Room temperature optomechanical squeezing

TL;DR

This paper demonstrates room-temperature optomechanical squeezing of light in the audio-frequency range, achieving sub-shot-noise quantum noise reduction, which is significant for enhancing precision measurements like gravitational wave detection.

Contribution

The work presents the first measurement of optomechanical squeezing at room temperature in a broad audio-frequency band relevant to gravitational wave detectors, using two independent measurement methods.

Findings

Achieved 0.7 dB quantum noise reduction below shot noise at 45 kHz.

Demonstrated room-temperature optomechanical squeezing in the 30-70 kHz range.

Used two independent measurement techniques, one calibration-free.

Abstract

The radiation-pressure driven interaction of a coherent light field with a mechanical oscillator induces correlations between the amplitude and phase quadratures of the light. These correlations result in squeezed light -- light with quantum noise lower than shot noise in some quadratures, and higher in others. Due to this lower quantum uncertainty, squeezed light can be used to improve the sensitivity of precision measurements. In particular, squeezed light sources based on nonlinear optical crystals are being used to improve the sensitivity of gravitational wave (GW) detectors. For optomechanical squeezers, thermally driven fluctuations of the mechanical oscillator's position makes it difficult to observe the quantum correlations at room temperature, and at low frequencies. Here we present a measurement of optomechanically (OM) squeezed light, performed at room-temperature, in a broad band near audio-frequency regions relevant to GW detectors. We observe sub-poissonian quantum noise in a frequency band of 30 kHz to 70 kHz with a maximum reduction of 0.7 $\pm$ 0.1 dB below shot noise at 45 kHz. We present two independent methods of measuring this squeezing, one of which does not rely on calibration of shot noise.

Figures

Tables

Table 1. Table 1 : Experimental parameters determined from measurements in the lab

${\mathrm{BS}}_1$ reflectivity	85 %
${\mathrm{BS}}_2$ reflectivity	96.5 %
Input coupler transmission	50 ppm
Cantilever mirror transmission	250 ppm
Cavity losses	250 $\pm$ 20 ppm
Cavity linewidth HWHM ($\gamma$)	650 kHz
Cavity detuning	0.33$\gamma$
Homodyne visibility	0.93
Intracavity power	260 $\pm$ 30 mW
Signal power $\left({\left|t{\overrightarrow{E}}_S\right|}^2\right)$	58 $\pm$ 4 µW
LO power $\left({\left|r{\overrightarrow{E}}_{\mathrm{LO}}\right|}^2\right)$	0 - 30 $\pm$ 3 µW
Detected power $\left({\left|E_{\mathrm{sqz}}\right|}^2\right)$	49 $\pm$ 3 µW
Detection inefficiency and extra losses	21 $\pm$ 8 %

Equations

$$a_1=\frac{e^{-r}{x}_1+c-y_1}{\sqrt{2}}$$

$$b_1=\frac{e^{-r}{x}_1+c+y_1}{\sqrt{2}},$$

$$⟨S_{ab}⟩=\frac{1}{2}\left(e^{-2r}+S_c-1\right)\alpha \beta ,$$

$$\frac{S_a}{{\alpha }^2}=\frac{S_b}{{\beta }^2}=\frac{e^{-2r}+S_c+1}{2}.$$

$$C=\frac{⟨S_{ab}⟩}{\sqrt{S_a{S}_b}}=\frac{e^{-2r}+S_c-1}{e^{-2r}+S_c+1}.$$

$$C=\frac{⟨S_{ab}⟩}{\sqrt{S_a{S}_b}}=\frac{R-1}{R+1}.$$

$$R=\frac{1+C}{1-C}.$$

$$S_a={\alpha }^2\frac{R+1}{2}+{\alpha }^2{S}_{da}$$

$$S_b={\beta }^2\frac{R+1}{2}+{\beta }^2{S}_{db},$$

$$C$$

$$=\frac{R-1}{R+1}{\left[\sqrt{1+\frac{S_{da}}{1+R}}\sqrt{1+\frac{S_{db}}{1+R}}\right]}^{-1/2}$$

$$=\eta \frac{R-1}{R+1},$$

$${\vec{E}}_{\mathrm{sqz}}=t{\vec{E}}_{\mathrm{S}}+r\mathbb{R}(\theta ){\vec{E}}_{\mathrm{LO}}$$

$$\tan {\varphi }_{\mathrm{S}}=\frac{r{E}_{\mathrm{LO}}\sin \theta }{r{E}_{\mathrm{LO}}\cos \theta +t{E}_{\mathrm{S}}}$$

$$\tan {\varphi }_{\mathrm{LO}}=\frac{-t{E}_{\mathrm{S}}\sin \theta }{r{E}_{\mathrm{LO}}+t{E}_{\mathrm{S}}\cos \theta }$$

$${\vec{E}}_{\mathrm{sqz}}=|{\vec{E}}_{\mathrm{sqz}}|\vec{U}({\varphi }_{\mathrm{S}})$$

$${\vec{e}}_{\mathrm{sqz}}=$$

$${\vec{E}}_{\mathrm{sqz}}.{\vec{e}}_{\mathrm{sqz}}=$$

$$S_{\mathrm{sqz}}({\varphi }_{\mathrm{S}})=$$

$$=$$

$$r^2\mathbb{R}(\theta ){\vec{e}}_{\mathrm{LO}}{\vec{e}}_{\mathrm{LO}}^{†}\mathbb{R}(\theta {)}^{†}).\vec{U}({\varphi }_{\mathrm{S}})$$

$$=$$

Full text

I Supplementary Material

I.1 Application to precision measurements

In addition to being all-wavelength and compact, an OM squeezer innately generates a bright squeezed field. This means an OM squeezed field comes with its own internal phase reference, hence eliminating the need for an extra coherent phase locking field (CLF) [36, 5]. For instance, if the state after $\mathrm{BS_{2}}$ was sent to a precision measurement setup, it would include the carrier field which provides a self-referenced signal to allow locking to the correct squeezing quadrature. In contrast, vacuum squeezed states generated by a nonlinear crystal have to be accompanied by an extra, usually frequency shifted optical beam that keeps track of the squeezing quadrature [36, 5].

While the current amount of squeezing in our setup is limited by extraneous technical noises, in the absence of those noises, the squeezing would be limited by intracavity losses. This expected squeezing in the absence of extraneous noise is shown in Fig. 5. After elimination of differential phase noise between the LO and cavity, the squeezing would extend to much lower frequencies. The differential phase noise injected into the system can be reduced by picking-off and recombining the LO inside the vacuum where the OM cavity is situated. The feedback noise can be subtracted by monitoring the actuator signal. A more optimized feedback scheme could also be implemented that injects lower noise, e.g. a feedback tuned for the measured quadrature’s transfer function with respect to an appropriate input quadrature that would couple minimum noise to the squeezing signal [28].

I.2 Noise budget

The measurement is also compared to a noise budget, shown in the bottom panel of Fig. 3. The total budgeted noise shown is the quadrature sum of individual contributions from measured thermal noise, quantum noise, classical laser noises, cavity-feedback noise, and differential phase noise between the signal and LO. It uses experimentally measured cavity parameters, thermal noise, $\mathrm{BS_{1}}$ and $\mathrm{BS}_{2}$ reflectivities and homodyne visibility, listed in Table 1. The quadratures for which squeezing is obtained depend on the various OM parameters of the cavity, such as the detuning, circulating power, losses, as well as the thermal noise. We measure the cavity detuning, intracavity power and losses by measuring transfer functions from amplitude modulations on input to transmitted light. The thermal noise is measured by a cross-spectrum measurement in the amplitude quadrature without the local oscillator [37]. We have also separately calibrated all the beam splitters, the mirror reflectivities, and the homodyne visibility. These measured quantities are then used to predict the squeezing using a numerical model based on Ref. [30]. In this model, we also include the effect of the unbalanced homodyne with an imperfect visibility and common-mode laser noises.

We then characterize the impact of technical noises by measuring their contribution. First, we measure the contribution of noise injected by the cavity locking system. The dominant source of this noise is shot noise at $\mathrm{PD}_{\rm lock}$ due to 15% transmission of $\mathrm{BS_{1}}$. In order to measure this feedback noise, we measure the coherence between $\mathrm{PD}_{\rm sqz}$ and the amplifier output that is fed to the phase modulator at input. This coherence when multiplied with the spectrum of $\mathrm{PD}_{\rm sqz}$ gives us the contribution of feedback noise. We do this at all measurement quadratures independently. We find that the impact of feedback noise is minimized at 17 degrees, akin to other intracavity displacement noises like thermal noise.

In principle, this cavity-feedback noise could be subtracted from the final result, as it is a measured quantity, but we choose not to do so for the sake of simplicity. Instead, we chose to pick-off the LO beam just after the cavity-feedback phase modulator, so that there is common mode rejection of this locking loop phase noise at the homodyning stage at $\mathrm{BS}_{2}$. The common mode rejection by the homodyne detection also allows us to cancel frequency noise originating from the NPRO laser, without requiring a frequency stabilization servo. Any scheme to measure squeezing not purely in the amplitude quadrature requires mixing the signal beam with an LO that is phase coherent with it, and so one always has the ability to reject common mode noise in this fashion. Also note that there is no risk of generating apparent squeezing after $\mathrm{BS}_{2}$ by deriving the LO from the cavity locking field (e.g. from feedback-squashing of the in-loop field), because the LO and signal fields are both out-of-loop [29].

Additionally, displacement fluctuations that are relative between the LO and the signal path cause an effective phase noise in the measurement. We refer to this as the differential phase noise, and we measure it by analyzing the measured noise at 17°. At this quadrature, all displacement noises including the feedback noise are canceled, and the quantum noise contribution is at the shot noise level. So we attribute all noise above shot noise at 17 degrees to this relative phase noise. We calculate the contribution of phase noise in all other quadratures by assuming that it is maximum at 90°quadrature and scaling it sinusoidally.

Finally, we are left with excess loss in the detection path. We fit this loss by adding a frequency and quadrature independent loss to the noise budget. We find an excess loss of 22 $\pm$ 1%, which agrees with the measured loss of 21 $\pm$ 8%. Note that optical loss is the only effect where a single scalar would be sufficient to explain the measurement over all quadratures and frequencies. All of the above contributions to the noise budget are shown in Fig. 6: as a function of measurement quadrature in Fig. 6(a), and as a function of frequency in the squeezing quadrature in Fig. 6(b).

I.3 Correlations

Consider splitting an intensity squeezed beam onto two photodetectors. For convenience, let’s split it as 50%. The amplitude quadrature of the two fields hitting the photodetectors may then be written as

[TABLE]

where $x_{1}$ is the vacuum that has been squeezed by the factor $e^{-r}$, $c$ represents any classical noise that might be present, and $y_{1}$ is the vacuum that enters at the beamsplitter.

If we measure the the averaged cross power spectrum of the resulting photocurrents, but don’t take the absolute value, we find

[TABLE]

where we have normalized shot noise to 1, and assumed detector a has a relative gain of $\alpha$, and detector b has a relative gain $\beta$, and $S_{c}$ is the power spectrum of the classical noise scaled to shot noise. All the cross terms between $x_{1}$, $y_{1}$ and $c$ will average to 0, as they are uncorrelated. If the original field is squeezed, then that requires $e^{-2r}+S_{c}<1$, which would then imply $\left<S_{ab}\right><0$. Note that if this is not satisfied, such that we have classical noise that destroys the squeezing, then $e^{-2r}+S_{c}>1$, which requires $\left<S_{ab}\right>>0$. Therefore, by looking at the sign of the average cross power spectrum, one can definitively prove whether squeezing is present or not.

To interpret this, when the beam is limited by classical noise, the power fluctuations hitting both PDs are identical and positively correlated. If the beam is exactly shot noise limited, the power fluctuations hitting the two PDs are uncorrelated. With a perfectly amplitude squeezed beam, the power fluctuations are exactly anti-correlated.

We may write the individual power spectra as

[TABLE]

Then define the normalized correlation as

[TABLE]

This is convenient because it supplies a unitless measure of the nature of the noise, and is independent of the relative gain of the photodetectors. This $C$ is similar to the square root of coherence, but retains phase information. In fact, the coherence may be written as $CC^{*}$.

We can see that if the field is entirely classical so that $S_{c}$ dominates, then $C=+1$. Likewise if the beam is exactly shot noise limited without squeezing, then $C=0$. Finally, for an infinitely squeezed field with no classical noise, $C=-1$.

To simplify, let’s call the total noise PSD of the original beam relative to shot noise $R=e^{-2r}+S_{c}$, in which case

[TABLE]

This leads to

[TABLE]

Thus, by measuring $C$, we have a method to measure the amount of noise relative to shot noise independent of our ability to calibrate shot noise.

This treatment is simplified by not propagating the DC carriers of the fields. The final physical result becomes invariant of the beam splitter convention if one keeps track of the DC carrier fields. The cross spectrum $\left<S_{ab}\right>$ is negative for a squeezing beam $x$, irrespective of the beam splitter convention, as long as the carrier of the field $y$ is smaller than the carrier of the field $x$. This condition is trivially satisfied in our measurement, since $y$ is coming from vacuum fluctuations, with a zero DC field.

To include the effects of uncorrelated electronics noise on the photodetectors, we may rewrite the power spectrums for each detector as

[TABLE]

where $S_{da}$ and $S_{db}$ are the PSDs of each detector from electronics noise. The resulting normalized correlation is

[TABLE]

where $\eta\leq 1$ is an effective efficiency of the measurement. In our experiment, since the dark noise is far below shot noise (Fig. 7), the efficiency $\eta$ is close to 1. If instead one had a lower efficiency, we can see from Eq. 10 that electronics noise will make observed correlations trend towards 0, and the inferred quantum noise level to shot noise.

I.4 Tunable homodyne detector

In our experiment, we opt to use a single-photodiode homodyne detection. Instead of combining the signal beam with the local oscillator (LO) on a 50-50 beam splitter, we combine it on a highly asymmetric beam splitter. We measure on the output port which transmits the larger signal fraction and reflects the smaller fraction of the LO. While this scheme introduces some loss of signal, it works with just a single photodetector and eliminates the need for performing perfect subtraction that is needed in a balanced homodyne. Since the amount of squeezing expected in this experiment is relatively low, the reduction in squeezing due to this beam splitter loss is small.

We show below that the signal quadrature in which the measurement is performed is given by the angle made by the resultant of vector addition of the carrier of signal and LO with respect to the signal, as displayed in Fig. 8. Similarly, the LO quadrature measured is given by the angle this resultant makes with the LO.

[TABLE]

Here $\vec{E}$ represents the strength of the carrier of the signal(S), LO and the resultant (sqz). $t$ is amplitude transmitivity ($\sqrt{0.965}$) and $r$ is amplitude reflectivity ($\sqrt{0.035}$) of the beam splitter. We define a unit vector $\vec{U}(\phi_{\rm S})$ which represents a vector in the direction of the resultant, and determines the measured quadrature. Using [30], we can also calculate the loss effect of the beam splitter. We define $\vec{e}$ as the fluctuations on the field, normalized such that the shot noise is $|\vec{E}|^{2}$ [28]. We then propagate these fluctuations from the signal and the LO as they interfere on the beam splitter :

[TABLE]

where we have assumed that the LO is shot noise limited, and defined $S_{\mathrm{S}}(\phi_{\rm S})=\vec{U}(\phi_{\rm S})^{\dagger}\cdot\vec{e}_{\rm S}\vec{e}^{\dagger}_{\rm S}\cdot\vec{U}(\phi_{\rm S})$ as the spectral density of the signal if measured perfectly with a balanced homodyne detector at the quadrature $\phi_{\rm S}$. The above equations show that the total spectral density measured, $S_{\mathrm{sqz}}(\phi_{\rm S})$ is a combination of $S_{\mathrm{S}}(\phi_{\rm S})$ and 1, in the ratio of the beam splitter’s reflectivity.