Route to Chaos in Optomechanics

We establish the emergence of chaotic motion in optomechanical systems. Chaos appears at negative detuning for experimentally accessible values of the pump power and other system parameters. We describe the sequence of period-doubling bifurcations that leads to chaos and state the experimentally observable signatures in the optical spectrum. In addition to the semiclassical dynamics, we analyze the possibility of chaotic motion in the quantum regime. We find that quantum mechanics protects the optomechanical system against irregular dynamics, such that simple periodic orbits reappear and replace the classically chaotic motion. In this way observation of the dynamical signatures makes it possible to pin down the crossover from quantum to classical mechanics.

Figure 1

(a),(c) Amplitudes of cantilever oscillation limit cycles given by the sinusoidal ansatz (black line) and by the full SC equations of motions [Eq. (6)] (red diamonds). The inset in (c) displays a zoom into the first PDB around $\mathrm{\Delta }=-0.75$. (b),(d) Initial dynamics of the cantilever converging to a period-1 (period-2) limit cycle. Here, as in all figures, we give $\mathrm{\Delta },\tau$ in units of ${\mathrm{\Omega }}^{(-1)}$ and use the dimensionless parameters $P$, $\sigma$ from Eq. (4).

Figure 2

Bifurcation diagram of the limit cycle amplitude (top) and corresponding maximal Lyapunov exponent ${\lambda }_{\max }$ (bottom) at $P=1.4$. Vertical dashed lines mark PDBs, signaled by ${\lambda }_{\max }=0$.

Figure 3

Bifurcation diagrams of the limit cycle amplitude and maximal Lyapunov exponents ${\lambda }_{\max }$ for larger pump power $P=1.5,1.6$.

Figure 4

Schematic picture of the regular and chaotic regimes of the optomechanical system in the bad-cavity limit $\sigma \ll 1$, plotted in the $\mathrm{\Delta }\text{−}P$ plane. Here, as everywhere, $\kappa =1$ and $\mathrm{\Gamma }={10}^{-3}$. Dots represent numerical data extracted from bifurcation diagrams (such as Figs. 2 and 3), dashed lines interpolate between the numerical data.

Figure 5

Fourier spectrum of the classical photon field amplitude $\alpha$ for $P=1.5$. The figures show spectra corresponding to cantilever dynamics on a simple periodic limit cycle (a), a period-2 limit cycle (b), a period-4 limit cycle (c), and a chaotic limit cycle (d).

Figure 6

Stroboscopic ($x,p$)-phase space plot of a single quantum trajectory (red points) for $P=1.5$, initial conditions $(x_0,p_0)=(0,0)$, and decreasing quantum-classical scaling parameter $\sigma$, approaching a simple periodic limit cycle (a), a period-2 limit cycle (b), and a chaotic limit cycle (c) (black curves) in the limit $\sigma \to 0$. We show points for $\tau /(2\pi )\ge 20$ only to omit the earliest transient dynamics.

Figure 7

Quantum dynamics of the cantilever from the ensemble average of 5000 quantum trajectories ($\sigma >0$) in comparison to the SC dynamics ($\sigma =0$), all for $P=1.5$. The figures depict the case of a classical period-2 orbit ($\mathrm{\Delta }=-0.85$, left) and a chaotic orbit ($\mathrm{\Delta }=-0.7$, right).