Multistable particle-field dynamics in cavity-generated optical lattices

Polarizable particles trapped in a resonator-sustained optical-lattice potential generate strong position-dependent backaction on the intracavity field. In the quantum regime, particles in different energy bands are connected to different intracavity light intensities and optical-lattice depths. This generates a highly nonlinear coupled particle-field dynamics. For a given pump strength and detuning, a factorizing mean-field approach predicts several self-consistent stationary solutions of strongly distinct photon numbers and motional states. Quantum Monte Carlo wave-function simulations of the master equation confirm these predictions and reveal complex multimodal photon-number and particle-momentum distributions. Using larger nanoparticles in such a setup thus constitutes a well-controllable playground to study nonlinear quantum dynamics and the buildup of macroscopic quantum superpositions at the quantum-classical boundary.

Figure 1

A particle within a driven optical cavity. The longitudinal cavity pump $\eta$ builds up an intracavity field that drives the particle motion. The particle's motional state affects the cavity detuning (dynamic refractive index), which in turn influences the intracavity photon number. Photons leak through the cavity mirrors at a rate $2\kappa$.

Figure 2

Self-consistent photon-number contours for a harmonically trapped particle as a function of the cavity-pump detuning for different pump amplitudes $\eta$. The four plots show contours in the ${\mathrm{\Delta }}_{\mathrm{C}}\text{−}\langle n\rangle$ plane for different harmonic-oscillator eigenstates $|n_{\text{ho}}\rangle$, where Eq. (11) holds self-consistently. All excited states, $n_{\text{ho}}\ge 1$, yield the possibility of two self-consistent solutions for a certain range of detuning ${\mathrm{\Delta }}_{\mathrm{C}}$ and therefore may allow for optomechanical bistability. Parameters: $U_0=-10{\omega }_{\mathrm{R}}$ and $\kappa ={\omega }_{\mathrm{R}}$.

Figure 3

Same as Fig. 2 for particles in localized Wannier states. Photon numbers above the dashed lines for $w_{m\ge 1}$ correspond to bound Wannier states $\left(E<0\right)$. Higher bands exhibit up to three solutions of Eq. (14) for a given value of ${\mathrm{\Delta }}_{\mathrm{C}}$. Parameters: $U_0=-10{\omega }_{\mathrm{R}}$ and $\kappa ={\omega }_{\mathrm{R}}$.

Figure 4

Spatial particle distribution in a fourth band Wannier state for different photon numbers: (a) Free particle: The wave function is barely localized, $b_4\lesssim 0.5$. (b) Transition to a bound state: The wave function is localized at potential maxima, $b_4<0.5$. (c) Tight-binding regime: The wave function is strongly localized in a single potential well, $b_4>0.5$.

Figure 5

Contour plot for the right-hand side of Eq. (14) for the fourth band Wannier state. Solid (dotted) lines mark stable (unstable) self-consistent solutions according to Eq. (16). Parameters: $\eta =6{\omega }_{\mathrm{R}},U_0=-10{\omega }_{\mathrm{R}}$, and $\kappa ={\omega }_{\mathrm{R}}$.

Figure 6

Expectation values on single quantum trajectories for $\eta =6{\omega }_{\mathrm{R}},{\mathrm{\Delta }}_{\mathrm{C}}=-7.5{\omega }_{\mathrm{R}},U_0=-10{\omega }_{\mathrm{R}}$, and $\kappa ={\omega }_{\mathrm{R}}$. The mean values jump between the stable values predicted by the self-consistency equation (14) (dashed lines). (a) and (b) Trajectories with several jumps from very high to very low photon numbers. (c) Trajectory with only a few jumps. (d) Trajectory with a large increase of $\langle E_{\mathrm{kin}}\rangle$ during evolution in a low-photon-number state. The kinetic energy mean value of the 12th and 14th excited bands are shown as black solid lines. Although numerical values indicate that the trajectory could be in a definite band, the neat picture of correlated photon number and momentum jumps clearly breaks down at this point. Note the different scaling of the $\langle E_{\mathrm{kin}}\rangle$ axis.

Figure 7

Self-consistent solutions of Eq. (14) for the lowest Wannier states $\left(m=0\right)$ compared to ensemble-averaged quantum simulation results. Solid lines mark self-consistent contours; data points are quantum simulation results. The black line separates the regions where ${\mathrm{\Delta }}_{\mathrm{eff},0}<-{\mathrm{\Delta }}_{\mathrm{eff},2}$ (left) and ${\mathrm{\Delta }}_{\mathrm{eff},0}>-{\mathrm{\Delta }}_{\mathrm{eff},2}$ (right) along the self-consistent contours of the zeroth band; cf. Eq. (18). In the left region quantum simulations are in accordance with the self-consistent solutions of Eq. (14) for the lowest band, while deviations arise in the right region due to population of the $m=2$ Wannier state. Parameters: $U_0=-10{\omega }_{\mathrm{R}}$ and $\kappa ={\omega }_{\mathrm{R}}$.

Figure 8

Combined photon-number and momentum-state occupation probability after a time interval of $\mathrm{\Delta }t=310{\omega }_{\mathrm{R}}^{-1}$, starting from a state with zero momentum and one cavity photon, $k=0$ and $n=1$, for $U_0=-10{\omega }_{\mathrm{R}},\kappa ={\omega }_{\mathrm{R}},\eta =6{\omega }_{\mathrm{R}}$, and different values of the detuning. The density matrix is approximated via quantum simulations with ensemble averages over 1000 trajectories for each parameter set. The numbers in each box give the detuning ${\mathrm{\Delta }}_{\mathrm{C}}$ in units of ${\omega }_{\mathrm{R}}$.

Figure 9

Quantum simulation results showing the particle's average kinetic energy $\langle E_{\mathrm{kin}}\rangle$ as a measure of its temperature as a function of the cavity detuning for different pump amplitudes. For $\eta =5{\omega }_{\mathrm{R}}$ an interval where the final temperature decreases with increasing detuning is clearly visible. The plot shows only a detail of the set of data points for better visibility of the effect. In order to eliminate temporal fluctuations at single time instances the plotted points are time averages of $\langle E_{\mathrm{kin}}\rangle$ of 100 values in the interval ${\omega }_{\mathrm{R}}t\in (305,310]$. The dashed lines are interpolations between data points and merely serve as a guide to the eye. The other parameters are $U_0=-10{\omega }_{\mathrm{R}}$ and $\kappa ={\omega }_{\mathrm{R}}$.