Observability of radiation-pressure shot noise in optomechanical systems

We present a theoretical study of an experiment designed to detect radiation-pressure shot noise in an optomechanical system. Our model consists of a coherently driven optical cavity mode that is coupled to a mechanical oscillator. We examine the cross-correlation between two quadratures of the output field from the cavity. We determine under which circumstances radiation-pressure shot noise can be detected by a measurement of this cross-correlation. This is done in the general case of nonzero detuning between the frequency of the drive and the cavity resonance frequency. We study the qualitative features of the different contributions to the cross-correlator and provide quantitative figures of merit for the relative importance of the radiation-pressure shot noise contribution to other contributions. We also propose a modified setup of this experiment relevant to the “membrane-in-the-middle” geometry, which potentially can avoid the problems of static bistability and classical noise in the drive.

Figure 1

Schematic overview of the model. The optical cavity mode $\mathrm{â}$ is coupled to three input modes and to the position fluctuations $\mathrm{ẑ}$ of the mechanical oscillator, represented in this case by a membrane in the middle. We imagine the cavity being coherently driven by a laser on the left-hand side (see Fig. 2) such that the input mode ${\mathrm{â}}_{\mathrm{in},\mathrm{L}}$ is the sum of a mean amplitude, classical laser noise, and quantum noise. The mode ${\mathrm{â}}_{\mathrm{in},\mathrm{R}}$ simply represents quantum noise entering the cavity from the right-hand side, whereas ${\mathrm{â}}_{\mathrm{in},\mathrm{M}}$ represents quantum noise associated with other types of decay.

Figure 2

Schematic overview of the experiment considered. The cavity mode is coherently driven by a laser from the left-hand side. One quadrature ${\mathrm{X̂}}_{{\theta }_R}$ of the transmitted beam and one quadrature ${\mathrm{Ŷ}}_{{\theta }_L}$ of the reflected beam are measured through homodyne detection. The RPSN felt by the mechanical oscillator can be detected through the cross-correlation of the fluctuations in the two quadratures.

Figure 3

Modified version of the experiment considered, where the transmitted intensity fluctuations are measured by a photodetector.

Figure 4

The critical angle ${\theta }_c$ as a function of detuning $\Delta$. The angle ${\theta }_c$ is multivalued, since $\theta \to \theta +\pi$ gives $\delta {\mathrm{Ŷ}}_{\theta }(t)\to -\delta {\mathrm{Ŷ}}_{\theta }(t)$ and hence $S[\omega ]\to -S[\omega ]$.

Figure 5

The contributions $R_{\mathrm{q},\mathrm{z}}[\omega ]$ (upper panel) and $R_{\mathrm{z},\mathrm{z}}[\omega ]$ (middle panel) and their sum $R[\omega ]$ (lower panel). We have used $\Delta =-0.01\kappa$, ${\omega }_{\mathrm{M}}/\kappa =3.2$, ${\omega }_M/\gamma ={10}^6$, $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$, and $\theta =\pi /2$. The temperature is 4.2 K and ${\omega }_{\mathrm{M}}/(2\pi )=1.2$ MHz. The mean number of photons in the cavity ($n_{\mathrm{photon}}=\left|\mathrm{ā}\right|{}^2$) is ${10}^{10}$ (left panel) and ${10}^{12}$ (right panel). In the first case, the total signal is dominated by $R_{\mathrm{z},\mathrm{z}}[\omega ]$, whereas in the second case $R_{\mathrm{q},\mathrm{z}}[\omega ]$ dominates. In the absence of classical laser noise, a measured signal as in the lower-right panel is a signature of RPSN.

Figure 6

The contributions $R_{\mathrm{q},\mathrm{z}}[\omega ]$ (upper panel) and $R_{\mathrm{z},\mathrm{z}}[\omega ]$ (middle panel) and their sum $R[\omega ]$ (lower panel). We have used $\Delta =-0.5\kappa$, ${\omega }_{\mathrm{M}}/\kappa =3.2$, ${\omega }_{\mathrm{M}}/\gamma ={10}^6$, and $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$. The temperature is 4.2 K and ${\omega }_{\mathrm{M}}/(2\pi )=1.2$ MHz. The mean number of photons in the cavity ($n_{\mathrm{photon}}=\left|\mathrm{ā}\right|{}^2$) is ${10}^{10}$. The angle $\theta$ is ${\theta }_c-\pi /10$ (left panel) and ${\theta }_c-\pi /100$ (right panel). In the first case, the total signal is dominated by $R_{\mathrm{z},\mathrm{z}}[\omega ]$, whereas in the second case, $R_{\mathrm{q},\mathrm{z}}[\omega ]$ dominates.

Figure 7

The ratio between the peak-to-peak value $P_{\mathrm{q}}$ of $R_{\mathrm{q},\mathrm{z}}[\omega ]$ and the height $M$ of $R_{\mathrm{z},\mathrm{z}}[\omega ]$ for ${\omega }_{\mathrm{M}}/\kappa =1.6$ (left panel) and ${\omega }_{\mathrm{M}}/\kappa =6.4$ (right panel). The deviation from the critical angle ${\theta }_c$ is $\delta \theta =\pi /100$, ${\omega }_M/\gamma ={10}^6$, $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$, $n_{\mathrm{photon}}={10}^{10}$, ${\omega }_{\mathrm{M}}/(2\pi )=1.2$ MHz, and the temperature is $300$ K.

Figure 8

The ratio between the peak-to-peak value $P_{\mathrm{q}}$ of $R_{\mathrm{q},\mathrm{z}}[\omega ]$ and the height $M$ of $R_{\mathrm{z},\mathrm{z}}[\omega ]$ for ${\omega }_{\mathrm{M}}/\kappa =1.6$ (left panel) and ${\omega }_{\mathrm{M}}/\kappa =6.4$ (right panel). The deviation from the critical angle ${\theta }_c$ is $\delta \theta =\pi /100$, ${\omega }_{\mathrm{M}}/\gamma ={10}^6$, $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$, $n_{\mathrm{photon}}={10}^{10}$, and the temperature is $4.2$ K.

Figure 9

The contributions $R_{\mathrm{q},\mathrm{z}}[\omega ]$ (upper panel), $R_{\mathrm{cl}}[\omega ]$ (middle panel), and their sum (lower panel). The detuning is $\Delta =-0.01\kappa$, ${\omega }_{\mathrm{M}}/\gamma ={10}^6$, $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$, $\theta =\pi /2$, and $n_{\mathrm{photon}}={10}^{10}$. The ratio between mechanical frequency and cavity linewidth is ${\omega }_{\mathrm{M}}/\kappa =0.64$ (left panel) and ${\omega }_{\mathrm{M}}/\kappa =6.4$ (right panel). Note the qualitative difference between the two contributions in the latter case.

Figure 10

The contributions $R_{\mathrm{q},\mathrm{z}}[\omega ]$ (upper panel), $R_{\mathrm{cl}}[\omega ]$ (middle panel), and their sum (lower panel). The detuning is $\Delta =-\kappa /2$, ${\omega }_{\mathrm{M}}/\gamma ={10}^6$, $A/{\omega }_{\mathrm{M}}=1.15\times {10}^{-6}$, $n_{\mathrm{photon}}={10}^{10}$, and $\theta ={\theta }_c$. The ratio between mechanical frequency and cavity linewidth is ${\omega }_{\mathrm{M}}/\kappa =3.2$ (left panel) and ${\omega }_{\mathrm{M}}/\kappa =6.4$ (right panel).