Time-Reversal Test for Stochastic Quantum Dynamics

The calculation of quantum dynamics is currently a central issue in theoretical physics, with diverse applications ranging from ultracold atomic Bose-Einstein condensates to condensed matter, biology, and even astrophysics. Here we demonstrate a conceptually simple method of determining the regime of validity of stochastic simulations of unitary quantum dynamics by employing a time-reversal test. We apply this test to a simulation of the evolution of a quantum anharmonic oscillator with up to $6.022\times {10}^{23}$ (Avogadro’s number) of particles. This system is realizable as a Bose-Einstein condensate in an optical lattice, for which the time-reversal procedure could be implemented experimentally.

Figure 1

Time-reversal test for a stochastic gauge simulation of the $X$ quadrature of the anharmonic oscillator. (a) An initial coherent state with mean boson number $\overline{n}=100$ and ${10}^5$ stochastic trajectories. (b) An initial coherent state with $\overline{n}=6.022\times {10}^{23}$ and ${10}^7$ stochastic trajectories, with the calculation performed in the rotating frame. The solid lines are the simulation result for $⟨\widehat{X}⟩$, with the statistical error bars (representing 1 standard deviation) shown by the gray shading. For both cases the time reversal was implemented at ${\tau }_R=0.5$ by negating the sign of the Hamiltonian and the initial state is recovered to within statistical error at $\tau =2{\tau }_R$. The forward time evolution to $2{\tau }_R$ demonstrating the collapse to $⟨\widehat{X}⟩=0$ is also shown in both (a) and (b).

Figure 2

A representation of the broadening of the gauge distribution function for $\overline{n}=100$. The plots are histograms of the quantity ${log}_{10}|\alpha \Omega |$ for the stochastic trajectories at a number of points in time. (a) The time-reversed case with the Hamiltonian negated at ${\tau }_R=0.5$ and further evolved to $\tau =2{\tau }_R$. (b) The forward time evolution to $\tau =2{\tau }_R$. The initial distribution at $\tau =0$ for both graphs is a delta function; however, it broadens with time to span several orders of magnitude at $\tau =2{\tau }_R$, despite being another representation of the initial quantum state in (a). Note that the logarithm of $|\alpha \Omega |$ is binned.