Ultimate limits for quantum magnetometry via time-continuous measurements
Francesco Albarelli, Matteo A. C. Rossi, Matteo G. A. Paris, Marco G. Genoni
UUltimate limits for quantum magnetometry via time-continuous measurements
Francesco Albarelli, ∗ Matteo A. C. Rossi, † Matteo G. A. Paris,
1, 2, ‡ and Marco G. Genoni § Quantum Technology Lab, Dipartimento di Fisica, Universit`a degli Studi di Milano, 20133 Milano, Italy INFN, Sezione di Milano, I-20133 Milano, Italy
We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjectedto collective transverse noise. By preparing an initial spin coherent state, for any measurement performed afterthe evolution, the mean-square error of the estimate is known to scale as /J , i.e. no quantum enhancementis obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclu-sively show that strategies based on time-continuous non-demolition measurements followed by a final strongmeasurement may achieve Heisenberg-limited scaling /J and also a monitoring-enhanced scaling in termsof the interrogation time. We also find that time-continuous schemes are robust against detection losses, aswe prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, weanalytically prove the optimality of our strategy. I. INTRODUCTION
Recent developments in the field of quantum metrologyhave shown how quantum probes and quantum measurementsallow one to achieve parameter estimation with precision be-yond that obtainable by any classical scheme [1, 2]. The es-timation of the strength of a magnetic field is a paradigmaticexample in this respect, as it can be mapped to the problemof estimating the Larmor frequency for an atomic spin ensem-ble [3–9].As a matter of fact, if the system is initially prepared ina spin coherent state, the mean-squared error of the field esti-mate scales, in terms of the total spin number J , as /J , whichis usually referred to as the standard quantum limit (SQL) toprecision. If quantum resources, such as spin squeezing orentanglement between the atoms of the spin ensemble, are ex-ploited, one observes a quadratic enhancement and achievesthe so-called Heisenberg scaling , i.e. /J [10, 11]. Onthe other hand, it has been proved that such ultimate quan-tum limit may be easily lost in the presence of noise [12]and that typically a SQL-like scaling is observed, with thequantum enhancement reduced to a constant factor. Theseobservations have been rigorously translated into a set of no-go theorems [13, 14], which fostered several attempts to cir-cumvent them. In particular, it has been shown how one canrestore a super-classical scaling in the context of frequencyestimation, for specific noisy evolution and/or by optimizingthe strategy over the interrogation time [15–19], or by ex-ploiting techniques borrowed from the field of quantum error-correction [20–22].In this manuscript, we put forward an alternative approach:we assume to start the dynamics with a classical state that ismonitored continuously in time via the interacting environ-ment [23, 24]. The goal is to recover the information on theparameter leaking into the environment and simultaneously toexploit the back action of the measurement to drive the system ∗ [email protected] † [email protected] ‡ matteo.paris@fisica.unimi.it § marco.genoni@fisica.unimi.it into more sensitive conditional states [25–34]. This approachhas received much attention recently [35–42] also in the con-text of quantum magnetometry [43–48].Here we rigorously address the performance of these pro-tocols, depicted in Fig. 1, taking into account the informa-tion obtained via the time-continuous non-demolition mea-surements on the environment, as well as the information ob-tainable via a strong (destructive) measurement on the con-ditional state of the system. In particular, in the limit oflarge spin, we derive an analytical formula for the ultimatebound on the mean-squared error of any unbiased estimator,and conclusively show that, for experimentally relevant valuesof the dynamical parameters, one can observe a Heisenberg-like scaling. J x J y J z probe detector unobserved environment B FIG. 1.
Atomic magnetometry via time-continuous measure-ments - an atomic ensemble, prepared in a spin-coherent statealigned to the x -direction and placed in a constant magnetic field B pointing in the y -direction, is coupled to train of probing fields thatare continuously monitored after the interaction with the sample. Remarkably, at variance with most of the protocols pro-posed for quantum magnetometry, and in general for fre-quency estimation, one does not need to prepare an initialspin-squeezed state. The Heisenberg scaling is in fact ob-tained also for an initial classical spin coherent state, thanksto the spin squeezing generated by time-continuous measure-ments’ back-action. Finally, we analytically prove that the ul-timate quantum limit for noisy magnetometry in the presenceof collective transverse noise [36] is in fact saturated by ourstrategy, i.e. one does not need to implement more involvedstrategies, e.g. jointly measuring the conditional state of the a r X i v : . [ qu a n t - ph ] D ec system and the output modes of the environment at differenttimes.The paper is organized as follows. In Sec. II, wepresent the quantum Cram´er-Rao bounds that hold for noisymetrology, with emphasis on estimation strategies based ontime-continuous, non-demolition measurements and a final strong measurement on the corresponding conditional quan-tum states. In Sec. III, we introduce the physical setting forthe estimation of a magnetic field via a continuously moni-tored atomic ensemble. In particular, we focus on the case oflarge total spin, where a Gaussian picture is able to describethe whole dynamics. In Sec. IV, we present the main results:we first calculate the classical Fisher information correspond-ing to the photoccurent obtained via the time-continuous mon-itoring of the environment, and we discuss how to attain thecorresponding bound via Bayesian estimation. We then ad-dress the possibility of performing also a strong measurementon the conditional state of the atomic ensemble, and derivethe ultimate limit on this kind of estimation strategy, quanti-fied by an effective quantum Fisher information. Upon study-ing this quantity, we observe how, in the relevant parameters’regime, the Heisenberg limit can be effectively restored, alsodiscussing the effects of non-unit monitoring efficiency, corre-sponding to the loss of photons before the detection. Finally,we also prove the optimality of our measurement strategy inthe case of ideal detectors. Section V closes the paper withsome concluding remarks. II. QUANTUM CRAM ´ER-RAO BOUNDS FORTIME-CONTINUOUS HOMODYNE MONITORING
A classical estimation problem consists in inferring thevalue of a parameter θ from a number M of measurementoutcomes χ = { x , x , . . . , x M } and their conditional distri-bution p ( x | θ ) . We define an estimator ˆ θ ( χ ) a function fromthe measurement outcomes to the possible values of θ andwe dub it asymptotically unbiased when, in the limit of largenumber of repetitions of the experiment M , its average isequal to the true value, i.e. (cid:82) dχ p ( χ | λ )ˆ θ ( χ ) = θ , where p ( χ | λ ) = Π Mj =1 p ( x j | θ ) . The Cram´er-Rao theorem states thatthe variance of any unbiased estimator is lower bounded as Var ˆ θ ( θ ) = ( M F [ p ( x | θ )]) − , where F [ p ( x | θ )] = (cid:90) dx p ( x | λ )( ∂ θ log p ( x | λ )) (1)denotes the classical Fisher information (FI).In the quantum realm, the conditional probability distri-bution reads p ( x | θ ) = Tr [ (cid:37) θ Π x ] , where (cid:37) θ is the quantumstate of the system labeled by the parameter θ , and Π x is aPOVM operator describing the quantum measurement. Onecan prove that the FI corresponding to any POVM is upperbounded F [ p ( x | θ )] ≤ Q [ (cid:37) θ ] , where Q [ (cid:37) θ ] = Tr [ (cid:37) θ L θ ] is thequantum Fisher information (QFI), and L θ is the so-calledsymmetric logarithmic derivative, which can be obtained bysolving the equation ∂ θ (cid:37) θ = L θ (cid:37) θ + (cid:37) θ L θ [49–51]. TheQFI depends on the quantum state (cid:37) θ only, and thus poses the ultimate bound on the precision of the estimation of θ .Moreover, in the single parameter case the bound is alwaysachievable, that is, there exists a (projective) POVM such thatthe corresponding classical FI equals the QFI.In this manuscript we consider a quantum system evolv-ing according to a given Hamiltonian ˆ H θ characterised bythe parameter we want to estimate, and coupled to a bosonicenvironment at zero temperature described by a train ofinput operators ˆ a in ( t ) , satisfying the commutation relation [ˆ a in ( t ) , ˆ a † in ( t (cid:48) )] = δ ( t − t (cid:48) ) , via an interaction Hamiltonian ˆ H int ( t ) = ˆ c ˆ a † in ( t ) + ˆ c † ˆ a in ( t ) ( ˆ c being a generic operator inthe system Hilbert space) [23]. By tracing out the environ-ment, the unconditional dynamics of the system is describedby the Lindblad master equation d(cid:37)dt = L (cid:37) = − i [ ˆ H θ , (cid:37) ] + D [ˆ c ] (cid:37) , (2)where D [ c ] (cid:37) = ˆ c(cid:37) ˆ c † − (ˆ c † ˆ c(cid:37) + (cid:37) ˆ c † ˆ c ) / .If one performs a homodyne detection of a quadrature ˆ x out ( t ) = ˆ a out ( t ) + ˆ a † out ( t ) on the output operators, i.e. onthe environment just after the interaction with the system, oneobtains that the dynamics of the system quantum state (cid:37) ( c ) conditioned on the measurement results (we will omit the de-pendence of the measured photocurrent y t ), is described bythe stochastic master equation [23] d(cid:37) ( c ) = − i [ ˆ H θ , (cid:37) ( c ) ] dt + D [ˆ c ] (cid:37) ( c ) dt + √ η H [ˆ c ] (cid:37) ( c ) dw t . (3)Here η denotes the efficiency of the detection, dw t is astochastic Wiener increment (s.t. dw t = dt ), and H [ c ] (cid:37) ( c ) =ˆ c(cid:37) ( c ) + (cid:37) ( c ) ˆ c † − Tr [ (cid:37) ( c ) (ˆ c + ˆ c † )] (cid:37) ( c ) (notice that in princi-ple one could consider other measurement strategies differentfrom homodyne, yielding a different superoperator). The cor-responding measurement record during a time step t → t + dt is given by the infinitesimal current dy t = √ η Tr [ (cid:37) ( c ) (ˆ c + ˆ c † )] dt + dw t . (4)With the help of such measurement strategies, one can esti-mate the value of the parameter θ both from the measuredphotocurrent y T = (cid:82) T dy t , and from a final strong (destruc-tive) measurement on the conditional state (cid:37) ( c ) . In this case,as we explicitly show in A (in general both for the classicaland quantum case), the proper quantum Cram´er-Rao boundreads Var ˆ θ ( θ ) ≥ M (cid:0) F [ p ( y T )] + E p ( y T ) (cid:2) Q [ (cid:37) ( c ) ] (cid:3)(cid:1) , (5)where the first term at the denominator F [ p ( y T )] is the FI cor-responding to the classical photocurrent y T , while the secondterm is the average of the QFI for the conditional state Q [ (cid:37) ( c ) ] over all the possible trajectories, i.e. on all the possible mea-surement outcomes for the photocurrent.The classical FI F [ p ( y T )] can be calculated as described in[35] by evaluating F [ p ( y T )] = E p ( y T ) [ Tr [ τ ] ] , (6)where the operator τ evolves according to the stochastic mas-ter equation dτ = − i [ ˆ H θ , τ ] dt − i [( ∂ θ ˆ H θ ) , (cid:37) ] dt + (7) + D [ˆ c ] τ dt + (ˆ cτ + τ ˆ c † ) dw t . (8)The conditional states (cid:37) ( c ) at time T can be obtained by in-tegrating (3), for a certain stream of outcomes y T ; then onecan first calculate the corresponding quantum Fisher informa-tion Q [ (cid:37) ( c ) ] , and, numerically or when possible analytically,its average over all the possible trajectories explored by thequantum system due to the homodyne monitoring.A more fundamental quantum Cram´er-Rao bound that ap-plies in this physical setting has been derived in [36], by con-sidering the QFI obtained from the unitary dynamics of theglobal pure state of system and environment. This QFI is ob-tained by optimizing over all possible POVMs, i.e. one alsoconsiders the possibility of performing non-separable (entan-gled) measurements over the system and all the output modes ˆ a out ( t ) at different times. On the other hand, in the previoussetting the estimation strategies were restricted to the moreexperimentally friendly case of sequential/separable measure-ments on the output modes and on the final conditional stateof the system.The QFI expressing this ultimate QCRB is by definition Q L ( θ ) = 4 ∂ θ ∂ θ log ( |(cid:104) ψ ( θ ) | ψ ( θ ) (cid:105)| ) (cid:12)(cid:12) θ = θ = θ , (9)where (cid:104) ψ ( θ ) | ψ ( θ ) (cid:105) is the fidelity between the global stateof system and environment for two different values of theparameter, and where we have highlighted its dependenceon the superoperator L that defines the unconditional mas-ter equation (2). The key insight is that this fidelity canbe determined by using operators acting on the system only[36, 52] and it can be expressed as the trace of an operatorTr [¯ ρ ] = (cid:104) ψ ( θ ) | ψ ( θ ) (cid:105) , which obeys the following general-ized master equation d ¯ ρdt = − i (cid:16) ˆ H θ ¯ ρ − ¯ ρ ˆ H θ (cid:17) + D [ˆ c ] ¯ ρ . (10)As before, we already assumed that the dependence on the pa-rameter lies only in the system Hamiltonian ˆ H θ and that wehave a single jump operator ˆ c . We remark that the operator ¯ ρ is not a proper density operator representing a quantum state,except in the limit case θ → θ , where we recover the stan-dard master equation (2). III. QUANTUM MAGNETOMETRY: THE PHYSICALSETTING
We address the estimation of the intensity of a static andconstant magnetic field B acting on a ensemble of N two-level atoms that are continuously monitored [43–46], as de-picted in Fig. 1. The atomic ensemble can be described as asystem with total spin J = N/ with collective spin opera-tors defined as ˆ J α = (cid:80) Ni =0 σ iα , where α = x, y, z and σ iα denotes the Pauli matrices acting on the i -th spin. The col-lective operators obey the same angular momentum commu-tation rules [ ˆ J i , ˆ J j ] = iε ijk ˆ J k , where ε ijk is the Levi-Civitasymbol. We remark that in the present manuscript we chooseunits such that (cid:126) = 1 .We assume that the atomic sample is coupled to a electro-magnetic mode a in ( t ) corresponding either to a cavity modein a strongly driven and heavily damped cavity [26], or analo-gously to a far-detuned traveling mode passing through theensemble [46]. By considering an interaction Hamiltonian ˆ H int = √ κ ˆ J z (ˆ a in ( t ) + ˆ a † in ( t )) and if these environmental light modes are left unmeasured, the evolution of the systemis expressed by (2), which in this case corresponds to a collec-tive transverse noise on the atomic sample, d(cid:37)dt = L tn (cid:37) = − iγB [ ˆ J y , (cid:37) ] + κ D [ ˆ J z ] (cid:37), (11)where the constants κ and γ represent respectively the strengthof the coupling with the noise and with the magnetic field, thatis directed on the y -axis and thus perpendicular to the noisegenerator. At t = 0 we consider the system prepared in aspin coherent state, i.e. a tensor product of single spin states(qubits) directed in the positive x direction, | ψ (0) (cid:105) = N (cid:79) k =0 | + (cid:105) k = | J, J (cid:105) x , (12)where | + (cid:105) is the eigenstate of σ x with eigenvalue +1 . We thushave that the spin component on the x direction attains themacroscopic value (cid:104) ˆ J x (0) (cid:105) = J . The unconditional dynamicsof (cid:104) ˆ J x (cid:105) is obtained by applying the operator ˆ J x to both sides ofEq. (11) and then taking the trace. The result is the followingequation describing damped oscillations d (cid:104) ˆ J x ( t ) (cid:105) dt = γB (cid:104) ˆ J z ( t ) (cid:105) − κ (cid:104) ˆ J x ( t ) (cid:105) , (13)where we observe how the the dissipative and unitary partsof the dynamics are respectively shrinking the spin vector (cid:104) (cid:126) ˆ J (cid:105) and causing its Larmor precession around the y -axis. In thefollowing we will assume to measure small magnetic fields,such that γBt (cid:28) and we can approximate the solution ofthe previous equation as (cid:104) ˆ J x ( t ) (cid:105) ≈ (cid:104) ˆ J x (0) (cid:105) e − κt/ = Je − κt/ . (14)If the light modes are continuously monitored via homodynemeasurements at the appropriate phase, one allows a continu-ous “weak” measurement of ˆ J z ; the corresponding stochasticmaster equation (3) for finite monitoring efficiency η reads d(cid:37) ( c ) = − iγB [ ˆ J y , (cid:37) ( c ) ]d t + κ D [ ˆ J z ] (cid:37) ( c ) dt + √ ηκ H [ ˆ J z ] (cid:37) ( c ) dw t , (15)while the measurement result at time t corresponds to an in-finitesimal photocurrent dy t = 2 √ ηκ Tr [ (cid:37) ( c ) ˆ J z ] dt + dw t . Itis important to remark how the collective noise characteriz-ing the master equation (11) describes the dynamics also inexperimental situations where no additional coupling to theatomic ensemble, with the purpose of performing continuousmonitoring, is engineered [53–55]. In this respect, assuminga non-unit efficiency η corresponds to considering both ho-modyne detectors that are not able to capture all the photonsthat have interacted with the spin, and environmental degreesof freedom, causing the same kind of noisy dynamics, thatcannot be measured during the experiment.Let us now consider the limit of large spin J (cid:29) . Inthis case, the dynamics may be effectively described with theGaussian formalism as long as (cid:104) ˆ J x ( t ) (cid:105) ≈ J , i.e. for times t small enough to guarantee that κt (cid:46) . We define the effec-tive quadrature operators of the atomic sample, satisfying thecanonical commutation relation [ ˆ X, ˆ P ] = i , as [46, 47] ˆ X = ˆ J y / (cid:112) ¯ J t ˆ P = ˆ J z / (cid:112) ¯ J t , (16)where ¯ J t ≡ |(cid:104) ˆ J x ( t ) (cid:105)| (notice that in the limit of large spin J we can safely consider the unconditional average value (cid:104) ˆ J x ( t ) (cid:105) , as the stochastic correction obtained via (15) wouldbe negligible). In the Gaussian description the initial state | ψ (0) (cid:105) corresponds to the vacuum state ( ˆ X + i ˆ P ) | (cid:105) = | (cid:105) ,which is Gaussian. As the stochastic master equation (15) be-comes quadratic in the canonical operators (and thus preservesthe Gaussian character of states) d(cid:37) ( c ) = − iγB (cid:112) ¯ J t (cid:104) ˆ X, (cid:37) ( c ) (cid:105) dt ++ κ ¯ J t D [ ˆ P ] (cid:37) ( c ) dt + (cid:113) ¯ J t ηκ H [ ˆ P ] (cid:37) ( c ) dw t , (17)the whole dynamics can be equivalently rewritten in terms offirst and second moments only [56, 57] (see B for the equa-tions describing the whole dynamics in the Gaussian picture).As it will be clear in the following, due to the nature of thecoupling, in order to address the estimation of B , we onlyneed the behaviour of the mean and the variance of the atomicmomentum quadrature ˆ P calculated on the conditional state (cid:37) ( c ) , which follows the equations d (cid:104) ˆ P ( t ) (cid:105) c = − Bγ (cid:112) Je − κt dt + 2Var c [ ˆ P ( t )] (cid:113) ηκJe − κt dw t , (18) d Var c [ ˆ P ( t )] dt = − ηκJe − κt (cid:16) Var c [ ˆ P ( t )] (cid:17) . (19)The differential equation for the conditional second momentis deterministic and can be solved analytically. For an initialvacuum state, i.e. with Var[ ˆ P (0)] = , we obtain the follow-ing solution Var c [ ˆ P ( t )] = 18 ηJ (cid:16) − e − κt (cid:17) + 2 , (20)that shows how the conditional state of the atomic sample isdeterministically driven by the dynamics into a spin-squeezedstate. IV. RESULTS
Here we will present our main results, that is the derivationof ultimate quantum limits on noisy magnetometry via time-continuous measurements of the atomic sample. We will firstevaluate the classical Fisher information F [ y t ] correspondingto the information obtainable from the photocurrent, andwe will also show how the corresponding bound can beachieved via Bayesian estimation. We will then evaluatethe second term appearing in the bound, corresponding tothe information obtainable via a strong measurement on theconditional state of the atomic sample. This will allow usto discuss the ultimate limit on the estimation strategy viathe effective quantum Fisher information: we will focus onthe scaling with the relevant parameters of the experiment, i.e. with the total spin number J and the monitoring timecharacterizing each experimental run t , and we will addressthe role of the detector efficiency η . A. Analytical FI corresponding to the time-continuousphotocurrent
As discussed before, the measured photocurrent y t ob-tained via continuous homodyne detection can be used to ex-tract information about the system and to estimate parameterswhich appear in the dynamics. The ultimate limit on the pre-cision of this estimate is quantified by the FI F [ p ( y t )] . Giventhe Gaussian nature and the simple dynamics of the problemwe can compute it analytically in closed form, by applying theresults of [58]. As we describe in more detail in B, one obtainsthe formula F [ p ( y t )] = 2 ηκJe − κt/ E p ( y t ) (cid:20)(cid:16) ∂ B (cid:104) ˆ P ( t ) (cid:105) c (cid:17) (cid:21) . (21)By considering (18) and remembering that d w t = d y t − (cid:113) ηκJe − κt (cid:104) ˆ P ( t ) (cid:105) c d t , one obtains that the time evolution ofthe derivative of the conditional first moment (cid:104) ˆ P ( t ) (cid:105) c w.r.t. tothe parameter B , can be written as d (cid:16) ∂ B (cid:104) ˆ P ( t ) (cid:105) c (cid:17) dt == − γ (cid:112) Je − κt/ − c [ ˆ P ( t )] ηκJe − κt/ (cid:16) ∂ B (cid:104) ˆ P ( t ) (cid:105) c (cid:17) . (22)where Var c [ ˆ P ( t )] is obtained from Eq. (20). We thus observethat the evolution is deterministic and one can easily derive itsanalytical solution. By applying Eq. (21), as the average overthe trajectories is not needed, we readily obtain the followinganalytical formula for the FI F [ p ( y t )] = 64 γ ηJ e − κt (cid:16) e κt − (cid:17) κ (cid:104) (4 ηJ + 1) e κt − ηJ (cid:105) ·· [ − ηJ − ηJe κt + 3(4 ηJ + 3) e κt + (4 ηJ + 3) e κt ] . (23)As intuitively expected, this is a monotonically increasingfunction of t , since the partial derivative is always positive.To get some insight into this expression we first report theleading term for t → F [ p ( y t )] ≈ J γ κt , (24)where we explicitly see both Heisenberg scaling J and amonitoring-enhanced time scaling t . We can get further in-tuition about this expression by expanding it around J = ∞ ,the limit in which the Gaussian approximation becomes ex-act. The leading order in this other expansion is quadratic in J , thus showing again Heisenberg scaling, irregardless of t : F [ p ( y t )] ≈ γ ηJ e − κt (cid:16) e κt − (cid:17) (cid:16) e κt + e κt + 1 (cid:17) κ (cid:16) e κt + 1 (cid:17) ; (25)this last approximations actually reproduces the behavior ofthe function quite well in the range of parameters we will con-sider in the following.We now want to show that one can achieve this classicalCram´er-Rao bound from the time-continuous measurementoutcomes obtained via an appropriate estimator. In Figure 2we indeed show the posterior distribution as a function of timefor a single experimental run, obtained after a Bayesian anal-ysis (see C for details). We observe how the distribution getsnarrower in time around the true value and we also explicitlyshow that its standard deviation σ est converges to the one pre-dicted by the Cram´er-Rao bound σ CR ( t ) = F [ p ( y t )] − / . Inthe initial part of the dynamics the values of σ est are smallerthan the corresponding σ CR : this is due to the choice of theprior distribution, being narrower than the likelihood and thusimplying some initial knowledge on the parameter which islarger than the one obtainable for small monitoring time. B. Quantum Cram´er-Rao bound for noisy magnetometry viatime-continuous measurements
In order to evaluate the quantum Cram´er-Rao boundin Eq. (5) we now need to consider the second term E p ( y T ) (cid:2) Q [ (cid:37) ( c ) ] (cid:3) , corresponding to the information obtain-able via strong quantum measurement on the conditional stateof the system. The conditional state (cid:37) ( c ) is Gaussian andhas a dependence on the parameter B only in the first mo-ments. Therefore the corresponding QFI can be evaluated as σ e s t / σ C R κt B ( G ) FIG. 2.
Bayesian estimation of B from a single simulated exper-iment - the data shown in the plots are obtained as a function of κt ,for γ/κ = 1 G − , J = 10 and η = 1 ; the prior distribution of theparameter B is uniform in the interval [ − . , .
01] G , and the truevalue is B true = 0 G . In the top panel we show the ratio betweenthe standard deviation of the posterior distribution and the standarddeviation predicted by the Cram´er-Rao bound. In the bottom panelwe show the posterior distribution as a function of time, the constantwhite dashed line marks the value B true . prescribed in [59] (see B for more details) obtaining, Q [ (cid:37) ( c ) ] = (cid:16) ∂ B (cid:104) ˆ P ( t ) (cid:105) c (cid:17) Var c [ ˆ P ( t )] . (26)Since, as we proved before, the evolution of both ∂ B (cid:104) ˆ P ( t ) (cid:105) c and Var c [ ˆ P ( t )] is deterministic, the average over all pos-sible trajectories is also in this case trivial and we have E p ( y T ) (cid:2) Q [ (cid:37) ( c ) ] (cid:3) = Q [ (cid:37) ( c ) ] . By exploiting the analytical so-lution for both quantities, the QFI reads Q [ (cid:37) ( c ) ] = 32 γ J (cid:16) ηJ − ηJe − κt − (8 ηJ + 3) e κt + 3 (cid:17) κ (cid:104) (4 ηJ + 1) e κt − ηJ (cid:105) . (27)As expected, for no monitoring of the environment ( η = 0 ),one obtains that Q [ (cid:37) ( c ) ] ∼ J , i.e. corresponding to the SQLscaling. This function is also monotonically increasing with t and we can expand it around J = ∞ to study the leadingterm, which shows again a quadratic scaling in J Q [ (cid:37) ( c ) ] ≈ γ ηJ e − κt (cid:16) − e κt + 2 e κt + 1 (cid:17) κ (cid:16) e κt − (cid:17) . (28)We also remark that the QFI is equal to the classical FI for ameasurement of the quadrature ˆ P , thus showing that a strongmeasurement of the operator ˆ J z on the conditional state ofthe atomic sample is the optimal measurement saturating thecorresponding quantum Cram´er-Rao bound.By combining Eqs. (23) and (27), we can now define the effective quantum Fisher information (cid:101) Q = F [ p ( y t )] + E p ( y T ) (cid:104) Q [ (cid:37) ( c ) ] (cid:105) = F [ p ( y t )] + Q [ (cid:37) ( c ) ] , (29)which represent the inverse of the best achievable variance ac-cording to the quantum Cram´er-Rao bound (5). The resultingexpression can be simplified to get the following simple ana-lytical formula ˜ Q = K J + ηK J (30)where K = 32 γ κ (cid:16) − e − κt/ (cid:17) , (31) K = 64 γ κ (cid:18) − e − κt/ + 2 e − κt/ − e − κt (cid:19) . (32)We start by studying how this quantity scales with the totalspin: in Fig. 3 we plot (cid:101) Q as a function of J in the appropri-ate regions of parameters. We remark that the plots will bepresented by using /κ as a time unit so that the strength ofthe interaction becomes γ/κ and is always fixed to − inthe following. We observe that, within the validity of our ap-proximation ( κt (cid:46) ), it is possible to obtain the Heisenberg-like scaling J for the effective QFI. There is a transition be-tween SQL-like scaling and Heisenberg scaling depending onthe relationship between J and κt showing how the quantumenhancement is observed for J (cid:29) /κt .The same conclusions are drawn if we look at the behaviourof (cid:101) Q as a function of the interrogation time t , plotted in Fig. 4:a transition from a t -scaling to a monitoring-enhanced t -scaling is observed for J (cid:29) /κt . We remark here that thetypical scaling obtained in quantum metrology for unitaryparameters is of order t . The observed t -scaling is due tothe continuos monitoring of the system. A similar scalingof the Fisher information would be in fact obtained for anequivalent classical estimation problem, where a continuouslymonitored classical system is estimated via a the Kalmanfilter [58]. Notice that there are also few recent examplesin the literature where a t -scaling can be observed. Thisis obtained in noiseless quantum metrology problems withtime-dependent Hamiltonian and by exploiting open-loopcontrol [60–63]. In particular in [63], it was also shown thata t -scaling can be achieved without additional control, butby performing repeated (stroboscopic) measurement on thesystem, analogously to our strategy.The previous results were both shown by considering per-fect monitoring of the environment, i.e. for detectors withunit efficiency η . In Fig. 5 we plot the behaviours of (cid:101) Q as afunction of J and t , varying the detector efficiency η ; we ob-serve how the quantum enhancements can be obtained for allnon-zero values of η . The effect of having a non-unit monitor-ing efficiency is simply to imply larger values of J to witnessthe transition between SQL to Heisenberg-scaling, as one can - - J ˜ κ t = - κ t = - κ t = - κ t = - FIG. 3.
J scaling - effective QFI (cid:101) Q as a function of J for differ-ent vales of κt , for unit efficiency η and effective coupling strength γ/κ = 1 G − ; axes are in logarithmic scale. The solid curves arefor increasing values of κt (shown in the legend) from top to bottom.The two regimes appearing in the plots are ∼ J (steeper slope) forhigher values of κt and higher values of J and ∼ J (gentler slope)for the opposite parameters’ regions. For visual comparison we showa dashed line at the top ∝ J and a dotted line at the bottom ∝ J . - - - - - - κ t ˜ J = J = J = J = FIG. 4.
Time scaling - effective QFI (cid:101) Q as a function of κt for differ-ent values of J , for unit efficiency η and effective coupling strength γ/κ = 1 G − ; axes are in logarithmic scale. The solid curves arefor increasing values of J (shown in the legend) from top to bottom.The two regimes appearing in the plots are ∼ ( κt ) (steeper slope)for higher values of κt and higher values of J and ∼ ( κt ) (gentlerslope) for the opposite parameters’ regions. For visual comparisonwe show a dashed line at the top ∝ ( κt ) and a dotted line at thebottom ∝ ( κt ) . also understand by looking at the role played by η and J inEq. (30).We remind that if we consider only the classical FI F [ p ( y t )] ,the Heisenberg scaling in terms of J and t -scaling are al-ways obtained for κt (cid:46) and for every η , as shown by theexpansion (24). However, if the contribution of this term, aswell as the contribution of conditioning to the QFI, are toosmall then the QFI of the unconditional state, i.e. (27) with η = 0 , dominates (the term ηK J in (30) is negligible) andwe observe SQL scaling for ˜ Q . We finally mention that the inthe regimes where we observe Heisenberg scaling of ˜ Q , theclassical FI F [ p ( y t )] amounts to a relevant part of the total,namely around 25% . - J ˜ κ t = - η = η = η = - η = - η = - - - - - - κ t ˜ J = η = η = η = - η = - η = FIG. 5.
Effect of non unit efficiency - effective QFI (cid:101) Q as a functionof J (top panel) and κt (bottom panel) for different values of η andeffective coupling strength γ/κ = 1 G − . The two regimes appear-ing in the plots are ∼ J (top panel) and ∼ ( κt ) (bottom panel) forhigher values of κt and higher values of J while ∼ J (top panel)and ∼ ( κt ) (bottom panel) for the opposite parameters’ regions. Forvisual comparison we show a dashed line at the top ∝ J (top panel)and ∝ ( κt ) (bottom panel) and also a dotted line at the bottom ∝ J (top panel) and ∝ ( κt ) (bottom panel). C. Optimality of time-continuous measurement strategy fornoisy quantum magnetometry
As explained before, the ultimate limit for quantum mag-netometry, in the presence of Markovian transversal noise asthe one described by the master equation (11), is given by theQFI Q L in Eq. (9). The generalized master equation (10) inthis case (considering the large-spin approximation) reads d ¯ (cid:37)dt = − iγ (cid:112) ¯ J t (cid:16) B ˆ X ¯ (cid:37) − B ¯ (cid:37) ˆ X (cid:17) + κ ¯ J t D (cid:104) ˆ P (cid:105) ¯ (cid:37) . (33)In D we show how this equation can be solved in a phasespace picture, since the equation contains at most quadraticterms in ˆ X and ˆ P and thus preserves the Gaussian characterof the operator ¯ (cid:37) .The final result is Q L tn = ˜ Q ( η = 1) = K J + K J , (34) i.e. , we exactly obtain the effective QFI ˜ Q defined in Eq. (29)in the limit of unit efficiency η = 1 . This result remarkablyproves that our strategy, not only allows to obtain the Heisen-berg limit, but also corresponds to the optimal one, given acollective transversal noise master equation (11) and in thepresence of perfectly efficient detectors. Indeed, any other more experimentally complicated strategy, based on entangledand non-local in time measurements of the output modes andthe system, would not give better results in the estimation ofthe magnetic field B . V. CONCLUSION AND DISCUSSION
We have addressed in detail estimation strategies for a staticand constant magnetic field acting on an atomic ensemble oftwo-level atoms also subject to transverse noise. In particular,we have evaluated the ultimate quantum limits to precision forstrategies based on time-continuous monitoring of the lightcoupled the atomic ensemble.After deriving the appropriate quantum Cram´er-Rao bound,we have calculated the corresponding effective quantumFisher information in the limit of large spin, posing the ul-timate limit on the mean-square error of any unbiased esti-mator. Our results conclusively show that both Heisenberg J -scaling in terms of spin, and a monitoring-enhanced t -scaling in terms of the interrogation time, are obtained for J (cid:29) /κt , confirming what was discussed in [43, 46]. Wehave remarkably demonstrated that these quantum enhance-ments are also obtained for not unit monitoring efficiency, i.e.even if one cannot measure all the environmental modes or fornot perfectly efficient detectors. Finally we have analyticallyproven the optimality of our strategy, i.e. that given the masterequation describing the unconditional dynamics of the systemand ideal detectors, no other measurement strategy would givebetter results in estimating the magnetic field.We remark that Heisenberg scaling, or at least a super-classical scaling, can be obtained in the presence of collectiveor individual (independent) transversal noise, by preparing ahighly entangled or spin-squeezed state at the beginning of thedynamics and, for individual noise, by optimizing on the inter-rogation time t [17–19]. In this respect, the advantage of ourprotocol lies in the fact that it achieves the Heisenberg scal-ing even for an initial classical spin-coherent state, exploitingthe dynamical spin squeezing that is generated by the weakmeasurement.In conclusion, we have shown that time-continuous mea-surements represent a resource for noisy quantum magnetom-etry [43, 46, 47]. Indeed, the information leaking into theenvironment, here represented by light modes coupled to theatomic sample, obtained via homodyne detection, and the cor-responding measurement back-action on the atomic sample,may be efficiently (and optimally) exploited in order to obtainthe promised quantum enhanced estimation precision. ACKNOWLEDGMENTS
MGG would like to thank A. Doherty and A. Ser-afini for discussions and acknowledges support from MarieSkłodowska-Curie Action H2020-MSCA-IF-2015 (projectConAQuMe, grant nr. 701154). This work has been supportedby EU through the collaborative H2020 project QuProCS(Grant Agreement 641277) and by UniMI through the H2020Transition Grant.
Appendix A: Classical and quantum Cram´er-Rao bounds forsequential non-demolition measurements
Here we will show how to derive the quantum Cram´er-Raobound for time-continuous homodyne monitoring reported inEq. (5).We start by considering a (classical) estimation problem of aparameter θ described by a conditional probability p ( z, y T | θ ) ,where the vector y T = ( y , y , . . . , y T ) T contains the out-comes of sequential measurements performed up to time T ,while z corresponds to a final measurement performed on thestate of the system that has been conditioned on the previousmeasurement results y T . The corresponding classical Fisherinformation can be evaluated as F [ p ( z, y T | θ )] = (cid:90) d y dz p ( z, y T | θ ) ( ∂ θ log p ( z, y T | θ )) = (cid:90) d y dz p ( z | y T , θ ) p ( y T | θ ) (cid:104) ( ∂ θ log p ( z | y T , θ )) + 2( ∂ θ log p ( z | y T , θ )) ( ∂ θ log p ( y T | θ ))+ ( ∂ θ log p ( y T | θ )) (cid:105) (A1)where the second expression has been obtained by means ofthe Bayes rule p ( z, y T | θ ) = p ( z | y T , θ ) p ( y T | θ ) . In the following, we will omit the dependence on the pa-rameter θ and we will denote by E p ( x ) [ · ] the average over aprobability distribution p ( x ) . By considering each term insidethe integral separately one obtains E p ( z, y T ) (cid:104) ( ∂ θ log p ( z | y T )) (cid:105) = E p ( y T ) [ F [ p ( z | y T )]] (A2) E p ( z, y T ) [ ∂ θ log p ( z | y T ) ∂ θ log p ( y T )] = (A3) = 2 (cid:90) d y ( ∂ θ p ( y T )) (cid:90) dz ( ∂ θ p ( z | y T )) = 0 E p ( z, y T ) (cid:104) ( ∂ θ log p ( y T )) (cid:105) = F [ p ( y T )] (A4)where we have used the property (cid:82) dz ( ∂ θ p ( z | y T )) = ∂ θ (cid:82) dz p ( z | y T ) = ∂ θ (1) = 0 . As a consequence, any un-biased estimator ˆ θ based on M experiments, i.e. obtained col-lecting M series of measurement outcomes ( y T , z ) , satisfiesthe generalized Cram´er-Rao bound Var ˆ θ ( θ ) ≥ M (cid:0) F [ p ( y T )] + E p ( y T ) [ F [ p ( z | y T )]] (cid:1) (A5)where the first term F [ p ( y T )] is the Fisher informationcorresponding to the sequential measurements with outcomes y T , while the second term is the average of the Fisher infor-mation F [ p ( z | y T )] , corresponding to the final measurement over all the possible trajectories conditioned on the previousmeasurement results y T . The bound in Eq. (A5) bears someformal similarity to the Van Tree’s inequality [64], whichhowever applies in a quite different situation, i.e. the casewhere the parameter to be estimated θ is a random variabledistributed according to a given probability distribution p ( θ ) .The estimation strategy here described is of particular inter-est when we deal with quantum systems, given the back-actionof quantum measurement on the state of the system itself. Wecan in fact associate each measurement outcome y k to a Krausoperator M y k such that the conditional quantum state, for thesystem initially prepared in a state (cid:37) and after obtaining thestream of outcomes y T , reads (cid:37) ( c ) y T = ˜ M y T (cid:37) ˜ M † y T Tr [ ˜ M y T (cid:37) ˜ M † y T ] . (A6)where ˜ M y T = M y T . . . M y M y and the probability of ob-taining the outcomes y T reads p ( y T | θ ) = Tr [ ˜ M y T (cid:37) ˜ M † y T ] [65]. One can then also perform a strong (destructive) mea-surement described by POVM operators { Π z } on the condi-tional state, and the whole measurement strategy is describedby the conditional probabilities p ( z | y T , θ ) = Tr [ (cid:37) ( c ) y T Π z ] ,p ( z, y T | θ ) = p ( z | y T , θ ) p ( y T | θ )= Tr [ ˜ M y T (cid:37) ˜ M † y T Π z ] (A7)Typically the parameter to be estimated θ enters in the the dy-namics described by the Kraus operators M y k . For this reasonwe will start by considering these operators fixed, while wesuppose we can optimize over the final measurement { Π z } .We can then apply the quantum Cram´er-Rao bound for theconditional states (cid:37) ( c ) y T , stating that F [ p ( z | y T )] ≤ Q [ (cid:37) ( c ) y T ] .One then obtains a more fundamental quantum Cram´er-Raobound for our estimation strategy Var ˆ θ ( θ ) ≥ M (cid:16) F [ p ( y T )] + E p ( y T ) (cid:104) Q [ (cid:37) ( c ) y T ] (cid:105)(cid:17) . (A8)Clearly this bound can be readily applied to the time-continuous case discussed in the main text, where the vector ofoutcomes y T corresponds to a measured homodyne photocur-rent, and where the conditional state (cid:37) ( c ) y T can be obtained viaa stochastic master equation as the one in Eq. (3).We should also remark that a bound of this kind has alreadybeen considered in [39], in a similar physical situation where n probes, that may be prepared in a quantum correlated ini-tial state, are coupled to n independent environments and oneperforms sequentially n measurement on the respective envi-ronments and a final measurement on the conditional state ofthe probes. Appendix B: Gaussian dynamics and Gaussian Fisherinformation
Here we will provide the formulas for the dynamics ofthe atomic ensemble described by the stochastic master equa-tion (15). As we mentioned in the text, the whole dynam-ics preserves the Gaussian character of the quantum state andthus can be fully described in terms of the first moments vec-tor (cid:104) ˆr (cid:105) c and of the covariance matrix σ of the quantum state (cid:37) ( c ) . These are defined in components as (cid:104) ˆ r j (cid:105) c = Tr (cid:2) ˆ r j (cid:37) ( c ) (cid:3) and σ jk = Tr (cid:2) { ˆ r j − (cid:104) ˆ r j (cid:105) c , ˆ r k − (cid:104) ˆ r k (cid:105) c } (cid:37) ( c ) (cid:3) for the operatorvector ˆ r = ( ˆ X, ˆ P ) T . In formulae one obtains [56, 57]: d (cid:104) ˆr (cid:105) c = u dt + σ M d w √ , (B1) d σ dt = D − σ M M T σ , (B2)where D = (cid:18) κJe − κt/
00 0 (cid:19) , (B3) M = (cid:18) (cid:112) ηκJe − κt/ (cid:19) , (B4) u = (0 , − γB (cid:112) Je − κt/ ) T , (B5)and d w is a vector of Wiener increments such that dw j dw k = δ jk dt , related to the photocurrent via the equation d y t = − M T (cid:104) ˆr (cid:105) c dt + d w √ . (B6)The Eqs. (18), (19) and (22) can be obtained fromthe ones above, remembering that for our definitions σ = 2Var c [ ˆ P ( t )] .The method to calculate the Fisher information correspond-ing to the time-continuous measurement in the case of linearGaussian system has been described in [58]. One has to eval-uate the formula F [ p ( y t )] = E p ( y t ) (cid:2) ∂ B (cid:104) ˆr (cid:105) T c ) M M T ( ∂ B (cid:104) ˆr (cid:105) c ) (cid:3) , (B7)that, by plugging in the matrices describing our problem, iseasily simplified to Eq. (21).As the conditional state is Gaussian, also the calculation ofthe corresponding QFI can be easily obtained, in this case byapplying the results presented in [59]. Moreover, as only thefirst moments of the state depend on the parameter B , the cal-culation is further simplified and one has Q [ (cid:37) ( c ) ] = 2 ( ∂ B (cid:104) ˆr (cid:105) T c ) σ − ( ∂ B (cid:104) ˆr (cid:105) c ) . (B8)By noticing that the only non-zero entry of the vector ∂ B (cid:104) ˆr (cid:105) c is the one corresponding to (cid:104) ˆ P ( t ) (cid:105) c , one easily obtainEq. (26). Appendix C: Bayesian analysis for continuously monitoredquantum systems
Bayesian analysis has proven to be an efficient tool for es-timation in continuously monitored quantum systems [35, 38,42, 48]. The goal is to reconstruct the posterior distribution of B given the observed current y t , by Bayes rule: p ( B | y t ) = L ( B | y t ) p ( B ) p ( y t ) , (C1)where p ( B ) is the prior distribution, L ( B | y t ) ≡ p ( y t | B ) isthe likelihood and p ( y t ) serves as a normalization factor. TheBayesian estimator is the mean of the posterior distribution ˆ B ( y t ) = E p ( B | y t ) [ B ] and it is proven that the correspond-ing variance Var ˆ B ( B ) = E p ( B | y t ) [ B ] − ( E p ( B | y t ) [ B ]) isasymptotically optimal, i.e. tends to saturate the Cram´er-Raobound when the length of the vector y T is large.The simulated experimental run is obtained by numericallyintegrating the stochastic differential equation (18) with theEuler-Maruyama method for the “true” value of the parame-ter B true . Time is discretized with steps of length ∆ t , i.e. toget from time to time T we perform n T = T / ∆ t steps.Experimental data is represented by the observed measure-ment current y T = (∆ y t , . . . , ∆ y t nT ) T , which correspondsto an n T -dimensional vector. The outcome at every time step ∆ y t i is sampled from a Gaussian distribution with variance ∆ t and mean ∆ y t i ( B ) = (cid:113) ηκJe − κt (cid:104) ˆ P ( t i ) (cid:105) c ∆ t . Noticethat ∆ y t i ( B ) depends explicitly on the parameter B via thequantum expectation value (cid:104) ˆ P ( t i ) (cid:105) c on the conditional state.Since we are estimating only one parameter the posteriorcan be obtained on a grid on the parameter space, while formore complicated problems Markov chain Monte Carlo meth-ods might be needed to sample from the posterior [35]. Inpractical terms we need to solve Eqs. (18) and (19) for everyvalue of the parameter B on the grid, assuming to perfectlyknow all the other parameters; then we need to calculate thelikelihood for each value via L ( B | y T ) ∝ n T (cid:89) i =0 exp (cid:34) − (cid:0) ∆ y t i − ∆ y t i ( B ) (cid:1) t (cid:35) , (C2)by considering the outcomes as independent random vari-ables, i.e. multiplying the corresponding probabilities. Wethen apply Bayes rule, Eq. (C1), assuming a flat prior distri-bution p ( B ) on a finite interval. The same analysis is triviallyapplied to more than one experiment by simply multiplyingthe likelihood obtained for every different observed measure-ment current. Appendix D: Ultimate quantum Fisher information viageneralized master equation in phase space
Here we explicitly show how to solve Eq. (33). The char-acteristic function for a generic operator ˆ O is defined as χ [ ˆ O ]( s ) = Tr (cid:104) ˆ D − s ˆ O (cid:105) , (D1)0where the displacement operator is defined as ˆ D − s = exp (cid:0) i s (cid:62) Ωˆ r (cid:1) . (D2)In particular we will work in the phase space of a single modesystem, so that ˆ r (cid:62) = ( ˆ X, ˆ P ) is the vector of quadrature oper-ators and s (cid:62) = ( x, p ) is the vector of phase space coordinates.The action of operators in the Hilbert space corresponds to dif-ferential operators acting on the characteristic function via thefollowing mapping [56, 66] ˆ Xρ ↔ (cid:16) − i∂ p − x (cid:17) χ ( s ) (D3) ρ ˆ X ↔ (cid:16) − i∂ p + x (cid:17) χ ( s ) (D4) ˆ P ρ ↔ (cid:16) i∂ x − p (cid:17) χ ( s ) (D5) ρ ˆ P ↔ (cid:16) i∂ x + p (cid:17) χ ( s ) . (D6)If we now define the characteristic function associated tothe operator ¯ (cid:37) introduced in Eq. (10) ¯ χ ( s , t ) ≡ χ [¯ (cid:37) ] ( s ) , (D7)the quantity of interest in order to compute the QFI is thenTr ¯ (cid:37) = ¯ χ (0 , t ) , as evident from the definition (D1).By applying the phase space mapping, from the generalizedmaster equation (33) we get to the following partial differen-tial equation for the characteristic function d ¯ χ ( s , t ) dt == (cid:20) iγ (cid:112) ¯ J t B + B x − κ ¯ J t p − γ (cid:112) ¯ J t ( B − B ) ∂ p (cid:21) ¯ χ ( s , t ) . (D8) This equation can be solved by performing a Gaussian ansatz,similarly to [67], i.e. assuming that at every time the charac-teristic function can be written in the following form ¯ χ ( s , t ) = C ( t ) exp (cid:34) − s (cid:62) Ω (cid:62) σ ( t )Ω s ++ i s (cid:62) Ω (cid:62) s m ( t ) (cid:35) . (D9)The dependence on time and on the parameters B / is com-pletely contained in the covariance matrix σ ( t ) , in the firstmoment vector s m ( t ) (cid:62) = ( x m ( t ) , p m ( t )) and in the function C ( t ) = ¯ χ (0 , t ) , which is the final result we are seeking.By plugging (D9) into (D8) and equating the coefficientsfor different powers of x and p , one obtains a system of differ-ential equations; the relevant ones are the equations comingfrom the coefficients of order one, and multiplying p and p : ˙ σ , ( t ) = 2 κJe − κt (D10) ˙ x m ( t ) = − i γ (cid:112) Je − κt ( B − B ) σ , ( t ) (D11) ˙ C ( t ) = − iγ (cid:112) Je − κt ( B − B ) x m ( t ) C ( t ) . 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