Boltzmann-Loschmidt dispute reloaded quantum 150 years later

Abstract

The Boltzmann-Loschmidt dispute of 1876 questioned the possibility of a statistical irreversible description by time reversible classical equations of motion of atoms. Here we show analytically and numerically that the quantum chaos diffusion of cold atoms, or ions, in a harmonic trap and pulsed optical lattice can be inverted back in time with up to 100\% efficiency. This is in sharp contrast to classical evolution where exponentially small errors break time reversibility. We argue that the existing experimental skills allow highlighting the Boltzmann-Loschmidt dispute from a quantum perspective.

Figures (7)

• Figure 1: Time dependence of the mean energy $E(t)$ for the classical system (\ref{['eq1']}) at $K=3$ (top panel) and $K=8$ (bottom panel). Time reversal is performed at $t_r=30$ (solid curves) and $t_r=40$ (dashed curves) in the presence of noise with amplitude $\varepsilon$. Values of $E$ are averaged over $10^6$ classical trajectories. At $t=0$, the initial distribution is a Gaussian centered at the unstable fixed point $(x_0=\pi, p_0=0)$ with a standard deviation $\sigma = \sqrt{2}/2$. The recovery time $t_d$ is defined as the time during which the energy decays after time reversal, as illustrated in the bottom panel. For $\varepsilon=0$, no artificial noise is added, leaving only computer round-off errors at the double-precision level ($\sim 10^{-16}$).
• Figure 2: Dependence of the ratio of recovery and reversal times $f=t_d/t_r$ on noise amplitude $\varepsilon$ and reversal time $t_r$ shown by color for $K=3$ (left) and $K=8$ (right). The values of $f$ are averaged over $10^6$ trajectories for each color cell, where $t_d$ values are obtained as it is shown in Fig. \ref{['fig1']}.
• Figure 3: Time evolution of the mean energy $E(t) = \langle n \rangle$ (top panel) and the quantum fidelity $F(t) = \mid \langle \psi(t=0,\varepsilon_q=0) \mid \psi(t,\varepsilon_q) \rangle \mid^2$ (bottom panel) for $K=3$, with $\hbar=q=1$. Solid curves correspond to different quantum noise amplitudes $\varepsilon_q$: $0$ (black), $0.05$ (red), $0.1$ (green), and $0.2$ (blue). Shaded areas represent the standard deviation computed over $1000$ noise realizations of the quantum map in Eq. (\ref{['eq2']}). Dashed curves show the classical mean energy $E(t)$ averaged over $10^6$ trajectories for noise amplitude $\varepsilon=10^{-5}$ (orange) and $10^{-3}$ (purple). The initial classical and quantum state distributions, centered at $x_0=\pi, p_0=0$, are shown in the top panels of Fig. \ref{['fig4']}.
• Figure 4: Phase-space comparison of the time-reversal dynamics for classical probability densities (left column) and quantum Husimi distributions haakefrahm (right column). The rows, from top to bottom, correspond to the initial state at $t=0$, the state at the reversal time $t=t_r=30$, and the final state at $t=2t_r=60$. Classical distributions are computed from an ensemble of $10^6$ trajectories with a noise amplitude $\varepsilon=10^{-3}$. The quantum Husimi distributions are shown for one noise realization with amplitude $\varepsilon_q=0.1$. All initial states are centered at $(x_0, p_0) = (\pi, 0)$. The system parameters are $K=3$ and $\hbar=1$, with non-linearity exponent $q=1$; here red color is for maximal density, blue for zero.
• Figure 5: Dependence of the quantum fidelity $F(t=2t_r)$ on the quantum noise amplitude $\varepsilon_q$ for $K=3$ (blue) and $K=8$ (red) at a fixed reversal time $t_r=30$. Values of $F$ are averaged over $1000$ quantum noise realizations.