Spectrum of single-photon emission and scattering in cavity optomechanics

We present an analytic solution describing the quantum state of a single photon after interacting with a moving mirror in a cavity. This includes situations when the photon is initially stored in a cavity mode as well as when the photon is injected into the cavity. In addition, we obtain the spectrum of the output photon in the resolved-sideband limit, which reveals spectral features of the single-photon strong-coupling regime in this system. We also clarify the conditions under which the phonon sidebands are visible and the photon-state frequency shift can be resolved.

Figure 1

(a) Schematic diagram of a Fabry-Perot-type optomechanical cavity formed by a fixed end mirror and a moving end mirror. (b) The energy-level structure of the optomechanical cavity (limited in the zero- and one-photon subspaces). Here we denote $|\tilde{n}{\left(0\right)\rangle }_b={|n\rangle }_b$ and $|\tilde{n}{\left(1\right)\rangle }_b={|\tilde{n}\rangle }_b$.

Figure 2

Spectrum $S\left({\Delta }_k\right)$ (red curves) of single-photon emission versus ${\Delta }_k$ for various $g$. The mirror's initial state is ${|0\rangle }_b$ and ${\gamma }_c/{\omega }_M=0.2$. The dark gray curve is a Lorentzian spectrum for comparison.

Figure 3

Spectrum $S\left({\Delta }_k\right)$ of single-photon emission versus ${\Delta }_k$ when the mirror is initially in (a) displaced ground state $|\tilde{0}{\rangle }_b$, (b) ground state ${|0\rangle }_b$, (c) coherent state ${|\beta \rangle }_b$ with $\beta =3$, and (d) thermal state ${\rho }_b^{\text{th}}$ with ${\overline{n}}_b=2$. Here $g/{\omega }_M=0.8$ and ${\gamma }_c/{\omega }_m=0.2$.

Figure 4

Spectrum $S\left({\Delta }_k\right)$ of single-photon scattering versus ${\Delta }_k$ for various $g$. The initial state of the mirror is ${|0\rangle }_b$. Other parameters are set as ${\gamma }_c/{\omega }_M=0.2$, $\epsilon /{\omega }_M=2$, and ${\Delta }_0/{\omega }_M=0$.

Figure 5

Spectrum $S\left({\Delta }_k\right)$ of single-photon scattering versus ${\Delta }_k$ for various $g$ and ${\Delta }_0$. The initial state of the mirror is ${|0\rangle }_b$. Here ${\gamma }_c/{\omega }_m=0.2$ and $\epsilon /{\omega }_M=0.05$.