Classical demonstration of frequency-dependent noise ellipse rotation using optomechanically induced transparency

Cavities with an extremely narrow linewidth of $10–100$ Hz are required for realizing frequency-dependent squeezing to enable gravitational wave detectors to surpass the free mass standard quantum limit over a broad frequency range. High-finesse cavities on the scale of tens of meters have been proposed for this purpose. Optomechanically induced transparency (OMIT) enables the creation of optomechanical cavities in which the linewidth limit is set by the extremely narrow linewidth of a high-$Q$-factor mechanical resonator. Using an 85-mm OMIT cavity with a silicon nitride membrane, we demonstrate a tunable linewidth from 3 Hz up to several hundred hertz and frequency-dependent noise ellipse rotation using classical light with squeezed added noise to simulate quantum squeezed light. The frequency-dependent noise ellipse angle is rotated in close agreement with predictions.

Figure 1

Configuration schematics (left) and frequency relationships. The signal light with squeezed added noise having frequency $\omega$ with respect to the cavity resonance ${\omega }_c$ is injected into an optical cavity with a high-$Q$-factor membrane in the middle that acts as an oscillator at the resonant frequency ${\omega }_m$. The position of the membrane is chosen to introduce a linear optomechanical coupling. The radiation pressure from the beating between the signal light ${\omega }_s={\omega }_p+\Omega$ and the control laser ${\omega }_p$ coherently drives the mechanical oscillator, which in turns creates sidebands destructively interfering with the signal light, which in effect rotates the noise ellipse angle.

Figure 2

Experimental setup. The 85-mm-long cavity sits in a vacuum chamber with a central silicon nitride membrane oscillator (1 mm $\times$ 1 mm $\times$ 50 nm, effective mass 40 ng). The green line represents the locking light for stabilizing the laser frequency to the cavity resonance using the Pound-Drever-Hall method [25]. The blue line represents the control light, with polarization orthogonal to the locking light. The broadband electro-optic modulator (BEOM) generates an upper sideband from the control light, which is our signal light (red line). The $\Delta \sim 400$ kHz for the control light was created using a pair of 80-MHz acousto-optic modulators (AOMs) in the locking path. PD-L, PD-R and PD-T: photo-detectors; HV: High-Voltage; FG: function generator; PBS: polarized beam splitter; FR: faraday rotator; Lock-in: Lock-in technique for signal readout.

Figure 3

Detected OMIT transmissivity. (a) Normalized transmissivity amplitude $|t_n\left(\Omega \right)|$ vs frequency offset $\Delta f$ (Hz), where $\Delta f\equiv \Omega /2\pi -402.5$ kHz. The control light powers were 0.5, 1.3, 2.7, and 4.0 mW, respectively. (b) Transmissivity amplitude vs frequency difference $\Omega /2\pi$ (kHz) in a span of 200 kHz. The coupled cavity full linewidth was 170 kHz in this measurement. (c) Normalized transmissivity amplitude dip depth value $|1-t_n(\Omega ={\omega }_m)|$ vs the control light input power. (d) The OMIT cavity full linewidth vs the control light input power. The full linewidth data correspond to the Lorentzian transmissivity of the OMIT cavity. In this measurement, the mechanical resonance frequency was ${\omega }_m=2\pi \times 402.7$ kHz.

Figure 4

(a) The OMIT cavity transmissivity amplitude $\left|t\right(\Omega \left)\right|$, (b) phase $\varphi \left(\Omega \right)$, and (c) rotation angle $\theta \left(\Omega \right)$ of the noise ellipse as a function of $\Delta f$ (kHz). The frequency offset is defined as $\Delta f\equiv \Omega /2\pi -390$ kHz. In this measurement, the mechanical resonance frequency was ${\omega }_m=2\pi \times 394.6$ kHz. (d) Contour plots of the normalized noise ellipse in the phasor diagram at different frequency offsets $\Delta f$ corresponding to (c). In each phasor diagram, the horizontal axis is the amplitude quadrature and the vertical axis is the phase quadrature. The frequency offset of the OMIT cavity resonance is 4.6 kHz.