Non-equilibrium quadratic measurement-feedback squeezing in a micromechanical resonator
Motoki Asano, Takuma Aihara, Tai Tsuchizawa, Hiroshi Yamaguchi
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Non-equilibrium quadratic measurement-feedbacksqueezing in a micromechanical resonator
Motoki Asano , Takuma Aihara , Tai Tsuchizawa , and HiroshiYamaguchi NTT Basic Research Laboratories NTT Device Technology LaboratoriesE-mail: [email protected]
Abstract.
Measurement and feedback control of stochastic dynamics has beenactively studied for not only stabilizing the system but also for generating additionalentropy flows originating in the information flow in the feedback controller. Inparticular, a micromechanical system offers a great platform to investigate suchnon-equilibrium dynamics under measurement-feedback control owing to its precisecontrollability of small fluctuations. Although various types of measurement-feedback protocols have been demonstrated with linear observables (e.g., displacementand velocity), extending them to the nonlinear regime, i.e., utilizing nonlinearobservables in both measurement and control, retains non-trivial phenomena in itsnon-equilibrium dynamics. Here, we demonstrate measurement-feedback control of amicromechanical resonator by driving the second-order nonlinearity (i.e., parametricsqueezing) and directly measuring quadratic observables, which are given by theSchwinger representation of pseudo angular momentum (referred as Schwinger angularmomentum). In contrast to that the parametric divergence occurs when the second-order nonlinearity is blindly driven, our measurement-feedback protocol enables usto avoid such a divergence and to achieve a strong noise reduction at the level of − . ± .
1. Introduction
Measurement-feedback control of fluctuation in mesoscopic systems has attracted largeinterest not only to precisely manipulate and stabilize the system but also to investigatestochastic dynamics itself under measurement and feedback. The presences of ameasurement-feedback controller modifies the balance of entropy flows (i.e., the secondlaw in the total system), and allows us to extract finite work to effectively heat up orcool down the system [1]. Such stochastic thermodynamics under the measurement andfeedback have been investigated in various types of mesoscopic systems with artificialsystems as well as natural biological ones [2]. In particular, micromechanical systems,such as an optically trapped nanoparticles [3, 4, 5] and micro/nanomechanical resonators[6, 7, 8, 9], have been widely used to investigate the stochastic thermodynamics becausefluctuation of their displacement (or velocity) can be precisely detected and controlledto implement measurement-feedback protocols.So far, the linear measurement-feedback protocols, which consists of measurementand feedback of linear observables in phase space, have been demonstrated inmicromechanical resonators to damp and amplify their displacement [10, 11, 12].Although such a linear measurement-feedback protocol can be simply implementedin a closed-loop setup, recently, non-trivial entropy production in that system witha finite delay has been unveiled in both theory [13, 14] and experiment [15]. As anatural but further extension, bringing nonlinear observables to the measurement andfeedback would extend these frameworks to be more general and non-trivial becausenonlinearity naturally contains unique dynamical (e.g., instability) and stochastic (e.g.,non-Markovianity) properties [16]. Although micro/nanomechanical resonators havebeen individually used to investigate both nonlinear control [17, 18, 19] and nonlinearmeasurement [20, 21, 22], combining them to develop a nonlinear measurement-feedbackprotocol has not been reported yet.In this study, we propose and demonstrate continuous measurement-feedbackcontrol of a micromechanical resonator based on a quadratic observable, which isreferred as “Schwinger angular momentum” because it is the quadratic form definedin the angular-momentum representation of bosons [23]. Because the Schwingerangular momentum holds quadratic and symmetric properties with SU(1,1) Lie algebra,we develop a continuous measurement-feedback protocol by combining parametricnonlinearity and measurement nonlinearity. The measurement nonlinearity enablesus to directly readout the component of Schwinger angular momentum via nonlinearoptomechanical transduction. The parametric nonlinearity enables us to drive theSchwinger angular momentum and squeeze the noise deviation (i.e., noise compressionalong a quadrature and noise amplification along an orthogonal one). In contrast to theblind parametric driving without measurement, our continuous measurement-feedbackprotocol enables us to achieve a non-equilibrium steady state (NESS) with strong noisesqueezing at the level of − . ± .
2. Theory
Parametric squeezing reduces noise deviation along a quadrature and amplifies italong the orthogonal quadrature in rotating-framed phase space. This parametricsqueezing with continuous drive can be represented with an effective Hamiltonian H eff = G ( p − q ) /
4, where q and p are the linear quadrature, and G is strengthof parametric driving (see Appendix A). Thus, the canonical equations for parametricsqueezing are given by˙ q = G p, ˙ p = G q. (1)This quadratic form ( p − q ) / K x = qp , K y = q − p , K z = q + p . (2)Note that we fixed the phase in parametric drive so that the effective Hamiltoniancontains only K y with θ drive = π/ H eff = − G ( K x cos θ drive + K y sin θ drive ). These three components satisfy K x + K y − K z = 0. Thisindicates that the dynamics of a mechanical resonator, which is commonly representedat a point in the phase space ( q, p ), is shown as a dynamics on a hyperboloid in theSchwinger-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 { K x , K y } = − K z , { K y , K z } = K x , { K z , K x } = K y , (3)where { A, B } denotes the Poisson bracket defined by { A, B } ≡ ∂A/∂q∂B/∂p − ∂A/∂p∂B/∂q . Since the relation { K x , K y } = − K z only contains a negative signcompared with the other two, the effective Hamiltonian for parametric squeezing, H eff = − G 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 inthe dynamics of each component:˙ K x = G K z , ˙ K z = G K x , ˙ K y = 0 . (4)Apparently, the general solutions of K x and K z are given by hyperbolic sine and cosinefunctions.The above formulation is valid to describe the stochastic dynamics of mechanicalresonators with a probability density function P ( q, p ) in its phase space and P ( K x , K y , K z ) in the Schwinger-angular-momentum space. In the phase-spacedescription, the parametric squeezing with H eff amplifies the deviation along a diagonalquadrature, q + ≡ ( q + p ) / √
2, and reduces the deviation along its orthogonal portion, q − ≡ ( q − p ) / √ H eff around the K y axis leads to abiased 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 noiseexpansion along K z + K x . By taking into account that h K z ± K z i = σ ( q ± ) / , (5)the compression (expansion) along the K z − K x ( K z + K x ) is regarded as the noisereduction (amplification) in the phase space. The divergence, which intrinsically limitsthe 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 drivestrength G C is equivalent to the mechanical damping factor Γ. Figure 1. (a) Conceptual illustration of the phase space spanned by q and p , andthe Schwinger-angular-momentum space spanned by K x , K y , and K z . The color mapshows correspondence between the phase space and the Schwinger-angular-momentumspace. The blue vectors show the force field of the parametric squeezing with H eff .(b) and (c) Schematic of probability distribution of thermal equilibrium (green) andsteady state with continuous squeezing (red) in the phase space and the Schwinger-angular-momentum space, respectively. Because the divergent contribution in parametric drive can be distinguished with thesign of K x , to avoid the divergence, our measurement-feedback protocol is derived as aswitching operation of the parametric drive with respect to this sign. Switching on theparametric drive (i.e., G takes a non-zero value) only when K x < G = 0) when K x > K x to construct a feedback loop, we can utilize nonlinearoptomechanical transduction in which higher harmonic signals are generated thanks to adispersive 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 protocolis continuously repeated with K x directly measured via nonlinear optomechanicaltransduction and the parametric drive switched on or off with respect to the sign ofmeasured K x [see Fig. 2(a)]. Figure 2. (a) Schematic of the measurement-feedback protocol in the Schwinger-angular-momentum space. Initially, the sign of K x is measured via nonlinearoptomechanical transduction (i). Then, only if K x <
0, the parametric drive isswitched on (ii). By repeating these two processes, we achieve a non-equilibriumsteady state (NESS) that is partially distributed around only K x > S bath , there exists entropyproduction called “entropy pumping” with the rate ˙ S pump due to the existence of thefeedback controller. The vectors describe the actual directions of both fluxes whenthe measurement-feedback protocol succeeds, where ˙ S bath ≥ S pump ≥ The parametric squeezing is intrinsically a heating operation because noise amplificationlevels along a quadrature are always larger than noise reduction levels along theorthogonal ones, which can be readily confirmed by taking into account the Shannonentropy in NESS (Appendix B). Thus, avoiding the heating part of pseudo rotation[i.e., K x > X and momentum P in the laboratory frame with a measurement-feedback operationgiven by the Langevin equations˙ X = P, (6)˙ P + Γ P + Ω X − G Ω f ( M ) cos 2Ω tX = F th , (7)where Γ is the damping rate, and Ω 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 eff provides us the exact expressions with the effective massof mechanical resonator, m eff ). The Langevin force F th satisfies h F th ( t ′ ) F th ( t ) i =2 k B T Γ δ ( t − t ′ ) 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 aHeaviside function θ ( · ) as f ( M ) = θ ( M ). Analytical difficulty in the discontinuity ofthe Heaviside function is avoided by taking into account the finite measurement noise,which is assumed as Gaussian white noise. Thus, the conditional probability withoutfeedback delay, P ( M |M ( X, P )) = 1 q πσ M exp " − ( M ( X, P ) − M ) σ M , (8)is suitable for modeling our measurement-feedback loop, where M ( X, P ) is the targetobservable in the measurement, and σ M is the standard deviation of the Gaussian noise.By multiplying Eq. (8) by Eq. (7) and integrating both sides with respect to M , acoarse-grained dynamics is represented as follows:˙ P + Γ P + Ω X − G Ωerfc M ( X, P ) √ σ M ! cos 2Ω tX = F th , (9)where erfc( · ) is a complementary error function reflecting the switching operation withfinite measurement noise. The entropy production in the continuous measurement andfeedback is formulated from this coarse-grained Langevin equation. Here, note thatwe 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 confirmingthe former results (Appendix D). The two formalisms result in the same expression ofthe entropy production rates with respect to the Schwinger angular momentum. Theformulated entropy production rate is decomposed to two contributions: the entropyproduction in the thermal bath, ˙ S bath , due to the heat flux from the system, andthe entropy production thanks to the existence of the feedback controller, ˙ S pump , i.e.“entropy pumping” owing to the measurement and feedback as follows: D ˙ S bath E ≈ Γ k B T h K z i − k B T ! , (10) D ˙ S pump E ≈ s π ˜ G Γ σ M * K z exp − K x σ M !+ , (11)where h·i denotes the stochastic average, and ˜ G = G / Γ is the effective driving strengthas a dimensionless quantity. Note that the entropy production rates in the thermal bath,˙ S bath , is defined to be positive when the heat flux flows out of the system; and ˙ S pump isdefined 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 verticalcomponent of Schwinger angular momentum K z from its value in the equilibriumbecause K z directly corresponds to the oscillation energy (phonon number) in theresonator. 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 σ M . Because of K z ≥ h ˙ S pump i ≥ h ˙ S bath i + h ˙ S pump i ≥ . (12)This inequality implies that ˙ S bath may take a negative value in contrast to the case ofno-feedback operation ( h ˙ S bath i ≥
3. Experiment
Our measurement-feedback protocol was implemented on a measurement-feedback loopwith a micromechanical resonator [see Fig. 3(a)]. A doubly-clamped silicon nitridemechanical resonator (150 µ m-long, 5- µ m-wide, and 525-nm-thick) was fabricated viathermal chemical vapor deposition, and placed in a vacuum environment ( ∼ − Pa).The resonator showed a high quality factor of 3 . × in its fundamental flexuralmode at the frequency of Ω = 2 π ×
510 kHz at room temperature. Linear quadratures( q and p ) and a component of Schwinger angular momentum ( K Mx ) in its mechanicalmotion were extracted from a laser Doppler interferometer (LDI). Here, we denote thedirectly measured Schwinger-angular-momentum component with K Mx to distinguishit from the one, K Px = 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 frequencyof Ω for the linear quadratures and that of 2Ω for the Schwinger-angular-momentumcomponents. Note that we induced additional white noise via an piezoelectric sheetattached on the resonator substrate to improve the signal-to-noise ratio for K Mx (theeffective temperature was estimated to be T eff ≈ K). By once connecting the outputfrom LDI to a spectrum analyzer, spectrum at 2Ω, which reflects the signal of K Mx , wasobserved as well as that at Ω [see Fig. 3(b)]. The linewidth at 2Ω becomes twice of thatat Ω because K Mx is the quadratic observable in q and p . The measured component, K Mx , was used to switch on (off) the parametric pump with an oscillation frequencyof 2Ω when the component K Mx is negative (positive) with a radio-frequency switch.To pump the mechanical resonator along the K y -direction, the phase of the parametricpump was 90-degrees shifted from that for the reference signal to the lock-in amplifierfor measuring K Mx . The parametric pump was fedback to the mechanical resonator viathe piezoelectric sheet. A typical temporal response in this measurement-feedback loopis shown in Fig. 3(c) where the parametric pump (red curve) with 2Ω was turned onwhen K Mx (green plot) was negative. Note that the time constant of the lock-in amplifierwas fixed at τ L = 100 µ sec to achieve all information on mechanical motion with themechanical lifetime τ M = 59 msec. An initial equilibrium probability distribution was observed without any feedback driveas an equally-distributed Gaussian distribution. To verify how our protocol affects theprobability distribution, we demonstrated both random pumping (i.e., the parametricdrive was randomly switched on or off) and our measurement-feedback protocol, andevaluated noise reduction and amplification levels in their non-equilibrium steady states.Here, the noise reduction level, ξ red , is defined as min θ σ ( q θ, NESS ) /σ ( q INIT ), where σ ( · )is the standard deviation, q INIT is the quadrature in the initial equilibrium, and q θ = q cos θ + p sin θ is the quadrature with an arbitrary angle θ . The noise amplification level, ξ amp , is defined as the deviation along the orthogonal part, i.e., min θ σ (¯ q θ, NESS ) /σ ( q INIT ),where ¯ q θ = − q sin θ + p cos θ . In the case of the random pumping [see Fig. 4(a)], asqueezed Gaussian distribution with ξ red ≤ ξ amp was achieved in the same way as instandard noise squeezing. However, the noise reduction level was limited to about -3 dBaround the drive voltage of 150 mV [see Fig. 4 (b)]. Drive voltages larger than 150 mVinduced parametric instability where both ξ red and ξ amp increased. On the other hand,once our measurement-feedback protocol was demonstrated, a non-equilibrium steadystate with ξ red ≥ ξ amp was observed with a non-Gaussian probability distribution [seeFig. 4(c)]. This non-Gaussianity directly reflects the non-Gaussian properties of ourquadratic observables [29]. Moreover, the noise reduction level finally reached − . ± . . ± . Figure 3. (a) Schematic of experimental setup with a high-Q silicon-nitride doubly-clamped beam. A radio-frequency (rf) oscillator with twice of the mechanical frequencywas connected to an rf switch for the parametric driving. This switching operationwas determined with respect to the measurement outcome of K Mx from a laser Dopplerinterferometer (LDI). The mechanical excitation was done by the parametric drivesignal from the rf oscillator and white noise signal via a piezoelectric signals. Here,note that this interferometer also yields temporal data of the phase quadratures q and p recorded on the oscilloscope. (b) Frequency spectra at Ω (measurementof phase quadratures) and 2Ω (measurement of Schwinger angular momenta). (c)Typical temporal sequence of our measurement-feedback protocol. The parametricdrive signal with 2Ω frequency (red curve) was sent to the piezoelectric sheet only ifthe measurement outcome (the signal of K Mx ) took negative values. The inset showsthe enlarged data which explicitly indicate the period of 1 / (2Ω). between our measurement-feedback protocol and the random protocol. In contrast to K Mx , we use the dataset of q and p to evaluate Schwinger angular momentum K Pi ( i = x, y, z ) via the post-processing to reconstruct the probability density function in theSchwinger-angular-momentum space. Note that K Pi has a more accurate value than K Mi Figure 4. (a) and (c) Probability density functions in the phase space with therandom protocol and our measurement-feedback protocol, respectively. (b) and (d)The maximum and minimum standard deviations of these distributions are evaluatedwith respect to the parametric drive amplitudes The vertical error-bar corresponds tothe standard deviation in ten trials. because the signal-to-noise ratio in Ω is better than that in 2Ω, although K Pi was onlyavailable in the post-processing without any fast feedback processor. Thus, we used K Mx in the measurement-feedback control and K Px in the analysis to, for instance, calculatethe stochastic average of K x . Figure 5 shows marginal probability density functions inthe space spanned by K Px and K Pz 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 therandom protocol [see Fig. 5(b) and (c)] and our measurement-feedback protocol [seeFig. 5(e) and (f)] show biased distributions along K x > K x side is almost completelykept thanks to the switch-off operation.1 Figure 5.
Probability density functions with the random protocol with the driveamplitudes of (a) 0, (b) 150, and (c) 400 mV, and with the feedback protocol in thedrive amplitude of (d) 0, (e) 150, and (f) 400 mV. Note that the axes scales are totallydifferent between the two protocols.
To unveil the net cooling effect hidden in our measurement-feedback experiment, weevaluate the entropy production rate from the heat bath, ˙ S bath , and the entropy pumpingrate ˙ S pump from the theoretical formulation in Eqs. (10) and (11). Note that thestochastic averages in Eqs. (10) and (11) are determined by using K Pi . ˙ S bath isexperimentally determined by h ˙ S bath i = Γ h K Pz i − h K Pz i h K Pz i , (13)where h·i denotes the stochastic average in the initial equilibrium state. In the samemanner, we can obtain h ˙ S pump i = s π Γ ˜ G σ M * K Pz exp " − ( K Px ) σ M . (14)To estimate the entropy production rates, the effective drive strength ˜ G andmeasurement noise deviation σ M are required. The ˜ G was estimated from the noisereduction level in the random protocol shown in Fig. 4(b) (yellow circles). From thedivergence condition ˜ G = 2 in the random protocol, at which the noise reduction levelchanges from decreasing to increasing, we can determine ˜ G /V drive = 10 . − . The σ M was estimated to be 0 . ± .
07 from both K Mx and K Px by assuming the Gaussian noisein measurement (see Appendix E).2The entropy production rates normalized by the mechanical damping rate Γ areshown in Fig. 6 (a). With increasing driving strength ˜ G , the entropy pumpingrate, ˙ S pump , naturally increases with positive values. On the other hand, the entropyproduction rate in the thermal bath, ˙ S bath , decreases with negative values. This indicatesthat our measurement-feedback protocol successfully induces a net cooling effects wherethe 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 instronger squeezing in its NESS because it allows us to inject stronger parametric drivingover the limitation due to the divergence. Moreover, we can confirm the second-law-like inequality h ˙ S bath i + h ˙ S pump i ≥ h ˙Σ i ≡ h ˙ S bath i + h ˙ S pump i , apparently increases with increasing ˜ G . Thisis because the smaller ˜ G purely induces the parametric squeezing with the net cooling[see Fig. 6(b)], while the larger ˜ G intrinsically induces an additional heating in itspseudo rotation despite successful operation of the protocol [see Fig. 6(c)]. Figure 6. (a) Entropy production rate (EPR) normalized by the mechanical dampingrate Γ in our measurement-feedback protocol. The red diamonds show the entropyproduction rates in the thermal bath, h ˙ S bath i / Γ, and the blue dots show that in thefeedback controller, h ˙ S pump i / Γ, (i.e., entropy pumping rate). The blue shaded areacorresponds to the cooling regime where the system operates as a cooler (conceptualimage is shown in the inset). The error bars show the standard deviation in ten trials.(b) and (c) Schematic of pseudo rotation with (b) smaller G and (c) larger G in theSchwinger angular momentum space. It is intuitive that entropy production rates between our measurement-feedbackprotocol and the random protocol can be continuously related with respect to themeasurement noise deviation σ M (i.e., σ M → ∞ corresponds to the random protocol).Thus, we numerically evaluate the entropy production rates h ˙ S bath i and h ˙ S pump i withdifferent σ M = { − , , } (see Fig. 7), and compare them with the analyticalexpression of h ˙ S bath i in continuous and random driving (see Appendix F). As themeasurement noise deviation increases, the slope of increment of h ˙ S bath i become steepwhile the slope of increment of h ˙ S pump i becomes gentle. This is because the measurement3error induces the heating from parametric driving with the pseudo rotation in K x ≥ Figure 7.
Numerically evaluated entropy production rates of (a) h ˙ S bath i / Γ and (b) h ˙ S pump i / Γ. The dots, diamonds, and squares correspond to the noise deviation of σ M =, 10 − , 10 , and 10 , respectively. The error bars show the standard deviationwithin fifty trials. The blue shaded area in (a) corresponds to the cooling regime.
4. Discussion
Our proof-of-principle experiment for the continuous measurement feedback control withSchwinger angular momentum was performed with additional white noise to improvethe signal-to-noise ratio in the quadratic measurement in the Doppler interferometer.The quadratic measurement can be extended to pure thermal fluctuation in mechanicalresonators with Doppler interferometry by increasing the mechanical Q factor anddecreasing effective mass in the mechanical modes [22]. Although the mechanical Qfactor simply contributes to the signal-to-noise ratio as √ Q dependence, the inverse ofeffective mass linearly contributes to it. Thus, mechanical resonators with small effectivemass (e.g. graphene drum resonators [30, 31]) are suitable for performing our protocolwith pure thermal motion with the Doppler interferometry. As an alternative approach,cavity optomechanical coupling in the unresolved sideband regime is also available formeasuring higher order harmonics in mechanical modes, and has been demonstrated forobserving them in pure thermal motion [20, 21]. Furthermore, extension to more higherorder observables and intermodal observables would open the way to more functionallycontrol in the nonlinear measurement-feedback frameworks.4It is important to emphasize that experimental verification of the second-law-like inequality with the entropy pumping rate has been demonstrated only in linearmeasurement-feedback schemes [15]. In contrast, we investigated entropy productionunder fully nonlinear measurement-feedback control of stochastic dynamics with acertain symmetry (i.e., certain geometry of variable space). Utilizing such rich intrinsicand external (measurement) nonlinearity in mechanical resonators might furtherpromote the experimental verification of various types of thermodynamic limitationswith information resources [32, 33].
5. Conclusion
In conclusion, we have demonstrated measurement-feedback control of Schwingerangular momentum using a high-Q silicon nitride mechanical resonator by Dopplerinterferometry. A strong noise reduction level − . ± . Acknowledgement
We thank Kensaku Chida for fruitful discussions. This work was partly supportedby a MEXT Grant-in-Aid for Scientific Research on Innovative Areas (Grants No.JP15H05869).5
Appendix A. Dynamics in the rotating frame with parametric squeezing
The dynamics of displacement X in a mechanical resonator with parametric force F p isgiven by ¨ X + Γ ˙ X + (Ω + F p ) X = F th (A.1)where Γ is the damping factor, Ω is the mechanical angular frequency, and the Langevinforce F th is given by h F th ( t ) F th ( t ′ ) i = 2 k B T Γ δ ( t − t ′ ). To induce the parametricsqueezing, the parametric force has a double period of mechanical oscillation, i.e., F p = − G Ω cos 2Ω t with the drive strength of G . The dynamics in rotating phasespace spanned by linear quadratures q and p ( X = q cos Ω t + p sin Ω t ) is approximatedby ¨ z ≪ Ω ˙ z ( z = q, p ) and Γ / Ω ≪
1, that is the linear quadrature in a high-Q modemore slowly varies than the mechanical frequency as follows: − Ω(2 ˙ q + Γ q ) sin Ω t + Ω(2 ˙ p + Γ p ) cos Ω t, + G Ω [ q (cos Ω t + cos 3Ω t ) + p (sin 3Ω t − sin Ω t )] = F th . (A.2)To take into account the rotating term with Ω, the Langevin force is split as F th = − f q sin Ω t − f p cos Ω t where h f z ( t ) f z ( t ′ ) i = 2 k B T Γ δ ( t − t ′ ) is satisfied. This leads to theLangevin equations for each quadrature with the parametric squeezing as follows:˙ q = − Γ2 q + G p + q ˜ β Γ ξ q , (A.3)˙ p = − Γ2 p + G q + q ˜ β Γ ξ p , (A.4)where ˜ β ≡ k B T / Ω . Note that the effective rotating-framed Hamiltonian is given by H eff = G ( p − q ) / Appendix B. Heating effect in parametric squeezing
Heating in continuous parametric squeezing can be simply seen in the change in theShannon entropy between the initial and final equilibrium states, which is given by∆ H S = 12 ln | Σ f || Σ i | , (B.1)where | Σ i | and | Σ f | are determinants of covariant matrices in the initial equilibriumstate and final squeezed state, respectively. From Langevin equations for continuousparametric squeezing given by˙ q = − Γ2 q + G C p + q Γ ˜ βξ q , (B.2)˙ p = − Γ2 p + G C q + q Γ ˜ βξ p , (B.3)where G C is the strength of continuous parametric drive, the determinant of thecovariant matrix in the final squeezed state is given by | Σ f | = 16 ˜ β Γ(Γ − G C ) . (B.4)6This leads to ∆ H S = ln 11 − G C / Γ . (B.5)Since the Shannon entropy monotonically increases in the stable squeezing regime G < Γ, the system (i.e., mechanical resonator) is totally heated up due to theparametric squeezing.
Appendix C. Entropy production in our protocol by means of path integral
Total entropy production Σ, which is always non-negative, is defined by the Kullback-Leibler divergence between the forward probability distribution and the inverseprobability distribution asΣ = ln P fwd P inv ≥ . (C.1)In the case of the continuous measurement-feedback control, entropy production hasbeen investigated in a coarse-grained dynamics, where the memory degree of freedomin measurement is coarse-grained in its equation of motion [25, 13, 14]. The inverseprobability in the coarse-grained dynamics was defined as a probability with “conjugate”dynamics, in which the time-reversal parity of feedback cooling forces is defined to bepositive [25, 13]. From the path integral formalism, the entropy production in thenon-equilibrium steady state (i.e., change in the Shannon entropy is zero) is expressedby Σ = Z d s ˙ S bath ( s ) + Z d s ˙ S pump ( s ) (C.2)where ˙ S bath and ˙ S pump are the entropy production rates in thermal bath and controller.The later has been referred to as “entropy pumping” [25, 13], which gives second-lawlike inequality including the influence of information extraction as˙ S bath ≥ − ˙ S pump . (C.3)To derive the actual expression of entropy production in our quadratic measurementfeedback, we start from the Langevin equation in the laboratory frame with thedisplacement X and momentum P as follows:˙ X = P, (C.4)˙ P + A ( X, P, t ) = F th , (C.5)where A ( X, P, t ) is the term of the equation of motion specified in Eq. (7). From theFokker-Planck equation, ∂ t P = LP (C.6)with an operator L ≡ ∂ P A ( X, P, t ) + k B T ∂ P − ∂ X ( P/m ) , (C.7)7we achieve the forward transition probability in stochastic path from the initial condition( X , P , t ) as P ( X, P, t | X , P , t ) = B exp (cid:20) − k B T Γ Z tt d s (cid:16) ˙ P ( s ) + A ( X ( s ) , P ( s ) , s ) (cid:17) (cid:21) × exp " Z tt d s ∂A ( X, P, t ) ∂P , (C.8)where B is a constant. The first exponential term corresponds to the Onsager-Machlupfunction, and the second term is derived from the Ito formula [26]. Thus, the conjugatedynamics is given by P ∗ ( X , P , t | X, P, t ) = B exp (cid:20) − k B T Γ Z tt d s (cid:16) ˙ P ( s ) + A ∗ ( X ( s ) , P ( s ) , s ) (cid:17) (cid:21) × exp " Z tt d s ∂A ( X, P, t ) ∂P ! ∗ , (C.9)where ∗ denotes a time-reversal operation. Because the entropy production is given bythe ratio between Eqs. (C.8) and (C.9), the time-reversal parity of A ( X, P, t ) is crucial.From Eq. (7), A ( X, P, t ) is given by A ( X, P, t ) = Γ P + Ω X − G Ωerfc M ( X, P ) √ σ M ! cos 2Ω tX = Γ P + Ω X − G Ω " − erf M ( X, P ) √ σ M ! cos 2Ω tX, (C.10)where the complementary error function is decomposed to a constant and a odd function(error function). This decomposition is crucial for calculating its conjugate dynamics as A ∗ ( X, P, t ) = Γ P ∗ + Ω X ∗ − G Ω " − t P erf M ∗ ( X, P ) √ σ M ! cos 2Ω tX ∗ , = − Γ P + Ω X − G Ω " − erf M ( X, P ) √ σ M ! cos 2Ω tX. (C.11)Here, t p = {− , } is determined by the time-reversal parity of the target observable M ( X, P ), where t p = 1 ( −
1) when the M ∗ ( X, P ) = M ( X, P ) [ M ∗ ( X, P ) = −M ( X, P )]. Thus, regardless of the time-reversal parity of the target observable, thefeedback force is treated as a reversible force in the conjugate dynamics [13]. By usingthis probability in the conjugate dynamics,exp[Σ] = P ( X, P, t | X , P , t ) P ∗ ( X , P , t | X, P, t )= exp " − k B T Z d sP ◦ ˙ P + Ω X − G Ω " − erf M ( X, P ) √ σ M ! cos 2Ω tX ! × exp G Ω √ πσ M Z d s exp − M ( X, P ) √ σ M ! ∂ M ( X, P ) ∂P ◦ X cos 2Ω t , (C.12)8where ◦ explicitly denotes Stratonovich integral. The first exponential term correspondsto the entropy production in the thermal bath, and the second exponential termcorresponds to the entropy production in the controller. The entropy production rate,which is directly achieved by taking the time derivative in Eq. (C.12), can be expressedas follows: ˙Σ = ˙ S bath + ˙ S pump , (C.13)˙ S bath = − k B T P ◦ ( − Γ P + F th ) , (C.14)˙ S pump = − G Ω √ πσ M exp − M ( X, P ) √ σ M ! ∂ M ( X, P ) ∂P ◦ X cos 2Ω t. (C.15)Because the target observable corresponds to K x which is given in the rotating frame,we perform the rotating wave approximation to linearize the transformation from thelaboratory frame ( X, P ) to the rotating frame ( q, p ) as X = q cos Ω t + p sin Ω t, (C.16) P/ Ω ≈ − q sin Ω t + p cos Ω t. (C.17)Thus, the momentum derivative of the target observable in Eq. (C.15) is evaluated by ∂ M ( X, P ) ∂P ≈ ∂q∂P ∂ M ( q, p ) ∂q + ∂p∂P ∂ M ( q, p ) ∂p = 12Ω ( q cos Ω t − p sin Ω t ) . (C.18)By using the following approximation, P ≈ K z , (C.19)( q cos Ω t − p sin Ω t ) X cos 2Ω t/ (2Ω) ≈ K z / , (C.20)we obtain the expressions of stochastic average of entropy production, D ˙ S bath E ≈ Γ k B T h K z i − k B T ! , (C.21) D ˙ S pump E ≈ s π ˜ G Γ σ M * K z exp − K x σ M !+ . (C.22) Appendix D. Entropy production in our protocol from probability currents
Although the path integral formalism shown in Appendix C provides us a completeexpression of entropy production with exact physical meaning, attempting to calculateit via the another simple formalism directly from the coarse-grained Fokker-Planckequation via probability currents [27, 28, 14] is worthwhile to confirm our formula inEqs. (C.21)-(C.22). The Fokker-Planck equation is re-expressed by probability currents J z ( z = X, P ) as ∂ t P = − X z ∂ z J z , (D.1)9 J X = − P P , (D.2) J P = − Γ P − Ω X + G Ω " − erf M ( X, P ) √ σ M ! cos 2Ω tX − k B T Γ ∂ P ! P . (D.3)Here, we split the momentum current J P into two in terms of the time-reversal parity,the same as discussed in Appendix C, in which the feedback force is regarded as areversible force, J rev P = " − Ω X + G Ω " − erf M ( X, P ) √ σ M ! cos 2Ω tX P , (D.4) J irr P = [ − Γ P − k B T Γ ∂ P ] P . (D.5)Introducing Shannon entropy hSi ≡ − R d X d P P ln P using a relationship ∂ t hSi = − R d X d P ( ∂ t P ) ln P = − R d X d P ( ∂ t P ) (1 + ln P ), ∂ hSi ∂t = Z d X d P X z ∂ z J z ! (ln P + 1) (D.6)= − Z d X d P X z J z ∂ z PP . (D.7)Here, the second equation is derived by using a partial integral and removing theboundary integral because the probability density function on the boundary is assumedto take zero. Note that Z d X d P J x ∂ X PP = − Z d X d P P ∂X ( − P ) = 0 . (D.8)Moreover, by using the relationship ∂ P P = − k B T Γ (cid:16) J irr P + Γ P (cid:17) from Eq. (D.5), itreduces to ∂ hSi ∂t = 1 k B T Γ Z d X d P P h ( J irr P ) + Γ P J irr P P i − Z d X d P P J rev P ∂ P P = h ˙ S tot i − h ˙ S bath i − h ˙ S pump i . (D.9)The first term in Eq. (D.9) corresponds to the non-negative entropy production, h ˙ S tot i ≡ k B T Γ Z d X d P ( J irr P ) P ≥ , (D.10)which obviously posses the second-law-like inequality, ∂ hSi ∂t + h ˙ S bath i + h ˙ S pump i ≥ . (D.11)The second term in Eq. (D.9) corresponds to the entropy production rate due to theexistence of irreversible currents. It can be expanded to h ˙ S bath i ≡ k B T Γ Z d X d P Γ P ◦ (Γ P + k B T Γ ∂ P ) P , = 1 k B T Γ Γ h P i − k B T Γ ! , ≈ Γ k B T h K z i − k B T ! , (D.12)0where ˜ G ≡ G / Γ is notated. The approximation in Eq. (D.12) is equivalent to thatin Eqs. (C.21) and (C.22). Apparently, we can confirm that Eq. (D.12) completelycorresponds to the entropy production in the thermal bath, Eq. (C.21), derived in thepath integral formalism.The third term in Eq. (D.9) is regarded as the entropy production rate thanks tothe presence of measurement and feedback, i.e., the entropy pumping rate, simplified as h ˙ S pump i Z d X d P P J rev P ∂ P P , = G Ω cos 2Ω t * ∂ P " erf M ( X, P ) √ σ M ! ◦ X , ≈ s π ˜ G Γ σ M * K z exp − K x σ M !+ . (D.13)This expression is also equivalent to that in Eq. (C.22) derived in the path integralformalism. Appendix E. Estimation of σ M To estimate a noise deviation in the measurement σ M , an observation is modelled by K Mx = a ( K x + σ M ξ M ) , (E.1)where a shows an arbitrary coefficient in measurement, and ξ shows a Markovian noisewith h ξ ( t ) ξ ( t ′ ) i = δ ( t − t ′ ). Because the true value of K x can be approximated by K Px ,which is the post-processed value, σ M can be determined by σ M = s h ( K Mx ) i a − h ( K Px ) i (E.2) a = h K Mx K Px ih ( K Px ) i (E.3)from the experimental data without any driving (i.e., h K Mx i = h K Px i = 0). As a result, σ M is determined to be 0 . ± . Appendix F. Entropy production in continuous driving and random driving
In the case of continuous driving (i.e., the feedback function becomes unity, f ( m ) = 1),the expression of entropy production is straightforwardly derived because it onlycontains the contribution of the entropy production in the thermal bath, ˙ S Cbath . Thus, h ˙ S Cbath i is achieved with the same definition given in Eq. (C.21). In the same manner, theentropy production with the random protocol, in which the feedback function is givenby f ( m ) = ξ R with the random integer ξ R ≡ { , } , can be formulated by taking intoaccount the contribution from the thermal bath h ˙ S Rbath i . Because entropy productionis defined as the ratio between the forward and backward probability, we consider theminimum entropy production as that under the random switching. In other words, theforce by the random switching is regarded as reversible in its conjugate dynamics, and as1a result the total entropy production just consists of the entropy production in thermalbath [25, 13]. Consequently, this means h ˙ S Rbath i can be calculated from Eq. (C.21). h ˙ S Cbath i and h ˙ S Rbath i can be analytically calculated by solving the following Langevinequation in the rotating frame: h ˙ K x i = − Γ h K x i + αG h K z i , (F.1) h ˙ K z i = − Γ h K z i + αG h K x i + Γ K , (F.2)where K ≡ k B T / , and α is a factor defined by α = 1 or α = 1 / h K z i = K − α ˜ G , h K x i = α ˜ G K − α ˜ G . (F.3)By substituting them into Eq. (C.21), the entropy production rates under the randomprotocol are analytically expressed as follows: h ˙ S Cbath i = Γ ˜ G − ˜ G , (F.4) h ˙ S Rbath i = Γ ˜ G / − ˜ G / G = 1 ( ˜ G = 2). This divergence occurs because the thermal bathcannot absorb the heat from the parametric driving due to G ≥ Γ (or ˜ G / ≥ Γ).
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