1. Introduction

Precision measurement of weak forces is crucial not only for testing and extending our understanding of fundamental physics but also for industrial applications such as geological surveys [1]. In recent efforts towards more sensitive and accurate measurements, exploitation of quantum technologies has led to breakthroughs in many areas of metrology, for example, accelerometry [2], gravimetry [3,4], and gravitational wave detection [5]. In a short period, some realizations of these technologies have already reached precision levels comparable to their best classical counterparts, and further reduction of the estimation error by utilizing various quantum resources coupled to the measurement system is of much interest [6].

Among numerous types of quantum systems, cavity optomechanics is an ideal platform for displacement/force measurement owing to the radiation pressure aptly mediating elemental quantum entities, photons and phonons, in a coherent manner [7,8]. To date, technical development and ongoing research in this platform have led to the successful preparation of nonclassical phonons such as quantum ground states and squeezed states in optomechanical systems [9].

In quantum estimation theory, as the metrological gain of precision beyond the classical limit, the Quantum Cramer-Rao bound is widely used to quantify the ultimate precision obtained from measurements on a quantum system [10,11]. Constituting this bound, the quantum Fisher information (QFI) imposes a fundamental limit to the accuracy of the relevant parameter estimation. A related scaling behavior, the standard quantum limit (SQL), was immediately derived for phase estimation: the uncertainty of a phase $\delta \theta$ in an interference-based measurement scheme is inversely related to the number of involved quanta $N$, that is, $\delta \theta \sim 1/\sqrt {N}$. Soon after, theoretical studies focused on the ultimate quantum bound surpassing the SQL and found the so-called Heisenberg limit (HL) represented by $\delta \theta \sim 1/{N}$, which is restricted only by the Heisenberg uncertainty principle [12]. The most prominent ideas for beating the SQL utilize highly non-classical states, for instance, squeezed vacuum states [13].

Optomechanical sensors have been explored extensively in terms of a noise power spectrum and a signal-to-noise ratio (SNR) [14]. While the noise power spectrum of a system highly depends on an observable from which the precision is inferred as well as the measurement scheme of which one makes use, the QCRB of a system is obtained by maximizing classical Fisher information over all possible generalized measurements, indicating that the QCRB does not rely on an observable from which the precision is inferred as well as the measurement scheme on which one relies. It is thus the case that the QCRB is an ultimate upper bound of the precision of a quantum sensor.

In previous studies, the QFI in cavity-optomechanical systems has been theoretically investigated for estimation of the optomechanical coupling strength [15,16] as well as gravitational acceleration [4,17]. We notice, however, that these studies assume small optical and mechanical dissipation which can be difficult to achieve with current experimental parameters. Furthermore, the time window of the measurements is highly restricted to short intervals during which the optical and mechanical subsystems are effectively decoupled.

Here we investigate the QFI and the related bound of precision for the steady state of an optomechanical system under realistic dissipation conditions. In particular, we identify the role of quantum noise by comparing the measurement error obtained for a coherent state driving field and a squeezed vacuum state driving field. Although squeezed states have been exploited as a means to reduce the quantum noise, the corresponding QFI from the optomechanical system has not been thoroughly discussed in the literature to our knowledge. We found a range of parameters for Heisenberg-limited measurements in the case of using a squeezed vacuum input. Our findings suggest a way to estimate the ultimate performance beyond the standard quantum limit of gravimetry based on optomechanical systems adopting quantum optical resources.

2. Model system

We consider a generic optomechanical system in which a single-mode mechanical oscillator of mass $m$ and resonant frequency $\Omega$ is coupled to a single-mode optical field of frequency $\omega _c$ in a high-Q optical resonator. When the cavity field is driven by an external monochromatic field of pump frequency $\omega _p$ and power ${\cal P}$, the mechanical oscillator is subject to radiation pressure force from the cavity field in conjunction with an external force that is the object of measurement. The resulting Hamiltonian in a frame rotating at the pump frequency is given by

The first two terms account for the cavity field pumped by an external field. The pump field detuning from the cavity resonance is $\Delta _c = \omega _p - \omega _c$ and the pump rate is $\sqrt {\kappa _{\rm in}}\alpha _{\rm in}$ where $\kappa _{\rm in}$ is the loss rate of the cavity input port mirror, and $\alpha _{\rm in} = \sqrt {{\cal P}/(\hbar \omega _p)}$ is the coherent amplitude of the external field. The last three terms describe the mechanical oscillator subject to both the radiation pressure force and the external dc force. The single-photon radiation pressure force and the external force are denoted by $g_0$ and $F$, respectively. The bosonic annihilation operator for the cavity field is denoted by $\hat a$, with $[\hat a, \hat a^{\dagger }]=1$. The dimensionless position and momentum operators of the mechanical oscillator, which are normalized by the natural length $q_0 =\sqrt {\hbar /(m\Omega )}$ and momentum $p_0=\sqrt {\hbar m\Omega }$, are denoted by $\hat q$ and $\hat p$ giving the relation $[\hat q, \hat p]=i$. Note that both the single-photon radiation pressure force and external force have units of ${\rm Hz}$.

The mechanical oscillator described above can be realized in various forms depending on the physical system, e.g., an optical cavity end mirror [18], the acoustics of a microresonator [19], a levitated mesoscopic crystal, or trapped atoms in a cavity. The analytical expression for single-photon radiation pressure force $g_0$ varies with the type of system. For example, we can set $g_0 = {\omega _c q_0}/{L_c}$ in the case of an archetypal Fabry-Pérot cavity of length $L_c$ with an moving end mirror [20,21]. In the case of a levitated nano- or micro-crystal $g_0 = {k_c \omega _c P q_0}/{(4 \epsilon _0 V_c)}$, where $k_c$ is the wavevector of the laser, $V_c$ is the volume of the cavity, $\epsilon _0$ is the vacuum permittivity, and $P=3 \epsilon _0 V_l {(\epsilon -1)}/{(\epsilon +2)}$ is the polarizability of the crystal. Here, $V_l$ and $\epsilon$ are the volume and the relative electric permittivity of the crystal [22,23]. In the case of a trapped atom we can write $g_0 = {g^{2} k_c q_0}/{\Delta _a}$, where $g$ is the atom-cavity QED coupling coefficient and $\Delta _a$ is the cavity field detuning from the atomic resonance [24,25]. An external force perturbing the mechanical oscillator can be measured in a specific manner tailored for a given system. For instance, a trichromatic lattice potential can provide an external force for ultracold atoms in a cavity [26]. Gravitational force can also affect the detectable motion of an inclining optomechanical cavity [4]. In this study, we assume that $F$ is not a quantum operator but a scalar $c$-number, i.e., the system parameter to be measured.

As a representative example of an optomechanical force sensor, a Fabry-Pérot resonator with trapped atoms is depicted in Fig. 1. The cavity supports a well-defined single-mode optical field between two fixed end mirrors: one being a perfectly reflecting mirror and the other being a partially reflecting input port mirror through which a driving field is fed and cavity field photons escape at a rate $\kappa$. For simplicity, we assume that the cavity photons leak only through the input port mirror, $\kappa =\kappa _{\rm in}$. It is also assumed that the cavity field is detuned far from the atomic resonance so that the internal state of the atom is not affected by its motional dynamics. It follows that the degree of freedom for the atom is only external in a manner where the atom experiences the radiation pressure force from the cavity field as well as the external dc force. The formalism developed in this manuscript is generic, i.e., it is not limited to this specific example but can be handily applied to other optomechanical systems.

 

Fig. 1. Trapped atoms in an optical resonator driven by light field in a coherent or squeezed vacuum state. The atoms can also in effect represent a nano-/micro-crystal or moving mirror. The center-of-mass position of the mechanics is displaced from its original equilibrium position by an external force which is measured by the system output.

Download Full Size | PDF

In order to describe cavity field dissipation and mechanical fluctuations in a quantum regime, the interaction between the optomechanical system with the optical and mechanical reservoirs should be appropriately taken into consideration. We assume that the cavity field decays to a squeezed optical reservoir with rate $\kappa$ and the mechanical oscillator is coupled to a finite temperature thermal reservoir at a rate $\gamma$. Assuming that the subsystems are weakly coupled to their reservoirs, the master equation for the density operator $\hat \rho$ of the optomechanical system reads

3. Quantum metrology

3.1 Brief introduction to quantum Cramer-Rao bound and quantum Fisher information

In parameter estimation theory the precision for the value of an unbiased estimator $F$ which determines a probability distribution $p(x|F)$ for a measurement outcome $x$ given the real parameter $F$ is limited to a lower bound called the Cramer-Rao bound

Here, $N_\textrm {rep}$ is the number of the measurement repetitions and the Fisher information $I(F)$ is given by

Therefore, the Fisher information is the variance of the score and depends on the probability distribution $p(x|F)$, providing information on the precision achievable for the relevant measurement process.

Noting that $I (F)$ relies highly on the observable $x$ that we have chosen, there is a possibility that $I (F)$ can be further improved when maximized over all possible generalized measurements. This supreme of $I (F)$ is referred to as the quantum Fisher information (QFI)

In quantum mechanics, the generalized measurement is described by a positive operator-valued measure (POVM) $\{\hat {\Pi }_x \}$: a set of positive Hermitian operators which is parameterized by $x$ and also constitutes the resolution of identity, $\sum _{x} \hat {\Pi }_x = \mathbb {1}$. It is therefore a central task to collect an optimal POVM maximizing the precision to which one can estimate $F$ in the measurement processes. Fortunately, even if we do not have knowledge of an optimal POVM leading to the supremum of $I$, there is a route to finding a closed expression for QFI [10,11]. Following Refs. [10,11], one can obtain the QFI by calculating the expectation value of the $F$-dependent Hermitian operator, the so-called symmetric logarithmic derivative (SLD) $\hat L(F)$,

Note that the SLD in Eq. (8) is an operator associated with the score in Eq. (4). Furthermore, QFI can be simplified by making use of Eq. (9) and resorting to the spectral decomposition of $\hat \rho$, i.e., $\hat \rho = \sum _i \lambda _i \! \left |{i}\rangle \!\langle {i}\right |$ as

This expression shows that one can readily obtain the QFI if both $\hat \rho$ and $\partial \hat \rho /\partial F$ are given. For example, if the quantum state is pure, i.e., $\hat \rho =|\psi \rangle \langle \psi |$, the QFI is simply

In the case where a physical system described by a density operator $\hat \rho _0$ undergoes a unitary transformation generated by a Hermitian operator $\hat {\cal H}$, the QFI for the transformed density operator $\hat \rho (F) = e^{-i \hat {\cal H} F} \hat \rho _0 e^{i \hat {\cal H} F}$ is given by

If $\rho _0$ is pure and hence $\rho (F) = \left |{\psi (F)}\rangle \!\langle {\psi (F)}\right |$ is also pure, Eq. (12) can be reduced to

Notice that the QFI for $\rho (F) = \left |{\psi (F)}\rangle \!\langle {\psi (F)}\right |$ is simply the variance of the Hermitian operator $\hat {\cal H}$ with a prefactor of four.

4. Obervables and QFI of the steady-state optomechanical system

As aforementioned, the optomechanical system of interest is coupled to optical and mechanical reservoirs where the optomechanical system undergoes an unavoidable nonunitary evolution governed by Eq. (2). In this situation, one should resort to the original definition Eq. (9) or Eq. (10) in order to find the QFI for the steady state of the optomechanical system.

Two challenges must be overcome when computing the SLD/QFI for open quantum systems. The first originates from the fact that the SLD is state-dependent. In such a situation where the Hilbert space for the optomechanical system is large, we may employ numerical methods to diagonalize the density operator to obtain the SLD. The second, which can be a more serious challenge, comes from the fact that the SLD, and hence the Fisher information, depends not only on the state $\rho$ but also on its derivative $\partial \hat \rho /\partial F$. It is often hard to numerically evaluate $\partial \hat \rho /\partial F$ since a numerical differentiation based on high-order symmetric formulas are typically ill-conditioned procedures. More importantly, numerical error propagates uncontrollably as the density operator evolves under the master equation. To get around this issue and to faithfully obtain $\partial \hat \rho /\partial F$, we follow the methods in Ref. [4]. Differentiating the original master equation for $\rho$ with respect to $F$ results in

In order to get the steady state density matrices $\hat \rho _\mathrm {ss}$ and $(\partial \hat \rho /\partial F)_\mathrm {ss}$, one should numerically solve Eq. (2) as well as Eq. (14). Thereafter, one can evaluate SLD $L_\mathrm {ss} (F) = L[\hat \rho _\mathrm {ss},\,(\partial \hat \rho /\partial F)_\mathrm {ss}]$ and the desired steady-state QFI $(I_\mathrm {Q})_\mathrm {ss} (F) = \mathrm {Tr} \left [\hat \rho _\mathrm {ss} \hat {L}_\mathrm {ss}^{2} (F) \right ]$ accordingly. In this study, we use QuTiP [27] for solving Eq. (2) but develop a home-made code for solving Eq. (14) and evaluating the SLD/QFI.

Before exploring a variety of physical observables and the QFI in the steady state, it would be instructive to discuss the unit system that we use for the calculations. Since $F$ has dimensions of ${\rm Hz}$ in this manuscript, $I_\mathrm {Q}$ and $\Delta F_{\rm QCR}$ have dimensions of $1/({\rm Hz})^{2}$ and ${\rm Hz}$, respectively. Now that we have normalized all the relevant frequencies with the mechanical frequency $\Omega$, and the mechanical momenta with the natural momentum $\sqrt {m\hbar \Omega }$, one can make use of ${\cal {I}}_Q=I_Q/(m\hbar \Omega )$ and $\Delta {\cal {F}}_{\rm QCR}=\Delta F_{\rm QCR} \sqrt {m\hbar \Omega }$ when comparing the calculations with experimental measurements in the laboratory. Finally, it should be mentioned that throughout the manuscript the mechanical oscillator has relatively low quality factor for fast convergence to a steady state.

4.1 Steady-state of the optomechanical system driven by a coherent field

We first consider the case where the optomechanical system is resonantly driven by a coherent field with $r=0$. The system is initially prepared in a product state of the vacuum state for the cavity field and the motional ground state for the mechanics. Figure 2(a) shows the time evolution of the mean photon number for the cavity field and the mean dimensionless position of the mechanical oscillator, indicating that both quantities converge so that the optomechanical system eventually becomes stable converging to a steady state. Figure 2(b) and (c) depict the steady-state Wigner quasi-probability distribution of the mechanical oscillator and the cavity field, respectively, showing that the steady-states are displaced symmetric Gaussian states. Before exploring the QFI of the optomechanical system, it is instructive to consider the QFI of the system with no optomechanical coupling. As the mechanical system is prepared in the motional ground state and the mechanical bath is at zero temperature, one can consider the mechanical oscillator to be in the vacuum state at all times. It follows from Eq. (13) that the QFI for the system is $I_\mathrm {Q}= 4 \sigma ^{2}_q=2$, and thus the single-shot QCRB is $\Delta F_{\rm QCR}/\Omega =1/\sqrt {2}$. Fig. 3 shows the steady-state behavior of the optomechanics as a function of $\Delta _c$, $\kappa$, $\alpha _{\rm in}$, and $g_0$. Specifically, we plotted the mean photon number $n_{\rm cav}$, the variance of the cavity field photon number $\sigma ^{2}_{n_{\rm cav}}$, the mean mechanical position $q$, the position fluctuations for the mechanical oscillator $\sigma _q$, and the single-shot QCRB $\Delta F_{\rm QCR}=1/\sqrt {I_\mathrm {Q}}$. The calculations show that the radiation pressure force increases both the mean mechanical position and the position fluctuations of the mechanics.

In Fig. 3 the position variance of the mechanics has the same tendency as the mean mechanical position but with a smaller scale because the cavity detuning is assumed to be zero, leading to insignificant radiation pressure cooling or amplification [23], and the mechanical oscillator is coupled to a zero-temperature mechanical bath [7]. It is evident from Fig. 3(a)-(c) that the mean photon number and its variance along with the mechanical displacement all increase when (a) approaching the cavity resonance, (b) decreasing the cavity linewidth, and (c) increasing the external field coherent amplitude. Moreover, in Fig. 3(a)-(c) the mean cavity photon number is almost identical to the variance of the cavity photon numbers, indicating that the cavity field is coherent, in other words, the second-order correlation function for the cavity field

The tendencies of the optomechanical system described above do not hold in Fig. 3(d). The mechanical displacement has opposite response to the mean cavity photon number as the single-photon radiation pressure force is increased. As is well-known, large single-photon optomechanical coupling can produce a photon blockade effect resulting in antibunching of the cavity field [28]. As the single-photon optomechanical coupling is increased, the degree of the antibunching of the cavity field becomes larger such that $g^{(2)}(0)<1$, and the variance of the cavity photon number becomes smaller than the mean cavity photon number; see Eq. (15). The reason why the mechanical displacement and the position fluctuations of the mechanics increase while the cavity photon number decreases is because in this situation the mean radiation pressure force and the radiation pressure noise are enhanced. Notice that when $g_0$ is large enough the QCRB is reduced despite the fact that the cavity photon number is smaller, indicating that the radiation pressure backaction noise manifests itself in the force measurement.

 

Fig. 2. (a) Time evolution of the mean photon number for the cavity field and the mean dimensionless position of the mechanical oscillator. (b) Steady-state Wigner quasi-probability distribution of the mechanics. (c) Steady-state Wigner quasi-probability distribution of the cavity field. The parameters we used for (a)-(c) are $\Delta _c=0,\,g_0/\Omega =0.02,\,F/\Omega =0.1,\,\kappa /\Omega =0.2,\,\gamma /\Omega =0.02,\,n_{\rm th}=0,\,\alpha _{\rm in}/\sqrt {\Omega }=0.3167,\,r=0$. (d) Steady-state Wigner quasi-probability distribution of the cavity field in the case where the optomechanical system is driven by a squeezed vacuum field. Parameters are identical to those above except for the squeezing strength and angle which are $r=1.148$ and $\theta =0$, respectively.

Download Full Size | PDF

 

Fig. 3. Steady-state observables of the optomechanical system driven by a coherent field. The mean photon number $n_{\rm cav}$ (red solid curve), the variance of the photon numbers in the cavity field $\sigma ^{2}_{n_{\rm cav}}$ (red dashed curve), the mean mechanical position $q$ (blue dotted curve), the position fluctuations for the mechanical oscillator $\sigma _q$ (blue dot-dashed curve), and the single-shot QCRB $\Delta F_{\rm QCR}/\Omega$ (blue solid curve) are plotted as functions of (a) the cavity detuning $\Delta _c/\Omega$, (b) the cavity linewidth $\kappa /\Omega$, (c) the coherent amplitude of the external field $\alpha _{\rm in}/\sqrt {\Omega }$, and (d) the single-photon radiation pressure force $g_0/\Omega$. The parameter set used here is based on that for Fig. 2.

Download Full Size | PDF

Notice in Fig. 3(b) when the optical decay rate is comparable to the single-photon optomechanical coupling coefficient, the photon number variance of the steady-state cavity field becomes marginally smaller than the mean photon number. This deviation from the Poissonian distribution is due to strong coupling between the optical field and the mechanical oscillator. Still, both the single-photon optomechanical coupling coefficient and the optical decay rate are far smaller than the mechanical frequency so the optomechanical system is not in the single-photon strong coupling regime. Hence, the second-order correlation function for the cavity field is nearly unity and the QCRB presented in Fig. 3(b) is in the regime where the optical shot noise is dominant.

In summary, it is clear that the QCRB tends to be small when the mean photon number is large in the case where the cavity field is nearly coherent. However, the QCRB can behave in an opposite fashion, i.e., decreasing along with the cavity photon number in the regime where the cavity field is not coherent and the radiation pressure backaction noise can be large. For the precision measurements of a force, one should exploit resonant fields and large external field coherent amplitudes when the optomechanical system is in a weak-coupling and resolved-sideband regime [29]. It should also be noted that the mean cavity photon number is small enough that the radiation pressure backaction noise does not manifest itself in Fig. 3(a)-(c).

4.2 Steady-state of the optomechanical system driven by a squeezed vacuum field

Now we show the main results of this study which is the case where the optomechanical system is resonantly driven by a squeezed vacuum field. One of the quadratures for the cavity field is squeezed and the direction of squeezing depends on the phase of the squeezed vacuum field as shown in Fig. 2(d). It should be kept in mind that although a squeezed cavity field can in principle mediate a squeezed heat bath for the mechanical oscillator [30], for the parameter regime used in this manuscript the squeezing of the cavity field is not transferred to the mechanical mode. Hence, the steady-state Wigner quasi-probability distribution of the mechanics should be similar to that shown in Fig. 2(b). For comparison, the coherent amplitude of the external field $\alpha _{\rm in}$ used in Fig. 2(a)-(c) and the squeezing strength $r$ used in Fig. 2(d) are set such that the mean steady state photon numbers are equal.

Figure 4 shows the steady-state behavior when the system is driven by a squeezed vacuum field, i.e., $\alpha _{\rm in}=0$. The plots can be directly compared to those for the coherent field input in Fig. 3. It is found that the QFI, and thus $\Delta F_{\rm QCR}$, is independent of the squeezing direction implying that the QCRB depends only on phase-insensitive observables such as the mean photon number, the cavity photon fluctuation, the mechanical displacement, the position fluctuation of the mechanics, etc. Since the cavity field is not driven by a coherent external field, the cavity field is directly coupled to its squeezed reservoir and indirectly coupled to the mechanical thermal reservoir through the optomechanical coupling.

 

Fig. 4. Steady-state observables of the optomechanical system driven by the squeezed vacuum field. Plotted values and variables are identical to those in Fig. 3. The insets show the fluctuations of the quadratures for the cavity field, $\sigma _X$ and $\sigma _Y$, as functions of the relative parameters.

Download Full Size | PDF

It is instructive to note that the steady-state mean photon number of the cavity field does not depend on the cavity detuning if the cavity is not driven by a coherent field and is coupled to a broad-band optical reservoir [31]. In addition, the mean photon number and the variance of the photon number of the squeezed optical reservoir with squeezing strength $r$ are given by $N=\sinh ^{2}(r)$ and $\sigma ^{2}_N=\frac {1}{2}\sinh ^{2}(2r)$, respectively. With a negligible optomechanical interaction, the mean photon number and the photon number variance of the steady-state cavity field are

Note that $\sigma ^{2}_N=\tilde \sigma _N^{2}$ for zero cavity detuning. In other words, the cavity field is balanced with the optical reservoir at the cavity resonance.

Figure 4(a) shows that the QCRB is minimum at the cavity resonance, similar to that in Fig. 3(a). On the other hand, the behavior of the optical and mechanical variables are quite different. First, the mean cavity photon number is independent of the cavity detuning and is identical to the mean photon number of the squeezed optical bath $N$, as expected. Furthermore, the degree of squeezing in one of the quadratures for the cavity field has a Lorentzian dependence on the cavity detuning. When the squeezing direction of the optical bath is $\theta =0$, the fluctuations in the phase quadrature of the cavity field fall below those of the vacuum state, $\sigma _Y<1/\sqrt {2}$, and the uncertainty product of the quadratures of the cavity field $\sigma _X \sigma _Y$ is minimized (see the inset in Fig. 4(a)). Second, the mechanical displacement is also independent of the cavity detuning, implying that the constant cavity photon number and thus constant radiation pressure force gives rise to the same displacement of the mechanical oscillator. Finally, the variance of the cavity photon number becomes resonant at zero cavity detuning, increasing the position fluctuations of the mechanical oscillator owing to an enhanced radiation pressure force noise.

In Fig. 4(b), the QCRB monotonically increases with respect to the cavity decay rate, which is a general behavior of the QCRB in systems with losses [32]. Again, the behavior of the QCRB is in line with that in Fig. 3(b) while the optical and mechanical variables display divergent results. Similar squeezing behavior occurs in the cavity field, but in this case, increased squeezing and decreased uncertainty product of the quadratures of the cavity field takes place as the cavity linewidth increases (see the inset in Fig. 4(b)). Notice that the variance of the cavity photon number exhibits a Lorentzian response with respect to the cavity detuning in Fig. 4(a), and monotonically increases with respect to the cavity linewidth in Fig. 4(b).

Now that the variance of the cavity photon number is always larger than the mean photon number, the cavity field photon distribution is super-Poissonian, $g^{(0)}>1$. When the cavity field is driven by a coherent field, the mean cavity photon number can be used as a good quantity to determine whether the QCRB is limited by the photon counting noise or the radiation pressure backaction noise. In other words, the QCRB reacts to parameters with an opposite tendency to the mean cavity photon number and is eventually limited by the photon counting noise if the cavity field is Poissonian; see Fig. 3(a)-(c). However, if the cavity field is sub-Poissonian, the QCRB shows the same tendencies to parameters as the mean cavity photon number and the radiation pressure backaction noise is what limits the QCRB; see Fig. 3(d). This kind of interpretation can be modified as follows in order to account for the discrepant behavior of the QCRB in Fig. 4 where the cavity field is driven by a squeezed vacuum. First, note that the mean cavity photon number is no longer a good measure for examining the QCRB. Rather, the variance of the cavity photon number can be used as a good quantity to determine whether the QCRB is limited by the photon counting noise or the radiation pressure backaction noise. If the variance of the cavity photon number follows that of the case where $g_0=0$ (gray dotted curve) and shows the opposite tendency with respect to the QCRB, the QCRB is limited by the photon counting noise as shown in Fig. 4. On the other hand, the QCRB is limited by the radiation pressure backaction noise if the variance of the cavity photon number is suppressed compared to when $g_0=0$ and shows the same tendency with respect to the QCRB, as shown in Fig. 4. This is clearly shown in Fig. 4(c) and (d) where the QCRB decreases when the variance of the cavity photon numbers follows that for $g_0=0$, i.e., when the squeezing strength is small for (c) or when the single-photon optomechanical coupling is small for (d). The QCRB can also be observed to increase when the cavity photon variance deviates from that for $g_0=0$ when squeezing strengths and optomechanical couplings are large. Finally, it is clear that the QCRB is minimum when the photon counting noise and the radiation pressure backaction noise are balanced. In quantum metrology literature, the QCRB typically represents the ultimate bound for the photon shot noise or photon counting noise. However, this result provides a clear example of where the QCRB can include backaction effects in open quantum systems. To our knowledge, this is the first theoretical presentation showing the balance between the photon counting noise and the radiation pressure backaction noise in terms of the QCRB.

4.3 QCRB versus the mean cavity photon number

We now investigate the close relation between the QCRB and the mean cavity photon number when the system parameters are fixed except for the input power. To compare the precision force measurement performance of two types of resources, a coherent field and a squeezed vacuum field, one needs a common quantity according to which we can quantitatively discuss their differences. For this purpose, we make use of the mean mechanical displacement and the mean cavity photon number in Fig. 3(c) and Fig. 4(c). As Fig. 5(a) shows the mean mechanical displacement $q$ is nearly identical for both resources as long as the mean cavity photon numbers $n_{\rm cav}$ are the same. This indicates that $n_{\rm cav}$ can serve as a common-ground quantity to assess their relative performance, particularly for comparing the scaling behaviors of the QCRBs. Figure 5(b) shows the QCRB as a function of the steady-state mean cavity photon number $n_{\rm cav}$ in the case where the optomechanical system is driven by a coherent field (blue solid curve) or a squeezed vacuum field (red dashed curve). Exploiting two fitting curves, we find that in the current setting, the QCRB for the system driven by a coherent field follows the standard quantum limit

 

Fig. 5. (a) Steady-state mean mechanical displacement $q$ as a function of the steady-state mean cavity photon number $n_{\rm cav}$ in the case where the optomechanical system is driven by the coherent field (blue dotted curve) or the squeezed vacuum field (red solid curve). The parameter set used here can be referenced from Fig. 2. (b) The dimensionless QCRB for the external force $\Delta F_{\rm QCR}/\Omega$ as a function of the steady-state mean cavity photon number $n_{\rm cav}$ in the case where the optomechanical system is driven by a coherent field (blue solid curve) or a squeezed vacuum field (red dashed curve). Two fitting curves are presented; $\Delta F_{\rm QCR}/\Omega = \frac {c_1}{\sqrt {n_{\rm cav}+2c_1^{2}}},\,c_1=1.8543$ (blue dotted curve), $\Delta F_{\rm QCR}/\Omega = \frac {c_2}{n_{\rm cav}+\sqrt {2}c_2},\,c_2=7.1193$ (red dot-dashed curve).

Download Full Size | PDF

5. Conclusion

We studied a scenario where a weak force exerted on a mechanical oscillator can be measured using the response of an optical cavity output. We evaluated the QFI as a performance indicator of the steady state of the optomechanical system taking into account realistic loss rates for the optical cavity and the mechanical oscillator. We compared two cases of the cavity-driving fields, a coherent field and a squeezed vacuum field, and explored various ranges of the system parameters for performance assessment. We also showed that the QCRB for the system driven by the coherent field follows the standard quantum limit while the QCRB for the system driven by the squeezed vacuum field can surpass the standard quantum limit and even the Heisenberg limit within some parameter window. We expect that this knowledge will be of help in the near future when designing an experimental setup for weak-force measurement using cavity optomechanics based on quantum enhanced metrology.