Single-photon scattering in an optomechanical Jaynes-Cummings model

We investigate an optomechanical system which realizes the Jaynes-Cummings (JC) model known in cavity QED. Such a system consists of a single photon and an optomechanical cavity with two optical cavity modes and one mechanical mode. Under the resonance condition when the mechanical frequency is close to the frequency difference between the optical modes, the photon and phonons can be strongly coupled. We present an analytic solution of single-photon scattering and show that the spectrum of the scattered photon exhibits excitation-number-dependent Rabi splitting of the JC model. In addition, we examine the response of the mechanical mode to a sequence of single photons, with one photon in the cavity at a time. We show that sequential photon scattering can efficiently excite the mechanical mode and generate sub-Poisson phonon statistics.

Figure 1

A schematic drawing of a two-mode optomechanical cavity and its coupling to outside modes. A single-photon wave packet is injected (via $\left\{c_k\right\}$ modes) into the $(+)$ cavity mode. In the long time limit, the photon escapes the cavity in the $\left\{c_k\right\}$ and $\left\{d_k\right\}$ modes.

Figure 2

Phonon excitation probability $P_{n_0\to n_0+1}$ by a single-photon scattering as a function of ${\gamma }_c$ and $\epsilon$.

Figure 3

Spectra of the scattered photon at various coupling parameters. The coupling strengths are (a, b) $g_0={\gamma }_c/2$, and (c, d) $g_0=2{\gamma }_c$. Other parameters are set as ${\gamma }_c=2\times {10}^{-2}{\mathrm{\Omega }}_M,\epsilon =5\times {10}^{-2}{\mathrm{\Omega }}_M$.

Figure 4

Plots of $S_d\left({\mathrm{\Delta }}_k\right)$ as a function of ${\mathrm{\Delta }}_k$ with mean phonon number ${\left|\alpha \right|}^2=1$. The Rabi splitting ${\nu }_m=g_0\sqrt{m+1}$ depends on the square root of the excitation number. The decay rate is $g_0=5{\gamma }_c$. Other parameters are ${\gamma }_c=0.5\times {10}^{-2}{\mathrm{\Omega }}_M,\epsilon =5\times {10}^{-2}{\mathrm{\Omega }}_M$.

Figure 5

Plot of the mean phonon number in the mechanical mode against the number of single photons injected. The insets show the phonon number distribution after $N=2$ and $N=10$ photon scattering. For all the $N$ in the figure, the corresponding phonon Mandel $Q$ parameter are negative. The parameters are $g_0={\gamma }_c/2,{\gamma }_c=2\times {10}^{-2}{\mathrm{\Omega }}_M,\text{and}\epsilon ={10}^{-3}{\mathrm{\Omega }}_M$.