Dynamics of entanglement between two harmonic modes in stable and unstable regimes
aa r X i v : . [ qu a n t - ph ] A p r Dynamics of entanglement between two harmonic modes in stable and unstableregimes
L. Reb´on, N. Canosa, R. Rossignoli
Departamento de F´ısica-IFLP, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
The exact dynamics of the entanglement between two harmonic modes generated by an angularmomentum coupling is examined. Such system arises when considering a particle in a rotatinganisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field,and exhibits a rich dynamical structure, with stable, unstable and critical regimes according to thevalues of the rotational frequency or field and trap parameters. Consequently, it is shown thatthe entanglement generated from an initially separable gaussian state can exhibit quite distinctevolutions, ranging from quasiperiodic behavior in stable sectors to different types of unboundedincrease in critical and unstable regions. The latter lead respectively to a logarithmic and lineargrowth of the entanglement entropy with time. It is also shown that entanglement can be controlledby tuning the frequency, such that it can be increased, kept constant or returned to a vanishingvalue just with stepwise frequency variations. Exact asymptotic expressions for the entanglemententropy in the different dynamical regimes are provided.
PACS numbers: 03.67.Bg,03.65.Ud,05.30.Jp
I. INTRODUCTION
The investigation of entanglement dynamics andgrowth in different physical systems is of great currentinterest [1–3]. Quantum entanglement is well known tobe an essential resource for quantum teleportation [4] andpure state based quantum computation [5], where its in-crease with system size is necessary to achieve an expo-nential speedup over classical computation [6, 7]. And alarge entanglement growth with time after starting froma separable state indicates that the system dynamics can-not be simulated efficiently by classical means [8], turningit suitable for quantum simulations.The aim of this work is to examine the dynamics ofthe entanglement between two harmonic modes gener-ated by an angular momentum coupling, and its abilityto reproduce typical regimes of entanglement growth inmore complex many body systems, when starting from aninitial separable gaussian state. The latter can be chosen,for instance, as the ground state of the non-interactingpart of the Hamiltonian, thus reproducing the typicalquantum quench scenario [1, 2, 8]. The present systemcan be physically realized by means of a charged parti-cle in a uniform magnetic field within a harmonic po-tential or by a particle confined in a rotating harmonictrap [9–12], where the field or rotational frequency pro-vides an easily controllable coupling strength. Accord-ingly, it has been widely used in quite different physicalcontexts, such as rotating nuclei [11, 12], quantum dotsin a magnetic field [13] and fast rotating Bose-Einsteincondensates within the lowest Landau level approxima-tion [14–19]. In spite of its simplicity, the model is ableto exhibit a rich dynamical structure [20], with both sta-ble and distinct types of unstable regimes, characterizedby bounded as well as unbounded dynamics, when con-sidering all possible values of the field or frequency ina general anisotropic potential. Nonetheless, being a quadratic Hamiltonian in the pertinent coordinates andmomenta, the dynamics can be determined analyticallyin all regimes, and the entanglement between modes canbe evaluated exactly through the gaussian state formal-ism [21–25]. For the same reason, the Hamiltonian is alsosuitable for simulation with optical techniques [26].The main result we will show here is that due its non-trivial dynamical properties, the entanglement dynamicsin the previous model can exhibit distinct regimes, in-cluding a quasiperiodic evolution in dynamically stablesectors, different types of logarithmic growth at the bor-der between stable and unstable sectors (critical regime)and a linear increase in dynamically unstable sectors.The model is then able to mimic the three typical regimesfor the entanglement growth with time after a quantumquench, arising in spin 1 / II. MODEL AND EXACT DYNAMICSA. Hamiltonian
We consider two harmonic systems with coordinatesand momenta Q µ , P µ , µ = x, y , coupled through theirangular momentum L z = Q x P y − Q y P x . The Hamilto-nian is H = H − Ω L z , (1) H = P x + P y m + 12 ( K x Q x + K y Q y ) . (2)Eq. (1) describes, for instance, the motion in the x, y plane of a particle of charge e and mass m within a har-monic trap of spring constants ˜ K µ in a uniform field H along the z axis [11, 12], if Ω = e | H | mc stands for half thecyclotron frequency and K µ = ˜ K µ + m Ω .It also determines the intrinsic motion of a particle in aharmonic trap with constants K µ which rotates aroundthe z axis with frequency Ω. In this case [11, 12], theactual Hamiltonian is H ( t ) = R ( t ) H R † ( t ), with R ( t ) = e − i Ω L z t/ ~ the rotation operator, but averages of rotatingobservables O ( t ) = R ( t ) OR † ( t ) evolve like those of O under the time-independent “cranked” Hamiltonian (1).Replacing Q µ = q µ / p m Ω / ~ , P µ = p µ √ ~ m Ω ,with q µ , p µ dimensionless coordinates and momenta([ q µ , p ν ] = iδ µν , [ q µ , q ν ] = [ p µ , p ν ] = 0) and Ω a ref-erence frequency, we have H = ~ Ω h , with h = h − ωl z , h = 12 ( p x + p y + k x q x + k y q y ) , (3) l z = q x p y − q y p x = − i ( b † x b y − b † y b x ) , (4)where k µ = K µ / ( m Ω ) and ω = Ω / Ω are dimensionless(Ω can be used to set | k x | = 1) and b µ = q µ + ip µ √ arethe boson annihilation operators associated with q µ , p µ .The l z coupling (4) is then seen to conserve the associ-ated total boson number N = P µ = x,y b † µ b µ , being in factthe same as that describing the mixing of two modes ofradiation field passing through a beam splitter [5]. No-tice, however, that [ h , N ] = 0 unless k x = k y = 1 (stableisotropic trap). B. Exact evolution
The Heisenberg equations of motion ido/dt = − [ h, o ]for the operators q µ , p µ (with t = Ω T and T the actualtime) become dq x dt = p x + ωq y , dq y dt = p y − ωq xdp x dt = − k x q x + ωp y , dp y dt = − k y q y − ωp x , (5) and can be written in matrix form as i ddt O = HO , (6) O = q x q y p x p y , H = i ω − ω − k x ω − k y − ω . (7)The system dynamics is then fully determined by thematrix H . We may write the general solution of (6) as O ( t ) = U ( t ) O , U ( t ) = exp[ − i H t ] , (8)where O ≡ O (0).In spite of their simplicity, Eqs.(5) can lead to quitedistinct dynamical regimes according to the values of ω and k µ , as the eigenvalues of H , which is in generala non-hermitian matrix, can become imaginary or com-plex away from stable regions [20]. Moreover, H can alsobecome non-diagonalizable at the boundaries betweendistinct regimes, exhibiting non-trivial Jordan canonicalforms [20]. Nonetheless, as H = k x + ω − ω k y + ω ω ω ( k x + k y ) k x + ω − ω ( k x + k y ) 0 0 k y + ω , (9)the eigenvalues of H are determined by 2 × λ ± and − λ ± , with λ ± = p ε + + ω ± ∆ , , (10)where ε ± = k x ± k y and ∆ = q ε − + 4 ω ε + .We can then write the solution (8) explicitly as q x ( t ) q y ( t ) p x ( t ) p y ( t ) = u xx u xy v xx v xy u yx u yy − v xy v yy w xx w xy u xx − u yx − w xy w yy − u xy u yy q x q y p x p y , (11)where u xxyy = (∆ ± ε − ) cos λ + t +(∆ ∓ ε − ) cos λ − t ,u xyyx = ± ω (∆ ∓ ε − +2 ε + ) sin λ + tλ + +(∆ ± ε − − ε + ) sin λ − tλ − ,v xxyy = (∆ ± ε − +2 ω ) sin λ + tλ + +(∆ ∓ ε − − ω ) sin λ − tλ − ,v xy = ω ( − cos λ + t +cos λ − t )∆ , w xy = − ε + v xy ,w xxyy = − (∆ ± ε − )(∆ ± ε − +2 ε + ) sin λ + tλ + +(∆ ∓ ε − )(∆ ∓ ε − − ε + ) sin λ − tλ − . (12)The matrix U ( t ) is real for any real values of ω , k µ and t , including unstable regimes where ∆ and/or λ ± can beimaginary or complex [20]. It represents always a linearcanonical transformation of the q µ , p µ , satisfying U ( t ) MU t ( t ) = M , M = i (cid:18) I − I (cid:19) , (13)( I denotes the 2 × O i , O j ] = M ij ).It corresponds to a proper Bogoliubov transformation ofthe associated boson operators.For ω = 0, we recover from Eqs. (11)–(12) thedecoupled harmonic evolution q µ ( t ) = q µ cos ω µ t + ω − µ p µ sin ω µ t , p µ ( t ) = p µ cos ω µ t − q µ ω µ sin ω µ t , where ω µ = p k µ for µ = x, y . Off-diagonal terms u xy , u yx , v xy , w xy in (12) are O ( ω ) for small ω .On the other hand, in the isotropic case k x = k y = k (where ∆ = 2 ω √ k and | λ ± | = |√ k ± ω | ), [ l z , h ] = 0and the evolution provided by Eqs. (11)–(12) is just therotation of identical single mode evolutions: U ( t ) = exp[ iω L z t ] exp[ − i H t ] , (14)exp[ iω L z t ] = (cid:18) R † ( t ) 00 R † ( t ) (cid:19) , R † ( t ) = (cid:18) cos ωt sin ωt − sin ωt cos ωt (cid:19) . In particular, the Landau case (free particle in a magneticfield) corresponds to k x = k y = ω , where λ + = 2 ω and λ − = 0. C. Dynamical regimes
The distinct dynamical regimes exhibited by this sys-tem for ω = 0 are summarized in Fig. 1. Let us firstconsider the standard stable case k x > k y > λ ± are here both real andnon-zero in sectors A and B , defined by ω < Min[ k x , k y ] (sector A ) , (15) ω > Max[ k x , k y ] (sector B ) , (16)when k x > k y > A is the full stable sector where h is positive definite, whereas B is that where the sys-tem, though unstable, remains dynamically stable [20](see also Appendix). If ω lies between these values (sec-tor D ), λ − becomes imaginary (with λ + remaining real),leading to a frequency window where the system be-comes dynamically unstable (unbounded motion), withsin( λ − t ) /λ − = sinh( | λ − | t ) / | λ − | in Eqs. (12).At the border between D and A or B ( ω = k x or ω = k y ), λ − = 0 (with λ + >
0) and H becomes non-diagonalizable if k y = k x , although H remains diagonal-izable. The system becomes here equivalent to a stableoscillator plus a free particle [20] (see Appendix), andwe should just replace sin( λ − t ) /λ − by its limit t in Eqs.(12), which leads again to an unbounded motion.Considering now the possibility of unstable potentials( k x < k y <
0, remaining quadrants), the dy-namically stable sector B extends into this region pro-vided k x > > k y > − k x (or viceversa) andMax[ k x , k y ] < ω < − ε − / (4 ε + ) , (17)where the upper bound applies only when ε + < − k x < k y < − k x or viceversa). Eq. (17) defines afrequency window where the unstable system becomes Λ - = Λ + = Λ - = Λ + =Λ - Λ ± = Λ - imaginary Λ - imaginary AD DE LL CC B F Λ ± complex Λ ± real - - k y (cid:144) Ω - - k x (cid:144) Ω FIG. 1. Dynamical phase diagram of the system describedby Hamiltonian (1). The evolution of the operators q µ , p µ isquasiperiodic in the dynamically stable sectors A , B , wherethe eigenfrequencies λ ± are both real, but unbounded in theremaining sectors, with λ − imaginary in D , both λ ± imagi-nary in C , and λ ± complex conjugates in E . At the bordersbetween these regions (except from the Landau point F ) thematrix H is non-diagonalizable and the evolution is also un-bounded, with λ − = 0 at the borders between D and A or B , λ + = 0 at the border between D and C , λ + = λ − at thecurve separating E from B and C and λ ± = 0 at the criticalpoints L . Dashed lines indicate the path described as ω isincreased at fixed k µ , showing that the system dynamics willbecome unbounded (bounded) in a certain frequency windowwhen starting at 1 (2). The black line indicates the isotropiccase k x = k y , where entanglement will be periodic (see sec.III). dynamically stable ( λ ± real). Beyond this sector, either λ − becomes imaginary (sectors D ) or both λ ± becomeimaginary (sectors C ) or complex conjugates (sector E ,where ∆ is imaginary), and the dynamics becomes againunbounded. This is also the case at the borders between D and B ( λ + > λ − = 0) and also D and C ( λ + = 0, λ − imaginary) where H is non-diagonalizable (see Ap-pendix for more details).The critical curve ∆ = 0, i.e., ω = − ε − / (4 ε + ) , (18)where ε + <
0, separates sectors B and C from E and deserves special attention. At this curve, λ ± = λ = p ε + + ω and both H and H become non-diagonalizable , with λ real at the border between B and E and imaginary at that between C and E . The eval-uation of U ( t ) in Eq. (8) can in this case be obtainedthrough the pertinent Jordan decomposition of H (two2 × → u xxyy = cos λt ∓ tε − sin λt λ ,u xyyx = ± ω tλ ( ε + ∓ ε − /
2) cos λt +( ω ± ε − /
2) sin λtλ ,v xxyy = tλ ( ω ± ε − /
2) cos λt +( ε + ∓ ε − /
2) sin λtλ ,v xy = ωt sin λtλ , w xy = − ε + v xy ,w xxyy = ε + tλ ( ω ∓ ε − /
2) cos λt − ( ε + +2 ω )( ε + ± ε − /
2) sin λtλ , (19)which contain terms proportional to t . The evolution is,therefore, always unbounded along this curve.Finally, if both ∆ and λ = p ε + + ω vanish, whichoccurs when ε + = − ω = −| ε − | /
2, i.e., ω = k x = − k y / , (20)(or ω = k y = − k x / critical point (points L in Fig. 1), where λ ± = 0 andsectors B , C , D and E meet. Here both H and H arenon-diagonalizable, with H represented by a single 4 × λ → polynomial (and hence also unbounded)evolution, involving terms up to the third power of t : Theelements of U ( t ) become u xxyy = 1 ∓ ω t u xy = ωt (1 + ω t ) , u yx = − ωt ,v xx = t (1 − ω t ) , v yy = t ,v xy = ωt , w xy = ω t ,w xx = − ω t, w yy = ω t (3 + ω t ) . (21)Nonetheless, we remark that Eq. (13) remains satisfied(in both cases (19) and (21)). III. DYNAMICS OF ENTANGLEMENT INGAUSSIAN STATESA. Exact evaluation
Let us now consider the evolution of the entanglementbetween the x and y modes, starting from an initially sep-arable pure gaussian state. Since the evolution is equiva-lent to the linear canonical transformation (8), the statewill remain gaussian ∀ t , which entails that entanglementwill be completely determined by the pertinent covari-ance matrix [22, 23].We may then assume that at t = 0, h q µ i = h p µ i = 0for µ = x, y ( hOi = 0), such that these mean values willvanish ∀ t ( hOi t = hO ( t ) i = 0, as implied by Eq. (11)).We may then define the covariance matrix as C = hOO t i − M = h q x i h q x q y i h q x p x + p x q x i h q x p y ih q x q y i h q y i h q y p x i h q y p y + p y q y i h q x p x + p x q x i h q y p x i h p x i h p x p y ih q x p y i h q y p y + p y q y i h p x p y i h p y i , (22) which, according to Eqs. (8) and (13), will evolve as C ( t ) = U ( t ) C (0) U t ( t ) . (23)The entanglement between the two modes will now bedetermined by the symplectic eigenvalue ˜ f ( t ) = f ( t )+1 / C µ ( t ) = hO µ O tµ i t − M , submatrix of (23), where O µ = ( q µ , p µ ) t . Here f ( t )is a non-negative quantity representing the average bosonoccupation h a † µ ( t ) a µ ( t ) i of the mode ( a µ ( t ) is the localboson operator satisfying h a µ ( t ) i = 0), which is the samefor both modes ( f x ( t ) = f y ( t )) when the global state isgaussian and pure. It is given by f ( t ) = q h q µ i t h p µ i t − h q µ p µ + p µ q µ i t / − . (24)Eq. (24) is just the deviation from minimum uncertaintyof the mode, and can be directly determined from theelements of (23).The von Neumann entanglement entropy between thetwo modes becomes S ( t ) = − Tr ρ µ ( t ) ln ρ µ ( t )= − f ( t ) ln f ( t ) + [1 + f ( t )] ln[1 + f ( t )] , (25)where ρ µ ( t ) denotes the reduced state of the mode. Eq.(25) is an increasing concave function of f ( t ). For futureuse, we note that for large and small f ( t ), S ( t ) ≈ ln f ( t ) + 1 + O ( f − ) , (26) S ( t ) ≈ f ( t )[ − ln f ( t ) + 1] + O ( f ) . (27)Other entanglement entropies, like the Renyi entropies S α ( t ) = ln Tr ρ αµ ( t )1 − α , α >
0, and the linear entropy S ( t ) =1 − Tr ρ µ ( t ) (of experimental interest as Tr ρ and in gen-eral Tr ρ n can be measured without performing a full statetomography [3, 36]), are obviously also determined by f ( t ), since Tr ρ αµ = [(1 + f µ ) α − f αµ ] − ( α > C (0) will be here assumedof the form C (0) = 12 α − x α − y α x
00 0 0 α y , (28)where α µ = 2 h p µ i , such that α µ = p k µ if the systemis initially in the separable ground state of h , as in thetypical quantum quench scenario [1]. For fixed isotropicinitial conditions we will just take α x = α y = 1.For these initial conditions, we first notice that forsmall t , Eqs. (12) and (24) yield f ( t ) ≈ ( α x − α y ) α x α y ( ωt ) + O ( t ) , (29)which indicates a quadratic initial increase of f ( t ) withtime for any anisotropic initial covariance. Eq. (29) is A , B Quasiperiodic Quasiperiodic C f ( t ) ∝ e ( | λ − | + | λ − | ) t S ( t ) ≈ ( | λ − | + | λ − | ) t D f ( t ) ∝ e | λ − | t S ( t ) ≈ | λ − | t E f ( t ) ∝ e | Im( λ ) | t S ( t ) ≈ | Im( λ ± ) | t Borders A - D , B - D f ( t ) ∝ t S ( t ) ≈ S ( t ) + ln t Border B - E f ( t ) ∝ t S ( t ) ≈ S + 2 ln t Points L f ( t ) ∝ t S ( t ) ≈ S + 4 ln t Line k x = k y Periodic ( α x = α y ) Periodic ( α x = α y ) TABLE I. The asymptotic evolution of the average occupation(24) and the entanglement entropy (25) in the different dy-namical sectors indicated in Fig. 1. Entanglement is boundedin the stable sectors A , B , but increases linearly (with t ) inthe unstable sectors C , D , E , and logarithmically at the bor-der between stable and unstable sectors, provided k x = k y .In the isotropic case k x = k y = k it remains periodic for anyvalue of k and anisotropic initial conditions. independent of the oscillator parameters k µ and propor-tional to ω . However, for isotropic initial conditions α x = α y , quadratic terms vanish and we obtain in-stead a quartic initial increase, driven by the oscillatoranisotropy ε − : f ( t ) ≈ ε − ω α x t + O ( t ) . (30)Eq. (27) implies a similar initial behavior (except for afactor ln t ) of the entanglement entropy.Next, in the isotropic case k x = k y = k , the exact ex-pression for f ( t ) becomes quite simple, since the rotationis decoupled from the internal motion of the modes (Eq.(14)), and entanglement arises solely from rotation andinitial anisotropy . We obtain f ( t ) = 12 s α x − α y ) α x α y sin (2 ωt ) − . (31)Entanglement will then simply oscillate with frequency4 ω if α x = α y , being independent of the trap parameter k , since the latter affects just a local transformation de-coupled from the rotation. Eq. (31) holds in fact even if k becomes negative (unstable potential) or vanishes.In the general case, the previous decoupling no longerholds and the explicit expression for f ( t ) becomes quitelong. The main point we want to show is that the differ-ent dynamical regimes lead to distinct behaviors of f ( t ),and hence of the generated entanglement entropy S ( t ),which are summarized in Table I. We now describe themin detail. B. Evolution in stable sectors
In the dynamically stable sectors A and B of Fig. 1,both λ ± are real and non-zero, implying that the evolu-
10 20 30 40 50 Ω x t S A A A-DBD D FIG. 2. The evolution of the entanglement entropy (25) be-tween the two modes for k y = 0 . k x > ω/ω y = 0 . A ), 0 .
95 ( A ), 1 ( A-D ), 1 .
05 ( D ), 1 . D )and 1 .
95 ( B ), where ω µ = p k µ and the label indicates thecorresponding sector in Fig. 1. S ( t ) is quasiperiodic in curves A , A and B , but increases logarithmically (on average) in A - D , and linearly in D , D . The initial state is the separableground state of H (uncoupled oscillators). tion of f ( t ) and S ( t ) will be quasiperiodic, as seen in Fig.2 (curves A , A and B ). The initial state was chosenas the ground state of h ( α µ = p k µ in (28)). Startingfrom point 1 in sector A (Fig. 1), the generated entangle-ment S ( t ) remains small when ω is well below the firstcritical value ω y = p k y (curve A ). As ω increases, S ( t ) will exhibit increasingly higher maxima, showing atypical resonant behavior for ω close to ω y (border withsector D ), where λ − vanishes. Near this border, S ( t ) willessentially exhibit large amplitude low frequency oscilla-tions determined by λ − , with maxima at t ≈ t m = mπ λ − ( m odd), plus low amplitude high frequency oscillationsdetermined by λ + , as seen in curve A .As ω increases, the system enters dynamically unsta-ble sectors for ω y ≤ ω ≤ ω x = √ k x , and the evolutionbecomes unbounded (curves A-D , D and D , describedin next subsection). For ω > ω x , the system reenters thedynamically stable regime and exhibits again the previ-ous behaviors, with an oscillatory resonant type evolutionfor ω above but close to ω x (curve B in Fig. 2).Close to instability but still within the stable regime,the maximum entanglement reached is of order ln | ω − ω µ | : For ω close to ω µ ( µ = x, y ) on the stable side, andfor the initial conditions (28), f ( t ) will be maximum at t ≈ t m , with f ( t m ) ≈ ω | ε − | λ λ − r ( αxαy + ω λ + ) sin mπλ +2 λ − + α µ cos mπλ +2 λ − α x α y , (32)where λ + ≈ q ε + + ω µ ) and λ − ≈ s ω µ | ε − || ω µ − ω | ε + + ω µ , (33) ab k y (cid:144) k x S ba k y (cid:144) k x S FIG. 3. The maximum entanglement S ( t m ) reached in stablesectors close to instability, as a function of the anisotropyratio k y /k x (see Eq. (32)). Top: Vicinity of border A - D ( ω =0 . ω y ). Bottom: Vicinity of border B - D ( ω = 1 . ω x ).The initial state is the separable ground state of H ( α µ = ω µ )in curves a and a separable isotropic state ( α µ = 1) in curves b . implying f ( t m ) = O ( | ω µ − ω | − / ) and hence S ( t m ) = O ( − ln | ω µ − ω | ).Expression (32) (and hence S ( t m )) will tend to de-crease for decreasing anisotropy, i.e., increasing ratio k y /k x ≤
1, as seen in Fig. 3 for m = 1, vanishing inthe isotropic limit k y /k x → f ( t m ) = O ( | k x − k y | ) / ). On the other hand, the behavior for k y /k x → H ( α µ = ω µ , curves a ), f ( t m ) will vanish at thefirst border ω ≈ ω y (top panel), where f ( t m ) = O ( √ ω y ),but diverge at the second border ω = ω x (bottom panel),where f ( t m ) = O (1 / √ ω y ), as obtained from Eq. (32). Ifthe initial state is fixed, however, f ( t m ) will approacha finite value for k y /k x →
0, and exhibit a monotonousdecrease on average with increasing ratio k y /k x in bothborders (curves b in Fig. 3), as also implied by (32). Wealso mention that the high frequency oscillations in f ( t m )and S ( t m ) observed in Fig. 3 stem from the λ + /λ − ratioin the arguments of the trigonometric functions in Eq.(32). For ω close to ω µ , this ratio is minimum around k y /k x ≈ /
5, which leads to the observed decrease in theoscillation frequency of S ( t m ) in the vicinity of this ratio(top panel). Ω (cid:144) Ω x S FIG. 4. The entanglement entropy between the two modesattained at fixed time ω x t = 40, as a function of the (constant)frequency ω , for the oscillator parameters and initial state ofFig. 2. Entanglement is bounded for ω < ω y (sector A ) and ω > ω x (sector B ), but is proportional to t in the instabilitywindow ω y < ω < ω x (sector D ). C. Evolution in unstable sectors
Let us now examine in detail the evolution of S ( t ) inthe dynamically unstable regimes. At the critical fre-quencies ω = ω µ , µ = y, x (borders A - D and B - D ), λ − vanishes and Eqs. (12) and (24) lead, for large t and theinitial conditions (28), to the critical evolution f ( t ) ≈ t ω | ε − | λ r ( αxαy + ω λ + ) sin λ + t + α µ cos λ + tα x α y , (34)where λ + = q ε + + ω µ ) >
0. This entails a linear increase, on average, of f ( t ) in this limit, and hence, a logarithmic growth of S ( t ), according to Eq. (26): S ( t ) ≈ S ( t ) + ln t , (35)where S ( t ) = 1 + ln[ f ( t ) /t ] is a bounded function os-cillating with frequency λ + . This behavior (curve A - D in Fig. 2) is the ω → ω µ limit of the previous resonantregime.On the other hand, in the unstable sector D ( ω y < ω <ω x ), λ − becomes imaginary. This leads to an exponential term in f ( t ) ( sin λ − tλ − → sinh | λ − | t | λ − | ), which will dominatethe large t evolution: In this sector Eqs. (12), (24) and(27) imply, for large t , f ( t ) ∝ e | λ − | t , S ( t ) ≈ | λ − | t , (36)and hence, a linear growth (on average) of the entan-glement entropy with time (curves D , D in Fig. 2).Therefore, in the unstable window ω y ≤ ω ≤ ω x , thereis an unbounded growth with time of the entanglemententropy, which will originate a pronounced maximum inthe generated entanglement at a given fixed time andanisotropy as a function of ω , as appreciated in Fig. 4. S + S + S H t L + ln t0 50 10001020 Ω x t S FIG. 5. Critical evolution of the entanglement entropy at theborder between sectors with distinct dynamics, for isotropicinitial conditions ( α µ = 1). The lower, middle and uppercurve correspond respectively to the border A - D (at k y =0 . k x , with ω = p k y ), B - E (at k y = − . k x , with ω givenby (18)) and the critical points L (Eq. (20)). The asymptoticbehavior for large t (Eqs. (35), (38), (40)) is indicated. We now examine the behavior at the other sectors ofFig. 1. In the unstable sectors C and E , where one orboth of the constants k µ are negative, λ ± are imaginaryor complex (Fig. I). This implies an exponential increaseof f ( t ), as indicated in table I, entailing again a lin-ear asymptotic growth of the entanglement entropy withtime: S ( t ) ≈ ( | λ + | + | λ − | ) t in C and S ( t ) ≈ | Im( λ ± ) | t in E , neglecting constant or bounded terms.On the other hand, at the border between sectors B and E , which corresponds to the critical curve ∆ = 0between both points L in Fig. 1, we obtain, for large t and k x = k y (with the initial conditions (28)), the asymptoticbehavior f ( t ) ≈ | ε − | ω α x α y + ε − ωλ √ α x α y t , (37)where λ = p ε + + ω >
0. This leads to S ( t ) ≈ S + 2 ln t , (38)with S ≈ | ε − | ω α x α y + ε − ωλ √ α x α y ]. Hence, the unboundedgrowth of f ( t ) and S ( t ) is here more rapid than thatat the previous borders A - D and B - D ( ω = ω y or ω x )(quadratic instead of linear increase of f ( t )). At the bor-der E - C the asymptotic behavior of f ( t ) is still exponen-tial (i.e., linear growth of S ( t )).Finally, a further remarkable critical behavior arises atthe special critical points L , obtained for condition (20),where all sectors B, C, D and E meet. We obtain herea purely polynomial evolution of ( f ( t ) + 1 / , as impliedby Eqs. (21). For large t , this leads to a quartic increaseof f ( t ): f ( t ) ≈ α x α y + ω √ α x α y ω t , (39) ab Ω x t S FIG. 6. Evolution of the entanglement entropy for a stepwisevarying frequency ω , starting from the separable ground stateof H (with k y = 0 . k x > a we have set ω/ω x =0 .
5, 0 .
7, 0 and 0 .
21 for successive time intervals of length ω x ∆ t = 30, such that the system is close to the first instabilityat the second interval (0 . ω x ≈ . ω y , with ω µ = p k µ ),while in curve b the only change is ω = 0 . ω x ≈ . ω y in the second interval, such that the system enters there theunstable regime leading to a linear entanglement growth. Thisplot shows that entanglement can be increased, kept constantand returned to a vanishing value just by tuning the frequency ω . implying the following logarithmic increase of S ( t ): S ( t ) ≈ S + 4 ln t , (40)where S ≈ α x α y + ω √ α x α y ω ]. Hence, the increase is herestill more rapid than at both previous borders. Thesecritical behaviors are all depicted in Fig. 5. D. Entanglement control
We finally show in Fig. 6 the possibilities offered by thismodel for controlling the entanglement growth througha stepwise time dependent frequency, starting from theseparable ground state of H . After applying a “low” ini-tial frequency ω = 0 . ω x for ω x t <
30, which leads to aweak quasiperiodic entanglement, by tuning ω to a valueclose to the first instability ω y = p k y for a finite time(30 < ω x t < a ). Then, by setting ω = 0 (i.e.,switching off the field or rotation), entanglement is kepthigh and constant, since the evolution operator becomesa product of local mode evolutions. Finally, by turningthe frequency on again up to a low value, entanglementcan be made to exhibit strong oscillations, practicallyvanishing at the minimum if ω is appropriately tuned.Thus, disentanglement at specific times can be achievedif desired. The entanglement increase at the second in-terval can be enhanced by allowing the system to enterthe instability region for a short time, as shown in curve b . f X l z \ - Ω x t f , X l z \ FIG. 7. Evolution of the average occupation number f ( t ) (Eq.(24)) and angular momentum h l z i for case a of Fig. 6. The growth of the average occupation f ( t ) (and hencethe entanglement entropy S ( t )) in the second interval isstrongly correlated with that of the average angular mo-mentum h l z i t , i.e., with the entangling term in H , asseen in Fig. 7 for case a of Fig. 6. Nonetheless, whilethe evolution of f ( t ) is similar to S ( t ), the average an-gular momentum exhibits pronounced oscillations when ω is switched off, since l z is not preserved in the presentanisotropic trap ([ l z , h ] = 0). These oscillations persistin the last interval, although shifted and partly atten-uated. Here the vanishing of h l z i t provides a check forthe vanishing entanglement, since in a separable state h l z i = h q x ih p y i − h q y ih p x i and for the present initial con-ditions h q µ i = h p µ i = 0 for all times. Thus, f ( t ) = 0implies here h l z i t = 0, although the converse is not valid.Though initially correlated, we remark that h l z i t andthe average occupation f ( t ) do not have a fixed asymp-totic relation in the whole plane. For instance, at thestability borders ω = ω µ , h l z i t increases on average as t for high t , i.e., as f ( t ) (Eq. (34)), and the same relationwith f ( t ) holds in the unstable sector D ( ω y < ω < ω x ),where h l z i t ∝ e | λ − | t . Nonetheless, in an unstable poten-tial at the critical curve ∆ = 0, h l z i t ∝ t (on average)for large t , increasing then as f ( t ) (Eq. (37)), while at thecritical points L we obtain h l z i t ∝ t , i.e., h l z i t ∝ f / ( t )asymptotically (Eq. (39)). IV. CONCLUSIONS
We have analyzed the entanglement generated byan angular momentum coupling between two harmonicmodes, when starting from a separable gaussian state.The general treatment considered here is fully analyticand valid throughout the entire parameter space, in-cluding stable and unstable regimes, as well as criticalregimes where the system cannot be written in termsof normal coordinates or independent quadratic systems(non-diagonalizable H ). Hence, in spite of its simplicity,the present model is able to exhibit different types of en- tanglement evolution, including quasiperiodic evolution,linear growth, and also logarithmic growth of the entan-glement entropy with time, which can all be reached justby tuning the frequency. The model is then able to mimicthe typical evolution regimes of the entanglement en-tropy encountered in more complex many-body systems.Even distinct types of critical logarithmic growth can bereached when allowing for general quadratic potentials.The system offers then the possibility of an easily con-trollable entanglement generation and growth, throughstepwise frequency changes, which can also be tuned inorder to disentangle the system at specific times. Themodel can therefore be of interest for continuous vari-able based quantum information.The authors acknowledge support from CONICET(LR,NC) and CIC (RR) of Argentina. We also thankProf. S. Mandal for motivating discussions during hisvisit to our institute. A. Appendix
In order to highlight the non-trivial character of thepresent model when considered for all real values of theconstants k µ and frequency ω >
0, we provide here somefurther details [20]. With the sole exception of the criti-cal curve ∆ = 0 (Eq. (18)), the Hamiltonian (3) can bewritten as a sum of two quadratic Hamiltonians, h = 12 ( α + p + β + q ) + 12 ( α − p − + β − q − ) , (41)where p ± , q ± are related with q x,y , p x,y by the linearcanonical transformation q ± = q x,y − ηp y,x (1+ γη ) , p ± = p x,y + γq y,x ,γ = (∆ − ε − ) / (2 ω ) , η = γ/ε + , (42)such that [ q r , p s ] = iδ rs , [ q r , q s ] = [ p r , p s ] = 0 for r, s = ± , and α ± = 12 + ε − ± ω , β ± = ∆ ω (∆ α ± − ε − ) , with α ± β ± = λ ± (Eq. (10)). Nonetheless, the coefficients α ± , β ± can be positive, negative or zero, and may be-come even complex, according to the values of k x , k y and ω . We may obviously interchange p ± with q ± in (42) bya trivial canonical transformation p ± → q ± , q ± → − p ± .This freedom in the final form will be used in the follow-ing discussion.In sector A of Fig. 1, α ± , β ± in Eq. (41) are both real and positive , and the system is equivalent to twoharmonic modes. Here λ ± are both real. In sector B , α + , β + are positive but α − , β − are both negative, so thatthe system is here equivalent to a standard plus an “in-verted” oscillator. Nevertheless, λ ± remain still real. Insector C , α ± β ± <
0, and the effective quadratic poten-tial becomes unstable in both coordinates (i.e., α ± > β ± < λ ± are both imaginary. In sector D , α + and β + are positive but α − β − <
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