Enhanced optomechanical interaction in the unbalanced interferometer

TL;DR

The paper proposes a modification to the Michelson-Sagnac interferometer by introducing an imbalance at the central beam-splitter to balance dissipative and dispersive optomechanical couplings. This unbalance, quantified by $ε$, enhances the dispersive coupling $ξ$ while tuning the dissipative coupling $η$, enabling strong optical rigidity and parametric cooling even at cavity resonance. The analysis addresses two configurations, power-recycling and signal-recycling, and derives optimal imbalance $ε_{opt}$ to maximize cooling and sensitivity, demonstrating potential to observe QRPN and ponderomotive squeezing in table-top setups with heavy test masses. The results offer a versatile route to bring macroscopic mechanical systems toward the quantum regime, with implications for fundamental tests of quantum mechanics and quantum gravity. The approach provides a platform for future detuned, recycled, or squeezed-light-enhanced implementations that can surpass the standard quantum limit in large-scale sensors.

Abstract

Quantum optomechanical systems enable the study of fundamental questions on quantum nature of massive objects. For that a strong coupling between light and mechanical motion is required, which presents a challenge for massive objects. In particular large interferometric sensors with low frequency oscillators are difficult to bring into quantum regime. Here we propose a modification of the Michelson-Sagnac interferometer, which allows to boost the optomechanical coupling strength. This is done by unbalancing the central beam-splitter of the interferometer, allowing to balance two types of optomechanical coupling present in the system: dissipative and dispersive. We analyse two different configurations, when the optomechanical cavity is formed by the mirror for the laser pump field (power-recycling), and by the mirror for the signal field (signal-recycling). We show that the imbalance of the beam splitter allows to dramatically increase the optical cooling of the test mass motion. We also formulate the conditions for observing quantum radiation-pressure noise and ponderomotive squeezing. Our configuration can serve as the basis for more complex modifications of the interferometer that would utilize the enhanced coupling strength. This will allow to efficiently reach quantum state of large test masses, opening the way to studying fundamental aspects of quantum mechanics and experimental search for quantum gravity.

Figures (5)

• Figure 1: SRM cavity (left) and PRM cavity (right). Michelson-Sagnac interferometer with movable mirror $M$ (which is a mass of probe oscillator) is a generalized input mirror (GM) of FP cavity. Central beam splitter can be imbalanced and has a the amplitude reflection and transmission coefficients $r_{BS}, t_{BS}$. Partially transmissive movable mirror (M) moves under the effect of thermal and radiation-pressure noise. Its average displacement $x_0$ from the center of the interferometer influenced the balance between the dissipative and dispersive optomehcanical coupling. Output wave $C_1$ contains information about the motion of the test mirror and can be read out with balanced homodyne detection.
• Figure 2: Cooling of a micro-mechanical membrane in the MSI. The plot shows the dependence of the thermal photons number $n_T$\ref{['Teff']} of a mechanical oscillator on the MSI bandwidth $\gamma_0$ for SRM and PRM schemes at $\epsilon=0$ (symmetric beam splitter) and $\epsilon_{opt}$ (optimally asymmetric beam splitter). To construct the dependencies, we used the following values of the system parameters: the bandwidth of the SRM (PRM) $\gamma_1/2\pi = 10^6 Hz$, the reflection coefficient of the membrane $r_m^2 = 0.5$ for $\text{SRM}_0, \text{PRM}_0$ and $r_m^2 = 0.98$ for $\text{SRM}_{\text{opt}}, \text{PRM}_{\text{opt}}$. The values of other parameters are taken from the Table \ref{['Table']}.
• Figure 3: Cooling of a micro-mechanical membrane in the MSI. The plot shows the dependence of the thermal photons number $n_T$\ref{['Teff']} of a mechanical oscillator on the MSI bandwidth $\gamma_0$ for SRM and PRM schemes at $\epsilon=0$ (symmetric beam splitter) and $\epsilon=0.15$. To construct the dependencies, we used the following values of the system parameters: Input power $P_{in} = 1 W$, Quality factor is $10^7$, the bandwidth of the SRM (PRM) $\gamma_1/2\pi = 10^6 Hz$, the reflection coefficient of the membrane $r_m^2 = 0.5$ for $\text{SRM}_0, \text{PRM}_0$ and $r_m^2 = 0.98$ for $\text{SRM}_{\text{opt}}, \text{PRM}_{\text{opt}}$, 6dB squeezing of the field $C$ quadrature (for a scheme with SRM it is \ref{['SRMct2b']}, and for a scheme with PRM it is \ref{['PRMct2b']}). The values of other parameters are taken from the Table \ref{['Table']}.
• Figure 4: Spectral density normalized to shot noise of optical and thermal fluctuations when measuring the phase quadrature of the output field $C_1$ at $\epsilon=\epsilon_{max}$\ref{['emax']}. Thermal noise $S_{\rm th}$ dominates the sensitivity for the quality factor of the mechanical oscillator equal to $10^6$. QRPN $S_{CC}$ caused by fluctuations of the field C is just below the thermal noise, and the contribution of laser fluctuations $S_{BB}$ is significantly lower (field $B$ is assumed to be a coherent state). By introducing 10 dB anti-squeezing in the field C, QRPN $S_{CC}^{\rm sqz}$can be increased above the thermal noise level and thus become observable. Alternatively, the quality factor of the mechanical oscillator could be increased by a factor of 10 to reach the same sensitivity. Here we assumed the bandwidth of the SRM $\gamma_1/2\pi = 10^6 Hz$ and the MSI $\gamma_0/2\pi = 10^5 Hz$ , the reflection coefficient of the membrane $r_m^2 = 0.98$, see Table \ref{['Table']} for further details.
• Figure 5: Demonstration of ponderomotive squeezing for different quality of mechanical oscillator. The power spectral densities are normalized to shot noise of a quadrature $c_{1\theta}$\ref{['SRMct']}, \ref{['PRMct']} of the output field $C_1$ for SRM (top) and PRM (bottom) schemes. $S_{out}$ is the total noise, including the contribution of thermal noise, as well as the fluctuations of the laser field, and $S_{CC}$ is a contribution of the fluctuation of the field C. Thermal noise prevents observation of strong squeezing, and thus has to be reduced, e.g. by increasing the quality factor of the oscillator. The higher it is, the higher is ponderomotive squeezing, due to reduced contribution of thermal noise. For these plots we used the following parameters: (SRM, top) the bandwidth of the SRM $\gamma_1/2\pi = 10^6 Hz$ and the MSI $\gamma_0/2\pi = 3*10^5 Hz$; (PRM, bottom) the bandwidth of the PRM $\gamma_1/2\pi = 3*10^5 Hz$ and the MSI $\gamma_0/2\pi = 10^6 Hz$. The beam splitter asymmetry coefficient $\epsilon=\epsilon_{opt}$\ref{['coolingmax']}, the reflection coefficient of the membrane $r_m^2 = 0.98$ and the values of other parameters are taken from the Table \ref{['Table']}.