Emergence of Chaos in Quantum Systems Far from the Classical Limit

The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation—as all experimental systems must be—their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.

Figure 1

Position distribution for the Duffing oscillator with measurement strengths $k=0.01$ (red) and $k=10$ (green), demonstrating measurement-induced localization ($k=10$). The momentum distribution behaves similarly.

Figure 2

Phase-space stroboscopic maps shown for 4 different measurement strengths, $k=5\times {10}^{-4}$, 0.01 (top), and 1, 10 (bottom). Contour lines are superimposed to provide a measure of local point density at relative density levels of 0.05, 0.15, 0.25, 0.35, 0.45, and 0.55.

Figure 3

Finite-time Lyapunov exponents $\lambda (t)$ for measurement strengths $k=5\times {10}^{-4},0.01,10$, averaged over 32 trajectories for each value of $k$ (linear scale in time, top, and logarithmic scale, bottom; bands indicate the standard deviation over the 32 trajectories). The (analytic) $1/t$ falloff at small $k$ values, prior to the asymptotic regime, is evident in the bottom panel. The unit of time is the driving period.

Figure 4

The emergence of chaos: The Lyapunov exponent $\lambda$ as a function of measurement strength $k$. Error bars follow those of Fig. 3, taken at the final time.