Transition to classical chaos in a coupled quantum system through continuous measurement

Continuous observation of a quantum system yields a measurement record that faithfully reproduces the classically predicted trajectory provided that the measurement is sufficiently strong to localize the state in phase space but weak enough that quantum backaction noise is negligible. We investigate the conditions under which classical dynamics emerges, via a continuous position measurement, for a particle moving in a harmonic well with its position coupled to internal spin. As a consequence of this coupling, we find that classical dynamics emerges only when the position and spin actions are both large compared to $\hslash$. These conditions are quantified by placing bounds on the size of the covariance matrix which describes the delocalized quantum coherence over extended regions of phase space. From this result, it follows that a mixed quantum-classical regime (where one subsystem can be treated classically and the other not) does not exist for a continuously observed spin-$\frac{1}{2}$ particle. When the conditions for classicality are satisfied (in the large-spin limit), the quantum trajectories reproduce both the classical periodic orbits as well as the classically chaotic phase space regions. As a quantitative test of this convergence, we compute the largest Lyapunov exponent directly from the measured quantum trajectories and show that it agrees with the classical value.

Figure 1

(Color online) (a) Mean position of the measured spin-$\frac{1}{2}$ system (solid) in a single quantum trajectory with $\Delta z\approx 22z_g$, $c=200E_g/J$, and $k=\omega ∕20z_g^2$. Outer solid curves show the variance of the wave function. The measurement backaction causes the quantum trajectory to diverge from the classical (dotted, black) trajectory. This is because part of the wave function moves along the upper adiabatic potential while the rest moves along the lower adiabatic potential [dashed red curves in (b)]. Eventually, the measurement collapses the wave function into the upper or lower potential [solid blue line in (b)]. The classical motion is along the dashed-dotted potential in (b).

Figure 2

Regular and chaotic classical trajectories [(a) and (b)] are recovered in the quantum trajectories [(c) and (d)] with $J=200\hslash$, $\Delta z\approx 45z_g$, and $k=\omega ∕8z_g^2$.

Figure 3

(Color online) Quantum trajectories (asterisks) follow the stable islands of the classical surface of section (black dots). Shown are quantum trajectories that reproduce the regular motion around different period-1 fixed points (blue, green), a period-3 orbit (red), and a higher period orbit (magenta).

Figure 4

Solutions of ${\tilde{C}}_{zz}\left(t\right)=C_{zz}\left(t\right)∕z_g^2$ for the regular and chaotic quantum trajectories of Fig. 2. The maximum cumulant is smaller than the total phase space covered by the motion by a factor of about 100.

Figure 5

Distribution of the largest Lyapunov exponent obtained from (a) classical dynamics using 500 fiducial trajectories at $E=0.58E_0$ with $I_0∕J=5$ and $c=200E_g∕J$. (b) Continuously measured quantum dynamics using 100 fiducial quantum trajectories with $J=200\hslash$, $\Delta z\approx 45z_g$, and $k=\omega ∕8z_g^2$. For numerical efficiency, the SSE was integrated by truncating the cumulants at second order.