Entanglement and parametric resonance in driven quantum systems
aa r X i v : . [ qu a n t - ph ] O c t Entanglement and parametric resonance in driven quantum systems
V. M. Bastidas , ∗ J. H. Reina , C. Emary , and T. Brandes Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany and Universidad del Valle, Departamento de F´ısica, A. A. 25360, Cali, Colombia (Dated: November 11, 2018)We study the relationship between entanglement and parametric resonance in a system of twocoupled time-dependent oscillators. As a measure of bipartite entanglement, we calculate the linearentropy for the reduced density operator, from which we study the entanglement dynamics. Inparticular, we find that the bipartite entanglement increases in time up to a maximal mixing scenario,when the set of auxiliary dynamical parameters are under parametric resonance. Moreover, weobtain a closed relationship between the correlations in the ground state, the localisation of theWigner function in phase space, and the localisation of the wave function of the total system.
PACS numbers: 03.67.Bg, 03.65.Ud, 03.65.Yz, 03.67.-a
I. INTRODUCTION
A quantum system composed of two or more entangledsubsystems has the intriguing property that although thestate of the total system can be well defined, it is impos-sible to identify individual properties for each one of itsparts. Understanding the properties displayed by the en-tanglement of physical systems is one of the fundamentalpurposes of quantum information theory [1, 2]. In thispaper we focus on the relationship between entanglementand parametric resonance in a composite quantum sys-tem.In classical dynamics, parametric resonance can occurwhen an appropriate parameter of a system is varied pe-riodically in time [3]. Under parametric resonance, stablepoints of the undriven system become unstable for spe-cific values of the period of the parameter variation. Onthe other hand, an unstable point may become a stableone — a phenomenon known as parametric stabilization.In the quantum regime, it is well known that a periodi-cally driven system follows a unitary dynamics. The dy-namics is not closed, however, since the external drivingchanges the physical parameters of the system, alteringits total energy. The usual technique for treating suchsystems is the Floquet Formalism [4, 5]. Perelomov andPopov [6], and most recently Weigert [7], have studiedquantum manifestations of classical parametric resonancein noncomposite quantum systems in the context of Flo-quet theory for the Schr¨odinger equation. They found,for example, that for parameters in the stable regions, adiscrete spectrum of quasienergies exist. In contrast, inthe unstable zones, there is a continuous spectrum.The question that naturally arises is: what are the quan-tum signatures of classical parametric resonance? Thereis no classical behavior analogous to entanglement, it isunique to quantum mechanics. In this paper we showthat this intrinsically quantum property shows strong sig-natures of classical parametric resonance in a system of ∗ Electronic address: [email protected] two driven coupled harmonic oscillators. Furthermore,we find that the entanglement dynamics is related to thedynamical behaviour of the Wigner function and the to-tal system’s ground-state wave packet. This paper is or-ganized as follows: In section II we describe the model.In section III we introduce the required basic aspectsof the Lewis-Riesenfeld canonical method and the singleparametric oscillator. In section IV we find the exactsolution of the Schr¨odinger equation and discuss the sta-bility properties of the auxiliary dynamical parametersand their relation to the phenomenon of parametric res-onance. In section V we present the calculation of theentanglement dynamics and the Wigner function. Fi-nally, discussion of the results and conclusive remarksare presented in sections VI and VII respectively.
II. MODEL
Time-periodic quadratic Hamiltonians are known toprovide examples of parametric resonance. This type ofHamiltonian has the general formˆ H ( t ) = 12 X i =1 ( µ i ( t )ˆ p i + ν i ( t )ˆ x i ) + γ ( t )ˆ x ˆ x . (1)In this work we are interesting in studying the particularmodel described by the time-periodic Hamiltonianˆ H ( t ) = 12 X i =1 (ˆ p i + ω ˆ x i ) + γ ( t )ˆ x ˆ x , (2)where γ ( t ) = g + ∆ g cos Ω t , ω is a time-independent fre-quency, g is called the static coupling, and ∆ g is a frac-tion of g .Further motivation for studying this type of Hamilto-nian comes from the time-dependent Dicke model. In1954, Dicke showed, in a now celebrated paper [8], thecoherent spontaneous emission arising from many atomsemitting collectively. The single mode Dicke Hamiltonianmodels the interaction of N atoms with a single modebosonic field via dipole interactions within an ideal cav-ity. The time-dependent generalization of Dicke model isdescribed by the Hamiltonianˆ H ( t ) = ω ( t ) J z + ω ( t )ˆ a † ˆ a + λ ( t ) √ N ( J + + J − ) (cid:0) ˆ a + ˆ a † (cid:1) . (3)By considering the Holstein-Primakoff transformation ofthe angular momentum algebra and taking the limit j → ∞ in the Hamiltonian Eq. (3), one obtains aneffective Hamiltonian [9, 10], which corresponds to theHamiltonian of two time-dependent oscillators linearlycoupled through the interaction strength λ ( t ). In termsof the field coordinate x and the atoms coordinate y , thisHamiltonian becomesˆ H ( t ) = ω ( t )2 ω (cid:0) ˆ p x + ω ˆ x (cid:1) + ω ( t )2 ω (cid:0) ˆ p y + ω ˆ y (cid:1) +2 λ ( t ) √ ωω ˆ x ˆ y − E , (4)where ω ( t ) = ω , ω ( t ) = ω , and E ( t ) = ω ( t )2 + ω ( t )2 + jω ( t ). In the next section we describe the Lewis-Riesenfeld canonical method and thus establish the non-perturbative technique used in this work. III. THE LEWIS–RIESENFELD CANONICALMETHOD AND THE SINGLE PARAMETRICOSCILLATOR
In order to study the nonequilibrium dynamics de-scribed by time-dependent Hamiltonians, we resort tothe Lewis-Riesenfeld canonical method. This techniquedeals with the relation between the eigenstates of an in-variant operator and the solutions of the correspondingSchr¨odinger equation [11]. Using this method, it is pos-sible to find the exact solution for a single parametricquantum oscillator [11, 12, 13].
A. The Lewis–Riesenfeld canonical method
An explicitly time-dependent invariant operator ˆ O should satisfy the condition ddt ˆ O ( t ) = ∂∂t ˆ O ( t ) + i [ ˆ H ( t ) , ˆ O ( t )] = 0 . (5)We write the system’s dynamics by means of the standardSchr¨odinger equation i ∂∂t | ψ, t i = ˆ H ( t ) | ψ, t i . (6)A non-equilibrium system evolves towards a final statewhich, in general, differs from the initial one. Lewis andRiesenfeld observed that any operator ˆ O ( t ) satisfying Eq.(5) can be used to construct the exact quantum states of the Schr¨odinger equation. In general, if | ψ, t i is a solu-tion, this can be written in terms of the eigenstates ofthe operator ˆ O ( t ), | ψ, t i = X n c n exp( i θ n ( t )) | λ n , t i , (7)where ˆ O ( t ) | λ n , t i = λ n | λ n , t i , and the time dependentphase is θ n ( t ) = R t h λ n , t | i ∂∂t − ˆ H ( t ) | λ n , t i dt . B. The single quantum parametric oscillator
We consider the Hamiltonianˆ H ( t ) = 12 ˆ p + ω ( t )2 ˆ q , (8)where ω ( t ) is an arbitrary (not necessarily real),piecewise-continuous function of time [11]. For this sys-tem, one can derive a set of time dependent invariantswhich are homogeneous, quadratic expressions in the dy-namical variables p and q . For this purpose, we introducetime dependent ladder operators ˆ a ( t ) and ˆ a † ( t ), that sat-isfy Eq. (5). For example, the annihilation operator isgiven by ˆ a ( t ) = − ˙ B ( t )ˆ q + B ( t )ˆ p , (9)where the auxiliary dynamical parameter B ( t ) satisfy theclassical equation of motion¨ B ( t ) + ω ( t ) B ( t ) = 0 , (10)and the Wronskian condition˙ B ( t ) B ∗ ( t ) − B ( t ) ˙ B ∗ ( t ) = i . (11)The last condition ensure that the operators are ladderoperators at all times [12, 13]. In the context of Lewis-Riesenfeld’s canonical method, we construct the Hermi-tian invariant operator ˆ n ( t ) = ˆ a † ( t )ˆ a ( t ) and the Fockspace consisting of time-dependent number statesˆ n ( t ) | n, t i = n | n, t i , (12)where | n, t i = (ˆ a † ( t )) n √ n ! | , t i , (13)and the vacuum state | , t i is anihilated by ˆ a ( t ).The dynamical behaviour of the system is fully deter-mined through the solution of the corresponding classicalproblem Eq. (10). IV. THE LEWIS–RIESENFELD CANONICALMETHOD FOR COUPLED OSCILLATORS ANDCLASSICAL PARAMETRIC RESONANCE
In this section we find the solution of the Schr¨odingerequation for the Hamiltonian Eq. (2) and study the sta-bility properties of the auxiliary dynamical parameters,which are related to the phenomenon of parametric res-onance in classical mechanics [3].
A. Exact solution of the Schr¨odinger equation
In the case of the Schr¨odinger equation for the Hamil-tonian Eq. (2), there is a transformation that allows us tomap this Hamiltonian into a Hamiltonian of two uncou-pled parametrically driven quantum harmonic oscillators.We define the time-independent unitary transformationˆ U acting over the system’s state | Ψ , t i , to obtain | Ψ , t i R = ˆ U | Ψ , t i . (14)We choose the unitary operator ˆ U such that the system’sHamiltonian ˆ H ( t ) is taken to a new operator which canbe associated to that of two uncoupled time dependentquantum harmonic oscillators, ˆ H R ( t ) = ˆ U † ˆ H ( t ) ˆ U , suchthatˆ H R ( t ) = 12 (cid:0) ˆ p − + ( ǫ − ( t )) ˆ q − + ˆ p + ( ǫ + ( t )) ˆ q (cid:1) , (15)where the energies of the two independent oscillatormodes ǫ ∓ ( t ) are defined in terms of the other control pa-rameters as ( ǫ ∓ ( t )) = ω ∓ ω | γ ( t ) | . The solution of theSchr¨odinger equation for the Hamiltonian of Eq. (15) isobtained as a linear combination of product of quantumstates | Ψ n − ,n + , t i R = exp i ( θ n − ( t ) + θ n + ( t )) | n − , t i R | n + , t i R , (16)where | n ∓ , t i R are the quantum states corresponding tothe oscillator mode with excitation energy ǫ ∓ ( t ). Thefundamental excitations of the system are given by theenergies ǫ ∓ ( t ). The general solution of the Schr¨odingerequation is given by | Ψ , t i R = X n − ,n + c n − ,n + | Ψ n − ,n + , t i R . (17)In order to find the wave functions corresponding to thequantum states | n ∓ , t i R , we now go back to the Lewis-Riesenfeld’s canonical method. The first step in the con-struction of the solution is to find the ground state of thesystem, which must satisfy the conditionˆ a ∓ | , , t i R = 0 , (18)where ˆ a ∓ ( t ) = − ˙ B ∓ ( t )ˆ q ∓ + B ∓ ( t )ˆ p ∓ are annihilation op-erators corresponding to the oscilator mode with energy ǫ ∓ ( t ) that satisfy Eq. (5), and the auxiliary dynamicalparameters B ∓ ( t ) satisfy the Mathieu equation [14]¨ B ∓ ( t ) + ( ǫ ∓ ( t )) B ∓ ( t ) = 0 , (19)subject to the Wronskian condition˙ B ∓ ( t ) B ∗∓ ( t ) − B ∓ ( t ) ˙ B ∗∓ ( t ) = i . (20) We next solve the differential equations subjected to aninitial thermal equilibrium condition [9, 10] satisfying theWronskian constraint; this requires that B ± ( t ) = i p ǫ ± ( t )) ∗ , ˙ B ± ( t ) = − r ǫ ± ( t )2 . (21)In the coordinate representation, the operator Eq. (18)is equivalent to the differential equation (cid:18) − ˙ B ∓ ( t ) q ∓ + 1 i B ∓ ( t ) ∂∂q ∓ (cid:19) Φ ∓ ( q ∓ , t ) = 0 . (22)The solution to this equation is the wave functionΦ ∓ ( q ∓ , t ) = (cid:18) π | B ∓ ( t ) | (cid:19) / exp i ˙ B ∓ ( t )2 B ∓ ( t ) q ∓ ! , (23)thus obtaining the normalized ground state wave func-tion Φ R , ( q − , q + , t ) = Φ − ( q − , t )Φ +0 ( q + , t ). The construc-tion of the Fock space consisting of time-dependent num-ber states is direct, as any number state is obtained byapplying the creation operators to the vacuum state | n − , n + , t i R = (ˆ a †− ( t )) n − p ( n − )! (ˆ a † + ( t )) n + p ( n + )! | , , t i R . (24)In the coordinate representation, the number stateis given by Φ Rn − ,n + ( q − , q + , t ) = Φ − n − ( q − , t )Φ + n + ( q + , t ),where the quantum states Φ ∓ n ∓ can be written in termsof the Hermite polynomials H n [13] asΦ ∓ n ∓ ( q ∓ , t ) = (cid:18) − n ∓ ) π | B ∓ ( t ) | ( n ∓ )! (cid:19) / (cid:18) B ∗∓ ( t ) B ∓ ( t ) (cid:19) n ∓ × H n ( Q ∓ ( t )) exp i ˙ B ∓ ( t )2 B ∓ ( t ) q ∓ ! , (25)where we have introduced the parameter Q ∓ ( t ) =(2 | B ∓ ( t ) | ) − / q ∓ . In analogy to the harmonic oscilla-tor case, we define a characteristic system length scale interms of the auxiliary dynamical parameters as l ∓ ( t ) = √ | B ∓ ( t ) | . (26)The solution of the Schr¨odinger equation for the originalHamiltonian is obtained by applying the inverse of theunitary transformation ˆ U to the quantum state Eq. (25).Finally, the state of the system h x , x | Ψ n − ,n + , t i can bewritten as a function of the coordinates x and x asfollows FIG. 1: Stability of the Mathieu’s equation: i) shaded areas correspond to a real Floquet exponent, and the white areas to acomplex FE; ii) the upper curve shows the real part of the FE for the solution B + ( t ) whereas the lower curve gives the realpart of the FE for the solution B − ( t ); iii) imaginary part of the Floquet exponent for the solution B − ( t ) in the gap zone of thelower curve in ii). Ψ n − ,n + ( x , x , t ) = exp( i θ n − ( t )) exp( i θ n + ( t ))Φ − n − (cid:18) x √ − x √ , t (cid:19) Φ + n + (cid:18) x √ x √ , t (cid:19) . (27) B. Mathieu’s equation and parametric resonance
Mathieu’s equation [14] is given by¨ f ( t ) + ( a − b cos(2 t )) f ( t ) = 0 , (28)the behaviour of its solution depends strongly on the pa-rameters a and b , the driving can stabilize or destabilizethe undriven oscillation, this means that the solution canbe bounded or increasing with time respectively. In gen-eral, parametric resonance can occur if the parameters ofa classical dynamical system vary periodically with time.Stable fixed points of the flow in phase space becomeunstable for specific values of certain parameters [3]. In-terestingly, the set of differential equations (19) has thegeneral form of Mathieu’s equation [14]¨ B ∓ ( t ) + [( ω ∓ ωg ) ∓ (2 ω ∆ g ) cos Ω t ] B ∓ ( t ) = 0 . (29)If we compare this with the canonical form of the Math-ieu’s equation of Eq. (28), it is clear that the parameters a and b are functions of the static coupling g , the fre-quency ω , and the external driving frequency Ω: a ∓ ( g, ω, Ω) = 4( ω ∓ ωg ) / Ω , (30) b ∓ ( g, ω, Ω) = ± ω ∆ g/ Ω . (31)In order to explore the relevant characteristics of the so-lutions of Eq. (29), we apply the Floquet theorem for second order differential equations with time periodic co-efficients [15]. For this, the solutions of Eq. (29) have thegeneral form B ∓ ( t ) = exp( iF ∓ t ) φ ∓ ( t ) , (32)where φ ∓ ( t + T ) = φ ∓ ( t ) and F ∓ ( a ∓ ( g, ω, Ω) , b ∓ ( g, ω, Ω))is the Floquet exponent (FE), which depends on theshape of the driving. The stability zones of the Math-ieu’s equation are well known: for driving functions forwhich F ∓ is complex, the solution becomes unstable, andin the stable regime, F ∓ is a real number [14, 16]. InFig. 1 ( i ) we show the stability zones of the Mathieu’sequation in the a − b plane: within the shaded areas theFE is real, whereas within the white areas this becomescomplex. Once we have established the parameters ∆ g , ω , and Ω, the map ζ ∓ ( g ) = ( a ∓ ( g, ω, Ω) , b ∓ ( g, ω, Ω)) de-scribes a straight line in the a − b plane, as can be ob-served in Fig. 1 ( i ). The real part ℜ ( F ∓ ( ζ ∓ ( g ))) of theFE for the auxiliary dynamical parameters B ∓ ( t ) as afunction of the static coupling, exhibits band-like regionsthat correspond to a complex FE of the solution, whichmeans that this solution is unstable within the “ gaps ”,where the Floquet exponent is imaginary. In order tostudy such features, we first investigate the particularcase ∆ g = 0 . g , ω = 1, and Ω = 1. Fig. 1 ( ii ) depictthe real part of the FE for the solution B ∓ ( t ) and Fig. 1( iii ) the imaginary part of FE for the solution B − ( t ). Inthese figures, we find that for this particular case only ex- FIG. 2: Phase space trajectories of the auxiliary dynamical parameters B ± ( t ) for ∆ g = 0 . g , and a value of the static couplingin a) the unstable region, g = 0 .
38, b) the stable region, g = 0 .
4, and c) the stable region, g = 0 . ist one band-like region for the solution B − ( t ). In Fig. 2,we plot the phase space representation of the trajectoriesof the auxiliary dynamical parameters B ∓ ( t ). Figure 2( a ) shows that when the static coupling belongs to theinstability zone ( g = 0 .
38) of B − ( t ), the solution B + ( t )is bounded, while the solution B − is unbounded in thephase space. In contrast, in Figs. 2( b ) and 2 ( c ), it isclear that for values of g that belong to a mutual stabilityzone ( g = 0 .
4, and g = 0 .
46) the solutions are boundedin the phase space.We can find an exact solution to the Schr¨odinger equationfor the general problem described by the Hamiltonian Eq.(1) through the Lewis and Riesenfeld canonical method[11, 12, 13], the general solution is given in Appendix A.
V. QUANTUM DYNAMICS ANDENTANGLEMENT QUANTIFIERS
In this section we establish a connection between thenon-equilibrium entanglement generated by the time-dependent coupling and the behaviour of the auxiliarydynamical parameters. Furthermore, in order to studythe correlations in the ground state and their relationwith the localisation of the total system’s wave function,we derive analytical expressions for the reduced densityoperator of one oscillator and the corresponding Wignerfunction.
A. The reduced density operator
When studying a composed quantum system whose dy-namics is unitary, it is often more interesting to study asubsystem of the whole system, which is described byits reduced system dynamics. In contrast to the dy-namics of the whole system, the dynamics of the sub- system is not unitary. It is possible, however, to makesuch a description of the subsystem through the reduceddensity operator. We describe the dynamics of the to-tal system in terms of the density operator, where thepure state of the universe is represented by the operatorˆ ρ G ( t ) = | Ψ − , + , t ih Ψ − , + , t | . In the coordinate repre-sentation, the density matrix takes the form ρ G ( x ′ , x ′ ; x , x , t ) = Ψ ∗ − , + ( x , x , t )Ψ − , + ( x ′ , x ′ , t ) . (33)At this stage, we develop a bipartite decomposition of theuniverse, which lets us study the reduced dynamics of oneoscillator through the reduced density matrix (RDM). Inso doing, we calculate the partial trace over the physicalfield coordinate x , ρ ( R ) G ( x ′ , x , t ) = Z + ∞−∞ Ψ ∗ − , + ( x , x , t )Ψ − , + ( x , x ′ , t ) dx . (34)The ground state of the universe Ψ − , + ( x , x , t ) givenin Eq. (27) is Gaussian, which facilitates the integrationof Eq. (34). After some algebraic calculations, we obtainthe following result for the reduced density matrix ρ ( R ) G ( x ′ , x , t ) = Λ exp( −ℜ ( α )(( x ′ ) + x ) + βx x ′ ) × exp( i ℑ ( α )(( x ′ ) − x ))) , (35)where the parameters α , β , and Λ are defined as α = ( ξ − ) ∗ s + ( ξ + ) ∗ c − c s [( ξ − − ξ + ) ∗ ] ℜ ( ξ − ) c + ℜ ( ξ + ) s ) ,β = c s ( ξ − − ξ + ) ∗ ( ξ − − ξ + )2( ℜ ( ξ − ) c + ℜ ( ξ + ) s ) , Λ = (cid:18) ℜ ( ξ − ) ℜ ( ξ + ) π ( ℜ ( ξ − ) c + ℜ ( ξ + ) s ) (cid:19) / , (36)where c = s = 1 / √ ℜ ( ξ ∓ ) is the real part of thefunction ξ ∓ ( t ) = − i ˙ B ∓ ( t ) B ∓ ( t ) . In the particular case γ ( t ) = FIG. 3: Linear entropy as a function of time for ∆ g = 0 . g ; a) g = 0 .
38, b) g = 0 .
40, and c) g = 0 .
46. For the case of periodicatom-field coupling. g , this function is independent of time and becomes ξ ∓ ( t ) = ǫ ∓ = p ( ω ∓ ωg ) , (37)and corresponds to the result previously reported in theliterature for the atom’s reduced density matrix for thesingle mode Dicke model in thermal equilibrium [9, 10].The reduced density operator Eq. (35) has the samestructure as the density operator of an ensemble of time-dependent oscillators in the coordinate representation[13]. B. The linear entropy
A density operator ˆ ρ describes a pure state if and onlyif it satisfies ˆ ρ = ˆ ρ , i.e., the density operator is idem-potent. We consider a pure state of a composed system(the universe), say system AB , represented by the den-sity operator ˆ ρ AB . The linear entropy for the reduceddensity operator ˆ ρ A = tr B (ˆ ρ AB ) of subsystem A is de-fined by L A = 1 − tr A (ˆ ρ A ) [1]. This gives a measureof purity of the reduced density operator ˆ ρ A (one partof the total system, in a bipartite decomposition of theuniverse). If the pure state of the universe is separable,the reduced density operator of one part of the systemrepresents a pure state and, as a result, its linear entropymust be zero. Similarly, if the pure state of the universeis a maximally entangled state [17], the linear entropyequals 1 /
2. In the context of our model, it is possible touse the linear entropy as a measure of bipartite entan-glement for the oscillators system. The linear entropyfor the oscillator’s time dependent reduced density oper- ator, Eq. (35), is given by L ( t ) = 1 − tr (cid:20)(cid:16) ˆ ρ ( R ) G ( t ) (cid:17) (cid:21) . By using the density matrix representation of Eq. (35),we explicitly calculate the linear entropy as L ( t ) = 1 − Z + ∞−∞ Z + ∞−∞ ρ ( R ) G ( x ′ , x , t ) ρ ( R ) G ( x , x ′ , t ) dx dx ′ , which gives, after some algebra, the result L ( t, g ) = 1 − π Λ p ℜ ( α )) − β . (38)In Fig. 3 we plot this quantity as a function of time: it isclear that in the case of a time periodic atom-field cou-pling there exist a direct correspondence between the sta-bility of the auxiliary dynamical parameters and the en-tanglement dynamics, which is reflected in the graph 3(a)for a value of the static coupling in the unstable zone,it is observed that the linear entropy oscillates, beforereaching the stationary state with diverging linear en-tropy L = 1. We consider the linear entropy in Figs. 3( b ), and 3 ( c ) for values of the static coupling in the sta-ble zone. Our choice of the initial conditions for Eq. (19)implies that the system starts in thermal equilibrium,but this is described by a non-separable quantum state.The dynamics exhibits a behaviour whereby the systemexperiences a disentanglement process, before reachinga maximum value of entanglement, and then successivecollapses and revivals. The frequency of the oscillationsin the “stable zones” depends strongly on the value ofthe static coupling g . C. The Wigner function
The Wigner function is a particular representation ofthe density operator for a pure or a mixed state of aquantum system. In principle, the density operator ismore fundamental than its Wigner representation, how-ever, the Wigner function is often a useful tool in deco-herence studies for investigating possible correlations be-tween position and momentum [5, 18, 19]. The Wignerrepresentation of a given density operator ˆ ρ is given by W ( q, p ) = 12 π Z ∞−∞ exp ( ips ) h q − s/ | ˆ ρ | q + s/ i ds, (39)where we have considered ~ = 1. The Wigner functioncan take on negative as positive values. This means that it cannot be strictly interpreted as a phase space proba-bility density. In the special case when the density opera-tor represents a pure Gaussian state, the Wigner functionis Gaussian and hence is positive definite. The reducedWigner function represents the partial trace of a densityoperator over a subsystem and so contains all informa-tion about a given subsystem. Here we calculate theWigner representation of the oscillator’s reduced densityoperator given by Eq. (35). Such Wigner function reads W ( R ) ( q, p, t ) = 12 π Z ∞−∞ e ips ρ ( R ) G ( q − s/ , q + s/ , t ) ds . (40)By performing the Gaussian integral we obtain W ( R ) ( q, p, t ) = Λ s π (2 ℜ ( α ) + β ) exp (cid:20) ℑ ( α ) qp ℜ ( α ) + β (cid:21) exp (cid:20) − q (4 | α | − β ) + p ℜ ( α ) + β (cid:21) , (41)where the parameters α , β , and Λ are defined as in Eq.(36). If we specialize to the case γ ( t ) = g , the Wignerfunction becomes time-independent, W ( R ) ( q, p ) = (cid:18) √ ǫ − ǫ + π ( ǫ − + ǫ + ) (cid:19) exp (cid:20) − ǫ − + ǫ + ( ǫ − ǫ + q + p ) (cid:21) , (42)where ǫ ∓ are defined in Eq. (37). This function exhibitssome interesting features: for values of the coupling nearto the critical coupling g c = ω , the Wigner function be-comes stretched along the position axis with a consequentcontraction along the momentum axis in the phase space;this results in the Wigner function being delocalized inphase space. This behaviour is due to the existing entan-glement, because the oscillator’s state becomes a mixedstate. Such a behaviour is corroborated by a divergenceof the von Neumann entropy near the critical coupling,signaling a “maximal mixing” scenario, as reported inRef. [10] in the context of the time-independent Dickemodel. Figures 4 and 5 show the evolution of the Wignerfunction in the case of a periodic atom-field coupling forvalues of the static coupling that belong to unstable andstable zones, respectively. In our dynamical coupling sce-nario (∆ g = 0), there is a direct relationship betweenthe atom-field entanglement and the localisation of theWigner function in phase space: the Wigner functionevolves in a delocalised region of the phase space for val-ues of the static coupling in the unstable zone; as shownin Fig. 4, the function evolves in such a way that it isstretched along a dynamical rotating axis with a conse-quent contraction in the perpendicular axis, in contrastto the time-independent behaviour. The behaviour ex-hibited by the Wigner function for a value of the static coupling in the stable zone is quite different: the Wignerfunction also becomes stretched along a dynamical axisbut it is now localised as the system evolves, as shown inFig. 5. VI. DISCUSSION
From our analytical calculations, we have found thatin contrast to the time independent case, the linear en-tropy exhibits a time dependence that is not determinedby the global phase of universe’s ground state wave func-tion, Eq. (27), but by the auxiliary dynamical param-eters B ∓ ( t ). Numerical results suggest that when theset of auxiliary dynamical parameters are under para-metric resonance, the trace of (cid:16) ˆ ρ ( R ) G ( t ) (cid:17) decays expo-nentially with time with periodic modulations. A care-ful exploration of the instability zone, shows that theimaginary part of the Floquet exponent grows from zerowithin a certain interval and then falls to zero as shownin Fig. 1( iii ). This fact involves a transition between os-cillatory and diverging linear entropy and is related withthe upper quantum Lyapunov exponent [20, 21]. Thecharacteristic length of Eq. (26) is bounded in the stablezones and unbounded in the unstable zones. The prod-uct of the two characteristic length scales in the system isproportional to the Gaussian normalisation factor of theground state, a result that tells us about the relative vol-ume in coordinate space that the wave packet occupies.The numerical simulations of the ground state probabil-ity density show that for values of the static couplingthat belong to the common stability zones, the proba- FIG. 4: Time evolution of the Wigner function in p − q space for the “unstable zone” g = 0 .
38, and ∆ g = 0 . g ; a) Ω t = 0, b)Ω t = 32, and c) Ω t = 50.FIG. 5: Time evolution of the Wigner function in p − q -space for the “stable zone” g = 0 .
4, and ∆ g = 0 .
1; a) Ω t = 0, b)Ω t = 32, and c) Ω t = 50. bility is localized in the x − x space and exhibits anoscillatory behaviour. In contrast, for values of the staticcoupling that belong to the unstable zones, the probabil-ity density has also an oscillatory behaviour but with adifferent ingredient: when the system evolves, the den-sity is systematically stretched in a fixed direction, witha consequent dilation in the perpendicular direction, this behavior results in a massive delocalisation of the wavefunction of the total system. This behaviour is very sug-gestive, because the Wigner function for one oscillatorin our case exhibits a similar pattern, but with a cor-responding change in the stretching direction in phasespace as time evolves, the Wigner function of one oscil-lator ensemble results in a delocalisation in phase space,a fact that we associate with the “maximal mixing” sce-nario.Recently, Blume-Kohout and Zurek [22] analyzed amodel in which they considered a harmonic oscillatorcoupled to an unstable environment, which consisted ofan inverted harmonic oscillator. They demonstrated thatthis unstable environment could produce decoherence inthe system of interest more effectively than a bath of har-monic oscillators. They used the von Neumann entropyas a measure of bipartite entanglement, and surprisingly,entropy increases linearly over time with periodic modu-lations. In contrast to this result, when the environmentis a stable oscillator, the von Neumann entropy oscil-lates, but does not increase over time. We can think ofour system from another perspective: assuming that theuniverse is composed of two coupled oscillators, we con-sider that one oscillator is the system of interest, and theother oscillator is the environment. In contrast to themodel [22], in our case, both systems are stable, how-ever, control over the dynamics is established throughthe time-dependent coupling between the oscillators. Nu-merical results suggest that the von Neumann entropy forthe system’s reduced density operator exhibits the samebehavior described above. Comparing our results withthose found by Blume-Kohout and Zurek, we find thatthe effect of external control over the system is to pro-duce instability in the universe, which is related to thephenomenon of parametric resonance. This instability isreflected in a loss of coherence in the system of interest. VII. CONCLUSIONS
We have obtained exact results for the oscillator’s re-duced density operator and the linear entropy. Theresults allow us to study the entanglement dynamicsthrough the calculation of the auxiliary dynamical pa-rameters. Our results for the Wigner function are to becontrasted with the time independent case, where the en-tanglement does not exhibit dynamics because the tem-poral dependence of the ground state of the universeis given by a global phase of the wave function. Wehave found that near the critical coupling, the oscillator’sWigner function in the time-independent case results ina delocalisation in phase space, a fact that we associatewith the “maximal mixing” scenario reported in [10] viathe divergence of the von Neumann entropy. We havegiven a detailed prescription for the new entanglementfeatures associated to the system in terms of the linearentropy and its relation to the Wigner function. The aux-iliary dynamical parameters and its stability propertiesplay a crucial role when determining the entanglementproperties of the system and the behaviour of the uni-verse ground state wave packet.
Acknowledgments
We acknowledge partial financial support fromCOLCIENCIAS under contract 1106-45-221296, andthe scientific exchange program PROCOL (DAAD-Colciencias).
APPENDIX A: SOLUTION OF THESCHR ¨ODINGER EQUATION FOR ATIME–DEPEDENT QUADRATICHAMILTONIAN
We use the Lewis–Riesenfeld canonical method to findthe exact solution of the Schr¨odinger equation for a sys-tem described by the Hamiltonian Eq. (1). With thepurpose of building the invariant operator, we define thetime–dependent ladder operatorsˆ a i ( t ) = X k =1 ( A ik ( t )ˆ x k + B ik ( t )ˆ p k ) , (A1)and ˆ a † i ( t ) = X k =1 ( A ∗ ik ( t )ˆ x k + B ∗ ik ( t )ˆ p k ) , (A2)which satisfy the standard bosonic commutation rela-tions [ˆ a m ( t ) , ˆ a n ( t )] = [ˆ a † m ( t ) , ˆ a † n ( t )] = 0 , (A3)and [ˆ a m ( t ) , ˆ a † n ( t )] = δ mn . (A4)The operators ˆ a i ( t ) and ˆ a † i ( t ) must satisfy the equationEq. (5), a condition which implies that the auxiliarydynamical parameters A ik and B ik satisfy the followingdifferential equations¨ B i ( t ) = ˙ µ ( t ) µ ( t ) ˙ B i − µ ( t ) ν ( t ) B i ( t ) − µ ( t ) γ ( t ) B i ( t ) , (A5)¨ B i ( t ) = ˙ µ ( t ) µ ( t ) ˙ B i − µ ( t ) ν ( t ) B i ( t ) − µ ( t ) γ ( t ) B i ( t ) , (A6)and ˙ B ik = − µ k ( t ) A ik ( t ) . (A7)As a consequence of these requirements, the auxiliarydynamical parameters B ik ( t ) should satisfy X k ˙ B mk ( t ) B nk ( t ) µ k ( t ) − ˙ B nk ( t ) B mk ( t ) µ k ( t ) ! = 0 , (A8) X k ˙ B ∗ mk ( t ) B ∗ nk ( t ) µ ∗ k ( t ) − ˙ B ∗ nk ( t ) B ∗ mk ( t ) µ ∗ k ( t ) ! = 0 , (A9) X k ˙ B mk ( t ) B ∗ nk ( t ) µ k ( t ) − B mk ( t ) ˙ B ∗ nk ( t ) µ ∗ k ( t ) ! = iδ mn . (A10)0Using Eqs. (A5) and (A6), one can easily show that Eqs.(A8), (A9), and (A10) must be time independent [12, 13].Interestingly, the auxiliary dynamical parameters B ik ( t )are solutions of the classical equations of motion for anon-conservative Hamiltonian system [3, 13], as is clearlyseen in Eqs. (A5) and (A6). This fact is related to theEhrenfest theorem, because for a particle in a parabolicpotential well, the motion of the center of the wavepacket rigorously obeys the laws of classical mechanics[23]. The previous procedure allows us to define the Her-mitian invariant operator ˆ O ( t ) = ˆ n ( t ) + ˆ n ( t ), where ˆ n k ( t ) = ˆ a † k ( t )ˆ a k ( t ) are time–dependent number opera-tors. We construct the Fock space consisting of time–dependent number states | n , n , t i , where the vacuumstate | , , t i is the only one that is annihilated by ˆ a ( t )and ˆ a ( t ). The number state is obtained by applyingˆ a † ( t ) and ˆ a † ( t ) to the vacuum state | n , n , t i = (ˆ a † ( t )) n (ˆ a † ( t )) n p ( n )!( n )! | , , t i . (A11) [1] M. B. Plenio and S. Virmani, Quant. Inf. Comp. , 1(2007).[2] C. H. Bennett and D. P. DiVincenzo, Nature , 247(2000).[3] V. I. Arnold, Mathematical Methods of Classical Mechan-ics (Springer-Verlag, New York, 1978).[4] J. H. Shirley, Phys. Rev. , B979 (1965).[5] T. Dittrich, P. H¨anggi, G. Ingold, B. Kramer, G. Sch¨onand W. Zwerger,