Single-photon nonlinearities in two-mode optomechanics

We present a detailed theoretical analysis of a weakly driven, multimode optomechanical system, in which two optical modes are strongly and near-resonantly coupled to a single mechanical mode via a three-wave mixing interaction. We calculate one- and two-time intensity correlations of the two optical fields and compare them to analogous correlations in atom-cavity systems. Nonclassical photon correlations arise when the optomechanical coupling $g$ exceeds the cavity decay rate $\kappa$, and we discuss signatures of one- and two-photon resonances as well as quantum interference. We also find a long-lived correlation that decays slowly with the mechanical decay rate $\gamma$, reflecting the heralded preparation of a single-phonon state after detection of a photon. Our results provide insight into the quantum regime of multimode optomechanics, with potential applications for quantum information processing with photons and phonons.

Figure 1

(a) Optomechanical system consisting of two tunnel-coupled optical cavity modes $c_1$ and $c_2$, and a mechanical oscillator $b$ coupled to one of the cavity modes by radiation pressure. The coupled optical modes are diagonalized in terms of symmetric and antisymmetric modes, $c_s$ and $c_a$. (b) Level diagram showing the relevant zero-, one- and two-photon states at zero temperature and under the three-mode resonance condition ${\omega }_s-{\omega }_a={\omega }_m$. States are labeled by $|n_a{n}_s{n}_b\rangle$ denoting the number $n_a$ ($n_s$) of antisymmetric (symmetric) photons and the number of phonons $n_b$. The optomechanical coupling $g$ splits the degeneracy of states $|n_a{n}_s{n}_b\rangle$ and $|n_a-1,n_s+1,n_b+1\rangle$.

Figure 2

(a) Normalized average photon number (green “$+$”) and photon-photon correlation function (blue “$\circ$”) for driven mode $c_a$ as a function of laser detuning at zero temperature. Solid lines are calculated from the analytic model [see Eqs. (18, 19, 20, 21, 22, 23)] and points show full numerical calculation. The average photon number is normalized by $n_0={(\Omega /\kappa )}^2$. (b) Same as (a) for the undriven mode $c_s$, and (c) for the reflected field $c_R$. In all plots, we took $g/\kappa =20$ and $\gamma /\kappa =0.2$. Dots mark features seen at values of detuning (A) $\Delta /g=0$, (B) $\frac{1}{\sqrt{8}}$, (C) $\frac{1}{2}$, and (D) $\frac{\sqrt{6}}{4}$. The small discrepancy between the analytic and numerical results in (c) is due to the approximation $g/\kappa \gg 1$ to simplify the expressions in Eqs. (18, 19, 20, 21, 22, 23). Bottom panels A–D illustrate the origin of these features as explained in the text. Suppression (enhancement) of the steady-state population of a specific level is indicated by a red X (green circle).

Figure 3

Average number (green dashed line) and intensity correlation (blue solid line) of total photon number in the coupled OM system. Parameters are the same as in Fig. 2, and dots mark the same detunings. One- and two-photon resonances are seen at C and D, but we see no bunching in the total photon number at $\Delta =0$ (point A). This reflects the lack of two-photon resonance due to destructive interference [see Eq. (24)].

Figure 4

Correlation functions for finite temperature. (a) Level diagram showing the six states populated by the drive from level $|0,0,n\rangle$ (left), and the associated eigenmodes (right) with $n$-dependent splittings. (b) Driven mode correlation function $g_{aa}^{\left(2\right)}\left(0\right)$ for thermal mechanical occupation $N_{\mathrm{th}}=0,1,2$. Solid lines show the analytic calculation with $\gamma \to 0$, and dots show the full numerical results for $N_{\mathrm{th}}=2$ only. The inset shows the minimal $g_{aa}^{\left(2\right)}\left(0\right)$ as a function of $N_{\mathrm{th}}$ for several coupling strengths. (c) Same as (b) for the undriven mode correlation function $g_{ss}^{\left(2\right)}\left(0\right)$. Parameters are $g/\kappa =20$ and (for numerics) $\gamma /\kappa ={10}^{-3}$.

Figure 5

(a) Finite time delay intensity correlation function $g_{aa}^{\left(2\right)}\left(\tau \right)$ for detunings (A) $\Delta /g=0$ and (E) $\frac{1}{\sqrt{2}}$ . Detuning for curve A is the same as marked in Fig. 2, while E shows a new effect not seen at equal times. Thin dashed line indicates the classical bound [see Eq. (36)] for curve E. (b) Evolution of amplitude $A_{100}$ (normalized by its steady-state value) at detuning A, $\Delta /g=0$, after detecting a driven $c_a$ photon at $\tau =0$. Vertical dash-dotted lines mark delay times (${\tau }_1,{\tau }_2$) where this amplitude vanishes, resulting in the vanishing of $g_{aa}^{\left(2\right)}\left(\tau \right)$ in (a). Parameters are $g/\kappa =8$ and $\gamma /\kappa$ $=$ 0.02.

Figure 6

Finite time delay intensity correlation function $g_{ss}^{\left(2\right)}\left(\tau \right)$ for detunings (A) $\Delta /g=0$, (D) $\sqrt{6}/4$, and (E) $\frac{1}{\sqrt{2}}$. Thin dashed lines indicate the classical bounds [see Eq. (36)]. Labels A, D, and E correspond to the same detunings marked in Figs. 2 and 5. Parameters are $g/\kappa =8$ and $\gamma /\kappa$ $=$ 0.02.

Figure 7

Effect of detection of a $c_s$ photon at detuning E ($\Delta /g=\frac{1}{\sqrt{2}}$). In the steady state (gray region on left), the drive is far off-resonant. However, after detection of a $c_s$ photon, the system jumps into the conditional subspace (pink region on right). Due to the presence of an extra phonon in this subspace, the drive is resonant and the probability to detect a second photon is much higher than in the steady state. This increased probability persists as long as the extra phonon, which decays slowly at rate $\gamma$.