Experimental Observation of a Time-Driven Phase Transition in Quantum Chaos

We report the first experimental observation of the time-driven phase transition in a canonical quantum chaotic system, the quantum kicked rotor. The transition bears a firm analogy to a thermodynamic phase transition, with the time mimicking the temperature and the quantum expectation of the rotor’s kinetic energy mimicking the free energy. The transition signals a sudden change in the system’s memory behavior: before the critical time, the system undergoes chaotic motion in phase space and its memory of initial states is erased in the course of time; after the critical time, quantum interference enhances the probability for a chaotic trajectory to return to the initial state, and thus the system’s memory is recovered.

Figure 1

The time-driven phase transition at the critical time $t_E^{*}$ (top) signals a sudden change in system’s memory behavior (middle). For $t<t_E^{*}$, the system wanders randomly in phase space, and the initial state ${\gamma }_0$ and the evolving state ${\gamma }_t$ are uncorrelated (i). For $t>t_E^{*}$, two trajectories (blue solid and red dashed lines) departing from Planck’s cell centered at ${\gamma }_0$, due to dynamical instability (bottom), can interfere constructively and meet again with a significant probability in the same cell (iii) or in the cell (ii) centered at ${\overline{\gamma }}_0$ (which is the time-reversed conjugate of ${\gamma }_0$) after long wandering; thus the memory of the initial state is recovered.

Figure 2

Experiments on the standard QKR ($N=1$, $K=5.5$, $\sout{k}=1$), that displays $T_c$ symmetry, confirm a time-driven phase transition, manifesting in the time profile of $\delta ⟨p^2(t)⟩-2D_{0}t$. Experimental data (symbols) are well described by Eq. (8), with $t_{\mathrm{loc}}$ and $t_E^{*}$ treated as fitting parameters, and show a time singularity at $t_E^{*}=4.7$ (green solid line). The data are incompatible with $t_E^{*}=0$ (pink dotted line). Inset: Same as main panel, showing $\delta ⟨p^2(t)⟩$ instead. The black dashed line represents the linear growth due to diffusion.

Figure 3

Experiments measuring $\delta ⟨p^2(t)⟩-2D_{0}t$ with the periodically shifted QKR with broken $T_c$ symmetry ($N=4$, $K=3.9$, $\sout{k}=1$). Experimental data (circles) are well described by Eq. (9), with $t_{\mathrm{loc}}$ and $t_E^{*}$ treated as fitting parameters, and show a time singularity at $t_E^{*}=10.2$ (green full line). The data are incompatible with $t_E^{*}=0$ (pink dotted line). Inset: Same as main panel, showing $\delta ⟨p^2(t)⟩$ instead. The black dashed line represents the linear growth corresponding to diffusion.