Classical Instability Effects on Photon Excitations and Entanglement
CClassical Instability Effects on Photon Excitationsand Entanglement
Radouan Hab-arrih a , Ahmed Jellal ∗ a,b and Abdeldjalil Merdaci ca Laboratory of Theoretical Physics, Faculty of Sciences, Choua¨ıb Doukkali University , PO Box 20, 24000 El Jadida, Morocco b Canadian Quantum Research Center, 204-3002 32 Ave Vernon,BC V1T 2L7, Canada c Facult´e des Sciences, Universit´e 20 Aoˆut 1955 Skikda,BP 26, Route El-Hadaiek 21000, Algeria
Abstract
The Schr¨odinger dynamics of photon excitation numbers together with entanglement in twonon-resonant time-dependent coupled oscillators is investigated. By considering π − periodicallypumped parameters and using suitable transformations, we obtain the coupled Meissner oscillators.Consequently, our analytical study shows two interesting results, which can be summarized asfollows. (i): Classical instability of classical analog of quantum oscillators and photon excitationaverages (cid:104) N j (cid:105) are strongly correlated. (ii): Photon excitation’s and entanglement are connected toeach other. These results can be used to shed light on the link between quantum systems and theirclassical counterparts. Also it allow to control entanglement by engineering only classical systemswhere the experiments are less expensive. PACS numbers : 03.65.Fd, 03.65.Ge, 03.65.Ud, 03.67.Hk
Keywords : Classical instability maps, time-dependent coupled oscillators, photon excitation’s,entanglement, Ermakov equation, Meissner equation. ∗ [email protected] a r X i v : . [ qu a n t - ph ] F e b Introduction
Since the emergence of quantum theory (QT) in the beginning of 20 th century, entanglement wasused to refute the basic QT’s principles. In the early stages, Einstein, Podolsky and Rosen (EPR) [1](known as EPR paper), have attacked violently QT by remarking that wave functions can be entangled,which entails in their point of view the existence of hidden variables. In other part, entanglementwas considered as a necessary complement of QT because without it, it is impossible to interpretand confirm the previsions of QT [2]. Actually, entanglement plays an important role in quantuminformation processing protocols and it is considered a necessary resource to go beyond the classicalcommunications and technologies.In the last years, controlling entanglement in time-dependent coupled harmonic oscillators wasextensively studied, especially when oscillator systems in contact with environment. For instance, Itwas shown that the possibility to generate entanglement by phasing control in two [3] and three [4]isotropic harmonic oscillator’s sinusoidally coupled to each other by c ( t ) = c cos( ωt ) and weeklycoupled to an harmonic bath. It was found that the survival of entanglement for a large simulationtime is due to instability of decoupled (from the bath) normal oscillator [5]. More recently, it was shownthat the vacuum | G (cid:105) of two time-independent resonant oscillators contains virtual excitation’s [6], i.e. (cid:104) G | a + a | G (cid:105) = (cid:104) G | b + b | G (cid:105) (cid:54) = 0, in the range of strong coupling, which a consequence of the counter-rotating terms appearing in the Hamiltonian. A a result, the presence of these excitation’s maintainsentanglement between oscillators.Motivated by the above studies, we address to the question: how classical instabilities affectphoton excitation’s and consequently entanglement in the vacuum of two time-dependent non-resonantcoupled harmonic oscillators. Our response will be given in the framework of an assumption based onthe fact that our harmonic oscillators are connected by a periodically quenched coupling parameter J ( t ) = J Θ( t ) and having perturbed frequencies ω , ( t ) = ω ± (cid:15) Θ( t ), with Θ is a π -periodic functionΘ( t ) = Θ( t + π ) and (cid:15) is the quench amplitude. As a result, we end up with an integrable model calledtwo coupled Meissner oscillators [7]. This kind of oscillators can be seen, for instance, as LC oscillatorwith the parametric frequency ω = ( LC ) − where the capacitance C is pumped by a voltage V ( t )such that C −→ C ( t ) = C + C p Θ( t ) ( C p < C ) [8], or as a charged pendulum in alternating, piece-wise constant, homogeneous electric field [9]. The resolution of the Schr¨odinger dynamics allows usto find Ermakov equations [10] and utilization of suitable transformations leads to get two Meissnerdifferential equations corresponding to classical counterparts of the decoupled Hamiltonian. Then,we study the instabilities of derived differential equations and show instability/stability diagrams.With these, we be able to investigate the link between two strongly different features: the photonexcitation’s and classical instabilities. Additionally, by computing logarithmic negativity we establisha bijection between entanglement and photon excitation’s.The layout of our paper is given as follows. In Sec. 2, we present our model and show how to exactlydecouple the Hamiltonian system by using suitable transformations. The instabilities of emergedequations of both Ermakov and Meissner will be discussed in Sec. 3. We compute entanglement byusing logarithmic negativity and quantifying excitation in both oscillators by averaging the numberoperators over the vacuum in Sec. 4. We show our numerical results and present different discussionsin Sec. 5. Finally, we give an exhaustive conclusion to our work.1 Model and Schr¨odinger dynamics
The main concern in the present work is to answer the question asked in our introduction. Mainlyabout how the classical instabilities affect photon excitation’s and therefore entanglement in the vac-uum of two time-dependent non-resonant coupled harmonic oscillators (Figure 1) described by theHamiltonian H (ˆ x , ˆ x ) = ˆ p p ω ( t )ˆ x + 12 ω ( t )ˆ x − J ( t )ˆ x ˆ x (1)where ω j ( t ) are the frequencies and J ( t ) is a coupling parameter, with j = 1 ,
2. For simplicity, weassume that the masses are unit (we set the masses as equal to one, because, as Macedo and Guedesshowed [11], a simple transformation may be applied that makes the assumption valid) and (cid:126) = 1.
Figure 1 – (color online) The schematic shows two coupled oscillators via a position-position coupling type x x . Theparticles still in their vacuum. The dynamics generates virtual excitation’s between oscillators that affect quantum quantities. Since the Hamiltonian (1) is involving an interacting term, then a straightforward diagonalizationis not an easy task. To overcome such situation, we introduce the time-dependent rotation with anangle α ( t ) R α ( t ) = exp[ − iα ( t ) ˆ L z ] , α ( t ) = 12 arctan (cid:18) J ( t ) ω ( t ) − ω ( t ) (cid:19) (2)in terms of the angular momentum ˆ L z = ˆ x ˆ p − ˆ x ˆ p . Consequently, the transformed Hamiltonian isgiven by ˜ H (ˆ x , ˆ x ) = R α ( t ) H (ˆ x , ˆ x ) R − α ( t ) − i R α ( t ) ∂ t R − α ( t ) (3)and after some algebras, we obtain˜ H (ˆ x , ˆ x ) = ˆ p p ( t )ˆ x + 12 Ω ( t )ˆ x + ˙ α ( t )(ˆ x ˆ p − ˆ x ˆ p ) (4)where we have defined the frequencies Ω j Ω , ( t ) = 12 (cid:18) ω ( t ) + ω ( t ) ± (cid:113)(cid:2) ω ( t ) − ω ( t ) (cid:3) + 4 J ( t ) (cid:19) . (5)2or the boundness of Hamiltonian, the physical parameters point P ( ω , ω , J ) should belongs to thephysical 3 D -space E B = (cid:8) P / ω ω > J (cid:9) (6)It is clearly seen from (4) that the separation of variables is possible for ˙ α ( t ) = 0, which is equivalentto have tan(2 α ( t )) = 2 J ( t ) ω ( t ) − ω ( t ) = constant (7)which has been used also in different occasions, one may see for instance [11,12]. Then the Hamiltonianis decoupled and canonically is equivalent to the following time-dependent harmonic oscillators˜ H (ˆ x , ˆ x ) = ˆ p ( t )ˆ x + ˆ p ( t )ˆ x = ˜ H ( ˆ x ) + ˜ H ( ˆ x ) (8)which can easily be solved to extract the solutions of energy spectrum and then solve different issuesrelated to our system. The commutativity (cid:104) ˜ H , ˜ H (cid:105) = 0 implies that the solutions of time-dependent Schr¨odinger equationhave the forms ˜Ψ( x , x ; t ) = ˜ φ ( x ; t ) ⊗ ˜ φ ( x ; t ) (9)where each ˜ φ ( x j ; t ) satisfies (cid:18) − ∂ x j + 12 Ω j ( t )ˆ x j (cid:19) ˜ φ j ( x j ; t ) = − i∂ t ˜ φ j ( x j ; t ) , j = 1 , . (10)This was earlier studied in [13] an then the general solution of the Schr¨odinger equation is a superposi-tion of orthonormal expanding modes ˜ ψ ( x j ; t ) = (cid:80) n =0 p n j ( t ) ˜ φ j ( x j , t ), with (cid:80) n j =0 | p n j ( t ) | = 1. It followsthat for a single mode and Ω j (0) >
0, we have˜ φ j ( x j ; t ) = exp (cid:20) − i (cid:18) n j + 12 (cid:19) (cid:90) t (cid:36) j ( τ ) dτ (cid:21) χ n j ( x j ; t ) (11)and the orthogonal Hermite polynomials are χ n j ( x j ; t ) = 1 (cid:112) n j n j ! (cid:18) (cid:36) j ( t ) π (cid:19) exp (cid:20) − (cid:36) j ( t ) x j (cid:21) H n j (cid:18)(cid:113) (cid:36) j ( t ) x j (cid:19) (12)where we have defined the scaling frequencies as (cid:36) j ( t ) = Ω j (0) h j ( t ) . The functions h j ( t ) are solutions ofErmakov equations (dots stand for time derivatives hereafter)¨ h j + Ω j ( t ) h j = Ω j (0) h j (13)and satisfy the initial conditions h j (0) = 1 , ˙ h j (0) = 0. It is worthy to note that the energy spectrumis time-independent E n j = (cid:18) n j + 12 (cid:19) Ω j (0) (14)3hereas the average of energy is time-dependent because we have (cid:68) ˜ H ( x j ; t ) (cid:69) n j = 2 n j + 14Ω j (0) (cid:32) ˙ h j + Ω j ( t ) h j + Ω j (0) h j (cid:33) . (15)Consequently the eigenfunctions of decoupled Hamiltonian are given by˜Ψ n ,n ( x , x ; t ) = 1 √ n + n n ! n ! (cid:18) (cid:36) ( t ) (cid:36) ( t ) π (cid:19) H n (cid:16)(cid:112) (cid:36) ( t ) x (cid:17) H n (cid:16)(cid:112) (cid:36) ( t ) x (cid:17) × exp (cid:20) − i (cid:18) n + 12 (cid:19) (cid:90) t (cid:36) ( τ ) dτ − i (cid:18) n + 12 (cid:19) (cid:90) t (cid:36) ( τ ) dτ (cid:21) (16) × exp (cid:34) i (cid:32) ˙ h h − i(cid:36) ( t ) (cid:33) x + i (cid:32) ˙ h h − i(cid:36) ( t ) (cid:33) x (cid:35) . Now by performing the reciprocal rotation R − α ( t ), we end up with the single mode solution of theHamiltonian (1), which isΨ n ,n ( x , x ; t ) = R − α ( t ) ˜ ψ n ,n ( x , x ; t )= ˜Ψ n ,n ( x cos α − x sin α, x cos α + x sin α ; t ) . (17)In the forthcoming analysis, we only consider the following vacuum solutionΨ , ( x , x ; t ) = (cid:18) (cid:36) ( t ) (cid:36) ( t ) π (cid:19) exp (cid:20) − i (cid:90) t (cid:36) ( τ ) dτ − i (cid:90) t (cid:36) ( τ ) dτ (cid:21) × exp (cid:20) − A ( t ) x − A ( t ) x + A ( t ) x x (cid:21) (18)where the involved time-dependent parameters read as A ( t ) = (cid:36) ( t ) cos α + (cid:36) ( t ) sin α − i (cid:32) ˙ h h cos α + ˙ h h sin α (cid:33) (19) A ( t ) = (cid:36) ( t ) sin α + (cid:36) ( t ) cos α − i (cid:32) ˙ h h sin α + ˙ h h cos α (cid:33) (20) A ( t ) = sin α cos α (cid:34) (cid:36) ( t ) − (cid:36) ( t ) + i (cid:32) ˙ h h − ˙ h h (cid:33)(cid:35) . (21)These results will be used to compute the number of occupation and discuss entanglement trough thelogarithmic negativity. As we have seen above the vacuum state (18) is strongly depending on the functions h and h solutionsof Ermarkov equations (13). Then it is of interest to discuss the classical stability and instability relatedto Ermakov equation by deriving the conditions of classical stabilities. In the beginning, let us performthe following transformation on h and h [14] X ( t ) = h ( t ) e i (cid:82) t (cid:36) ( s ) ds (22) Y ( t ) = h ( t ) e i (cid:82) t (cid:36) ( s ) ds (23)4o obtain the Hill system ¨ X + Ω ( t ) X = 0 (24)¨ Y + Ω ( t ) Y = 0 (25)describing the two decoupled classical time-dependent harmonic oscillators of frequencies Ω ( t ) andΩ ( t ). Now it is clear that the stability of Hill system solutions leads to find that of Ermakov one andtherefore h j ( t ) can be expressed as [10] h ( t ) = x ( t ) + Ω (0) W − [ x , x ] x ( t ) (26) h ( t ) = y ( t ) + Ω (0) W − [ y , y ] y ( t ) . (27)such that x j and y j are independent solutions of (24) and (25), respectively, satisfying the initialconditions x (0) = y (0) = 1 and x (0) = y (0) = 0. Both Wronskian W [ x , x ] = x ˙ x − x ˙ x and W [ y , y ] = y ˙ y − y ˙ y are constant.To proceed further, we require some conditions on frequencies and coupling parameter [8,9]. Indeed,let us modulate them as ω , ( t ) = ω ± (cid:15) Θ( t ) , J ( t ) = J Θ( t ) (28)where ω , J , and (cid:15) are the parametric frequency, coupling amplitude and quench amplitude, respec-tively. The involved π − periodic quencher Θ( t ) is defined byΘ( t ) = (cid:40) +1 0 ≤ t < π − π ≤ t < π (29)and then the frequencies (5) reduce to the followingΩ , ( t ) = ω ± Θ( t ) (cid:113) (cid:15) + J . (30)It is worthy to note that with the modulation (28), the Hill system (24-25) reduces to the Meissnerequations [9]. Now, we discuss the boundness of Hamiltonian because the solutions presented in (18)are only valid for Ω j (0) >
0, with j = 1 , ω − (cid:15) > J , which is equivalent to an open disc of center ( J = 0 , (cid:15) = 0) and radius R = ω . In Figure 2, we give the physical maps for boundness of Hamiltonian in two differentconfigurations ( (cid:15), J ) and ( ω , J ). The maps show that the unbound regions are very large thanbound ones, which limits our choices. Note that, the edges of boundness present great importance forexample in generating important entanglement and leading to inverse engineering of time-dependentcoupled harmonic oscillators [16].To study the classical instability of (24-25), we will use the discrete transition matrix formalismor Floquet exponents technique [8]. Then, after some algebra we show that of the stability conditionof two oscillators can be written as S := max [1 − Λ(Ω , Ω ) , > , Ω ) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) Ω π (cid:19) cos (cid:18) Ω π (cid:19) − (cid:18) Ω Ω + Ω Ω (cid:19) sin (cid:18) Ω π (cid:19) sin (cid:18) Ω π (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (32)5n Figure 3, we numerically show the stability diagram in the physical configuration ( (cid:15), ω ). It isclearly seen that the stability diagram is very sensitive to the physical parameters because a smallchange produces important configuration variation. We notice that the instability region increases aslong as the coupling parameter increases. This in fact tells us that why one has to investigate theeffect of instabilities on the quantum features. Figure 2 – (color online) The boundness maps in two space configurations. (a): Configuration ( (cid:15), J ) for ω = 1 , aspredicted the physical points that bound the Hamiltonian (1) form an open disc of radius R = ω = 1 . (b): Configuration ( ω , J ) for (cid:15) = 1 . Figure 3 – (color online) The stability diagram of Meissner system (24-25) with modulation (28) in the space configuration ( (cid:15), ω ) for two cases (a): J = 0 and (b): J = 1 . The yellow area stands for instable region ( S = 0) , red area for stableregion ( S > , white dots for forbidden parameters ( Ω (0) = 0 ). Entanglement and photon excitation’s
According to Peres-Horodecki criterion, [17, 18], the necessary and sufficient condition for the separa-bility of two Gaussian mode states is the positivity of the partially transposed state. Since the vacuumstate (18) is pure and symmetric, then separability can be realized by switching (21) to zero, namelyhaving the condition A ( t ) = 0 (33)which is necessary and sufficient for separability and obviously it is achieved in two subordinate cases,(i): for J = 0 the dynamics can not generate entanglement and the oscillators still separable duringthe dynamics, (ii): for the Wronskian W [ h ( t ) , h ( t )] = 0 and (cid:36) ( t ) = (cid:36) ( t ) where the geometricalmeaning of W is substraction of the rectangular phase space areas S = h ˙ h and S = h ˙ h [16]. Thelast case indicates that the dynamics can extinct entanglement and then to avoid its extinction, theengineering of initial and final normal frequencies Ω j ( t f ) , Ω j (0) ( j = 1 ,
2) deserves a suitable tuning.Regarding our case, the vacuum state is completely described by the marginal purities µ j , ( j = 1 , µ ( t ) = µ ( t ) := µ ( t ) = (cid:18) (cid:36) ( t ) (cid:36) ( t )) (cid:36) ( t ) (cid:36) ( t ) + |A ( t ) | (cid:19) . (34)Since our state is pure and Gaussian, then all quantum correlation can be derived from the secondmoment of it, that is the covariance matrix (CM) V ( t ). Such CM can be transformed via a localsymplectic transformation S = S ⊕ S to a particular form called standard form V sf ( t ) [17] V sf ( t ) = µ − (cid:112) µ − − µ − − (cid:112) µ − − (cid:112) µ − − µ − − (cid:112) µ − − µ − = (cid:32) A CC A (cid:33) . (35)By performing the partial transposition (PT) prescription, det( A ) −→ det( A ) and det( C ) −→ − det( C ),then the minimal symplectic eigenvalue of the PT covariance matrix ˜ V is λ min ( t ) = 12 (cid:18) ∆( ˜ V ) − (cid:113) ∆ ( ˜ V ) − (cid:19) (36)and after evaluation, we find λ min ( t ) = (cid:18) |A ( t ) | (cid:36) ( t ) (cid:36) ( t ) (cid:19) (cid:16) − (cid:112) − µ ( t ) (cid:17) − V ) = 2(det( A ) − det( C )). Consequently, the logarithmic negativityis given by E N ( t ) = max (0 , − log [ λ min ( t )]) (38)which is monotonically increasing with |A | . From its expression (21) it appears that the main con-tributions of the time-dependent Hamiltonian is the emergence of an imaginary part, that is (cid:16) ˙ h h − ˙ h h (cid:17) and initial normal mode scaling, i.e. Ω j (0) −→ (cid:36) j ( t ) = Ω j (0) h j ( t ) .7 .2 Photon excitation’s Using of the phase space prescription [19], we will analyze photon excitation’s by computing theaverage of photon numbers (cid:104) a + j a j (cid:105) in the vacuum state. The case of two resonant time-independentcoupled oscillators was analyzed in [6] where the creation a + j, and annihilation a j, operators aresimply mapped as ( a + j, ) + = a j, = (cid:114) ω j x j + i (cid:112) ω j p j . (39)However for time-dependent Hamiltonian, the situation is not obvious because the realization can bedone as follows [20, 21]( a + j ) + = a j = e i (cid:82) s η j ( s ) ds (cid:112) η j (cid:20) η j (cid:18) − i ˙ ν j ν j η j (cid:19) x j + ip j (cid:21) (40)where (cid:104) a j , a + j (cid:105) = I , η j ( t ) = ω j (0) ν j and the functions ν j satisfy the Ermakov equations¨ ν j + ω j ( t ) ν j = ω j (0) ν j , j = 1 , . (41)After a straightforward algebra, one can compute the average of photon number operators N j = a + j a j to end up with (cid:104) N j (cid:105) ( t ) = 12 ω j (0) (cid:32) ω j (0) ν j + ˙ ν j (cid:33) (cid:104) ˆ x j (cid:105) t + ν j ω j (0) (cid:104) ˆ p j (cid:105) t − ν j ˙ ν j ω j (0) (cid:104) ˆ x j ˆ p j (cid:105) t −
12 (42)and different averages are explicitly given by (cid:104) ˆ x j (cid:105) t = (cid:36) j ( t ) sin α + (cid:36) k ( t ) cos α (cid:36) j ( t ) (cid:36) k ( t ) (43) (cid:104) ˆ p j (cid:105) t = 12 (cid:36) j ( t ) cos α + (cid:36) k ( t ) sin α + 1 (cid:36) j ( t ) (cid:32) ˙ h j h j (cid:33) cos α + 1 (cid:36) k ( t ) (cid:32) ˙ h k h k (cid:33) sin α (44) (cid:104) ˆ x j ˆ p j (cid:105) t = − (cid:36) k ( t ) cos α ˙ h j h j + (cid:36) j ( t ) sin α ˙ h k h k (cid:36) j ( t ) (cid:36) k ( t ) (45)where k = 1 , k (cid:54) = j .We emphasis that the time-dependent Hamiltonian (1) generates important effects such that thescalings η j ( t ) = ω j (0) ν j and (cid:36) j ( t ) = Ω j (0) h j ( t ) , the shiftings in positions (43) and momenta (44), whichare due to the dilatation functions ν j ( t ) and h j ( t ). In addition, the existence of the term (cid:104) ˆ x j ˆ p j (cid:105) t ispurely a consequence of time-dependence, which of course disappears by setting time to zero, namelyhaving constant parameters. Note that, for time-independent Hamiltonian, we have ν j = h j = 1 and˙ ν j = ˙ h j = 0, then the virtual photon excitation’s become (cid:104) N (cid:105) = 14 cos α (cid:18) Ω ω + ω Ω (cid:19) + 14 sin α (cid:18) Ω ω + ω Ω (cid:19) −
12 (46) (cid:104) N (cid:105) = 14 sin α (cid:18) Ω ω + ω Ω (cid:19) + 14 cos α (cid:18) Ω ω + ω Ω (cid:19) − . (47)8n the case of resonant oscillators, i.e. ω = ω = ω r , the rotation angle reduces to α = π . Furthermore,by setting r a,b = ln (cid:16) Ω , ω r (cid:17) , we retain the results derived in [6] (cid:104) N (cid:105) = (cid:104) N (cid:105) = 12 (cid:0) sinh r a + sinh r b (cid:1) . (48)We mention that the study in [6] was confined on the resonant case because it was not easy todisentangle the rotation operator (2) in the frame of creation a + j and annihilation a j representation.Now it becomes clear that from our analysis how the phase space picture can be used to overcomesuch situation. In addition, it is interesting to note that for ω (cid:54) = ω and J = 0, the vacuum state(18) contains excitation’s, i.e. (cid:104) N (cid:105) = (cid:104) N (cid:105) (cid:54) = 0, even if the oscillators are decoupled, such phenomenadoes not exist in the frame of resonant oscillators. Before numerically presenting and discussing the main results derived so far, it is convenient for ourtask to define dimensionless parameters t −→ t Ω , ω j −→ Ω j Ω , ω j −→ ω j Ω , (cid:15) −→ (cid:15) Ω , J −→ J Ω (49)where Ω is an arbitrary frequency. For a numerical study of classical instabilities effects on generationof photon excitation’s and hence entanglement between oscillators, we start by giving the dilatationfunctions h j and ν j . Indeed, by using (27) one can solve (13) to obtain the solutions in first period[0 , π ], which are h ( t ) = (cid:40) , ≤ t < π − Ω cos (cid:0) ( t − π ) (cid:1) + Ω +Ω , π ≤ t < π (50) h ( t ) = (cid:40) , ≤ t < π − Ω cos (cid:0) ( t − π ) (cid:1) + Ω +Ω , π ≤ t < π. (51)Note that the solutions of ν j will be immediately obtained from (50-51) only under the substitution (cid:112) (cid:15) + J −→ (cid:15) . Now, we notice that the freezing dynamics will be occurred in odd half periods whiledynamics evolution in even ones, which is due to the periodic quench, and the continuity of solutions h j and ν j .In Figure 4, we remark that vacuum state does not contain virtual excitation’s when the classicalanalog of quantum oscillators are stable and vice verse. Another point that deserves attention isthe emergence of excitation’s even if the oscillators are decoupled ( J = 0) and beyond resonance ω (cid:54) = ω ( (cid:15) (cid:54) = 0), amazingly the classical oscillators are unstable. Such phenomena is not observed forresonant oscillators [6]. Generally, the virtual photons are originated from counter-rotating (CR) terms( a +1 a +2 and a a ) appearing in the Hamiltonian. However, these terms disappear when the coupling isswitched-off and the non-resonance oscillations has no relation with CR. Indeed, the resonance affectsthe potential energy operator, V ( J = 0) = ω ( t )ˆ x + ω ( t )ˆ x , and this does not mix the quadraturesˆ x j and ˆ p k , then CR terms does not appear. This phenomena can be seen as follows, when the classicaloscillators are stable then the virtual excitation’s will be suppressed, but for unstable case it will be9reated. Note that (cid:15) and J have the same dimension but different effects on excitation’s generation.In order to compare their effects on excitation’s amount and hierarchy, we also plot in Figure 4, thedynamics of excitation’s (cid:104) N (cid:105) and (cid:104) N (cid:105) . The numerical results show that the increasing of (cid:15) increasesthe amount of excitation’s as well as the hierarchy (cid:104) N (cid:105) (cid:54) = (cid:104) N (cid:105) and changes the topological behaviorin the simulate time (linear/sinusoidal). Now, if the values of (cid:15) and J ( C ↔ D ) are interchanged, weobtain different aspects, which are due to contributions of (cid:15) and J in the frequencies ω j and Ω j .In this way, we can affirm that virtual photons are correlated with the instability of classical coun-terpart’s solutions. In our best of knowledge, this phenomena is the first time that it has been studied,and this will lead to control virtual excitation’s in quantum systems only by engineering their classicalcounterparts, where the financial requirements are not enormous. Also it will leads to understand thenature of the connection between quantum systems and theirs classical counterparts. Figure 4 – (color online) The golden surface indicates instability region ( S = 0 ) and the stability area ( S > ) for thefrequency ω = 1 . . Panels A, B, C and D present the virtual photon excitation’s (cid:104) N (cid:105) (blue line) (cid:104) N (cid:105) (red dashed) versusthe time t . If we use the notation M ( (cid:15), J ) , it turns out that the physical points are A (0 , , B (0 , . , C (0 . , . , and D (0 . , . . In Figure 5, we plot the link between the entanglement and photon excitation’s when the oscillatorsare resonant ω = ω = 1 .
01 ( (cid:15) = 0) for three values of coupling J = 0 . , . , .
3. We observe thatentanglement and vaccum excitation’s are freezed during the half period [0 , π ]. This is due to the factthat dilatation functions (50-51) still constant in this interval. For t > π , the both quantities exhibitthe same topological behavior ( (cid:104) N (cid:105) ∝ E N ). 10 igure 5 – (color online) Left panel presents the dynamics of entanglement quantified by Logarithmic negativity E N andright panel shows the photon excitation’s (cid:104) N (cid:105) = (cid:104) N (cid:105) = (cid:104) N (cid:105) for the parameters (cid:15) = 0 , ω = 1 . and J = 0 . , . , . . In Figure 6, we investigate the effects of non-resonance on the geometric average of excitation’s M ( t ) = ( (cid:104) N ( t ) (cid:105) (cid:104) N ( t ) (cid:105) ) and entanglement. When t ∈ [0 , π ] a tiny and constant amount of excita-tion’s and entanglement are detected, but since t > π the excitation’s dramatically increase comparedto the entanglement generation. We mention, although the excitation’s are important the oscilla-tors still weekly entangled and the maximum of entanglement is obtained when the excitation’s aremore hierarchical (i.e. | (cid:104) N ( t ) (cid:105) ) − (cid:104) N ( t ) (cid:105) | reaches its maximal value). To conclude, the extinctionof entanglement requires the condition J = 0, but that of excitations requires resonance togetherwith J = 0. The photon excitation’s generate important amounts of entanglement when oscillatorsare resonant ( (cid:15) = 0) and strongly coupled. This is similar to what has been found by consideringtime-independent coupled oscillators [6]. It turns out this phenomenon can be seen as the Casimireffect because the virtual photon exchange plays a vital role in mediating the coupling between os-cillators [22]. Then they generate entanglement, but their contribution is perhaps limited by theirquantum destructive interference, which becomes more important when the system moves away fromthe resonance ( (cid:15) −→ ω ). Figure 6 – (color online) Left panel presents the dynamics of entanglement quantified by logarithmic negativity E N ( t ) . Rightpanel shows the geometric average of photon excitation’s M ( t ) = ( (cid:104) N (cid:105)(cid:104) N (cid:105) ) . The configuartions are taken ω = 1 . ,(blue line) for ( (cid:15) = 0 . , J = 0 . and (red line) for ( (cid:15) = 0 . , J = 0 . .
11n Figure 7, we show the effects of coupling on entanglement E N ( t ), geometric average of photon M ( t ) = ( (cid:104) N ( t ) (cid:105)(cid:104) N ( t ) (cid:105) ) , ( t ∈ [0 , π/ Ins := Λ(Ω , Ω ) − J , they are monotonically increasing, which entails the importance of stong couplingphysics [6]. It is worthy to note that Ins ( J = 0 , (cid:15) = 0 . ∼ .
005 (oscillators are unstable), E N = 0and M ( t ) (cid:54) = 0, similarly, Ins ( J = 0 , (cid:15) = 0) ∼ − . . − (oscillators are stable), E N = 0 and M ( t ) = 0. Finally, those similarities show that classical instabilities affect the generation of excita-tion’s and hence entanglement between oscillators. Figure 7 – (color online) Effects of coupling on entanglement E N ( π ) , geometric average M = ( (cid:104) N (cid:105)(cid:104) N (cid:105) ) and Ins =Λ(Ω , Ω ) − , left panel: (cid:15) = 0 and right panel: (cid:15) = 0 . with ω = 1 . . We have studied two non resonant coupled harmonic oscillators connected by a periodically pumpedcoupling J ( t ) = J × Θ( t ) and frequencies. We have solved exactly the Schr¨odinger dynamics, whichleads to Ermakov differential equations. After a suitable transformations, we have showed that theyare equivalent to classical counterparts of the decoupled quantum Hamiltonian. By studying theinstability of the classical analog and computing the photon excitation’s in the vacuum state, we havefound that the excitation’s will be created if the classical oscillators are unstable.We have studied the dynamics of entanglement by computing logarithmic negativity together withphoton excitation’s beyond resonance by using phase space picture. We have analyzed the effects ofcoupling and quench amplitude on entanglement dynamics and photon excitation’s. Consequently,it was found that photon excitation’s present the same behavior when the oscillators are resonantand strongly coupled J ∼ ω , , and when the oscillators are non resonant and weakly coupled thephoton’s excitation’s have a tiny contribution on entanglement generation. We have showed alsothat extinction of excitation’s entails the suppression of entanglement. However, it does not implynecessarily the suppression of excitation’s if the oscillators are separated. This allows us to concludethat the excitation’s generate and maintain entanglement.12 eferences [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).[2] J. S. Bell, Physics 1, 195 (1964).[3] J. C. Gonzalez-Henao, E. Pugliese, S. Euzzor, S. F. Abdalah, R. Meucci and J. A. Roversi,Scientific Reports 5, 13152 (2015).[4] J. C. Gonzalez-Henao, E. Pugliese, S. Euzzor, R. Meucci, J. A. Roversi and F. T. Arecchi,Scientific Reports 7, 9957 (2017).[5] T. Figueiredo Roque and J. A. Roversi, Phys. Rev. A 88, 032114 (2013).[6] J.-Y. Zhou, Y.-H. Zhou, X.-L. Yin, J.-F. Huang and J.-Q. Liao, Scientific Reports 10, 12557(2020).[7] E. Meissner, Schweizer Bauzeitung 72, 95 (1918).[8] J. A. Richards, Analysis of periodically time-varying systems (Springer-Verlag, Berlin, 1983).[9] A. A. Burov and V. I. Nikonov, Int. J. Non-Linear Mech. 110, 26 (2019).[10] E. Pinney, Proc. Am. Math. Soc. 1, 681, (1950).[11] D. X. Macedo and I. Guedes, J. Math. Phys. 53, 052101 (2012).[12] S. Menouar, M. Maamache and J. R. Choi, Physica Scripta 82, 6 (2010).[13] H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).[14] K. E. Thylwe and H. J. Korsch, J. Phys. A 31, L279–L285 (1998).[15] Xi Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gu´ery-Odelin and J. G. Muga, Phys.Rev. Lett. 104, 063002 (2010).[16] A. Tobalina, E. Torrontegui, I. Lizuain, M. Palmero and J. G. Muga, Phys. Rev. A 102, 063112(2020).[17] G. Adesso and F. Illuminati, Phys. Rev. A 72, 032334 (2005).[18] G. Adesso, A. Serafini and F. Illuminati, Phys. Rev. A 73, 032345 (2006).[19] Y. S. Kim and M. E. Noz,