Probabilistic hysteresis from a quantum-phase-space perspective

Probabilistic hysteresis is a manifestation of irreversibility in a small dissipationless classical system [R. Bürkle et al., Sci. Rep. 9, 14169 (2019)]: After a slow cyclic sweep of a control parameter, the probability that an initial microcanonical ensemble returns to the neighborhood of its initial energy is significantly below one. A similar phenomenon has recently been confirmed in a corresponding quantum system, when the particle number $N$ is not too small. Quantum-classical correspondence has been found to be nontrivial in this case however; the rate at which the control parameter changes must not be extremely slow and the initial distribution of energies must not be too narrow. In this paper we directly compare the quantum and classical forms of probabilistic hysteresis by making use of the Husimi quantum-phase-space formalism. In particular, we demonstrate that the classical ergodization mechanism, which is a key ingredient in classical probabilistic hysteresis, can lead to a breakdown of quantum-classical correspondence rather than to quantum ergodization. Such quantum failure of ergodization leads to strong quantum effects on the long-term evolution even when the quantum corrections in the equations of motion, which are proportional to $1/N$, would naively seem to be small. We also show, however, that quantum ergodization can be restored by averaging over energies, so that for not-too-narrow initial energy width and not-too-slow parameter change the classical results are recovered after all at large $N$. Finally, we show that the formal incommutability of the classical and adiabatic limits in our system, which is responsible for the breakdown of quantum-classical correspondence in the quasistatic limit, is due to macroscopic quantum tunneling through a large energetic barrier. This explains the extremely slow sweep rates needed to reach the quantum adiabatic limit that were reported in our previous work. The formal incommutability therefore has no consequences for any realistically slow sweeps unless $N$ is quite small ($N\lesssim 20$).

Figure 1

Evolution of a classical ensemble consisting of 2000 points (black dots) in phase space: (a) initial state, (b) end of forward sweep, and (c) final state. The gray lines show adiabatic energy contours; the dashed black line is the separatrix that divides phase space into the regions $A_u, A_l$, and $A_o$. Because adiabaticity breaks down when the separatrix is crossed between (a) and (b) and between (b) and (c), only a finite fraction of the ensemble returns to the initial energy shell, so probabilistic hysteresis occurs. For a clearer graphical presentation we have chosen the canonical coordinates $q^{\prime }=arctan(p/\sqrt{p_0^2-p^2}cosq)$ and $p^{\prime }=-\sqrt{p_0^2-p^2}sinq$ here. (Figure has been reproduced from Fig. 1 of Ref. [3].)

Figure 2

Shown on the right is the evolution of the Husimi function corresponding to a single initial state (37th adiabatic energy eigenstate) compared to the classical Liouville dynamics with the same initial phase-space distribution on the left. After the separatrix is crossed the two evolutions are quite different, despite the large total particle number. While the classical phase-space density is grainy due to swirling (see the text), the Husimi function is smooth, but far from ergodic, in that it has bright and dark spots. The system parameters are $N=1000, u=-3, {\mathrm{\Delta }}_0/\mathrm{\Omega }=-{\mathrm{\Delta }}_I/\mathrm{\Omega }=2$, and $T=5000{\mathrm{\Omega }}^{-1}$.

Figure 3

Higher-resolution image of Fig. 2 at $t=0$ showing the swirling of (a) the classical phase-space density and (b) the Husimi function. The region shown here corresponds to approximately one pixel in Fig. 2 and is much smaller than $\hslash$.

Figure 4

Dependence of the return probability $P_{\mathrm{ret}}$ on the total sweep time $2T$ for $N=1000, u=-3, {\mathrm{\Delta }}_0/\mathrm{\Omega }=-{\mathrm{\Delta }}_I/\mathrm{\Omega }=2$, and a single initial state. The red line shows the quantum return probability if the initial state is the 37th eigenstate of the initial Hamiltonian. The black line shows the corresponding classical return probability for an initial phase-space density equal to the initial Husimi function. The slight variation of the classical return probability is due to our finite phase-space resolution (sampling error).

Figure 5

Shown on the right is the evolution of the Husimi function corresponding to a microcanonical ensemble of 20 initial states compared to the classical Liouville dynamics with the same initial phase-space distribution on the left. The oscillations of the Husimi function seen in Fig. 2 are strongly suppressed, so we observe much better quantum-classical correspondence in the return probability. The parameters are the same as in Fig. 2. The 20 quantum states are the initial 28th to 47th states, so the mean energy of the ensemble is essentially the same as in Fig. 2.

Figure 6

Dependence of the return probability $P_{\mathrm{ret}}$ on the total sweep time $2T$ for $N=1000, u=-3, {\mathrm{\Delta }}_0/\mathrm{\Omega }=-{\mathrm{\Delta }}_I/\mathrm{\Omega }=2$, and an ensemble of 20 initial states (28th to 47th energy eigenstates) with mean energy similar to the 37th state (red line). The black line shows the corresponding classical return probability for an initial phase-space density equal to the initial Husimi function. The oscillations of the return probability are much smaller than in Fig. 4 and are expected to vanish completely for larger initial energy width.

Figure 7

Wehrl entropy $S_W$ corresponding to the quantum evolutions shown in Fig. 2 (black line) and Fig. 5 (blue line). While the oscillation of the Husimi function in the case of a single initial state leads to a strong oscillation of the Wehrl entropy, the oscillations of the Wehrl entropy in the case of an initial ensemble of states are suppressed. The red line shows the coarse-grained entropy of the classical simulation of Fig. 5 for comparison. For further comparison, the magenta line shows the von Neumann entropy of the time-averaged quantum density matrix (see the text). Since the definitions of the Wehrl and classical entropies allow an arbitrary constant shift from the phase-space measure, we have used this to set all three entropies equal for the mixed initial state.

Figure 8

Evolution of the Husimi function in the reversible quantum adiabatic limit for $N=10, T={10}^8{\mathrm{\Omega }}^{-1}, u=-3$, and ${\mathrm{\Delta }}_0/\mathrm{\Omega }=-{\mathrm{\Delta }}_I/\mathrm{\Omega }=2$ during the forward sweep, where the initial state is the ground state. The panels for the backward sweep are essentially identical to the panels shown, but in reverse order. Reversibility in this extreme case is restored by macroscopic quantum tunneling: All atoms tunnel collectively through the separatrix, which is indicated by the dashed white line, in the forward and backward sweep. Accordingly, the Husimi function also tunnels through the separatrix: Instead of continuously flowing through the separatrix as in Fig. 2 and Fig. 5, the Husimi function simply fades away on one side of the separatrix and grows on the other side, without ever having visible support on the separatrix itself. Because the system always stays in a single adiabatic eigenstate, interference effects like those in Fig. 2 are absent.