Force sensing with an optomechanical system at room temperature

We present an alternative approach to force sensing with optomechanical systems. The operation is based on a nonlinear dynamical mechanism, which locks the mechanical oscillation and the associated cavity field pattern of a system when two external drive tones satisfy a frequency match condition. Under a weak force that adds a slight detuning to the external driving fields, the cavity field will undergo a transition between two locked patterns while the locked mechanical oscillation is well preserved, thus having the small modifications to its sidebands. The force sensing is realized by detecting the intensity changes of the cavity field sidebands in such process. With the currently available optomechanical systems, the sensitivity of force detection can reach the level of attonewton and can be further improved with improved system parameters and longer detection time. One important feature of the applied dynamical scenario is that thermal noise insignificantly affects the cavity field sidebands of the locked states, so that the scheme can work well even at room temperature. This method is hopeful to reduce the difficulty in the practical applications of force sensing.

Figure 1

The setup of the force sensor. A laser beam with two frequency components of different frequencies drives an optical cavity with a movable mirror. When an external constant force is applied to this mirror, the size of the cavity will be slightly changed. Through the detection of the changed amplitude of the cavity sidebands, this external force can be measured. Here the real displacement is related to the dimensionless displacement $X_m$ as $x_m\left(t\right)=\sqrt{\hslash /\left(m{\omega }_m\right)}{X}_m\left(t\right)$, where $m$ is the effective mass of the moving mirror.

Figure 2

The stabilized cavity field intensity as the photon number ${\left|a\right|}^2$ and the stabilized pattern of the field quadrature $X_c$ in the time domain. While the cavity oscillation pattern is locked under the frequency match condition for two driving fields, the ${\left|a\right|}^2$ for the exemplary drive amplitudes $E={10}^6\kappa$ to $E=1.1\times {10}^6\kappa$ become similar. A frequency shift ${\delta }_F=0.1\kappa$ induced by the external force modifies the field spectrum to deviate the ${\left|a\right|}^2$ from the original ones. It is more obvious to see the modified quadrature $X_c$ (in the system rotating at the resonant cavity frequency) due to the shift ${\delta }_F$. The parameters are chosen as $g_m={10}^{-5}\kappa$, ${\omega }_m=10\kappa$, which are scaled with the field damping rate $\kappa$ and $Q={\omega }_m/{\gamma }_m={10}^4$.

Figure 3

The variations of the zeroth- and first-order sideband with the detuning induced by an external force. The system parameters are the same as those in Fig. 2, and the drive amplitude is kept to be $E=1.1\times {10}^6\kappa$.

Figure 4

The first-order cavity field sideband under two different detuning combinations. Here the different combinations $({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2$) indicate the frequencies of the two used driving fields and the sideband at the position ${\omega }_1$, which is defined in Eq. (10), is measured for both combinations. After the shift ${\delta }_F=0.1\kappa$, the sideband peak shifts from the red solid one to the blue dashed one, associated with a decrease of the amplitude (the green dash-dotted lines delineate the amplitude change). The parameters are chosen to be the same as those in Fig. 2 and the drive amplitude is set to be $E={10}^6\kappa$.

Figure 5

The relations between the sideband intensity changes and the induced shifts ${\delta }_F$ at different temperature. The mechanical quality factor is fixed to be $Q={10}^4$ and the sideband intensity difference is slightly modified by the increased temperature. The combination of the two driving frequencies is ${\mathrm{\Delta }}_1=-5\kappa$, ${\mathrm{\Delta }}_2=-15\kappa$, while the drive amplitude is $E=1.1\times {10}^6\kappa$. We here measure the sideband at ${\omega }_2$ defined in Eq. (10). The fixed parameters are the same as those in Fig. 2.

Figure 6

(a) The SNR versus the drive amplitude $E/\kappa$. The frequency shift is assumed to be ${\delta }_F=5\times {10}^{-3}\kappa$ and the difference in the drive amplitude between two neighboring points in the figure is fixed to be $2\times {10}^4\kappa$. These results indicate the SNR for three different combinations $({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2$), where $|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2|={\omega }_m$. For all these combinations we measure the sideband at the locations of ${\omega }_2$ defined in Eq. (10). (b) The SNR versus the frequency shift ${\delta }_F/\kappa$. The drive amplitude is fixed to be $E=1.1\times {10}^6\kappa$, and the difference of the frequency shift between two neighboring points in the figure is ${\delta }_F={10}^{-4}\kappa$. The SNR also increases linearly with the increasing of frequency shift (a logarithmic scale is used for the vertical axis). The fixed parameters are the same as those in Fig. 2.

Figure 7

The evolved intracavity photon number ${\left|a\right|}^2$ and the corresponding correction ${\delta \left|a\right|}^2$ by the thermal noise at the room temperature $T=300\mathrm{K}$. In the left frames, the red curve is for $T=0$ and the blue dashed one is for $T=300\mathrm{K}$. In (a) the system is driven by a single field detuned at $\mathrm{\Delta }/\kappa =-5$, while in (b) the system is driven by two fields with ${\mathrm{\Delta }}_1=-5\kappa$, ${\mathrm{\Delta }}_2=-15\kappa$, respectively. The mechanical quality factor is fixed to be $Q={10}^4$ and the drive amplitude is $E={10}^6\kappa$. The other fixed parameters are the same as those in Fig. 2.