Entanglement tongue and quantum synchronization of disordered oscillators
The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.
Citation Lee, Tony E., Ching-Kit Chan, and Shenshen Wang. “Entanglement Tongue and Quantum Synchronization of Disordered Oscillators.” Phys. Rev. E 89, no. 2 (February 2014). © 2014 American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevE.89.022913
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/89028
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW E 89, 022913 (2014)
Entanglement tongue and quantum synchronization of disordered oscillators
Tony E. Lee,1,2 Ching-Kit Chan,1,2 and Shenshen Wang3 1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Physics and Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 16 December 2013; published 12 February 2014) We study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when each oscillator is near its quantum ground state. Two quantum oscillators with different frequencies exhibit an entanglement tongue, which is the quantum analog of an Arnold tongue. It means that the oscillators are entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling increases with detuning. An ensemble of many oscillators with random frequencies still exhibits a synchronization phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the frequency disorder. Our results can be experimentally observed with trapped ions or neutral atoms.
DOI: 10.1103/PhysRevE.89.022913 PACS number(s): 05.45.Xt, 42.50.Lc, 03.65.Ud
I. INTRODUCTION (a) (b) 0.16 Synchronization is a fascinating phenomenon at the in- 5 50 terface of statistical physics and nonlinear dynamics [1,2]. 0.12
) 4 ) 40 1 1 It is a collective behavior that arises among a group of κ locked κ entangled self-sustained oscillators, each with a random intrinsic fre- 3 30 0.08 quency. The interaction between the oscillators overcomes the 2 20 frequency disorder and causes them to oscillate in unison. V (units of V (units of 0.04 Synchronization of biological cells plays an important role in 1 10 heart beats [3], circadian rhythm [4], and neural networks [5]. unlocked unentangled 0 0 0 It is also important in active hydrodynamic systems [6], −10 −5 0 5 10 −100 −50 0 50 100 Δ (units of κ ) Δ (units of κ ) such as the beating of flagella [7] and arrays of cilia [8]. 1 1 Applications of synchronization include the self-organization of laser arrays [9], improving the frequency precision of FIG. 1. (Color online) (a) Arnold tongue for phase locking of oscillators [10], and stabilizing atomic clocks with each two classical oscillators. V is the coupling strength, and is the other [11]. difference of the intrinsic frequencies. (b) Entanglement tongue for There has been much theoretical work on synchronization two oscillators in the quantum limit. Concurrence is plotted using of classical oscillators. Each oscillator is usually modeled as color scale on right. The dashed line marks the edge of the entangled a nonlinear dynamical system with a limit-cycle solution that region. oscillates with its own intrinsic frequency. Then due to the mutual interaction, the oscillators spontaneously synchronize with each other in steady state. losing individual phonons [27]. The second effect is that the Synchronization is usually studied in two scenarios: two oscillators can be quantum mechanically entangled with each oscillators and a large ensemble of oscillators with all-to-all other [28]. The general question is whether synchronization coupling. In the case of two oscillators, phase locking occurs survives in the quantum limit, and how quantum mechanics when the coupling strength is above a critical value [1,12]. qualitatively changes the behavior. This critical coupling increases with the oscillators’ frequency To study quantum synchronization, it is useful to consider detuning. The “Arnold tongue” refers to the set of coupling and quantum van der Pol oscillators [22,23], since there has been detuning values for which phase locking occurs [Fig. 1(a)]. a lot of work on the synchronization of classical van der Pol In the case of a large ensemble of oscillators with random oscillators [1,12,16–19]. Recently, one of us showed that when frequencies, there is a nonequilibrium phase transition from the quantum oscillators are reactively coupled (via a term in the unsynchronized phase to the synchronized phase [13–19]. the Hamiltonian), there is no synchronization or entanglement The critical coupling for the phase transition depends on the in the quantum limit, when the oscillators are confined to the frequency disorder. ground state |0 and the single-phonon state |1 [22]. This is There has been growing interest in synchronization of because quantum noise washes out the phase correlations. quantum systems [20–26]. In this case, each oscillator is a In this paper, we study the synchronization of dissipatively quantum harmonic oscillator with driving, dissipation, and coupled van der Pol oscillators in the quantum limit, and we nonlinearity. The classical limit corresponds to when each find significant differences with the reactive case. We first oscillator has many phonons (or photons), while the quantum consider the case of two quantum oscillators. We find that limit corresponds to when each oscillator is near its quantum the oscillators still exhibit phase correlations in the quantum ground state. Quantum mechanics introduces two effects. The limit, when each oscillator occupies only |0 and |1 . Also, the first is quantum noise, which is due to the oscillator gaining or oscillators exhibit entanglement, which is a genuinely quantum
1539-3755/2014/89(2)/022913(10) 022913-1 ©2014 American Physical Society TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG PHYSICAL REVIEW E 89, 022913 (2014) property and can be quantified using concurrence [29]. In order and (3) become for entanglement to exist in steady state, the coupling strength 2 r˙1 = r1 κ1 − 2κ2r − V (r1 − r2 cos θ), (4) must be larger than a critical value, and the critical coupling 1 increases with the detuning of the oscillators [Fig. 1(b)]. This = − 2 − − r˙2 r2 κ1 2κ2r2 V (r2 r1 cos θ), (5) “entanglement tongue” is the quantum analog of the Arnold r r tongue. θ˙ =− − V 1 + 2 sin θ, (6) Then we consider a large ensemble of quantum oscillators r2 r1 with random frequencies and all-to-all coupling. The synchro- where θ = θ − θ is the phase difference, and = ω − ω nization phase transition still occurs in the quantum limit, but 2 1 2 1 | is the detuning. Equation (6) shows that the coupling causes each oscillator must occupy at least the two-phonon state 2 . the oscillator phases to attract each other, while the detuning We analytically calculate the critical coupling by doing a linear pulls them apart. If V is larger than a critical value Vc,the stability analysis of the unsynchronized phase and find good oscillators phase lock with each other, i.e., there is a stable agreement with numerical simulations. The dependence of the fixed-point solution with r1,r2,θ constant in time. If V< critical coupling on the frequency disorder is different in the V , the oscillators are unlocked and |θ| increases over time. quantum limit than in the classical limit. c Vc increases with | |, which reflects the fact that the more In Ref. [26], it was found that linear oscillators can exhibit different the oscillators are, the harder it is to synchronize collective oscillations and entanglement when all normal them. The “Arnold tongue” is the region in V, space such modes except one are damped. In contrast, we consider non- that V>Vc( ), as shown in Fig. 1(a) [1,12]. linear oscillators in order to directly compare with well-known Then consider the generalization to N oscillators with all- classical synchronization models. Also, Ref. [23] considered to-all coupling: the synchronization of one quantum van der Pol oscillator with an external drive and found that quantum noise introduces a N 2 V nonzero threshold of driving strength, below which there is α˙ n =−iωnαn + αn(κ1 − 2κ2|αn| ) + (αm − αn), (7) N only weak frequency locking. In contrast, we consider two or m=1 more oscillators and focus on phase locking. where the frequencies ωn are randomly drawn from a distri- In Sec. II, we describe the model of quantum van der bution g(ω), and we let N →∞. This model has also been Pol oscillators, as well as experimental implementation with studied previously [16,17,32,33]. The competition between the cold atoms. In Sec. III, we consider two quantum oscillators interaction and the frequency disorder results in a continuous and characterize their phase correlations and entanglement. phase transition when the coupling is at a critical value In Sec. IV, we consider a large ensemble of quantum Vc (which is not the same as for two oscillators). When oscillators and characterize the synchronization transition. In V>Vc, the interaction overcomes the frequency disorder, Sec. V, we conclude and suggest future directions of research. and there is macroscopic synchronization. The order parameter The Appendix provides details on the two-mode Wigner | | (1/N) n αn is zero in the unsynchronized phase and greater function. than zero in the synchronized phase. The synchronized phase iβ breaks the U(1) symmetry present in the system (αn → αne ). II. MODEL Of interest is how Vc depends on the frequency disorder g(ω). In the limit of small V ,Eq.(7) is equivalent to the well-known A. Classical regime Kuramoto model [13–15]. A common starting point for studying classical synchro- nization is to consider an oscillator whose complex amplitude B. Quantum regime α obeys We are interested in what happens to the synchronization 2 α˙ =−iωα + α(κ1 − 2κ2|α| ). (1) behavior of the above models in the quantum limit, when the oscillators are near the ground state. To study this, we This is the amplitude equation for the van der Pol oscilla- quantize the above classical models. First, we review how tor [1,30]. It is also the normal form for a Hopf bifurcation [31]. to quantize the van der Pol oscillator in Eq. (1). We let the κ1 and κ2 correspond to negative damping and nonlinear oscillator be a quantum harmonic oscillator, meaning that it damping, respectively. They are assumed to be positive, so the exists in the Hilbert space of Fock states {|n }, where n is steady-state solution is a limit cycle with amplitude |α|= κ1 2κ2 the number of phonons. The quantum van der Pol oscillator is and frequency ω. The phase of the oscillations is free, since described in terms of a master equation for the density matrix Eq. (1) is time-translationally invariant. ρ [22,23]: Now consider two coupled oscillators with linear dissipa- † † † ρ˙ =−i[H,ρ] + κ (2a ρa − aa ρ − ρaa ) tive coupling: 1 2 †2 †2 2 †2 2 + κ2(2a ρa − a a ρ − ρa a ), (8) α˙ =−iω α + α (κ − 2κ |α |2) + V (α − α ), (2) 1 1 1 1 1 2 1 2 1 H = ωa†a, (9)
=− + − | |2 + − where = 1. There are two dissipative processes: the oscilla- α˙ 2 iω2α2 α2(κ1 2κ2 α2 ) V (α1 α2). (3) † tor gains one phonon at a time with rate 2κ1aa , and it loses †2 2 This model is well-known in nonlinear dynamics [1,12]. It is two phonons at a time with rate 2κ2a a . These correspond iθn convenient to use polar coordinates: αn = rne . Then Eqs. (2) to negative damping and nonlinear damping, respectively.
022913-2 ENTANGLEMENT TONGUE AND QUANTUM . . . PHYSICAL REVIEW E 89, 022913 (2014) √ This model has also been studied in the context of polariton |A =(|01 −|10 )/ 2 corresponds to the antiphase state. condensates [34,35]. Other dissipative models were similarly (The reason for this correspondence is discussed in the quantized in Refs. [36–40]. Appendix.) |S is a dark state with respect to c, since c|S =0. The quantum limit corresponds to when κ2 is large relative Thus, c dissipatively pumps the system into |S , leading to to κ1, since then the oscillator is near the ground state [22]. in-phase locking [42–44]. We analyze this model in Sec. III. → Conversely, the classical limit corresponds to when κ2 0, Then consider the quantum version of the all-to-all model since then the oscillator has an infinite number of phonons. To in Eq. (7)forN oscillators: see the connection with the classical model, one notes from =− + † − † − † Eq. (8) that ρ˙ i[H,ρ] κ1 (2anρan ananρ ρanan) n da =− + − † 2 iω a κ1 a 2κ2 a a . (10) + 2 †2 − †2 2 − †2 2 dt κ2 2anρan an anρ ρan an In the classical limit, one can replace the operator a with a n V complex number α denoting a coherent state, and Eq. (10) + (2c ρc† − c† c ρ − ρc† c ), (14) N mn mn mn mn mn mn becomes Eq. (1). m
022913-3 TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG PHYSICAL REVIEW E 89, 022913 (2014) system into |A , leading to antiphase locking. In addition, 1 (a) there are other methods (with Rydberg atoms [45] or trapped (b) ions [46]) to dissipatively pump the system into |A , although 0.2 0.8 the resulting c˜ is more complicated. 0.6 W W 0.15 0.4 III. TWO QUANTUM OSCILLATORS 0.2 In this section, we study the quantum model defined by 0.1 0 Eqs. (12) and (13). There are several factors at work here. −π −π/2 0 π/2 π −π −π/2 0 π/2 π θ − θ θ − θ As in the classical model, the dissipative coupling causes the 2 1 2 1 phases to attract and lock with each other, while the detuning inhibits phase locking. But the quantum model introduces two FIG. 2. (Color online) Two-mode Wigner function in (a) the →∞ → new features. The first new feature is quantum noise due to the quantum limit (κ2 ) and (b) the classical limit (κ2 0), plotted as a function of phase difference with V = 3κ : = 0 (solid, black stochastic dissipation that adds one phonon (κ1) and removes 1 line) and = 4κ (dashed, red line). two phonons (κ2) at a time. Quantum noise is quantitatively 1 equivalent to classical white noise when an oscillator has many phonons, but not when the oscillator is near the ground r1,r2,θ1 + θ2 so that W depends only on θ: state, since then the addition or loss of a phonon has a large 1 κ V (κ + V )[2(3κ + V ) cos θ − sin θ] effect on the system [27]. We are interested in whether phase W(θ) = + 1 1 1 . attraction survives in the quantum limit, when there is a lot 2π N of quantum noise. The second new feature is entanglement, (23) which results from the fact that the dissipative coupling tries to Details of this calculation are provided in the Appendix. pump the system into the entangled state |S . We are interested Figure 2(a) shows example Wigner functions in the quantum in whether entanglement exists in the quantum limit despite limit. The Wigner function is peaked at some value of θ, decoherence from quantum noise and frequency detuning. meaning that the relative phase tends toward some value. The peak is quite broad, indicating that the phase correlation is = A. Phase correlations imperfect. If the oscillators are identical ( 0), then the peak is at θ = 0, which means the oscillators tend to be in-phase We want to find the steady-state density matrix of Eq. (12). = →∞ with each other. However, if they are nonidentical ( 0), the Since we only care about the quantum limit (κ2 ), we peak is offset from θ = 0: when >0, then θ<0. The height can use a truncated Hilbert space that contains only |0 and | →∞ of the peak increases with V due to stronger phase attraction. 1 . Then the steady-state density matrix in the limit κ2 This behavior is actually similar to the classical model is [Eqs. (2) and (3)]. When classical oscillators are phase locked, κ (5κ + 2V )[ 2 + 4(3κ + V )2] the Wigner function has a delta-function peak at a certain value |ρ| = − 1 1 1 , of θ, as seen in Fig. 2(b).If >0, then the peak would be at 00 00 1 N (18) θ<0, just like in Eq. (23).1 Then suppose one added white 2 2 κ1(2κ1 + V )[ + 4(3κ1 + V ) ] noise to Eqs. (2) and (3). The noise would broaden the peaks in 01|ρ|01 =10|ρ|10 = , N the Wigner function, just like in Fig. 2(a). The oscillator would (19) no longer be strictly phase locked due to noise-induced phase slips, but there would still be a tendency toward locking [1]. κ2[ 2 + 4(3κ + V )2] 11|ρ|11 = 1 1 , (20) There would also no longer be an Arnold tongue due to the N absence of phase locking. 2κ V (κ + V )(6κ + 2V − i ) Thus, the phase correlations in the quantum limit are still |ρ| = |ρ| ∗ = 1 1 1 , 01 10 10 01 N qualitatively consistent with finite classical noise. This is a (21) rather surprising result, since na¨ıvely one would expect quan- tum noise to completely wash out the phase correlations when N = (3κ + V ) 3κ 2 + 36κ2 1 1 1 κ2 →∞. In fact, when the coupling is reactive [via a term † † + 2 + 2 + 2 in the Hamiltonian, V (a a2 + a1a )] instead of dissipative, 108κ1 V 32κ1V , (22) 1 2 there are indeed no phase correlations in this limit [22]. So and the other matrix elements are zero. [This result is most the behavior in the quantum limit depends a great deal on the easily derived using the spin model in Eqs. (16) and (17).] type of coupling: phase attraction survives with dissipative The fact that there are off-diagonal matrix elements like coupling but not with reactive coupling. |01 10| means that there are still phase correlations between the oscillators in the quantum limit. The phase correlations can be better understood from 1Equations (2)and(3) are usually written with the opposite sign ∗ ∗ the two-mode Wigner function W(α1,α1 ,α2,α2 ), which is a convention for ωn [1,12]. With the usual convention, when the quasiprobability density for the two-oscillator system [47]. oscillators are locked, the oscillator with the larger frequency is ahead iθn Let us use radial coordinates: αn = rne . Since we only in phase. Then using the sign convention of Eqs. (2)and(3), if >0, care about the relative phase θ = θ2 − θ1, we integrate out the locked state has θ<0.
022913-4 ENTANGLEMENT TONGUE AND QUANTUM . . . PHYSICAL REVIEW E 89, 022913 (2014)
There is another similarity between the quantum and disorder g(ω). As in Sec. III, there are several factors at classical behavior. The classical model can exhibit “amplitude work. The coupling tries to synchronize the oscillators with death” (or “oscillator death”), which means that the trivial each other, but the frequency disorder and quantum noise solution α1 = α2 = 0isstable[1,12]. This occurs when | | inhibit synchronization. In this section, we do not discuss and V are large and is due to the fact that the dissipative bipartite entanglement, since the all-to-all coupling leads coupling increases the overall dissipation each oscillator to a concurrence that scales as ∼ 1/N, but it would be feels. Interestingly, the quantum model exhibits something interesting, as a future work, to see whether there is multipartite akin to amplitude death: in the limit | |,V →∞, we find entanglement. † † → The strategy for calculating V is similar to that for classical a1a1 , a2a2 0. c models [15,17,19]. First we find the mean-field equations. Then we find the unsynchronized state. Then we do a linear B. Entanglement tongue stability analysis around the unsynchronized state. The onset Now we check whether the quantum model exhibits of instability of the unsynchronized state signals a continuous entanglement, which is a property unique to quantum systems. phase transition to the synchronized state. The key difference Two particles are entangled if and only if their density matrix with classical models is that the stability analysis here is based cannot be written as a sum of separable states [28]. Since on the master equation for the density matrix instead of a each oscillator is limited to |0 and |1 in the quantum partial differential equation for the probability density. limit, we can quantify the entanglement by calculating the With dissipative coupling, the phase transition turns out to concurrence for the density matrix ρ in Eqs. (18)–(21). be continuous. In contrast, with reactive coupling, the phase ∗ Considerρ ˜ = (σy ⊗ σy )ρ (σy ⊗ σy ). Concurrence is defined transition is discontinuous [22]. as C ≡ max(0,λ1 − λ2 − λ3 − λ4), where λ1,λ2,λ3,λ4 are the square roots of the eigenvalues of ρρ˜ in decreasing order [29]. A. Mean-field equations C ∈ [0,1] by definition and the oscillators are entangled if and only if C>0. Also, larger C implies more entanglement. We want to rewrite Eq. (14) in a way such that each Whether entanglement exists is the result of several com- oscillator interacts with the mean field. We first make the peting processes. The dissipative coupling tries to entangle mean-field ansatz that the density matrix is a product state of = the oscillators since it pumps the system into |S . However, density matrices for each site: ρ n ρn [43]. (This ansatz N the negative damping (κ1) and nonlinear damping (κ2) lead to is exact when is infinite, as can be shown rigorously using decoherence since they act on individual oscillators. Also, the a phase-space approach [48].) Then we plug this ansatz into detuning leads to dephasing between the oscillators and thus Eq. (14). Using the fact that reduces the entanglement. ρ˙ = ρ˙ ⊗ ρ ⊗ ρ ⊗···+ρ ⊗ ρ˙ ⊗ ρ ⊗··· The concurrence for Eqs. (18)–(21) is plotted in Fig. 1(b). 1 2 3 1 2 3 There is indeed entanglement, but only when V is greater than + ρ1 ⊗ ρ2 ⊗ ρ˙3 ⊗···+··· , (24) a critical value V that depends on . When the oscillators are c tr ρ = 1, and trρ ˙ = 0, we obtain the equation of motion for identical ( = 0), V = 8.664κ . This is nonzero because the n n c 1 ρ by tracing out all other sites: coupling must be strong enough to overcome decoherence n | | =− † + † − † − † from the onsite dissipation (κ1 and κ2). As increases, ρ˙n i[ωnanan,ρn] κ1(2anρnan ananρn ρnanan) V increases, because the coupling must overcome additional c + 2 †2 − †2 2 − †2 2 decoherence from detuning. This is reminiscent of how the κ2 2anρnan an anρn ρnan an critical coupling for phase locking in the classical model † † † † † + V [2anρna − a anρn − ρna an + A(a ρn − ρna ) increases with | |, giving rise to an Arnold tongue. Thus, n n n n n − A∗(a ρ − ρ a )], (25) we call the entanglement region defined by V>Vc( )the n n n n “entanglement tongue.” It reflects the fact that the more 1 A = a . (26) different the oscillators are, the harder it is to entangle them. N m | | = | | m In the limit of large , Vc 2 . Also, the maximum C 1 = =∞ These are self-consistent equations, since each oscillator is 4 , which occurs when 0 and V . To summarize the results for two oscillators, the quantum depends on the mean field A, which itself depends on the behavior resembles the classical behavior in terms of phase oscillators. There are N such equations. correlations, but the quantum model exhibits entanglement, Now we write out Eq. (25) in terms of matrix elements. which is a genuinely quantum feature. Although there is no Unlike in Sec. III,weletκ2 be large but finite, and we include critical coupling for phase locking in the quantum limit, there |2 in addition to |0 and |1 . Inclusion of |2 is crucial to have a is a critical coupling for entanglement. As such, the Arnold phase transition, as we show below. Using the notation ρn,jk ≡ tongue is replaced by the entanglement tongue. j|ρn|k , the equations of motion for the diagonal elements are
IV. MANY QUANTUM OSCILLATORS ρ˙n,00 =−2κ1ρn,00 + 4κ2ρn,22 ∗ Now we study the all-to-all model defined by Eqs. (14) − V (Aρn,01 + A ρn,10 − 2ρn,11), (27) and (15). The goal is to see whether a synchronization ρ˙ = 2κ ρ − 4κ ρ + V [−2ρ + 4ρ transition occurs in the quantum limit, and if so, how the n,11 1 n,00 √1 n,11 n,11 √ n,22 ∗ critical coupling Vc for the transition depends on the frequency + A(ρn,01 − 2ρn,12) + A (ρn,10 − 2ρn,21)], (28)
022913-5 TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG PHYSICAL REVIEW E 89, 022913 (2014)
ρ˙n,22 = 4κ1ρn,11 − 4κ2ρn,22 2κ κ + V (κ + V ) √ √ ρ¯ = 1 2 2 , (34) + − + + ∗ 00 2 + + + + V ( 4ρn,22 2Aρn,12 2A ρn,21). (29) κ1 κ1(3κ2 V ) V (κ2 V )
The equations of motion for the off-diagonal elements κ (κ + V ) ρ¯ = 1 2 , (35) are 11 2 + + + + κ1 κ1(3κ2 V ) V (κ2 V ) √ = − − + − + ρ˙n,10 ( 3κ1 iωn)ρn,10 V [ ρn,10 2 2ρn,21 κ2 ρ¯ = 1 , (36) √ 22 2 + + + + ∗ κ1 κ1(3κ2 V ) V (κ2 V ) − 2A ρn,20 + A(ρn,00 − ρn,11)], (30) = √ and all off-diagonal elements are zero so that A 0. Note that † → →∞ ρ˙n,21 = (−2κ2 − iωn)ρn,21 + 2 2κ1ρn,10 − 2κ1ρn,21 ρ¯ exhibits amplitude death since a a 0asV . √ + − + ∗ + − V [ 3ρn,21 A ρn,20 2A(ρn,11 ρn,22)], (31) C. Stability analysis
ρ˙ = (−κ − 2κ − 2iω )ρ + V [−2ρ Now we do a linear stability analysis of the unsynchronized n,20 1 2 n n,20 n,20 state. If the unsynchronized state becomes unstable for certain √ parameter values, that signals a continuous phase transition to + A( 2ρ − ρ )], (32) n,10 n,21 the synchronized state since then |A| > 0. We consider small perturbations δρn around the fixed point: ρn(t) = ρ¯ + δρn(t). with similar equations for ρn,01, ρn,12, and ρn,02. The mean We expand Eqs. (27)–(33) around the fixed point, keeping field can be written as only terms to linear order in δρn. It turns out that Eqs. (30) and (31) decouple from the other equations after linearization, √ 1 so we only have to consider them since A depends only on A = (ρm,10 + 2ρm,21). (33) N ρn,10 and ρn,21: m √ ˙ = − − − + δρ n,10 ( 3κ1 V iωn)δρn,10 2 2Vδρn,21 We emphasize that Eqs. (27)–(33) are accurate only to O(1/κ2) + − due to truncating the Hilbert space. VA(¯ρ00 ρ¯11), (37) √ ˙ = + − − − − δρ n,21 2 2κ1δρn,10 ( 2κ1 2κ2 3V iωn)δρn,21 B. Unsynchronized state √ + 2VA(¯ρ − ρ¯ ), (38) iβ 11 22 The model in Eq. (14) has a U(1) symmetry: an → ane . √ When the coupling V is larger than a critical value Vc,this 1 | | A = (δρm,10 + 2δρm,21). (39) symmetry is broken. A acts as an order parameter for the N phase transition: |A|=0 in the unsynchronized state, and m |A| > 0 in the synchronized state. We want to find whether the unsynchronized state is stable We now find the unsynchronized state, denoted byρ ¯n, or not, i.e., whether the perturbations δρn grow or decay. To λt which is a fixed point of the mean-field equations but with do this, we write δρn(t) = e bn, where λ is an eigenvalue and no off-diagonal elements, since this implies the lack of bn is a 2N-dimensional eigenvector. But we only care about phase coherence among the oscillators. It turns out thatρ ¯n when an eigenvalue λ = 0, since that corresponds to when the 2 is independent of ωn,sowewrite¯ρ ≡ ρ¯n. This state is easily unsynchronized state just becomes unstable ; i.e., V = Vc.So found by solving for the diagonal elements in Eqs. (27)–(29): we solve for bn,10 and bn,21 when λ = 0:
AVc[(2κ1 + 2κ2 + 3Vc + iωn)¯ρ00 − (2κ1 + 2κ2 − Vc + iωn)¯ρ11 − 4Vcρ¯22] bn,10 =− , (40) 8κ1Vc − (3κ1 + Vc + iωn)(2κ1 + 2κ2 + 3Vc + iωn) √ 2AVc[2κ1ρ¯00 + (κ1 + Vc + iωn)¯ρ11 − (3κ1 + Vc + iωn)¯ρ22] bn,21 =− , (41) 8κ1Vc − (3κ1 + Vc + iωn)(2κ1 + 2κ2 + 3Vc + iωn) 1 √ A = (b + 2b ). (42) N m,10 m,21 m
2There is the possibility that the imaginary part of λ is nonzero when the real part is zero. However, numerically we have found that this is never the case.
022913-6 ENTANGLEMENT TONGUE AND QUANTUM . . . PHYSICAL REVIEW E 89, 022913 (2014)
Now, we require self-consistency by plugging Eqs. (40) and (41) into Eq. (42). This provides an implicit expression for Vc in terms of the other parameters. At this point, it is more convenient to move to a continuum description in order to calculate Vc analytically.
D. Continuum description
So far, we have let oscillator n have frequency ωn and density matrix ρn. Since we are interested in the limit N →∞,we can just label the density matrices by frequency, ρn → ρ(ω), where ρ(ω) should be viewed as the average density matrix for all oscillators with frequency ω. Then bn → b(ω) and AV [(2κ + 2κ + 3V + iω)¯ρ − (2κ + 2κ − V + iω)¯ρ − 4V ρ¯ ] b (ω) =− c 1 2 c 00 1 2 c 11 c 22 , (43) 10 8κ V − (3κ + V + iω)(2κ + 2κ + 3V + iω) √ 1 c 1 c 1 2 c 2AV [2κ ρ¯ + (κ + V + iω)¯ρ − (3κ + V + iω)¯ρ ] b (ω) =− c 1 00 1 c 11 1 c 22 , (44) 21 8κ V − (3κ + V + iω)(2κ + 2κ + 3V + iω) 1 c 1 c 1 2 c ∞ √ A = [b10(ω) + 2b21(ω)]g(ω)dω, (45) −∞
where we use the notation bjk(ω) ≡j|b(ω)|k . Equa- Delta function (identical oscillators): tion (45) shows how the frequency disorder g(ω) comes in. = For self-consistency, we plug Eqs. (43) and (44)into g(ω) δ(ω), (51) Eq. (45). We assume that g(ω) is even: g(−ω) = g(ω). Then 10κ2 since the imaginary parts of b (ω) and b (ω) are odd in ω, Vc = . (52) 10 21 3 only their real parts contribute to the integral. Thus, Eq. (45) becomes We see that Vc →∞as κ2 →∞due to too much quantum noise, so there is no phase transition when the oscillators are ∞ 2 confined to |0 and |1 . But there is a transition when κ is large z1ω + z2 2 1 = g(ω)dω, (46) but finite so that the oscillators also occupy |2 . (Recall from −∞ ω4 + z ω2 + z 3 4 Sec. III that for two oscillators, phase correlations do survive with only |0 and |1 .) where we have defined the constants It is interesting to note that the phase transition in this case is due only to dissipative dynamics, instead of a combination of dissipative and coherent dynamics as in other = − + + + + z1 Vc[( κ1 Vc)¯ρ00 (5κ1 4κ2 Vc)¯ρ11 models [20,43,49,50]. − (4κ + 4κ + 2V )¯ρ ], (47) Uniform distribution: 1 2 c 22 = + + + + 2 z2 Vc 6κ1(κ1 κ2) Vc(3κ1 2κ2) 3Vc g(ω) = 1/(2 )forω ∈ [− , ], (53)
×[(6κ1 + 2κ2 + 3Vc)¯ρ00 + (−2κ2 + 3Vc)¯ρ11 + 2 + 2 2 + 2 + 4 − (6κ1 + 6Vc)¯ρ22], (48) 10κ1κ2 100κ1 κ2 28κ1κ2 V = , (54) = 2 + + 2 + + + 2 c z3 13κ1 8κ1κ2 4κ2 34κ1Vc 12κ2Vc 10Vc , 6κ1 (49) where is the half width of g(ω). Clearly, Vc increases as = + + + + 2 2 increases, reflecting the fact that the greater the disorder is, z4 6κ1(κ1 κ2) Vc(3κ1 2κ2) 3Vc . (50) the harder it is to synchronize the oscillators. Figure 3(a) plots Eq. (54). Equation (46) is one of the main results of this paper. It provides Lorentzian distribution: an implicit expression for V in terms of the other parameters c 1 and g(ω). g(ω) = , (55) π ω2 + 2 ⎧ + ⎨ 2κ2(5κ1 ) 3(κ − ) <κ1 E. Results for different disorder distributions = 1 Vc ⎩ , (56) After plugging in a function for g(ω) into Eq. (46) and ∞ κ1 doing the integral via contour integration, we can then solve explicitly for Vc. In general, the result of the integral is very where is the half width at half maximum of g(ω). complicated, but since we only care about the limit of large Figure 4(a) plots Eq. (56). Interestingly, synchronization never κ2 and thus large Vc, we can simplify it by expanding in 1/κ2 occurs when κ1 due to the long tails of the Lorentzian and 1/Vc. Here we state the results for different types of g(ω), distribution. If the tails are cutoff beyond some value, Vc no valid for large κ2: longer diverges, as also shown in Fig. 4(a).
022913-7 TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG PHYSICAL REVIEW E 89, 022913 (2014)
340 6 0.06 (a) (b) ) ) (a) 1 338 1 κ κ 0.04 4 |A| 336 0.02 334 2 (units of (units of
c c 0 V 332 V 300 350 400 450 500 550 κ 330 0 V (units of ) 0 1 2 3 4 0 1 2 3 4 1 Γ (units of κ ) Γ (units of κ ) 1 1 0.06 (b) FIG. 3. Phase diagram for uniform frequency disorder, showing |A| 0.04 critical coupling Vc vs. disorder width . (a) Quantum model with 0.02 κ2 = 100κ1 using Eq. (54). (b) Classical model using Eq. (58). 0 300 350 400 450 500 550 V (units of κ ) There is an important difference between the uniform and 1 Lorentzian cases. In the limit of large κ2, Vc( ) for the uniform 0.06 case is independent of , because quantum noise is more (c) 0.04 important than the disorder. But Vc( ) for the Lorentzian case |A| is always dependent on , because the long tails cause the disorder to be as important as quantum noise. 0.02 To check these predictions, we have simulated the mean- 0 field equations Eqs. (27)–(32) using fourth-order Runge-Kutta 300 350 400 450 500 550 = = × −4 V (units of κ ) integration for N 3000 with a step size of dt 5 10 /κ1 1 3 for a time of t = 10 /κ1. Figure 5 shows that the simulations agree well with the analytical predictions. FIG. 5. (Color online) Synchronization phase transition found by simulating mean-field Eqs. (27)–(32)withN = 3000 and κ2 = 100κ1. Order parameter |A| is plotted as a function of coupling strength V for different frequency distributions. (a) Delta-function F. Comparison with classical results distribution (identical oscillators). (b) Uniform distribution with We want to compare the dependence of Vc on g(ω)inthe = 20κ1. (c) Lorentzian distribution with = 0.7κ1 and cutoff quantum limit with that in the classical limit. Since the classical at |ω|=100 . Arrows point to the critical coupling predicted by model [Eq. (7)] has been studied previously [17], below we Eq. (46). quote the known results for different g(ω), using the definitions in Eqs. (51), (53), and (55). Uniform distribution: Delta function (identical oscillators): − 2 −1 2(Vc κ1) π = π + tan < κ1 Vc 2 Vc = 0. (57) π = tan−1 κ . (58) V V − κ 2 1 Due to the lack of disorder and noise, there is always c c 1 synchronization as long as V>0. Vc is found by solving these implicit expressions. It is plotted in Fig. 3(b). Lorentzian distribution: ⎧ √ ⎪ κ +3 − κ2−2κ +5 2 2000 2 ⎨ 1 1 1 <κ (a) (b) 2 1 ) ) = 1 1 Vc . (59) κ 1500 κ 1.5 ⎩⎪ ∞ κ1 1000 1 This is plotted in Fig. 4(b). (units of (units of
c 500 c 0.5 Clearly, the classical results are different from the quantum V V results. However, there are also some notable similarities. In 0 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 both quantum and classical limits, there cannot be synchro- Γ (units of κ ) Γ (units of κ ) nization with Lorentzian disorder when κ1.Also,for 1 1 uniform disorder with large , both quantum and classical 2 FIG. 4. Phase diagram for Lorentzian frequency disorder, show- limits have Vc = /3κ1. ing critical coupling Vc vs. disorder width . (a) Quantum model In fact, the quantum results are similar to what one would with κ2 = 100κ1. The solid line is without a cutoff [Eq. (56)]. The expect by adding a lot of white noise to the classical model dashed line is with a cutoff at |ω|=100 . (b) Classical model using [Eq. (7)]. The curve Vc( ) has nonanalytic points in the Eq. (59). classical limit, but they are smoothed out by quantum noise.
022913-8 ENTANGLEMENT TONGUE AND QUANTUM . . . PHYSICAL REVIEW E 89, 022913 (2014)
Also, Vc in the quantum limit is a lot larger than in the classical momentum. If the state of the oscillator is given by a density limit, because the system needs to overcome substantial matrix ρ, the corresponding Wigner function is [47] quantum noise. ∞ 1 W(x,p) = dy x − y|ρ|x + y e2ipy , (A1) π −∞ V. CONCLUSION = m|ρ|n Wmn(x,p), (A2) We have studied the synchronization of dissipatively mn coupled van der Pol oscillators in the quantum limit. Syn- where m and n denote Fock states, and we define chronization survives all the way down to the quantum limit ∞ (|1 for two oscillators, and |2 for many oscillators), and the 1 2ipy Wmn(x,p) = dy ψm(x − y)ψn(x + y)e . (A3) synchronization behavior is qualitatively consistent with noisy π −∞ classical oscillators. However, the quantum model exhibits ψ (x) is the Fock state |n in the position basis: entanglement, which is absent in the classical model. n 1 Given that synchronization and entanglement can exist in 1 4 x2 ψ (x) = exp − H (x). (A4) the quantum limit, there are numerous directions for future n π4n(n!)2 2 n work. One possibility is to go beyond the all-to-all coupling and consider low-dimensional lattices with short-range cou- Hn(x) is the Hermite polynomial of degree n. pling. Classical oscillators with short-range coupling exhibit Then a two-mode Wigner distribution W(x1,p1,x2,p2) universal scaling: the correlation length has a power-law for two oscillators can be thought of as a quasiprobability dependence on the disorder width [51]. There can also be phase distribution in the four-dimensional phase space. It is defined transitions in low dimensions [52,53]. One should investigate by ∞ how quantum mechanics affects these classical results. A 1 + = 2i(p1y1 p2y2) convenient technique for studying disordered oscillators in W(x1,p1,x2,p2) dy1dy2 e π 2 −∞ low dimensions is real-space renormalization group, which × − − | | + + was used in the classical regime [51,54,55]. x1 y1,x2 y2 ρ x1 y1,x2 y2 , (A5) Another promising direction is to characterize the entangle- ment when there are more than two oscillators. For example, = m1m2|ρ|n1n2 Wm n (x1,p1)Wm n (x2,p2). (A6) how does entanglement depend on the frequency disorder? 1 1 2 2 m1n1m2n2 Does entanglement reach a maximum or exhibit a diverging length scale at a phase transition [56,57]? Is there multipartite Now we change variables a few times. First,√ we write the = + ∗ = entanglement [28]? Wigner function√ in terms of αn (xn ipn)/ 2 and αn Aside from synchronization, classical oscillators also (xn − ipn)/ 2 instead of xn and pn, where αn corresponds exhibit a variety of collective behavior: glassiness [58], to a coherent state. Then we move to polar coordinates: αn = iθn chimeras [59], phase compactons [60], and topological de- rne . Then since we only care about the relative phase θ = fects [53]. One should see what happens to these behaviors in θ2 − θ1, we integrate out r1, r2, and θ1 + θ2. After doing all the quantum limit. Furthermore, it would be interesting to put this, we find that if the density matrix is quantum oscillators on a complex network and see how the ⎛ ⎞ f 000 network topology affects the dynamics [61]. 1 ⎜ 0 f g + ih 0 ⎟ ρ = ⎝ 2 ⎠ (A7) 0 g − ih f3 0 00 0f ACKNOWLEDGMENTS 4 in the basis {|00 ,|01 ,|10 ,|11 }, the corresponding two-mode This work was supported by the NSF through a grant to Wigner function is ITAMP. C.K.C. is supported by the Croucher Foundation. 1 g cos θ + h sin θ W(θ) = + . (A8) 2π 2 APPENDIX: TWO-MODE WIGNER DISTRIBUTION This is how we derived Eq. (23). √ Here, we review what a two-mode Wigner distribution is Now suppose the system is in |S =(|01 +|10 )/ 2. The and provide details on how to derive Eq. (23). First, recall corresponding√W(θ) is peaked at θ = 0. In contrast, |A = that the one-mode Wigner function W(x,p) for an oscillator (|01 −|10 )/ 2 has W(θ) peaked at θ = π.Thisiswhy can be thought of as a quasiprobability distribution in the |S and |A correspond to in-phase and antiphase locking, two-dimensional phase space, where x is position and p is respectively.
[1] A. S. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: [3]D.Cai,Y.-C.Lai,andR.L.Winslow,Phys.Rev.Lett.71, 2501 A Universal Concept in Nonlinear Science (Cambridge Univer- (1993). sity Press, New York, 2001). [4] D. Gonze, S. Bernard, C. Waltermann, A. [2] S. H. Strogatz, Sync: The Emerging Science of Spontaneous Kramer, and H. Herzel, Biophys. J. 89, 120 Order (Hyperion, New York, 2003). (2005).
022913-9 TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG PHYSICAL REVIEW E 89, 022913 (2014)
[5] D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Scholl,¨ Phys. [32] G. Ermentrout, Physica D 41, 219 (1990). Rev. Lett. 110, 104102 (2013). [33] R. E. Mirollo and S. H. Strogatz, J. Stat. Phys. 60, 245 (1990). [6] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, [34] P. Schwendimann, A. Quattropani, and D. Sarchi, Phys. Rev. B J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 82, 205329 (2010). (2013). [35] L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Phys. Rev. [7] R. E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett. 103, Lett. 110, 195301 (2013). 168103 (2009). [36] M. I. Dykman and M. A. Krivoglaz, Phys. Status Solidi 68, 111 [8] R. Golestanian, J. M. Yeomans, and N. Uchida, Soft Matter 7, (1975). 3074 (2011). [37] R. Grobe and F. Haake, Z. Phys. B 68, 503 (1987). [9] S. Peles,ˇ J. L. Rogers, and K. Wiesenfeld, Phys.Rev.E73, [38] D. Cohen and S. Fishman, Phys.Rev.A39, 6478 (1989). 026212 (2006). [39] T. Dittrich and R. Graham, Europhys. Lett. 4, 263 (1987). [10] M. C. Cross, Phys. Rev. E 85, 046214 (2012). [40] T. Dittrich and R. Graham, Ann. Phys. (NY) 200, 363 [11] S. Droste, F. Ozimek, T. Udem, K. Predehl, T. W. Hansch,¨ (1990). H. Schnatz, G. Grosche, and R. Holzwarth, Phys. Rev. Lett. [41] T. E. Lee and M. C. Cross, Phys.Rev.A88, 013834 (2013). 111, 110801 (2013). [42] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Buchler,¨ and [12] D. Aronson, G. Ermentrout, and N. Kopell, Physica D 41, 403 P. Zoller, Nature Phys. 4, 878 (2008). (1990). [43] S. Diehl, A. Tomadin, A. Micheli, R. Fazio, and P. Zoller, Phys. [13] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence Rev. Lett. 105, 015702 (2010). (Springer-Verlag, Berlin, 1984). [44] P. Schindler, M. Muller,¨ D. Nigg, J. T. Barreiro, E. A. Martinez, [14] J. A. Acebron,´ L. L. Bonilla, C. J. Perez´ Vicente, F. Ritort, and M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt, Nature R. Spigler, Rev. Mod. Phys. 77, 137 (2005). Phys. 9, 361 (2013). [15] S. H. Strogatz and R. E. Mirollo, J. Stat. Phys. 63, 613 (1991). [45] A. W. Carr and M. Saffman, Phys.Rev.Lett.111, 033607 (2013). [16] P. C. Matthews and S. H. Strogatz, Phys.Rev.Lett.65, 1701 [46] Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. (1990). Sørensen, D. Leibfried, and D. J. Wineland, Nature 504, 415 [17] P. C. Matthews, R. E. Mirollo, and S. H. Strogatz, Physica D 52, (2013). 293 (1991). [47] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge [18] M. C. Cross, A. Zumdieck, R. Lifshitz, and J. L. Rogers, Phys. University Press, Cambridge, 1997). Rev. Lett. 93, 224101 (2004). [48] T. E. Lee and C.-K. Chan, Phys.Rev.A88, 063811 (2013). [19] M. C. Cross, J. L. Rogers, R. Lifshitz, and A. Zumdieck, Phys. [49] T. E. Lee, H. Haffner,¨ and M. C. Cross, Phys. Rev. A 84, 031402 Rev. E 73, 036205 (2006). (2011). [20] M. Ludwig and F. Marquardt, Phys. Rev. Lett. 111, 073603 [50] T. E. Lee, S. Gopalakrishnan, and M. D. Lukin, Phys. Rev. Lett. (2013). 110, 257204 (2013). [21] A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, [51] T. E. Lee, G. Refael, M. C. Cross, O. Kogan, and J. L. Rogers, Phys. Rev. Lett. 111, 103605 (2013). Phys.Rev.E80, 046210 (2009). [22] T. E. Lee and H. R. Sadeghpour, Phys. Rev. Lett. 111, 234101 [52] H. Hong, H. Park, and M. Y. Choi, Phys. Rev. E 72, 036217 (2013). (2005). [23] S. Walter, A. Nunnenkamp, and C. Bruder, arXiv:1307.7044 [53] T. E. Lee, H. Tam, G. Refael, J. L. Rogers, and M. C. Cross, (2013). Phys.Rev.E82, 036202 (2010). [24] M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. [54] O. Kogan, J. L. Rogers, M. C. Cross, and G. Refael, Phys. Rev. Holland, arXiv:1307.5891 (2013). E 80, 036206 (2009). [25] I. H. de Mendoza, L. A. Pachon,´ J. Gomez-Garde´ nes,˜ and [55] A. Amir, Y. Oreg, and Y. Imry, Phys.Rev.Lett.105, 070601 D. Zueco, arXiv:1309.3972 (2013). (2010). [26] G. Manzano, F. Galve, G. L. Giorgi, E. Hernandez-Garc´ ´ıa, and [56] A. Hu, T. E. Lee, and C. W. Clark, Phys.Rev.A88, 053627 R. Zambrini, Sci. Rep. 3, 1439 (2013). (2013). [27] H. J. Carmichael, Statistical Methods in Quantum Optics 1 [57] C. Joshi, F. Nissen, and J. Keeling, Phys.Rev.A88, 063835 (Springer-Verlag, Berlin, 1999). (2013). [28] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. [58] H. Daido, Phys. Rev. Lett. 68, 1073 (1992). 80, 517 (2008). [59] E. A. Martens, C. R. Laing, and S. H. Strogatz, Phys. Rev. Lett. [29] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). 104, 044101 (2010). [30] A. H. Nayfeh, Perturbation Methods (Wiley, New York, 2000). [60] P. Rosenau and A. Pikovsky, Phys. Rev. Lett. 94, 174102 (2005). [31] A. H. Nayfeh, Method of Normal Forms (Wiley, New York, [61] A. Arenas, A. Daz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, 1993). Phys. Rep. 469, 93 (2008).
022913-10