Non-equilibrium quadratic measurement-feedback squeezing in a micromechanical resonator

Parametric squeezing reduces noise deviation along a quadrature and amplifies it along the orthogonal quadrature in rotating-framed phase space. This parametric squeezing with continuous drive can be represented with an effective Hamiltonian $\mathcal{H}_{\mathrm{eff}}=G_{0}(p^{2}-q^{2})/4$, where $q$ and $p$ are the linear quadrature, and $G_{0}$ is strength of parametric driving (see Appendix A). Thus, the canonical equations for parametric squeezing are given by

This quadratic form $(p^{2}-q^{2})/4$ in the effective Hamiltonian is proportional to one of the Schwinger-angular-momentum components:

Note that we fixed the phase in parametric drive so that the effective Hamiltonian contains only $K_{y}$ with $\theta_{\mathrm{drive}}=\pi/2$ from its general representation given by $\mathcal{H}_{\mathrm{eff}}=-G_{0}(K_{x}\cos\theta_{\mathrm{drive}}+K_{y}\sin\theta_{\mathrm{drive}})$. These three components satisfy $K_{x}^{2}+K_{y}^{2}-K_{z}^{2}=0$. This indicates that the dynamics of a mechanical resonator, which is commonly represented at a point in the phase space $(q,p)$, is shown as a dynamics on a hyperboloid in the Schwinger-angular-momentum space [see Fig 1(a)].

In addition to the geometrical property, they also satisfy a Lie algebra in SU(1,1) group such that

where $\{A,B\}$ denotes the Poisson bracket defined by $\{A,B\}\equiv\partial A/\partial q\partial B/\partial p-\partial A/\partial p\partial B/\partial q$. Since the relation $\left\{K_{x},K_{y}\right\}=-K_{z}$ only contains a negative sign compared with the other two, the effective Hamiltonian for parametric squeezing, $\mathcal{H}_{\mathrm{eff}}=-G_{0}K_{y}$, leads a pseudo rotation around the $K_{y}$ axis in the Schwinger-angular-momentum space [see arrows in Fig. 1(a)]. This pseudo rotation is also confirmed in the dynamics of each component:

Apparently, the general solutions of $K_{x}$ and $K_{z}$ are given by hyperbolic sine and cosine functions.

The above formulation is valid to describe the stochastic dynamics of mechanical resonators with a probability density function $\mathcal{P}(q,p)$ in its phase space and $\mathcal{P}(K_{x},K_{y},K_{z})$ in the Schwinger-angular-momentum space. In the phase-space description, the parametric squeezing with $\mathcal{H}_{\mathrm{eff}}$ amplifies the deviation along a diagonal quadrature, $q_{+}\equiv(q+p)/\sqrt{2}$, and reduces the deviation along its orthogonal portion, $q_{-}\equiv(q-p)/\sqrt{2}$ direction [see Fig. 1(b)]. On the other hand, in the Schwinger-angular-momentum description, the pseudo rotation with $\mathcal{H}_{\mathrm{eff}}$ around the $K_{y}$ axis leads to a biased probability distribution along the $K_{z}+K_{x}$ direction [see Fig. 1(c)]. Importantly, this pseudo rotation can be decomposed into noise compression along $K_{z}-K_{x}$ and noise expansion along $K_{z}+K_{x}$. By taking into account that

the compression (expansion) along the $K_{z}-K_{x}$ ($K_{z}+K_{x}$) is regarded as the noise reduction (amplification) in the phase space. The divergence, which intrinsically limits the noise reduction level at -3 dB [17], appears as an infinite large noise amplification (i.e., noise expansion along $K_{z}+K_{x}$ direction in the pseudo rotation) when the drive strength $G_{C}$ is equivalent to the mechanical damping factor $\Gamma$.

Because the divergent contribution in parametric drive can be distinguished with the sign of $K_{x}$, to avoid the divergence, our measurement-feedback protocol is derived as a switching operation of the parametric drive with respect to this sign. Switching on the parametric drive (i.e., $G_{0}$ takes a non-zero value) only when $K_{x}<0$ leads a suppression of the probability density function in the Schwinger-angular-momentum space. On the other hand, switching it off ($G_{0}=0$) when $K_{x}>0$ can avoid the divergence in parametric driving.

To read out the sign of $K_{x}$ to construct a feedback loop, we can utilize nonlinear optomechanical transduction in which higher harmonic signals are generated thanks to a dispersive modulation of optical phase via mechanical motion [20, 21, 22]. In particular, the sine and cosine parts in the second-order harmonics are regarded as $K_{x}$ and $K_{y}$, respectively. To achieve steady-state squeezing, our measurement-feedback protocol is continuously repeated with $K_{x}$ directly measured via nonlinear optomechanical transduction and the parametric drive switched on or off with respect to the sign of measured $K_{x}$ [see Fig. 2(a)].

The parametric squeezing is intrinsically a heating operation because noise amplification levels along a quadrature are always larger than noise reduction levels along the orthogonal ones, which can be readily confirmed by taking into account the Shannon entropy in NESS (Appendix B). Thus, avoiding the heating part of pseudo rotation [i.e., $K_{x}>0$ in Fig. 1(c)] in Schwinger-angular-momentum space with our measurement-feedback protocol effectively induces a net cooling effect. To unveil this net cooling effect hidden in our protocol, we take into account entropy production in our setup, which consists of a system (a mechanical resonator), a heat bath, and a feedback controller [see Fig.2(b)]. Stochastic thermodynamics provides us insights into heating and cooling in a non-equilibrium steady state in terms of the balance among entropy production rates, where additional entropy production exists due to the presence of a feedback controller [24, 25, 14]. In particular, entropy production with a continuous measurement-feedback controller has been investigated by coarse-graining methods, where the degree of feedback memory is coarse-grained [25, 13, 14]. Such investigations have indicated the existence of an additional entropy production called “entropy pumping” due to the existence of a feedback controller. Because previous studies on the entropy pumping have focused on a linear operation (i.e., observable and controlled quantities directly correspond to a quadrature in its phase space) [25, 13, 14], we attempt to formulate the entropy production rate in our quadratic measurement-feedback protocol with respect to the Schwinger angular momentum as follows.

To achieve the expression of entropy productions with the rotating wave approximation, we start from the stochastic dynamics of the mechanical displacement $X$ and momentum $P$ in the laboratory frame with a measurement-feedback operation given by the Langevin equations

where $\Gamma$ is the damping rate, and $\Omega$ is the angular frequency of the mechanical resonator. Note that we set the effective mass to be unity in the following discussion for simplicity (replacing $k_{B}$ by $k_{B}/m_{\mathrm{eff}}$ provides us the exact expressions with the effective mass of mechanical resonator, $m_{\mathrm{eff}}$). The Langevin force $F_{\mathrm{th}}$ satisfies $\langle F_{\mathrm{th}}(t^{\prime})F_{\mathrm{th}}(t)\rangle=2k_{B}T\Gamma\delta(t-t^{\prime})$ with the Boltzmann constant $k_{B}$ and the temperature of the environment $T$. The protocol of measurement and feedback is expressed by the feedback function $f(M)$ with the memory value $M$. In our protocol, the feedback function is given by a Heaviside function $\theta(\cdot)$ as $f(M)=\theta(M)$. Analytical difficulty in the discontinuity of the Heaviside function is avoided by taking into account the finite measurement noise, which is assumed as Gaussian white noise. Thus, the conditional probability without feedback delay,

is suitable for modeling our measurement-feedback loop, where $\mathcal{M}(X,P)$ is the target observable in the measurement, and $\sigma_{M}$ is the standard deviation of the Gaussian noise. By multiplying Eq. (8) by Eq. (7) and integrating both sides with respect to $M$, a coarse-grained dynamics is represented as follows:

where $\mathrm{erfc}(\cdot)$ is a complementary error function reflecting the switching operation with finite measurement noise. The entropy production in the continuous measurement and feedback is formulated from this coarse-grained Langevin equation. Here, note that we formulate the entropy production rates by utilizing both a path integral formalism [26, 25, 13], which allows us to formulate them with clear physical meaning in our model (Appendix C), and probability currents [27, 28, 14], which might be better for confirming the former results (Appendix D). The two formalisms result in the same expression of the entropy production rates with respect to the Schwinger angular momentum. The formulated entropy production rate is decomposed to two contributions: the entropy production in the thermal bath, $\dot{\mathcal{S}}_{\mathrm{bath}}$, due to the heat flux from the system, and the entropy production thanks to the existence of the feedback controller, $\dot{\mathcal{S}}_{\mathrm{pump}}$, i.e. “entropy pumping” owing to the measurement and feedback as follows:

where $\langle\cdot\rangle$ denotes the stochastic average, and $\tilde{G}_{0}=G_{0}/\Gamma$ is the effective driving strength as a dimensionless quantity. Note that the entropy production rates in the thermal bath, $\dot{\mathcal{S}}_{\mathrm{bath}}$, is defined to be positive when the heat flux flows out of the system; and $\dot{\mathcal{S}}_{\mathrm{pump}}$ is defined to be positive when the feedback controller pulls the entropy from the system [please see the vectors in Fig. 2(b)].

The entropy production in the thermal bath is given by the shift of the vertical component of Schwinger angular momentum $K_{z}$ from its value in the equilibrium because $K_{z}$ directly corresponds to the oscillation energy (phonon number) in the resonator. On the other hand, the entropy pumping rate is given by a function of $K_{x}$, $K_{z}$, and the measurement noise characterized by its deviation of $\sigma_{M}$. Because of $K_{z}\geq 0$, $\langle\dot{\mathcal{S}}_{\mathrm{pump}}\rangle\geq 0$ always holds, which indicates that the entropy pumping operates to pull the entropy from the system. These three contributions satisfy the second-law-like inequality

This inequality implies that $\dot{\mathcal{S}}_{\mathrm{bath}}$ may take a negative value in contrast to the case of no-feedback operation ($\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle\geq 0$). In this negative regime, the system can operate as a cooler where heat is pumped out of the thermal bath to the feedback controller.

Our measurement-feedback protocol was implemented on a measurement-feedback loop with a micromechanical resonator [see Fig. 3(a)]. A doubly-clamped silicon nitride mechanical resonator (150 $\mathrm{\mu m}$-long, 5-$\mathrm{\mu}$m-wide, and 525-nm-thick) was fabricated via thermal chemical vapor deposition, and placed in a vacuum environment ($\sim 10^{-4}$ Pa). The resonator showed a high quality factor of $3.0\times 10^{4}$ in its fundamental flexural mode at the frequency of $\Omega=2\pi\times 510$ kHz at room temperature. Linear quadratures ($q$ and $p$) and a component of Schwinger angular momentum ($K_{x}^{M}$) in its mechanical motion were extracted from a laser Doppler interferometer (LDI). Here, we denote the directly measured Schwinger-angular-momentum component with $K_{x}^{M}$ to distinguish it from the one, $K_{x}^{P}=qp/2$, calculated via post-processing with the measured $q$ and $p$. The output of LDI was connected to lock-in amplifiers with a reference frequency of $\Omega$ for the linear quadratures and that of $2\Omega$ for the Schwinger-angular-momentum components. Note that we induced additional white noise via an piezoelectric sheet attached on the resonator substrate to improve the signal-to-noise ratio for $K_{x}^{M}$ (the effective temperature was estimated to be $T_{\mathrm{eff}}\approx 10^{5}$ K). By once connecting the output from LDI to a spectrum analyzer, spectrum at $2\Omega$, which reflects the signal of $K_{x}^{M}$, was observed as well as that at $\Omega$ [see Fig. 3(b)]. The linewidth at $2\Omega$ becomes twice of that at $\Omega$ because $K_{x}^{M}$ is the quadratic observable in $q$ and $p$. The measured component, $K_{x}^{M}$, was used to switch on (off) the parametric pump with an oscillation frequency of $2\Omega$ when the component $K^{M}_{x}$ is negative (positive) with a radio-frequency switch. To pump the mechanical resonator along the $K_{y}$-direction, the phase of the parametric pump was 90-degrees shifted from that for the reference signal to the lock-in amplifier for measuring $K_{x}^{M}$. The parametric pump was fedback to the mechanical resonator via the piezoelectric sheet. A typical temporal response in this measurement-feedback loop is shown in Fig. 3(c) where the parametric pump (red curve) with $2\Omega$ was turned on when $K^{M}_{x}$ (green plot) was negative. Note that the time constant of the lock-in amplifier was fixed at $\tau_{L}=100$ $\mathrm{\mu sec}$ to achieve all information on mechanical motion with the mechanical lifetime $\tau_{M}=59$ msec.

An initial equilibrium probability distribution was observed without any feedback drive as an equally-distributed Gaussian distribution. To verify how our protocol affects the probability distribution, we demonstrated both random pumping (i.e., the parametric drive was randomly switched on or off) and our measurement-feedback protocol, and evaluated noise reduction and amplification levels in their non-equilibrium steady states. Here, the noise reduction level, $\xi_{\mathrm{red}}$, is defined as $\mathrm{min}_{\theta}\sigma(q_{\theta,\mathrm{NESS}})/\sigma(q_{\mathrm{INIT}})$, where $\sigma(\cdot)$ is the standard deviation, $q_{\mathrm{INIT}}$ is the quadrature in the initial equilibrium, and $q_{\theta}=q\cos\theta+p\sin\theta$ is the quadrature with an arbitrary angle $\theta$. The noise amplification level, $\xi_{\mathrm{amp}}$, is defined as the deviation along the orthogonal part, i.e., $\mathrm{min}_{\theta}\sigma(\bar{q}_{\theta,\mathrm{NESS}})/\sigma(q_{\mathrm{INIT}})$, where $\bar{q}_{\theta}=-q\sin\theta+p\cos\theta$. In the case of the random pumping [see Fig. 4(a)], a squeezed Gaussian distribution with $\xi_{\mathrm{red}}\leq\xi_{\mathrm{amp}}$ was achieved in the same way as in standard noise squeezing. However, the noise reduction level was limited to about -3 dB around the drive voltage of 150 mV [see Fig. 4 (b)]. Drive voltages larger than 150 mV induced parametric instability where both $\xi_{\mathrm{red}}$ and $\xi_{\mathrm{amp}}$ increased. On the other hand, once our measurement-feedback protocol was demonstrated, a non-equilibrium steady state with $\xi_{\mathrm{red}}\geq\xi_{\mathrm{amp}}$ was observed with a non-Gaussian probability distribution [see Fig. 4(c)]. This non-Gaussianity directly reflects the non-Gaussian properties of our quadratic observables [29]. Moreover, the noise reduction level finally reached $-5.1\pm 0.2$ dB over the -3 dB, whereas the noise amplification level was suppressed at $1.7\pm 0.2$ dB [see Fig. 4(d)].

Because we attempt to perform fully quadratic measurement-feedback protocol on the Schwinger-angular-momentum space, the NESS in that space is totally different between our measurement-feedback protocol and the random protocol. In contrast to $K_{x}^{M}$, we use the dataset of $q$ and $p$ to evaluate Schwinger angular momentum $K_{i}^{P}$ ($i=x,y,z$) via the post-processing to reconstruct the probability density function in the Schwinger-angular-momentum space. Note that $K_{i}^{P}$ has a more accurate value than $K_{i}^{M}$ because the signal-to-noise ratio in $\Omega$ is better than that in $2\Omega$, although $K_{i}^{P}$ was only available in the post-processing without any fast feedback processor. Thus, we used $K_{x}^{M}$ in the measurement-feedback control and $K_{x}^{P}$ in the analysis to, for instance, calculate the stochastic average of $K_{x}$. Figure 5 shows marginal probability density functions in the space spanned by $K_{x}^{P}$ and $K_{z}^{P}$ for both the random protocol and our measurement-feedback protocol. Apparently, the initial equilibrium state was isotropically distributed [see Fig.5 (a) and (d)]. On the other hand, the non-equilibrium steady states in the random protocol [see Fig. 5(b) and (c)] and our measurement-feedback protocol [see Fig. 5(e) and (f)] show biased distributions along $K_{x}>0$ directions due to the pseudo rotation in parametric squeezing. We emphasize that the axes scales are totally different between the random protocol and our measurement-feedback protocol. Although the distribution is broadened in the random protocol because the parametric drive is blindly injected, the distribution in our protocol in the positive $K_{x}$ side is almost completely kept thanks to the switch-off operation.

To unveil the net cooling effect hidden in our measurement-feedback experiment, we evaluate the entropy production rate from the heat bath, $\dot{\mathcal{S}}_{\mathrm{bath}}$, and the entropy pumping rate $\dot{\mathcal{S}}_{\mathrm{pump}}$ from the theoretical formulation in Eqs. (10) and (11). Note that the stochastic averages in Eqs. (10) and (11) are determined by using $K_{i}^{P}$. $\dot{\mathcal{S}}_{\mathrm{bath}}$ is experimentally determined by

where $\langle\cdot\rangle_{0}$ denotes the stochastic average in the initial equilibrium state. In the same manner, we can obtain

To estimate the entropy production rates, the effective drive strength $\tilde{G}_{0}$ and measurement noise deviation $\sigma_{M}$ are required. The $\tilde{G}_{0}$ was estimated from the noise reduction level in the random protocol shown in Fig. 4(b) (yellow circles). From the divergence condition $\tilde{G}_{0}=2$ in the random protocol, at which the noise reduction level changes from decreasing to increasing, we can determine $\tilde{G}_{0}/V_{\mathrm{drive}}=10.4$ $\mathrm{V}^{-1}$. The $\sigma_{M}$ was estimated to be $0.52\pm 0.07$ from both $K_{x}^{M}$ and $K_{x}^{P}$ by assuming the Gaussian noise in measurement (see Appendix E).

The entropy production rates normalized by the mechanical damping rate $\Gamma$ are shown in Fig. 6 (a). With increasing driving strength $\tilde{G}_{0}$, the entropy pumping rate, $\dot{\mathcal{S}}_{\mathrm{pump}}$, naturally increases with positive values. On the other hand, the entropy production rate in the thermal bath, $\dot{\mathcal{S}}_{\mathrm{bath}}$, decreases with negative values. This indicates that our measurement-feedback protocol successfully induces a net cooling effects where the system pulls the heat from the thermal bath to the feedback controller [see Fig.6 (b)]. Thus, this net cooling effect avoids heating due to the parametric driving, and results in stronger squeezing in its NESS because it allows us to inject stronger parametric driving over the limitation due to the divergence. Moreover, we can confirm the second-law-like inequality $\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle+\langle\dot{\mathcal{S}}_{\mathrm{pump}}\rangle\geq 0$ from the experimental results. The total entropy production rate, $\langle\dot{\Sigma}\rangle\equiv\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle+\langle\dot{\mathcal{S}}_{\mathrm{pump}}\rangle$, apparently increases with increasing $\tilde{G}_{0}$. This is because the smaller $\tilde{G}_{0}$ purely induces the parametric squeezing with the net cooling [see Fig. 6(b)], while the larger $\tilde{G}_{0}$ intrinsically induces an additional heating in its pseudo rotation despite successful operation of the protocol [see Fig. 6(c)].

It is intuitive that entropy production rates between our measurement-feedback protocol and the random protocol can be continuously related with respect to the measurement noise deviation $\sigma_{M}$ (i.e., $\sigma_{M}\to\infty$ corresponds to the random protocol). Thus, we numerically evaluate the entropy production rates $\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle$ and $\langle\dot{\mathcal{S}}_{\mathrm{pump}}\rangle$ with different $\sigma_{M}=\{10^{-1},10^{0},10^{1}\}$ (see Fig. 7), and compare them with the analytical expression of $\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle$ in continuous and random driving (see Appendix F). As the measurement noise deviation increases, the slope of increment of $\langle\dot{\mathcal{S}}_{\mathrm{bath}}\rangle$ become steep while the slope of increment of $\langle\dot{\mathcal{S}}_{\mathrm{pump}}\rangle$ becomes gentle. This is because the measurement error induces the heating from parametric driving with the pseudo rotation in $K_{x}\geq 0$, which decreases the influence of measurement and feedback (it becomes close to the random protocol). We emphasize again that our measurement-feedback scheme yields a noise-squeezed NESS in the cooling regime, whereas the conventional parametric scheme (continuous or random) yields it in the heating regime.