Light scattering in an optomechanical cavity coupled to a single atom

We theoretically analyze the light scattering of an optomechanical cavity which strongly interacts with a single two-level system and couples simultaneously to a mechanical oscillator by radiation forces. The analysis is based on the assumptions that the system is driven at low intensity, and that the mechanical interaction is sufficiently weak, permitting a perturbative treatment. We find quantum interference in the scattering paths, which allows one to suppress the Stokes component of the scattered light. This effect can be exploited to reduce the motional energy of the mechanical oscillator.

Figure 1

An optomechanical setup, consisting of a cavity coupled to a mechanical oscillator by radiation pressure, interacts with a single two-level system at fixed position $x_0$ via dipole interaction. We consider two scenarios: (a) The cavity is driven; (b) the atom is driven. Both the atom and the cavity can decay with rates $\gamma$ and $2\kappa$, respectively. The frequency of the mechanical oscillator is $\nu$.

Figure 2

The Raman scattering processes leading to the transition matrix elements ${\mathcal{T}}_{j,\pm }$: The system, initially in the stable state $|g,0;m\rangle$, is excited by the pump interaction $W_{\mathrm{L}}$ into a superposition of the dressed state $|\pm ;m\rangle$. It passes through these two intermediate states, with an energy defect $E_i-\hslash {\omega }_{\pm }-\hslash (m+1/2)\nu$. The mechanical interaction $Fx$ then brings the $|g,1\rangle$ component of the dressed states to an adjacent vibrational level, where the system again passes through the dressed states $|\pm ;m\pm 1\rangle$, this time with an energy defect of $E_i-\hslash {\omega }_{\pm }-\hslash (m\pm 1+1/2)\nu$. Finally, due to their instability, the dressed states can either decay due to (a) cavity loss or (b) spontaneous decay, in that way closing the cooling and heating cycles. After many such cycles a stationary occupation of the vibrational levels can be reached, where cooling and heating transitions are balanced.

Figure 3

Population in the single-photon state of the cavity when either (a) the fixed cavity is directly driven, or (b) indirectly via the atom. The two peaks correspond to the dressed states $|\pm \rangle$ of the atom-cavity system. In (a) a dark resonance appears at $\delta =\Delta +{\delta }_{\mathrm{ca}}=0$, that is, when the pump is resonant with the atomic transition. This effect is due to coherent population trapping in the $\Lambda$-type configuration shown in the inset. (Parameters are as follows: $\gamma =0.1\kappa$, $g=2\kappa$, ${\delta }_{\mathrm{ca}}=-2\kappa$, $\Omega =\kappa$.)

Figure 4

Cooling rate $\Gamma$ as a function of the laser-cavity detuning $\Delta$ and the laser-atom detuning $\delta$ for the cases of (a) driven cavity and (b) driven atom. Bright shaded regions represent large cooling rates; in the hatched area the rate is negative which corresponds to heating the mechanical oscillator. Shown is also the resonance condition ${\omega }_{\pm }(\Delta ,\delta )=0,-\nu ,\nu$, where the laser is in resonance with the dressed state $|\pm \rangle$ (solid curve), or with the blue (dashed) or red (broken) sidebands of the dressed states, respectively. (Parameters are as follows: $\kappa =7\nu ,\gamma =0.05\nu ,\Omega =\nu ,\chi =\nu /10,g=2\nu$.)

Figure 5

The mean vibrational occupation number $\langle m\rangle$ as a function of the laser-cavity detuning $\Delta$ and the laser-atom detuning $\delta$, being equal for the driven atom and the driven cavity. Darker shading represents lower temperatures; in the hatched regions the theory does not provide a stationary state. The curves are connected with the dressed states' frequencies and their sidebands as explained in Fig. 4.

Figure 6

Scattering rate of photons into the Stokes (red) and anti-Stokes (blue) component of the cavity output (solid lines) and fluorescence signal (dashed lines) for $\kappa \gg \gamma$ and $\Delta =0$ at steady state. The black dashed line being the sum of the blue or red curves gives the total rate $R_{\pm }$ of scattered photons into the motional sidebands, which in steady state are equal for heating and cooling transitions. In (a) the cavity is driven; (b) shows the case of a driven atom. (Parameters are as follows: $\kappa =5\nu$, $\gamma =0.1\nu$; other parameters like in Fig. 4.)