Equivalence between an optomechanical system and a Kerr medium

We study the optical bistability of an optomechanical system in which the position of a mechanical oscillator modulates the cavity frequency. The steady-state mean-field equation of the optical mode is identical to the one for a Kerr medium, and thus we expect it to have the same characteristic behavior with a lower, a middle, and an upper branch. However, the presence of position fluctuations of the mechanical resonator leads to a new feature: the upper branch will become unstable at sufficiently strong driving in certain parameter regimes. We identify the appropriate parameter regime for the upper branch to be stable, and we confirm, by numerical investigation of the quantum steady state, that the mechanical mode indeed acts as a Kerr nonlinearity for the optical mode in the low-temperature limit. This equivalence of the optomechanical system and the Kerr medium will be important for future applications of cavity optomechanics in quantum nonlinear optics and quantum information science.

Figure 1

Optomechanical setup (upper panel) and Kerr medium in a cavity (lower panel). The main part of the paper investigates in detail whether and in which way the two systems are equivalent.

Figure 2

Optical bistability in the semiclassical regime. Typical curves for the mean-field cavity occupation $\overline{n}$ as a function of the dimensionless driving power $z$ (a) and the dimensionless detuning $y$ (b), obtained from the condition $p\left(\chi \overline{n}\right)=0$ [see Eq. (9)]. According to the stability criteria $c_{1,2}>0$ [see Eqs. (13)], Gaussian fluctuations lead to stable (solid black) or unstable (dotted blue and dashed red) mean-field solutions. As in the case of the Kerr medium, the first criterion $c_1>0$ always yields an unstable middle branch (dotted blue), while the additional criterion for the optomechanical system $c_2>0$ can turn part of the upper or only branch unstable (dashed red). In (b) we also show the critical mean-field occupation $n_c$ (dash-dotted gray) obtained from the condition $c_2=0$. In (c) we summarize the behavior of the mean-field solution as a function of the parameters $y$ and $z$. In regions $II$ and $III$, between the curves $z_{-}$ and $z_{+}$, Eqs. (10) and (11) are satisfied and there are three distinct mean-field solutions; the middle branch is always unstable. In region $II$ (blue) the lower and upper branches are stable. In region $III$ (purple) the second stability criterion shows the upper branch to be unstable ($c_2<0$) and only the lower branch is stable. In regions $I$ and $IV$ the mean-field equations (MFEs) have only one solution. Below the $z_c$ curve in region $I$ (gray) this unique branch is stable, while in region $IV$ (red) the second criterion again shows that this solution is unstable ($c_2<0$). The values of the detuning $y$ and driving power $z$ used in (a) and (b) are indicated by the orange and green dashed lines. Note that none of these features depends on the nonlinearity parameter $\chi$, due to appropriate scaling of the axes. The threshold detuning $\tilde{y}$ and driving power $\tilde{z}$ indicate the minimal values of $y$ and $z$ needed for the MFEs to have three solutions. The sideband parameter and mechanical quality factor chosen to show the influence of the second stability criterion $c_2>0$ are ${\omega }_m/\kappa =10$ and $Q_m=1000$.

Figure 3

Critical cavity occupation $n_c$ in units of $n_{\Delta }$, as a function of the sideband parameter ${\omega }_m/\kappa$ and the mechanical quality factor $Q_m$. At $n_c$ the mean-field solution $\overline{n}$ leads to unstable linear dynamics for the optomechanical system. The cavity occupation $n_{\Delta }=y/\left(2\chi \right)$ marks the point at which the effective detuning $\Delta$ becomes positive. We find $n_c$ from the second stability criterion, Eq. (13b). The bare detuning is $y=-{\Delta }_0/\kappa =1.5$. Note that the ratio $n_c/n_{\Delta }$ does not depend on the nonlinearity parameter $\chi$. The black cross indicates the parameters used in Fig. 4.

Figure 4

Optical bistability in the quantum regime. (a) Mean-field cavity occupation $\overline{n}$, with stable (black solid line) and unstable (black dotted line) branches, steady-state photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ (red), and cavity amplitude $|\langle \widehat{a}\rangle {|}^2$ (purple) of the optomechanical system, as a function of the dimensionless driving power $z$. The upper branch turns unstable outside the range of $z$ parameters we plot, beyond $z_c\simeq 92$ and $n_c\simeq 42$. For comparison we also show $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ (black dashed line) and $|\langle \widehat{a}\rangle {|}^2$ (black dash-dotted line) for the equivalent Kerr medium. For both systems, $y=-{\Delta }_0/\kappa =1.5$ and $\chi =0.08$. The parameters of the optomechanical system are ${\omega }_m/\kappa =30$, $Q_m=300$ (indicated by the black cross in Fig. 3), and $k_{B}T=0$ (dots) or $k_{B}T={\omega }_m$ (crosses). Inset (b) shows the second-order correlation function $g^{\left(2\right)}\left(0\right)=\langle {\widehat{a}}^{†}{\widehat{a}}^{†}\widehat{a}\widehat{a}\rangle /{\langle {\widehat{a}}^{†}\widehat{a}\rangle }^2$ for the optomechanical system with $k_{B}T=0$ (green solid line) as well as $k_{B}T={\omega }_m$ (green dashed line) and for the Kerr medium (black dash-dotted line). The first and third curves are indistinguishable. Inset (c) shows the overlap $F({\widehat{\rho }}_{\text{opt}},{\widehat{\rho }}_K)$, as defined in Eq. (18), between the density matrices of the pure Kerr medium ${\widehat{\rho }}_K$ and the reduced density matrix of the optomechanical system ${\widehat{\rho }}_{\text{opt}}$, obtained by tracing out the mechanical degree of freedom from $\widehat{\rho }$. The temperatures chosen are $k_{B}T=0$ (solid line) and $k_{B}T={\omega }_m$ (dashed line). (d) Wigner function $W_{\text{opt}}\left(\alpha \right)$ of the optical mode of the optomechanical system for six different driving powers $z$ and two different temperatures. The white crosses indicate the mean-field amplitudes $\overline{a}$ of the stable branches. The values of $z$ are indicated by blue dots and lines in (a).