Time-dependent pointer states of the quantized atom-field model in a nonresonance regime and consequences regarding the decoherence of the central system
aa r X i v : . [ qu a n t - ph ] A p r Time-dependent pointer states of the quantizedatom-field model in a nonresonance regime andconsequences regarding the decoherence of thecentral system
Hoofar Daneshvar and G W F Drake
Department of Physics, University of Windsor, Windsor ON, N9B 3P4, CanadaE-mail: [email protected] and [email protected]
Abstract.
We consider the quantized atom-field model and for the regime thatˆ H E ≪ ˆ H S ≪ ˆ H ′ (but ˆ H E = 0 and ˆ H S = 0); where ˆ H E , ˆ H S and ˆ H ′ respectivelyrepresent the self Hamiltonians of the environment and the system, and the interactionbetween the system and the environment. Considering a single-mode quantized field weobtain the time-evolution operator for the model. Using our time-evolution operator wecalculate the time-dependent pointer states of the system and the environment (whichare characterized by their ability not to entangle with states of another subsystem)by assuming an initial state of the environment in the form of a Gaussian package inposition space. We obtain a closed form for the offdiagonal element of the reduceddensity matrix of the system and study the decoherence of the central system in ourmodel. We will show that for the case that the system initially is not prepared in one ofits pointer states, the offdiagonal element of the reduced density matrix of the systemwill decay with a decoherence time which is inversely proportional to the square rootof the mass of the bosonic field particles. Keywords : Foundation of quantum mechanics, Atom-field model, Decoherence, Pointerstates of measurement. ime-dependent pointer states of the quantized atom-field model in a nonresonance regime
1. Introduction
In our previous paper [1] we discussed the pointer states of measurement ‡ and wepresented a general method for obtaining the pointer states of a two-level system and itsenvironment, for a given total-Hamiltonian defining the system-environment model § . Aswe discussed in this paper, time-independence of pointer states by no means should betaken for granted; since time-independent pointer states can be realized only under somespecific conditions [1]. We used our method in order to rederive the time-dependentpointer states of the system and the environment (initially prepared in the coherentstate) in the Jaynes-Cummings model (JCM) of quantum optics and for the exactresonance regime; verifying the previous results for the JCM [2, 3]. Also, to furtherdemonstrate the generality and usefulness of our method of obtaining pointer states, inanother paper [4] we obtained the time-dependent pointer states of the system and theenvironment for the generalized spin-boson model and in the exact resonance regime.In this paper we study the quantized atom-field model without the assumption ofresonance between the splitting of the states of the two-level atom ω and the cavityeigenmode frequency ω . Our quantized atom-field model basically is consisted of atwo-level atom, with upper and lower levels that can respectively be represented by | a i and | b i , interacting with a single-mode quantized bosonic field (such as photons)inside an ideal cavity, represented by creation and annihilation operators ˆ a † and ˆ a . TheHamiltonian for the total composite system can be written as [5]ˆ H = 12 ω ˆ σ z + ω ˆ a † ˆ a + g χ ˆ σ x ˆ x, (1)Where g = − ̺ . ǫ q ω hε ◦ V is the atom-field coupling constant, with ̺ = e h a | r | b i as theatomic electric-dipole transition matrix element. ( ǫ is the field polarization vector andV is the cavity mode volume). Also χ = √ mω ; so that χ ˆ x = ˆ a + ˆ a † . k The main purpose of this paper is to obtain the time-dependent pointer states ofthe system and the environment, as well as expressions for the evolution of the reduceddensity matrix of the system in the regime that ˆ H E ≪ ˆ H S ≪ ˆ H ′ , but ˆ H S = 0 and ‡ The pointer states of a subsystem are characterized by their ability not to entangle with the states ofanother subsystem and appear in the diagonal state of the total composite system after premeasurementby the environment. As we elaborately described in [1], generally we should distinguish between thepointer states of a system and the preferred basis of measurement. We proved that the pointer states ofa subsystem generally are time-dependent and a preferred basis of measurement does not exist, unlessunder some specific conditions (discussed there in [1]) for which the pointer states of measurementbecome time-independent. Moreover, the pointer states of a system necessarily are not orthonormalamongst themselves at all times. Therefore, necessarily they cannot form a basis for the Hilbert spaceof the system at all times. § Although a reference to our other works ([1] and [4]) can be useful for the interested reader, in writingthis paper we have tried to make it self-contained; so that the reader can well understand this workwithout a need to refer to the other two works. k Here in this paper we use the atomic units wherein ¯ h = 1. ime-dependent pointer states of the quantized atom-field model in a nonresonance regime H E = 0. In other words, to demonstrate how our formulation for obtaining time-dependent pointer states can be used in practice, here we consider a very specificregime of the parameter space and will obtain the corresponding pointer states of thesystem and the environment within that specific regime; as pointer states ( if they existin certain regimes of a system-environment model) generally depend on the specificregime of the parameter space which we are considering and generally acquire differentforms in different regimes of the parameter space, even for a specifically given system-environment Hamiltonian.For the Hamiltonian of equation (1), as we will show here, the special regime ofˆ H E ≪ ˆ H S ≪ ˆ H ′ is valid only and only if we have1 ≪ r ω ω ≪ | ̺ . ǫ | × s m ¯ hε ◦ V , ω = 0 and ω = 0 . (2)To show this, note that the condition ˆ H S ≪ ˆ H ′ requires that q ω ω ≪ | ̺ . ǫ | × q m ¯ hε ◦ V ;while the condition ˆ H E ≪ ˆ H S requires that 1 ≪ q ω ω . Also, we should emphasize thatwe must have ω = 0 ( ˆ H E = 0) and ω = 0; as otherwise we would have zero couplingg χ , and we cannot have ˆ H S ≪ ˆ H ′ (since we have g χ = − ̺ . ǫ q mωω ¯ hε ◦ V ). Therefore, as wesee, the regime that we are considering and the results of this article are valid only inthe specific part of the parameter space where the inequalities of equation (2) are valid.This paper is organized as follows:After this foreword we review our method for obtaining the pointer states of thesystem and the environment; and in section 3 we exploit it in order to calculate thetime-dependent pointer states of the quantized atom-field model represented by theHamiltonian of equation (1).In order to be able to exploit our method and obtain the pointer states of thesystem and the environment in our model, we need to know the time-evolution operatorof our model in the regime that we are considering. This task is done in section 2.In section 4 we exploit the pointer states of the system and the environment (whichwe obtain in section 3) in order to study the decoherence of the central system in ourmodel. Finally, in section 5 we further discuss the significance of our results and theconclusions. In order to be able to obtain the pointer states of the system and the environment for anarbitrary total Hamiltonian defining our system-environment model we first need to findthose probable initial states of the system which do not entangle with the states of theenvironment throughout their evolution with time; and then we should obtain their timeevolution. Finally, we should obtain their corresponding states from the environmentwhich in fact, are the pointer states of the environment. As we saw in [1], existenceof pointer states may require having a sufficiently large environment which contains alarge number of degrees of freedom. In other words, pointer states characterized by ime-dependent pointer states of the quantized atom-field model in a nonresonance regime S with two arbitrary basis states | a i and | b i , initiallyprepared in the state | ψ S ( t ) i = α | a i + β | b i with | α | + | β | = 1; (3)and an environment initially prepared in the state | Φ E ( t ) i = ∞ X n =0 c n | ϕ n i , (4)where {| ϕ n i} ’s are a complete set of basis states for the environment. For the two-statesystem with the two basis states | a i and | b i we can take the set of any four linearlyindependent operators in the Hilbert space of the system as a complete set of basisoperators, which can induce any change to the initial state of the two-state systemgiven by equation (3). For example, we can take the Pauli operators in addition to theidentity operator ˆ I = | a ih a | + | b ih b | as our complete set of basis operators; or equivalentlywe can take the four operators | a ih a | , | a ih b | , | b ih a | and | b ih b | as our complete set of basisoperators. So, the time evolution operator for the global state of the system and theenvironment, which (for a two-state system) generally is of the formˆ U tot ( t ) = X α =1 ˆ S α ⊗ ˆ E α , (5)Can be considered asˆ U tot ( t ) = ˆ E | a ih a | + ˆ E | a ih b | + ˆ E | b ih a | + ˆ E | b ih b | . (6)In the above equation ˆ E i ’s are operators acting on the Hilbert space of the environment,and depend on the total Hamiltonian defining the system-environment model. Forexample, for the Jaynes-Cummings model and for exact resonance and in the rotatingwave approximation (RWA), it can be shown [5] that the ˆ E i ’s are given by the followingrelations ˆ E = cos(g t q ˆ a † ˆ a + 1) , ˆ E = − i sin(g t √ ˆ a † ˆ a + 1 √ ˆ a † ˆ a + 1 ˆ a )ˆ E = − i ˆ a † sin(g t √ ˆ a † ˆ a + 1 √ ˆ a † ˆ a + 1 ) , ˆ E = cos(g t √ ˆ a † ˆ a ) . (7)Using equations (3) to (6) we can write the global state of the system and theenvironment as follows | Ψ tot ( t ) i = ˆ U tot ( t ) . ( α | a i + β | b i ) ⊗ ( ∞ X n =0 c n | ϕ n i )= A ( t ) | a i + B ( t ) | b i with A ( t ) = ∞ X n =0 c n { α ˆ E + β ˆ E } | ϕ n i (8)and B ( t ) = ∞ X n =0 c n { α ˆ E + β ˆ E } | ϕ n i . ime-dependent pointer states of the quantized atom-field model in a nonresonance regime G ( t ) as the operator in the Hilbert spaceof the environment which relates the vectors A ( t ) and B ( t ) to each other A ( t ) = ˆ G ( t ) B ( t ) or X n c n { α ˆ E + β ˆ E } | ϕ n i = ˆ G ( t ) X n c n { α ˆ E + β ˆ E } | ϕ n i . (9)Now, for the global state of the system and the environment, which is given by | Ψ tot ( t ) i = A ( t ) | a i + B ( t ) | b i = ˆ G ( t ) B ( t ) | a i + B ( t ) | b i = { ˆ G ( t ) | a i + | b i} × ( ∞ X n =0 c n { α ˆ E + β ˆ E } | ϕ n i ) , (10)we observe that if for some initial states of the system and the environment ˆ G ( t ) turnsout to become in the formˆ G ( t ) = G ( t ) × ˆ I E , (11)with G ( t ) as a scalar (rather than an operator) and ˆ I E representing the identity operatorin the Hilbert space of the environment, then those initial states of the system and theenvironment will not entangle with each other, and hence they can represent the initialpointer states of the system and the environment. This result simply is because ofthe fact that if for some initial states of the system and the environment ˆ G ( t ) turnsout to become a scalar in the form of equation (11), G ( t ) will be independent of theindices of the environment (i.e. independent of n ); as in this case all components of B ( t ) will be mapped into their corresponding components from A ( t ) through the same scalar function G ( t ) (which will keep the two vectors A ( t ) and B ( t ) parallel to eachother). Therefore, in this case ˆ G ( t ) will not enter the summation in the expression P n c n { α ˆ E + β ˆ E } | ϕ n i of equation (10); and (as one can see from equation (10)) thestates of the system and the environment respectively represented by { G ( t ) | a i + | b i} and P n c n { α ˆ E + β ˆ E } | ϕ n i will not entangle to each other.In another word, if for some initial states of the system and the environment A ( t ) = ˆ G ( t ) B ( t ) is equal to G B ( t ), it means that for those initial states of the systemand the environment B ( t ) becomes an eigenstate of the operator ˆ G ( t ); and the twovectors A ( t ) and B ( t ) will stay parallel with each other throughout their evolution withtime; and as we discussed, in this case the states of the system and the environmentwill not entangle with each other and (as one can see from equation (10)) pointer statescan be realized for the system and the environment given by | ± ( t ) i = N ± { G ( t ) | a i + | b i} and | Φ ± ( t ) i = N − ± ( ∞ X n =0 c n { α ˆ E + β ˆ E } | ϕ n i ) . (12)In the above equation we have represented the pointer states of the system by | ± ( t ) i and those of the environment by | Φ ± ( t ) i . Also, N ± is the normalization factor for thepointer states of the system (clearly N ± = √ if | G ( t ) | = 1, as for the example of theJCM in the exact-resonance regime). ime-dependent pointer states of the quantized atom-field model in a nonresonance regime necessary condition forobtaining pointer states; since unless ˆ G ( t ) turns out to become a scalar, the two vectors A ( t ) = ˆ G ( t ) × B ( t ) and B ( t ) will not be parallel at all times and the operator ˆ G ( t ) willenter the summation over the environmental degrees of freedom (i.e. the summation over n ) in equation (10), in which case no longer the states of the system and the environmentwill be separable in a tensor product form; and pointer states cannot be realized for thestates of the system and the environment. Also, as we discussed in [4], generally there isno guaranty for the condition (11) to be satisfied; and satisfaction of this condition oftenmay require having a sufficiently large environment which contains a large number ofdegrees of freedom. However, if in some regime and for a given Hamiltonian defining asystem-environment model we can find initial states for the system and the environmentwhich satisfy this condition, we do know that pointer states can be realized for thesystem and the environment and those initial states would correspond to the initialpointer states of the system and the environment.In essence, in order to find the pointer states of the system and the environmentfor a given total Hamiltonian defining our system-environment model, and for a giveninitial state of the environment, our main goal would be finding those possible initialstates of the system for which ˆ G ( t ) (which is defined through equation (9)) is of the formof equation (11). In section 3 considering the quantized atom-field model represented bythe Hamiltonian of equation (1) and for the regime that ˆ H E ≪ ˆ H S ≪ ˆ H ′ (but ˆ H E = 0and ˆ H S = 0), we exploit this method to obtain the time-dependent pointer states ofthe system and the environment; by assuming an initial state of the environment inthe form of a Gaussian package in position space. As we will see, once we have thetime-evolution operator for our system-environment model in the form of equation (6)and the ˆ E i operators, this task can be done quite easily for our model.
2. Calculation of the time-evolution operator
In order to calculate the time-evolution operator in the interaction picture for theHamiltonian of equation (1), first we need to have the Hamiltonian in the interactionpicture, which is defined through the following equationˆ H int ( t ) = e i ˆ H t ˆ H ′ e − i ˆ H t . (13)Here ˆ H = ω ˆ σ z + ω ˆ a † ˆ a is the sum of the self Hamiltonians of the system and theenvironment; and ˆ H ′ = g χ ˆ σ x ˆ x is the Hamiltonian for the interaction between the systemand the environment. So, now we must calculateˆ H int ( t ) = g ( e iω ˆ σ z t/ ˆ σ x e − iω ˆ σ z t/ ) ⊗ ( e iω ˆ a † ˆ at χ ˆ x e − iω ˆ a † ˆ at ) , (14)where χ ˆ x = ˆ a + ˆ a † . However, ˆ σ x = ˆ σ + + ˆ σ − ; and e iω ˆ σ z t/ ˆ σ ± e − iω ˆ σ z t/ = ˆ σ ± e ± iω t . Alsoˆ a ( t ) = ˆ ae − iωt . Soˆ H int ( t ) = g(ˆ σ + e iω t + ˆ σ − e − iω t ) ⊗ (ˆ ae − iωt + ˆ a † e iωt )= g { ˆ σ + (ˆ a e i ∆ t + ˆ a † e i ( ω + ω ) t ) + c.c. } , with ∆ = ω − ω. (15) ime-dependent pointer states of the quantized atom-field model in a nonresonance regime i ∂∂t ˆ U ( t ) = ˆ H int ˆ U ( t ) , (16)we have i ˙ˆ E ˙ˆ E ˙ˆ E ˙ˆ E = ˆ H int ( t ) ˆ E ˆ E ˆ E ˆ E ! = g a e i ∆ t + ˆ a † e i ( ω + ω ) t ˆ a † e − i ∆ t + ˆ a e − i ( ω + ω ) t ! ˆ E ˆ E ˆ E ˆ E ! (17)= g (ˆ a e i ∆ t + ˆ a † e i ( ω + ω ) t ) ˆ E (ˆ a e i ∆ t + ˆ a † e i ( ω + ω ) t ) ˆ E (ˆ a † e − i ∆ t + ˆ a e − i ( ω + ω ) t ) ˆ E (ˆ a † e − i ∆ t + ˆ a e − i ( ω + ω ) t ) ˆ E ! . Now, we assume ω ≪ ω ; so that ∆ ≈ ω and ω + ω ≈ ω . In otherwords, in the Hamiltonian of our total composite system, given by equation (1), weassume that the self-Hamiltonian of the system dominates the self-Hamiltonian of theenvironment. Therefore, equation (17) for the evolution of the time-evolution operatorcan be simplified to the following set of four equations i ˙ˆ E = g χ ˆ x e iω t ˆ E ,i ˙ˆ E = g χ ˆ x e iω t ˆ E ,i ˙ˆ E = g χ ˆ x e − iω t ˆ E , (18) i ˙ˆ E = g χ ˆ x e − iω t ˆ E . In order to solve the above set of coupled differential equations, we proceed asfollows. First, we take derivative with respect to time of the first equation. By replacing˙ˆ E from the third equation in the resulting equation we find¨ˆ E = − (g χ ˆ x ) ˆ E + (g χ ˆ xω e iω t ) ˆ E . (19)Similarly, by doing the same procedure on the third equation for ˙ˆ E we find¨ˆ E = − (g χ ˆ x ) ˆ E − (g χ ˆ xω e − iω t ) ˆ E . (20)One can easily verify that if ω ≪ (g χ ) (i.e. if ˆ H S ≪ ˆ H ′ ), so that (g χ ˆ x ) + ω / ≈ (g χ ˆ x ) , the following solutions will satisfy the differential equations given by equations(19) and (20) for ˆ E and ˆ E :ˆ E = cos(g χ ˆ xt ) e iω t/ and ˆ E = − i sin(g χ ˆ xt ) e − iω t/ . (21)In quite the same manner we can calculate ˆ E and ˆ E as followsˆ E = − i sin(g χ ˆ xt ) e iω t/ and ˆ E = cos(g χ ˆ xt ) e − iω t/ . (22)The above operators together with equation (6) make the time-evolution operatorof the quantized atom-field model and for the regime that ˆ H E ≪ ˆ H S ≪ ˆ H ′ , but ˆ H S = 0and ˆ H E = 0. One can easily verify that the above set of operators satisfies the unitarity of ime-dependent pointer states of the quantized atom-field model in a nonresonance regime U † ˆ U = ˆ U ˆ U † = ˆ I (with ˆ I representing the identityoperator). Moreover, ˆ E (0) = ˆ E (0) = 1 and ˆ E (0) = ˆ E (0) = 0. So, these operators dosatisfy the initial condition for the time-evolution operator given by ˆ U tot ( t ) = ˆ I .
3. Calculation of the time-dependent pointer states of the system and theenvironment
Using the time-evolution operator which we already obtained for our model and forthe regime that ˆ H E ≪ ˆ H S ≪ ˆ H ′ (but ˆ H S = 0 and ˆ H E = 0), now we canobtain the corresponding pointer states of the system and the environment in thisregime. For this purpose we assume that the system initially is prepared in the state | ψ S ( t ) i = α | a i + β | b i . Moreover, let us assume that the initial state of the environmentcan be represented by a Gaussian package in the position space | Φ E ( t ) i = N Z ∞−∞ dx e − α ◦ x | x i , (23)where N = (2 α ◦ /π ) / is the normalization factor for this state. Now, the condition fordetermining the pointer states of the system and the environment, given by equations(9) and (11), reads( α ˆ E + β ˆ E ) | Φ E ( t ) i = ˆ G ( t ) × ( α ˆ E + β ˆ E ) | Φ E ( t ) i ; with ˆ G ( t ) being proportional to the unit matrix. (24)(In other words, for an initial state of the system corresponding to one of its pointerstates at t = t , the operator ˆ G ( t ) must be independent of the indices of the environment.i.e. x ). Inserting the ˆ E i ’s from equations (21) and (22) into the above condition it reads Z ∞−∞ dx [ α cos(g χxt ) − iβ sin(g χxt )] e − α ◦ x + iω t/ | x i = ˆ G ( t ) × Z ∞−∞ dx [ − iα sin(g χxt ) + β cos(g χxt )] e − α ◦ x − iω t/ | x i (25) and ˆ G ( t ) be proportional to the unit matrix. For pointer states ˆ G ( t ) must satisfy the condition (11) for obtaining the pointer statesof the system and the environment, i.e. ˆ G ( t ) = G ( t ) × ˆ I E . Therefore, since the set {| x i} is a complete set of basis states for the environment, for initial pointer states we cansimply equalize those terms from the two sides of equation (25) which correspond to thesame | x i state and obtain G ( t ) = α cos(g χxt ) − iβ sin(g χxt ) − iα sin(g χxt ) + β cos(g χxt ) e iω t . (26)The above result for G ( t ), which generally depends on x , would contradict our initialassumption of ˆ G ( t ) being proportional to the unit matrix unless if we can find certaininitial states for the system for which G ( t ) turns out to become independent of x ; since,as we discussed, for pointer states, all components of the vector A ( A x ′ s ) must be relatedto their corresponding components from B ( B x ′ s ) through the same scalar factor G (see ime-dependent pointer states of the quantized atom-field model in a nonresonance regime ¶ So now we should seek for those particular initial states ofthe system which can make G ( t ) independent of the variable x of the states of theenvironment.From equation (26) we easily see that for α = ± β , G ( t ) turns out to become G ( t ) = ± e iω t (27)which clearly is independent of the variable x of the states of the environment.The above result simply means that for the initial states of the system obtainedfrom α + = β + = 1 √ α − = − β − = 1 √ , or | ± ( t ) i = 1 √ | a i ± | b i ) , (28)(which correspond to the initial conditions for the state of the system given by α = ± β )the states of the system and the environment will not entangle with each other.Moreover, using equation (12), which gives us the general time evolution of the pointerstates of the system, and G ( t ) of equation (27) (which is independent of the variable x of the states of the environment) we can find the time evolution of the pointer states ofthe system as follows | ± ( t ) i = N { G ( t ) | a i + | b i} = 1 √ e iω t | a i ± | b i ) . (29)As we observe, in the regime that we are considering ( ˆ H E ≪ ˆ H S ≪ ˆ H ′ , with ˆ H E = 0and ˆ H S = 0), G ( t ) and the time evolution of the pointer states of the system arecharacterized by ω of the self-Hamiltonian of the system; unlike the exact-resonancewith the rotating wave approximation regime where the evolution of the pointer statesof the system is characterized by the atom-field coupling constant g and the averagenumber of photons ¯ n , through the factor g / √ ¯ n [2, 3].Next, we obtain the corresponding pointer states of the environment. Usingequations (12), (23) and (28) we have | Φ ± ( t ) i = N − ( α ± ˆ E + β ± ˆ E ) | Φ E ( t ) i = N ( ˆ E ± ˆ E ) Z ∞−∞ dx e − α ◦ x | x i ; (30)since N − α ± = 1 and N − β ± = ±
1. Therefore, | Φ ± ( t ) i = ( 2 α ◦ π ) Z ∞−∞ dx e − α ◦ x ∓ i (g χx ± ω / t | x i . (31)Also, the overlap between the pointer states of the environment can be calculated as h Φ − ( t ) | Φ + ( t ) i = e − (g χt ) / α ◦ . (32)We should also mention that the pointer states of the system at t = t (seeequation (28)) are orthonormal and hence, they form a complete basis set for the state ¶ We would like to see if the condition can be satisfied for any initial state of the system and theenvironment with G ( t ) becoming independent of the variable x of the states of the environment. So, iffinally we can find any specific set of initial states for the system and the environment which satisfiesthis condition with G ( t ) independent of the indices of the environment, then we have reached our goal. ime-dependent pointer states of the quantized atom-field model in a nonresonance regime | ψ S ( t ) i = α ′ | + ( t ) i + β ′ | − ( t ) i with an initial field | Φ E ( t ) i , in the form of equation(23), can be expressed as a linear combination of the evolution of | + ( t ) i| Φ E ( t ) i and | − ( t ) i| Φ E ( t ) i ( α ′ | + ( t ) i + β ′ | − ( t ) i ) | Φ E ( t ) i → α ′ | + ( t ) i | Φ + ( t ) i + β ′ | − ( t ) i | Φ − ( t ) i , (33)where in the above equation the evolution of the pointer states of the system | ± ( t ) i is given by equation (29) and the evolution of the pointer states of the environment | Φ ± ( t ) i is given by equation (31).
4. Consequences regarding the decoherence of the central system
In this section first we use the time-evolution operator, already obtained in section 2, toobtain the general time evolution of the total composite system for our model and foran initial state of the environment in the form of a Gaussian package in position space,such as that of equation (23). After that, we will calculate the offdiagonal element ofthe reduced density matrix of the system (i.e. ρ ( S )12 ( t )) by tracing over the environmentaldegrees of freedom. Then, we will also obtain the coherences of the reduced densitymatrix of the system in another way by using the pointer states of the system and theenvironment obtained in section 3. As we will see, the two results will be in perfectagreement with each other. Finally, we will discuss some interesting features which canbe observed in our study of the decoherence of the central system.Using equations (8), (21) and (22) to obtain | Ψ tot ( t ) i , we can write | Ψ tot ( t ) i = A ( t ) | a i + B ( t ) | b i = ( α cos(g χ ˆ xt ) e iω t/ − iβ sin(g χ ˆ xt ) e iω t/ ) | Φ E ( t ) i| a i +( − iα sin(g χ ˆ xt ) e − iω t/ + β cos(g χ ˆ xt ) e − iω t/ ) | Φ E ( t ) i| b i . (34)In the above equation | Φ E ( t ) i is the initial state of the environment, represented by theGaussian package of equation (23).For the state of the total composite system in our model, which is given by equation(34), we can do the trace operation over the basis states of the environment (i.e. the {| x i} which make a complete basis for the state of the environment) to obtain thereduced density matrix of the system S ˆ ρ S ( t ) = Z ∞−∞ dx h x | ˆ ρ tot ( t ) | x i = Z ∞−∞ dx h x | Ψ tot ( t ) ih Ψ tot ( t ) | x i = Z ∞−∞ dx ( | ψ a ( x, t ) | | a ih a | + | ψ b ( x, t ) | | b ih b | + ψ a ( x, t ) ψ ∗ b ( x, t ) | a ih b | + c.c. ) . (35)where ψ a ( x, t ) = ( 2 α ◦ π ) [ α cos(g χxt ) e iω t/ − iβ sin(g χxt ) e iω t/ ] e − α ◦ x and ψ b ( x, t ) = ( 2 α ◦ π ) [ − iα sin(g χxt ) e − iω t/ + β cos(g χxt ) e − iω t/ ] e − α ◦ x . (36)Using equations (35) and (36), after doing the integrations we easily find ρ S aa ( t ) = 1 − ρ S bb ( t ) = Z ∞−∞ dx | ψ a ( x, t ) | = 12 [1 + ( | α | − | β | ) e − (g χt ) / α ◦ ] and ime-dependent pointer states of the quantized atom-field model in a nonresonance regime ρ S ab ( t ) = Z ∞−∞ dx ψ a ( x, t ) ψ ∗ b ( x, t ) = 12 [( αβ ∗ + βα ∗ ) + ( αβ ∗ − βα ∗ ) e − (g χt ) / α ◦ ] e iω t . (37)(In the above equation we used the notation ρ ab = h a | ˆ ρ S ( t ) | b i and etc.) As we see fromthe above equations, for the initial pointer states of the system, for which | α | = | β | (see equation (28)), and also for very large times t → ∞ , the diagonal elements of thereduced density matrix of the system will be equal to the constant number of . Also,for the initial pointer states of the system we have ρ S ab ( t ) = ( αβ ∗ + βα ∗ ) e iω t . Thismeans that for the initial pointer states of the system | ρ S ab ( t ) | will always be equal tothe constant value of ; while for most of the other states (for whom αβ ∗ = βα ∗ ) onlyat sufficiently large times | ρ S ab ( t ) | will converge to the constant value of ( αβ ∗ + βα ∗ ),with a decoherence time given by τ dec = ¯ h g √ α ◦ χ = ¯ h g r α ◦ mω . (38)The reduced density matrix of a two-level system ˆ ρ S ( t ) generally can be expressedin terms of the Bloch vector R ( t ) ≡ ( R x , R y , R z ) [6] as followsˆ ρ S ( t ) = 12 ( ˆ I + R ( t ) . ˆ σ ) = 12 ( ˆ I + R x σ x + R y σ y + R z σ z ); (39)from which one can easily verify that the Bloch vector components must be defined by R x = ρ ab + ρ ba R y = i ( ρ ab − ρ ba ) and R z = ρ aa − ρ bb . (40)So now, using our expressions for the elements of the reduced density matrix of thesystem, given by equation (37), we can also calculate the components of the Blochvector, which are a measure for the polarization of the state of the two-level system[1, 7]. One would easily find R x ( t ) = ρ ab + ρ ∗ ab = ( αβ ∗ + βα ∗ ) cos( ω t ) + i ( αβ ∗ − βα ∗ ) sin( ω t ) e − (g χt ) / α ◦ ,R y ( t ) = i ( ρ ab − ρ ∗ ab ) = − ( αβ ∗ + βα ∗ ) sin( ω t ) + i ( αβ ∗ − βα ∗ ) cos( ω t ) e − (g χt ) / α ◦ , (41) R z ( t ) = ρ aa − ρ bb = ( | α | − | β | ) e − (g χt ) / α ◦ . For t → ∞ and χ = 0 we have R x ( t ) → ( αβ ∗ + βα ∗ ) cos( ω t ) ,R y ( t ) → − ( αβ ∗ + βα ∗ ) sin( ω t ) and (42) R z ( t ) → . The above result simply means that at t → ∞ and if χ = √ mω = 0 the pointer statesof the system will evolve between the eigenstates of the ˆ σ x and ˆ σ y Pauli matrices; andtherefore, a preferred basis of measurement is not determined in the regime that we areconsidering; although the eigenstates of ˆ σ z are excluded from being realized at t → ∞ .One can easily obtain the coherences of the reduced density matrix of the systemin another way by using the pointer states of the system and the environment whichwe obtained in section 3. As we saw, for a two-state system S in contact withan environment E after determination of the pointer states of the system and theenvironment, the state of the total composite system generally can be represented by ime-dependent pointer states of the quantized atom-field model in a nonresonance regime | Ψ tot ( t ) i = α ′ | + ( t ) i | Φ + ( t ) i + β ′ | − ( t ) i | Φ − ( t ) i . For | Ψ tot ( t ) i givenby equation (33) the reduced density matrix of the system ˆ ρ S ( t ) can be calculated bytracing over the environmental degrees of freedom to obtainˆ ρ S ( t ) = | α ′ | × | + ( t ) ih +( t ) | + | β ′ | × | − ( t ) ih− ( t ) | + α ′ β ′∗ ×| + ( t ) ih− ( t ) | × h Φ − ( t ) | Φ + ( t ) i + β ′ α ′∗ × | − ( t ) ih +( t ) | × h Φ + ( t ) | Φ − ( t ) i . (43)So, in an arbitrary basis | a i and | b i of the state of the two-level system generally wehave ρ S aa ( t ) = 1 − ρ S bb ( t ) = | α ′ | × h a | + ( t ) ih +( t ) | a i + | β ′ | × h a | − ( t ) ih− ( t ) | a i + α ′ β ′∗ ×h a | + ( t ) ih− ( t ) | a i × h Φ − ( t ) | Φ + ( t ) i + β ′ α ′∗ × h a | − ( t ) ih +( t ) | a i × h Φ + ( t ) | Φ − ( t ) i (44)and ρ S ab ( t ) = | α ′ | × h a | + ( t ) ih +( t ) | b i + | β ′ | × h a | − ( t ) ih− ( t ) | b i + α ′ β ′∗ ×h a | + ( t ) ih− ( t ) | b i × h Φ − ( t ) | Φ + ( t ) i + β ′ α ′∗ × h a | − ( t ) ih +( t ) | b i × h Φ + ( t ) | Φ − ( t ) i . The expansion coefficients α ′ and β ′ for the state of the two-level system in thebasis of the | ± ( t ) i states are related to the corresponding coefficients in the | a i and | b i basis + through α ′ = √ ( α + β ) and β ′ = √ ( α − β ). So now, for our quantizedatom-field model and in the regime that we are considering one can use equations (29)and (32) to calculate the expressions in equation (44) for the elements of the reduceddensity matrix of the system; obtaining exactly the same results as those of equation(37).One could similarly study the decoherence of the state of the system in the basis ofthe | ± ( t ) i states. As one can see from equation (29), for t ≪ ω − the pointer statesof the system approximately can be represented by | ± ( t ) i . Therefore, in the basis ofthe | ± ( t ) i states the short-time evolution of the off-diagonal element of the reduceddensity matrix of the system should be given by ρ S ( t ) ≈ α ′ β ′∗ h Φ − ( t ) | Φ + ( t ) i = α ′ β ′∗ e − (g χt ) / α ◦ (45)Hence, in the basis of the | ± ( t ) i states the short-time decoherence of the state ofthe system is characterized by the decaying factor e − (g χt ) / α ◦ , when the system initiallyis not prepared in one of its pointer states ( α ′ β ′∗ = 0); while in this basis the pointerstates of the system (for whom α ′ β ′ = 0) almost do not decohere within short times;and ρ S ( t ) ≈ t ≪ ω − for which | ± ( t ) i ≈ | ± ( t ) i ).Finally, let us study whether the short-time decay of ρ S ( t ), given by equation (45),might be reversible or not. As we will show here, the coherences of the reduced densitymatrix of the system, may revive at a later time. In such cases, of course we cannothave irreversible decoherence.Using equation (44) for the offdiagonal element of the reduced density matrix ofthe system and equations (29) and (32), we can calculate the all-time evolution of ρ S ( t )for the regime that we are considering and in the basis of the initial pointer states of + Now by | a i and | b i we mean the upper and lower levels of the two-level system; i.e. | a i and | b i nolonger are some arbitrary basis states for the state of the two-level system. ime-dependent pointer states of the quantized atom-field model in a nonresonance regime | ± ( t ) i as follows ρ S ( t ) = ( | β ′ | − | α ′ | ) × [ i ω t )] + α ′ β ′∗ × cos ( ω t/ × e − (g χt ) / α ◦ + β ′ α ′∗ × sin ( ω t/ × e − (g χt ) / α ◦ ; (46)which its short time evolution ( t ≪ ω − ) is the same as equation (45).Now, clearly for t → ∞ we have ρ S ( t ) = ( | β ′ | − | α ′ | ) × [ i ω t )] . (47)Therefore, except for | α ′ | = | β ′ | , in the basis of the initial pointer states of the systemand for t → ∞ the offdiagonal element of the reduced density matrix of the system,ˆ ρ S , will be oscillating with the frequency of ω . As a result, we should note that theshort-time decay, represented by equation (45), can be reversible; as ρ S ( t ) may reviveat a later time.
5. Summary and conclusions
Considering the quantized atom-field model of quantum optics, we obtained the time-evolution operator for the regime that ˆ H E ≪ ˆ H S ≪ ˆ H ′ (but ˆ H S = 0 and ˆ H E = 0). Usingthis time-evolution operator then we calculated the corresponding pointer states of thesystem and the environment, which are characterized by their ability not to entanglewith each other, by assuming an initial state of the environment in the form of a Gaussianpackage in position space. Most importantly, we observed that for our model representedby the Hamiltonian of equation (1) the pointer states of the system turn out to become time-dependent , as opposed to the pointer states of some simpler models, which often arecited in the context of quantum information and quantum computation [8-15]. However,in most of the practical situations different noncommutable perturbations may exist inthe total Hamiltonian of a realistic system-environment model, which would result inhaving time-dependent pointer states for the system [1]. Indeed, the authors believe thatthe fact that the pointer states of a system generally are time-dependent and may evolvewith time has not been seriously acknowledged in the context of quantum computationand quantum information. In specific, in the context of quantum error correction [11, 12]it is often assumed that the premeasurement by the environment does not changethe initial pointer states of the system. In other words, quantum “nondemolition”premeasurement by the environment often is assumed [11, 12]; as is also assumed inthe Von Neumann scheme of measurement [16, 7]. Also, in the context of Decoherence-Free-Subspaces (DFS) theory the models which often are studied either contain self-Hamiltonians for the system which commute with the interaction between the systemand the environment, or it is assumed that we are in the quantum measurement limit ∗ ∗ In the quantum measurement limit the interaction between the system and the environment isso strong as to dominate the evolution of the system ˆ H ≈ ˆ H int . Also in the quantum limit ofdecoherence the Hamiltonian for the system almost dominates the interaction between the systemand the environment as well as the self-Hamiltonian of the environment ˆ H ≈ ˆ H S . ime-dependent pointer states of the quantized atom-field model in a nonresonance regime quantum limit of decoherence [8, 13, 14, 15]. However, all of these assumptionsare in fact a simplification of the problem; since, as we discussed in [1], they completelyexclude the possibility of having pointer states for the system which may depend ontime.Using the time-evolution operator obtained in section 2, we also obtained a closedform for the elements of the reduced density matrix of the system, and studied thedecoherence of the central system in our model. We showed that for the case that thesystem initially is not prepared in one of its pointer states and in the basis of the initialpointer states of the system (i.e. the | ± ( t ) i states), the short time ( t ≪ ω − ) evolutionof the offdiagonal elements of the reduced density matrix of the system will demonstratedecoherence, with a decoherence factor given by e − (g χt ) / α ◦ ; and a decoherence timewhich is inversely proportional to the square root of the mass of field particles.It will be interesting to generalize this study to the case that the environment is notmerely represented by a single-mode bosonic field; and consider some classes of spectraldensities for the environment. Also, for the model represented by the Hamiltonian ofequation (1) at least in principle one should be able to obtain the pointer states of thesystem and the environment in some other regimes of the parameter space. References [1] Daneshvar H and Drake G W F submitted to
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