1. Introduction

Sensing has played an important role in cutting-edge technologies, associated with the-state-of-the-art metrology platforms for measurements of various physical quantities [1], ranging from magnetic and electric fields [2,3], to time and frequency [4], to minute force [5], acceleration [6], and mass [7,8]. In contrast to the conventional Hermitian systems, where the sensing of weak perturbations hinges on the linear response of resonant spectrum shifts or splitting to a perturbation parameter, recent advances have found that a new type of sensing schemes can be built on the non-Hermitian Hamiltonians with Parity-Time (PT) [9–12] or anti-PT symmetry [13–18]. The spectra for such a kind of non-Hermitian Hamiltonians typically exhibit exceptional point (EP) singularities, at which the eigenvalues and the corresponding eigenvectors coalesce [8,19–22]. As a result, an external perturbation on the system parameter across the EPs leads to phase transitions between the PT (anti-PT) phase with a purely real (imaginary) spectra and the broken PT (anti-PT) phase with complex eigenvalues. Moreover, the response to linear perturbations shows a power-law divergence behavior in the first-order derivative with respect to the perturbation parameter [14], which makes the PT and anti-PT symmetric systems the good candidates for sensitivity enhancement [17,23–25]. The EP physics has been experimentally demonstrated in different settings for realizing nonreciprocal light transport [26,27], single-mode lasers [28,29], time-asymmetric topological operations [30,31], enhanced sensing [9,20,32,33], as well as energy-difference conserving dynamics [34].

While early EP studies have focused on sensing linear perturbations with non-Hermitian systems [17], recent attentions have been paid to the detection of anharmonic perturbations, which are essential for quantum state engineering and creation of non-Gaussian bosonic fields. As an example, Nair et al. have recently proposed an anti-PT symmetry sensing enhancement scheme in the context of a weakly anharmonic yttrium iron garnet sphere interacting with a cavity via a fiber waveguide in the vacuum [18]. In contrast to the gain-loss balanced system in the PT symmetric phase [9–11], which has purely real eigenvalues, the system in the anti-PT symmetric phase may have a unique singular point corresponding to the emergence of only one real eigenvalue, which is referred to as the linewidth suppression caused by the vacuum induced coherence (VIC) [35–40]. It was shown that the nonlinear response of the cavity intensity around the singularity can efficiently quantify the strength of the anharmonicity explicitly included in the Kittel mode of spins [41]. Besides the cavity-magnonics [24], many other physical settings, which have been tailored to exhibit anti-PT symmetry [42–45] and to sense linear perturbations [15,17,46], may have potentials to detect fundamental nonlinear oscillators, manifesting themselves in many modern fields of physics, e.g., quantum electrical circuits [47], cold atoms [48], levitated nanoparticles [49,50], and optomechanical systems [51,52]

In this paper, we study enhanced sensing of optomechanical interactions with a coupled cavity-waveguide system. The system consists of an optomechanical cavity and an auxiliary cavity, which are coupled via a one-dimensional waveguide in the vacuum. The Hamiltonian of the system can be anti-PT symmetric if the two cavities are dissipatively coupled, which is valid only when the phase accumulation of light propagation from one cavity to the other is precisely a multiple of $2\pi$. When a phase deviation is introduced, a coherent coupling between the cavity modes is induced and the anti-PT symmetry can not hold. Then, there does not exist the eigenvalue with vanishing imaginary part, and the eigenmodes are subject to a finite loss to the waveguide, which may be detrimental to sensing of anharmonicity in the cavity modes [18]. On the other hand, it is well known that optical bistability can appear in the presence of a Kerr nonlinearity with appropriate cavity drivings. However, the standard anti-PT symmetric system shows bistability only when the frequency detuning between the cavity modes is large enough. This condition is related to the broken anti-PT phase, where the VIC induced linewidth suppression can not be found [53]. Here, by considering the optomechanically induced Kerr-type nonlinearity and introducing a phase deviation to the field propagation, we find that optical bistability can occur at the cavity resonance, where the system Hamiltonian does not have the purely real eigenvalue, but the VIC induced linewidth suppression still holds, corresponding to a long-lived eigenmode. By combining the advantages of the resonant linewidth suppression and the optical bistability, a strong bistable response of the cavity intensity to the OMIN can be found. By comparing the responses to two optomechanical strengths with a factor of 3 difference, we identify two working regions for sensing of the OMIN. In the first region, the system may display monostability (or optical bistability) for both coupling strengths, where the sensitivity measured by the enhancement factor can be double compared to that of the anti-PT symmetric model. In the second region, the system displays monostable behavior for one strength and optical bistability for the other, then a remarkably high enhancement factor of a few hundreds can be achieved. Moreover, the sensing scheme is resistant to a weak cavity decay, and effectively functions with the enhancement factor remaining a few tens. As an example, we envision a cascaded optomechanical setup where an optically levitated nanodiamond couples to two waveguide-coupled microspheres [54]. The model can be potentially generalized to other integrated optomechanical cavity-waveguide systems, such as hybrid cavity-magnonic systems [55,56], optical crystal circuits [57], and microwave optomechanical circuits [58].

The remainder of the paper is organized as follows. In Sec. II, we first derive the coherent and dissipative couplings between two optical cavities coupled via a vacuum waveguide. In Sec. III, we recall the anti-PT symmetric system constructed by the coupled cavity-waveguide setup, and discuss the VIC induced linewidth suppression significantly for sensing of the OMIN. In Sec. IV, we examine the coherence induced bistability around the cavity resonance, based on which, the sensitivity enhancement can be realized, as studied in Sec. V. We discuss the experimental feasibility of the scheme in Sec. VI and summarized our results in Sec. VII.

2. Theoretical model

As schematically shown in Fig. 1, we start by considering the general model where two cavity modes $\hat {a}_{1}$ and $\hat {a}_{2}$ of frequencies $\omega _{1}$ and $\omega _{2}$, respectively, are coupled via a waveguide, and have the decays rates $\kappa _{1}$ and $\kappa _{2}$. The cavity mode $\hat {a}_{1}$ is coupled to a mechanical mode of the frequency $\omega _{m}$ via the standard optomechanical interaction. For simplicity, we assume that the waveguide has a linear dispersion relation $\omega (k)=v_{G}|k|$ for the left- and right-going waveguide modes of wavevector $k$. Working at the rotating frame with respect to the central frequency $\omega _{0}\equiv (\omega _{1}+\omega _{2})/2$ of the two coupled cavities, the Hamiltonian of the system is then given by ($\hbar =1$) [59–62]

 

Fig. 1. Schematic of the optomechanical cavity-waveguide system. A cavity optomechanical system (consisting of an optical mode $\hat {a}_{1}$ and a mechanical mode $\hat {b}$) is coupled to the auxiliary cavity mode $\hat {a}_{2}$ through an optical waveguide with a linear dispersion relation. $\hat {c}_{L,\text {in}}(x_{j},t)$ and $\hat {c}_{R,\text {in}}(x_{j},t)$ denote the left- and right-moving fields which interact with the cavity modes at the position $x_{j}$ with the strength $\Gamma$. The cavity modes $\hat {a}_{j}$ decay into the surrounding environments with the rates $\kappa _{j}$.

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The optical cavities can be dissipatively coupled by reservoir engineering the waveguide field, and can be further designed as an anti-PT symmetric system [18]. To see this, we first leave the optomechanical interaction out and include it later as the probe. Then, the Heisenberg equations of motion for the waveguide modes are given by [60]

By integrating the field equations across the discontinuity at $x_{j}$, one obtains the input-output relations between the cavity modes and the left- (right-) going fields [60]

For clarity, we can further define $\hat {c}_{L,\text {in}}(x_{j},t)\equiv \hat {c}_{L}(x_{j}^{+},t)$ and $\hat {c}_{R,\text {in}}(x_{j},t)\equiv \hat {c}_{R}(x_{j}^{-},t)$ as the input fields to the cavities from the waveguide, and $\hat {c}_{L,\text {out}}(x_{j},t)\equiv \hat {c}_{L}(x_{j}^{-},t)$ and $\hat {c}_{R,\text {out}}(x_{j},t)\equiv \hat {c}_{R}(x_{j}^{+},t)$ as the output fields. This leads to the familiar form of the standard input-output relations:

In addition, since the waveguide field propagates freely between the cavities and the waveguide dispersion is linear over all frequencies, we have [59,63,64]

3. VIC induced linewidth suppression

For the propagating phase being a multiple of $2\pi$, i.e. $\Phi =2n\pi$ ($n\in \mathit {Z}$), the coherent coupling with the amplitude $(\Gamma /2)\text {sin}\Phi$ between the cavity modes vanishes [59], leading to a purely dissipative hopping interaction of the strength $\Gamma /2$. The matrix $\mathcal {H}$ reduces to the standard anti-PT symmetric form

For $\Phi =2n\pi +\phi$ with a finite phase deviation $\phi \neq 0$, the matrix $\mathcal {H}$ becomes $\phi$ dependent. A weak coherent coupling with the amplitude $(\Gamma /2)\text {sin}\Phi \sim \Gamma \phi /2$ for $\phi \ll \pi$ is introduced to the coupled cavity modes, and the anti-PT symmetry breaks down. The eigenvalues of the matrix $\mathcal {H}$ now read

Compared with the case of the completely dissipative coupling regime, both the real and imaginary parts of $\lambda _{\pm }$ become nondegenerate and nonzero regardless of the value of $\delta$, see the solid lines in Figs. 2(a)-(b). In other words, there does not exist the EPs which strictly correspond to the degeneracy points in the spectra. In this case, we note that $\theta =2\phi$ at $\delta =0$, and thus $\text {Im}\lambda _{+}\approx -\frac {1}{2}(\kappa +\frac {\Gamma }{2}\phi ^{2})$, as indicated by the arrow in Fig. 2(b). It implies that the system is now subject to a weak loss induced by the waveguide-mediated coherent coupling, namely, there does not exist the real singularity with a completely vanishing linewidth (i.e. the VIC induced zero linewidth) even though the cavity decay $\kappa$ is set to zero. When the system is engineered to be anti-PT symmetric, a dissipative eigenmode, corresponding to the eigenvalue $\lambda _{+}$ with a nonvanishing imaginary part $\text {Im}\lambda _{+}\neq 0$, may reduce the sensitivity for sensing of weak anharmonicity in the cavity modes [18]. However, the imaginary part of $\lambda _{+}$ remains negligible compared to $\Gamma$ (i.e. $\text {Im}\lambda _{+}/\Gamma \sim 10^{-3}$), as the result of the coherence induced symmetry breaking, see the inset in Fig. 2(b). Indeed, we find that a weak coherent coupling $(\Gamma /2)\text {sin}\phi$ still allows for a long-lived cavity resonance, and remarkably, it can induce optical bistability around the cavity resonance at weak cavity driving power, which can not happen for the anti-PT symmetric Hamiltonian Eq. (18) with $\delta =0$ [53]. By combining the linewidth suppression with the optical bistability, the sensitivity of detecting the OMIN can be greatly enhanced, and moreover, can be resistant to a finite inherent cavity linewidth, see the details in Sec. V.

 

Fig. 2. (a) Real parts (eigenfrequencies) and (b) imaginary parts (linewidths) of the eigenvalues of $\mathcal {H}$ versus the detuning $\delta /\Gamma$ for $\phi =0$ (dashed) and $\phi =-0.03\pi$ (solid), respectively. The cavity decay rates are set to zero such that the VIC induced zero linewidth ($ \textrm{Im} {\lambda _{+}/\mathrm{\Gamma} =0}$) arises at the resonance ($\delta =0$) for the anti-PT symmetric regime $\phi =0$. The EPs disappear for $\phi \neq 0$, but the VIC induced linewidth suppression (with $\textrm{Im} {\lambda _{+}/\mathrm{\Gamma} \rightarrow 10^{-3}}$) remains effective for $\delta \rightarrow 0$ and $\phi \ll \pi$, as can be seen from the inset. Here, we consider a finite cavity linewidth with $\kappa /\Gamma =0.002$.

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4. Coherence induced bistability

To demonstrate enhanced sensing of the OMIN with the coupled cavity-waveguide system, we now include the optomechanical coupling term as a perturbative probe, and consider the classical Langevin equation of motions for the mean (steady-state) values of the field and mechanical operators $\alpha _{j}\equiv \langle \hat {a}_{j}\rangle$, $\langle \hat {q}\rangle$ and $\langle \hat {p}\rangle$ under the mean-field approximation $\langle \hat {q}\hat {a}_{1}\rangle \approx \langle \hat {q}\rangle \langle \hat {a}_{1}\rangle$ and $\langle \hat {a}_{1}^{\dagger }\hat {a}_{1}\rangle \approx |\alpha _{1}|^{2}$. It follows that

In the steady state for Eqs. (23)–(26), i.e., $\dot {\alpha }_{1}=\dot {\alpha }_{2}=\langle \dot {\hat {q}}\rangle =\langle \dot {\hat {p}}\rangle =0$, and by eliminating the mechanical degree of freedom, we find the modified relations for the field amplitudes

Moreover, by inserting Eq. (28) into Eq. (27), the cavity intensity $\beta \equiv \left |\alpha _{1}\right |^{2}$ is found to satisfy a cubic relation

For the completely dissipative coupling regime $\Phi =2n\pi$, Eq. (29) entails a bistable response to a strong driving $I$ only when $|\delta |>\sqrt {3}(\kappa +\Gamma )$, namely, a bistability behavior can not be found in the anti-PT symmetry phase corresponding to $|\delta /\Gamma |<1$ . However, we find that a bistable response to both $\phi$ and $\delta$ can happen by including the weak coherent coupling $\sim \frac {\Gamma }{2}\text {sin}\Phi$, at low driving power and small detuning $|\delta /\Gamma |\ll 1$. For the driving power dependence of the cavity intensity $\beta$, the bistability turning points can be derived from Eq. (29) by inspecting the solutions of $\text {d}I/\text {d}\beta =0$, which turn out to be

Thus, the necessary conditions for observing a bistable signature are $A<0$ and $A^{2}>3B$, which are shown in Fig. 3 by plotting them against $\delta$ and $\phi$. The boundaries $A=0$ and $A^{2}=3B$ between the monostable and the bistable regions are indicated by the dashed lines. It can be seen that the bistable region embraces the small detuning regime $|\delta /\Gamma |\rightarrow 0$ for a finite negative phase $\phi <0$, namely, the VIC induced linewidth suppression and the static optical bistability can co-exist. By combining the two effects, the response of the cavity intensity $\beta$ to optomechanical interactions around $|\delta /\Gamma |\ll 1$ can be strong, which is important for the sensing scheme. However for $\phi >0$, only a monostable behavior can be found.

 

Fig. 3. The necessary conditions for bistability. (a) Dimensionless $A/\Gamma$ and (b) $\left (A^{2}-3B\right )/\Gamma ^2$ plotted against $\phi$ and $\delta$. The black dashed lines indicate $A=0$ in (a) and $A^{2}=3B$ in (b).

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5. Sensing optomechanical interaction enhanced by bistability

Since the weak single-photon optomechanical coupling $g/\omega _{m}$ is typically on the order ranging from $10^{-7}$ to $10^{-4}$ [52,65–69], the optomechanical interaction induced Kerr nonlinearity $\chi =g^{2}/\omega _{m}$ can only be measured with a reasonably strong laser driving. Its effect is reflected in the response of the cavity intensity $\beta$ of the cavity mode $\hat {a}_{1}$. For the standard anti-PT symmetric system with $\phi =0$ and $\kappa =0$, the response to $g$ at $\delta =0$ can be captured by the functional dependence $\beta \propto g^{-4/3}$ for $A=B=0$, according to which the sensitivity to $g$ is given by $|\delta \beta /\delta g|\propto g^{-7/3}$. Thus, a triple decrease in $g$ (or approximately a tenfold decrease in $\chi$) scales up the peak value of $\beta$ by a factor of $\sim 4$.

For $\phi \neq 0$, the response of $\beta$ to $g$ can be examined by solving Eq. (29), which may have one or two stable solutions. When optical bistability occurs, a simple functional dependence for $\delta \beta /\delta g$ can not be analytically given anymore since the terms with respect to $A$ and $B$ can not be ignored. It is just the case considered in this work, and thus, the sensitivity to variations in $g$ should be defined otherwise. Here, $\beta$ is plotted against the detuning $\delta$ and the phase deviation $\phi$ in Fig. 4(a) for the typical set of parameters: $g/2\pi =3$ Hz, $\Gamma /2\pi =100$ MHz, $\kappa /\Gamma =2\times 10^{-3}$, and $P_{\text {in}}\approx 8$ mW with $\lambda _{d}=2\pi c/\omega _{d}=1550$ nm [70,71]. As expected, the bistability occurs only for negative $\phi$. The asymmetry in the diagram with respect to $\phi$ arises from the fact that the stability condition Eq. (32) does depend on the sign of the coherent coupling strength $\frac {\Gamma }{2} {\sin {\phi }}$. Note that for weak cavity driving, the detuning and the weak coherent coupling associated with the bistable region, which is encircled by the white dashed line, are limited to $|\delta /\Gamma |<0.14$ and $-0.012\Gamma <\frac {\Gamma }{2} {\sin {\phi }}<-0.003\Gamma$, respectively. We emphasize that, for $\phi =0$, optical bistability can occur only for a strong driving and a sufficiently large detuning $|\delta |>\sqrt {3}(\kappa +\Gamma )$, corresponding to the case of the completely dissipative coupling regime.

 

Fig. 4. (a) Cavity intensity $ { {\beta }}$ plotted against $\phi$ and $\delta$ for optomechanical coupling strength $g/2\pi =3$ Hz. The white dashed line encircles the bistable region. (b) Cavity intensity $ {{\beta }}$ on resonance $\delta =0$ versus $\phi$ for optomechanical coupling strengths $g_{1}/2\pi =1$ Hz and $g_{2}/2\pi =3$ Hz, respectively. $\phi _{1}$ and $\phi _{2}$ denote the upper turning points of the curves, which are the boundaries related to the two highlighted regions I and II. (c) Cavity intensity $ {{\beta }}$ versus $\delta /\Gamma$ with $\phi =0$ (the left panel) and $\phi =-0.008\pi$ (the right panel). (d) The enhancement factor $\eta$ defined by Eq. (33) plotted against $\phi$ and $\delta$. (e) $\eta$ versus $\phi$ for $\delta /\Gamma =0$. (f) $\eta$ versus $\delta /\Gamma$ with $\phi =0,\text { }-0.008\pi$. Other parameters are: $\Gamma /2\pi =100\text { MHz}$, $\omega _{m}/2\pi =10$ kHz, $\kappa /\Gamma =2\times 10^{-3}$, $P_{\text {in}}=8.06$ mW with $\lambda _{d}=1550$ nm.

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For studying the sensitivity to $g$, we look into the modified bistability behavior of $\beta$ by comparing its responses to two different optomechanical coupling strengths $g=g_{1}=2\pi \times 1$ Hz and $g=g_{2}=2\pi \times 3$ Hz. As shown in Fig. 4(b), we plot the cavity intensity $\beta$ as a function of the phase deviation $\phi$ with $\delta =0$ for $g=g_{1}$ and $g=g_{2}$, respectively. In both cases, $\beta$ has two stable branches (solid) and one unstable branch (dashed). Recalling that the perfect anti-PT symmetric system relies on the fulfillment of the condition $\phi =0$, here when $\phi$ is slowly swept backward, the cavity intensity $\beta$ responses to the weak nonlinearity $\chi \propto g^{2}$ in a counter-intuitive way: the system shows bistability for both the optomechanical coupling strengths $g=g_{1}=2\pi \times 1$ Hz and $g=g_{2}=2\pi \times 3$ Hz, but the larger $g$ (or the nonlinearity $\chi$) is, the smaller peak value of the cavity intensity $\beta$ has. For comparison with the anti-PT symmetric model, we show in Fig. 4(c) the cavity intensity $\beta$ versus $\delta$ for $\phi =0$ (the left panel) and $\phi =-0.008\pi$ (the right panel), corresponding to the monostable and bistable regimes, respectively. In both cases, $\beta$ shows flat central peak around $\delta =0$, benefiting from the VIC induced linewidth suppression. But for $\phi =-0.008\pi$, $\beta$ has a peak value about four (two) times greater than that for $\phi =0$ with $g=g_{1}$ ($g=g_{2}$). It proves that the optical bistability enables a stronger response of $\beta$ to $g$.

Considering the sensing of the OMIN $\chi$, there are generally two working regions [as highlighted in Fig. 4(b)], corresponding to $\phi \in (\phi _{1},0)$ (region I) and $\phi \in (\phi _{2},\phi _{1})$ (region II), respectively. Here $\phi _{1}$ and $\phi _{2}$ are the phase deviations corresponding to the turning points of the curves, where the upper branch starts from and the middle unstable branch ends. In both regions I and II, we observe optical bistability in one of the responses and thus the sensitivity can not be given by a simple functional dependence for $\delta \beta /\delta g$. Instead, we introduce the parameter $\eta (\phi,\delta )$, which is the ratio of the cavity intensities $\beta (g_{1})$ and $\beta (g_{2})$ at the upper branches with respect to the two optomechanical coupling strengths $g_{1}/2\pi =1\text { Hz}$ and $g_{2}/2\pi =3\text { Hz}$, as the measure of the sensitivity, namely

We call $\eta (\phi,\delta )$ the enhancement factor if ${\beta (g_{1})}/{\beta (g_{2})}>1$, distinguishing from the conventional definition of sensitivity $\delta \beta /\delta g$ and being appropriate for region I. With that in hand, the responses of the cavity intensity to $g_{1}$ and $g_{2}$ with $g_{2}/g_{1}=3$ [or $\chi (g_2)/\chi ({g_1})=9$] may give rise to $\eta (\phi,\delta )\gg 1$. This is to be contrasted with the $ { {\mathrm{\Phi} =2n\pi }}$ case, where the enhancement factor is $\eta (0,0)\sim 4$ for an otherwise identical set of parameters. Thus, the enhancement factor can, thus, serve as an efficient measure for sensing of the OMIN. While for region II, $\eta ^{-1}$ is used as the enhancement factor instead since $\eta \ll 1$. For an overview, we show $\eta (\phi,\delta )$ versus $\phi$ and $\delta$ in Fig. 4(d), where the red region corresponds to the increased enhancement factor with the maximum $\eta _{\text {max}}\sim 7.5$ for a triple decrease in $g$. In region I, for both coupling strengths the system displays monostability near $\phi =0$ and optical bistability near $\phi =\phi _{1}$. The enhancement factor $\eta (\phi,0)$ at $\delta =0$ is gradually increased for a backward sweep of the phase deviation from $\phi =0.003\pi$ to $\phi =\phi _{1}\approx -0.008\pi$, as shown in Fig. 4(e). It achieves the maximum $\eta (\phi _{1},0)\sim 7.3$, which is about twice that with $\phi =0$. Moreover, a slight detuning allows for a better enhancement factor for $\phi \approx -0.008\pi$, as manifested by the $\delta$ dependence of $\eta$ shown in Fig. 4(f). The optimal $\eta (\phi =-0.008\pi,\delta )$ turns up at $\delta _{\text {opt}}\approx -0.025\Gamma$, and is resistant to a detuning deviation $\sim 0.03\Gamma$ from the optimal detuning $\delta _{\text {opt}}$.

Next, we turn to region II in Fig. 4(b), where $\beta$ becomes monostable for $g=g_{1}$ while maintains the bistability feature for $g=g_{2}$. In this regime, the enhancement factor $\eta$ approaches zero as $\phi$ sweeping backward and across $\phi _{1}$, but the inverse of it (as the new enhancement factor) $\eta ^{-1}$ can be more than thirty times larger than the maximal $\eta$ at region I. As shown in Fig. 5(a), we plot $\eta ^{-1}$ as a function of $\phi$ with $\delta /\Gamma =-0.08$. Remarkably, $\eta ^{-1}$ increases from 0.3 to 240 in an almost monotonic manner when $\phi$ is swept backward from $\phi =0$ to $\phi =\phi _{2}=-0.024\pi$, except a weak dip at $\phi =-0.003\pi$ corresponding to the maximal $\beta (g_{1})$. Furthermore, Fig. 5(b) shows that the high enhancement factor (the maximum $\eta ^{-1}$) remains flat within a certain frequency range, which can be referred to as the bandwidth for the measurement of the OMIN. Here the bandwidth is about $0.06\Gamma$ ($0.13\Gamma$) for $\phi =-0.023\pi$ ($\phi =-0.018\pi$). Therefore, the coupled cavity-waveguide system can act as a fluctuation-resistant and highly sensitive sensor.

 

Fig. 5. (a) The enhancement factor $\eta ^{-1}$ plotted against $\phi$ at $\delta =-0.08\Gamma$; (b) $\eta ^{-1}$ plotted against $\delta$ at $\phi =-0.018\pi$, $-0.023\pi$; (c) Cavity intensity $ { {\beta }}$ on resonance $\delta =0$ versus $\phi$ for optomechanical coupling strengths $g_{1}/2\pi =1$ Hz and $g_{2}/2\pi =3$ Hz, where the cavity decay rates are set to $\kappa /\Gamma =0.008$ and 0.03. (d) The enhancement factors $\eta$ and $\eta ^{-1}$ versus $\phi$ on resonance $\delta =0$ with $\kappa /\Gamma =0.008$. Other parameters are the same as those in Fig. 4.

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Finally, we consider the effect of the cavity decay by increasing $\kappa$ and meanwhile keeping the cavity driving power invariant. In Fig. 5(c), we show $\beta$ as a function of $\phi$ with $\kappa /\Gamma =8\times 10^{-3}$ and $\kappa /\Gamma =0.03$, respectively [72]. For $\kappa /\Gamma =0.03$, the responses of $\beta$ to $g_{1}/2\pi =1$ Hz and $g_{2}/2\pi =3$ Hz both show a strong quenching and become monostable. The intensities decrease sharply because a large cavity dissipation will inhibit the system from achieving long-lived resonance and amplitude accumulation. Since the profiles of the two dotted curves (with respect to $\kappa /\Gamma =0.03$) are almost overlapped, the sensing scheme becomes invalid. While for a weaker inherent decay $\kappa /\Gamma =8\times 10^{-3}$, the behavior observed at region II [shown in Fig. 4(b)] replays, namely for $\phi \in (-0.018\pi,-0.011\pi )$, the response of $\beta$ to $g_{1}/2\pi =1$ Hz becomes monostable, while the response to $g_{2}/2\pi =3$ Hz remains bistable [highlighted in Fig. 5(c)]. Thus, our model can still work as a high-performance sensor in this region. As shown in Fig. 5(d), the new enhancement factor $\eta ^{-1}$ can still achieve a few tens. In comparison, the enhancement factor $\eta$ can only reach about 3 for the anti-PT symmetric model with respect to $\phi =0$.

6. Discussions on experimental realization

To address the experimental feasibility, we consider an optomechanical setup with a levitated nanosphere [73–76], e.g. the silicon-vacancy (SiV) centers, coupling to the evanescent field of a microsphere cavity with frequency $\omega _{c}$ [54]. The dielectric nanosphere with a nanodiamond structure contains $N$ two-level quantum emitters, which are driven by a bichromatic field of frequencies symmetrically red- and blue-detuned from the atomic transition frequency $\omega _{0}$, forming an optical trap for the nanosphere at a distance $z$ from the cavity surface [54]. The emitter-cavity coupling $\Omega _{c}(\hat {z})$ then depends on the mechanical displacement of the nanosphere along the $z$ axis, with the mechanical eigenfrequency $\omega _{m}$. In the dispersive interaction regime, where the emitter-cavity detuning $\Delta _{c}\equiv \omega _{c}-\omega _{0}$ is much larger than $\Omega _{c}$, and by considering that the induced atomic level Stark shift is much less than the excited-state linewidth, one can finally obtain the effective optomechanical interaction $\hat {H}_{\text {om}}=g\hat {a}^{\dagger }\hat {a}\hat {q}_{z}$ between the cavity field and the mechanical motion, where [54]

The sensing device can then build on an optomechanical cavity-waveguide system with two microcavities [32,72,77,78], one of which couples to an optically trapped nanosphere. For the typical parameters [54]: $p_{e}\approx 3.2\times 10^{-5}$, the density of quantum emitters $\rho _{q}\approx 1.4\text { nm}^{-3}$, the radius of the nanosphere $R=15$ nm, the mechanical frequency $\omega _{m}/2\pi \sim 10$ kHz, and $\gamma _{c}^{-1}=283$ nm, one obtains the single-photon coupling strength $g/2\pi \sim$ 1 Hz (or $\chi /2\pi \sim 100\text { }\mu$Hz) with the distance between the nanosphere and the cavity surface being $z=783$ nm. Moreover, the inherent decay rate for the microcavity is typically $\kappa /2\pi \sim 10$ kHz, which is much less than the waveguide induced coupling ($\sim$100 MHz) between the cavity modes [71].

To observe optical bistability and implement the sensing scheme, we can in principle tune the propagating phase $\Phi$ by varying the distance between the two cavity-waveguide coupling points. Alternatively, the propagating phase $\Phi$ can be tuned with optical waveguide couplers, such as silicon photonic phase shifters, which have been experimentally demonstrated based on thermo-optic effect [79], electro-mechanical actuation [80,81], and opto-electro-mechanical effects [82,83]. On the other hand, the detuning $\delta$ can be tuned by varying the frequency difference between the two coupled cavities. An appropriate way to reach this end is by tuning the resonant frequency of the auxiliary cavity, which can be further realized by varying the temperature [84,85], by applying mechanical strain [86] to the resonator or by using electro-optics effects [87]. Therefore, the coupled cavity-waveguide device can be used for sensing weak perturbations on the set of parameters with respect to the single-photon optomechanical coupling strength (and the OMIN), such as the radius of the nanosphere and the steady-state excitation number of the quantum emitters for the density of emitters being fixed.

Finally, the bistable behavior can also be observed with two directly coupled cavities (one of which is optomechanical, as considered in this paper), where the coupling phase is tuned to be an odd multiple of $\pi$. However, it requires the cavities to interface each other at short distances. By contrast, the waveguide-coupled device alternatively has the merit for long-distance control of the sensing of the OMIN, and moreover, the two coupled cavities may have an interaction of the non-Hermitian form with the propagating phase deviated from $n\pi$, which is fundamentally different from the coherent case under direct tunneling coupling.

7. Conclusion

In summary, we have studied the sensitivity enhancement for measuring the OMIN with two cavities coupled via a vacuum optical waveguide. When the phase accumulation of light propagation from one cavity to the other is precisely a multiple of $2\pi$, the two cavities are dissipatively coupled. In this regime, the system can be anti-PT symmetric, and its Hamiltonian shows a singular point around the cavity resonance ($\delta =0$), corresponding to the VIC induced zero linewidth. This singularity has been exploited for sensing of a weak Kerr nonlinearity [18], but can be fragile to a nonzero coherent coupling and a finite cavity decay rate. In addition, the standard anti-PT symmetric system shows bistability only at the broken anti-PT phase with $|\delta /\Gamma |>1$ [53]. Although the standard anti-PT symmetry cannot be held when a small phase deviation is introduced, we find that both the VIC induced linewidth suppression and optical bistability can be observed around the cavity resonance $\delta /\Gamma \sim 0$. The joint effect allows us to greatly increase the enhancement factor for measuring the OMIN, which can reach a few hundreds for a triple reduction of the single-photon optomechanical coupling strength. The high enhancement factor can be retained for a bandwidth of a few megahertz, and can remain a few tens for a cavity decay rate of hundreds of kHz. While we consider the setup consisting of an optically levitated nanodiamond coupled to waveguide-coupled microspheres [54], the model can be generalized to other integrated optomechanical cavity-waveguide systems, such as hybrid cavity-magnonic systems [55,56], optomechanical crystal circuits [57], and microwave optomechanical circuits [58].