Quantum Metrology via Repeated Quantum Nondemolition Measurements in "Photon Box"
aa r X i v : . [ qu a n t - ph ] A p r Quantum Metrology via Repeated Quantum Nondemolition Measurements in “Photon Box”
Yu-Ran Zhang, Jie-Dong Yue, and Heng Fan
1, 2, ∗ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100190, China (Dated: July 13, 2018)In quantum metrology schemes, one generally needs to prepare m copies of N entangled particles, such asentangled photon states, and then they are detected in a destructive process to estimate an unknown parameter.Here, we present a novel experimental scheme for estimating this parameter by using repeated indirect quantumnondemolition measurements in the setup called “photon box”. This interaction-based scheme is able to achievethe phase sensitivity scaling as /N with a Fock state of N photons. Moreover, we only need to prepare oneinitial N -photon state and it can be used repetitively for m trials of measurements. This new scheme is shown tosustain the quantum advantage for a much longer time than the damping time of Fock state and be more robustthan the common strategy with exotic entangled states. To overcome the π/N periodic error in the estimationof the true parameter, we can employ a cascaded strategy by adding a real-time feedback interferometric layout. PACS numbers: 06.20.-f, 03.65.Wj, 42.50.Dv, 42.50.Pq
I. INTRODUCTION
Quantum parameter estimation, the emerging field of quan-tum technology, aims to use entanglement and other quantumresources to yield higher statistical precision of a parameter θ than purely classical approaches [1, 2]. The precision of esti-mation of θ will depend on the available resources used in themeasurement. It has been shown that standard quantum limit(SQL) or called shot noise limit scaling as δθ ≃ / √ N tot with N tot the number of particles can be surpassed by using coher-ent light with squeezed vacuum [3]. It is also commonly con-sidered that using non-Gaussian states like NOON states [4]and quantum entanglement allows one to achieve a sub-shotnoise accuracy. Heisenberg limit scaling as δθ ≃ /N tot isthe ultimate limit set by quantum mechanics. Recently, someworks [5–7] have shown that, without prior information, sub-Heisenberg estimation strategies are ineffective. There arealso some papers showing that the Heisenberg limit can besaturated without the use of any exotic quantum entangledstates [8, 9]. Interferometric strategies with nonlinear phaseencoding are investigated in Refs. [10, 11]. Practical quantummetrology considering the impact of noise has been consid-ered and studied in Refs. [12–14]. The technique of quan-tum parameter estimation figures in several metrology plat-forms, including optical interferometry [15–18], atomic sys-tems [19, 20], and Bose-Einstein condensates [21–26]. Inaddition, it is at the heart of many modern technologies andresearches, such as quantum clock synchronization [27, 28],quantum imaging [29], and gravitational wave observation[30].General parameter estimation procedure can be divided intothree distinct sections: probe preparations, interaction be-tween the probe and the system, and the probe readouts [2].These three sections will be repeated many times before thefinal construction of the estimation of θ . Most of the quan-tum parameter estimation strategies require preparation of m ∗ [email protected] copies of entangled states ( m is large enough). However,these states are extremely difficult to generate and fragile tothe impact of decoherence. Therefore, the method of quan-tum nondemolition (QND) measurements [31] initially withentanglement-free states may be a suitable and practical wayto overcome these challenges. The QND measurements dat-ing back to as early as the 1920s realize ideal projective mea-surements that leave the system in an eigenstate of the mea-sured observable [32]. With these ideal projective measure-ments performed on an initial coherent state, Fock states and“Schr¨odinger cat” states can be prepared and reconstructed[33]. Moreover, with appropriate feedback loops, it is pos-sible to prepare on demand photon states and subsequentlyreverses the effects of decoherence [34]. With these merits,we can foresee the widespread applications of this techniquesin quantum information and quantum metrology.In this paper, we present a practical proposal for realiz-ing quantum parameter estimation in “photon box” [35–38]via QND measurements. We show that, with single-modeFock state of N photons in the “photon box”, this proposalcan estimate the parameter θ within a scaling of /N . Un-like other quantum metrology strategies, our proposal has thisadvantage-the state of photons can be used circularly. Thus,our scheme performs better than the strategy with NOON statewhen the total resource is taken into consideration. We alsoinvestigate our QND metrology scheme with cavity damping.It is shown that our scheme can sustain the quantum advan-tage for a longer time than the damping time of Fock state andis more robust than the interferometric strategy with exoticentangled states. An improved cascaded estimation schemeis also proposed by adding a real-time feedback interferomet-ric layout [34], with which the common π/N periodic errorcan be handled. The experimental feasibility of our proposalscan be justified with current laboratory parameters [37]. Wealso discuss the possible applications of our QND metrologyscheme. Atom BeamAtom Source(S) Detector (D)
Ramsey cavity(R )Ramsey Cavity(R ) Cavity (C) x m p r o b e a t o m s (a)(b) FIG. 1. (color online). Setup and experimental sequence. (a) Thecavity QED Ramsey interferometer for phase estimation. The Ryd-berg atoms, prepared in state | ↑ z i , are generated in the atom sourceS. The interaction between the Rydberg atoms and microwave pulsein auxiliary cavities R and R perform Hadamard gate operation.The unknown parameter is imprinted by the interaction of the atomand the superconducting cavity C. After crossing the R -C-R inter-ferometric arrangement, the states of atoms are detected in the de-tector D. (b) Diagram for our sequential strategy. m probe atomsare used. The Fock state of photons in C stays unchanged after eachQND measurement. II. PARAMETER ESTIMATION VIA QNDMEASUREMENTS IN “PHOTON BOX”
In our QND metrology proposal, the experimental setup issimilar to the one discussed in Refs. [35–38] and is shown inFig. 1(a). The core of this setup is a “photon box”, whichis an open high Q cavity C made up of two superconductingmirrors facing each other (Fabry-P´erot configuration). QNDprobe atoms, generated form the atomic resource S, are pre-pared in circular Rydberg states and travel along the trans-verse direction of the cavity axis. The atoms cross the cav-ity C sandwiched between two auxiliary low Q cavities R and R before being detected in the detector D. The R -C-R structure can be regarded as a Ramsey interferometry. The mi-crowave field stored in the cavity C is with frequency ω C / π .The atomic frequency is ω/ π and is detuned from the cavitymode by δ/ π ( δ = ω C − ω ).Suppose that the state of photons in the cavity C is in asuperposition of Fock states with different photon numbers | ψ i S = P n c n | n i . One Rydberg atom is prepared in states | ϕ i P = | ↑ z i ; and afterwards, for simplicity, we replace | ↑ z i and | ↓ z i with | i and | i . Both pulses in Ramsey cavitiesR and R are acting as an Hadamard operation on each atomwhich is written as H = (
11 1 − ) / √ and transforms | i and | i to | + i = ( | i + | i ) / √ and |−i = ( | i − | i ) / √ , re-spectively. The interaction between the probe atom and pho-tons contains an unknown parameter θ (see Appendix A fordetails) and can be expressed as a unitary operator: [38, 39] ˆ U SP ( θ ) = exp [ iθ (ˆ n S + 1 / σ zP / to the lowest order forsmall θ , where ˆ n S is the photon number operator in the cav-ity C and ˆ σ zP = (
10 0 − ) is the Pauli operator. The final state of photons and one probe atom after the probe atom passingthrough the R -C-R is expressed as | Φ f i SP = H P ˆ U SP ( θ ) H P | ψ i S | ϕ i P . (1)We then perform the ˆ σ z measurement on the atom in thedetector, and the output is i = 1 or − with probability p ( i | θ ) = P n c n cos [( n + 1 / θ/ i − π/ . Then, thephoton state in C is affected by this measure due to differentoutputs: | ψ ( i | θ ) i S = X n c n cos[( n + 1 / θ/ i − π/ p p ( i | θ ) | n i S . (2)It is easy to verify that [ˆ σ zP , H P ˆ U SP ( θ ) H P ] | i P = 0 , whichis the general necessary and sufficient condition [31] that theQND probe must satisfy.Next, we present the experimental procedure for estimatinga parameter, see Fig. 1(b). We consider that the state of pho-tons in the cavity is prepared as a Fock state of N photons, c n = δ n,N . Although generating a single mode Fock stateof N photons in the cavity C is a challenging task, it seemsnowadays experimentally available [38]. A general theoreticreview of this method is given in Appendix B. It is also possi-ble to prepare and lock the field to on-demand photon numberstates by the real-time quantum feedback techniques reportedin Refs. [34, 40]. Preparing the single-mode number squeez-ing state is also helpful and urgent for ultrasensitive two-modeinterferometry [41]. It is easy to verify that, after the QNDmeasurements procedure discussed above, a Fock state staysunchanged [42].Because h σ z i = cos[( N + 1 / θ ] , the parameter θ can beestimated from the readouts of σ z measurements performedon a sequence of probe atoms interacting with the light in C.For each probe, the probability for the readout i = 1 or − is p ( i | θ ) = cos [( N + 1 / θ/ i − π/ . With the assump-tion that the estimation is asymptotically unbiased, we canutilize Fisher information (FI) F θ = P i p ( i | θ )[ ∂ θ ln p ( i | θ )] and Cram´er-Rao bound δθ = 1 / ( √ m F θ ) [2] to calculatethe statistical precision of the estimation. FI is calculated as F θ = ( N + 1 / which leads to a lower bound: δθ ≥ √ m ( N + 1 / . (3)Therefore, by using this QND metrology technique, our pro-posal is able to achieve the /N scaling accuracy of parameterestimate with only one initial N -photon Fock state for m trialsof measurements.The advantages of this QND metrology strategy are: theinitial photon state is entanglement-free state which is morerobust than the exotic states (e.g. NOON states), and the Fockstate stays unchanged after QND measurements and can beused repeatedly. Technically, for our scheme, the total re-source can be written as N tot = N + m ∼ m for a suffi-ciently large m , and the lower bound is expressed as δθ & / ( N √ N tot ) . When m copies of NOON states are used toachieve the same accuracy, the total resource are N tot = mN and the lower bound is δθ en & / √ N N tot . Although bothstrategies do not achieve the Heisenberg limit /N tot , giventhe same total resources, our scheme gives / √ N advantagecompared with the strategy using NOON states and /N ad-vantage over SQL / √ N tot . III. FEASIBILITY ANALYSIS IN THE REALEXPERIMENT
Since the Bohr-Einstein photon box thought experiment,experiments with circular Rydberg atoms and Fabry-P´erothave become closest to this goal. They have also led to fun-damental tests of quantum theory and various demonstrationsof quantum information procedures [37]. Here, based on thedevelopments and advances made in the cavity quantum elec-trodynamics in the microwave domain, we discuss the feasi-bility of our QND metrology scheme using current laboratoryparameters.It is reported in Ref. [43] an ultrahigh finesse Fabry-P´erotresonator ω C / π = 51 . GHz with cavity damping time T C = 0 . ± . s and cavity quality factor Q = 4 . × at 0.8 K (mean number of blackbody photons n b =0 . ). The damping rate is given as Γ C = 1 /T C = ω C /Q .In Ref. [40], Rydberg atoms are prepared by a pulsed processrepeated at τ a = 82 µ s time intervals with selected atomicvelocity v = 250 m/s. For m trials of measurement, the totaltime is t = mτ a and the photon-loss intensity in the cavityis written as η ( t ) = 1 − exp( − Γ C t ) . In fact, if we shortenthe interval τ a , the number of measurement m can be largewith a low photon-loss intensity. This long damping time andQND detection technique can stabilize the Fock state in thecavity C and make our metrology scheme practicable. In thenext section, we will discuss the effect of cavity damping indetail. The technologies of other experimental procedures,such as generation of Rydberg atoms and polarizing measure-ment on the atoms, should be feasible and mature referring toRef. [37]. IV. QND METROLOGY WITH CAVITY DAMPING
The quantum metrological bounds in noisy systems havebecome a focus of attentions because in real experiments therewill always be some degree of noise and limitation. TheFock state prepared in the cavity C mainly suffers from cavitydamping. Given a certain the damping rate Γ C , the interactionpicture of reduced density operator for the field in the cavityC under the Born-Markov obeys the master equation [44]: ˙ ρ S = − Γ C n b (ˆ a ˆ a † ρ S − a † ρ S ˆ a + ρ S aa † ) / − Γ C ( n b + 1)(ˆ a † ˆ aρ S − aρ S ˆ a † + ρ S ˆ a † ˆ a ) / , (4)where ˆ a ( ˆ a † ) is the annihilation (creation) operator for fieldin cavity. Considering the radiation field with a reservoir atnearly zero temperature n b ≪ , we approximately expressthe density operator for the field by the well-known photonloss model: ρ S = P Nk =0 ( Nk )(1 − η ) k η N − k | k i S h k | .The initial probe state is still prepared as ρ P = | i P h | .The density operator form of the final state can be written as θ / π F I (b) θ / π p ( | θ ) (a) Γ C t (c) η =0 η =0.1 η =0.2 F o ( t )/ F o (0)Decay of Fock stateQFI, η =0 FI, η =0.1QFI, η =0.1 FI, η =0.2QFI, η =0.2 (d) m F I F o (0) F a ( m τ a ) F o ( m τ a ) FIG. 2. (color online). The photon number for the field in cavityC is set as N = 8 and cavity damping time is T C = 0 . s. (a)Probability p (0 | θ ) against parameter θ for η = 0 , . and . . (b) FIand QFI against parameter θ for η = 0 , . and . . (c) The decay ofthe optimal QFI F o compared with the decay of Fock state. (d) Theaverage QFI F a compared with the optimal QFI F o against numberof measurements m with measurement interval τ a = 82 µ s. ρ fSP = H P U SP H P ρ S ⊗ ρ P H P U † SP H P and the final reducedstate for probe atom is ρ fP = 12 (cid:18) r N cos( N ϕ ) − ir N sin ( N ϕ ) ir N sin ( N ϕ ) 1 − r N cos( N ϕ ) (cid:19) (5)where we set r = 1 − η (1 − η ) sin θ and ϕ = θ N +arctan (1 − η ) sin θη +(1 − η ) cos θ . Then we perform the ˆ σ z measurementon the atom and obtain the results and with proba-bilities p (0 , | θ ) = [1 ± r N cos( N ϕ )] / . Given photonnumber N = 8 , p (0 | θ ) is shown in Fig. 2(a) for differ-ent values of lossy intensities. We can therefore calcu-late FI as F θ = { ∂ θ [ r N cos( N ϕ )] } / [1 − r N cos ( N ϕ )] where we have used ∂ θ r = − η (1 − η ) sin θ/r and ∂ θ ϕ = η (1 − η ) cos θ +(1 − η ) [ η +(1 − η ) cos θ ] +sin θ (1 − η ) + N . For N = 8 , FI is θ depen-dent for a nonzero η , see Fig. 2(b).By choosing the optimal measurement, we can obtain themaximum FI which is also called quantum Fisher information(QFI). Given the spectral decomposition of final reduced statefor probe atom, ρ fP = P i p i | i ih i | , QFI can be written withcondition p i + p j = 0 as F Q = 2 X ij |h i | ∂ ω ρ | j i| p i + p j = N r N (cid:20) | ∂ θ (ln r ) | − r N + | ∂ θ ϕ | (cid:21) (6)where we have used p , = (1 ± r N ) / and | , i = ( | i ± e iNϕ | i ) / √ . Comparing FI and QFI in Fig. 2(b), we canachieve the optimal FI and QFI as we carefully choose θ → : F o ≡ lim θ → F Q = [(1 − η ) N + 1 / + η (1 − η ) N (7)where for large N , we obtain that F o → [(1 − η ) N +1 / . Therefore, with the two-step adaptive method basedon Bayesian estimation [45], the optimal QFI can be achievedwith the same measurement on the probe atoms used in thenoiseless case.Although the lifetime of the Fock state | N i is / ( N Γ C ) ,much less than the cavity damping time T C , the optimalQFI for this QND strategy decays much slower as shownin Fig. 2(c). It has been recognized that photon losses ininterferometers gradually blur the gain yielded by the spe-cial quantum states for parameter estimation; even with thebest strategy, asymptotically the improvement with respectto standard light sources is not by a scale change but onlyby a limited constant factor [12]. Unlike the interferomet-ric strategy with exotic states, e.g. NOON states, the QNDmetrology scheme with Fock state will sustain the quantumadvantage for a longer time than the damping time of Fockstate. For instance, given photon number N = 8 and cav-ity damping time T C = 0 . s, the quantum advantage re-mains until t ≃ . s for F o ( t ) ≥ N + 1 / and the num-ber of trials can be m ≃ for time interval τ a = 82 µ s. We can also define the average QFI for m trials of mea-surement as F a = P m − i =0 F o ( iτ a ) /m , with which we canwrite the lower bound of estimate accuracy for decoherencescenarios as δ dec θ ≥ /mF a . Since we show in Fig. 2(d)that the quantum-enhanced estimation against decoherencedoes not limit the number of measurements to be too small( m . ), we can conclude that this QND metrologyscheme is expected to be robust against the cavity damping. V. QND METROLOGY IN CASCADED SCHEME
Let us assume that the phase to be estimated lies in the in-terval θ ∈ [ − π, π ) . One common but intractable problem inthe quantum-enhanced metrology is the π/N periodic errorin the estimation of the true phase if N θ / ∈ [ − π, π ) [28, 46].To address this problem we will next extend the cascaded pro-tocol reported in Ref. [47] to our QND method. It is realizablewith the help of the mature technology of state control in the“photon box” [37].Our cascaded scheme employs L successively larger Fockstates of , , · · · , L − photons. We use m Rydberg atomsas the QND probe for each Fock state. The total resourceused in this cascaded scheme is mN = m P L − j =0 j ≃ m L .The interaction with the Fock state consisting of j photonspicks up the phase Θ j = 2 j θ mod [ − π, π ) , where j =0 , · · · , L − . The real phase to be estimated can be written inan exact binary representation θ = 2 π P L − k =1 d k k − π + Θ L − L − ,with digits d k ∈ { , } . By distinguishing whether the phaseis shifted by π or not, we can determine the value of the bit d k according to the relation d k = [2(Θ k − + π ) − (Θ k + π )] / π .We should note that the rounding error [40] that occurs when-ever | Θ estj − Θ j | > π/ can be neglected given a large numberof trials m . The last group ( j = L − then yields a Heisen-berg type limited estimate of the parameter with accuracy δθ cas ≥ √ m (2 L − + 1 / ≃ √ m ( N + 1 / , (8)which is merely less sensitive by a constant compared withEq. (3). FIG. 3. (color online). Layout of cascaded estimation scheme. Inaddition to the QED Ramsey interferometer for phase estimation in x axis, we need another real-time feedback interferometric setup toprepare the target Fock state in y axis: R ′ -C-R ′ interferometric ar-rangement and detector D ′ . The detection results from detector D ′ are sent to the computer based controller K. The controller K an-alyzes each detection result and determines the the real translationamplitude α applied by actuator A . To realize this cascaded QND metrology scheme, it is nec-essary to prepare and lock the field to different photon num-ber states during each QND measurement. This requirementcan be fulfilled by the real-time quantum feedback techniquesreported in Refs. [34, 40]. The experimental layout of ourproposal is shown in Fig. 3. The layout is supposed to workin two modes: phase estimation mode and target state prepa-ration mode. Since only one mode works at the same time,we need two interferometric setups. In addition to the setupused for the phase estimation mode, another interferometricsetup is performed to prepare and stabilize the successivelylarger target Fock states. The computer based controller Kcontrols the conversion between those two modes (more de-tails are shown in Appendix B).
VI. APPLICATIONS
In our scheme, the unknown parameter expressed as θ ( v, z ) = √ π Ω w cos ( ω C z/c ) /vδ is determined by theatom velocity v and the position z in the cavity C, see Ap-pendix A for more details. Here, we use the cylindrical co-ordinates ( R, z ) , c is the speed of light in vacuum, w is thewaist at center (0 , and Ω is the vacuum Rabi frequencyat center. Therefore, this high precision quantum-enhancedmeasurement can be used to detect and measure the mini-displacement of the cavity C along z axis. On the practicalperspective, a high sensitivity in θ leads to the high sensitivityin the displacement z when we measure the small displace-ment around the maximum slope point z = cπ/ (4 ω C ) . Theaccuracy can be obtained by straightforward error propaga-tion, δz = δθ | dθ/dz | ≥ z √ mN (9)where z = δvc/ ( √ π Ω wω C ) . Using the current labora-tory parameters shown in Sec. III and Ω / π = 49 kHz, w = 6 mm and δ/ π = 245 kHz in Ref. [34], one obtainsthat δz & . / ( √ mN ) mm. With m = 1000 and N = 8 ,the sensitivity is δz & . µ m and it can be improved byreducing the atom-cavity detuning and velocity of atom or in-creasing N and m . However, these methods for improvementmay seem challenging in the real experiment, for instance,reducing the wavelength of the light in the cavity will thenmake it difficult to place the atoms at the maximum slopepoint. Although this sensitivity by now seems several ordersof magnitude worse than the sensitivity needed for gravita-tional wave observation, our scheme will inspire future exper-iments demonstrating quantum-enhanced metrology and maybe helpful to prospective applications in other experimentalplatforms. VII. DISCUSSIONS
In this Letter, we have presented an experimental proposalfor estimating an unknown parameter in “photon box” by us-ing the method of QND measurements. We have shown thatinitially with Fock state of N photons, the /N scaling ac-curacy of the estimation can be achieved. Moreover, we donot need to prepare m copies of initial state as other metrol-ogy schemes, which will give a / √ N advantage comparedwith the scheme using NOON states when the total resourceis taken into consideration. We also show that this sub-shot-noise estimation scheme is robust against cavity damping.The feasibility of our scheme can be met by the current lab-oratory achievements and it can be improved via a cascadedscheme to overcome the π/N periodic error. Furthermore,this proposal with the help of QND measurements will also bean inspiration to other experimental platforms [48] for quan-tum metrology and quantum information techniques. In addi-tion, our results should be of broad interest as many applica-tions, such as clock synchronization and phase imaging. Sincegenerating and using the NOON states with more than pho-tons [18] for quantum metrology is still an arduous task, re-searchers have been able to generate Fock state with or evenmore photons in the cavity. That is to say if our scheme canbe realized in the experiments, it will be a great advance in theresearch area of quantum metrology and quantum physics. ACKNOWLEDGMENTS
We would like to thank Augusto Smerzi and Mehdi Ah-madi for useful discussions. This work was supported by the“973” Program (2010CB922904), NSFC (11175248) grantsfrom the Chinese Academy of Sciences.
Appendix A: Atom-Light Interaction in Cavity
We describe in this section the interaction between the Ry-dberg atom and photons in the cavity in a concise form. Thissimple case will provide us the phase shift linearly given byper photon. The interaction can be described via the Hamiltonian ofJaynes-Cummings model [49] ˆ H ~ = ω ˆ σ zP ω C (cid:18) ˆ a † ˆ a + 12 (cid:19) + g (ˆ a ˆ σ + P + ˆ a † ˆ σ − P ) (A1)where ˆ a ( ˆ a † ) is the photon annihilation (creation) operator inthe cavity C. Let the cavity C contain n photons. For large de-tuning frequency, the atom-field states | , n i and | , n i evolveinto dressed states and are shifted in angular frequency unitsby [50] ∆( r , n ) ≃ ~ ( n + 1 / ( r ) /δ. (A2)where Ω( r ) = Ω exp( − R /w ) cos( ω C z/c ) is the vac-uum Rabi frequency following the Gaussian distributionat cavity center z = 0 ; here we use cylindrical coor-dinates ( R, z ) where z is the position of the atom alongthe beam axis. c is the speed of light in vacuum, w isthe waist at center (0 , and Ω is the vacuum Rabi fre-quency at center. The difference of the phase imprinted onatomic states should be expressed as ( n + 1 / θ ( v, z ) where θ ( v, z ) = √ π Ω (0) w cos ( ω C z/c ) /vδ with w the waistat center and v the atom velocity. Therefore, the interac-tion between the probe atom and photons can be expressedwith the parameter θ ( v, z ) as a unitary operator: ˆ U SP ( θ ) =exp [ iθ (ˆ n S + 1 / σ zP / to the lowest order for large detun-ing frequency. The value of the parameter θ ( v, z ) is deter-mined by the atom velocity and the position in the cavity C,which will be of great value in scientific and engineering ap-plications. Appendix B: Preparation of Fock States and Cascaded Scheme
In this section, we review two different methods of prepara-tion of Fock state in the “photon box”. The stochastic methodis expected to require less equipments and be suitable for ademonstrative experiment of quantum-enhanced metrology.The deterministic method needs additional experimental de-vices and is able to generate a Fock state with on-demandphoton number, which is helpful to the cascaded strategy.
1. Stochastic Approach
We start with | ψ i S = P n c n | n i and set the interactionparameter θ s at an appropriate and definite value such that p ( i | n ) = p ( i | n ′ ) for all possible photon number n = n ′ ,where p ( i | n ) = cos [( n + 1 / θ s / − ( i − π/ . Sup-pose that M atoms cross the R -C-R interferometric layoutand are measured with operator σ z . The sequence of measure-ment results for M probe atoms, called a event, is expressed as ω M = ( i , · · · , i M ) , where i µ = 1 , − and µ = 1 , , · · · , M .The photon number distribution of the final state in C can becalculated as P ( n | ω M ) = | c n cos η [( n + ) θ s /
2] sin ξ [( n + ) θ s / | Z ( ω M ) , (B1) P ( n | M ) P ( n | M ) P ( n | M ) P ( n | M ) (a) (b) (c) (d) FIG. 4. (color online). Numerical simulation of the indirected measurement procedures. For each simulation, the convergence event is obtainedvia Monte Carlo method and considers totally 100 atoms. The state of light in the cavity is initially chosen as a coherent state |√ i S . M is the number of atoms interacting with the light in the cavity, n represents the photon number and P ( n | M ) denotes the photon numberprobability distribution after the M th atom flies through the cavity. The 2D diagram shows the detection result of M atom in the sequence.Two parameters are considered: θ (1) s = 0 . for (a) and (b); θ (2) s = π/ for (c) and (d). where Z ( ω M ) = P n | c n cos η [ ( n +1 / θ s ] sin ξ [ ( n +1 / θ s ] | ; η and ξ are the number of and − in the event ω M , re-spectively. It has been proved in Ref. [39] that ( i ) this pho-ton number distribution converges as M becomes infinity: lim M →∞ P ( n | ω M ) = δ n,N , ( ii ) the probability for the statein cavity converges to a Fock state | N i is | c N | , and ( iii ) theconvergence for δ n,N is exponentially fast. Therefore, we canobtain a single mode Fock state | N i by this approach withprobability | c N | when M is large enough. The final photonnumber N can be determined via analyzing the spin measure-ment results of the probe atoms ω M . After generating a Fockstate of a nonzero and known photon number, we can performthe parameter estimation without adjusting the experimentalapparatus.Most commonly the initial state of the light in cavity C isa coherent state | ψ i S = | α i S . The QND measurement pro-cedures are numerically simulated via Monte Carlo methodand plotted in Fig. 4. We observe the converging events ofdifferent photon numbers in Fig. 4(a) and 4(b). In Fig. 4(c)and 4(d), we present the situation that the special condition q ( i | n ) = q ( i | n ′ ) is saturated and the convergent states aresuperposed Fock states. We also numerically simulate theprobability for the coherent state converging to a Fock state,in Fig. 5, which conforms with the experimental results inRef. [38].The average Fisher information for all possible Fock statesfrom the initial coherent state | α i can be written as F α = ∞ X i =0 e − ¯ n ¯ n i i ! (cid:18) i + 12 (cid:19) = (cid:18) ¯ n + 12 (cid:19) + ¯ n (B2)which leads to the Heisenberg-type lower bound: δ coh θ ≥ / [(¯ n + 1 / + ¯ n ] . Instead of the coherent state, an effi-cient method for improving this strategy is to use the squeezedstate | α, ζ i S which may be generated by first acting with thesqueeze operator ˆ S ( ζ ) on the vacuum followed by the dis-placement operator ˆ D ( α ) [51]. We can obtain via squeezedstate a higher success rate for generating a useful Fock statefor sub-shot-noise metrology due to its super-poissonian andnarrower photon number statistics, see Fig. 5.
2. Deterministic Approach
In order to prepare and lock the field to different photonnumber states during each QND measurement, it is neces-sary to use real-time quantum feedback techniques reportedin Refs. [34, 40] to fulfill this requirement. The experimentallayout of our proposal is shown in Fig. 3 in the main text. Inaddition to the estimation interferometric setup ( x direction)shown in Fig. 3, a quantum feedback setup is put in the y di-rection: another atom source S ′ generates test Rydberg atoms,and two auxiliary cavities R ′ and R ′ act as two Hadamardgates. The information on the interaction between the testatom and the field is assumed to be known. The measurementresults of the test Rydberg atoms obtained by detector D ′ aresent to the computer based controller K. By analyzing eachdetection result, the controller K updates the photon distribu-tion p ( n ) = | c n | and controls actuator A to feed cavity C bydiffraction on the mirror edges [34]. The controller K analyzeseach detection result to determine the real translation ampli- n P r ( n ) Numerical result, CSNumerical result, SSTheoretical result, CSTheoretical result, SS
FIG. 5. (color online). Reconstructed photon number distributionfor coherent state and squeezed state. Photon number distributionsfor coherent state (CS) |√ i S and squeezed state (SS) |√ , . i S are plotted by dashed blue line and dash-dotted magenta line, re-spectively. convergence events are simulated to figure out thephoton number distribution probability numerically. Γ C t P r o b a b ili t y Γ C t F i d e li t y n =8 n =7 n =6 n =0 n =1== n =2 n =3 n =5 nn n =4 FIG. 6. (color online). Probabilities of ρ exact S and ρ S against time forphoton number N = 8 . Solid lines are for ρ exact S and dashed lines arefor ρ S . Inset: Fidelity of ρ exact S and ρ S against time. tude α applied by actuator A which minimizes the distance d (ˆ ρ t , ˆ D ( α )ˆ ρ ˆ D † ( α )) [34] between the target ˆ ρ t = | n t ih n t | and the field estimation ˆ ρ . Here ˆ D ( α ) is the displacementoperator. When the controller K finds that α → , it stopsatom source S ′ and the target Fock state has been prepared.It is reported that Fock states with photon numbers n t up to7 can be prepared with number photon number distributionpeaked p ( n ) = 0 . ∼ . [40]. Then, controller K activatesanother atom source S and m probe states are sent and de-tected to estimate the unknown phase. We will next show thatthis technique based on real-time quantum feedback is alsonecessary to realize the cascaded scheme [47].
3. QND Metrology in Cascaded Scheme
One common but intractable problem in the quantum-enhanced metrology is the π/N periodic error [46] in theestimation of the true phase if N θ / ∈ [ − π, π ) . This problemcan not be settled by simply adding an ancillary phase so thatthe cascaded protocol is proposed to solve it. To realize thecascaded QND metrology scheme, it is necessary to prepareand lock the field to different photon number states duringeach QND measurement.The first target state starts with n t = 2 for ˆ ρ t = | n jt ih n jt | .When the target Fock state n jt = 2 j for j = 0 , · · · , L − has been prepared, controller K activates another atom sourceS and m probe states are sent and detected to estimate theunknown phase with value Θ j . After the estimation using thisFock state | n jt i is finished, controller K activates the feedbacksetup and the target state is changed to the next Fock state | n j +1 t i . The last target Fock state that we consider has n max t =2 L − photons. When all Θ j for j = 0 , · · · , L − have beenobtained, we can calculate digits d k for k = 1 , · · · , L − andretrieve the true value of phase θ within a accuracy presentedin Eq. (8).As a brief summary, the layout is supposed to work in twomodes: target state preparation mode and phase estimationmode. Since only one mode works at the same time, we needtwo interferometric setups. In addition to the setup used forthe phase estimation mode, another interferometric setup isperformed to prepare and stabilize the successively larger tar-get Fock states. The computer based controller K controls theconversion between those two modes. Appendix C: Approximation in QND Metrology with CavityDamping
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