Quantum reservoir engineering through quadratic optomechanical interaction in the reversed dissipation regime

We explore an electromagnetic field coupled to a mechanical resonator via quadratic optomechanical interaction in the reversed dissipation regime where the mechanical damping rate is much higher than the cavity-field dissipation rate. It is shown that in this regime, the cavity field effectively acquires an additional reservoir which is conditioned by the temperature of the mechanical bath as well as the mechanical damping rate. We analytically find the steady-state mean photon number and the critical temperature of the mechanical oscillator to cool or heat the coupled electromagnetic field. We also show that in the case of quadratic coupling, the temperature of the mechanical oscillator can be estimated in the quantum regime by observing the noise spectrum of the cavity field.

Figure 1

Magnitude of the steady-state cavity-field amplitude $\left|\alpha \right|$ as functions of the cavity detuning ${\mathrm{\Delta }}_c$ and pumping rate $\left|\eta \right|$ normalized with the mechanical frequency ${\omega }_m$. The thick solid line indicates the resonance condition ${\mathrm{\Delta }}_c=-2{\omega }_c$. The parameters we use are $g_0^{\left(2\right)}/{\omega }_m=5\times {10}^{-7}$, $\kappa /{\omega }_m=0.0025$.

Figure 2

Mean cavity photon number as a function of time for different mean phonon numbers: ${\overline{n}}_m=0$ (solid red curve), ${\overline{n}}_m=0.5$ (dashed blue curve), and ${\overline{n}}_m=1.2$ (dotted green curve). Other parameters are ${\overline{n}}_o=1$ and $C_2=1$.

Figure 3

Steady-state mean photon number of the cavity field as functions of the mean phonon number of the mechanical oscillator and the quadratic optomechanical cooperativity when ${\overline{n}}_o=1.0$.

Figure 4

Noise spectra of the cavity field for several representative mean phonon numbers of the mechanics: ${\overline{n}}_m=0$ (dashed blue curve), ${\overline{n}}_m=0.5$ (dot-dashed green curve), ${\overline{n}}_m=2.414$ (dotted orange curve), and ${\overline{n}}_m=3.6$ (solid red curve). Other parameters are $\kappa /{\omega }_m^{\prime }=0.0025$, ${\overline{n}}_o=1.0$, $C_2=1.0$. Inset: Plot of ${\overline{n}}_{\mathrm{ss}}$ as a function of ${\overline{n}}_m$ for the parameters ${\overline{n}}_o=1.0$, $C_2=1.0$, describing the temperature of the cavity field with respect to that of the mechanical bath.

Figure 5

Effective decay rate of the cavity field coupled to the mechanics linearly (solid red line) and quadratically (blue lines) as a function of the mean phonon number of the mechanics. For the linear coupling we have used $C_1=1.0$, and for the quadratic coupling we have used $C_2=0.1$ (dot-dashed blue line), $C_2=0.5$ (dashed blue line), and $C_2=1.0$ (dotted blue line).

Figure 6

Steady-state mean photon number of the cavity field coupled to the mechanics via a linear coupling as functions of the mean phonon number of the mechanical oscillator and the linear optomechanical cooperativity. The parameter we have used is ${\overline{n}}_o=1.0$.