Open Access
Add to BibSonomy
Abstract
Application of frequency-dependent squeezed vacuum improves the force sensitivity of an optomechanical interferometer beyond the standard quantum limit by a factor of e−r, where r is the squeezing parameter. In this work, we show that the application of squeezed light along with quantum back-action nullifying meter in an optomechanical cavity with mechanical mirror in middle configuration can enhance the sensitivity beyond the standard quantum limit by a factor of e−reff, where reff = r + ln(4Δ/ζ)/2, for 0 < ζ/Δ < 1, with ζ as the optomechanical cavity decay rate and Δ as the detuning between cavity eigenfrequency and driving field. The technique described in this work is restricted to frequencies much smaller than the resonance frequency of the mechanical mirror. We further studied the sensitivity as a function of temperature, mechanical mirror reflectivity, and input laser power.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The quest to detect gravitational waves [1–3] has revolutionized precision measurements using an optical interferometer. The laser interferometer gravitational wave detector is based on the coupling of optical modes with mechanical modes, which is known as optomechanics [4–7]. With the miniaturization [8–10] of mechanical mirrors, optomechanics has emerged as one of the best physical systems to design ultra-precise sensors [11–14]. Such a sensor is usually designed by embedding optomechanical cavities into the arms of an optical interferometer [15,16].
Shot noise and radiation pressure noise (RPN) [17–20] are two major noises in optomechanics. Shot noise arises from the randomness in the photon counting, while the RPN arises because of the randomness in the radiation pressure force exerted on the mechanical mirror. The effect of shot noise can be decreased by increasing the laser power, however, this leads to an increase in RPN. This trade-off between shot noise and RPN imposes standard quantum limit (SQL) [21,22]. Several techniques [23–34] were developed to overcome SQL. One of the most popular methods is to use squeezed light [35–44]. The Squeezed light [45–50] is a special quantum state in which the uncertainty in one quadrature is decreased at the expense of increased uncertainty in the other. Frequency-dependent squeezing [51–53] can improve the force sensitivity of optomechanical interferometer beyond SQL by a factor [54] of $e^{-r}$ with $r$ being the squeezing parameter. But it is experimentally difficult to synthesize squeezed state with large $r$ value. To our knowledge, the highest squeezing reported experimentally to date is $15$ dB [55]. In this article, we propose a method to improve the squeezed light optomechanical interferometer sensitivity beyond SQL by a factor of ${e^{-r_{eff}}}$, for $\zeta /\Delta <1$, here $\zeta$ is the cavity decay rate and $\Delta$ is detuning between cavity eigenfrequency and driving field. We are able to enhance the squeezed light optomechanical interferometer sensitivity factor from $e^{-r}$ to $e^{-r_{eff}}$ by combining quantum back-action nullifying meter [56] (QBNM) technique with vacuum squeezing technique in an optomechanical cavity with membrane in middle. Owing to the difficulty associated with synthesizing squeezed state with large $r$, the technique described in this manuscript provides an alternate way to improve squeezed light optomechanical interferometer sensitivity.
2. Model
Consider an optomechanical cavity with a perfectly reflective mechanical mirror in the middle [57] as shown in Fig. 1. The mechanical mirror divides the total cavity into two sub-cavities (sub-cavity-a and sub-cavity-c), each with length $l$ and eigenfrequency $\omega _{e}$. The annihilation operators for optical fields inside the sub-cavities are given by $\hat {a}$ and $\hat {c}$ as shown in Fig. 1. There is no tunnelling of $\hat {a}$ into $\hat {c}$ and vice-versa as the mechanical mirror is perfectly reflective. A co-sinusoidal classical force $f\cos (\omega _{f} t)$, with $\omega _{f}$ as frequency and $t$ as time, changes the position $\hat {z}$ of the mechanical mirror. The total Hamiltonian $\hat {H}$ of the optomechanical cavity [58] is given as
Fig. 1. Interferometer with a membrane in the middle. The optomechanical membrane is perfectly reflective. The optical fields in sub-cavity-a and sub-cavity-c are synthesized such that optical restoring force counters the fluctuations induced by radiation pressure force.
There is no $f\cos (\omega _{f}t)$ term in Eq. (6) as it is treated like a small perturbation and is included in the fluctuations. The $\bar {z}$ in $\bar {a}$ and $\bar {c}$ leads to optomechanical bi-stability [62] which can be avoided by choosing $|\bar {a}|^{2}=|\bar {c}|^{2}$. Then the average radiation pressure force on the mechanical mirror from both the sub-cavities is equal but opposite in direction, and $\bar {z}$ is zero. We assume the beam-splitters in Fig. 1 are 50:50. The input fields $\hat {b}$ and $\hat {d}$ are phase adjusted such that, $\hat {b}=\left (\hat {E}+i\hat {V}\right )e^{-i\phi }/\sqrt {2}$ and $\hat {d}=\left (\hat {F}+i\hat {U}\right )e^{-i\phi }/\sqrt {2}$, where $\phi =\tan ^{-1}\left (-2\Delta /\zeta \right )$, $\hat {E}$ and $\hat {F}$ are the laser field annihilation operators while $\hat {V}$ and $\hat {U}$ are the vacuum field annihilation operators. Then the steady state cavity fields $\bar {a}=\sqrt {\zeta }\bar {E}/\sqrt {2\left (\Delta ^{2}+\zeta ^{2}/{4}\right )}$ and $\bar {c}=\sqrt {\zeta }\bar {F}/{\sqrt {2\left (\Delta ^{2}+\zeta ^{2}/{4}\right )}}$ can be set to be real by taking $\bar {E}$ and $\bar {F}$ as real, where $\bar {E}$ and $\bar {F}$ are the mean values of $\hat {E}$ and $\hat {F}$, respectively. By using Eq. (5) and Eq. (6), we can write
We assume $\bar {E}=\bar {F}$ while writing Eq. (7). As there are no external losses in the sub-cavities, Eq. (7) imply that the average optical power of the output field and input field are equal. As $\bar {z}=0$, Eq. (7) is not influenced by optomechanical interaction but the $\phi$ appears because of the input field detuning from the cavity resonance. The equations of motion for fluctuations are given as
Note that we used the relation $\bar {a}=\bar {a}^{*}=\bar {c}$ in writing Eq. (8) to Eq. (10). The superfix symbols $``*"$ and $``\dagger "$ represent complex conjugate and adjoint operations, respectively. It is useful to write Eq. (8) to Eq. (10) in a simplified version as
We have used the relation $\bar {B}= \bar {D}$ in writing Eq. (16). The fluctuations $\hat {\delta }_{R_{1}}$ and $\hat {\delta }_{r_{1}}$ in the reference fields are given as $\hat {\delta }_{R_{1}}(\omega )= H[i\hat {\delta }_{E}(\omega )+\hat {\delta }_{V}(\omega )]/\sqrt {2}$, $\hat {\delta }_{r_{1}}(\omega )=H[i\hat {\delta }_{F}(\omega )+\hat {\delta }_{U}(\omega )]/\sqrt {2}$. The cavities in arm-a and arm-b have rigidly fixed mirrors and they do not have any external losses. Hence the steady state reference fields are given as $\bar {R}_{1}=\bar {r}_{1}=i\bar {E}/\sqrt {2}$ (because $\bar {E}=\bar {F}$). The noise spectral density $S_{QQ}$ is given by Eq. (16) according to the relation $\langle [\hat {\delta }_{Q}(\omega )]^{\dagger }\hat {\delta }_{Q}(\omega _{1})\rangle =S_{QQ}(\omega )\delta (\omega +\omega _{1})$.
The action of $f \cos (\omega _{f} t)$ changes the equilibrium position of the mechanical mirror leading to signal $\bar {Q}$ as
As the classical force $f\cos (\omega _{f}t)$ drives the mechanical mirror at frequency $\omega _{f}$, the $\bar {Q}$ is also oscillating at the same frequency. Hence the force sensitivity $F_{s}$ at $\omega _{f}$ is given as
3. Results
The objective of this article is not only to break SQL but also to go beyond the squeezed light optomechanical interferometer sensitivity limit. To achieve this objective, we combine QBNM technique with vacuum squeezing technique as described below.
First, we establish different varieties of noises and how they influence the force sensitivity given in Eq. (18). For this, we set $\Delta =0$, $\omega =\omega _{f}$ in Eq. (18) and estimate the force sensitivity $F_{o}$ as
The first term on the right-hand side (RHS) of Eq. (19) gives the shot noise, while the second and third term gives the RPN and mechanical mirror noise, respectively. In Eq. (19), the mechanical mirror is assumed to be in its quantum mechanical ground state and hence the mechanical mirror noise arises from the zero point fluctuations. The contributions from the shot noise and the RPN compete in Eq. (19) leading to SQL at an input intensity $I_{opt}$. Using Eq. (19), the $I_{opt}$ can be calculated as
A prominent property of Eq. (19) is its dependence on $|\bar {E}|^{2}$. With increase of $|\bar {E}|^{2}$ above $I_{opt}$, the shot noise contribution decreases but the RPN contribution increases. Similarly with decrease of $|\bar {E}|^{2}$ below $I_{opt}$, the RPN contribution decreases but the shot noise contribution increases. Hence in Eq. (19), for best sensitivity, we must set $|\bar {E}|^{2}=I_{opt}$ which enforces SQL. Substituting Eq. (20) into Eq. (19) gives the force sensitivity as $F_{1}=\sqrt [4]{2}F_{sql}$, where $F_{sql}=\sqrt {\hbar m \omega _{m}^{2}}$.
Eq. (19) establishes the presence of shot noise, RPN, and mechanical mirror noise. According to the QBNM technique, the RPN can be suppressed in the low frequency regime ($\omega \ll \omega _{m}$) by imposing the condition $\Delta =\mathcal {R}(\alpha )$, where $\mathcal {R}(\alpha )$ is real part of $\alpha$. The effectiveness of this condition can be understood by realising that $\alpha$ is the only variable with optomechanical coupling $g$ in Eq. (15). Hence any contribution to RPN must come from $\alpha$. Thus setting $\Delta -\alpha =0$ completely eliminates the RPN, but that is impossible as $\Delta$ is real while $\alpha$ is complex.
There is no RPN in Eq. (22) as it is suppressed by setting $\Delta =\mathcal {R}({\alpha })$ and $\gamma \omega _{f}/\omega _{m}^{2}\ll 1$. We simplified Eq. (22) by assuming that $1>\zeta /\Delta >\zeta ^{2}/\Delta ^{2}\gg \gamma \omega _{f}/\omega _{m}^{2}$. The condition that $\zeta /\Delta$ should lie between 1 and $\epsilon$ is not necessary for RPN suppression but required for improving force sensitivity beyond SQL. The first term on the RHS of Eq. (22) gives shot noise contribution while the second term gives the mechanical mirror noise contribution. The input intensity $|\bar {E}|^{2}$ in Eq. (22) is constrained by the condition $\Delta =\mathcal {R}(\alpha )$ as
Substituting Eq. (23) in Eq. (22) gives the best force sensitivity $F_{2}$ achievable as
As $\gamma \omega _{f}/\omega _{m}^{2}\ll \zeta /4\Delta$, $F_{2}$ is better than $F_{sql}$ by a factor of $\sqrt {\zeta /4\Delta }$. The intensity in Eq. (23) is larger than $I_{opt}$ by a factor of $(4\Delta ^{2}/\zeta ^{2}+1)2\sqrt {2}\Delta /\zeta$. With suppression of RPN, we are able to increase the intensity beyond $I_{opt}$. However the signal in Eq. (17) is reduced by a factor of $1/\sqrt {4\Delta ^{2}/\zeta ^{2}+1}$. Combining these two factors, we observe an improvement by a factor of $\sqrt {\zeta /4\Delta }$ beyond $F_{sql}$.
A plot of Eq. (18) as a function of input laser power, under the condition $1>\zeta /\Delta \gg \gamma \omega _{f}/\omega _{m}^{2}$ and $\Delta \simeq \mathcal {R}(\alpha )$ is shown in Fig. 2. The plotting parameters are chosen from the experimental work Ref. [63]. Hence the best force sensitivity $F_{2}$ in Fig. 2 is improved beyond $F_{1}$. The best force sensitivity in Fig. 2 is equal to $F_{2}$ which is given in Eq. (24).
Fig. 2. Variation of force sensitivity as a function of optical power. RPN is suppressed by setting $\mathcal {R}(\alpha )\simeq \Delta$. The lowest point of the curve gives the best force sensitivity $F_{2}$ ($3.28\times 10^{-19}$N/$\sqrt {\mbox {Hz}}$) at optical power $58.5$ mW. The $F_{2}$ represents the force sensitivity is improved by a factor of $\sqrt {\zeta /4\Delta }$ over $F_{sql}$. The simulation parameters are : $m=1.1\times 10^{-10}$Kg, $\omega _{m}/2\pi =9.7\times 10^{3}$Hz, $\omega _{f}=100$Hz, $\omega _{l}/2\pi =2.8\times 10^{14}$Hz, $\Delta =100\zeta$, $\zeta /2\pi =4.7\times 10^{5}$Hz, $\gamma /2\pi =1.3\times 10^{-2}$Hz, $g/2\pi =7.8\times 10^{15}$Hz/m. The optical power corresponding to the lowest point in Fig. 2 is given by $\hbar \omega _{l}|\bar {E}|^{2}$, where $|\bar {E}|^{2}$ is given by Eq. (23).
3.1 Squeezing
A squeezed light is a non-classical state [64,65] which allows reducing uncertainty in one quadrature at the expense of increased uncertainty in the corresponding conjugate quadrature [66,67]. On the other hand, the competitive behaviour between shot noise and RPN in optomechanics arises because of interplay between canonically conjugate quadratures. Hence by eliminating RPN from Eq. (16), we eliminated the interplay between the canonically conjugate quadratures. This allows us to use squeezed states to further enhance the force sensitivity without any problem from the increased uncertainty from its conjugate quadrature.
The force sensitivity in Eq. (24) is derived by assuming that the input fields are vacuum and laser fields. Now lets squeeze the vacuum field [68–71] entering through the empty port of the interferometer so that
The final result in Eq. (26) is obtained by considering squeezing such that the squeezing angle $\theta =2\phi$. As the squeezing is implemented only on the vacuum field, which makes no contribution to $\bar {Q}$, $\bar {Q}$ remains same as given in Eq. (17). Hence, with the squeezed vacuum, the force sensitivity $F_{\xi }$ is given as
Substituting Eq. (23) into Eq. (27) gives the force sensitivity $F_{3}$ as
The RHS of Eq. (28) shows that the sensitivity is improved by a factor of $\sqrt {e^{-2r}\zeta /4\Delta }$ beyond $F_{sql}$. In Eq. (28), the squeezed light leads to the improvement factor $e^{-r}$ while the QBNM leads to the improvement factor $\sqrt {\zeta /4\Delta }$. Hence using squeezed light in combination with QBNM can enhance the interferometer performance beyond the squeezed light limit by a factor of $\sqrt {\zeta /4\Delta }$. It is worth mentioning that a big challenge with squeezed states is generating them with large $r$ value. To our knowledge, experimentally, the highest squeezing realized so far is $15$ dB. In this scenario Eq. (28) presents an alternate approach to improve the squeezed light interferometer sensitivity not only by increasing $r$ but also by minimizing the $\zeta /\Delta$ factor. This point can be further illustrated by rewriting Eq. (28) as
where $r_{eff}=r+\mbox {ln}\left (4\Delta /\zeta \right )/2$ is the effective squeezing parameter. Hence for an input squeezing of $r$, the sensitivity of the optomechanical interferometer is improved by an effective squeezing parameter of $r_{eff}$. At the same time, it must be noted that this method is limited to frequencies much smaller than the resonance frequency of the mechanical mirror.The Eq. (28), Eq. (26), Eq. (24), and Eq. (23) are analytically simplified results from Eq. (18). These equations together illustrate the improved sensitivity in the squeezed light optomechanical interferometer. To obtain a broader understanding on variation of sensitivity under various parameters, we plot Eq. (18) as a function of various parameters such as input laser power, $r$, and $\theta$ in Fig. 3 and Fig. 4. The results and conclusion from the plots agree with the simplified results (Eq. (28), Eq. (26), Eq. (24), and Eq. (23)). Figure 3 and Fig. 4 shows the enhanced force sensitivity beyond squeezed light limit at two different cases.
Fig. 3. Variation of force sensitivity as a function of optical power with different squeezing parameter $r$. The squeezing angle $\theta$ is fixed at $2\phi$. RPN is suppressed by setting $\mathcal {R}(\alpha )\simeq \Delta$. For each curve the lowest point gives the best force sensitivity $F_{3}$ which is given in Eq. (29). For convenience we only marked $F_{3}$ for $r=3$. For $r=0$ curve, $F_{3}$ is same as $F_{2}$. All the other simulation parameters are same as Fig. 2.
Fig. 4. Variation of force sensitivity as a function of optical power with different squeezing angles. RPN is suppressed by setting $\Delta \simeq \mathcal {R}(\alpha )$. Squeezing parameter $r$ is fixed to $3$. Best sensitivity is achieved when $\theta =2\phi$. All the other simulation parameters are same as in Fig. 2.
Case (i) : Figure 3 shows the plot of Eq. (18) when the input vacuum is squeezed for different $r$ values $i.e.,$ $r=0,1,2,3$. The plotting parameters are chosen such that $1>\zeta ^{2}/\Delta ^{2}\gg \gamma \omega _{f}/\omega _{m}^{2}$ and $\mathcal {R}(\alpha )\simeq \Delta$, so that RPN is suppressed. The squeezing angle $\theta$ is fixed at $2\phi$ and the squeezing parameter $r$ is varied for each plot. Increasing $r$ value increases the overall force sensitivity. The best sensitivity in each curve corresponds to $F_{3}$ which is given by Eq. (28) (for convenience we only marked $F_{3}$ for $r=3$ in Fig. 3). The optical power $58.5$mW, corresponding to $F_{3}$, is equal to $\hbar \omega _{l}|\bar {E}|^{2}$, where $|\bar {E}|^{2}$ is given in to Eq. (23). As an example, for $r = 3$ curve, the plot shows that $F_{3}=1.63\times 10^{-20}$N/$\sqrt {\mbox {Hz}}$ which matches with the theoretical value given in Eq. (28). It shows that the force sensitivity is improved by a factor of $20.12$ from the force sensitivity with no squeezing. For $r=0$ curve, the best sensitivity is same as $F_{2}$ which is the best possible sensitivity when input states are not squeezed. If $r \to \infty$ then Eq. (28) gives a force sensitivity of $3.08\times 10^{-21}$N/$\sqrt {\mbox {Hz}}$ which is determined by the mechanical mirror noise.
Case (ii) : In Fig. 4, the squeezing parameter is fixed at $r = 3$ and five curves are simulated by varying the squeezing angle $\theta$. The parameters are same as in Fig. 2 so that $1>\zeta ^{2}/\Delta ^{2}\gg \gamma \omega _{f}/\omega _{m}^{2}$ and $\mathcal {R}(\alpha )\simeq \Delta$. Figure 4 indicates that best sensitivity is achieved when $\theta =2\phi$ which is in agreement with Eq. (26). Again the best sensitivity for $\theta =2\phi$ curve is exactly equal to $F_{3}$. For all other values of $\theta$ the best sensitivity is smaller than $F_{3}$.
Overall both the Fig. 3 and Fig. 4 agree with the results in Eq. (28) and Eq. (23) and show that the improved force sensitivity is $\sqrt {e^{-2r}\zeta /4\Delta } F_{sql}$. The optical power corresponding to the best sensitivity $F_{3}$ for $\theta =2\phi$ curve is related to Eq. (23) as $\hbar \omega _{l}|\bar {E}|^{2}$.
In this paragraph, we briefly describe some prominent techniques for overcoming SQL. Quantum non-demolition measurements [72] (QND) describe the underlying physics to go beyond SQL. The sufficient condition for the QND measurement is that the measured variable has to commute with the Hamiltonian. Such Hamiltonians are rare and hence several alternate techniques are proposed. Even though all these techniques are different, they all use the QND physics at some point in their methodology. Coherent quantum noise cancellation [73,74] technique overcomes SQL by adding an auxiliary system to the main system. The auxiliary system parameters are tuned such that the quantum back-action in the main system is cancelled. Variational measurement technique is used in Ref. [75] to overcome SQL by using frequency dependent homodyne. Depending on the frequency region of interest, the phase of the reference beam is adjusted to induce correlations between the amplitude and phase quadratures. Then these correlations are exploited to overcome SQL. The technique of quantum mechanics free subsystems [76] overcomes SQL by measuring combined quadratures like sum of two amplitude quadratures and the difference of the two phase quadratures. The commutation relation of such combined quadratures is zero, leading to sub-SQL measurements. Signal recycling [77–79] is another technique where some of the output from the interferometer is recycled back into the interferometer. An additional mirror achieves the recycling of output. This recycling mirror leads to additional resonances in the interferometer, which can be adjusted to overcome SQL.
4. Temperature dependence
We assumed that the mechanical mirror is in its quantum mechanical ground state in all the plots and in all the results until Eq. (29). In experimental scenario, the mechanical mirror will be subject to non-zero temperature which can cause thermal excitations. These thermal effects can be taken account by using the mechanical mirror correlation function [80] $\left \langle {\hat \varpi }(\omega ){\hat \varpi }(\omega ') \right \rangle =\hbar m \omega \gamma [1+\coth (\hbar \omega /2k_{B}T)]\delta (\omega +\omega ')$. Hence for a non-zero temperature $T$ Eq. (28) will become
The thermal excitations are important when the second term in the square root of Eq. (30) is atleast comparable to the first term. Hence to judge the strength of each term, we define a ratio $A$ as
A plot of Eq. (31) is shown in Fig. 5 as a function of thermal phonon occupancy number $n$. The phonon occupancy number is related to temperature as $n=1/(e^{\hbar \omega _{f}/k_{B}T}-1)$. The value of $A$ is equal to one when $n\approx 1408$. When $A=1$, $F_{3}$ is given as $F_{sql}\sqrt {\zeta e^{-2r}/2\Delta }$ which still shows enhancement in the force sensitivity of the squeezed light optomechanical interferometer by a factor of $\sqrt {\zeta /2\Delta }$. Hence, we can safely conclude that the method described in this work is applicable when $A\le 1$. The curve in Fig. 5 becomes flat for small $n$ values. This flat region corresponds to the scenario where the thermal energy is so low that it can not excite any phonons and the mechanical mirror remains in its quantum ground state. Given the current experimental status, [81–83] which can cool the mechanical mirror to quantum ground state, achieving $A\le 1$ is experimentally feasible.
Fig. 5. Variation of the parameter $A$ in Eq. (31) as a function of phonon occupancy number, $n=1/(e^{\hbar \omega _{f}/k_{B}T}-1)$ of the mechanical mirror. Squeezing parameter $r=3$, while all the other parameters are same as in Fig. 2.
5. Transmittance of mechanical mirror
We assumed that the mechanical mirror is perfectly reflective in the Hamiltonian Eq. (1). If the mechanical mirror is not perfectly reflective, photons in sub-cavity-a tunnel into sub-cavity-c, and vice-versa. Such tunnelling can be accounted by adding $\hbar J\left (\hat {a}^{\dagger }\hat {c}+\hat {c}^{\dagger }\hat {a}\right )$, where $J$ is photon tunnelling rate, term to the Hamiltonian given in Eq. (1). The reflectivity $r_{m}$ of the mechanical mirror is related to $J$ as $J=\sqrt {2\left (1-r_{m}\right )}c/2l$[84]. This modifies the steady state solutions in Eq. (7) as
where $\phi _{J}=\tan ^{-1}(-2\Delta _{1}/\zeta )$ with $\Delta _{1}=\Delta +J$. Similarly the fluctuation equation in Eq. (11) and Eq. (12) becomesBy comparing Eq. (30) and Eq. (35), we conclude that $J$ should be as small as possible for achieving best force sensitivity. By imposing the condition $J < \Delta = 100\zeta$, we find that $r_{m}$ should be greater than $0.9975$ for the simulation parameters used in this article.
6. Intuitive picture
It is known that vacuum entering through the empty port of the interferometer is the reason for noise in the optical interferometer [18]. Hence squeezing the vacuum leads to improved sensitivity. In optomechanical interferometers, it was shown in Ref. [85] that vacuum squeezing at squeezing angle $\theta =0$ leads to suppression of noise in one quadrature by a factor of $e^{-2r}$ while the noise in its conjugate quadrature increases as $e^{2r}$. Later, it was proposed in Ref. [86] that by using a generalized squeezed light, both shot noise and RPN can be simultaneously reduced by a factor of $e^{-2r}$. Overall, the sensitivity of optomechanical interferometer is improved by a factor of $e^{-r}$ beyond SQL because of squeezed light.
Coming to QBNM part, by using Eq. (8) and Eq. (9), we can write
While SQL is imposed by the competition between shot noise and RPN as described in Eq. (19), QND principles [23] describes the underlying theoretical mechanism for this is the interplay between the measured variable and its canonically conjugate variable. A QND measurement eliminates this inter-dependence between the conjugate variables thus paving way for sub-SQL measurements. In Eq. (36) $\hat {Y}_{a}(\omega )-\hat {Y}_{c}(\omega )$ is a function of its conjugate variable $\hat {X}_{a}(\omega )-\hat {X}_{c}(\omega )$. Setting $\Delta =\alpha$ is one way to eliminate the interdependence between the conjugate variables, thus leading to QND situation. However as described Eq. (21), we can only set $\Delta = \mathcal {R}(\alpha )$. The remaining contribution from the imaginary part of $\alpha$ can be approximated to first order of $\gamma \omega /\omega _{m}^{2}$ by using a high quality mechanical mirror ($\gamma \ll \omega _{m}$) in the low frequency ($\omega \ll \omega _{m}$) regime. The frequency symmetrization in Eq. (18) further cancels whatever the tiny RPN coming from the first order of $\gamma \omega /\omega _{m}^{2}$. Overall, in the final result, we actually implemented QND measurement which led to sub-SQL sensitivity.
As the inter-dependence between the conjugate quadratures is eliminated, only one quadrature determines the final sensitivity. Squeezing that particular quadrature can further enhance the sensitivity without any problem from the anti-squeezing in the other quadrature. The unique feature of the technique described in this manuscript is its compatibility with squeezing as well as QBNM. Both squeezing and QBNM relies on adjusting the parameters of laser field to improve the sensitivity. Squeezing relies of reducing the uncertainty in one quadrature at the expense of increased uncertainty in the other quadrature, while QBNM relies on using the restoring force [34] of the optical field to counter the perturbation coming from the conjugate quadrature. The mutual compatibility of both these techniques led to improvement of overall sensitivity to Eq. (28). The importance of this work is demonstrated in Eq. (29) in which Eq. (28) is rewritten in terms of effective squeezing parameter $r_{eff}$. Even though the input vacuum’s squeezing parameter is $r$, the final sensitivity is given as if $r_{eff}$ is the squeezing parameter of the input vacuum. This is important owing to the difficulty in preserving and synthesizing squeezed states with large $r$.
7. Simulation parameters
For simulation, we use the following optomechanical parameters: $m=1.1\times 10^{-10}$Kg, $\omega _{m}/2\pi =9.7\times 10^{3}$Hz, $\omega _{f}=100$Hz, $\omega _{l}/2\pi =2.8\times 10^{14}$Hz, $\Delta =100\zeta$, $\zeta /2\pi =4.7\times 10^{5}$Hz, $\gamma /2\pi =1.3\times 10^{-2}$Hz, $g/2\pi =7.8\times 10^{15}$Hz/m.
8. Conclusions
We discussed a new method to improve the sensitivity of squeezed light optomechanical interferometer for classical force detection. We eliminated the RPN in an optomechanical cavity by using the QBNM technique and then the squeezed vacuum technique is combined with QBNM. Under appropriate conditions, combination of QBNM with vacuum squeezing improved the sensitivity of the squeezed light optomechanical interferometer. Generally the squeezed light leads to classical force detection with sensitivity $e^{-r}F_{sql}$. In our method, the sensitivity is improved to $\sqrt {e^{-2r}\zeta /4\Delta }F_{sql}$, $1>\zeta /\Delta >0$, which is better than the sensitivity achieved by vacuum squeezing alone. Given the challenges associated with synthesizing squeezed states with large squeezing parameter, the method described in this work will improve the squeezed light effectiveness without requiring a relatively large squeezing parameter. We further studied dependence of force sensitivity on various parameters like laser power, squeezing parameters, temperature, and reflectivity of the mechanical mirror. The method described in this article is applicable to frequencies much smaller than the resonance frequency of the optomechanical mirror.
Funding
Science and Engineering Research Board (SRG/2020/001167), India; National Natural Science Foundation of China (12074030, 12274107).
Disclosures
The authors declare no conflicts of interest
Data availability
No data were generated or analyzed in the presented research.
References
1. A. Giazotto, “Interferometric detection of gravitational waves,” Phys. Rep. 182(6), 365–424 (1989). [CrossRef]
2. C. Bond, D. Brown, A. Freise, and K. A. Strain, “Interferometer techniques for gravitational-wave detection,” Living Rev. Relativ. 19(1), 3 (2016). [CrossRef]
3. M. Liu and B. Gong, “A three-dimensional laser interferometer gravitational-wave detector,” Sci. Rep. 10(1), 16285 (2020). [CrossRef]
4. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013). [CrossRef]
5. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
6. T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef]
7. S. Barzanjeh, A. Xuereb, S. Gröblacher, M. Paternostro, C. A. Regal, and E. M. Weig, “Optomechanics for quantum technologies,” Nat. Phys. 18(1), 15–24 (2022). [CrossRef]
8. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Français, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97(13), 133601 (2006). [CrossRef]
9. M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304(5667), 74–77 (2004). [CrossRef]
10. J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015). [CrossRef]
11. B.-B. Li, L. Ou, Y. Lei, and Y.-C. Liu, “Cavity optomechanical sensing,” Nanophotonics 10(11), 2799–2832 (2021). [CrossRef]
12. J. Moser, J. Güttinger, A. Eichler, M. J. Esplandiu, D. E. Liu, M. I. Dykman, and A. Bachtold, “Ultrasensitive force detection with a nanotube mechanical resonator,” Nat. Nanotechnol. 8(7), 493–496 (2013). [CrossRef]
13. M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,” Rev. Mod. Phys. 68(3), 755–799 (1996). [CrossRef]
14. C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. i. issues of principle,” Rev. Mod. Phys. 52(2), 341–392 (1980). [CrossRef]
15. K. Komori, T. Kawasaki, S. Otabe, Y. Enomoto, Y. Michimura, and M. Ando, “Improving force sensitivity by amplitude measurements of light reflected from a detuned optomechanical cavity,” Phys. Rev. A 104(3), L031501 (2021). [CrossRef]
16. J. Cripe, T. Cullen, Y. Chen, P. Heu, D. Follman, G. D. Cole, and T. Corbitt, “Quantum backaction cancellation in the audio band,” Phys. Rev. X 10(3), 031065 (2020). [CrossRef]
17. W. A. Edelstein, J. Hough, J. R. Pugh, and W. Martin, “Limits to the measurement of displacement in an interferometric gravitational radiation detector,” J. Phys. E: Sci. Instrum. 11(7), 710–712 (1978). [CrossRef]
18. C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980). [CrossRef]
19. K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4(7), 561–564 (2008). [CrossRef]
20. J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568(7752), 364–367 (2019). [CrossRef]
21. S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016). [CrossRef]
22. R. S. Bondurant, “Reduction of radiation-pressure-induced fluctuations in interferometric gravity-wave detectors,” Phys. Rev. A 34(5), 3927–3931 (1986). [CrossRef]
23. V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209(4456), 547–557 (1980). [CrossRef]
24. J. B. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A. Clerk, and K. C. Schwab, “Back-action-evading measurements of nanomechanical motion,” Nat. Phys. 6(3), 213–217 (2010). [CrossRef]
25. J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke, P. Meystre, A. A. Clerk, and K. C. Schwab, “Mechanically detecting and avoiding the quantum fluctuations of a microwave field,” Science 344(6189), 1262–1265 (2014). [CrossRef]
26. K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, “Quantum nondemolition measurements of harmonic oscillators,” Phys. Rev. Lett. 40(11), 667–671 (1978). [CrossRef]
27. A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10(9), 095010 (2008). [CrossRef]
28. S. Vyatchanin and E. Zubova, “Quantum variation measurement of a force,” Phys. Lett. A 201(4), 269–274 (1995). [CrossRef]
29. M. Tsang and C. M. Caves, “Coherent quantum-noise cancellation for optomechanical sensors,” Phys. Rev. Lett. 105(12), 123601 (2010). [CrossRef]
30. C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, “Quantum back-action-evading measurement of motion in a negative mass reference frame,” Nature 547(7662), 191–195 (2017). [CrossRef]
31. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004). [CrossRef]
32. T. Xie, Z. Zhao, X. Kong, W. Ma, M. Wang, X. Ye, P. Yu, Z. Yang, S. Xu, P. Wang, Y. Wang, F. Shi, and J. Du, “Beating the standard quantum limit under ambient conditions with solid-state spins,” Sci. Adv. 7(32), eabg9204 (2021). [CrossRef]
33. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007). [CrossRef]
34. S. Davuluri, “Quantum optomechanics without the radiation pressure force noise,” Opt. Lett. 46(4), 904–907 (2021). [CrossRef]
35. L. F. Buchmann, S. Schreppler, J. Kohler, N. Spethmann, and D. M. Stamper-Kurn, “Complex squeezing and force measurement beyond the standard quantum limit,” Phys. Rev. Lett. 117(3), 030801 (2016). [CrossRef]
36. C. F. Ockeloen-Korppi, E. Damskägg, G. S. Paraoanu, F. Massel, and M. A. Sillanpää, “Revealing hidden quantum correlations in an electromechanical measurement,” Phys. Rev. Lett. 121(24), 243601 (2018). [CrossRef]
37. M. T. Jaekel and S. Reynaud, “Quantum limits in interferometric measurements,” Europhys. Lett. 13(4), 301–306 (1990). [CrossRef]
38. Y. Ma, H. Miao, B. H. Pang, M. Evans, C. Zhao, J. Harms, R. Schnabel, and Y. Chen, “Proposal for gravitational-wave detection beyond the standard quantum limit through epr entanglement,” Nat. Phys. 13(8), 776–780 (2017). [CrossRef]
39. R. Schnabel, “Squeezed states of light and their applications in laser interferometers,” Phys. Rep. 684, 1–51 (2017). Squeezed states of light and their applications in laser interferometers. [CrossRef]
40. B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, “Quantum sensing with squeezed light,” ACS Photonics 6(6), 1307–1318 (2019). [CrossRef]
41. C.-W. Lee, J. H. Lee, and H. Seok, “Squeezed-light-driven force detection with an optomechanical cavity in a mach–zehnder interferometer,” Sci. Rep. 10(1), 17496 (2020). [CrossRef]
42. N. Aggarwal, T. J. Cullen, J. Cripe, G. D. Cole, R. Lanza, A. Libson, D. Follman, P. Heu, T. Corbitt, and N. Mavalvala, “Room-temperature optomechanical squeezing,” Nat. Phys. 16(7), 784–788 (2020). [CrossRef]
43. A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013). [CrossRef]
44. K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, “Deterministic squeezed states with collective measurements and feedback,” Phys. Rev. Lett. 116(9), 093602 (2016). [CrossRef]
45. U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scr. 91(5), 053001 (2016). [CrossRef]
46. M. J. Yap, J. Cripe, G. L. Mansell, T. G. McRae, R. L. Ward, B. J. J. Slagmolen, P. Heu, D. Follman, G. D. Cole, T. Corbitt, and D. E. McClelland, “Broadband reduction of quantum radiation pressure noise via squeezed light injection,” Nat. Photonics 14(1), 19–23 (2020). [CrossRef]
47. D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983). [CrossRef]
48. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387(6632), 471–475 (1997). [CrossRef]
49. Y. Zhang, M. Menotti, K. Tan, V. D. Vaidya, D. H. Mahler, L. G. Helt, L. Zatti, M. Liscidini, B. Morrison, and Z. Vernon, “Squeezed light from a nanophotonic molecule,” Nat. Commun. 12(1), 2233 (2021). [CrossRef]
50. Z. Vernon, N. Quesada, M. Liscidini, B. Morrison, M. Menotti, K. Tan, and J. Sipe, “Scalable squeezed-light source for continuous-variable quantum sampling,” Phys. Rev. Appl. 12(6), 064024 (2019). [CrossRef]
51. H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65(2), 022002 (2001). [CrossRef]
52. P. Kwee, J. Miller, T. Isogai, L. Barsotti, and M. Evans, “Decoherence and degradation of squeezed states in quantum filter cavities,” Phys. Rev. D 90(6), 062006 (2014). [CrossRef]
53. A. Dutt, S. Miller, K. Luke, J. Cardenas, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “Tunable squeezing using coupled ring resonators on a silicon nitride chip,” Opt. Lett. 41(2), 223–226 (2016). [CrossRef]
54. S. Davuluri and Y. Li, “Absolute rotation detection by coriolis force measurement using optomechanics,” New J. Phys. 18(10), 103047 (2016). [CrossRef]
55. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 db squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016). [CrossRef]
56. S. Davuluri and Y. Li, “Light as a quantum back-action nullifying meter,” J. Opt. Soc. Am. B 39(12), 3121–3127 (2022). [CrossRef]
57. R. Burgwal, J. del Pino, and E. Verhagen, “Comparing nonlinear optomechanical coupling in membrane-in-the-middle and single-cavity systems,” New J. Phys. 22(11), 113006 (2020). [CrossRef]
58. C. K. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian formulation,” Phys. Rev. A 51(3), 2537–2541 (1995). [CrossRef]
59. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012). [CrossRef]
60. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010). [CrossRef]
61. H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103(10), 100402 (2009). [CrossRef]
62. A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51(17), 1550–1553 (1983). [CrossRef]
63. D. Kleckner, B. Pepper, E. Jeffrey, P. Sonin, S. M. Thon, and D. Bouwmeester, “Optomechanical trampoline resonators,” Opt. Express 19(20), 19708–19716 (2011). [CrossRef]
64. A. I. Lvovsky, Squeezed Light (John Wiley and Sons, Ltd, 2015), chap. 5, pp. 121–163.
65. R. Cernansky and A. Politi, “Nanophotonic source of quadrature squeezing via self-phase modulation,” APL Photonics 5(10), 101303 (2020). [CrossRef]
66. D. Stoler, “Equivalence classes of minimum uncertainty packets,” Phys. Rev. D 1(12), 3217–3219 (1970). [CrossRef]
67. D. Stoler, “Equivalence classes of minimum-uncertainty packets. ii,” Phys. Rev. D 4(6), 1925–1926 (1971). [CrossRef]
68. A. Otterpohl, F. Sedlmeir, U. Vogl, T. Dirmeier, G. Shafiee, G. Schunk, D. V. Strekalov, H. G. L. Schwefel, T. Gehring, U. L. Andersen, G. Leuchs, and C. Marquardt, “Squeezed vacuum states from a whispering gallery mode resonator,” Optica 6(11), 1375–1380 (2019). [CrossRef]
69. S. Ast, M. Mehmet, and R. Schnabel, “High-bandwidth squeezed light at 1550 nm from a compact monolithic ppktp cavity,” Opt. Express 21(11), 13572–13579 (2013). [CrossRef]
70. T. Aoki, G. Takahashi, and A. Furusawa, “Squeezing at 946nm with periodically poled ktiopo4,” Opt. Express 14(15), 6930–6935 (2006). [CrossRef]
71. Y. Zhao, Y. Okawachi, J. K. Jang, X. Ji, M. Lipson, and A. L. Gaeta, “Near-degenerate quadrature-squeezed vacuum generation on a silicon-nitride chip,” Phys. Rev. Lett. 124(19), 193601 (2020). [CrossRef]
72. V. B. Braginsky and F. Y. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68(1), 1–11 (1996). [CrossRef]
73. M. Tsang and C. M. Caves, “Evading quantum mechanics: Engineering a classical subsystem within a quantum environment,” Phys. Rev. X 2(3), 031016 (2012). [CrossRef]
74. K. Li, S. Davuluri, and Y. Li, “Improving optomechanical gyroscopes by coherent quantum noise cancellation processing,” Sci. China Phys. Mech. Astron. 61(9), 90311 (2018). [CrossRef]
75. N. S. Kampel, R. W. Peterson, R. Fischer, P.-L. Yu, K. Cicak, R. W. Simmonds, K. W. Lehnert, and C. A. Regal, “Improving broadband displacement detection with quantum correlations,” Phys. Rev. X 7(2), 021008 (2017). [CrossRef]
76. L. M. de Lépinay, C. F. Ockeloen-Korppi, M. J. Woolley, and M. A. Sillanpää, “Quantum mechanics–free subsystem with mechanical oscillators,” Science 372(6542), 625–629 (2021). [CrossRef]
77. A. Buonanno and Y. Chen, “Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors,” Phys. Rev. D 64(4), 042006 (2001). [CrossRef]
78. A. Buonanno and Y. Chen, “Scaling law in signal recycled laser-interferometer gravitational-wave detectors,” Phys. Rev. D 67(6), 062002 (2003). [CrossRef]
79. H. Rehbein, H. Müller-Ebhardt, K. Somiya, C. Li, R. Schnabel, K. Danzmann, and Y. Chen, “Local readout enhancement for detuned signal-recycling interferometers,” Phys. Rev. D 76(6), 062002 (2007). [CrossRef]
80. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum brownian motion,” Phys. Rev. A 63(2), 023812 (2001). [CrossRef]
81. M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99(7), 073601 (2007). [CrossRef]
82. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011). [CrossRef]
83. J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Sideband cooling beyond the quantum backaction limit with squeezed light,” Nature 541(7636), 191–195 (2017). [CrossRef]
84. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10(9), 095008 (2008). [CrossRef]
85. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981). [CrossRef]
86. W. G. Unruh, Quantum Noise in the Interferometer Detector (Springer US, Boston, MA, 1983), pp. 647–660.
