$ is the ground state of the initial Hamiltonian. The respective initial Wigner function $W_{\mathrm{i}}$ is plotted in grey. Video available from Ref. [55].

Figure 5

The same as in Fig. 4, but for the forward quench in the critical subspace $M=2j$.

Figure 6

The same as in Fig. 4, but for a noncritical backward quench from ${\lambda }_{\mathrm{i}}=5{\lambda }_c$ to ${\lambda }_{\mathrm{f}}={\lambda }_c$ in the subspace $M=2j$.

Figure 7

The same as in Fig. 4, but for a critical backward quench from ${\lambda }_{\mathrm{i}}=5{\lambda }_c$ to ${\lambda }_{\mathrm{f}}=1.544{\lambda }_c$ in the subspace $M=2j$.

Figure 8

Comparison between the classical and the full quantum evolution for the Husimi and the Wigner functions after the noncritical forward quench in the subspace $M=3j/2$. Other parameters are the same as in Fig. 4.

Figure 9

Comparison between the classical and the full quantum evolution for the Husimi and the Wigner functions after the critical forward quench in the subspace $M=2j$. Other parameters are the same as in Fig. 5.

Figure 10

Schematic picture of the phase space around the initial point $(x_0,p=0)$. The circle marks the domain of $W_{\mathrm{i}}$ and the blue dashed lines represent examples of the energy equivalue lines which are parallel with $p$ axis (approximation i.) and equidistant (approximation ii.) in that domain.