Average quantum dynamics of closed systems over stochastic Hamiltonians
AAverage quantum dynamics of closed systems overstochastic Hamiltonians
Li Yu and Daniel F. V. JamesOctober 24, 2018
Department of Physics, University of Toronto, 60 St. George Street, Toronto,Ontario M5S 1A7, Canada
Abstract
We develop a master equation formalism to describe the evolutionof the average density matrix of a closed quantum system driven bya stochastic Hamiltonian. The average over random processes gen-erally results in decoherence effects in closed system dynamics, inaddition to the usual unitary evolution. We then show that, foran important class of problems in which the Hamiltonian is propor-tional to a Gaussian random process, the 2nd-order master equationyields exact dynamics. The general formalism is applied to study theexamples of a two-level system, two atoms in a stochastic magneticfield and the heating of a trapped ion.PACS number(s): 03.65.Ca, 03.65.Yz, 05.40.-a, 03.67.Lx
The density operator encapsulates all the statistical information about the stateof a quantum system. The evolution of the density operator of a closed systemis governed by the Hamiltonian. In practice, the Hamiltonian can seldom bestrictly determined or precisely controlled – it fluctuates both in a temporalsense and between repeated realizations, which can be mathematically describedby random processes. Therefore, instead of treating any Hamiltonian as deter-ministic in an idealized manner, we would like to take such fluctuations intoaccount explicitly when studying quantum dynamics. Our goal is to obtain the average dynamics in the following sense: Suppose an ensemble of systems are1 a r X i v : . [ qu a n t - ph ] N ov repared in some initial state and subsequently evolve under a randomly fluc-tuating Hamiltonian, how does the density matrix that describes the ensembleas a whole evolve?Previous work on stochastic average dynamics was done by Budini [1] usinga variational calculus method and Novikov’s theorem, and by Guha et al. [2]using a non-perturbative cluster cumulant method. A different kind of averagedynamics over the time domain was studied by Gamel and James [3], assumingthe deterministic (i.e. non-stochastic) Hamiltonian but taking into account thefinite time-window of measurements. The master equation formalism is alsowidely used in the study of open systems dynamics [4]. It should be noted that,despite the formal similarity, our study is on the dynamics of closed quantumsystems and no environment is involved.In this paper, we will first adopt a series expansion approach and derive atime-local master equation that describes the ensemble-average dynamics of ageneral quantum system. The general formalism is then used to study a repre-sentative class of Hamiltonians obeying Gaussian statistics. Finally, we applythe master equation method to some physical examples and find interestingphenomena such as fluctuation-induced decoherence and decoherence-induceddisentanglement. Throughout, our results are compared to exact dynamics andthe validity of the master equation approach is discussed. Consider a closed, but not isolated, system for which the Hamiltonian is de-termined by some classical stochastic quantity x ( t ) . Suppose an experiment iscarried out repeatedly with each realization labelled by µ . The evolution of thedensity matrix ρ µ ( t ) that describes the quantum system in the µ -th realizationis governed by the Hamiltonian ˆ H µ ( t ) = ˆ H [ x µ ( t )] , and is given by ρ µ ( t ) = ˆ U µ ( t, t ) ρ ˆ U µ † ( t, t ) , (1)where ρ is the initial density matrix, which is assumed to be uncorrelated with x ( t ) and thus is the same in all realizations. The unitary evolution operator ˆ U µ ( t, t ) obeys the equation of motion, i (cid:126) ∂∂t ˆ U µ ( t, t ) = ˆ H µ ( t ) ˆ U µ ( t, t ) . (2)The average density matrix ρ ( t ) is defined as follows, ρ ( t ) ≡ lim N →∞ N N (cid:88) µ =1 ρ µ ( t ) . (3)It can be shown that ρ ( t ) is Hermitian, positive and of unit trace, which isensured by the properties of the individual density matrices ρ µ ( t ) . Thus the op-2rator ρ ( t ) is indeed a physical density matrix, describing the average statisticsof the ensemble of realizations as a whole.The equation of motion for ρ ( t ) is formally given by i (cid:126) ∂∂t ρ ( t ) = lim N →∞ N N (cid:88) µ i (cid:126) ∂∂t ρ µ ( t ) = lim N →∞ N N (cid:88) µ [ ˆ H µ ( t ) , ρ µ ( t )] = [ ˆ H ( t ) , ρ ( t )] . (4)However, since the right hand side cannot be written as a function of ρ ( t ) , theequation is not of a closed form and thus not very useful. With the goal ofobtaining a closed equation for ρ ( t ) , we resort to a series expansion approach. Following the standard recipe for perturbative expansion [5], the unitary oper-ator ˆ U µ ( t, t ) in a particular realization µ can be written as a power series in λ (a parameter controlling the “strength” of the Hamiltonian): ˆ U µ ( t, t ) = ∞ (cid:88) n =0 λ n ˆ U µn ( t, t ) (5)where ˆ U µ ( t, t ) = ˆ I, (6) ˆ U µn ( t, t ) = 1 i (cid:126) t ˆ t dt (cid:48) ˆ H µ ( t (cid:48) ) ˆ U µn − ( t (cid:48) , t ) , n (cid:62) . (7)Thus ρ ( t ) can be expressed in terms of ˆ U µn ( t, t ) and λ : ρ ( t ) = (cid:32)(cid:88) m λ m ˆ U m ( t, t ) (cid:33) ρ (cid:32)(cid:88) n λ n ˆ U † n ( t, t ) (cid:33) = ∞ (cid:88) k =0 λ k E k [ ρ ] ≡ E [ ρ ] , (8)where E k [ ρ ] is the time-dependent map defined as E k [ ρ ] ≡ k (cid:88) j =0 ˆ U k − j ( t, t ) ρ ˆ U † j ( t, t ) , (9)and E [ ρ ] is a completely positive linear map [6]. Although their argument is adensity matrix in this instance, E k and E can act on any operator in general. The map E is a linear transformation that maps ρ to ρ . Since both ρ and ρ are operators in the same Hilbert space, and thus of the same dimension, it is3atural to postulate that an inverse transformation F = E − exists that maps ρ to ρ . That is, ρ = E − [ ρ ] ≡ F [ ρ ] . (10)Note that the meaning of “inverse” is purely mathematical here: the map E − is not to be confused with an inverse dynamical evolution in the physical sense.According to the semigroup property, a completely positive, trace preserving(CPTP) linear map is physically invertible if and only if it is an unitary map(see Section 3.8 of Ref.[7]). Therefore, in general, a CPTP map E : ρ → ρ doesnot have a physical inverse, that is, we cannot find another CPTP map thatgives ρ → ρ . However, the mathematical inverse E − that serves our purposehere needs not be CPTP.Since the composition of a transformation and its inverse is the identitytransformation, the identity F [ E [ ρ ]] = I [ ρ ] holds for an arbitrary operator ρ .Following [3], we adopt the ansatz that F can be expanded in powers of λ , F [ ρ ] = ∞ (cid:80) m =0 λ m F m [ ρ ] . Then we have ∞ (cid:88) m =0 λ m F m [ ∞ (cid:88) n =0 λ n E n [ ρ ]] = ∞ (cid:88) k =0 λ k k (cid:88) j =0 F j [ E k − j [ ρ ]] = λ I [ ρ ] . (11)Collecting terms of like powers in λ , we obtain the set of equations involving F m and E n : F [ E [ ρ ]] = I [ ρ ] , (12) F [ E [ ρ ]] + F [ E [ ρ ]] = 0 , (13) F [ E [ ρ ]] + F [ E [ ρ ]] + F [ E [ ρ ]] = 0 , (14)and so on. Solving for F m in terms of E n , and making use of E = I as definedin Eq.(9), we have F [ ρ ] = E [ ρ ] = I [ ρ ] , (15) F [ ρ ] = −E [ ρ ] , (16) F [ ρ ] = −E [ ρ ] + E [ E [ ρ ]] , (17)and so on. Differentiating Eq.(8) with respect to time and making use of the inverse relationin Eq.(10), we obtain the following equation: i (cid:126) ∂∂t ρ ( t ) = i (cid:126) ˙ E [ ρ ] = i (cid:126) ˙ E [ F [ ρ ( t )]] . (18)4ere, the notation ˙ E [ ρ ] means first taking time-derivative of the time-dependenttransformation E to obtain a new transformation denoted by ˙ E and then let-ting ˙ E act on ρ ; the argument ρ is not differentiated whether or not it istime-dependent. Assuming the order of differentiation and summation can beswitched, we have ˙ E [ F [ ρ ( t )]] = ∞ (cid:88) n =0 λ n ˙ E n [ ∞ (cid:88) m =0 λ m F m [ ρ ( t )]] = ∞ (cid:88) k =0 λ k k (cid:88) j =0 ˙ E j [ F k − j [ ρ ( t )]] ; (19)thus the equation of motion can be written as i (cid:126) ∂∂t ρ ( t ) = ∞ (cid:88) k =0 λ k i (cid:126) k (cid:88) j =0 ˙ E j [ F k − j [ ρ ( t )]] ≡ ∞ (cid:88) k =0 λ k L k [ ρ ( t )] . (20)Evaluating F m and ˙ E n explicitly, we find L [ ρ ] = i (cid:126) ˙ E [ F [ ρ ]] = 0 , (21) L [ ρ ] = i (cid:126) ˙ E [ F [ ρ ]] + i (cid:126) ˙ E [ F [ ρ ]] = ˆ Hρ − ρ ˆ H, (22) L [ ρ ] = i (cid:126) ˙ E [ F [ ρ ]] + i (cid:126) ˙ E [ F [ ρ ]] + i (cid:126) ˙ E [ F [ ρ ]]= ˆ H ˆ U ρ − ˆ H ˆ U ρ + ˆ Hρ ˆ U † − ˆ Hρ ˆ U † − ρ ˆ U † ˆ H + ρ ˆ U † ˆ H − ˆ U ρ ˆ H + ˆ U ρ ˆ H, (23)and so on. Note again that the argument ρ is not to be averaged or differentiatedand that terms like ˆ H are time-dependent just as L k [ ρ ] are time-dependenttransformations.Keeping terms up to 2nd order and setting λ = 1 in Eq.(20), a time-localmaster equation is thus obtained for the evolution of ρ ( t ) : i (cid:126) ∂∂t ρ ( t ) = [ ˆ H, ρ ( t )] + ˆ Aρ ( t ) − ρ ( t ) ˆ A † + D [ ρ ( t )] , (24)where ˆ A ≡ ˆ H ˆ U − ˆ H ˆ U and D [ ρ ] ≡ ˆ Hρ ˆ U † − ˆ Hρ ˆ U † − ˆ U ρ ˆ H + ˆ U ρ ˆ H . Theeffective Hamiltonian responsible for unitary evolution is ˆ H eff ≡ ˆ H + 12 ( ˆ A + ˆ A † ) , (25)with which the master equation can be written in a more insightful way, i (cid:126) ∂∂t ρ ( t ) = [ ˆ H eff , ρ ( t )] + 12 (cid:110) ˆ A − ˆ A † , ρ ( t ) (cid:111) + D [ ρ ( t )] . (26)It can be shown that the right-hand side of the equation can be put into theLindblad form, which ensures Hermiticity, complete positivity and trace preser-vation of the evolution. 5his result is formally similar to a previous work on average dynamics [3].However, the physical meaning is different since the derivation in that case isfor a time-average density matrix in a single realization. Incidentally, our resultmay also be reminiscent of some master equations for the reduced density matrixof open quantum systems. But it should be emphasized that our derivation isfor a closed system and thus quantum entanglement with environment does notplay a role here.Note that the above results are formally applicable to an interaction-picturedensity matrix, though we implicitly assume the Schrà ¶ dinger picture in thederivation. The only difference is in the interpretation of the density matrix:When we use the interaction-picture density matrix ρ I , the expectation valueof an observable ˆ O is given by (cid:104) ˆ O (cid:105) = T r (cid:16) ˆ O I ρ I (cid:17) , where ˆ O I is the interaction-picture operator instead of the original operator in Schrà ¶ dinger picture. Let us first apply our general result to the simple case where the parametersin the Hamiltonian are time-independent. That is, ˆ H = ˆ H ( a ) , where a repre-sents random variable(s) instead of random process(es). Suppose further that ˆ H is of zero-mean, which implies a particular choice of “picture”: Any time-independent, deterministic part of the Hamiltonian plus the average compo-nent of the stochastic part can be removed by a gauge transformation, thatis, by switching to an suitably chosen interaction picture [8]. Note that, inthe case of time-independent random variables, ˆ U ( t, t ) = ( t − t ) ˆ H/i (cid:126) and ˆ U † ( t, t ) = − ˆ U ( t, t ) , thus [ ˆ U , ˆ H ] = [ ˆ U † , ˆ H ] = 0 . So we have ˆ A + ˆ A † = 0 andthus ˆ H eff = 0 . This result is special to the time-independent case, however. Aswe will see later, the effective Hamiltonian (to 2nd order) is in general non-zero,due to the non-commutativity of ˆ U and ˆ H in the time-dependent case. Aftersimplification, the 2nd-order master equation in this particular case is ∂∂t ρ ( t ) = − t (cid:126) { ˆ H , ρ ( t ) } + 2 t (cid:126) ˆ Hρ ( t ) ˆ H. (27)A class of problems of physical interest has a Hamiltonian of the form ˆ H = (cid:126) (cid:88) n a n ˆ h n + a ∗ n ˆ h † n , (28)where a n are jointly circular complex Gaussian random variables of zero mean. Substituting Eq.(28) into Eq.(27), we find According to the central limit theorem, Gaussian statistics is applicable when the randomvariables are due to the addition of many uncorrelated random sources. For those readers who might be concerned about the factor t on the right-hand side ofEq.(29): It is just a result of a time integral, as can be seen in the more general case of ∂t ρ ( t ) = t (cid:88) k,l { Γ kl (cid:16) − ˆ h k ˆ h † l ρ ( t ) − ρ ( t )ˆ h k ˆ h † l + 2ˆ h † l ρ ( t )ˆ h k (cid:17) + Γ ∗ kl (cid:16) − ˆ h † k ˆ h l ρ ( t ) − ρ ( t )ˆ h † k ˆ h l + 2ˆ h l ρ ( t )ˆ h † k (cid:17) } , (29)where Γ kl = a k a ∗ l are the correlation functions. Note that the equation is ofthe familiar Lindblad form, which can be further simplified to a diagonal formthrough a linear transformation of the coefficients. Now consider the case of a time-dependent Hamiltonian ˆ H ( t ) = (cid:126) a ( t )ˆ h, (30)where a ( t ) is a (real) Gaussian random process. This is representative of a wideclass of problems, for example, the Zeeman effect, where a ( t ) is proportionalto the external magnetic field and ˆ h is the z -component of the total angularmomentum [9]. The random process in this section is taken to be the mostgeneral case, that is, we do not assume any additional property like zero-meanor stationarity.The ensemble-average dynamics under this Hamiltonian is exactly solvable,so let us first work out the exact, analytic result. The unitary evolution operatorin a particular realization µ is ˆ U µ ( t, t ) = exp (cid:16) − iv µ ( t )ˆ h (cid:17) , (31)where v µ ( t ) ≡ t ˆ t dt (cid:48) a µ ( t (cid:48) ) . (32)For an initial state ρ ( t ) = (cid:80) k,l ρ kl ( t ) | k (cid:105)(cid:104) l | , where {| n (cid:105)} is the energy-eigenbasiswith ˆ h | n (cid:105) = E n | n (cid:105) , the evolution in a particular realization is ρ µ ( t ) = ˆ U µ ( t, t ) ρ ( t ) ˆ U µ † ( t, t ) = (cid:88) k,l ρ kl ( t ) exp {− iv µ ( t )( E k − E l ) } | k (cid:105)(cid:104) l | , (33)thus the ensemble-average is time-dependent Hamiltonians later. Note that the trace-preserving property of the equationis guaranteed by the Lindblad form. ( t ) = (cid:88) k,l ρ kl ( t )exp {− iv ( t )( E k − E l ) }| k (cid:105)(cid:104) l | . (34)Invoking the special properties of Gaussian statistics, it can be shown that exp {− iv ( t )( E k − E l ) } = exp (cid:26) − i ( E k − E l ) v ( t ) − ( E k − E l ) (cid:104) v ( t ) − v ( t ) (cid:105)(cid:27) . (35)Thus, the exact ensemble-average dynamics is given by the elements of theaverage density matrix: ρ kk ( t ) = ρ kk ( t ) , (36) ρ kl ( t ) = ρ kl ( t ) exp (cid:26) − i ( E k − E l ) v ( t ) − ( E k − E l ) v ( t ) − v ( t ) ] (cid:27) . (37)Now let us solve the same problem by the master equation approach. Usingthe results from Eqs.(25-26), the following expression is obtained, i (cid:126) ∂∂t ρ ( t ) = [ (cid:126) a ( t )ˆ h, ρ ( t )]+ i (cid:126) a ( t ) t ˆ t dt (cid:48) a ( t (cid:48) ) − t ˆ t dt (cid:48) a ( t ) a ( t (cid:48) ) (cid:16) ˆ h ρ ( t ) + ρ ( t )ˆ h (cid:17) + 2 i (cid:126) t ˆ t dt (cid:48) a ( t ) a ( t (cid:48) ) − a ( t ) t ˆ t dt (cid:48) a ( t (cid:48) ) ˆ hρ ( t )ˆ h, (38)which can be simplified to ∂∂t ρ ( t ) = − ia ( t )[ˆ h, ρ ( t )] + D ( t ) (cid:104) ˆ h, [ˆ h, ρ ( t )] (cid:105) , (39)where D ( t ) ≡ a ( t ) t ˆ t dt (cid:48) a ( t (cid:48) ) − t ˆ t dt (cid:48) a ( t ) a ( t (cid:48) ) . (40)To find the solution to this differential equation, we first write it down in termsof the matrix elements in the ˆ h -eigenbasis: Since v ( t ) is a linear filtered Gaussian random process, it is a Gaussian random processitself. (See page 83 of Ref.[10].) ∂t ρ kk ( t ) = 0 , (41) ∂∂t ρ kl ( t ) = (cid:104) − ia ( t )( E k − E l ) + D ( t )( E k − E l ) (cid:105) ρ kl ( t ) , ( k (cid:54) = l ) . (42)Now we have a set of (de-coupled) linear ordinary differential equations (ODE’s) , which is easily solvable, ρ kk ( t ) = ρ kk ( t ) , (43) ρ kl ( t ) = ρ kl ( t ) exp t ˆ t dt (cid:48) (cid:104) − ia ( t (cid:48) )( E k − E l ) + D ( t (cid:48) )( E k − E l ) (cid:105) = ρ kl ( t ) exp (cid:26) − i ( E k − E l ) v ( t ) − ( E k − E l ) v ( t ) − v ( t ) ] (cid:27) . (44)The second equality in Eq.(44) is obtained after some calculation, where v ( t ) isgiven by Eq.(32). Thus we find the dynamics generated by the 2nd-order masterequation coincides with the exact dynamics in this case.We observe that the energy population is conserved during the evolutionwhile the coherence between different energy levels decays. Thus the evolutionof the average dynamics is pure decoherence, with the “pointer basis” [12] beingthe energy-eigenbasis. Note that, although the Hamiltonian varies with time andacross different realizations, the energy-eigenbasis is the same throughout. Inthe case where some energy level is degenerate, we readily have a “decoherence-free subspace”, in which quantum information can be stably stored [14]. Inci-dentally, a derivation in the context of open systems also suggests that energyeigenstates can emerge as “pointer states” in the so-called “quantum limit ofdecoherence” [15]. In that case, however, the decoherence results from small en-vironmental perturbation to the system, not from the fluctuation of the systemHamiltonian itself.We could have worked out the higher-order terms (i.e. L n [ ρ ] for n (cid:62) )explicitly to see how accurate the 2nd-order approximation is. However, sincethe solution to the 2nd-order master equation coincides with the exact dynamics,we can readily conclude that all higher-order terms must sum up to zero withoutactually carrying out further calculations.Note that when deriving Eq.(39) we do not assume anything about the natureof the random process a ( t ) , not even the Gaussian statistics. In other words, thesolution to the 2nd-order master equation is given by Eqs.(43-44) in all cases.On the other hand, the exact dynamics Eqs.(36-37) is based on the assumptionof Gaussian statistics. If a ( t ) is not a Gaussian random process, then the exactdynamics will be different. The implication is that, for a ( t ) being non-Gaussian,the 2nd-order master equation is not exact. It may not be exactly solvable, but we know for sure that the solution is different fromthat in the Gaussian case. .3 Multiple jointly circular complex Gaussian randomprocesses Let us briefly present the results for the more general Hamiltonian ˆ H ( t ) = (cid:126) (cid:80) n (cid:16) a n ( t )ˆ h n + a ∗ n ( t )ˆ h † n (cid:17) , where a n ( t ) are jointly circular complex Gaussianrandom processes of zero mean. The 2nd-order master equation in Lindbladform is found to be ∂∂t ρ ( t ) = − (cid:88) k,l α kl ( t ) (cid:104) [ˆ h k , ˆ h † l ] , ρ ( t ) (cid:105) + (cid:88) k,l β kl ( t ) × (cid:110) − ˆ h k ˆ h † l ρ ( t ) − ρ ( t )ˆ h k ˆ h † l + 2ˆ h † l ρ ( t )ˆ h k − ˆ h † l ˆ h k ρ ( t ) − ρ ( t )ˆ h † l ˆ h k + 2ˆ h k ρ ( t )ˆ h † l (cid:111) , (45)where α kl ( t ) ≡ t ˆ t dt (cid:48) (cid:16) a k ( t ) a ∗ l ( t (cid:48) ) − a ∗ l ( t ) a k ( t (cid:48) ) (cid:17) , (46) β kl ( t ) ≡ t ˆ t dt (cid:48) (cid:16) a k ( t ) a ∗ l ( t (cid:48) ) + a ∗ l ( t ) a k ( t (cid:48) ) (cid:17) . (47)By comparing with Eq.(29) for the time-independent Hamiltonian case, we no-tice a major difference in this case is that the effective Hamiltonian is non-zerodespite ˆ H ( t ) = 0 . This effective unitary evolution results from the fact thatthe Hamiltonian operators at different times do not commute with each otherin general.The 2nd-order master equation yields exact dynamics only for the specialcase of a single real Gaussian random process. In this more general case, Eq.(45)does not lead to exact dynamics in general. This can be shown by explicitlyevaluating higher-order terms like L [ ρ ] to find that they do not vanish in gen-eral. Despite the lack of perfect agreement, the master equation is neverthelessof great use in such cases, because the exact dynamics is generally not obtain-able and the 2nd-order master equation serves as a good approximation whenthe higher-order terms (e.g. L [ ρ ] ) are small compared to L [ ρ ] . We will illustrate the general results by applying them to a few examples. Thefindings will then be used to gain physical insights, and the validity of the masterequation approach will be examined by comparing to the exact dynamics.10 .1 Two-level system
First consider a two-level system (e.g. spin-1/2) subject to the Hamiltonian ˆ H ( t ) = (cid:126) ω ( t ) ˆ Z , where ˆ Z is the z -component of Pauli operator and ω ( t ) astationary Gaussian random process of zero mean. This falls into the categoryof Hamiltonians (30). Using Eq.(39), the 2nd-order master equation is obtained, ∂∂t ρ ( t ) = − d ( t ) (cid:104) ˆ Z, [ ˆ Z, ρ ( t )] (cid:105) , (48)where d ( t ) ≡ ´ tt dt (cid:48) ω ( t ) ω ( t (cid:48) ) . Assuming an auto-correlation function of theform ω ( t ) ω ( t (cid:48) ) = ω exp ( −| t − t (cid:48) | /T ) , where ω ≡ ω ( t ) , we have d ( t ) =4 ω T (cid:0) − e − ( t − t ) /T (cid:1) for t > t .Written in the ˆ Z -eigenbasis {| (cid:105) , | (cid:105)} , the master equation becomes a set oflinear ODE’s: ∂∂t ρ ( t ) = 0 , (49) ∂∂t ρ ( t ) = 0 , (50) ∂∂t ρ ( t ) = − d ( t ) ρ ( t ) , (51) ∂∂t ρ ( t ) = − d ( t ) ρ ( t ); (52)the solutions to which are ρ ( t ) = ρ ( t ) , (53) ρ ( t ) = ρ ( t ) , (54) ρ ( t ) = ρ ( t ) exp (cid:26) − ω T (cid:18) t − t T + e − ( t − t ) /T − (cid:19)(cid:27) , (55) ρ ( t ) = ρ ( t ) exp (cid:26) − ω T (cid:18) t − t T + e − ( t − t ) /T − (cid:19)(cid:27) . (56)As already discussed in the general case of a single Gaussian random process,the energy population remains constant while the coherence decays. This canbe understood from a more physical perspective. Quantum coherence dependson the relative phase between the two components | (cid:105) and | (cid:105) . In an individualrealization, the relative phase factor is exp (cid:110) i ´ tt dt (cid:48) ω ( t (cid:48) ) (cid:111) . Since ω ( t ) is a ran-dom process, the quantity ´ tt dt (cid:48) ω ( t (cid:48) ) becomes increasingly randomized with thepassage of time. When the average is taken over an ensemble, these randomlydistributed relative phase factors cancel out, thus suppressing the coherence.This suggests that quantum interference is difficult to observe because randomfluctuation is ubiquitous. 11s has been shown in the more general case Eqs.(43-44), the 2nd-order mas-ter equation gives exact dynamics. When we work out the exact dynamicsdirectly, the result is indeed found to be consistent, though such a direct calcu-lation is more demanding. Clearly, calculational convenience is one advantageof the master equation approach. Next consider an example of two atoms in magnetic field, each atom beinga two-level system. The interaction of the spin with the B-field is given by ˆ H ( t ) = (cid:126) Ω( t ) (cid:16) ˆ Z I ⊗ ˆ I II + ˆ I I ⊗ ˆ Z II (cid:17) ≡ (cid:126) Ω( t ) ˆ Z total , where ˆ Z j is the usual Pauli z -operator of the j -th atom. Suppose that the frequency Ω( t ) , which is propor-tional to the B-field strength, is a stationary Gaussian random process of zeromean and that its auto-correlation is of a Markovian type Ω( t )Ω( t (cid:48) ) = γδ ( t − t (cid:48) ) ,where the constant γ has dimension of inverse-time. Applying Eq.(39), it canbe shown that the 2nd-order master equation is ∂∂t ρ ( t ) = − γ (cid:104) ˆ Z total , [ ˆ Z total , ρ ( t )] (cid:105) . (57)Suppose the system starts in an entangled state between two atoms | Ψ( t ) (cid:105) = √ ( | (cid:105) + | (cid:105) ) . Since it is an eigenstate of ˆ Z total , the right-hand side of Eq.(57)is identically zero, thus the system does not evolve (except possibly to an unob-servable global phase). So the two atoms remain entangled over time. Indeed,notice that | (cid:105) and | (cid:105) are degenerate eigenstates with the same energy. Thusany arbitrary superposition state of | (cid:105) and | (cid:105) will remain unchanged overtime; in particular, the coherence between them does not decay. Thus, any statein this degenerate subspace is immune to decoherence, making it a good placeto store quantum information [14].Let us see what happens if the 2-atom system starts in a different entangledstate like | Ψ( t ) (cid:105) = √ ( | (cid:105) + | (cid:105) ) . Writing the master equation in the energyeigenbasis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , we obtain a set of decoupled linear ODE’s asusual. The solutions are found to be ρ , ( t ) = ρ , ( t ) = 12 , (58) ρ , ( t ) = ρ , ( t ) = 12 exp {− γ ( t − t ) } , (59)while the rest of the matrix elements are identically zero. Note that decoherenceoccurs here, as is expected, since the initial state does not lie in the decoherence-free subspace. Furthermore, as t → ∞ , the coherence is suppressed to zero and ρ ( t ) → | (cid:105)(cid:104) | + | (cid:105)(cid:104) | = | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | . Interestingly,the two atoms become disentangled , as there is no quantum correlation betweenthem. In contrast to the general belief that entanglement leads to decoherence,as is widely studied for open quantum systems, here we find that decoherencecan result in disentanglement in the case of a closed system.12 .3 Heating of a trapped ion Consider an ion with mass M and charge e in a harmonic binding potentialwith characteristic frequency ω . The ion is driven by a classical electric field E ( t ) , which is a stationary Gaussian random process of zero mean. It is moreconvenient to work in the interaction picture, in which the easily solvable, de-terministic evolution induced by the harmonic potential is treated separately.The interaction-picture Hamiltonian is given by ˆ H ( t ) = i (cid:126) (cid:2) u ( t )ˆ a † − u ∗ ( t )ˆ a (cid:3) ,where u ( t ) = ieE ( t ) e iω t / √ M (cid:126) ω and ˆ a (ˆ a † ) being the zero-time annihilation(creation) operator for the harmonic oscillator. The evolution is exactly solvableand the analytic results are given in [11].Let us derive the 2nd-order master equation for this case. Note that it doesnot fall into the category of a single real Gaussian random process as in Eq.(30).Since ˆ H = 0 , applying Eq.(25), the effective Hamiltonian is found to be ˆ H eff = 12 t ˆ t dt (cid:48) (cid:16) u ( t ) u ∗ ( t (cid:48) ) − u ∗ ( t ) u ( t (cid:48) ) (cid:17) [ˆ a, ˆ a † ]= − e M ω t ˆ t dt (cid:48) E ( t ) E ( t (cid:48) ) sin [ ω ( t − t (cid:48) )] ˆ I. (60)In this case, the 2nd-order contribution to the effective Hamiltonian is non-zero, a consequence of the non-commutativity of ˆ H ( t ) and ˆ U ( t ) . However,since ˆ H eff is proportional to ˆ I , the unitary part of the equation of motionresults only in an unobservable global phase in this case. For more generalcases, ˆ H eff can be different from the identity ˆ I and can well lead to non-trivialdynamics. Evaluating the remaining terms in Eq.(23) for this example, we findthe following master equation: ∂∂t ρ ( t ) = −C ( t ) (cid:0) ˆ a † ˆ aρ ( t ) + ρ ( t )ˆ a † ˆ a − aρ ( t )ˆ a † (cid:1) − C ( t ) (cid:0) ˆ a ˆ a † ρ ( t ) + ρ ( t )ˆ a ˆ a † − a † ρ ( t )ˆ a (cid:1) − e iω t [ C ( t ) − i S ( t )] (cid:0) (ˆ a † ) ρ ( t ) + ρ ( t )(ˆ a † ) − a † ρ ( t )ˆ a † (cid:1) − e − iω t [ C ( t ) + i S ( t )] (cid:0) ˆ a ρ ( t ) + ρ ( t )ˆ a − aρ ( t )ˆ a (cid:1) , (61)where C ( t ) ( S ( t ) ) are proportional to the incomplete cosine (sine) transform of Throughout this section we work in the interaction picture. The subscripts to denoteinteraction-picture operators are dropped for notational simplicity. C ( t ) ≡ e M (cid:126) ω t ˆ t dt (cid:48) E ( t ) E ( t (cid:48) ) cos [ ω ( t − t (cid:48) )] , (62) S ( t ) ≡ e M (cid:126) ω t ˆ t dt (cid:48) E ( t ) E ( t (cid:48) ) sin [ ω ( t − t (cid:48) )] . (63)Assuming E ( t ) E ( t (cid:48) ) = E (0) exp ( −| t − t (cid:48) | /T ) and setting t = 0 for conve-nience, we find C ( t ) = (1 / τ ) (cid:8) e − t/T [ ω T sin( ω t ) − cos( ω t )] + 1 (cid:9) and S ( t ) = − (1 / τ ) (cid:8) e − t/T [sin( ω t ) + ω T cos( ω t )] − ω T (cid:9) , where τ is the heating timedefined as /τ = (cid:16) e E (0) /M (cid:126) ω (cid:17) (cid:0) T / (1 + ω T ) (cid:1) .Unlike the case of Eq.(30), the 2nd-order master equation does not generateexact dynamics in this case. To get an approximation of the heating from theground state (i.e. ρ (0) = 1 ) for a short period of time, let us write the masterequation in the energy eigenbasis of the harmonic oscillator, ∂∂t ρ ( t ) = − C ( t ) ρ ( t ) + 2 C ( t ) ρ ( t ) − √ e iω t [ C ( t ) − i S ( t )] ρ ( t ) − √ e − iω t [ C ( t ) + i S ( t )] ρ ( t ) . (64)Since ρ ( t ) , ρ ( t ) and ρ ( t ) are all negligibly small for t (cid:28) T, /ω , wehave ∂∂t ρ ( t ) ∼ = − C ( t ) ρ ( t ) to the lowest order. In the same manner, sincethe depopulation − ρ ( t ) is perturbatively small, we have ρ ( t ) ∼ = 1 to thelowest order on the right-hand side. Thus an approximate differential equationis obtained as ∂∂t ρ ( t ) ∼ = − C ( t ) . Solving this ODE, we find, to lowest order, − ρ ( t ) ∼ = 2 t ˆ dt (cid:48) C ( t (cid:48) ) ∼ = e E (0) M (cid:126) ω t , (65)which holds for short times and agrees with the analytic result in [11].To investigate the evolution of the system for longer times, we write downthe master equation in the same basis and solve it numerically. Since the Hilbertspace is of infinite dimensions, it is not possible to write down the complete setof ODE’s for the matrix elements. Instead, we truncate it to a set of × coupled ODE’s that includes only the matrix elements of the five lowest energy-eigenstates and their coherence. The numerical solutions of F ( t ) ≡ ρ ( t ) The same result can be obtained by working out the evolution of ρ ( t ) for t (cid:28) T, /ω using approximation to the same order. Since the system is of continuous nature, we could have solved the master equation inthe Wigner representation. However, that approach is not analytically solvable either and isnot computationally economical. Furthermore, even in that case, we still have to accept theimperfection of truncation since the numerics can only be done on a finite region of the “phasespace”.
The fidelity of the ground state as a function of dimensionless time ω t . Dash linesrepresent our numerical results, while solid lines are exact dynamics from [11]. (i.e. fidelity of the ground state) are shown in Figure 1 for different sets ofparameters. It can be seen that, as ω τ (i.e. the dimensionless heating time)increases with ω T (i.e. the dimensionless coherence time of E ( t ) E ( t (cid:48) ) ) fixed,the numerical result gives better approximation to the exact dynamics. Alsonote that, for larger values of ω T , the ground state population shows temporaryrevival against its general trend of decrease.Despite the artificial defect caused by the truncation of the set of ODE’s, itis of more interest to know the validity of the 2nd-order master equation itselfin approximating the exact dynamics. This is done by comparing the size ofhigher-order terms to that of the 2nd-order term. Using the Gaussian momenttheorem[10], it is easy to show that L n [ ρ ] = 0 for all odd numbers n , so we areinterested in the ratios between the even-number-order terms. Assuming ω T is fixed, it can be shown that L [ ρ ] ∝ /τ ω as opposed to L [ ρ ] ∝ /τ , so L [ ρ ] / L [ ρ ] ∝ /ω τ . The same ratio holds for L [ ρ ] / L [ ρ ] , etc. Therefore,as long as /ω τ is small, the higher-order terms become progressively small,lending legitimacy to the 2nd-order approximation. This is also consistent withthe previous observation from the numerical results. Physically, this can bebetter understood by switching to the Schrà ¶ dinger picture: The external field H field ∝ /τ is treated as a perturbation to the self-Hamiltonian of the sys-tem H self ∝ ω . Naturally, as the relative size of the perturbing Hamiltonian15 self /H field ∝ /ω τ becomes smaller, a perturbative method such as the2nd-order master equation gives better approximation to the exact dynamics. In this paper we have presented the derivation of a master equation for closedsystems driven by stochastic Hamiltonians from an ensemble-average perspec-tive. The principal result is given in Eqs.(25-26). The validity of this approach isexamined and 2nd-order master equation is found to yield either exact dynamicsor good approximations to exact dynamics.Applying the formalism to various physical examples, we find the ensemble-average dynamics usually contains decoherence terms in addition to the unitaryevolution. Decoherence plays an important role in the foundational problemsof quantum mechanics, as it gives insights in two aspects of the measurementproblem, namely the absence of observable superposition and the problem ofpreferred basis [12]. Extensive research has been done on how environmentalentanglement causes decoherence in open systems. However, as our findings sug-gest, decoherence could also be attributed to the random fluctuations of physicalquantities in closed systems. If this is true, then the tension between the classi-cality of our experience and the quantumness of the underlying laws of physicscould be reconciled in some degree by the ubiquitous random fluctuations. Fur-ther investigation is needed to find out (a) to what extent decoherence is actuallycaused by random fluctuations and (b) whether/how we can distinguish it fromthe usual entanglement-induced decoherence through physical observation.
Acknowledgements
The authors would like to thank O. Gamel for valuable discussions and C.-H.Chang for comments on the manuscript. This work is supported by Natural Sci-ences and Engineering Research Council of Canada (NSERC) through CREATEand by University of Toronto through UTEA-NSE.
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