1. Introduction

Precision measurement is an intriguing and essential task in quantum optics. Optomechanical system [1, 2] holds great promise for precision measurement due to the properties like small mass, large quality-factor and high frequency. Typically, an optomechanical system consists of a cavity field and a moveable mirror which oscillates under the radiation pressure of the cavity fields. Meanwhile, the resonance frequency of the cavity field depends on the position of the moveable mirror. Owing to this noise back action mechanism, optomechanical system has been a fascinating platform for realizing various ultrasensitive measurement, such as cavity optomechanical magnetometer [3], displacement measurement [4–7], force sensing [8, 9], gravity wave detection [10, 11] and mass detection [12].

Recently optomechanically induced transparency (OMIT) that allows the propagation of a probe field in the optomechanical system by manipulating another strong control field [13–15] has gained much attention due to its extensive applications in quantum information [16–25]. Since first theoretically proposed by Agarwal [14], the study of OMIT effect has been extended to different optomechanical systems, such as two coupled optomechanical resonators [26, 27], coupled-cavity optomechanical sytem [28–30] and atom-assisted optomechanical system [31, 32]. It is reported that OMIT with a squeezed probe field can be achieved even at a single-photon level [33].

OMIT has been exploited as an efficient tool for precision measurement [34–45]. Most of these schemes [34–43] are Coulomb-interaction-based and they are carried out in a hybrid optomechanical system with two charged objects. To carry out these schemes, the influence of other undesirable electromagnetical field must be excluded. What is more, system parameters, like electric charge quantity of the charged objects, require precise control. It is not easy to meet all these requirements. The uncertainty in the operation degrade the measurement accuracy and sensitivity. To solve the problems, some measurement schemes without Coulomb interaction are proposed. Wang et al. [44] investigated the homodyne spectra of the output field from the optomechanical system with a squeezed probe. They found that the width of two symmetric peaks in the output spectra is sensitive to the mass adsorbed by the nanomechanical resonator. Using this property, they proposed a measurement scheme for mass sensing. Lately, a precision measurement scheme for the environmental temperature is proposed based on the OMIT homodyne spectra in the optomechanical system [45]. These schemes utilize the properties of OMIT spectra in the optomechanical system to measure the concerned parameters like temperature. Thus the system setup and the experimental realization is easy to be implemented. However, the OMIT curves in these schemes are susceptible to the thermal noise and they can only be observed over a narrow temperature range. If not, the quantum signal will be buried by the thermal noise and then the measurement sensitivity is degraded.

This paper presents a wide-range precision temperature measurement scheme with double OMIT homodyne spectra in a double-cavity optomechanical system. The system concerned consists of two tunnelling-coupled resonators. One cavity driven by a strong pump field and a squeezed probe field. The other cavity is coupled to the mechanical oscillator via radiation pressure. It is found that double OMIT windows emerge in the homodyne detection spectra of the output field for appropriate tunneling strength. This effect is attributed to the quantum interference between three quantum paths. In addition, the height of central peak varies linearly with the environmental temperature. In view of this linear relationship, a precision measurement scheme for the environmental temperature is proposed, based on the double OMIT properties of the outfield homodyne spectra. Compared with the scheme in [35], the scheme proposed by this paper does not utilize the Coulomb interaction and it can be applied in a wide variety of environments. What is more, different from [45], this scheme is robust against the mechanical decay. Hence our scheme can be used in wide-range high-sensitivity detection of the environmental temperature.

The rest of this paper is organized as follows. Section 2 describes the model of a double-cavity optomechanical system. In section 3 the analytical equation of the output homodyne spectrum is derived and the influence of the system parameter on the homodyne spectra of the output field is analyzed. In section 4 a feasible scheme to measure the environmental temperature is proposed. Extended discussion is presented in section 5. The conclusion is given in section 6.

 

Fig. 1 Schematic of the system and the measurement. Two cavities are coupled via photon tunneling effect with strength J. The field a₁ in the left cavity is driven by a strong classical field with frequency ω_c and a squeezed vacuum at frequency ω_p. In combination with $a_{1,in}$, the output field $a_{1,out}$ turns out to be $a_{out}$. Then $a_{out}$ is mixed with a strong local field a_lo at a $50:50$ beam splitter (BS). The strong local field is centered around the probe frequency and can be depicted as $a_{lo}\left(t\right)=a_{lo}{e}^{-i{\omega }_{m}t}$. Based on the difference between the output signals of the two photodetectors (PD), the homodyne spectrum can be obtained from the spectrum analyzer (SA).

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2. Model for the optomechanical interaction

As shown in Fig. 1(a), the model consists of two cavities, where the fields are coupled via photon tunneling effect. The left cavity is driven by a strong coupling field at frequency ω_c and a weak quantized field at ω_p. The right mirror of the right cavity is assumed moveable due to the radiation pressure effect. In the frame rotating at the pump field frequency ω_c, the Hamiltonian of the system can be written as

Here a_i ($a_i^{†}$) denotes the annihilation (creation) operator of the ith cavity field. ${\mathrm{\Delta }}_i={\omega }_i-{\omega }_c$ represents the frequency detuning of the cavity field frequency ω_i from the strong pump field frequency ω_c. m and ω_m are the mass and the oscillating frequency of the moveable mirror. q and p are the position and momentum operator of the mechanical oscillator. J is the tunneling strength between the two cavities. g is the optomechanical coupling coefficient related to the nonlinear coupling between the cavity field a₂ and the mechanical resonator b. ${\epsilon }_c=\sqrt{\frac{2{\kappa }_1\wp }{\hslash {\omega }_c}}$ represents the pump field strength with frequency ω_c and pumping power $\wp$. κ₁ is the decay rate of cavity a₁. Taking the dissipation and system noise into consideration, we can derive the quantum Langevin equations for the system as below

The operators in Eqs. (2)–(5) can be rewritten as the sum of steady-state parts and the fluctuation parts, which reads $q=q_s+\delta q,p=p_s+\delta p,a_1=a_{1,s}+\delta a_1,a_2=a_{2,s}+\delta a_2$. As denoted in [33], the concerned component of the output field is near the probe frequency ω_p. Since the steady-state parts of the cavity field are at the frequency of ω_c, they does not contribute to the output field near ω_p. We focus on the influence of the fluctuation operators on the final output field. By line arising the operators and omitting higher order infinitesimal terms like $\delta a_1^{†}\delta a_1$, we can get the effective Langevin equations for the fluctuation operators as below

Using Fourier Transformation like $f\left(\omega \right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(t\right)e^{-i\omega t}dt$, $f^{†}\left(-\omega \right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}{f}^{†}\left(t\right)e^{-i\omega t}dt$, we can transform Eqs. (14)–(17) into the frequency domain in the matrix form as $AX=B$, where

By using the input-output relation $a_{1,out}\left(\omega \right)=\sqrt{2{\kappa }_1}{a}_1\left(\omega \right)-a_{1,in}\left(\omega \right)$, we can get the fluctuations of the output field $\delta a_{1,out}\left(\omega \right)$. For further measurement shown in Fig. 1, the output field $a_{1,out}$ is mixed with the input field $a_{1,in}$ and it can be expressed as $a_{out}\left(\omega \right)=a_{1,out}\left(\omega \right)+a_{1,in}\left(\omega \right)=\sqrt{2{\kappa }_1}{a}_1\left(\omega \right)$. From the Langevin equations and the input-output relation, we can obtain the outfield as follows

3. Homodyne spectrum of the double OMIT for identical resonators

In the following part we analyze the homodyne spectrum of the output field. As in [33], the homodyne spectrum $X\left(\omega \right)$ from the spectrum analyzer fulfills the following relation

Based on the correlation function as Eqs. (6)–(9), we can obtain the homodyne spectrum $X\left(\omega \right)$ as below

The terms involving N and M originate from the input squeezed field. The last two terms in Eq. (34) are from the thermal noise of the oscillating mirror. The rest of Eq. (34) are related to the spontaneous emission of the input vacuum noise.

Based on the analytic equation above, we present the characteristics of double OMIT using the homodyne spectra of the output fields. To simplify the model, we assume that the two cavity fields are identical in this part.

For numerical analyses of the homodyne spectra, we choose the system parameters as in [33]. The wave length of the strong coupling field $\lambda =2\pi c/{\omega }_c=775nm$, the mass of the moveable mirror $m=20ng$, the frequency of the movable mirror ${\omega }_m=2\pi \times 51.8MHz$, the coupling constant $g=2\pi \times 12MHz/nm\sqrt{\hslash /\left(2m{\omega }_m\right)}$, the cavity decay rate ${\kappa }_1={\kappa }_2=\kappa =2\pi \times 15MHz$, the mechanical damping rate ${\gamma }_m=2\pi \times 41kHz$, the mechanical quality factor $Q={\omega }_m/{\gamma }_m=1263$, the linewidth of the squeezed vacuum $\mathrm{\Gamma }=2\kappa$. Considering ${\mathrm{\Delta }}_{1,2}=\mathrm{\Delta }$ and ${\omega }_{1,2}={\omega }_m$, we present the plot of the homodyne spectra of the output field under the condition $\mathrm{\Delta }={\omega }_m$ in Fig. 2. In most of the calculation, we choose the temperature $T=100$ mK, the photon number N = 5, and the coupling field power $\wp =10$mW, the coupling strength $J=\kappa$.

As shown in Fig. 2(a), the homodyne spectrum displays sharp peak above the Lorentz line for $J=0.6\kappa$. In this case, the tunneling interaction is weak. The output field component with frequency ω_p mainly comes from cavity a₁, which is pumped by the probe field. When the coupling strength J is increased to $J=\kappa$, as denoted by the red dashed line in Fig. 2(a), central peak appears around ω = 0 and the spectrum shows double OMIT profile. This double OMIT profile is due to the quantum interference of three quantum paths to excite cavity a₁. The first one is from the direct coupling of the input probe field to cavity a₁. The second one is from a₂ to a₁ through the tunnelling interaction (the pathway of lightis: $a_1\to a_2\to a_1$). The third quantum path is originated from the interaction between the coupling field and the probe field. This nonlinear interaction leads to the anti-Stokes process and gives rise to anti-Stokes field. Through the tunnelling interaction between the two cavities, the anti-Stokes component of cavity a₂ is excited. Then the phonon in mechanical resonator will be excited by this anti-Stokes component. The produced phonon interacts with the field at the frequency ω_c and reproduces a field with frequency closed to cavity a₁. The interference of the former two quantum paths leads to the coupled resonator induced transparency. The emergence of the third quantum path splits the OMIT window and gives rise to the double OMIT curve. If the tunneling strength is increased to $J=2{\kappa }_1$, the heights of the central peak and the two nadirs shrink. This is due to the fact that the quantum interference among the three excited paths is enhanced with the increase of the coupling strength.

 

Fig. 2 Homodyne spectrum $X\left(\omega \right)$ versus the normalized frequency $\omega /{\omega }_m$ (a) for different coupling strength J with $T=100$mK, N = 5, (b)for different environmental temperature T with $J=\kappa ,N=5$, (c) for different photon number N with $J=\kappa ,T=100$mK.

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The dependence of the homodyne spectra of the output field versus the temperature is shown in Fig. 2(b). When T = 0, there is only one nadir around frequency ω = 0. With the increase of the temperature, a sharp peak emerges around frequency ω = 0. Different from double OMIT spectra in [35], the two nadirs in our scheme are almost fixed with the increase of T. Meanwhile, the value of the central peak increases with the environmental temperature. To reveal the influence of T on $X\left(\omega \right)$ more clearly, we plot $X\left(\omega \right)$ versus $\omega /{\omega }_m$ for g = 0 and g = 0 in Fig. 3(a). It is shown that the central peak is absent for g = 0. The nadir part of $X\left(\omega \right)$ for g = 0 almost overlaps with the nadir part of the case g = 0. From this comparison, we can infer that the nadir part of $X\left(\omega \right)$ is independent of the interaction between the light and the mechanical mode. It is mainly originated from the quantum interference between the first quantum path and the second quantum path. Since the thermal noise of the mechanical mode is only involved in the third quantum path, the value of the nadir part is scarcely affected by the thermal temperature T. Moreover, we also plot thermal-related homodyne spectrum $X^{\prime }\left(\omega \right)=X{(\omega )}_{g\ne 0}-X{(\omega )}_{g=0}$ versus $\omega /{\omega }_m$ in Fig. 3(b). We can see that $X\left(\omega \right)$ manifests a Lorentzian curve. With the increase of T, as shown in Fig. 3(b), the peak value increases further. The Lorentzian curve is attributed to the third quantum path. For larger T, the contribution of the thermal noise of mechanical mode to the homodyne spectrum is more obvious.

It is interesting to compare the effect of the thermal noise in conventional OMIT system and in our measurement system. For conventional optomechanical system, the interaction of the mechanical mode and the light field at control frequency leads to upconversion process and yields cavity field at probe frequency. The quantum interference between this upconversion process and the direct coupling of the probe field to the cavity field results in the OMIT phenomenon. Owing to the thermal noise in quantum interference, the nadir value in OMIT is influenced by the thermal noise of the mechanical mode. With the increase of the thermal temperature T, the nadir value approaches to or exceeds the peak values. Thus the OMIT curve is ruined. However, in our measurement scheme the quantum interference between the nadir value is determined by the quantum inference between the first quantum path and the second quantum path. The nadir value is independent of the thermal noise. The upconversion process that contains the thermal noise only influences the peak value. For our measurement scheme, the peak value increase further for higher T with the nadir value fixed. As a result, the double OMIT curve in our scheme is robust against the thermal noise.

 

Fig. 3 (a) Homodyne spectrum $X\left(\omega \right)$ as a function of $\omega /{\omega }_m$ for $J={\kappa }_1$,$T=100mK$,N = 5. (b) The thermal-related homodyne spectrum $X^{\prime }\left(\omega \right)=X{(\omega )}_{g\ne 0}-X{(\omega )}_{g=0}$ versus $\omega /{\omega }_m$. Other parameters are chosen as in Fig. 2(a).

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Figure 2(c) presents the plot of the homodyne spectrum $X\left(\omega \right)$ versus generalized frequency for different photon number N of input squeezed field. With the increase of N, the two nadirs and the two sideband peaks increase remarkably. As mentioned in the above part, the nadir part is related to the quantum interference between the first quantum path and the second quantum path. Thus the nadirs and the sidebands in our scheme are more sensitive to the influence of the input squeezed field. Moreover, the relative heights of the central peak to the nadirs are almost invariant. This result verifies the fact that central peak is dependent on the third quantum path and is more sensitive to the thermal noise.

 

Fig. 4 (a) Homodyne spectrum $X\left(\omega \right)$ as a function of $\omega /{\omega }_m$ for different temperature. (b) The rescaled central peak value versus the temperature T. Other parameters are chosen as in Fig. 2(a).

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4. Measurement of the environmental temperature

Based on the above properties of the homodyne spectrum $X\left(\omega \right)$, we intend to propose a precision measurement scheme of the environmental temperature. As shown in Fig. 4(a), the central peak at frequency ω = 0 increases with the environmental temperature over a wide temperature range. Introducing the normalized central peak value $X_p=X\left(0\right)/X_0$ with $X_0=22.11$ as the value of $X\left(0\right)$ at zero temperature, we can infer from Fig. 4(b) that normalized central peak height X_p varies with the environmental temperature T linearly. The linear dependence of $X\left(0\right)$ on T is due to the fact that the first two terms of Eq. (34) approximately equal to $4k_B{\gamma }_{m}m|V\left({\omega }_m\right){|}^{2}T$ for $\omega =0,T\to 0$. Thus the environmental temperature can be obtained by the measurement of X_p. What is more, with the increase of the environmental temperature, the relative height of the central peak to the nadirs rises. In contrast to the measurement in [45], the upper measurable limit of our proposal is not restricted by the resolution limit of the observation. In the meantime, the sensitivity of the measurement can be calculated from the slope $k=\frac{\partial T}{\partial X_0}=7.9\times {10}^{-3}K$. Assuming the measurement precision of the peak value $\delta X\left(0\right)=1\%$, the lower limit of the detectable temperature change can be as small as $7.9\times {10}^{-5}K$, which is of the same order with the detectable temperature in [35, 45].

 

Fig. 5 The quantum signal visibility V_QS as a function of the photon number N and the temperature T with $J={\kappa }_1$. Other parameters are chosen as in Fig. 2(a).

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For an accurate measurement, the central peak should be clearly distinguished from the nearby two nadirs. To analyze the measurement sensitivity, we introduce quantum signal visibility $V_{QS}=$(peak-nadir)/peak as in [35]. A reliable measurement requires large enough V_QS so that the central peak can be distincted from the nadirs. Here we assume that the minimum quantum signal visibility is 1%. The variation of V_QS versus environmental temperature T and photon number N of squeezed probe field is presented in Fig. 5. As shown by Fig. 5, V_QS is larger than $0.01$ for the presented phase space ($0<N<100$ and $0<T<1$K). The double OMIT curve can be observed for the parameters from this phase space. Especially, for T and N chosen from the right below part of Fig. 5, the quantum signal visibility V_QS can be larger than $0.5$. As noted in the former section, the values of the central peak and the two nadirs are mainly determined by the term involving T and the terms involving N respectively. For the values of T and N related to the right below part of Fig. 5, the value of central peak is evidently larger than that of the nadirs.

To conclude this part, we would like to compare our measurement scheme with those mentioned in [34–45]. For the measurement schemes in [34–45], the values of the nadir part is dependent on the quantum interference between two quantum paths to excite cavity field at probe frequency. Besides the direct coupling with the probe field, the cavity field can be generated from the anti-Stokes process between the coupling field and the mechanical mode. The thermal noise of the mechanical mode is involved in anti-Stokes process and can not be neglected for near resonance case. Thus the nadir part of the homodyne spectra in [34–45] is sensitive to the thermal noise. With the increase of T, the value of the thermal noise terms will exceed those of the quantum signal terms. As a result, the peak is buried by the nadirs and these measurement schemes lose efficacy. On the contrary, the nadir part of the homodyne spectra in our measurement scheme is exempted from the thermal noise of the mechanical mode. As mentioned in section 3, there are three quantum paths to excite cavity field at the probe frequency. The first one is the direct coupling from the probe field to the cavity field a₁. The second one is from cavity a₂ to cavity a₁. The third one is the anti-Stokes process related to the mechanical mode. The quantum interference between the former two paths leads to the nadir part of the homodyne spectra. The involvement of the third quantum path gives rises to the Lorentzian central peak. The thermal noise is only introduced in the third quantum path. With the increase of T, the value of the nadir part is fixed while the central peak increases. The central peak will not be buried by the nadirs for large thermal temperature T. Thus our measurement scheme can be applied for wide-range precision temperature measurement.

5. Extended discussion

The former analysis is based on the two-identical-cavity system. This part will focus on a more common case in which the decay rates and the frequency detunings of the two cavities are different.

 

Fig. 6 (a) Homodyne spectrum $X\left(\omega \right)$ as a function of $\omega /{\omega }_m$ for $J={\kappa }_1$,$T=100mK$,N = 5. (b) The rescaled central peak value X_p (in unit of $X_0=22.11$) as a function of T. Other parameters are chosen as in Fig. 2(a).

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First, we consider the case with different cavity decay rates of the two cavities. Figure 6(a) presents the homodyne spectrum $X\left(\omega \right)$ versus the generalized frequency $\omega /{\omega }_m$ for fixed κ₁ and different κ₂.The red dashedline in Fig. 6(a) corresponds to the case of identical decay rate. When ${\kappa }_2=0.5{\kappa }_1$ (shown by the red solid line), compared with the identical case, the height of the central peak and the nadirs shrink. Meanwhile, the profile of two sideband peaks becomes sharper. As κ₂ deviates from the value of κ₁, the quantum interference among the three quantum paths to excite a₁ is weakened. Thus the double OMIT effect is not as obvious as that in the identical cavities case. The blue dotted line in Fig. 6(a) corresponds to the case of ${\kappa }_2=2{\kappa }_1$. In this case, there exists a sharp central peak above a Lorentz-type curve. The field components in cavity a₂ will decay quickly and the formation of both the second and the third quantum path are prevented. As a result, for larger κ₂ quantum interference between the three quantum paths cannot be achieved and double OMIT curve disappears.

Figure 6(b) shows the the linear variation of the central peak value with respect to the temperature. The three lines are almost parallel. Straightforward calculations of the slopes shows that the measurement sensitivity of the three cases are approximately the same. This measurement scheme shows high sensitivity even for ${\kappa }_2=2{\kappa }_1$. Thus this scheme is applicable for the case of cavities with different decay rates.

 

Fig. 7 Homodyne spectrum $X\left(\omega \right)$ versus $\omega /{\omega }_m$ (a) for different Δ₂ with ${\mathrm{\Delta }}_2={\omega }_m$, (b) for different Δ₁ with ${\mathrm{\Delta }}_1={\omega }_m$.(c) The rescaled central peak value ${X^{\prime }}_p$ as a function of T for different detunings.

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The following part analyzes the case of cavities with different frequency detunings. We assume the two cavity detunings satisfy ${\mathrm{\Delta }}_i={\omega }_m,{\mathrm{\Delta }}_j=\left(1+s\right){\omega }_m,i,j=1,2,i\ne j,\left|s\right|<1$. Obviously, $s{\omega }_m$ is the frequency deviation from the resonant condition ${\mathrm{\Delta }}_{1,2}={\omega }_m$. Figure 7(a) corresponds to the case ${\mathrm{\Delta }}_2={\omega }_m,{\mathrm{\Delta }}_1=\left(1+s\right){\omega }_m$. For ${\mathrm{\Delta }}_1=0.9{\omega }_m$ and ${\mathrm{\Delta }}_1=1.1{\omega }_m$, there appears double OMIT curve and the two curves almost overlap. As for ${\mathrm{\Delta }}_1=0.8{\omega }_m$ and ${\mathrm{\Delta }}_1=1.2{\omega }_m$, the curves become asymmetric and both of the sideband peaks decrease. The quantum interference among the three paths has been reduced by the frequency deviation from the resonance.

Figure 7(b) is related to the case ${\mathrm{\Delta }}_1={\omega }_m,{\mathrm{\Delta }}_2=\left(1+s\right){\omega }_m$. The symmetrical double OMIT curves show up for relatively small frequency deviation. For ${\mathrm{\Delta }}_2=1.2{\omega }_m$ and ${\mathrm{\Delta }}_2=0.8{\omega }_m$, the double OMIT curves become asymmetric and the values of the nadirs increase.

Figure 7(c) presents plots of the central peak values versus the environmental temperature T in different cases. It shows that the central peak of the output homodyne spectra varies linearly with the environmental temperature T for different frequency detuning. For different values of ${\mathrm{\Delta }}_{1,2}$, the slopes are different. The slopes of the red dashed and blue dotted curves are flatter than the magenta short-dashed curves. The measurement sensitivity for ${\mathrm{\Delta }}_1=0.8{\omega }_m$ and ${\mathrm{\Delta }}_2=1.2{\omega }_m$ are better than the counterpart of the case ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\omega }_m$. Thus the measurement scheme is robust against the small frequency deviation from the resonance and the measurement sensitivity can be improved by adjusting the cavity detunings.

6. Conclusion

This paper proposes a wide-range temperature measurement scheme based on a double-cavity optomechanical system. From the homodyne spectra of the output field, we find that double OMIT curve can be observed by adjusting the coupling strength between the two cavities. In addition, the central peak value of double OMIT varies linearly with respect to the environmental temperature. The precision measurement of the environmental temperature is carried out by the measurement of the value of the central peak in double OMIT. Even for the cases of cavity with different decay rates and frequencies, this scheme is still efficient and it shows high measurement sensitivity. This measurement scheme is robust against the thermal noise of the mechanical resonator, thus it can be used as a wide-range quantum thermometer.