Table Of Content

Operators for the Aharonov-Anandan and Samuel-Bhandari Phases P.-L. Giscard† Department of Physics and College of Optical Sciences The University of Arizona, Tucson, Arizona 85721 (Dated: January 18, 2009) We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometricphaseandthetotallengthofevolution. Fromthisequation,wederiveanoperatorforthe SamuelandBhandariphase(SB-phase)fornoncyclicevolutions. FinallyweshowhowtheSB-phase canbeusedtoconstructanoperatorcorrespondingtoaquantumclockwhichcommutatorwiththe 9 Hamiltonian has a canonical expectation value. 0 0 PACSnumbers: 03.65.Vf,03.65.Ca,03.65.Ta 2 n Although experiments related to the geometric phase value, is then used to construct the corresponding oper- a J are known since Pancharatnam’s work [1], the concept atorandto deriveaselectionrulegivingtheallowedand 8 of geometric phase was first recognized by M.V. Berry forbidden geometric phases depending on the state and 1 [2]. In his seminal paper, Berry showed that an eigen- the Hamiltonian evolving it. In a second part, the role stateofaparameter-dependentHamiltonianvaryingadi- of the operator for the AA-phase is explored through an ] abatically and cyclically, acquires both the well known newequationofmotionforcyclicquantumsystemswhich h p dynamical phase and a gauge invariant phase that de- involves neither time nor any fundamental constant and - pends only on the geometry of the path taken by the fromwhichweobtaintheoperatorfortheSB-phase. We t n Hamiltonianinthespaceofparameters. The mathemat- continuebyderivinganoperatorforthedistancebetween a ical explanation to this geometric phase was soon given quantum states as measured by the Fubini-Study met- u by Simon [3] whereas a generalization to Berry’s phase ric along the curves solutions to Schrödinger equation q [ was discovered by Aharonov and Anandan a few years andthe correspondingtime-operator. We finishon some later [4]. The Aharonov-Anandangeometric phase (AA- propertiesofthetime-operatorandshowhowitisrelated 4 phase) arises when a quantum system undergoes cyclic to a quantum clock. v 3 evolutions with time. The concept of geometric phase The first step is the derivation of a new expression for 3 has since then been generalized to non cyclic, non uni- thegeometricAA-phase. Westartfromtheresultsfound 3 tary evolutions by Samuel and Bhandari [5] (SB-phase). in [6]. For a time independent Hamiltonian Hˆ, provided 0 Howeverthe conditionof cyclicalityrequiredby the AA- thatastate ψ undergoesacyclicmotion,thetotalAA- . | i 1 phase,highlyconstrainthevaluesofthegeometricphase phase it acquires after one complete period of evolution 0 itself,whicheventuallyledtoanewalgebraicmethodfor is given by 9 the calculation of the AA-phase [6]. 0 τ v: Oneofthefundamentalaxiomsofquantummechanics γ =φ+ ~ψhψ|Hˆ|ψi, (1) i is that to any measurable physical quantity, an observ- X φ and τ being respectively the total phase and the able,correspondsanoperator. Beyondtheproblemscur- ψ r period of the cyclic motion. These quantities depend a rentlyencounteredwithtime-relatedquantitiesforwhich on both the Hamiltonian Hˆ and the state ψ . Let the construction of the corresponding operator(s) is still B = φ be a basis in which Hˆ is diag|onial and an unsolved question, researchers have produced opera- {| ki} Λ = λ the corresponding set of its eigenvalues. Let tors for all other physical quantities. The AA-phase, as { k} B B be the smallest set of eigenvectors needed to well as the SB-phase have been observed and measured ψ ⊆ decompose the state ψ on B and let Λ Λ be the experimentally in a large variety of systems ([7] and ref- | i ψ ⊆ corresponding set of eigenvalues. Finally, let ∆E be erences therein), thus confirming their status of observ- ψ the set of non-zero energy spacings in Λ , i.e. ∆E = ables. Accordingly,thereshouldexistoperatorsforthese ψ ψ ∆E =λ λ . Then from [6] we have quantities which construction is the main goal of this { k,i k− i}λk,i∈Λψ, λk6=λi paper. We start by deriving a new expression for the AA-phasebasedonaprecedingworkbythesameauthor τψ = 2π~ LCM ∆Eψ−1 , (2) (cid:16) (cid:17) [6]. This expression, which appears as an expectation φ = 2π m λ LCM ∆E−1 , (3) h − (cid:16) ψ (cid:17)i where LCM means least common multiple and m is an †[email protected] integer depending on λ. Note that Eq.(3) is valid for 2 any λ Λ . For the sake of clarity, we will use L = or equivalently ψ ψ ∈ LCM ∆Eψ−1 in the following. Gˆ = 2πp φ φ . (13) To(cid:16)obtain (cid:17)an expression for the geometric phase γ, ψ Xi ψ,ij| iih i| we combine Eq.(1), Eq.(2) and Eq.(3). The expression FromEq.(1) andEq.(3), the relationto the Hamiltonian for the total phase φ is calculated using an arbitrary λ Λ . Moreover, the expectation value ψ Hˆ ψ is is the following j ψ eval∈uated on B to be h | | i Gˆ = τψ(Hˆ λ ), (14) ψ ~ − j ψ Hˆ ψ = λ φ 2, (4) i i with λ Λ . This operator depends on the considered h | | i | | j ψ λiX∈Λψ state ψ∈as both τψ and Λψ depend on ψ . To empha- | i | i size this state-dependence, we denote Gˆ the geometric where φ = φ ψ . By combining these expressions we ψ i i h | i operator for state ψ . This operator is necessarily the obtainthefollowingexpressionofthetotalAA-phasefor | i same for any state ϕ located on the path C solution a normalized state ψ undergoing cyclic evolutions | i | i to Schrödinger equation passing through ψ and could γ =2πm +2π (λ λ )L φ 2, (5) equivalently be denoted Gˆϕ∈C. Thus, al|thiough Gˆψ is ψ j i j ψ i − | | linear once ψ has been specified, an operator acting on λiX∈Λψ | i the whole space of states andwith expectationvalue the m being an integer that depends on λ . Let us now AA-phase is non-linear and accepts no matrix represen- j j define p = (λ λ )L . Independently of the state tation. Only the state dependent form Gˆ accepts one ψ,ij i j ψ ψ − under consideration, the coefficients p have the fol- (given in Eq.(13)). ψ,ij lowing properties : Before studying the role of the geometric-operator in the motion of cyclic quantum systems, let us consider pij = pji, (6) some consequences of Eq.(9). First, note that a simpli- − p = p p , (7) fication of Eq.(9) occur for a two-level states. As shown ij kj ki − in [6] two-level quantum states (i.e. Λ contains exactly p = m m . (8) ψ ij i j − twodifferent elements)alwaysundergocyclic evolutions, The last equality is obtained by equalizing Eq.(5) calcu- whatever the time independent Hamiltonian considered. lated with two different λ . From the above properties, Using the definition of the pj coefficients of Eq.(9), we j i we obtaina new expressionfor the totalAA-phase accu- see that p10 = −1 if λ0 < λ1 and p10 = 1 otherwise. This mulated by the state ψ after one cyclic evolution observation leads to | i γ = 2π φ 2 =2π φ 2 [2π], (15) γψ = 2π pψ,ij φi 2 [2π], (9) − | 0| | 1| | | Xi if λ0 < λ1 and all the signs are reversed otherwise. Be- γ = 2π m φ 2 [2π], (10) yond these signs, the only Hamiltonian dependence of ψ i i Xi | | this equation is contained in the |φi|2. This is consistent with the fact that the dynamics of any two-level withbothm andp integers. Furthermore,the abovefor- system can be mapped on the dynamics of a spin-1/2 mulas Eq.(9) and Eq.(10) are valid in the diagonal basis particlein afictitious magnetic field. Indeed this consid- of the considered Hamiltonian Hˆ and therefore, cannot eration entail that, given that we work in the eigenbasis besimultaneouslycorrectfortwononcommutingHamil- of the Hamiltonian, the geometric phase for a spin-1/2 tonians Hˆ and Hˆ′. particle is the same than for any other two-level Hamil- Let us now turn to the problem of finding an oper- tonian (up to a sign). The geometric-operator Gˆ for ψ ator for the total AA-phase. Interestingly, Eq.(9) (and ψ =eiΦcos(θ/2)φ +sin(θ/2)φ takes a particularly 0 1 equivalently Eq.(10)) can be used to define such an op- | i | i | i simple form erator, that we call the geometric-operator Gˆ (the no- ψ tvaatluioenfoisresxtaptlaesinuedndienrgtohiengfocllyocwliicnge)v,owluhtiicohnsexwpheecntastuiobn- Gˆψ =(cid:18)±02π 00(cid:19). (16) jectedtosometimeindependentHamiltonianisthetotal Then, Eq.(15) or Eq.(11, 16), immediately give the well AA-phase known result γψ = Gˆψ . (11) γ = π(1 cos(θ))[2π]. (17) h i ± − FromEq.(9),theexpressionofGˆ inthebasisB isgiven A secondnotable consequence ofEq.(9) givesa rule of ψ by selection for geometric-phases. Let Ξ be a set of time- independentHamiltoniansthatallcommutewithonean- G =2πp δ , (12) other and ψ a normalized state that can be expressed ψ,kl ψ,kj k,l | i 3 in the common eigenbasis B of Ξ as distance parametrizationofthe Schroo¨dingerequationis straightforward. In [8], J. Anandan and Y. Aharonov, 1 ψ = α eiθi φ , (18) showed indeed that the distance s as measured by the i i | i √ω X | i Fubini-Studymetricinthespaceofquantumstatessince |φii∈Bψ the beginningofthe unitaryevolutionalongacurveC is with all α integers, θ real numbers, and ω is the small- given by i i est positive integer satisfying Eq.(18). Note that such a nfourmmbeexrsis.tsTihfuasnwdeownliyll isfayallth|haφti|sΨtait|e2i∈s[0o,Nbe]yainregrEaqti.o(1n8a)l s=ZC ∆Hˆ(~t)ψ∈Cdt≡ZCϑψ(t)dt, (21) tahceceppjtacoreaffitiocineanltdseocfomEqp.o(s9i)tiaorneoinntBeg.eFrrso,mittfhoellofawcsttthhaatt which reduces to s = ∆Hˆ~ψ∈Ct for time independent i Hamiltonians, i.e. the time elapsed times the speed of statesacceptingarationaldecompositiononB,canonly evolution. This allows to write a simple equation of evo- acquire AA-phases of the form lution equivalent to the Schrödinger equation but using 2nπ the distance s as parameter γ = [2π], n integer, (19) ψ ω ∂ ψ ds i~ | i =Hˆ ψ . (22) whatever the Hamiltonian(s) of Ξ and/or the number of ∂s dt | i consecutivecyclicevolutionsconsidered. Thisshowsthat Eq.(21)gives ds = ∆Hˆψ and,forstatesundergoingcyclic only the geometric phases that are rational multiples of dt ~ evolution, we can use Eq.(14) to get ∆Gˆ = τ ∆Hˆ/~. π are allowed for state accepting a rational decompo- ψ ψ These manipulations lead to sition on B. This is an intrinsic property of both the state ψ and the set Ξ. Note that the rationality of ∂ ψ(s) λτ γψ/π |imiplies a finite number of cyclic evolution giving i∆Gˆψ |∂s i =(Gˆψ + ~ψ)|ψ(s)i. (23) different geometric phases. At the opposite, a state ϕ | i which yields, by dropping the λτ /~ constant term [9], that does not accept a rational decomposition on B can ψ only acquire AA-phases that are irrational multiples of ∂ ψ(s) π when subjected to cyclic evolution(s) by some Hamil- i∆Gˆψ | i =Gˆψ ψ(s) . (24) ∂s | i tonian(s) of Ξ. Interestingly this implies in principle the Now since the evolution is cyclic in time, it is also cyclic possibility of distinguishing any number of cyclic evo- in distance, i.e. there exists a period in length S such lutions from any other. Indeed since γ /π is irrational, ψ ϕ that, n being an integer, ψ(nS +s) = ψ(s) upto a (pγϕ)[2π]=(qγϕ)[2π],foranyintegersp=qrepresenting | ψ i | i 6 6 phase factor. Logically S = τ ϑ thus S ∆G and the number of cyclic evolutions elapsed. However,it can ψ ψ ψ ψ ≡ ψ the motion of a cyclic quantum system is governby be shown that there will always be a number q′ of cyclic evolutions so that the resulting geometric phase is arbi- ∂ ψ(s) iS | i =Gˆ ψ(s) , (25) trarily close to γ [2π] which means that distinguishing ψ ψ ϕ ∂s | i aninfinite number of cyclic evolutionswouldexperimen- ∂ ψ(t) tally require an infinite precision in the measurement of iτψ | i =Gˆψ ψ(t) . (26) ∂t | i the geometric-phase. in the two parametrizations. These two equations are Let us now focus on the role of the geometric oper- valid for any quantum state undergoing cyclic evolu- ator in an equation of motion for cyclic quantum sys- tions and only for them, or in other terms, their solu- tems. To know how evolves the state of a quantum sys- tions are the cyclic solutions to Schrödinger equation. tem, a parameter evolving along the path C solution to Note that Eq.(25) can be considered as a purely geo- Schrödinger equation in the Hilbert space is required. metric equation of motion as it involves only geomet- This parameter is generally chosen to be the time. But ric objects, the geometric-operator (or equivalently the in a paper about the evolution of quantum systems [8], geometric-phase), the distance between quantum state J.AnandanandY.Aharonov,gavethe expressionofthe as measured by the Fubini-Study metric and the total instantaneous speed of evolution of a quantum system length of a cyclic evolution. Now we can use Eq.(25) to ψ(t) in the space of quantum states as | i get another expression for the AA-phase as ∆Hˆψ(t) ∂ ϑψ(t)= ~ , (20) γψ =iSψ , (27) h∂si where∆Hˆ (t)=( ψ(t)Hˆ2 ψ ψ(t)Hˆ ψ(t) 2)1/2isthe andanewexpressionforthegeometric-operatorforstate ψ h | | i−h | | i ψ as uncertainty in energy of the state ψ(t) . For time inde- | i | i pendent Hamiltonians, this speed in a constant of mo- ∂ tion and the passage from a time parametrization to a Gˆψ =iSψ . (28) ∂s 4 A simple example of a solution to Eq.(25) is given Notethat,sincethegeometric-operatorisHermitianand by a two level system. The geometric operator is since S is real, Eq.(28) leads to (i∂/∂s)† = i∂/∂s. ψ given by Eq.(16) and let the initial state be ψ = Therefore, s, τ and γ being real, we conclude using ψ ψ | i eiΦcos(θ/2)φ0 +sin(θ/2)φ1 . Thestateatdistances is Eq.(34)thatTˆ isHermitian. Interestinglythecommuta- | i | i then torsofTˆwithHˆ andGˆ canbecalculatedandarefound ψ to be ψ(s) =e±i|φ0|s|φ1|eiΦcos(θ/2)φ0 +sin(θ/2)φ1 . (29) | i | i | i ∂ where we have used Sψ = ∆Gˆψ = 2π φ0 φ1 . The only [Hˆ,Tˆ(s)] = −~∂γ(s), (35) | || | Hamiltonian dependence is contained in the decomposi- ∂ tion of the initial state on the eigenbasis {|φ0i, |φ1i}. [Gˆψ,Tˆ(s)] = −τψ∂γ(s), (36) Again, this is consistent with the possibility of mapping the dynamics of any two level systems on the one of a with, as i ∂ =1, the following expectation values spin-1/2 in a magnetic field. h ∂γ(s)i To find an operator for the SB-phase, we now intro- [Hˆ,Tˆ(s)] = i~, (37) duce a third parametrization of the Schrödinger equa- h i tion based on the instantaneous geometric phase γ(s), [Gˆψ,Tˆ(s)] = iτψ, (38) h i whichisthe geometricphaseaccumulatedatagivendis- remarkably the first quantity is both time and state in- tance s. This quantity can be calculated following the dependent. Furthermore, at t = τ , the operator Tˆ(t) work by Samuel and Bhandari [5] who generalized the ψ becomes Tˆ(τ ) = iτ ∂ which, on using the relation notion of geometric phase to non necessarily closed evo- ψ ψ∂γψ lutions of quantum systems. It is found to be simply between geometric phase γ and energy ǫ = Hˆ λ, ψ γ(s) = sγ /S . This together with Eq.(28), lead us to τ ǫ=~γ (see Eq.(14)) leads to h i− ψ ψ ψ ψ propose the instantaneous geometric-operatorGˆ(s) ∂ Tˆ(τ )=i~ . (39) Gˆ(s)=is ∂ , (30) ψ ∂ǫ ∂s This operator has already been proposed for time in which is state independent. This is a generalization of quantum mechanics, see [10]. the operator for the AA-phase and corresponds to the We emphasize that the operators Sˆ and Tˆ proposed SB-phase in the case of time independent Hamiltonians. here correspond respectively to the distance and the InthisparametrizationwecancontructanoperatorSˆfor time elapsed since an arbitrarily chosen initial state. In the distance elapsed since an arbitrary point ψ(s = 0) other terms if the expectation value of Tˆ was obtained and so that s = ψ(s)Sˆψ(s) . Indeed, since t|he instani- on a ensemble of quantum systems at two different mo- h | | i taneous geometric phase fulfill γ(s) = sγψ/Sψ. Thus ments,thenthedifferencebetweenthoseexpectationval- s=γ(s)Sψ/γψ and we propose ues would be the time elapsed between the two series of measurements. This could therefore be used as quan- S Sˆ(s)=Gˆ(s) ψ. (31) tum clock measuring the time of unitary Schrödinger- γ ψ evolution. We thank M. Bhattacharya for starting our interest Upon using Eq.(30) and remarking that S /γ = ψ ψ in geometric phases and for helpful discussions and P. ∂s/∂γ(s), we obtain Meystre whose support allowed this work. ∂ Sˆ(s)=is . (32) ∂γ(s) This is consistent with the fact that i ∂ = 1, which h ∂γ(s)i [1] R. W. Ditchburn, Collected Works of S. Pancharatnam canbedemonstratedusingEq.(30). Itisnowstraightfor- (Oxford University Press for the Raman Research Insti- ward to obtain a similar expression for a time-operator tute, London, 1975), 1st ed. Tˆ with the parametert entering Schrödingerequationas [2] M. V. Berry, Proc. Roy. Soc. London, Ser. A 392, 45 expectation value. Indeed using the speed of quantum (1984). evolution we have Tˆ(s)=Sˆ(s)/ϑ , which gives on using [3] B. Simon, Phys.Rev.Lett. 51, 2167 (1983). ψ s/ϑ =t and s=γ(s)S /γ , [4] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58, 1593 ψ ψ ψ (1987). ∂ [5] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 Tˆ(t) = it , (33) (1988). ∂γ(ϑ t) ψ [6] P.-L. Giscard, arXiv:0810.0309 [quant-ph](2008). Tˆ(s) = isτψ ∂ . (34) [7] F.WilczekandA.Shapere,GeometricPhasesinPhysics γ ∂s (World ScientificPub. Co., Singapore, 1989), 1st ed. ψ 5 [8] J. Anandan, Y. Aharonov, Phys. Rev. Lett. 65, 1697 [10] A.S.Holevo,Probabilisticandstatisticalaspectsofquan- (1990). tum theory (ElsevierSciencePub.Co.,NewYork,1982), [9] This gauge change does not change the geometric-phase 1st ed. northe dynamicsof thesystem.