Entanglement in Coupled Harmonic Oscillators via Unitary Transformation
aa r X i v : . [ qu a n t - ph ] F e b ucd-tpg:1106.06 Entanglement in Coupled Harmonic OscillatorsStudied Using a Unitary Transformation
Ahmed Jellal a,b,c ∗ , Fethi Madouri a,d and
Abdeldjalil Merdaci a,ea
Physics Department, College of Science, King Faisal University,PO Box 380, Alahsa 31982, Saudi Arabia b Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia c Theoretical Physics Group, Faculty of Sciences, Choua¨ıb Doukkali University,PO Box 20, 24000 El Jadida, Morocco d IPEIT, Rue J. Lal Nehru, Montfleury, University of Tunis, Tunisia e D´epartement des Sciences Fondamentales, Universit´e 20.08.1955 Skikda,BP 26, DZ-21000 Skikda, Algeria
Abstract
We develop an approach to study the entanglement in two coupled harmonic oscillators. Westart by introducing an unitary transformation to end up with the solutions of the energy spec-trum. These are used to construct the corresponding coherent states through the standard way. Toevaluate the degree of the entanglement between the obtained states, we calculate the purity func-tion in terms of the coherent and number states, separately. The result is yielded two parametersdependance of the purity, which can be controlled easily. Interesting results are derived by fixingthe mixing angle of such transformation as π . We compare our results with already published workand point out the relevance of these findings to a systematic formulation of the entanglement effectin two coupled harmonic oscillators. PACS numbers: 03.65.Ud, 03.65.-w, 03.67.-aKeywords: harmonic oscillator, unitary transformation, purity, entanglement. ∗ [email protected] - [email protected] Introduction
Entanglement is one of the most remarkable features of quantum mechanics that does not have any clas-sical counterpart. It is a notion which has been initially introduced and coined by Schr¨odinger [1] whenquantum mechanics was still in its early stage of development. Its status has evolved throughout thedecades and has been subjected to significant changes. Traditionally, entanglement has been relatedto the most quantum mechanical exotic concepts such as Schr¨odinger cat [1], Einstein-Podolsky-Rosenparadox [2] and violation of Bell’s inequalities [3]. Despite its conventional significance, entanglementhas gained, in the last decades, a renewed interest mainly because of the development of the quantuminformation science [4]. It has been revealed that it lies at the heart of various communication andcomputational tasks that cannot be implemented classically. It is believed that entanglement is themain ingredient of the quantum speed-up in quantum computation [4]. Moreover, several quantumprotocols such as teleportation, quantum dense coding, and so on [5, 6, 7, 8, 9, 10, 11] are exclusivelyrealized with the help of entangled states. With this respect, many interesting works appeared dealingwith the development of a quantitative theory of entanglement and the definition of its basic measure.These concern the concurrence, entanglement of formation and linear entropy [12, 13, 14, 15].Entangled quantum systems can exhibit correlations that cannot be explained on the basis ofclassical laws and the entanglement in a collection of states is clearly a signature of non-classicality[16]. Furthermore, in the last few years it has become evident that quantum information may lead tofurther insights into other areas of physics [17]. This has led to a cross-fertilizing between differentareas of physics. It is worthy of note that the nonlinear Kerr effect [18] has been considered as themost famous source of physical realization of photon pairs of entangled polarization states. However,it raises a number of difficulties to the control of photons that are traveling at the speed of light. Thisis why so much attention has been paid recently to the entangled states of massive particles as theyare viewed to be much more easy to control [17, 19].On the other hand, the harmonic oscillator machinery plays a crucial role in many areas of physics.These are the Lee model in quantum field theory [20], the Bogoliubov transformation in superconduc-tivity [21], two-mode squeezed states of light [22, 23, 24], the covariant harmonic oscillator model forthe parton picture [25], and models in molecular physics [26]. There are also models in which one ofthe variables is not observed, including thermo-field dynamics [27], two-mode squeezed states [28, 29],the hadronic temperature [30], and the Barnet-Phoenix version of information theory [31]. Thesephysical models are the examples of Feynman’s rest of the universe. In the case of two coupled har-monic oscillator, the first one is the universe and the second one is the rest of universe. For sakeof the mathematical simplicity, the mixing angle (rotation of the coordinate system), in the abovementioned references, is taken to be equal π . This means that the system consists of two identicaloscillator coupled together by a potential term.In the context of the entangled massive particles, we cite the recently achieved investigation of aspecific realization of two coupled harmonic oscillator model by the authors of reference [19]. In fact,they calculated the interatomic entanglement for Gaussian and non-Gaussian pure states by usingthe purity function of the reduced density matrix. This allowed them to treat the cases of free andtrapped molecules and hetero- and homonuclear molecules. Finally, they concluded that when thetrap frequency and the molecular frequency are very different, and when the atomic masses are equal,1he atoms are highly-entangled for molecular coherent states and number states. Surprisingly, whilethe interatomic entanglement can be quite large even for molecular coherent states, the covarianceof atomic position and momentum observables can be entirely explained by a classical model withappropriately chosen statistical uncertainty.Motivated by the mentioned references above and in particular [19], we undertake to develop anew approach to study the entanglement in two coupled harmonic oscillators. It is based on a suitabletransformation having the merit of reducing the relevant physical parameters into two: the couplingparameter η and mixing angle θ . It turns out that we can easily derive the solutions corresponding tothe energy spectrum. Then, the obtained solutions are used to construct the coherent states throughthe standard method. In order to characterize the degree of entanglement, we calculate, within theframework of the coherent states, the purity function. Then the final form of the purity is cast interms of η and θ . Our finding shows two interesting results: the first one tells us that the presentsystem is not entangled at η = 0, as expected, and highly entangled at large η (Figure 1). The secondone is when we fix θ = π , the purity behaves like the inverse of cosh η and the corresponding plot(Figure 2) shows that the purity is ranging between 0 and 1. It is worthy of notice that, in this case,the purity becomes one parameter dependent, which means that it is easy to control.Subsequently, we evaluate the purity in terms of the number states. In doing so, we use the well-known relation to express the number states | n , n i as function of the corresponding coherent states | α, β i . Then after a lengthy but a straightforward algebra, we end up with the final form of purity. Tobe much more concrete, we restrict ourselves to some interesting cases that are ( n = 1 , n = 0) and( n = 1 , n = 1). For the first configuration, the obtained purity is simply a ratio of hyperbolic andsinusoidal functions, which tells us that the entanglement is maximal at large η for all θ (Figure 3). Inthe particular case when θ = π , the purity is typically a ratio of a hyperbolic cosine function, whichshows clearly that the purity is positive as it should be (Figure 4). The second configuration gives alsoa mixing dependence between the hyperbolic and sinusoidal functions where the corresponding plots(Figures 5 and 6) show some difference in the form with respect to the first one. In both cases, wenotice that the numerators are always hyperbolic cosine of even η and denominators are also power ofthe function cosh η .The present paper is organized as follows. In section 2, we review the derivation of the solutionsof the energy spectrum of two coupled harmonic oscillators [32]. These will be used to build thecorresponding coherent states and therefore evaluate the purity function of the reduced matrix elementsin section 3. The final form of purity function is subjected to different investigations where we underlineits dependence to two physical parameters η and θ . In section 4, we evaluate the purity in terms ofthe number states after a series of transformation. Two interesting case of the purity will be discussedin section 5. Finally, we give conclusion and perspective of our work. In doing our task, we consider a system of two coupled harmonic oscillators parameterized by theplanar coordinates ( X , X ) and masses ( m , m ). Accordingly, the corresponding Hamiltonian is2ritten as the sum of free and interacting parts [33] H = 12 m P + 12 m P + 12 (cid:0) C X + C X + C X X (cid:1) (1)where C , C and C are constant parameters. After rescaling the position variables x = µX , x = µ − X (2)as well as the momenta p = µ − P , p = µP (3) H can be written as H = 12 m (cid:0) p + p (cid:1) + 12 (cid:0) c x + c x + c x x (cid:1) (4)where the parameters are given by µ = ( m /m ) / , m = ( m m ) / , c = C r m m , c = C r m m , c = C . (5)As the Hamiltonian (4) involves an interacting term, a straightforward investigation of the basicfeatures of the system is not easy. Nevertheless, we can simplify this situation by a transformation tonew phase space variables y a = M ab x b , ˆ p a = M ab p b (6)where the matrix ( M ab ) = cos θ − sin θ sin θ cos θ ! . (7)is a unitary rotation with the mixing angle θ . Inserting the mapping (6) into (4), one realizes that θ should satisfy the condition tan θ = c c − c (8)to get a factorizing Hamiltonian H = 12 m (cid:0) ˆ p + ˆ p (cid:1) + k (cid:0) e η y + e − η y (cid:1) (9)where we have introduced two parameters k = q c c − c / , e η = c + c + p ( c − c ) + c k (10)under the reserve that the condition 4 c c > c must be fulfilled. The parameter η is actually measuringthe strength of the coupling.For later use, it is convenient to separate the Hamiltonian (9) into two commuting parts and thenwrite H as H = e η H + e − η H (11)where H and H are given by H = 12 m e − η ˆ p + k e η y , H = 12 m e η ˆ p + k e − η y . (12)3ne can see that the decoupled Hamiltonian H = 12 m ˆ p + k y + 12 m ˆ p + k y (13)is obtained for η = 0, which is equivalent to set c = 0.The Hamiltonian H can simply be diagonalized by defining a set of annihilation and creationoperators. These are a i = r k ~ ω e εη y i + i √ m ~ ω e − εη ˆ p i , a † i = r k ~ ω e εη y i − i √ m ~ ω e − εη ˆ p i (14)with the frequency ω = r km (15)and ε = ± i = 1 ,
2, respectively. They satisfy the commutation relations[ a i , a † j ] = δ ij (16)whereas other commutators vanish. Now we can map H in terms of a i and a † i as H = ~ ω (cid:16) e η a † a + e − η a † a + cosh η (cid:17) . (17)To obtain the eigenstates and the eigenvalues, one solves the eigenequation H | n , n i = E n ,n | n , n i (18)getting the states | n , n i = ( a † ) n ( a † ) n √ n ! n ! | , i (19)as well as the energy spectrum E ,n ,n = ~ ω (cid:0) e η n + e − η n + cosh η (cid:1) . (20)It is clear that these eigenvalues reduce to those of the decoupled harmonic oscillators, namely ~ ω ( n + n + 1). This shows clearly that the presence of the coupling parameter η will make dif-ference and allow us to derive interesting results in the forthcoming analysis.To show the correlation between variables, let us just focus on the ground state and write thecorresponding wavefunction in y -representation. This is ψ ( ~y ) ≡ h y , y | , i = r mωπ ~ exp n − mω ~ (cid:0) e η y + e − η y (cid:1)o (21)which can easily be used to deduce the ground state wavefunction in terms of the variables ( x , x ).Therefore, from the unitary representation we find ψ ( ~x ) ≡ h x , x | , i = r mωπ ~ exp ( − mω ~ " e η (cid:18) x cos θ − x sin θ (cid:19) + e − η (cid:18) x sin θ x cos θ (cid:19) . (22)We notice that (21) is separable in terms of the variables y and y , which is not the case for (22) interms of x and x . We close this part by claiming that the obtained results so far will be used tostudy the entanglement in the present system. 4 Entanglement in coherent states
As we claimed above, we implement our approach to study the entanglement of two coupled harmonicoscillators. Actually, it can be seen as another alternative method to recover the results obtainedin [19] not only in a simpler way but also with less physical parameters of control. To start let us firstintroduce the coherent states corresponding to the eigenstates | n , n i given in (19). As usual, we canuse the displacement operator to define the coherent states in terms of two complex numbers α and β . These are | α, β i = D ( a , α ) D ( a , β ) | , i (23)which gives the wavefunctionΦ αβ ( y , y ) = (cid:18) λ λ π (cid:19) / exp " − λ y − | α | − α √ αλ y − λ y − | β | − β √ βλ y (24)where we have set the quantities λ = e η (cid:18) mk ~ (cid:19) / , λ = e − η (cid:18) mk ~ (cid:19) / . (25)In terms of the original variables ( X , X ), (24) reads asΦ αβ ( X , X ) = (cid:16) λ λ π (cid:17) / exp (cid:20) − λ (cid:16) µ cos θ X − µ sin θ X (cid:17) − λ (cid:16) µ sin θ X + µ cos θ X (cid:17) (cid:21) × exp h √ αλ (cid:16) µ cos θ X − µ sin θ X (cid:17) + √ βλ (cid:16) µ sin θ X + µ cos θ X (cid:17)i × exp h − | α | − | β | − α − β i . (26)As it is clearly shown in the wavefunction (26), the non-separability of the variables will play in crucialrole in discussing the entanglement in the present system. This statement will be clarified later onwhen we will come to the analysis of the role of the involved parameters.At this level we have set all ingredients to study the entanglement in the present system. All weneed is to determine explicitly the purity function that is a trace of the density square correspondingto the obtained eigenstates. More precisely, we have P = Tr ρ (27)which in terms of the above coherent states reads as P αβ = Z dX dX ′ dX dX ′ Φ αβ ( X , X ) Φ ∗ αβ (cid:0) X ′ , X (cid:1) Φ αβ (cid:0) X ′ , X ′ (cid:1) Φ ∗ αβ (cid:0) X , X ′ (cid:1) . (28)Upon substitution, we obtain the form P αβ = (cid:18) λ λ π (cid:19) Z dX dX ′ dX dX ′ e − µ ( λ cos θ + λ sin θ )( X + X ′ ) − µ ( λ sin θ + λ cos θ )( X + X ′ ) × e ( λ − λ ) sin θ ( X ′ X + X X + X ′ X ′ + X X ′ ) e µ (cid:16) α + α ∗√ λ cos θ + β + β ∗√ λ sin θ (cid:17) ( X + X ′ ) (29) × e − µ (cid:16) α + α ∗√ λ sin θ − β + β ∗√ λ cos θ (cid:17) ( X + X ′ ) e − | α | − | β | − α − α ∗ − β − β ∗ . X X ′ X X ′ = 12 ω µ √ − a √ µ ω ω µ √ a ω µ √ − a − √ µ ω ω µ √ a − µω √ − a µω √ a √ µω − µω √ − a µω √ a −√ µω u u u u (30)where ω , ω and a are given by ω = 1 q λ cos θ + λ sin θ , ω = 1 q λ sin θ + λ cos θ , a = − (cid:0) λ − λ (cid:1) sin θω ω . (31)By showing that the determinant of such transformation is ω ω λ λ , it is easy to map P αβ in terms of thenew variables as P αβ = 1 π λ λ ω ω e − | α | − | β | − α − α ∗ − β − β ∗ Z + ∞−∞ du du e − u − u × Z + ∞−∞ du e − u + √ √ − a [ λ ( ω cos θ + ω sin θ ) ( α + α ∗ )+ λ ( ω sin θ − ω cos θ ) ( β + β ∗ ) ] u (32) × Z + ∞−∞ du e − u + √ √ a [ λ ( ω cos θ − ω sin θ ) ( α + α ∗ )+ λ ( ω sin θ + ω cos θ ) ( β + β ∗ ) ] u . Performing the integration to end up with the result P αβ ( η, θ ) = 1 q η sin θ cos θ + cos θ + sin θ . (33)This is among the interesting results derived so far in the present work. Indeed, it shows clearly thatthe purity depends on the physical parameters ( η, θ ) rather than the complex displacements ( α, β )and hereafter it will be denoted by P ( η, θ ). Furthermore, the obtained purity is two parametersdependent, which means that it can be controlled easily. If one requires the decoupling case ( η = 0), P ( η, θ ) reduces to one as expected and therefore there is no entanglement.To understand better the above results, we recall that the purity is related to linear entropy bythe simple form L = 1 − P (34)where P lies in the interval [0 , η and byconsidering θ ∈ [0 , π ]. From Figure 1, it is clear that the purity, as function of η , is symmetric withrespect to the decoupling case η = 0. It is maximal for η = 0, which really shows that the systemis disentangled. After that it decreases rapidly to reach zero and indicates that the entanglement ismaximal. More importantly, the purity becomes constant whenever θ takes the value zero or π . Thisbehavior of the purity traced in below tell us that one can easily play with two parameters to controlthe degree of entanglement in the present system.6 - Η Θ P H Η , Θ L Figure 1: Purity in terms of the coupling parameter η and the mixing angle θ . Specifically at θ = π , we obtain a simple form P (cid:16) η, θ = π (cid:17) = 1cosh η (35)which is one parameter dependent and can be adjusted only by varying the coupling η to control thedegree of the entanglement. To be much more accurate, we underline such behavior by plotting (35)in Figure 2: - - Η P H Η L Figure 2: Purity in terms of the coupling parameter η for the mixing angle θ = π . From the above figure, one can deduce two interesting conclusions. The first one tells as that P ( η, θ )is bounded, i.e. 0 ≤ P ≤
1, as expected. The second one shows clearly that the purity goes to zerofor a strong coupling, which indicates the entanglement is maximal.
To gain more information about the behavior of the present system, we evaluate the degree of theentanglement between inter states. For this, we consider the relation inverse to express the numberstates in terms of the coherent states. This is | n , n i = 1 √ n ! n ! ∂ n ∂α n ∂ n ∂β n e | α | e | β | | α, β i (cid:12)(cid:12)(cid:12)(cid:12) α =0 ,β =0 . (36)7n the y -representation, (36) leads to the wavefunction˜Φ n n ( y , y ) = ˜Φ n n (cid:18) µ cos θ X − µ sin θ X , µ sin θ X + 1 µ cos θ X (cid:19) ≡ ˜Φ n n ( X , X )= 1 √ n ! n ! ∂ n ∂α n ∂ n ∂β n e | α | e | β | Φ αβ ( X , X ) (cid:12)(cid:12)(cid:12)(cid:12) α =0 ,β =0 (37)where Φ αβ ( X , X ) is given in (26). This will be implemented to study the purity in terms of thenumber states and discuss different issues.Returning back to the purity definition, we have P n n = Z dX dX ′ dX dX ′ ˜Φ n n ( X , X ) ˜Φ ∗ n n (cid:0) X ′ , X (cid:1) ˜Φ n n (cid:0) X ′ , X ′ (cid:1) ˜Φ ∗ n n (cid:0) X , X ′ (cid:1) . (38)Using (37) to obtain the form P n n = Z dX dX ′ dX dX ′ (cid:18) n ! n ! (cid:19) × ∂ n ∂α n ∂ n ∂β n e | α | e | β | Φ α β (cid:18) µ cos θ X − µ sin θ X , µ sin θ X + 1 µ cos θ X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α ,β =0 × ∂ n ∂α ∗ n ∂ n ∂β ∗ n e | α | e | β | Φ ∗ α β (cid:18) µ cos θ X ′ − µ sin θ X , µ sin θ X ′ + 1 µ cos θ X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α ∗ ,β ∗ =0 (39) × ∂ n ∂α n ∂ n ∂β n e | α | e | β | Φ α β (cid:18) µ cos θ X ′ − µ sin θ x ′ , µ sin θ X ′ + 1 µ cos θ x ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α ,β =0 × ∂ n ∂α ∗ ∂ n ∂β ∗ n e | α | e | β | Φ ∗ α β (cid:18) µ cos θ x − µ sin θ x ′ , µ sin θ X + 1 µ cos θ X ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α ∗ ,β ∗ =0 . After some algebra, we show that the purity takes the form P n n = (cid:18) λ λ πn ! n ! (cid:19) Q i =1 ∂ n ∂α n i ∂ n ∂β n i Z dX dX ′ dX dX ′ e − ( α + β + α + β + α + β + α + β ) e − λ (cid:20)(cid:16) µ cos θ X − µ sin θ X (cid:17) + (cid:16) µ cos θ X ′ − µ sin θ X ′ (cid:17) + (cid:16) µ cos θ X ′ − µ sin θ X (cid:17) + (cid:16) µ cos θ X − µ sin θ X ′ (cid:17) (cid:21) e − λ (cid:20)(cid:16) µ sin θ X + µ cos θ X (cid:17) + (cid:16) µ sin θ X ′ + µ cos θ X ′ (cid:17) + (cid:16) µ sin θ X + µ cos θ X ′ (cid:17) + (cid:16) µ sin θ X ′ + µ cos θ X (cid:17) (cid:21) e √ µ [( λ ( α + α ) cos θ + λ ( β + β ) sin θ ) X + ( λ ( α + α ) cos θ + λ ( β + β ) sin θ ) X ′ ] e − √ µ [( λ ( α + α ) sin θ − λ ( β + β ) cos θ ) X + ( λ ( α + α ) sin θ − λ ( β + β ) cos θ ) X ′ ] . (40)This can be written, in a compact form, as P n n = (cid:18) λ λ πn ! n ! (cid:19) Q i =1 ∂ n ∂α n i ∂ n ∂β n i Z d Ze − Z t · A · Z + B t · Z + C (41)where z T = (cid:16) X X ′ X X ′ (cid:17) , the matrix A is given by A = A A A A A A A A A A A A (42)8uch that their components read as A = µ (cid:18) λ cos θ λ sin θ (cid:19) , A = 1 µ (cid:18) λ sin θ λ cos θ (cid:19) , A = λ − λ θ (43)and the matrix B takes the form B = √ µλ ( α + α ) cos θ + µλ ( β + β ) sin θ µλ ( α + α ) cos θ + µλ ( β + β ) sin θ µ λ ( β + β ) cos θ − µ λ ( α + α ) sin θ µ λ ( β + β ) cos θ − µ λ ( α + α ) sin θ . (44)To go further in evaluating the purity, we perform a method to simplify our calculation. This canbe done by introducing the change of variables X X ′ X X ′ = ω µ √ − a √ µ ω ω µ √ a ω µ √ − a − √ µ ω ω µ √ a − µω √ − a µω √ a √ µω − µω √ − a µω √ a − √ µω x x x x (45)where the corresponding measure is dX dX ′ dX dX ′ = J dx dx dx dx and the Jacobian reads as J = 1 λ λ q(cid:0) λ cos θ + λ sin θ (cid:1) (cid:0) λ sin θ + λ cos θ (cid:1) . (46)This performance allows us to map (41) as P n n = (cid:18) λ λ πn ! n ! (cid:19) J Q i =1 ∂ n ∂α n i ∂ n ∂β n i e − ( α i + β i ) Z d Qe − Q + D t Q (47)where Q t = (cid:16) x x x x (cid:17) and D t is the transpose of D , such as D = √ ω cos θ + ω sin θ √ − a λ ( α + α + α + α ) + ω sin θ − ω cos θ √ − a λ ( β + β + β + β ) √ ω λ ( α + α − α − α ) cos θ + √ ω λ ( β + β − β − β ) sin θ ω cos θ − ω sin ( θ ) √ a λ ( α + α + α + α ) + ω sin θ + ω cos θ √ a λ ( β + β + β + β ) − √ ω λ ( α + α − α − α ) sin θ + √ ω λ ( β + β − β − β ) cos θ . (48)Since the above integral is Gaussian, then after some algebra we end with the form P n n = (cid:18) λ λ πn ! n ! (cid:19) J Q i =1 ∂ n ∂α n i ∂ n ∂β n i exp (cid:20) uρ α + 2 vρ α α − uρ α α + 2 wρ α α + 2 sρ α β − tρ α β − sρ α β + 2 tρ α β + uρ α + 2 wρ α α − uρ α α − tρ α β + 2 sρ α β + 2 tρ α β − sρ α β + uρ α + 2 vρ α α − sρ α β + 2 tρ α β + 2 sρ α β − tρ α β (49)+ uρ α + 2 tρ α β − sρ α β − tρ α β + 2 sρ α β − uρ β + 2 wρ β β + 2 uρ β β + 2 vρ β β − uρ β + 2 vρ β β + 2 uρ β β − uρ β + 2 wρ β β − uρ β (cid:21) ρ = 4 mk ~ (cid:20) η ) + cot θ θ (cid:21) , u = 2 mk ~ sinh 2 ηv = 2 mk ~ (cid:20) cosh(2 η ) + tan θ (cid:21) , w = 2 mk ~ (cid:20) cosh(2 η ) + cot θ (cid:21) (50) t = 4 mk ~ cosh η sin θ , s = − mk ~ sinh η cos θ sin θ . We are still looking for the final form of the purity, which can be obtained by calculating the partialderivatives. These can be performed in different ways and may be it is easier to proceed step by step.Indeed, we factorize the exponential function and then map each factor into a series expansion. Thisoperation has been postponed to Appendix A and the yielded result is P n n ( η, θ ) = (cid:16) ρ (cid:17) n n ( n ! n !) sin( θ ) q η +tan ( θ ) +cot ( θ ) P i + j + k + l + r =2( n + n ) C n n ( i, j, k, l, r ) u i v j w k t l s r (51)where the coefficients C n n are given by C n n = Q e =1 i e − P i e =0 ! Q e =1 j e − P j e =0 ! Q e =1 k e − P k e =0 ! Q e =1 l e − P l e =0 l e − − l e )! ! (cid:18) Q e =1 r e − P r e =0 r e − − r e )! (cid:19) − i ( − l − l + l − l + l − l + r − r + r − r + r − r + i − c − c ( i − i )! ( i − i )! ( i − i )! ( i − i )! l ! r ! c ! c ! c ! c ! c ! c ! c ! c ! . (52)It is clear that the final form of the purity is actually only depending on two parameters, i.e. η and θ . On the other hand, it is easy to check that P n n is symmetric under the change of the quantumnumbers n and n . To be much more accurate let is illustrate some particular cases. With these we will be able to getmore information from the above purity about the degree of entanglement. In the beginning, let uschoose the configuration ( n = 0 , n = 1), which means that we are considering now the entanglementbetween the ground state of the first oscillator and the first excited state of the second one. In thiscase, (51) reduces to the form P ( η, θ ) = (cid:16) ρ (cid:17) sin θ q η )+tan θ +cot θ P l + r + j + k + i =2 C ( i, j, k, l, r ) u i v j w k t l s r (53)which can be evaluated to obtain P ( η, θ ) = (cid:16) ρ (cid:17) sin θ q η )+tan θ +cot θ (cid:0) u + v + w (cid:1) (54)and after replacing different parameters, one gets the final result P ( η, θ ) = 3 cosh (4 η ) + 4 (cid:0) tan θ + cot θ (cid:1) cosh (2 η ) + 2 tan θ + 2 cot θ + 1sin θ (cid:0) η ) + tan θ + cot θ (cid:1) . (55)This is a nice form, which can be worked more since it is only function of two physical parameters η and θ . Indeed, we plot it in Figure 3: 10 - Η Θ P H Η , Θ L Figure 3: Purity P as function of the coupling parameter η and mixing angle θ for the quantumnumbers ( n = 0 , n = 1) . Here we have the same conclusion as in Figure 1 except that the present plot is showing some de-formation at the point η = 0. Otherwise, for certain values of θ the purity is not always holding amaximum value at η = 0. More precisely, at this point it decreases to reach 1 / θ = π and thenincreases to attends 1 at θ = π . This is because in the present case the masses are equal and the sameconclusion is obtained in [19].Now let us look at some interesting situations by fixing the mixing angle θ and varying the couplingparameter η . In particular when θ = π , P reduces to the form P (cid:16) η, θ = π (cid:17) = 3 cosh(4 η ) + 8 cosh(2 η ) + 532 cosh η . (56)This can be plotted to obtain Figure 4: - - Η P H Η L Figure 4: Purity P as function of η measuring the entanglement between the ground state n = 0 andthe first excited state n = 1 for θ = π . Compared to Figure 2, we notice that the behavior of the purity in terms of the coupling parameter η is large. As long as η is large the entanglement is going to hold the maximum value. It shows clearly11he role playing by η and thus allows an easy control of the degree of the entanglement. This maygive some hint about an experiment realization of the present case.Now let us look at the case of the entanglement between the two first excited states of the twooscillators, i.e. n = n = 1. This result gives P ( η, θ ) = (cid:16) ρ (cid:17) sin θ q η )+tan θ +cot θ P i + j + k + l + r =4 C ( i, j, k, l, r ) u i v j w k t l s r (57)after lengthy but simple calculations, we find P = (cid:16) ρ (cid:17) sin θ q η )+tan θ +cot θ (58) × (cid:0) u + v + w + 2 s + 2 t + 2 s t + 2 u v + 2 u w + 2 v w + 2 ustv − ustw − u s − u t − t w − s w − t v − s v + 2 vws + 2 vwt (cid:1) . Finally, we obtain P ( η, θ ) = 14 sin θ (cid:2) η ) + tan θ + cot θ (cid:3) (cid:20) η ) + 16 (cid:18) tan θ θ (cid:19) cosh 6 η + (cid:18)
96 tan θ θ − (cid:19) cosh (4 η ) + 240 (cid:18) tan θ θ (cid:19) cosh (2 η )+8 tan θ θ −
64 tan θ −
64 cot θ (cid:21) . (59)Comparing this with (55), we notice that the numerator of both of them is containing a hyperboliccosine function of a even number of coupling parameter η and the denominators are power of cosh η .To go further, we plot (58) in Figure 5: - - Η Θ P H Η , Θ L Figure 5: Purity P as function of the coupling parameter η and mixing angle θ for the quantumnumbers ( n = 1 , n = 1) . Clearly, we see that for certain values of θ the purity is not always holding a maximum value atdecoupling case, i.e η = 0. At this point, the purity decreases to reach 1 / θ = π and then increasesto attends 1 at θ = π . 12urthermore, (59) can be worked much more to underline its behavior. The simplest way to do sois to fix the mixing angle θ and play with the coupling parameter η . For instance, by requiring θ = π we end up with the form P ( η, θ ) = 9 cosh(8 η ) + 32 cosh(6 η ) + 156 cosh(4 η ) + 480 cosh(2 η ) + 3472048 cosh η . (60)This shows clearly that P ( η, θ ) is one parameter dependent and therefore it can be manipulatedeasily. For more precision, we plot (60) in Figure 6: - - Η P H Η L Figure 6: Purity P as function of η measuring the entanglement between the first exited states ( n = 1 , n = 1) for θ = π . This is showing a difference with respect to Figure 4. It is clear that as long as η is small the purityincreases rapidly to reach its maximal value. Also it decreases rapidly to attend zero for large η , whichmeans that the system is strongly entangled. The present work is devoted to study the entanglement of two coupled harmonic oscillators by adopt-ing a new approach. For this, a Hamiltonian describing the system is considered and an unitarytransformation is introduced. With this latter, the corresponding solutions of the energy spectrumare obtained in terms of the coupling parameter η and the mixing angle θ . It is clearly seen that when η = 0, the system becomes decoupled and therefore nothing new except harmonic oscillator in twodimensions.To study the entanglement of the present system, we have introduced the purity function toevaluate its degree. In the beginning, we have realized the corresponding coherent states by using thestandard method based on the displacement operator. These are used to determine explicitly the formof the purity in terms of the physical parameters η and θ . Also, the obtained result confirmed therange of the purity that is 0 ≤ P ≤
1. Moreover, we have clearly shown that purity is easy to controland can also be cast in a simple form when we fix θ = π . In such case the purity is obtained as the13nverse of the hyperbolic function cosh η and the disentanglement simply corresponds to switching off η . Subsequently, we have used the relation inverse between the number of states and the coherentstates to determine the purity. After making different changes of variable, we have got a tractableGaussian form, which was integrated easily. The final result showed that the purity is two parametersdependent. This allowed us to illustrate our finding by restricting ourselves to two particular cases.In the first configuration, we have considered the entanglement between the ground state and excitedstate, i.e. ( n = 0 , n = 1) where the purity is exactly obtained. In the second configuration westudied the entanglement between the states ( n = 1 , n = 1). In both cases, we have analyzed thecase where θ = π , which showed that a strong dependence of the purity to the hyperbolic cosinefunction of even coupling parameter.On the other hand, the system of two coupled oscillators can serve as an analog computer for manyof the physical theories and models. Therefore, one can extend the method developed here to studythe entanglement in other interesting systems those illustrating the Feynman’s rest. Furthermore, oneimmediate extension is to consider the case of a coupled systems submitted to an external magneticfield. This work and related matter are actually under consideration. Acknowledgments
The authors acknowledge the financial support by King Faisal University. The present work is doneunder Project Number: 110135, ”Quantum information and Entangled Nano Electron Systems”. Theauthors would like to thank E.B. Choubabi for the numerical help and are indebted to the referee forhis constructive comment.
Appendix A: Final form of purity
In this appendix, we show how to derive the final form of the purity given in (51). Indeed from (49),we obtain the result P n n = (cid:18) λ λ πn ! n ! (cid:19) J ∞ P i,j,k,l,r =0 Q e =1 i e − P i e =0 i e − − i e )! ! Q e =1 " j e − P j e =0 j e − − j e )! k e − P k e =0 k e − − k e )! Q e =1 " l e − P l e =0 l e − − l e )! r e − P r e =0 r e − − r e )! uρ (cid:19) i (cid:18) vρ (cid:19) j (cid:18) wρ (cid:19) k (cid:18) tρ (cid:19) l (cid:18) sρ (cid:19) r i − i i ! j ! k ! l ! r ! (cid:16) ∂ n ∂α n α a (cid:17) (cid:16) ∂ n ∂α n α a (cid:17) (cid:16) ∂ n ∂α n α a (cid:17) (cid:16) ∂ n ∂α n α a (cid:17) (cid:16) ∂ n ∂β n β a (cid:17) (cid:16) ∂ n ∂β n β a (cid:17)(cid:16) ∂ n ∂β n β a (cid:17) (cid:16) ∂ n ∂β n β a (cid:17)(cid:12)(cid:12)(cid:12) ( α i ,β i )=(0 , (A1)14here different parameters are given by a = 2 i + i − i + l − l + l − l + r − r + r − r + j + k − k a = 2 i − i + i − i + l + l − l + r − r + r − r + j + k a = 2 i − i + i − i + l − l + l − l + r + r − r + j − j + k a = i − i + 2 i − i + j − j + l − l + l − l + k − k + r − r + r − r a = r − r + i − i + l − l + l − l + 2 i − i + r − r + j − j + k − k a = l − l + i − i + l − l + 2 i − i + r − r + r − r + j − j + k − k a = l − l + l + i − i + r + r − r + j − j + k − k + 2 i − i a = l − l + l − l + i − i + r − r + r − r + 2 i − i + k − k + j − j and for the coherence of notations, ( i , j , k , l , r ) ≡ ( i, j, k, l, r ) has to be under heard. Making useof the well-known formula ∂∂x n x l (cid:12)(cid:12)(cid:12) x =0 = n ! δ l,n (A2)we end up with the form P n n = (cid:18) λ λ π n ! n ! (cid:19) J ∞ P i,j,k,l,r =0 Q e =1 i e − P i e =0 1( i e − − i e )! ! Q e =1 j e − P j e =0 1( j e − − j e )! ! Q e =1 k e − P k e =0 1( k e − − k e )! ! × Q e =1 l e − P l e =0 1( l e − − l e )! ! (cid:18) Q e =1 r e − P r e =0 1( r e − − r e )! (cid:19) (cid:18) uρ (cid:19) i (cid:18) vρ (cid:19) j (cid:18) wρ (cid:19) k (cid:18) tρ (cid:19) l (cid:18) sρ (cid:19) r × i − i i ! j ! k ! l ! r ! δ b ,n δ b ,n δ b ,n δ b ,n δ b ,n δ b ,n δ b ,n δ b ,n . (A3)This shows clearly that a non vanishing purity should satisfy a set of constraint on different quantumnumbers. These are b − n = 2 i + i − i − l + l − r + r + j + k − k − n = 0 b − n = 2 i − i + i − i + l − r + r + j + k − n = 0 b − n = 2 i − i + i − i − l + l + r + r − r + j − j + k − n = 0 b − n = i − i + 2 i − i + j − j − l + l + k − k − r + r − n = 0 b − n = l − l + i − i + l − l + 2 i − i + r − r + r − r + j − j + k − k − n = 0 b − n = l − l + l + i − i + r + r − r + j − j + k − k + 2 i − i − n = 0 b − n = r − r + i − i + l − l + l − l + 2 i − i + r − r + j − j + k − k − n = 0 b − n = l − l + l − l + i − i + r − r + r − r + 2 i − i + k − k + j − j − n = 0 . (A4)We arrange the labels into two sets that we refer to them as the principals and secondary ones,respectively. The so-called secondary ones disappear upon summation of the 8 constraints and we get i + j + k + l + r = 2( n + n ) (A5)which is the constraint on the principal labels. The main result that emerges is that the purity is onlydepending on two parameters, such as P n n ( η, θ ) = (cid:16) ρ (cid:17) n n ( n ! n !) sin θ q η )+tan θ +cot θ P i + j + k + l + r =2( n + n ) C n n ( i, j, k, l, r ) u i v j w k t l s r . (A6)15he most important future of our result is that the function C n n ( i, j, k, l, r ) can now be derivedexactly for any n and n . This is C n n = Q e =1 i e − P i e =0 i e − − i e )! ! Q e =1 j e − P j e =0 j e − − j e )! ! Q e =1 k e − P k e =0 k e − − k e )! ! Q e =1 l e − P l e =0 l e − − l e )! ! × (cid:18) e =7 Q e =1 r e − P r e =0 r e − − r e )! (cid:19) − i ( − i − i ( − r − r + r − r + r − r ( − l − l + l − l + l − l i ! j ! k ! l ! r ! ! . (A7)Using the above constraints, we show that C n n can be reduced to the form C n n = Q e =1 i e − P i e =0 ! Q e =1 j e − P j e =0 ! Q e =1 k e − P k e =0 ! Q e =1 l e − P l e =0 l e − − l e )! ! (cid:18) Q e =1 r e − P r e =0 r e − − r e )! (cid:19) − i ( − l − l + l − l + l − l + r − r + r − r + r − r + i − c − c ( i − i )! ( i − i )! ( i − i )! ( i − i )! l ! r ! c ! c ! c ! c ! c ! c ! c ! c ! (A8)where the involved parameters are fixed as c = 12 [2 n − ( i − i ) − ( j − j ) − ( k − k ) − ( l − l ) − ( l − l ) − ( r − r ) − ( r − r )( i − i ) − ( r − r ) − ( j − j ) − ( l − l ) − ( l − l ) − r − k ] c = 12 [ n − ( i − i ) − ( r − r ) − ( r − r ) − l − j − k + ( i − i ) − ( l − l )( l − l ) − ( r − r ) − ( r − r ) − j − ( k − k )] c = (cid:18) n − ( i − i ) − ( r − r ) − ( j − j ) − ( l − l ) − ( l − l ) − r − k (cid:19) ! c = (cid:18) n − ( i − i ) − ( r − r ) − ( r − r ) − l − j − k (cid:19) ! c = (cid:18) n − ( i − i ) − ( l − l ) − ( l − l ) − ( r − r ) − ( r − r ) − j − ( k − k )2 (cid:19) ! c = (cid:18) n − ( i − i ) − ( j − j ) − ( k − k ) − ( l − l ) − ( l − l ) − ( r − r ) − ( r − r )2 (cid:19) ! c = (cid:18) n − ( r − r ) − ( i − i ) − ( l − l ) − ( l − l ) − ( r − r ) − ( j − j ) − ( k − k )2 (cid:19) ! c = (cid:18) n − ( l − l ) − ( i − i ) − ( l − l ) − ( r − r ) − ( r − r ) − ( j − j ) − ( k − k )2 (cid:19) ! c = (cid:18) n − ( l − l ) − ( l − l ) − ( i − i ) − ( r − r ) − ( r − r ) − ( k − k ) − ( j − j )2 (cid:19) ! c = (cid:18) n − ( l − l ) − ( i − i ) − ( r − r ) − ( j − j ) − ( k − k ) − r − l (cid:19) ! 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