Quantum metrology at level anti-crossing
Luca Ghirardi, Ilaria Siloi, Paolo Bordone, Filippo Troiani, Matteo G. A. Paris
QQuantum metrology at level anti-crossing
Luca Ghirardi, Ilaria Siloi, Paolo Bordone,
1, 3
Filippo Troiani, and Matteo G. A. Paris
4, 5, ∗ Dipartimento FIM, Universit`a di Modena e Reggio Emilia, I-41125 Modena, Italy Department of Physics, University of North Texas, 76201 Denton, TX, USA Centro S3, CNR-Istituto di Nanoscienze, I-41125 Modena, Italy Quantum Technology Lab, Dipartimento di Fisica dell’Universit`a degli Studi di Milano, I-20133 Milano, Italy INFN, Sezione di Milano, I-20133 Milano, Italy (Dated: November 11, 2018)We address parameter estimation in two-level systems exhibiting level anti-crossing and prove thatuniversally optimal strategies for parameter estimation may be designed, that is, we may find a pa-rameter independent measurement scheme leading to the ultimate quantum precision independentlyon the nature and the value of the parameter of interest. Optimal estimation may be achievablealso at high temperature depending on the structure of the two-level Hamiltonian. Finally, we showthat no improvement is achievable by dynamical strategies and discuss examples of applications.
I. INTRODUCTION
The avoided level-crossing theorem [1, 2], often referredto as the level anti-crossing theorem, describes a charac-teristic phenomenon occurring in systems with a parame-ter dependent Hamiltonian. It states that if the Hamilto-nian depends on n real parameters, then the eigenvaluescannot be degenerate, apart from a ( n − cross at all as a function of theparameter itself. Level anti-crossing, also referred to aslevel repulsion, plays a relevant role in several branchesof quantum physics and chemistry [3–7] and frequentlyarises in the study of condensed matter systems. In sys-tems with parameter dependent Hamiltonian, and thusanticrossing, small perturbations to the parameter mayinduce relevant changes in the system ground state [8, 9],which are possibly reflected in large variations of someaccessible observable. Level anti-crossings, which is alsoconnected to creation of resonances [10, 11] and the onsetof chaos [12–15], may thus represent a resource for thecharacterization of Hamiltonians and/or the estimationof parameters [16, 17].In this paper, we address in details metrological appli-cations of level anti-crossing and show that universallyoptimal strategies for parameter estimation may be de-signed. By this terminology we mean that we may finda parameter independent measurement scheme leadingto the ultimate quantum precision independently on thenature and the value of the parameter of interest. Inparticular, we address quantum estimation for parame-ter dependent two-level Hamiltonians [18–24] and showanalytically that universal optimal estimation is achiev-able, that is, the ultimate precision permitted by quan-tum mechanics may be obtained by a class of parameter-independent measurement schemes. This is of metrolog- ∗ Electronic address: matteo.paris@fisica.unimi.it ical interest since it is often the case that the descriptionand the dynamics of a metrological system may be re-stricted to the effective two-level system made of its twolowest energy levels.The paper is structured as follows. In Section II weintroduce notation and the basic tools to analyze two-level systems with parameter dependent Hamiltonian. InSection III we discuss the ultimate quantum bounds toprecision of parameter estimation, whereas in Section IVwe show how those limits may be achieved by parameterindependent measurement schemes, including estimationat finite temperature. We also show that dynamical esti-mation strategies cannot improve performances. SectionV is devoted to some examples and Section VI closes thepaper with some concluding remarks.
II. THE SYSTEM
Let us consider a two-level system governed by a pa-rameter dependent Hamiltonian of the form H = (cid:18) ω ( λ ) γ ( λ ) γ ∗ ( λ ) ω ( λ ) (cid:19) . (1)The parameter λ is the quantity of interest. It is initiallyunknown and we want to estimate its value by perform-ing measurements on the system. We assume that λ ∈ Λwhere Λ is a generic subset of the real field. The eigen-values of H are given by h ± ( λ ) = ω ( λ ) ± (cid:112) | γ ( λ ) | + ∆ ( λ ) (2) h + − h − = 2∆( λ ) (cid:112) | x | (3) x = γ ( λ )∆( λ ) , where ω ( λ ) = 12 [ ω ( λ ) + ω ( λ )] , (4)∆( λ ) = 12 [ ω ( λ ) − ω ( λ )] . (5) a r X i v : . [ qu a n t - ph ] O c t Upon looking at Eqs. (2) and (3) we see that h −
In order to gain information about the value of the pa-rameter λ , which may not correspond to an observable,one performs repeated measurements on the system andsuitably processes data. The optimal measurement corre-sponds to the spectral measure of the so-called symmetriclogarithmic derivative (SLD) L λ , which is defined by theLyapunov-like equation ∂ λ ρ λ = 12 ( L λ ρ λ + ρ λ L λ ) , (9)being ρ λ the (parameter-dependent) state of the system[25, 26]. At zero temperature the system stays in itsground state and the SLD reduces to L λ = 2 ( | ∂ λ ψ − (cid:105)(cid:104) ψ − | + | ψ − (cid:105)(cid:104) ∂ λ ψ − | ) (10)= ∂ λ x (1 + x ) ( σ − xσ ) . (11)By measuring L λ on repeated preparations of the sys-tem one collects data and then builds an estimator forthe unknown quantity λ , i.e. a function ˆ λ ( χ ) of thedata sample χ = { x , x , ..., x M } that returns the valueof the parameter when averaged over data. The preci-sion of the overall estimation strategy corresponds to thevariance of the estimator. An efficient estimator (e.g.the maximum-likelihood estimator or the Bayesian one)has variance saturating the quantum Cramer-Rao boundVarˆ λ = 1 /M H ( λ ) where M is the number of measure-ments and H ( λ ) = Tr (cid:2) ρ λ L λ (cid:3) = (cid:104) ψ − | L λ | ψ − (cid:105) is the so-called Quantum Fisher information (QFI) [27, 28, 30–43]. Notice that the optimal measurement, and the cor-responding precision, do explicitly depend of the value of λ . Using Eq. (11) one has H ( λ ) = ( ∂ λ x ) (1 + x ) , (12)= (cid:0) ∆ ∂ λ γ − γ∂ λ ∆ (cid:1) γ + ∆ (13)= 16 (cid:18) ∆ h + − h − (cid:19) ( ∂ λ x ) , (14)where the last expression well illustrates the connectionswith level anti-crossing. The same result may be ob-tained from the expression of the QFI in term of theground state fidelity [44–48], i.e. H ( λ ) = lim δλ → − (cid:12)(cid:12) (cid:104) ψ − ( λ + δλ ) | ψ − ( λ ) (cid:105) (cid:12)(cid:12) δλ . (15)Using Eq. (15) it may be proved that the QFI for any set of superposition states of the form | ψ θ (cid:105) = cos θ | ψ − (cid:105) +sin θ | ψ + (cid:105) is equal to the ground state one.As it is apparent from the above expressions, the QFI,and in turn the precision of any estimation scheme, doesnot depend on ω . Notice also that if either ∆( λ ) = 0or γ ( λ ) = 0, ∀ λ then H ( λ ) = 0 and no estimation strat-egy is possible. This behaviour may be understood bylooking at Eq. (7), which shows that for ∆( λ ) = 0 or γ ( λ ) = 0 the eigenstates of the system become P ± ( λ ) = [ I ∓ sgn( γ ) σ ] or P ± ( λ ) = [ I ± sgn(∆) σ ] respectively.In both cases the ground state is independent on λ (ex-cept for the crossing values), and no information may begained by performing measurements on the system. IV. UNIVERSALLY OPTIMAL ESTIMATIONBY PROJECTIVE MEASUREMENTS
Since the SLD does depend on the unknown value ofthe parameter a question arises on whether the ultimateprecision may be actually achieved without a priori in-formation. As we will see, universal estimation based ona single detector implementing a parameter independentmeasure may be indeed obtained.A generic (projective) measurement on a two-level sys-tem is described by the operatorial measure { Π , I − Π } where Π = 12 ( I + r · σ ) , (16)and | r | = 1. The distribution of the two possible out-comes is governed by the probability q ( λ ) ≡ q ( λ ) = Tr[ ρ λ Π] = 12 (cid:18) xr − r √ x (cid:19) (17) q ( λ ) = Tr[ ρ λ ( I − Π)] = 1 − q ( λ ) . (18)The variance of any estimator is now bounded by theclassical Cram´er-Rao bound Varˆ λ ≥ /M F ( λ ) [49] andefficient estimators are those saturating the bound, where F ( λ ) is the Fisher information of the distribution q k ( λ ),i.e. F ( λ ) = (cid:88) k ( ∂ λ q k ) q k = ( ∂ λ q ) q (1 − q ) (19)= H ( λ ) g λ ( r , r ) . (20)where g λ ( r , r ) ≡ g ( x, r , r )= ( r + xr ) x − ( xr − r ) . (21)As expected from the quantum Cram´er-Rao theorem wehave F ( λ ) ≤ H ( λ ), i.e. g λ ( r , r ) < ∀ λ, r , r (see Fig.1). On the other hand, we have equality, F ( λ ) = H ( λ ),either for r = 1 and r = 0 or for r = 0 and r =1, i.e. by measuring either σ or σ on the two-levelsystem. In addition, if r = 0 we have r = (cid:112) − r and F ( λ ) = H ( λ ), ∀ r , i.e. any observable of the form σ θ = σ sin θ + σ cos θ leads to optimal estimation.In other words, Eq. (20) and the following argumentsshow that universal optimal estimation, i.e. maximumprecision for any value of λ , may be achieved by param-eter independent measurements.Let us now discuss robustness of the estimation strat-egy. The discussion above has shown that the opti-mal (projective) measurement corresponds to the choice r = 0 and any pair ( r , r ) satisfying r + r = 1 forthe expression of the operator measure Π in Eq. (16).On the other hand, some class of pairs may be betterthan others in practical implementation, depending onthe relative values of γ ( λ ) and ∆( λ ), i.e. the value of x .Indeed, as it is apparent from the upper panels of Fig.1, if γ ( λ ) (cid:28) ∆( λ ) in the whole range of variation of λ ,then x (cid:28) F ( λ ) (cid:39) H ( λ ) also if some imperfectionslead to the measurement of a slightly perturbed observ-able corresponding to r (cid:38) , r (cid:38) , r (cid:46)
1, rather thanthe optimal ideal one σ θ . The situation is reversed if γ ( λ ) (cid:29) ∆( λ ) in the whole range of variation of λ , seethe lower panels of Fig. 1, and also from the symmetry g ( x, r , r ) = g (1 /x, r , r ) of the function g . A. Estimation at finite temperature
If the system is not at zero temperature the equilibriumstate is given by ρ λβ = p + P + + p − P − , (22)where β is the inverse temperature and the projectors P ± over the eigenvectors | ψ ± (cid:105) are given in Eq. (7). Theprobabilities p ± = e − βh ± /Z are obtained from the eigen-values h ± of Eq. (2) and from the partition function Z = e − βh + + e − βh − , i.e. Z = 2 e − βω cosh (cid:16) β (cid:112) γ + ∆ (cid:17) . (23) FIG. 1: Density plot of g λ ( r , r ) at fixed values of x as afunction of r and r . From top left to bottom right we show g λ ( r , r ) for x = 0 . , . , ,
100 respectively. The functionis defined only in the region r + r ≤
1. Darker regionscorrespond to lower values of g . Using the above expressions we arrive at p ± = 12 (cid:104) ± tanh (cid:16) β (cid:112) γ + ∆ (cid:17)(cid:105) , (24)and, in turn, to ρ λβ = 12 (cid:20) I − tanh (cid:16) β ∆ (cid:112) x (cid:17) σ − xσ √ x (cid:21) , (25)which is a mixed state with purity µ λβ = Tr (cid:2) ρ λβ (cid:3) = 12 (cid:104) (cid:16) β ∆ (cid:112) x (cid:17)(cid:105) . (26)The quantum Fisher information is now given by sum oftwo terms H β ( λ ) = H C ( λ )+ H Q ( λ ) usually referred to asthe classical and quantum part of the QFI. The classicalpart corresponds to the Fisher information of the spectraleigenmeasure i.e. H C ( λ ) = ( ∂ λ p + ) p + p − = ( γ∂ λ γ + ∆ ∂ λ ∆) γ + ∆ k C ( β, λ ) (27) k C ( β, λ ) = β cosh (cid:16) β (cid:112) γ + ∆ (cid:17) . (28)The quantum part H Q take into account the contributioncoming from the dependence of the eigenvectors on λ , wehave H Q ( λ ) = 2 (cid:88) j,k = ± |(cid:104) ψ j | ∂ λ ψ k (cid:105)| ( p j − p k ) p j + p k = H ( λ ) k Q ( β, λ ) (29) k Q ( β, λ ) = tanh (cid:16) β (cid:112) γ + ∆ (cid:17) , (30)where H ( λ ) is the zero-temperature QFI reported in Eq.(13). In the limit of low temperature we have k Q ( β, λ ) β (cid:29) (cid:39) k C ( β, λ ) β (cid:29) (cid:39) , (31)whereas for high temperature one may write k Q ( β, λ ) β (cid:28) (cid:39) β ( γ + ∆ ) k C ( β, λ ) β (cid:28) (cid:39) β . (32)Eqs. (31) and (32) say that the quantum part H Q domi-nates in the low temperature regime, whereas for high Tthe two contributions are of the same order.Given a generic projective measurement the distribu-tion of the outcomes is now governed by the quantity q β ( λ ) = Tr[ ρ βλ Π] = Tr[( p + P + + p − P − ) Π] (33)= 1 − p − + q ( λ )(2 p − −
1) (34)= 12 + (cid:20) q ( λ ) − (cid:21) tanh (cid:104) β (cid:112) γ + ∆ (cid:105) (35) β (cid:29) (cid:39) q ( λ ) + (cid:20) q ( λ ) − (cid:21) e − β √ γ +∆ , (36)where q ( λ ) is the zero temperature distribution given inEq. (17). The fast convergence of the exponential func-tion ensures that optimal estimation may be achievedalso for finite temperature, provided that β (cid:38) (cid:112) γ + ∆ .In the opposite limit of high temperature, i.e. β (cid:28) β F β ( λ ) = [ ∂ λ q β ( λ )] q β ( λ )[1 − q β ( λ )]= ( r ∂ λ γ − r ∂ λ ∆) β + O ( β ) . (37)The Fisher information of Eq. (37) should be comparedto the QFI H β ( λ ) = H C ( λ ) + H Q ( λ ) which, up to secondorder in β , reads as follows H β ( λ ) = (cid:104) ( ∂ λ γ ) + ( ∂ λ ∆) (cid:105) β + O ( β ) . (38)The two quantities coincides, i.e. universal optimal es-timation is achievable also at high temperature, whenonly the transverse, or only the diagonal, part of theHamiltonian does depend on the parameter λ , i.e. ifeither ∂ λ ∆ = 0 or ∂ λ γ = 0. In those cases, we have H β ( λ ) (cid:39) F β ( λ ) up to second order by performing aprojective measurement with r = 1 , r = r = 0, or r = 1 , r = r = 0 respectively. On the other hand, again from the expression inEq. (37) one finds that a projective measurement with r /r = γ/ ∆ or r /r = − ∆ /γ is globally optimal with-out restrictions on the form of the Hamiltonian. In thiscase, however, the optimal measurement is not universal,i.e. it depends on the value of the parameter itself. B. Dynamical estimation strategies
One may wonder whether having access to the initialpreparation of the system may improve precision for someclass of estimation strategies. In fact, general considera-tions about unitary families of states suggest the opposite[26], i.e. that no improvement may be achieved in thisway. In order to prove this explicitly for our system, letus now address a scenario in which we are able to initiallyprepare the system in any desired state | ψ θ (0) (cid:105) = cos θ | ψ − (cid:105) + e iφ sin θ | ψ + (cid:105) , (39)which then evolve according to the Hamiltonian in Eq.(6). The evolution operator U = exp {− iHt } may bewritten as U t = e − iω t (cid:104) cos (cid:16) t (cid:112) γ + ∆ (cid:17) σ − it sinc (cid:16) t (cid:112) γ + ∆ (cid:17) ( γσ − ∆ σ ) (cid:105) , (40)and the evolved state | ψ θ ( t ) (cid:105) = U t | ψ θ (0) (cid:105) as | ψ θ ( t ) (cid:105) = cos θ e − ih − t | ψ − (cid:105) + e iφ sin θ e − ih + t | ψ + (cid:105) . (41)The SLD L λ ( t ) = 2 ( | ∂ λ | ψ θ ( t ) (cid:105)(cid:104) ψ θ ( t ) | + | ψ θ ( t ) (cid:104) ∂ λ | ψ θ ( t ) | )may be easily evaluated, thanks to the covariant natureof the problem L λt = 2 U t (cid:104) | ∂ λ | ψ θ (0) (cid:105)(cid:104) ψ θ (0) | + | ψ θ (0) (cid:104) ∂ λ | ψ θ (0) | (cid:105) U † t (42)= U t L λ U † t , (43)where L λ ≡ L λ is given in Eq. (11). Finally, we have H t ( λ ) = (cid:104) ψ θ ( t ) | L λt | ψ θ ( t ) (cid:105) = (cid:104) ψ θ (0) | L λ | ψ θ (0) (cid:105) ≡ H ( λ ) , (44)where H ( λ ) ≡ H ( λ ) is given in Eq. (13). Notice thatthe above negative arguments hold when the Hamilto-nian is given by Eq. (1), i.e. it depends on the parameterof interest but it is time-independent. Improved perfor-mances, i.e. more precise estimation strategies may beinstead achieved if the two-level Hamiltonian is explicitlydepending on time [50–53]. V. EXAMPLESA. Level anti-crossing induced by a perturbation
Let us consider a two-level system with Hamiltonian H = H + λH where H is the bare Hamiltonian of thesystem, H represents a perturbation and λ , which is theparameter to be estimated, is the perturbation strenght[6, 54]. Without loss of generality we assume the follow-ing structure H = (cid:18) ω ω + δ (cid:19) H = R (cid:18) (cid:15) (cid:19) R T (45)where δ > (cid:15) > R is a rotation matrix R = (cid:18) cos φ − sin φ sin φ cos φ (cid:19) with φ ∈ [0 , π/ φ = 0, R = I and the two terms[ H , H ] = 0 commute. In this case, the eigenvalues of H are given by h − = ω , h + = ω + δ + λ(cid:15) and they arecrossing at λ c = − δ/(cid:15) . For φ (cid:54) = 0 this degeneracy isremoved since the two levels are coupled each other. Thetwo eigenvalues are now given by h ± = ω + 12 ( δ + λ(cid:15) ) ± (cid:112) δ + λ (cid:15) + 2 δλ(cid:15) cos 2 φ , (46)i.e. we have level anti-crossing, which may be exploitedfor the precise estimation of the perturbation coupling λ . The QFI may be evaluated using Eq. (13), where thequantities ∆ and γ are now given by∆ = δ + (cid:15)λ cos 2 φ (47) γ = − (cid:15)λ sin 2 φ . (48)The QFI H ( λ ) is maximised for φ = π/
4, i.e. when H and H are ”maximally non-commuting” and in this caseit is given by H ( λ ) = 1(1 + y λ ) y = (cid:15) δ , (49)where, rather intuitively, the dependence on the structureof the Hamiltonian terms H and H is summarised bythe ratio (cid:15)/δ . B. Driven double-well systems
It is often the case in condensed matter that double-well systems exhibit two lowest-energy levels well sepa-rated from the next pair by a large gap, i.e. larger thanthe other relevant energies, e.g. the tunnelling energyand the frequency of the driving field. In those cases, atwo-level approximation describes rather well the physics of the system, and the dynamics may be understood interms of the celebrated periodic
Rabi Hamiltonian H = 12 ω σ + λσ cos ωt , where the coupling λ is the quantity to be estimated and ω is the frequency of the driving field, which we assumeto be known to the experimenter. The model cannot besolved exactly [55], since the Hamiltonian is not com-muting with itself at different times. On the other hand,upon going to the appropriate interaction picture and ne-glecting the counter-rotating terms, the system may bedescribed by a two-level time-independent Hamiltonian[56] which, in the relevant subspace. reads as follows H eff = (cid:18) Ω γγ − Ω + 2 ω (cid:19) , (50)where γ = − λ
4Ω [Ω − ( ω − ω )] (51)Ω = (cid:112) λ + ( ω − ω ) . (52)The physics underlying this approximated Hamiltonian isthat of a system with avoided level crossing and a gap (cid:39) γ separating the otherwise crossing unperturbed levels.The quantity ∆ introduced in the previous Sections ishere given by ∆ = Ω − ω . Inserting this expression inEq. (13) we arrive at the QFI H ( λ ) ω (cid:39) ω (cid:39) ω − y + y ) y = λω , (53)where, for the sake of simplicity, we have reported onlythe expression close to resonance ω (cid:39) ω . The QFIis maximised for λ = ω , indicating that for anyvalue of λ optimisation may be achieved by tuning thenatural frequency of the well. As proved in the pre-vious Sections, those ultimate limits to precision maybe achieved by measuring any observable of the form σ θ = σ sin θ + σ cos θ , where σ is here the popula-tion of the unperturbed levels and σ the correspondingpolarisation. More general driven systems with level an-ticrossing [57] may be also addressed in the same way. C. Effective description of three-level systems
Level anti-crossing may also occur in systems withmore than two levels. In this case, the additional levelsmay influence the form of the eigenstates and, in turn,the behaviour of the QFI when the value of the parame-ter λ is perturbed. Let us consider a three-level systemwith two close energy levels and a third level being wellseparated in energy and weakly coupled to the first twolevels. The Hamiltonian for such a system reads as fol-lows H (3) = ω ( λ ) γ ( λ ) gγ ∗ ( λ ) ω ( λ ) gg g (cid:15) , (54)where we assume a large gap between the third level andthe others, i.e. (cid:15) (cid:29) ω k and a weak coupling g (cid:28)
1. Inthis regime, the system is amenable to an effective two-level description [47], with an effective Hamiltonian givenby H (2) eff = (cid:18) ω ( λ ) + g /(cid:15) γ ( λ ) + g /(cid:15)γ ( λ ) + g /(cid:15) ω ( λ ) + g /(cid:15) (cid:19) , (55)where we have also assumed γ ∈ R . Using this effectivedescription we may now exploit the approach of the pre-vious Sections in order to assess the performances of thissystem as a scheme to estimate the value of the λ . TheQFI may be evaluated using Eq. (13). Up to first orderin the quantity κ = g /(cid:15) we have H κ ( λ ) = H ( λ ) (56) − κ (cid:112) H ( λ ) 2 γ ∆ ∂ λ γ + ∂ λ ∆(∆ − γ )( γ + ∆ ) where H ( λ ) is the QFI of Eq. (13), corresponding to κ = 0, i.e. a genuine two-level system. The possibility ofenhancing estimation by coupling with additional levelsis thus depending on the explicit dependence on λ of thequantities γ and ∆. VI. CONCLUSIONS
Systems with Hamiltonian depending on a single pa-rameter exhibits level anti-crossing. In turn, small per- turbations to the value of the parameter may induce rel-evant changes in the system ground state, which may bedetected by measuring some accessible observable. Levelanti-crossings may thus represent a resource for the char-acterization of Hamiltonians and for parameter estima-tion.Here, we have addressed in details metrological appli-cations of level anti-crossing and have shown that univer-sally optimal strategies for parameter estimation may bedesigned, independently on the nature and the value ofthe parameter of interest. In particular, we have studiedquantum estimation for parameter dependent two-levelHamiltonians and show analytically that universal opti-mal estimation is achievable. We also found that univer-sally optimal estimation may be achievable also at hightemperature if only the transverse, or only the diagonal,part of the Hamiltonian depends on the parameter.We have also analyzed few examples, which confirm thegenerality of our approach and pave the way for furtherapplications.
Acknowledgments
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