Quantum parameter-estimation of frequency and damping of a harmonic-oscillator
QQuantum parameter-estimation of frequency and damping of a harmonic-oscillator
Patrick Binder
1, 2, 3 and Daniel Braun ∗ Institute for Theoretical Physics, Tbingen University , 72076 Tbingen, Germany BioQuant Center, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany Institute for Theoretical Physics, Heidelberg University, Philosophenweg 19, 69120 Heidelberg, Germany (Dated: May 22, 2019)We determine the quantum Cramr-Rao bound for the precision with which the oscillator frequencyand damping constant of a damped quantum harmonic oscillator in an arbitrary Gaussian state canbe estimated. This goes beyond standard quantum parameter estimation of a single mode Gaussianstate for which typically a mode of fixed frequency is assumed. We present a scheme throughwhich the frequency estimation can nevertheless be based on the known results for single-modequantum parameter estimation with Gaussian states. Based on these results, we investigate theoptimal measurement time. For measuring the oscillator frequency, our results unify previouslyknown partial results and constitute an explicit solution for a general single-mode Gaussian state.Furthermore, we show that with existing carbon nanotube resonators (see J. Chaste et al. NatureNanotechnology 7, 301 (2012)) it should be possible to achieve a mass sensitivity of the order of anelectron mass Hz − / . I. INTRODUCTION
The harmonic oscillator is one of the most importantmodel systems in all of physics. It is exactly solvable,both classically and quantum mechanically, and playsa fundamental role in quantum field theories, where itselementary excitations can be identified with e.g. pho-tons or phonons. The harmonic oscillator arises as low-amplitude limit of a much wider class of non-harmonicoscillators, and its regular motion is at the basis of time-and frequency measurements. Indeed, the most precisemeasurements of a physical quantity are often achievedwhen transducing their variations into frequency changes.It is therefore of utmost importance to figure out howprecisely the two characteristic quantities of a harmonicoscillator, namely its frequency and its damping can bemeasured in principle. A partial answer was provided in[1], where the quantum Cram´er-Rao bound (QCRB) forthe frequency measurement of an undamped harmonicoscillator in an arbitrary pure quantum state was calcu-lated. The QCRB is the ultimate lower bound for theuncertainty with which a parameter can be estimated.It is optimized over all possible (POVM-)measurements(POVM=positive operator-valued measure, a class ofmeasurements that includes but is more general than theusual projective von Neumann measurements), and overall data-analysis procedures (in the sense of unbiasedestimator functions of the measurement results alone).It becomes relevant when all technical noise sources areeliminated, and only the noise inherent in the quantumstate remains. Importantly, the QCRB can be saturatedin the limit of a large number of measurements.A damped harmonic oscillator leads, however, natu-rally to mixed quantum states, and for those the calcu-lation of the QCRB is much more difficult than for pure ∗ Author to whom correspondence should be addressed:[email protected] states, owing to the need to diagonalize the density op-erator in an infinitely dimensional Hilbert space. In [2]an attempt was made to obtain the QCRB for the fre-quency of a kicked and damped oscillator [3], by usingthe formulas for Gaussian states. Indeed, in [4] the ex-act QCRB was found for any of the five parameters thatuniquely fix an arbitrary Gaussian state of a harmonicoscillator. However, those formulas were derived for anoscillator of fixed frequency, and they cannot be directlyapplied for frequency estimation. Doing so would amountto considering the Hamiltonian H = (cid:126) ωa † a as a genera-tor of a phase shift, i.e. the unknown parameter ω mul-tiplies a hermitian generator, whose variance gives, upto a factor 4, the pure state quantum Fisher information(QFI). However, this ignores that the annihilation- andcreation operators depend themselves on ω . That they doso is most easily seen by writing them in the Fock-basisand realizing that the wave-functions corresponding tothe energy eigenstates depend on ω through the oscil-lator length. Physically, ignoring the ω − dependence of a, a † hence implies that one neglects the ω − dependenceof the energy-eigenstates, which is particularly importantat small times, i.e. much smaller than the period of theoscillator.One might then think that calculating the QCRB forthe damped harmonic oscillator is a hopeless endeavorif the formulas for the Gaussian states cannot be ap-plied, and the state is not already diagonalized. Herewe show, however, that there is a well-defined procedurethat allows one to use those formulas nevertheless forthe large and experimentally most relevant class of ini-tial Gaussian states, by carefully incorporating the conse-quences of a change of frequency. This allows us to fullysolve the problem of parameter estimation of a (weakly)damped harmonic oscillator, described by a Lindblad-master equation. a r X i v : . [ qu a n t - ph ] M a y II. GENERAL FRAMEWORK
We start by briefly describing the dynamics of adamped harmonic oscillator. Afterwards we review theclosed-form expression for the general quantum Fisherinformation (QFI) for single-mode Gaussian states [4].
A. Dynamics
We consider a quantum harmonic oscillator with barefrequency ω weakly coupled to a Markovian environment.Assuming the validity of the Born-Markov approximationand the rotating-wave approximation, the density matrix ρ of the oscillator evolves according to the master equa-tion (ME) [5, 6]d ρ d t = − iω (cid:2) ˆ a † ˆ a, ρ (cid:3) + γ n (cid:0) a † ρ ˆ a − ˆ a ˆ a † ρ − ρ ˆ a ˆ a † (cid:1) + γ n + 1) (cid:0) aρ ˆ a † − ˆ a † ˆ aρ − ρ ˆ a † ˆ a (cid:1) , (1)where we introduced the mean thermal photon numberof the bath ¯ n = (e x − − at frequency ω , dimensionlessinverse temperature x ≡ (cid:126) ω/k B T , the damping constant γ . By introducing the quadrature operator X = (ˆ q, ˆ p ) T , the three-dimensional vector S ( t ) = ( M ωσ qq , σ pp /M ω, σ pq ) T , where σ AB ≡ / × (cid:104) AB + BA (cid:105) − (cid:104) A (cid:105) (cid:104) B (cid:105) and by using the ME(1) one finds equations of motion [7]:d (cid:104) X (cid:105) ( t )d t = G (cid:104) X (cid:105) ( t ) , (2a)d S ( t )d t = K S ( t ) + S inh , (2b)where G = (cid:18) − γ/ /M − M ω − γ/ (cid:19) , K = − γ ω − γ − ω − ω ω − γ (2c)and S inh = γ (cid:126) (2¯ n + 1) / , , T . The solutions of thetime evolution of the first order moments are given by (cid:104) X (cid:105) ( t ) = exp( Gt ) (cid:104) X (cid:105) (0). For the second order momentswe get S ( t ) = exp( Kt ) S (0) + K − (exp( Kt ) − I ) S inh ,where I denotes the identity operator.The two phase-space coordinates ˆ q and ˆ p are linked tothe annihilation and creation operator ˆ a ω and ˆ a † ω of themode by ˆ q = (cid:114) (cid:126) M ω (cid:0) ˆ a † ω + ˆ a ω (cid:1) , (3a)ˆ p = i (cid:114) (cid:126) M ω (cid:0) ˆ a † ω − ˆ a ω (cid:1) . (3b)Summing up, ω , γ , and ¯ n are coded into a state bythe dynamics (1), but in addition a state specified ini-tially e.g. in the Fock basis acquires an ω -dependence due to the ω -dependence of the harmonic oscillator en-ergy eigenstates (oscillator length). B. QFI of single-mode Gaussian states
Gaussian state.
The Wigner function for an arbitrarydensity matrix ρ of a continuous variable system witha single degree of freedom (such as a single harmonicoscillator) is defined by [8] W ( q, p ) = 1 π (cid:126) (cid:90) ∞−∞ e − ipy/ (cid:126) (cid:104) q − y | ρ | q + y (cid:105) d y . (4)By definition, a Gaussian state is a state whose Wignerfunction is Gaussian. Thus, for a Gaussian state of asingle harmonic oscillator (such as a single mode of anelectro-magnetical field) the Wigner function takes thegeneral form [9] W ( q, p ) = Pπ exp (cid:20) −
12 ( X − (cid:104) X (cid:105) ) T Σ − ( X − (cid:104) X (cid:105) ) (cid:21) , (5)where X = (ˆ q, ˆ p ) T is the quadrature operator, Σ is thecovariance matrix, (cid:104) . . . (cid:105) ≡ tr( ρ . . . ) defines the expecta-tion value and P = tr ρ is the purity. For single-modeGaussian states the purity is completely described by thecovariance matrix and is given by [10] P = (cid:126) (cid:112) det(Σ) . (6)Next, we recall that a general single-mode Gaussianstate ρ can always be represented as a rotated squeezeddisplaced thermal state ν , i. e. [9, 11] ρ = R ( ψ ) D ( α ) S ( z ) νS † ( z ) D † ( α ) R † ( ψ ) , (7)where S ( z ) = exp (cid:2) (1 / (cid:0) z ˆ a † − z ∗ ˆ a (cid:1)(cid:3) is the squeezingoperator, R ( ψ ) = exp (cid:0) iψ ˆ a † ˆ a (cid:1) denotes the rotation op-erator and D ( α ) = exp (cid:0) α ˆ a † − α ∗ ˆ a (cid:1) introduces the dis-placement operator. By introducing N th = tr( ν ˆ a † ˆ a ),which denotes the number of initial thermal photons, and z = r e iχ the general Gaussian state can be parametrizedby five real parameters α, ψ, r, χ, N th ∈ R . Note that wekeep N th and ¯ n as independent parameters. Quantum Fisher information.
We start from a densityoperator ρ θ , which depends on an unknown real scalarparameter θ . To estimate this parameter, m independentmeasurements with the outcome ξ = ( ξ , ξ , . . . , ξ M ) T are taken. From the outcome we construct an estimatorˆ θ est . For unbiased estimators the sensitivity with whicha parameter θ can be measured has a lower bound, theso-called quantum Cramr-Rao bound (QCRB), given by[12–15] Var[ˆ θ est ] ≥ m I ( ρ θ ; θ ) , (8)where I ( ρ θ ; θ ) denotes the QFI. The fidelity, definedby F ( ρ , ρ ) = { tr[( √ ρ ρ √ ρ ) / ] } , for two arbitrarysingle-mode Gaussian states ρ and ρ is given by [16] F ( ρ , ρ ) = 2 exp (cid:104) − ( (cid:104) X − X (cid:105) ) T (Σ + Σ ) − (cid:104) X − X (cid:105) (cid:105)(cid:112) | Σ + Σ | + (1 − | Σ | )(1 − | Σ | ) − (cid:112) (1 − | Σ | )(1 − | Σ | ) . (9)This formula is valid generally for two Gaussian Wignerfunctions, regardless of the underlying physical system.It remains therefore valid if the two Wigner functionsrepresent states of two different harmonic oscillators, no-tably harmonic oscillators that can differ in frequency.Using further the fact that the fidelity is linked to theQFI through [4]I ( ρ θ ; θ ) = − ∂ F ( ρ θ , ρ θ + ε ) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 (10)one obtains the general QFI for Gaussian states of a sin-gle harmonic oscillator of fixed frequency [4]I ( ρ θ ; θ ) = 12 tr (cid:104)(cid:0) Σ − ∂ θ Σ (cid:1) (cid:105) P + 2 ( ∂ θ P ) − P +( ∂ θ (cid:104) X (cid:105) ) T Σ − ∂ θ (cid:104) X (cid:105) . (11)By following the approach adopted by Jiang in Ref. [17]the same result can be obtained [18]. III. UNDAMPED CASE
This section provides a scheme for the calculation ofand results for the QFI relevant for estimating the fre-quency ω in the case of no damping. A. Scheme for the estimation of the quantumFisher information
Firstly we will illustrate that by directly using Eq. (11)for a frequency measurement one does not get the fullQFI, instead one obtains just that part that correspondsto having ˆ a † , ˆ a as frequency-independent generator of thetime evolution. For this purpose, we use the known re-sults of the QFI for pure states, where one does not getthe full QFI if taking ˆ a † , ˆ a independent of ω . In par-ticular, this means that directly inserting the solutionof the dynamics into equation Eq. (11) will not providethe correct result, as the ω -dependence of the Fock basisis not considered. Lastly, we justify that one can stilluse Eq. (11) if one treats the squeezing due to frequencychange correctly, which leads to the scheme we propose.We consider the case that only the dynamics of thestate, and not the initial pure state ρ = | ψ (cid:105)(cid:104) ψ | it-self, depends on the frequency ω to be measured. Forgiven Hamiltonian H = (cid:126) ω (cid:0) ˆ a † ˆ a + 1 / (cid:1) , the dynamics ofthe system is described by ρ ( t ) = U ( t ) ρ (0) U † ( t ), where U ( t ) = exp( − it H / (cid:126) ) is the time evolution operator. By neglecting the ω -dependence of a, a † the QFI is given by[19] I ( ρ ω ( t ); ω ) = 4 Var (cid:2) t (cid:0) ˆ a † ˆ a + 1 / (cid:1) , | ψ (cid:105) (cid:3) , (12)where Var [ A, | ψ (cid:105) ] ≡ (cid:104) ψ | A | ψ (cid:105) − (cid:104) ψ | A | ψ (cid:105) denotesthe variance. For a general pure Gaussian state in theform of Eq. (7), i. e. | ψ (cid:105) = R ( ψ ) D ( α ) S (cid:0) r e iχ (cid:1) | (cid:105) , theQFI readsI ( ρ ω ( t ); ω ) = 4 α t [cosh(2 r ) + cos( χ ) sinh(2 r )]+2 t sinh (2 r ) . (13)Next, we determine the same QFI by directly using equa-tion (11). For this we first use that we can write thetime-evolved density operator in the following way: ρ ω ( t ) = R ( ζ ) D ( α ) S ( z ) | (cid:105)(cid:104) | S † ( z ) D † ( α ) R † ( ζ ) , (14)where ζ = ψ − ωt . Using σ = e − r , equation (16) from [4]can be rewritten as [20]I ( ρ ω ( t ); ζ ) = 4 α [cosh(2 r ) + cos( χ ) sinh(2 r )]+2 sinh (2 r ) . (15)Thus, with d / d ω = − t d / d ξ we get the same re-sult as obtained in equation (13), of which we havedemonstrated that by directly using equation (11) the ω -dependence of the basis is not considered.In order to consider all frequency dependencies cor-rectly, we have developed the following scheme for theestimation of the QFI:1. Start with an initial Gaussian state given in theFock basis (cid:8) | n (cid:105) ω (cid:9) .2. Perform a sudden change of frequency ω → ω ,which corresponds to a squeezing, at time t = 0.3. Evolve the quantum state with respect to the newfrequency ω .4. Estimate the QFI I ( ρ ω ( t ); ω ) by using Eq. (11).5. Take the limit ω → ω .The sudden change of frequency ω → ω at time t = 0ensures that also the frequency dependence of the basisis considered. Furthermore, it can be shown that thefrequency jump corresponds to squeezing (see Appendix.A), i. e. | n (cid:105) ω = S ω ( s ) | n (cid:105) ω , (16)where s = − tanh − ( y ) and y = ( ω − ω ) / ( ω + ω ).It should be noted that the introduced scheme is onlyneeded to determine the QFI for a frequency measure-ment using Eq. (11). For pure states, for example, theQFI can be determined directly from the overlaps of thestates propagated with slightly different frequency [1],or, equivalently, from the variance of the local generator,taking into account the ω -dependence of ˆ a ω , ˆ a † ω (see Ap-pendix. B). Furthermore, it should be noted that sincethe Fock basis does not depend on the damping constant,the introduced scheme is not needed for calculating the QFI for the estimation of γ . B. Result for QFI for vanishing damping
By using the introduced scheme we now determine theQFI for the estimation of ω for the general Gaussian stategiven in Eq. (7). For a time evolution of the Gaussianstate with the harmonic oscillator H = (cid:126) ω (cid:0) ˆ a † ω ˆ a ω + 1 / (cid:1) follows the result (see Appendix. C) ω I ( ρ ( τ ); ω ) = C + 2 C sin ( τ ) (cid:20) sinh (2 r ) cos ( χ + 2 ψ − τ ) + 1 + 2 C α (cosh(2 r ) + cos( χ + 4 ψ − τ ) sinh(2 r )) (cid:21) +2 C τ sin( τ ) (cid:20) C α cos(2 ψ − τ ) cosh(2 r ) + cos( χ + 2 ψ − τ ) (cid:0) C α sinh(2 r ) + sinh(4 r ) (cid:1)(cid:21) +2 C τ (cid:2) C α (cosh(2 r ) + cos χ sinh(2 r )) + sinh (2 r ) (cid:3) , (17)where τ = ωt and C = (1 + 2 N th ) N th (1 + N th ) , (18a) C = 1 C (1 + 2 N th ) , (18b) C = N th (1 + N th ) (cid:20) ln (cid:18) N th N th (cid:19)(cid:21) . (18c)The first term ( C ) of Eq. (17) results from the ω -dependence of the initial photon number N th , the secondterm is due to the ω -dependence of the Fock basis, andthe term ∝ t arises from ˆ a † , ˆ a as generator of the timeevolution.For an initial thermal state ρ (0) = ν , Eq. (17) reducesto I ( ν ( τ ); ω ) = 2 C sin ( τ ) + C ω . (19)Thus, a measurement with t > π/ ω does not provideany additional information regarding the frequency andthe QFI has an upper bound (2 C + C ) /ω —where C it-self is bounded by C ∈ [1 , ∀ N th and C is bounded by C ∈ [0 , ∀ N th . Furthermore, the result demonstratesthat one can measure the frequency of a mode of an e.m.field without any light at all, just from the vacuum fluc-tuations. The latter have been measured directly in [21].While our results from Eq. (17) agree with the obtainedQFI for a coherent state [1], our result in Eq. (19) con-tains an extra term C /ω due to the consideration ofthe ω -dependence of N th neglected in [1]. It should alsobe noted that our result agrees with the result by calcu-lating the QFI via the variance in the case of a generalpure Gaussian state (see Appendix. B). Optimal state . The QFI can be drastically increasedby displacing and/or squeezing the initial thermal state. In both cases, the QFI acquires a part proportional to t that always dominates at sufficiently large times. For aninitial state displaced with α ∈ R , the part proportionalto t has its maximum at χ = 0. We further point outthat the long-term behavior of the QFI for a squeezedthermal state also improves due to additional displacing.The optimal choice of thermal photons N th depends onthe initial state. If the QFI is dominated by the terms dueto the squeezing, a high number of photons is favorable.If, on the other hand, the terms due to the displacement,which are ∝ (1 + 2 N th ) − , dominate, the lowest possi-ble number of photons is desirable. The behavior canbe well explained by the Wigner function. A larger N th is equivalent to a wider distribution of the state. Thismeans that a small shift in the Wigner function of thedisplaced state, e. g. due to the time evolution, is lessmeasurable for larger N th . Consequently, the enlarge-ment of the thermal photons counteracts the additionalgain of the displacement. The benefits of squeezing, onthe other hand, increase with the thermal photon num-ber. This can be directly from eq.(17) seen, since its QFIis proportional to C , which is also the only term thatincreases with N th . IV. DAMPED CASE
In this section we will calculate the QFI for mixedGaussian states for the damped harmonic oscillator forestimating the oscillator frequency and damping con-stant. Furthermore, we determine the optimal measuringscheme and the optimal measuring time and we demon-strate that with existing carbon nanotube resonators itshould be possible to achieve a mass sensitivity of theorder of an electron mass Hz − / . FIG. 1. The long-term behavior of the dimensionless QFI, ω I ( ρ ∞ ; ω ), for a damped Gaussian state for measuring ω isshown as function of the thermal photon number of the bath.In the limit of validity of (1), the result is independent of thedamping constant. A. Measuring the oscillator frequency
By sticking to the scheme explained in Sec.III A, weobtain the exact expression for the QFI for a generalinitial Gaussian state by considering the time evolutiongiven by the ME (1), which can be found in the Ap-pendix, in Eq. (C7) to (C12). However, since the solutionis too heavy to report here, we will first look at the long-term behavior and then limit ourselves to specific initialstates—coherent state and squeezed state.
1. Long-term behavior
For longer periods, the solution of ME (1) relaxes tothe thermal equilibrium state, i. e. for t (cid:29) γ − , ρ t (cid:29) γ − −−−−→ e − (cid:126) ω/k B T / tr (cid:16) e − (cid:126) ω/k B T (cid:17) ≡ ρ ∞ . (20)It should be remembered that the thermal equilibriumstate as well as the mean thermal photon number ¯ n alsodepend on the oscillator frequency ω itself. It can there-fore be expected that the QFI does not vanish due tothe dependency of the final state on the frequency. Sinceboth first order moments vanish, i. e. lim t →∞ (cid:104) X (cid:105) = 0,only the first two terms of Eq. (11) contribute to QFIand calculation yieldsI ( ρ ∞ ; ω ) = 12 ω (cid:34) n (1 + ¯ n ) ln (cid:18) n ¯ n (cid:19) + 1 + 4¯ n (1 + ¯ n )1 + 2¯ n (1 + ¯ n ) (cid:35) . (21)This means that for large times, the QFI has an upperbound given by 2 /ω (see Fig. 1). The upper bound canbe reached in the high temperature limit. As a conse-quence, a longer measurement does not necessarily yield FIG. 2. The dimensionless QFI, ω I ( ρ α ; ω ), for an initial co-herent state (solid) is compared with the lower bound ω I α ( τ )(dashed) (see Eq. (22)) for measuring ω . Results are depictedfor ¯ n = 5 and g = 0 .
1: blue, α = 1 /
2; orange, α = 1. a better result for the experiment. In other words, thereis an optimal measurement (OMT) time in which thefrequency can be measured best, which is in accordancewith the physical expectations.
2. Optimal measurement time and maximal quantumFisher information
Coherent state . We start by considering an initial co-herent state ρ α (0) = D ( α ) | (cid:105)(cid:104) | D † ( α ). Recall, displac-ing the initial state is one of the possibilities to stronglyincrease the QFI in the undamped case. Since displacingthe ground state only affects the expectation values ofthe quadrature operators and not the covariance matrix,the QFI of the coherent state can be written asI ( ρ α ( τ ); ω ) = I ( ρ ( τ ); ω ) + I α ( τ ) , (22)where ρ ( τ ) denotes the time-evolved ground state andI α ( τ ) = ( ∂ ω (cid:104) X (cid:105) ) T Σ − ∂ ω (cid:104) X (cid:105) . The QFI of the groundstate is bounded by 2 . /ω (see Appendix. C). Thus,the upper bound of the QFI for the ground state is in-creased by introducing the system-bath coupling, whichcan be explained by the ω -dependence of ¯ n . I.e. alsoin the damped case, the frequency can be measuredwhen the system is initially prepared in the ground state.Straightforward calculation leads toI α ( τ ) = 4 α ω sin ( τ ) + τ sin(2 τ ) + τ (2¯ n + 1) e gτ − n , (23)where g = γ/ω introduces a dimensionless damping con-stant. Thus, for frequency measurements an as big aspossible displacement is recommended.Since the QFI of the ground state is bounded (andsmall), I ( ρ α ( τ ); ω ) ≈ I α ( τ ) applies for α (cid:29) g ¯ n (byassuming n (cid:29) g (cid:28) n (cid:29) α /g , I α ( τ ) becomes arbitrarily small FIG. 3. The dimensionless QFI, ω I ( ρ r ; ω ), for an initialsqueezed state (solid) is compared with the approximationfrom Eq. (30) (dashed) for measuring ω . Results are depictedfor ¯ n = 0 . g = 0 . r = 2 . and the QFI is then described by the QFI of the groundstate. In other words, displacement only improves fre-quency measurements for resulting mean energies largerthan the thermal energy.By neglecting small oscillations, maximization ofEq. (23) provides the maximal QFI I max ( ρ, θ ) ≡ max τ I( ρ, θ ) and the optimal measurement time τ max with I( ρ ( τ max ) , θ ) = I max ( ρ, θ ), i. e.I max ( ρ α ( τ ) , ω ) = − α ¯ ng ω W (cid:18) − n e (1 + 2¯ n ) (cid:19) × (cid:20) W (cid:18) − n e (1 + 2¯ n ) (cid:19)(cid:21) , (24) τ max = 1 g (cid:20) W (cid:18) − n e (1 + 2¯ n ) (cid:19)(cid:21) , (25)where W ( z ) denotes the Lambert W function defined by z = W ( z ) e W ( z ) , z ∈ C . The Taylor series for I ( ρ ( τ ); ω ) at ¯ n (cid:29) ρ ( τ ); ω ) = 2 ω (cid:20) − cos ( τ )(e gτ − n (cid:21) + O (cid:0) / ¯ n (cid:1) . (26)That means that for high temperatures ¯ n (cid:29) ∼ e − gτ ) than I α ( τ )( ∼ τ e − gτ ) and varies only slightly close to the time τ max . Consequently, the use of I α ( τ ) for estimating theoptimal measurement time leads, even in this range, toa good result of the OMT (see Fig. 2). Furthermore,it should be noted that the smaller g , the larger ¯ n canbe, so that the OMT is still very well described by I α ( τ ).By reducing the system-bath coupling, the maximal QFIincreases proportionally to ∝ g − . However, it shouldbe noted that the OMT also increases proportionally to ∝ g − .Thus, it is a natural to consider time as a resourceand to introduce the rescaled maximal QFI I ( t )max ( ρ, θ ) ≡ max t I( ρ, θ ) /t and the optimal measurement time τ ( t )max that maximizes it. For an initial coherent state we getI ( t )max ( ρ α ( t ) , ω ) = − α ¯ ngω W (cid:18) − n e(1 + 2¯ n ) (cid:19) , (27) τ ( t )max = 1 g (cid:20) W (cid:18) − n e(1 + 2¯ n ) (cid:19)(cid:21) . (28)Taking time into account as a resource leads to a reduc-tion of the OMT. Squeezed state . Besides displacement, squeezing theinitial state is another possibility to increase the QFI inthe undamped case. Therefore, we determine the QFIfor a squeezed state ρ r (0) = S ( r ) | (cid:105)(cid:104) | S † ( r ). For thecoherent state we have seen that reducing the temper-ature leads to an increase in the QFI. This behavior isreasonable, since increased temperature implies increaseddamping according to the master equation (1). A similarbehavior can be observed here with the squeezed state.The QFI for a vanishing bath temperature, i. e. ¯ n = 0,readsI( ρ r ( t ) , ω ) = (cid:2) ω (cid:0) e gτ sinh ( r ) + e gτ − cosh(2 r ) + 1 (cid:1)(cid:3) − × (cid:104) τ sinh(2 r ) sin(2 τ ) ( e gτ + cosh(2 r ) − − e gτ −
1) cosh(2 r )(2 cos(2 τ ) − e gτ ( e gτ −
1) + (cid:0) τ + 1 (cid:1) cosh(4 r ) − ( r ) cosh ( r ) cos(4 τ ) − τ − τ ) + 7 (cid:105) . (29)Alternatively, for high squeezing and low temperatures,i. e. r (cid:29) n (cid:28)
1, the QFI can be approximated as(see Fig. 3)I( ρ r ( τ ) , ω ) ≈ e r [2 τ + sin(2 τ )] ω (e gτ − n ) . (30)Thus, the QFI can be significantly increased by squeezing also for an initial thermal state. Neglecting the oscilla-tions, the OMT can be determined to τ max = 1 g (cid:2) W (cid:0) − / e (cid:1)(cid:3) ≈ . g . (31)For sufficiently high squeezing and low temperature, theOMT does not depend on the squeezing and temperatureanymore. B. Measuring the damping constant
Next we consider the QFI for the estimation of thedamping constant. First of all, the QFI disappears forlarge times, i.e. I( ρ ∞ , γ ) = 0. This can be seen directly from the fact that the final thermal state (for the masterequation approach) itself no longer depends on the damp-ing constant. In other words, there is again an OMT.After a straightforward but long and tedious calcula-tion we find for the QFI of a general Gaussian stateI ( ρ ( τ ); γ ) = P ( τ ) g τ γ e gτ (cid:40) α e gτ [ A (cosh(2 r ) − cos( χ ) sinh(2 r )) + a ,τ ] + 2 P ( τ )1 − P ( τ ) (cid:2) A + A ( a ,τ − a ) cosh(2 r ) − a a ,τ (cid:3) + P ( τ )1 + P ( τ ) (cid:20) A + A (cid:0) a + a ,τ (cid:1) cosh(4 r ) + 2 A ( a ,τ − a ) (cid:0) A − a a ,τ (cid:1) cosh(2 r ) − a a ,τ A + a a ,τ (cid:21)(cid:41) , (32)where a = 1 + 2¯ n , a ,τ = (e gτ − a , A = 1 + 2 N th and P ( τ ) = e gτ (cid:2) A + a ,τ + 2 a ,τ A cosh(2 r ) (cid:3) − / . (33)The result does not depend on the rotation angle ψ , butonly on the squeezing angle χ . In contrast to frequencymeasurement, the QFI for measuring γ is maximized for χ = π . This is in agreement with the physical expec-tation, as the relevant dynamic here is the relaxation of (cid:104) X (cid:105) . To illustrate the result, we again consider specificinitial states — thermal state, displaced thermal stateand squeezed state. Thermal state . The QFI of a thermal state ν can bewritten asI( ν ( τ ) , γ ) = (¯ n − N th ) g τ γ [(e gτ − n + N th ](e gτ (1 + ¯ n ) + N th − ¯ n ) . (34)The greater the deviation of the initial temperature fromthe bath temperature, the better γ can be measured. Inparticular, for a vanishing deviation, i.e. N th = ¯ n , theQFI vanishes, since in this case the state has no dynamicsat all. For ¯ n = 0, the OMT is given by τ max = 2 + W (cid:0) N th e − (cid:1) g . (35) Displaced thermal state . For an initial displaced ther-mal state ρ α,N th (0) = D ( α ) νD † ( α ) the QFI for measur-ing γ readsI( ρ α,N th ( τ ) , γ ) = I( ν ( τ ) , γ )+ α g τ γ [2 N th − n + e gτ (1 + 2¯ n )] . (36)Particularly for N th = ¯ n , the QFI simplifies toI( ρ α, ¯ n ( τ ) , γ ) = α g τ γ e gτ (1 + 2¯ n ) (37)and the OMT is given by τ max = 2 /g . For N th = ¯ n only the 3 rd part of equation (11) contributes to the QFI, i.e. the QFI results solely from the relaxation of (cid:104) p (cid:105) , (cid:104) q (cid:105) .By considering the rescaled QFI the OMT reduces to τ ( t )max = 1 /g . Squeezed state . The low temperature limit behavior,i. e. ¯ n = 0, of the QFI for an initial squeezed state ρ r isgiven byI ( ρ r ( τ ); γ ) = (cid:2) e gτ − gτ − (cid:3) g τ sinh ( r ) γ (e gτ − (cid:2) gτ −
1) sinh ( r ) + e gτ (cid:3) . (38)The sensitivity with which the damping parameter canbe measured improves by squeezing, displacing and/or atemperature deviation (see Fig. 4). C. Nano-mechanical resonators
In the following we apply the results obtained to nano-mechanical resonators, which function as precision mass
FIG. 4. The dimensionless QFI, γ I ( ρ ; γ ), for measuring γ is shown. Results are depicted for ¯ n = 1, g = 0 . χ = π and:blue, N th = 5, α = r = 0; orange, N th = 10, α = r = 0; red, N th = 10, α = 1, r = 0; purple, N th = 10, α = 1, r = 1 / sensors as their resonance frequency changes when ad-ditional mass is adsorbed. More precisely, we considercarbon nanotube resonators. Using the QCRB (8) and ω = (cid:112) D/M , where D is the effective spring constant ofthe harmonic oscillator, the smallest δM that can be re-solved from m measurements of the resonance frequencyis given by δM min = 2 Mω (cid:112) m I max ( ρ, ω ) . (39)Assuming a coherent state with oscillation amplitude ofabout 10 nm for the carbon nanotube resonator in [22]( M = 3 × − kg, ω = 2 π × .
865 GHz, T = 4 K and Q ∼ ), δM min according to (39) is slightly below oneproton mass. Using the OMT given by t max = 270 ns, thesensitivity corresponds to δM min √ t max = 0 . m e Hz − / ,which is less than 1 / δM min for the car-bon nanotube resonator in [23] ( M = 10 − kg, ω =2 π × . T = 300 K and Q ∼ ) was deter-mined to the order of a thousandth of an electron mass.Including the system-bath coupling, δM min increases toabout 74 proton masses, where the OMT is given by t max = 1 . µ s. This result is equivalent to 0 . / √ Hz,which approximately corresponds to one hundredth ofthe 78 u / √ Hz achieved in the experiment.
V. CONCLUSIONS
In summary, we have derived the quantum Cramr-Rao bound for measuring the oscillator frequency anddamping constant encoded in the dynamics of a gen-eral mixed single-mode Gaussian state of light, includingdamping through photon loss described by a Lindbladmaster equation. We first demonstrated that the knownsolution for the QFI for Gaussian states of a single har-monic oscillator of fixed frequency cannot be directly ap-plied to frequency measurements. Next, we presented ascheme through which the frequency estimation can nev-ertheless be based on the results of Pinel et al. [4].Furthermore, we have shown that displacing and/orsqueezing the initial state significantly increases the pre-cision with which ω and γ can be estimated. For measur-ing ω and r (cid:54) = 0, χ = 0 is optimal, whereas for measuring γ , χ = 0 maximizes the QFI.Our results can serve as important benchmarks for theprecision of frequency measurements of any harmonic os-cillator with given damping. In particular, we found op-timal measurement times that limit the sensitivity per √ Hz with which frequencies can be measured, in con-trast to the undamped case, where e.g. coherent stateslead to growing QFI for arbitrarily large times.
Appendix A: Change of basis
By presenting the scheme for the estimation of the QFIfor measuring ω we made use of the fact that the fre-quency jump corresponds to squeezing. Next we provethe statement, i. e. the following formula | n (cid:105) ω = S ω ( s ) | n (cid:105) ω , (A1)where s = − tanh − ( y ). For the sake of simplicity thetwo parameters y = ω − ωω + ω , y = 2 √ ω ωω + ω (A2)are introduced. A squeezed number state is given by [24] ω (cid:104) m | S ω ( s ) | n (cid:105) ω = √ n !cosh n +1 / | s | (cid:98) n/ (cid:99) (cid:88) j =0 ( − d (cid:63) ) j cosh j | s | ( n − j )! j ! × ∞ (cid:88) k =0 d k (cid:112) ( n − j + 2 k )! k ! ω (cid:104) m | n − j + 2 k (cid:105) ω (cid:124) (cid:123)(cid:122) (cid:125) = δ m,n − j +2 k , (A3)where d ≡ ( s/ | s | ) tanh | s | and (cid:98) n/ (cid:99) denotes the floorfunction. With m = n − j + 2 k and k ∈ N we get k = j + m − n ∈ N . This means in particular that m and n must be both even or both odd numbers, otherwise theoverlap disappears. If m and n satisfy this condition andby using cosh | s | = 1 /y and d = − y /
2, the expressioncan be further simplified as follows ω (cid:104) m | S ω ( s ) | n (cid:105) ω = (cid:112) y m ! n ! (cid:98) n/ (cid:99) (cid:88) j =0 ( − j + m − n (cid:0) y (cid:1) j + m − n ( n − j )! j ! (cid:0) j + m − n (cid:1) ! y n − j . (A4)By changing the index of summation to l = n − j we getthe new upper bound of min ( m, n ), where min ( m, n )denotes the smaller of the two integers m , n . l is alsobounded by m , since k = j − n − m = m − l ∈ N and thus l ≤ m . Using the new index of summation we get [25] ω (cid:104) m | S ω ( s ) | n (cid:105) ω = (cid:114) y m ! n !2 m + n min ( m,n ) (cid:88) l =0 , (2 y ) l l ! y ( m + n − l ) / ( − ( m − l ) / (cid:0) n − l (cid:1) ! (cid:0) m − l (cid:1) != R ωω ( m, n )= ω (cid:104) m | n (cid:105) ω , (A5)where R ωω ( m, n ) denotes the overlap matrix element be-tween energy eigenstates of the two oscillators with fre-quency ω and ω . Since this is true for all m , we haveproven the formula. Thus, for any density operator fol-lows ρ ω = S ω ( s )˜ ρ ω S † ω ( s ) , (A6)where s = − tanh − ( y ) and ˜ ρ ω corresponds to the initialstate ρ ω by replacing the frequency ω of the basis withthe new frequency ω . Thus, we have shown that theinitial frequency change corresponds to a squeezing. Itshould be noted that in the case of a vanishing frequencychange, i. e. ω = ω , y = 0, s = 0 and S ( s = 0) = I follow and thus ρ ω = ρ ω is ensured. Appendix B: QFI for pure states
In the following it will be shown that the introducedscheme provides the correct QFI for an undamped pureGaussian state. Therefore, the QFI is calculated anal-ogously to chapter III A, but this time also the ω -dependence of ˆ a † , ˆ a are taken into account.This means, we consider the case that only the dynam-ics of the state, and not the initial state ρ = | ψ (cid:105)(cid:104) ψ | , (B1)where | ψ (cid:105) = R ( ψ ) D ( α ) S (cid:0) r e iχ (cid:1) | (cid:105) , depends on the fre-quency ω to be measured. For given Hamiltonian H ω = (cid:126) ω (cid:0) ˆ a † ω ˆ a ω + 1 / (cid:1) , the dynamics of the system is describedby ρ ω = U ω ρ U † ω , where U ω = exp (cid:2) − iωt (cid:0) ˆ a † ω ˆ a ω + 1 / (cid:1)(cid:3) isthe time evolution operator. With the help of the localgenerator K = iU † ω ( t ) ∂U ω ( t ) ∂ω (B2)the QFI can be rewritten as follows [26]I ( ρ ω ; ω ) = 4 Var [ K , | ψ (cid:105) ] . (B3)If A is a Matrix depending on the parameter x , A = A ( x ), then [27] ∂∂x e A ( x ) = (cid:90) e α A ( x ) ∂ A ( x ) ∂x e − α A ( x ) d α e A ( x ) . (B4)Using this formula we can rewrite the local generator K as [28] K = t (cid:126) (cid:90) − V ( α ) ∂ H ω ∂ω V † ( α ) d α , (B5) where V ( α ) = exp( − iαt H ω / (cid:126) ). The derivative of theHamiltonian H ω with respect to the oscillator frequency ω reads ∂ H ω ∂ω = (cid:126) a † ω ˆ a ω + (cid:126) (cid:104)(cid:0) ˆ a † ω (cid:1) + ˆ a ω + 1 (cid:105) , (B6)where we made use of ∂ ω a † ω = ˆ a ω / ω and ∂ ω ˆ a ω = ˆ a † ω / ω ,which can be seen from their representation in the Fockstate basis | n (cid:105) ω . With the help ofe − iψ ˆ a † ω ˆ a ω ˆ a ω e iψ ˆ a † ω ˆ a ω = e iψ ˆ a ω (B7)we get V ( α ) ∂ H ω ∂ω V † ( α )= (cid:126) ˆ a † ω ˆ a ω + (cid:126) (cid:104) e − iαωt (cid:0) ˆ a † ω (cid:1) + e iαωt ˆ a ω + 1 (cid:105) . (B8)Insertion and subsequent integration provides the localgenerator K = t (cid:18) ˆ a † ω ˆ a ω + 12 (cid:19) − i ω (cid:2)(cid:0) − e − iωt (cid:1) ˆ a ω + (cid:0) − e iωt (cid:1) ˆ a † ω (cid:3) . (B9)Next, the QFI is calculated. The annihilation and cre-ation operator ˆ c ω and ˆ c † ω , defined byˆ c ω = S † ω (cid:0) r e iχ (cid:1) D † ω ( α ) R † ω ( ψ )ˆ a ω R ω ( ψ ) D ω ( α ) S ω (cid:0) r e iχ (cid:1) = e iψ (cid:0) cosh( r )ˆ a ω + e iχ sinh( r )ˆ a † ω + α (cid:1) (B10)and (ˆ c ω ) † = ˆ c † ω , can be used to rewrite the expectationvalues of K as follows (cid:104) ψ | K k | ψ (cid:105) = (cid:104) | K k (cid:12)(cid:12)(cid:12)(cid:12) ˆ a † =ˆ c † ˆ a =ˆ c | (cid:105) . (B11)Using the formulae ˆ a | n (cid:105) = √ n | n − (cid:105) and ˆ a † | n (cid:105) = √ n + 1 | n + 1 (cid:105) , we obtain the QFI after a straightforwardcalculation:I ( ρ ( t ); ω ) = 2 tω sin ωt (cid:20) α cos(2 ψ − ωt ) cosh 2 r + cos( χ + 2 ψ − ωt ) (cid:0) α sinh 2 r + sinh 4 r (cid:1)(cid:21) + 2 ω sin ωt (cid:20) sinh r cos ( χ + 2 ψ − ωt ) + 1 + 2 α (cosh 2 r + cos( χ + 4 ψ − ωt ) sinh 2 r ) (cid:21) +2 t (cid:2) α (cosh 2 r + cos χ sinh 2 r ) + sinh r (cid:3) . (B12)Comparison with Eq. (17) for a pure Gaussian state, i. e. N th = 0, shows that the results are identical.0 Appendix C: Calculation of QFI
Here we report the calculation of the QFI for mea-suring ω . First, the dynamics resulting from ME (1) isdetermined. Then the QFI for the undamped case is cal-culated. Finally, the exact QFI for the damped case isgiven.The solutions of ME (1) are given by [7] (cid:104) q (cid:105) t = e − γt (cid:20) cos( ωt ) (cid:104) q (cid:105) + 1 M ω sin( ωt ) (cid:104) p (cid:105) (cid:21) , (C1a) (cid:104) p (cid:105) t = e − γt/ (cid:20) cos( ωt ) (cid:104) p (cid:105) − M ω sin( ωt ) (cid:104) q (cid:105) (cid:21) (C1b)and σ qq ( t ) = (cid:126) M ω (1 + 2¯ n ) (cid:0) − e − γt (cid:1) + e − γt (cid:20) cos ( ωt ) σ qq (0) + sin ( ωt ) M ω σ pp (0) + sin(2 ωt ) M ω σ pq (0) (cid:21) , (C2a) σ pp ( t ) = (cid:126) M ω n ) (cid:0) − e − γt (cid:1) + e − γt (cid:20) cos ( ωt ) σ pp (0) + M ω sin ( ωt ) σ qq (0) − M ω sin(2 ωt ) σ pq (0) (cid:21) , (C2b) σ pq ( t ) = e − γt (cid:20) cos(2 ωt ) σ pq (0) + 1 M ω sin( ωt ) cos( ωt ) (cid:16) σ pp (0) − M ω σ qq (0) (cid:17)(cid:21) . (C2c)Indeed, the second equation (C1b) is an immediate con-sequence of p = M ∂ t q . For the general single-modeGaussian state in Eq. (7) the initial expectation valuesare given by (cid:104) q (cid:105) = α (cid:114) (cid:126) M ω cos( ψ ) , (C3a) (cid:104) p (cid:105) = α (cid:112) (cid:126) M ω sin( ψ ) . (C3b) σ qq (0) = (cid:126) M ω (2 N th + 1)[cosh(2 r ) + cos( χ + 2 ψ ) sinh(2 r )] , (C3c) σ pp (0) = (cid:126) M ω N th + 1)[cosh(2 r ) − cos( χ + 2 ψ ) sinh(2 r )] , (C3d) σ pq (0) = (cid:126) N th + 1) sin( χ + 2 ψ ) sinh(2 r ) . (C3e)Here we give the expectation values with respect to theinitial frequency ω . The time evolution of the ME (1),on the other hand, is with respect to the new frequency ω , as described in the scheme.
1. Undamped case
We start with the calculation for the QFI of the un-damped case of Eq. (17). The undamped dynamic cor- responds to the expectation values from Eq. (C1) andEq. (C2) for γ →
0. By using these results we calculatedthe five parameters of interest Σ − , ∂ ω Σ , P, ∂ ω P, ∂ ω (cid:104) X (cid:105) .For the sake of clarity we give the results after executingthe limit ω → ω and additionally use the dimensionlesstime τ = ωt . The derivative of quadrature operator isgiven by ∂ ω (cid:104) X (cid:105) = α (cid:114) M (cid:126) ω (cid:18) τ sin( ψ − τ ) − sin( ψ ) sin( τ ) Mω − τ cos( ψ − τ ) − cos( ψ ) sin( τ ) (cid:19) (C4)The purity and its derivative read P = 11 + 2 N th , (C5a) ∂ ω P = 2 N th (1 + N th ) ln(1 + 1 /N th ) ω (1 + 2 N th ) , (C5b)whereas the derivative of the covariance matrix is de-scribed by the following equations:2 M ω (cid:126) ∂ ω σ qq ( t ) = 1 ωP (cid:2) − r ) sin ( τ ) + (cos( χ + 2 ψ ) − cos(2 τ − χ − ψ ) − τ sin(2 τ − χ − ψ )) sinh(2 r ) (cid:3) − ∂ ω PP [cosh(2 r ) + cos(2 τ − χ − ψ ) sinh(2 r )] , (C6a)2 (cid:126) M ω ∂ ω σ pp ( t ) = 1 ωP (cid:2) r ) sin ( τ ) + (cos( χ + 2 ψ ) − cos(2 τ − χ − ψ ) + 2 τ sin(2 τ − χ − ψ )) sinh(2 r ) (cid:3) − ∂ ω PP [cosh(2 r ) − cos(2 τ − χ − ψ ) sinh(2 r )] , (C6b)2 (cid:126) ∂ ω σ pq ( t ) = ∂ ω PP sin(2 τ − χ − ψ ) sinh(2 r ) − ωP (sin(2 τ ) cosh(2 r ) + 2 τ cos(2 τ − χ − ψ ) sinh(2 r )) . (C6c)By inserting Eq. (C4)-(C6) into Eq. (11) one obtainEq. (17).
2. Damped case
By repeating the previous calculations with a non-vanishing γ , we estimate the QFI for damped Gaussianstates for measuring ω . Since the solution is too heavy, we specify the three terms of Eq. (11) separately, i. e.I ( ρ ( t ); ω ) = I ,ω + I ,ω + I ,ω , (C7)where I ,ω = (2(1 + P )) − tr (cid:104)(cid:0) Σ − ∂ θ Σ (cid:1) (cid:105) , (C8a)I ,ω = 2( ∂ θ P ) (1 − P ) − , (C8b)I ,ω = ( ∂ θ (cid:104) X (cid:105) ) T Σ − ∂ θ (cid:104) X (cid:105) . (C8c)Repeating the previous calculation for a non-vanishingdamping leads to the following exact result of the QFI:I ,ω = P ( τ )2 ω e gτ (1 + P ( τ )) (cid:40) A (cid:0) A + a ,τ + 2 a ,τ A C r (cid:1) S r τ +8 A (cid:0) A + a ,τ + 2 a ,τ A C r (cid:1) [ A sin( ξ ) C r + ( a ,τ + A C r ) sin(2 τ − ξ )] S r τ + A (cid:16) A (cid:0) S r + 2 (cid:1) + A A + 2 A A a ,τ a C r + a ,τ a (cid:0) S r + 1 (cid:1) − A (cid:2) A cos(2 τ ) + A (cos(2 ξ ) + cos(4 τ − ξ ) + 4 sin[ ξ ] sin[2 τ − ξ ]) S r − a ,τ a sin( τ ) sin( τ − ξ ) S r (cid:3) + A a ,τ (cid:16) A (cid:2) A C r sin ( τ ) (cid:0) ( τ − ξ ) S r (cid:1) + S r cos(2 τ − ξ )( A A − a ,τ a C r ) − A A S r cos( ξ ) (cid:3) + A A C r + 2 A A a ,τ a + a ,τ a C r (cid:17) + a ,τ (cid:16) A (7 + 6 S r ) + a ,τ a + A A (cid:2) a ,τ a C r + A A (1 + 2 S r ) (cid:3) + 2 A (cid:8) A S r [2(cos(2 τ − ξ ) − τ )) − cos(4 τ − ξ ) + cos(2 ξ )] − A cos(2 τ )+ 8 A A C r S r sin( τ ) sin( τ − ξ ) − a ,τ a S r cos( τ ) cos( τ − ξ ) (cid:9)(cid:17) + a ,τ [2 A A S r cos(2 τ − ξ ) − A C r (cos(2 τ ) − a ,τ (cid:41) , (C9)I ,ω = e − gτ P ( τ )2 ω (1 − P ( τ )) (cid:2) A A A + a ,τ a ,τ a + ( A a ,τ a + a ,τ A A ) C r − a ,τ A cos( ξ ) S r (cid:3) , (C10)I ,ω = 4 e − gτ α P ( τ ) ω (cid:8) [ a ,τ + A ( C r + cos( χ ) S r )] τ + [cos( τ − ψ )( a ,τ + A C r ) + A cos( τ − ξ ) S r ]2 τ sin( τ )+[ a ,τ + A ( C r + cos(2 τ − ξ − ψ ) S r )] sin ( τ ) (cid:9) , (C11)where we introduced a new angle ξ = χ + 2 ψ , C r =cosh(2 r ), S r = sinh(2 r ) and a = 1 + 2¯ n, A = 1 + 2 N th , (C12a) a = 4¯ n (1 + ¯ n ) , A = 4 N th (1 + N th ) , (C12b)2 a = ln(1 + 1 / ¯ n ) , A = ln(1 + 1 /N th ) , (C12c) a ,τ = (e gτ − a , a ,τ = (e gτ − a . (C12d) Finally the maximum of the QFI of an initial groundstate, which was used to approximate the QFI of thecoherent state, is determined. For an initial ground state,the QFI simplifies to ω I ( ρ ; ω ) = 1 + [e gτ (1 + 2¯ n ) − n ] − gτ (1 + 2¯ n ) − n ] cos(2 τ )2[2¯ n − gτ ¯ n (1 + 2¯ n ) + e gτ (1 + 2¯ n + 2¯ n )] + (e gτ − n (1 + ¯ n ) ln (1 + 1 / ¯ n )e gτ (1 + ¯ n ) − ¯ n . (C13)Numerical maximization of I ( ρ ; ω ) with respect to the three parameters τ, g, ¯ n returns the value 2 . /ω . [1] D. Braun, Ultimate quantum bounds on mass measure-ments with a nano-mechanical resonator, EPL (Euro-physics Letters) , 68007 (2011).[2] Q. Zheng, Y. Yao, and Y. Li, Optimal quantum parame-ter estimation in a pulsed quantum optomechanical sys-tem, Phys. Rev. A , 013848 (2016).[3] M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D.Giuseppe, and D. Vitali, Robust stationary mechanicalsqueezing in a kicked quadratic optomechanical system,Phys. Rev. A , 023849 (2014).[4] O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun,Quantum parameter estimation using general single-mode gaussian states, Phys. Rev. A , 040102 (2013).[5] G. S. Agarwal, Brownian motion of a quantum oscillator,Phys. Rev. A , 739 (1971).[6] S. Dattagupta, Brownian motion of a quantum system,Phys. Rev. A , 1525 (1984).[7] A. Isar and A. Sndulescu, Damped quantum harmonicoscillator, Romanian Journal of Physics (1992).[8] E. Wigner, On the quantum correction for thermody-namic equilibrium, Phys. Rev. , 749 (1932).[9] C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J.Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussianquantum information, Rev. Mod. Phys. , 621 (2012).[10] M. G. A. Paris, F. Illuminati, A. Serafini, andS. De Siena, Purity of gaussian states: Measurementschemes and time evolution in noisy channels, Phys. Rev.A , 012314 (2003).[11] G. Adam, Density matrix elements and moments for gen-eralized gaussian state fields, Journal of Modern Optics , 1311 (1995).[12] S. L. Braunstein and C. M. Caves, Statistical distanceand the geometry of quantum states, Phys. Rev. Lett. , 3439 (1994).[13] A. S. Holevo, Statistical structure of quantum theory ,Vol. 67 (Springer Science & Business Media, 2003).[14] A. S. Holevo,
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