Quantum Magnetometer with Dual-Coupling Optomechanics

Ultrasensitive magnetic detection has contributed immensely to a wide range of scientific areas from fundamental physics to advanced technologies, such as geological exploration, aerospace Bennett et al. (2021), biomedical imaging and diagnostics Hämäläinen et al. (1993); Lee et al. (2015); Murzin et al. (2020). Over the past few decades, various magnetometers have been developed, including the superconducting quantum interference devices (SQUID) based on superconducting effects Jaklevic et al. (1964); Erné et al. (1976); Kleiner et al. (2004), spin-exchange relaxation-free atomic magnetometers Allred et al. (2002); Kominis et al. (2003); Savukov et al. (2005); Xia et al. (2006), NV center magnetometers Taylor et al. (2008); Maze et al. (2008) and Hall-effect sensors Bending (1999). Normally, they require the elaborated operating conditions, such as the associated denoising technology and/or the complex signal read-out schemes Robbes (2006), which reduces their capability of on-chip integration.

Cavity optomechanical system (OMS) Aspelmeyer et al. (2014); Kippenberg and Vahala (2008); Aspelmeyer et al. (2012); Meystre (2013) offers an alternative platform for the precision measurements of mass Li and Zhu (2012); Lin et al. (2017); Bin et al. (2019), weak forces Clerk et al. (2010); Tsang and Caves (2010); Pontin et al. (2014); Armata et al. (2017); Qvarfort et al. (2018), and magnetic fields Forstner et al. (2012, 2014); Yu et al. (2016); Wu et al. (2017); Zhu et al. (2017); Li et al. (2018). Particularly, optomechanical magnetometer with high sensitivity has excellent quality of on-chip integration Li et al. (2021). Recently, the YIG sphere-based optomechanical magnetometer has been experimentally demonstrated in Ref. Colombano et al. (2020), which have attained extremely low sensitivity values. In such magnetometers, the existence of Joule and Villari effects of magnetostrictive transducer Olabi and Grunwald (2008), allows one to directly extract magnetic information by reading out the optical frequency shift (or transmission spectrum). Moreover, quantum metrology Giovannetti et al. (2004, 2006, 2011); Dowling and Seshadreesan (2015) points out that the quantum squeezing or entanglement Caves (1981); Ma et al. (2011); Baumgratz and Datta (2016); Degen et al. (2017); Engelsen et al. (2017); Nagata et al. (2007); Israel et al. (2014); Luo et al. (2017) could improve the accuracy of parameter estimation in physical systems from the shot-noise limit to the Heisenberg limit, i.e., the optimal precision scales from $1/\sqrt{N}$ to $1/N$ with $N$ being the number of resources employed in the measurements. This has stimulated enormous interests in exploiting quantum resources in the atoms (or spins) Jones et al. (2009); Tanaka et al. (2015); Kómár et al. (2014); Hou et al. (2020) and optical systems Holland and Burnett (1993); Boto et al. (2000); Anisimov et al. (2010); Joo et al. (2011) for high-precision physical quantity measurements.

By applying quantum metrology to the detection of a static magnetic field, here we propose a quantum magnetometer based on a dual-coupling OMS, which has a periodic decoupling behavior between the optical and mechanical modes. The OMS supports two optical modes coupled simultaneously to the same mechanical mode, with radiation-pressure and quadratic optomechanical interactions, respectively Thompson et al. (2008); Sankey et al. (2010); Bhattacharya et al. (2008); Zhu et al. (2018). The radiation-pressure coupling acts as a signal transducer, encoding the magnetic signal received by the mechanical oscillator into the optical phase. The quadratic optomechanical interaction amplifies both the signal to be measured and the signal transducing rate via inducing a periodic squeezing effect on the mechanical oscillator, whose maximum squeezing strength is determined by a controllable squeezing parameter $r$. By performing a homodyne detection on the optical phase within a wide time window around the first decoupled time $\tau_{1}$, the magnetic signal could be estimated with high precision.

To qualitatively characterize the precision of magnetometer, we define a displaced phase accumulation efficiency (PAE) that is experimentally observable via state tomography technique. By presenting the exponentially increased quantum Fisher information (QFI), i.e., $\mathcal{F}_{q}(\tau_{1})\propto e^{12r}$, we quantitatively demonstrate that the fundamental bound of measurement precision can be dramatically reduced even with a small squeezing parameter $r$. The periodic opto-mechanical decoupling makes the classical Fisher information (CFI) robust against the mechanical environment at the decoupled time, which in turn allows the sensitivity of a specific measurement to saturate the fundamental bound and reach to the order of $10^{-17}{\rm T/\sqrt{Hz}}$ in the presence of system dissipations. Moreover, the sensitivity to the order of $\sim 10^{-15}{\rm T/\sqrt{Hz}}$ is predicted even in the case of the thermal phonon number $\bar{n}_{\rm th}\sim 10^{3}$. Our work establishes a connection between quantum metrology and dual-coupling optomechanics, which is suitable for detecting various fields that linearly interact with the mechanical oscillator.

System and periodic mechanical squeezing.— We consider a dual-coupling optomechanical system depicted in Fig. 1 with Hamiltonian

where $a_{j}$ $(j=1,2)$ and $b$ are the annihilation operators of the cavity mode with frequency $\omega_{j}$ and the mechanical mode with frequency $\omega_{m}$, respectively. The mechanical membrane is placed at a node (antinode) of cavity mode $a_{1}$ ($a_{2}$), and then the fourth (fifth) term in Eq. (1) describes the radiation-pressure (quadratic optomechanical) interaction between modes $a_{1}$ ($a_{2}$) and $b$ with strength $\lambda_{1}$ ($\lambda_{2}$). In the presence of a static magnetic field ${B_{z}}$ (along $z$ direction), the field-sensitive Terfenol-D expands, which leads to the change of the equilibrium position of the mechanical oscillator, thus generating an effective magnetic potential on the Hamiltonian, i.e., the last term of Equation (1) SM . The mechanical motion modulates the optical cavity field via the nonlinear radiation-pressure coupling. Meanwhile, the phase shift of the mechanical motions encoded with magnetic signal is transferred to the optical field. By reading out the phase shift of optical field via homodyne detection, we can extract the original magnetic information. Here $z=\sqrt{\hbar/(2m\omega_{m})}(b^{{\dagger}}+b)$ is the position operator of the mechanical oscillator with mass $m$, and $c_{\rm act}=m\omega_{m}^{2}L{\alpha_{\rm mag}}/{E}$ is the magnetic actuation constant charactering how well the magnetic field is converted into a force applied on the oscillator. The symbol $L$ denotes the length of Terfenol-D rods, $\alpha_{\rm mag}$ is magnetostrictive coefficient and $E$ is the Young’s modulus SM .

By considering the ancillary mode $a_{2}$ in the coherent state $|\xi\rangle$, the number operator $a_{2}^{\dagger}a_{2}$ can be approximately replaced by an algebraic number $\mathcal{N}_{2}=|\xi|^{2}$ in the case of $\mathcal{N}_{2}\gg 1$ SM . Assuming the modes $a_{1}$ and $b$ are initially in the coherent state $|\Psi(0)\rangle=|\alpha\rangle\otimes|\beta\rangle$ with $\beta=\beta_{\rm Re}+i\beta_{\rm Im}$, the instantaneous state of system is given by SM

where $\eta=\lambda_{s}n/\omega_{s}-f_{s}/\omega_{s}$ with $\lambda_{s}=\lambda_{1}e^{r}$, $\omega_{s}=\omega_{m}e^{-2r}$, $f_{s}=fe^{r}=B_{z}c_{\rm act}\sqrt{1/(2m\hbar\omega_{m})}e^{r}$, $r=-(1/4)\ln(1-4\lambda_{2}\mathcal{N}_{2}/\omega_{m})$, and a rotating frame with ${\rm exp}(-i\omega_{1}a_{1}^{{\dagger}}a_{1}/\omega_{s})$ was adopted. The mechanical state reads $|\varphi_{s}(t)\rangle=S^{\dagger}(r)S(r^{\prime})|\varphi_{n}(t)\rangle$ with the defined squeezing operator $S(r)=\exp[r(b^{2}-b^{\dagger 2})/2]$ and the squeezing parameter $r^{\prime}=re^{-2i\omega_{s}t}$. Here $|\varphi_{n}(t)\rangle=|e^{-i\omega_{s}t}\beta+\eta\bar{\mu}\rangle$ is a displaced coherent state with $\bar{\mu}=(1-e^{-i\omega_{s}t})(\cosh r-e^{-i\omega_{s}t}\sinh r)$. The expression of $|\varphi_{s}(t)\rangle$ clearly shows that the mechanical mode is dynamically squeezed with the period $T=\pi/\omega_{s}$, whose squeezing degree of quadrature $X=1/\sqrt{2}(b+b^{\dagger})$ is defined by $S(t)=10\log_{10}(\delta X^{2}(t)/\delta X_{\rm min}^{2}){\rm dB}$. The maximum squeezing degree $S_{\rm max}=10\log_{10}(e^{4r}){\rm dB}$ occurs at $T/2$ with the state $S^{\dagger}(2r)|\beta\rangle$ SM . This periodic squeezing effect on the mechanical oscillator is induced by the quadratic optomechanical coupling, and can be qualitatively presented in the case of $\alpha=\eta=0$ [see Fig. 2(a)].

Periodic-squeezing-enhanced phase accumulation efficiency.— As shown in Eq. (Quantum Magnetometer with Dual-Coupling Optomechanics), the magnetic signal is transduced into the optical phase during the evolution of system via the radiation-pressure interaction. Interestingly, at time $\tau_{m}\!=\!2m\pi/\omega_{s}$ ($m=1,2,....$), Eq. (Quantum Magnetometer with Dual-Coupling Optomechanics) can be reduced to SM

which demonstrates that the optical and mechanical modes are periodically decoupled, meanwhile the magnetic signal to be measured is periodically encoded into the accumulated optical phase $\Phi_{n}(\tau_{m})=2m\pi(\lambda_{s}n/\omega_{s}-f_{s}/\omega_{s})^{2}$ for a Fock state $|n\rangle$. Here the opto-mechanical decoupled period is double of the period of dynamical squeezing, i.e., $\tau_{1}=2\pi/\omega_{s}=2T$, and hence the mechanical squeezing also disappears at the decoupled time $\tau_{m}$.

The above unique property allows us to estimate the magnetic field $B_{z}$ by performing a homodyne detection [see Fig. 1] on the optical mode $a_{1}$ at the first decoupled time $\tau_{1}$. To qualitatively describe the detection precision, here we define a displaced PAE

Obviously, the larger $\widetilde{\mathcal{P}}_{n}$, i.e., the faster phase accumulation rate on the optical Fock state $|n\rangle$, the higher measurement accuracy of magnetometer should be obtained. As shown in Fig. 2(c), the $\widetilde{\mathcal{P}}_{n}$ is exponentially enhanced by increasing the coherent amplitude $\sqrt{\mathcal{N}_{2}}$ of the ancillary mode $a_{2}$. This enhancement originally comes from the periodic squeezing of mechanical mode during one opto-mechanical decoupled period [see Fig. 2(a)]. The system Hamiltonian in the squeezed frame shown in Eq. (S17) of supplementary material SM clearly demonstrates that the mechanical squeezing effect not only accelerates the signal tranducing rate from phonon to photon (i.e., the radiation-pressure interaction $\lambda_{s}a^{\dagger}_{1}a_{1}(b+b^{\dagger})$) Lü et al. (2015); Lemonde et al. (2016), but also amplifies the magnetic signal to be measured (i.e., the magnetic potential $f_{s}(b^{{\dagger}}+b)$).

More importantly, this well-defined PAE is experimentally observable via state tomography in the proper basis vectors. Specifically, expanding the system state $|\Psi(\tau_{1})\rangle$ in the subspace $\{|\!\!\downarrow\rangle,|\!\!\uparrow\rangle\}$ with basis vectors $|\!\downarrow\rangle:=(1/\sqrt{2})(|0\rangle-|l\rangle)$ and $|\!\uparrow\rangle:=(1/\sqrt{2})(|0\rangle+|l\rangle)$ ($l=1,2,....$), we obtain four elements of density matrix SM , as shown in Fig. 2(b). The difference between two diagonal elements is denoted by $\sigma_{z}(\tau_{1})=\rho_{\uparrow\uparrow}(\tau_{1})-\rho_{\downarrow\downarrow}(\tau_{1})=2e^{-|\alpha|^{2}}(\alpha^{l}/\sqrt{l!})\cos(\Delta\Phi_{l}(\tau_{1}))$, where $\Delta\Phi_{l}(\tau_{1})=\Phi_{l}(\tau_{1})-\Phi_{0}(\tau_{1})$. Then one can easily obtain the values of phase difference $\Delta\Phi_{l}(\tau_{1})$ by directly measuring $\rho_{\downarrow\downarrow}(\tau_{1})$ and $\rho_{\uparrow\uparrow}(\tau_{1})$, which ultimately leads to $\widetilde{\mathcal{P}}_{l}=\Delta\Phi_{l}(\tau_{1})/\tau_{1}$ being experimentally observable. Moreover, the oscillation with increasing frequency of $\sigma_{z}(\tau_{1})$, shown in the insert of Fig. 2(c), is another evidence for the enhanced $\widetilde{\mathcal{P}}_{n}$ along with increasing the mechanical squeezing.

Quantum and classical Fisher information.— From a quantitative point of view, the fundamental bound to the sensitivity and the measurement-specific sensitivity of the proposed magnetometer are respectively decided by the QFI $\mathcal{F}_{q}$ and CFI $\mathcal{F}_{c}$ based on the Cramér-Rao inequality ${\Delta}B_{z}\geq 1/\sqrt{M\mathcal{F}_{j}}$ $(j=q,c)$, where $\Delta B_{z}$ is the standard deviation with respect to an unbiased estimator $(B_{z})_{\rm est}$, and $M$ is the number of repetition of the experiments Braunstein and Caves (1994).

Considering system state $|\Psi(t)\rangle$, the QFI with respect to the parameter $B_{z}$ reads $\mathcal{F}_{q}(t)\!=\!4(\langle\partial_{B_{z}}\Psi(t)|\partial_{B_{z}}\Psi(t)\rangle\!-\!\left|\langle\Psi(t)|\partial_{B_{z}}\Psi(t)\rangle\right|^{2})$, where $\partial_{B_{z}}=\partial/\partial_{B_{z}}$ Paris (2009); Tóth and Apellaniz (2014); Liu et al. (2019); Lu and Wang (2021); Liu et al. (2022). At the first decoupled time $\tau_{1}$, the QFI reduces to SM

where $\mathcal{N}_{1}=|\alpha|^{2}$ is the mean photon number of cavity mode $a_{1}$. It can be seen that the QFI is exponentially enhanced with power of $12r$, and hence increasing a small value of $r$ by changing $\mathcal{N}_{2}$ can give rise to a large enhancement of the QFI. Figure 3(a) shows an enhancement with 7 orders of magnitude for the QFI, corresponding to a dramatic reduction of the optimal sensitivity $\Delta\mathbb{B}_{z}(\tau_{1})=1/\sqrt{M\mathcal{F}_{q}{(\tau_{1})}}$. As shown in Fig. 3(b), the optimal sensitivity $\Delta\mathbb{B}_{z}(\tau_{1})$ can reach to the order of $10^{-15}-10^{-17}\rm{(T/\sqrt{Hz})}$ for a wide parameter region in terms of the mean photon numbers ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$. Equation (Quantum Magnetometer with Dual-Coupling Optomechanics) also indicates that the resources ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$ exert different influences on the ultimate lower bound of sensitivity SM . We note that the QFI at $\tau_{1}$ does not depend on the actual value of $B_{z}$.

Next, let us calculate the CFI related to a specific measurement on the quadrature $X_{\theta}=(a_{1}e^{-i\theta}+a_{1}^{{\dagger}}e^{i\theta})/\sqrt{2}$, where $\theta$ is the phase of local oscillator. At the first decoupled time $\tau_{1}$, the CFI is given by SM

Evidently, it consists precisely with the QFI shown in Eq. (Quantum Magnetometer with Dual-Coupling Optomechanics) when $\theta=\pi/2$ ($\theta=0$) and $\alpha$ is a real (imaginary) number, which means that the momentum (position) measurement can saturate the optimal sensitivity in the absence of system dissipation.

In the practical situation, the dissipation caused by the system-bath coupling should be taken into account. Then the dynamics of system is dominated by the master equation

where $\kappa(\gamma)$ is the cavity (mechanical) decay rate, $\bar{n}_{\rm th}$ is the thermal phonon number of the mechanical mode, and $\mathcal{D}[o]\rho=o\rho o^{{\dagger}}-(o^{{\dagger}}o\rho+\rho o^{{\dagger}}o)/2$. Here we have considered the cavity mode $a_{2}$ being in the coherent state $|\xi\rangle$ for Hamiltonian $H$. Performing a momentum homodyne measurement (i.e., $\theta=\pi/2$) on cavity $a_{1}$, in Fig. 3(c) and Fig. 4, we numerically demonstrate the influence of system dissipation on the sensitivity limit, i.e., $\Delta B_{z}(\tau_{1})=1/\sqrt{M\mathcal{F}_{c}{(\tau_{1})}}$, of magnetometer Johansson et al. (2012).

With the practical experimental parameters, Figure 3(c) shows that the specific sensitivity $\Delta B_{z}(\tau_{1})$ obtained in the presence of system decay and noise, still can fit well with the optimal one $\Delta\mathbb{B}_{z}(\tau_{1})$ from the QFI without system dissipation. This consistency is only broken weakly when one increases the squeezing parameter $r$ to a large value. This can be explained as follows. On the one hand, system dissipation has little effect on the CFI $\mathcal{F}_{c}(\tau_{1})$ in the case of weak squeezing parameter $r$ [see the inserts of Fig. 4]. More importantly, the periodic optomechanical decoupling makes the CFI robust against the mechanical thermal noise. As shown in Fig. 4(b), a high sensitivity reaching to the order of $\sim 10^{-15}{\rm T/\sqrt{Hz}}$ is allowed even when $\bar{n}_{\rm th}=10^{3}$. This means that the mechanical ground state cooling is not necessary for obtaining high-precision magnetometer in our proposal. On the other hand, the strong mechanical squeezing amplifies the effect of mechanical dissipation on CFI via effectively heating the environment (see the insert of Fig. 4(b) and Fig. S5 in supplementary material SM ), which leads to the weak disagreement between the measurement-specific sensitivity limit and the fundamental bound of sensitivity in the case of large values of $r$.

Discussion of experimental feasibility.— Regarding experimental implementations, while we have considered here a Febry-Pérot cavity with the membrane-in-the-middle configuration, our versatile proposal is not limited to this particular architecture. Based on the excellent controllability of the SQUID You and Nori (2011); Xiang et al. (2013), the transmission-line (TL) resonator coupled to a SQUID-terminated TL resonator is a promising platform to realize dual-coupling optomechanical system Johansson et al. (2014); Kim et al. (2015). Recently, the dual-coupling optomechanics was also demonstrated in photonic crystal cavities Kalaee et al. (2016); Brunelli et al. (2018) and whispering gallery microcavities Li et al. (2012). Based on recent optomechanical experiments Thompson et al. (2008); Sankey et al. (2010); Ockeloen-Korppi et al. (2018), here the system parameters can be chosen as $m=4\times 10^{-8}$ g, $\omega_{m}=2\pi\times 134$ kHz, $\lambda_{1}=8.4$ kHz, $\lambda_{2}=0.08$ Hz, $\kappa=8.4$ kHz, $\gamma=840$ Hz, and $\mathcal{N}_{1}=10^{6}$. Then, with a cycle time on tens of $\mu$s, our work theoretically predicts that the sensitivity in the range of $10^{-15}\!-\!10^{-17}\,{\rm T/\sqrt{Hz}}$ can be realized with the achievable squeezing parameter $r\in[0,0.9]$. Note that most of the above results are obtained in the case of performing the homodyne measurement at the first decoupled time $\tau_{1}$. Fortunately, our proposal is robust against the detection time, i.e., the CFI in the vicinity of $\tau_{1}$ is nearly flat as shown in Figure 4(a). In other words, our proposal exhibits a wide time-window to perform measurement with sensitivity reaching to the order of $10^{-17}{\rm T}/\sqrt{\rm Hz}$.

Moreover, the experimental implementation of our proposal relies on the thin membrane held by Terfenol-D rods, which gives an effective magnetic potential on the Hamiltonian. The conversion efficiency from the applied magnetic field to an effective force on the mechanical oscillator, is determined by the magnetic actuation constant $c_{\rm act}$. The precise experimental determination of the magnetic actuation constant is of key importance for measuring afterwards accurate values of the magnetic field. This magnetic constant depends heavily on the specific geometrical configuration of the rods with respect to the membrane. To optimize the magnetostrictive effect, the applied magnetic field needs to be paralleled to the magnetostricitive direction of the Terfenol-D rods. In addition, here we considered the Terfenol-D rods of the length $\sim 10^{-5}$ m, thus it is approximatively valid to consider the magnetostrictive material to be in the whole homogeneous magnetic field, which also allows the system to have a large actuation constant.

Despite here we focus on detecting a static magnetic field, our proposal, in principle, can also be applied to probe the alternating magnetic fields. The corresponding frequency response characteristics are discussed in Sec. VII of the supplementary material SM . Referring to the parameters used in optomechanical experiments  Li et al. (2021); Thompson et al. (2008), we numerically simulate the displacement noise power spectrum and the corresponding force sensitivity at different probe powers (see Fig. S6 in the supplementary material). We find that the thermal-noise-limited frequency range covers $0\sim 380$ kHz with the central resonant frequency 250 kHz in the case of the probe power at $20\,{\rm nW}$. Therefore, similar as a general optomechanical system Li et al. (2021), here the dual-coupling magnetometer also has a broad bandwidth, when it is used as a resonant sensor.

Conclusions.– We have presented a protocol to measure the weak magnetic field using dual-coupling optomechanics. The sensitivity could be enhanced to the order of $10^{-15}-10^{-17}\,{\rm T/\sqrt{Hz}}$ by adjusting the photon numbers of two cavity modes. This enhancement originally comes from the periodic mechanical squeezing, which greatly amplifies both the signal to be measured and the transducing rate of signal. We stress out that this periodic squeezing effect is self-sustained, which avoids the complicated process of preparing squeezed states. Our proposal, with wide time-window of detection, is robust against the mechanical thermal noise, and hence the ground state cooling of mechanical mode is not necessary. This work might inspire the studies of high-precision measurements of various physical quantities based on dual-coupling optomechanical systems.

Acknowledgments.–We thank Bei-Bei Li for fruitful discussions and valuable comments. This work is supported by the National Key Research and Development Program of China grant 2021YFA1400700 and the National Science Foundation of China (Grant Nos. 11822502, 11974125, 11875029, 12175075 and 11805073).

Supplementary Material for
“Quantum Magnetometer with Dual-Coupling Optomechanics”

Gui-Lei Zhu¹, Jing Liu^1,2, Ying Wu¹, and Xin-You Lü${}^{1,^{*}}$