Quantum manifestations of homogeneous and inhomogeneous oscillation suppression states
Biswabibek Bandyopadhyay, Taniya Khatun, Debabrata Biswas, Tanmoy Banerjee
QQuantum manifestations of homogeneous and inhomogeneous oscillation suppressionstates
Biswabibek Bandyopadhyay, Taniya Khatun, Debabrata Biswas, and Tanmoy Banerjee ∗ Chaos and Complex Systems Research Laboratory, Department of Physics,University of Burdwan, Burdwan 713 104, West Bengal, India Department of Physics, Bankura University, Bankura 722 155, West Bengal, India (Dated: November 20, 2020)We study the quantum manifestations of homogeneous and inhomogeneous oscillation suppressionstates in coupled identical quantum oscillators. We consider quantum van der Pol oscillators coupledvia weighted mean-field diffusive coupling and using the formalism of open quantum system we showthat depending upon the coupling and the density of mean-field, two types of quantum amplitudedeath occurs, namely squeezed and nonsqueezed quantum amplitude death. Surprisingly, we findthat the inhomogeneous oscillation suppression state (or the oscillation death state) does not occurin the quantum oscillators in the classical limit. However, in the deep quantum regime we discoveran oscillation death-like state which is manifested in the phase space through the symmetry-breakingbifurcation of Wigner function. Our results also hint towards the possibility of the transition fromquantum amplitude death to oscillation death state through the “quantum” Turing-type bifurcation.We believe that the observation of quantum oscillation death state will deepen our knowledge ofsymmetry-breaking dynamics in the quantum domain.
I. INTRODUCTION
The collective dynamics of coupled oscillators are ofgreat interest in the field of physics, chemistry, and bi-ology [1]. The two most important emergent behaviorsshown by a system of coupled oscillators are synchroniza-tion [2] and oscillation quenching [3]. While synchroniza-tion is predominantly governed by the phase dynamics,the oscillation quenching is a manifestation of the ampli-tude dynamics.In recent years synchronization in quantum regime hasattracted much attention: two seminal papers by Lee andSadeghpour [4] and Walter et al. [5] unravel the impor-tant aspects of quantum synchronization (i.e., the mani-festation of synchronization in the quantum regime) us-ing the paradigmatic quantum van der Pol oscillators.Later on, several studies explored the richness of quan-tum synchronization [6–10] and proposed techniques toimprove synchronization measures in the background ofquantum noise [11, 12]. Recent experimental observa-tions of synchronization in the quantum regime in spin-1limit-cycle oscillators [13] and IBM-Q system [14] estab-lished that quantum synchronization is a physical reality.Although much attention has been given in reveal-ing the quantum manifestation of synchronization, theoscillation quenching states is relatively a less exploredtopic. Ishibashi and Kanamoto [15] first explored thenotion of “quantum” amplitude death in quantum vander Pol oscillators under diffusive coupling. In the clas-sical sense, in the amplitude death (AD) state oscillatorsarrive at a common steady state which was unstable inthe absence of coupling, therefore, AD leads to a stablehomogeneous steady state (HSS). Unlike classical AD, ∗ [email protected] the authors showed that the presence of quantum noisehinders the genuine AD state, however, a sufficient de-crease in the mean phonon number was considered asthe indication of the quantum AD state. Later, Ami-tai et al. [16] reported quantum AD in the presence ofKerr type nonlinearity and showed that anharmonicityleads to true quantum effects in the oscillation suppres-sion phenomenon. In Ref. [15] parameter mismatch wasintroduced explicitly to induce AD and in Ref. [16] thepresence of Kerr type nonlinearity effectively introducesfrequency detuning between the oscillators that leads tonoise induced quantum AD. Therefore, parameter mis-match seems to be a necessary ingredient to induce quan-tum AD.Moreover, in the context of coupled oscillators, oscilla-tion quenching process is much more subtle. Apart fromAD there exists another oscillation quenching process,namely oscillation death (OD) [3]. In the OD state, os-cillators populate different coupling dependent nontrivialsteady states and thereby give rise to symmetry-breaking stable inhomogeneous steady states (IHSS). In this con-text, Koseska et al. [17] established that AD and ODmay occur in the same system and AD transforms intoOD through a symmetry-breaking bifurcation, which re-sembles the Turing-type bifurcation of spatially extendedsystem [18]. However, the quantum mechanical analog ofthe OD state has hitherto not been reported.Motivated by the above discussion, in this paper weask the following questions: (i) What are the differentmanifestations of quantum amplitude death state in cou-pled identical oscillators? (ii) Does OD occur in quan-tum oscillators? If yes, what is the quantum mechanicalanalog of an OD state? To answer these questions weconsider two quantum van der Pol (vdP) oscillators [19]coupled by weighted mean-field coupling. The paradig-matic quantum vdP oscillator has been chosen as the testbed to study several emergent dynamics in the quantum a r X i v : . [ n li n . C D ] N ov domain. More importantly, the quantum vdP oscillatorsare proposed to be realizable in experiment with trappedion and “membrane-in-the-middle” set up [4, 5]. Thechoice of weighted mean-field diffusive coupling as thecoupling scheme adopted in this study is motivated bythe fact that it is the simplest yet physically relevantmodel to distinctly observe AD and OD [20, 21]. Undernormal diffusive coupling AD appears under parametermismatch [22, 23], and OD generally coexists with limitcycle(s) making AD impossible and OD difficult to ob-serve in identical oscillators [3]. In the present paper, weuse two types of weighted mean-field diffusive coupling,namely nonscalar and scalar coupling [24]. The nonscalarcoupling is known to induce AD only (no OD state is pos-sible) and the scalar coupling is conducive to both ADand OD [25].At this point it is important to understand the diffi-culty of identifying OD in quantum systems. In the caseof AD, the oscillators populate the zero steady state,therefore, a pronounced reduction in the mean phononnumber or increased probability of ground Fock level arethe sufficient indicators of transition from oscillation toquantum AD state [15, 16]. However, in the case of classi-cal OD since two or more than two non-zero steady statesare created, therefore, the mean phonon number andFock level distribution can no longer distinguish quan-tum OD and oscillatory states unambiguously. Because,in the quantum OD state the mean phonon number doesnot reduce drastically and the ground state is no longerthe highest populated state. Therefore, we have to relylargely on the phase space representation: for the limit-ing case of two oscillators, in the classical OD state twosteady states are created, which are displaced from theorigin in phase space. Therefore, in the quantum ODstate it is instructive to observe the equivalent displace-ment in the Wigner distribution function in the phasespace.In this paper, using the formalism of open quantumsystems, we show that under the weighted mean-fielddiffusive coupling, identical quantum vdP oscillators ex-hibit quantum amplitude death. We identify two types ofquantum AD states, namely squeezed and non-squeezedquantum AD: the former AD state has not been observedin the previous studies [15, 16]. The quantum AD stateis explored using quantum master equation and com-pared with the AD state of the classical and semiclassicalcases. Further, we find that the quantum OD state doesnot occur in quantum oscillators in the classical limit .However, in the deep quantum region we discover anoscillation death-like state which emerges as the resultof the symmetry-breaking bifurcation of Wigner distri-bution function. Also, we see that the transition fromquantum AD to OD provides a qualitative indication ofthe quantum mechanical analog of the Turing-type bifur-cation.The rest of the paper is organized in the following man-ner. The next section describes the classical and quan-tum van der Pol oscillator. In Sec. III we describe the mathematical model of classical vdP oscillators coupledthrough weighted mean-field diffusive coupling. For aclear understanding of the classical dynamics we revisitthe bifurcation scenarios that lead to classical amplitudeand oscillation death. Section. IV presents the resultsof quantum amplitude death under nonscalar coupling;also, we compare the results with the noisy classicalmodel. Section V reports the appearance of squeezedquantum AD and the quantum manifestation of the os-cillation death state that appears under scalar coupling.Finally, we conclude the paper in Sec. VI discussing theimportance of the results. II. VAN DER POL OSCILLATOR: CLASSICALAND QUANTUM
A van der Pol oscillator has the following mathematicalform [19]: ¨ x = − ω x + k ˙ x − k x ˙ x, (1)where ω is the intrinsic frequency and k is the gain ratecorresponding to the linear pumping and k is the lossrate corresponding to the nonlinear damping ( k , k > α = x + iy (where ˙ x = ωy ) and the corresponding ampli-tude equation is given by (see Appendix A):˙ α = − iωα + ( k − k | α | ) α. (2)The oscillator shows a limit cycle oscillation with an am-plitude (cid:113) k k .The quantum van der Pol oscillator is represented bythe quantum master equation in density matrix ρ [4, 5]:˙ ρ = − i [ ωa † a, ρ ] + k D [ a † ]( ρ ) + k D [ a ]( ρ ) , (3)where D [ ˆ L ]( ρ ) is the Lindblad dissipator having the form D [ ˆ L ]( ρ ) = ˆ Lρ ˆ L † − { ˆ L † ˆ L, ρ } , where ˆ L represents an op-erator. Here and throughout the paper we take ¯ h = 1. a and a † are the Bosonic anihilation and creation oper-ators, respectively. k and k have the same meaning asthe classical case. In the classical limit, linear pumpingdominates over the nonlinear damping (i.e., k > k ) andone approximates (cid:104) a (cid:105) ≡ α , and starting from the mas-ter equation (3) one arrives at the classical amplitudeequation (2) by the following relation: ˙ (cid:104) a (cid:105) = Tr( ˙ ρa ) (seeAppendix B). III. CLASSICAL VDP OSCILLATORS:NONSCALAR AND SCALAR COUPLING
We consider two identical classical van der Pol oscilla-tors, which are coupled via weighted mean-field diffusivecoupling scheme. The mathematical model is given be-low,˙ x j = ωy j + ε (cid:32) q (cid:88) m =1 x m − x j (cid:33) , (4a)˙ y j = − ωx j + ( k − k x j ) y j + ε (cid:32) q (cid:88) m =1 y m − y j (cid:33) , (4b) j ∈ { , } . ε , are the coupling parameters ( ε , > ω .The control parameter q determines the density of theweighted mean-field. Originally, the parameter q was in-troduced in the context of quorum sensing in genetic os-cillators that controls the extracellular autoinducer con-centration in cell to cell communication [26, 27]. Later onits effect was investigated in physical systems [20, 21, 28]and ecological network (as an parameter controlling addi-tional mortality) [29]. The coupling scheme was also real-ized experimentally in electronic circuits [21, 28]. Fromphysical point of view q determines the degree of dis-sipation in the coupling path: lesser q implies greaterdissipation and vice versa. Generally, q acts as a dilution parameter in the limit 0 ≤ q ≤
1. However, this limit on q is not strict [30]: for q >
1, it acts as an amplification parameter.The coupling scheme of Eq. (4) can be categorized intotwo types. (i)
Nonscalar coupling : When ε , (cid:54) = 0 thecoupling is said to be nonscalar coupling. This type ofcoupling is conducive for the amplitude death state [25].(ii) Scalar coupling : If either ε = 0 or ε = 0 thecoupling is said to be scalar coupling. In this paper weconsider ε = ε (cid:54) = 0 and ε = 0 as this type of “real partcoupling” is known to induce oscillation death [25, 31]. A. Nonscalar coupling: Classical AD
The amplitude equation of Eq. (4) under nonscalarcoupling is given by,˙ α j = − iωα j + ( k − k | α j | ) α j + ε (cid:32) q (cid:88) m =1 α m − α j (cid:33) . (5)Here without any loss of generality we consider ε = ε = ε . The system represented in equation (5) has atrivial fixed point at the origin: F HSS ≡ (0 , , , ε HB,ns = k − q ) .Figure 1(a) illustrates this scenario in bifurcation di-agram of x , with ε for an exemplary parameter set( ω = 2, k = 1, k = 0 .
2, and q = 0 .
2) (using
XPPAUT [32]). In the AD state one has x , = y , = 0, i.e., boththe oscillators attain a common steady state F HSS whichis the origin.
FIG. 1. (a) Nonscalar coupling: Amplitude death (AD) oc-curs through inverse Hopf bifurcation (HB). (b) Scalar cou-pling: AD appears through HB and oscillation death (OD)emerges through a symmetry-breaking pitchfork bifurcation(PB). Solid (red) line: stable steady state, dotted (black) line:unstable steady state, and hollow circles (green): stable limitcycle. Parameters are q = 0 . ω = 2, k = 1, and k = 0 . B. Scalar coupling: Classical AD and OD
The amplitude equation corresponding to Eq. (4) un-der the scalar coupling is given by (by considering ε = ε and ε = 0)˙ α j = − iωα j + ( k − k | α j | ) α j + ε (cid:32) q (cid:88) m =1 α m − α j (cid:33) + ε (cid:32) q (cid:88) m =1 α ∗ m − α ∗ j (cid:33) . (6)Eq. (6) has the following fixed points: the trivialfixed point F HSS ≡ (0 , , , F IHSS ≡ ( x ∗ , y ∗ , − x ∗ , − y ∗ ) where x ∗ = − ωy ∗ ω + εy ∗ and y ∗ = (cid:113) ( ε − ω )+ √ ε − ω ε . The system shows a transition fromoscillatory state to amplitude death state through an in-verse Hopf bifurcation at ε HB,s = k (1 − q ) [20, 21]. An in-teresting transition from AD to OD state occurs througha symmetry-breaking pitchfork bifurcation at (cid:15) PB = ω k [20, 21]. This transition from AD to OD is analogous tothe Turing-type bifurcation in spatially extended system[17]. Figure 1(b) shows the corresponding bifurcation di-agram for an exemplary parameter set ( ω = 2, k = 1, k = 0 .
2, and q = 0 . x = − x and y = − y .Our main aim in this work is to explore the quantummanifestation of the above mentioned classical results.In particular, we try to reveal the quantum mechanicalanalog of the symmetry-breaking OD state. IV. QUANTUM VDP OSCILLATORS UNDERNONSCALAR COUPLING: QUANTUM ADA. Pure quantum oscillators
The quantum master equation of two nonscalar mean-field diffusively coupled identical quantum van der Poloscillators is given by,˙ ρ = (cid:88) j =1 (cid:16) − i [ H j , ρ ] + k D [ a † j ]( ρ ) + k D [ a j ]( ρ ) (cid:17) + qε D [( a + a ) † ]( ρ ) + 2 ε (cid:88) j =1 D [ a j ]( ρ ) , (7)where, H j = ωa † j a j and a j ( a † j ) is the annihilation(creation) operator corresponding to the j -th oscillator.In the classical limit ( k > k ) the master equation(7) is equivalent to the classical amplitude equation (5)through the following relation: ˙ (cid:104) a (cid:105) = Tr( ˙ ρa ).We numerically solve the master equation (7) using QuTiP [33]. To visualize and understand the systemdynamics we employ the Wigner function representa-tion in phase space since it provides a reliable repre-sentation of the quantum dynamical states. Moreover,Wigner function is also an experimentally observablequasi-probability distribution function that makes it ac-cessible [34]. We computed the mean phonon number (cid:104) a † a (cid:105) (= (cid:104) a † a (cid:105) ) and plot them in the ε/k − q parame-ter space (since the oscillators are identical, both have thesame mean phonon numbers). Figure 2(a) shows this incolor map. The solid line indicates the Hopf bifurcationcurve (HB,ns) obtained classically: below the Hopf curvethe classical AD occurs. It is interesting to note that themean phonon number also decreases appreciably underthe Hopf bifurcation curve and due to the hindrance fromquantum noise it does not reach zero but shows a mod-erate collapse in the oscillation. However, this moderatecollapse is stronger than the noisy classical oscillators(discussed in the next subsection). The correspondingsteady state Wigner function of two representative pointsare shown in Figs. 2(b c): Fig. 2(b) demonstrates the os-cillatory behavior for ε/k = 0 . , q = 0 . ε/k = 5 , q = 0 . (cid:104) a † a (cid:105) )with three different values of q is shown in Fig. 3. It isfound that quantum AD occurs more effectively in thelower q values. This is due to the fact that a lower(higher) q imposes stronger (weaker) dissipation in thecoupled system. In Fig. 3, the probability of occupationof the Fock levels are shown in the insets for two couplingstrengths at q = 0 .
2: for ε/k = 0 .
01 (left inset) the sys-tem shows oscillation and for ε/k = 5 .
75 (right inset)quantum AD appears. It is clear that in the quantumAD state the occupation near the quantum ground stateis much more prominent.We also explored the deep quantum region where
FIG. 2. (a) Two parameter diagram in the ε/k − q spaceshowing the mean phonon number (cid:104) a † a (cid:105) (= (cid:104) a † a (cid:105) ) of boththe oscillators. The solid line is the classical Hopf bifurca-tion curve (HB,ns). The steady state Wigner functions at (b) ε/k = 0 . ε/k = 5 for q = 0 . k = 1, k = 0 . ω = 2. q=0.2q=0.5q=0.8 P ( N ) N N 〈 a (cid:1) a 〉 (cid:2) /k (cid:0) /k =0.01 (cid:0) /k =5.75 FIG. 3. (a) The steady state mean phonon number (cid:104) a † a (cid:105) (= (cid:104) a † a (cid:105) ) for three different values of mean-field density ( q =0 . , . , . ε/k = 0 .
01 (left panel) and quantum ADat ε/k = 5 .
75 (right panel) with q = 0 .
2. Other parametersare k = 1, k = 0 . ω = 2. strong nonlinear damping rate dominates the linearpumping rate ( k (cid:29) k ) and got qualitatively similar re-sults (not shown here). In the deep quantum regime onlya few states are populated, which are near the quantumground state: note that in the limit k → ∞ the steadystate density matrix is given by [4] ρ ss = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ,i.e., the system oscillates between its two lowest lying en-ergy levels. Therefore, the notion of quantum AD is notobvious in the deep quantum regime as the mean phononnumber always remains very low irrespective of the cou-pling conditions. B. Noisy classical model
For the proper understanding of quantum AD it is in-structive to compare the results of the quantum systemwith the corresponding noisy classical model (or semiclas-sical model) [15]. In the noisy classical model the classicaldynamics is considered in the presence of a finite amountof noise whose intensity is equal to that of the quan-tum noise. To evaluate the amount of quantum noise in-tensity, a stochastic differential equation is derived fromthe quantum master equation following [15]. For this,the quantum master equation (7) is represented in phasespace using partial differential equation of Wigner distri-bution function ( W ( α )) [35]. ∂ t W ( α ) = (cid:88) j =1 (cid:20) − (cid:18) ∂∂α j µ α j + c.c. (cid:19) + 12 (cid:18) ∂ ∂α j ∂α j ∗ D α j α j ∗ + ∂ ∂α j ∂α j (cid:48) ∗ D α j α j (cid:48)∗ (cid:19) + k (cid:18) ∂ ∂α j ∗ ∂α j α j + c.c (cid:19)(cid:21) W ( α ) , (8)where the elements of the drift vector ( µ ) are: µ α j = (cid:20) − iω + k − k ( | α j | − − (cid:16) ε − εq (cid:17)(cid:21) α j + εq α j (cid:48) , and the elements of the diffusion matrix D are: D α j α j ∗ = k + 2 k (2 | α j | −
1) + εq + 2 ε, D α j α j (cid:48)∗ = εq. with j = 1 , j (cid:48) = 1 , j (cid:54) = j (cid:48) . In weak nonlinearregime ( k (cid:28) k ), Eq.8 reduces to the Fokker-Planckequation, which is given by ∂ t W ( X ) = (cid:88) j =1 (cid:20) − (cid:18) ∂∂x j µ x j + ∂∂y j µ y j (cid:19)(cid:19) + 12 (cid:18) ∂ ∂x j ∂x j D x j x j + ∂ ∂y j ∂y j D y j y j + ∂ ∂x j ∂x j (cid:48) D x j x j (cid:48) + ∂ ∂y j ∂y j (cid:48) D y j y j (cid:48) (cid:19)(cid:21) W ( X ) , (9) where X = ( x , y , x , y ). The elements of drift vectorare, µ x j = ωy j + (cid:20) k − k ( x j + y j − − (cid:16) ε − εq (cid:17)(cid:105) x j + εq x j (cid:48) , (10a) µ y j = − ωx j + (cid:20) k − k ( x j + y j − − (cid:16) ε − εq (cid:17)(cid:105) y j + εq y j (cid:48) . (10b)The diffusion matrix has the following form, D = 12 ν εq ν εq εq ν εq ν . (11)where ν j = k + k [2( x j + y j ) −
1] + εq + ε . FromEq.(9), the following stochastic differential equation canbe derived, d X = µ dt + σ d W t , (12)where σ is the noise strength and d W t is the Wienerincrement. As the diffusion matrix D (given inEq.(11)) is symmetric, we can analytically derive σ from it. The diagonal form of D is given by: D diag = U − D U = diag ( λ − λ − λ + λ + ). Here λ ± = (cid:104) ν + ν ± (cid:112) ( ν − ν ) + ( εq ) (cid:105) and U has the follow-ing form: U = u − u + u − u +
00 1 0 11 0 1 0 , (13)where u ± = εq (cid:104) ν − ν ± (cid:112) ( ν − ν ) + ( εq ) (cid:105) . Now, σ matrix can be evaluated from the equation σ = U (cid:112) D diag U − and it has the following form. σ = σ σ σ σ σ σ σ σ , (14)where σ = u + √ λ + − u − √ λ − u + − u − , σ = u + √ λ − − u − √ λ + u + − u − and σ = √ λ + − √ λ − u + − u − .By solving the stochastic differential equation(Eq. (12)) (using JiTCSDE module in Python [36]), wecompute the ensemble average of the squared steady-state amplitude of the first oscillator ( | α | nc ), averagedover 1000 realizations, starting from random initial con-ditions. To compare the scenarios of oscillation collapsefor each model, in Fig. 4 we plot the averaged ampli-tude of classical model ( | α | ), that of the noisy classi-cal model ( | α | nc ) and the mean phonon number of the noisy classicalclassicalquantum | ____ | , 〈 a a 〉 , | _____ | n c /k HB,ns /k FIG. 4.
Nonscalar coupling:
Comparison of the classical,quantum, and semiclassical results. At q = 0 .
2, the aver-age amplitude from the classical model ( | α | ), mean phononnumber from the quantum model ( (cid:104) a † a (cid:105) ) and the averagedamplitude from the noisy classical model ( | α | nc ) of the firstoscillator plotted together with coupling strength. Other pa-rameters are k = 1, k = 0 .
2, and ω = 2. quantum model ( (cid:104) a † a (cid:105) ) of the first oscillator with thecoupling parameter. The averaged classical amplitudeshows an abrupt jump from oscillatory state to deathstate at ε HB,ns . Whereas, the mean phonon number andthe averaged amplitude of noisy classical model do notshow a zero-amplitude death state, rather they show asignificant decrement in amplitude. It can be seen thatthe mean phonon number is always lesser than the aver-age amplitude of the noisy classical model. Therefor thequantum AD lies in between the classical AD and theAD in the noisy classical model.
V. QUANTUM VDP OSCILLATORS UNDERSCALAR COUPLING: QUANTUM OD STATE
The quantum master equation of two coupled identicalquantum van der Pol oscillators under scalar coupling isgiven by˙ ρ = − i (cid:20) ω ( a † a + a † a ) + i(cid:15) (cid:0) q ( a † a † − a a )+ ( q − a † + a † − a − a ) (cid:1) , ρ (cid:21) + k (cid:88) j =1 D [ a † j ]( ρ ) + k (cid:88) j =1 D [ a j ]( ρ )+ q(cid:15) D [( a + a ) † ]( ρ ) + (cid:15) (cid:88) j =1 D [ a j ]( ρ ) . (15)In the classical limit, (15) gives the classical amplitudeequation (6) using ˙ (cid:104) a (cid:105) = Tr( ˙ ρa ).We solve the master equation (15) numerically using QuTiP [33]. At first, similar to the nonscalar coupling
FIG. 5.
Scalar coupling:
Two parameter phase diagram ofthe mean phonon number (cid:104) a † a (cid:105) for ( k , k ) = (1 , . ε/k = 0 . q = 0 . squeezed quantum AD for ε/k = 5 and q =0 .
2. In the quantum AD state note the presence of squeezingin the quadrature space. In the Wigner plots axes and scalesare identical to Fig. 2. Other parameter: ω = 2. H B , s P B | ____ | , 〈 a a 〉 , | _____ | n c /k Classical AD Classical OD noisy classicalclassicalquantum
FIG. 6.
Scalar coupling:
Comparison of the classical, quan-tum, and semiclassical results. Plots of the average amplitudefrom the classical model ( | α | ), mean phonon number fromthe quantum model ( (cid:104) a † a (cid:105) ) and the averaged amplitude fromthe noisy classical model ( | α | nc ) of the first oscillator at q = 0 .
2. Red dashed line represents the shift of the stableinhomogeneous fixed points from the origin in the OD state.Other parameters are k = 1, k = 0 . ω = 2. case of the the previous section, we consider k = 1 and k = 0 .
2. The results are summarized in Fig. 5: it showsthe mean phonon number (cid:104) a † a (cid:105) (= (cid:104) a † a (cid:105) ) along withthe classical Hopf and pitchfork bifurcation curves in the ε/k − q parameter space (classical bifurcation curves aredrawn using the expressions derived in Sec. III B). Theinsets in Fig. 5 show the Wigner function in phase spaceof oscillatory and quantum AD state for ε/k = 0 . ε/k = 5, respectively at fixed q = 0 .
2. An interesting ob-servation from the Wigner function plot of the quantumAD is the presence of squeezing in the quadrature space.The squeezing gets stronger with increasing ε . This maybe the direct reflection of the classical case, where un-der scalar coupling x < y (for x, y (cid:54) = 0) for a nonzerocoupling strength. This type of squeezing is not presentin the case of nonscalar coupling or in diffusive coupling[15, 16] as coupling is symmetric there with respect to allthe variables.In Fig. 5 classical AD occurs below the Hopf bifur-cation curve (HB,s) and left to the PB line and clas-sical OD occurs below the HB,s curve right to the PBline. Although, the occurrence of classical and quantumAD agrees with each other in the parameter space, sur-prisingly, for no value of ε and q , inhomogeneous steadystates (OD) are observed; squeezed quantum AD appearseven beyond the PB line (below the HB,s curve). How-ever, a slight increase in mean phonon number (cid:104) a † a (cid:105) isobserved beyond the classical PB line. For better under-standing of the fact we study the noisy classical modelusing the formalism equivalent to Sec. IV B. Figure 6shows the plots of average amplitude and mean phononnumber in classical, semiclassical and quantum oscilla-tors at a fixed q = 0 .
2. Upto the PB line, the quantumAD scenario of Fig. 6 qualitatively matches with Fig. 4of the nonscalar case. Beyond the PB line, in the classi-cal case, oscillation ceases, and nonzero inhomogeneousfixed points are created that are shifted from the origin:the (red) dashed line shows the amount of shift of thefixed points in the classical OD state. However, in thequantum as well as noisy classical cases, noise tends tohomogenize the steady states around the origin. As aresult no quantum OD is observed here, rather, it resultsin a slight increase in the mean phonon number (in quan-tum case) or average amplitude (noisy classical case).Next, we search for any possible symmetry-breakingdynamics in the deep quantum regime ( k > k ). Fol-lowing [11] we choose k = 1 and k = 3. In this regimeonly a few Fock states are populated (near the quantumground state) and quantum noise becomes much moreprominent. Therefore, it is inconclusive to distinguishbetween the oscillatory state and the quantum AD statebased on the mean phonon number. However, qualitativechanges in the Wigner function provides distinction be-tween them. Figure 7(a) shows the Wigner function rep-resentation of oscillation in the uncoupled case ( ε = 0)and Fig. 7(c) shows that for quantum AD at ε/k = 20and q = 0 . q , at a moderate ε we observean interesting symmetry-breaking bifurcation that gov-erns the creation of inhomogeneous steady states, i.e.,the quantum oscillation death state. The quantum ODstate emerges as the Wigner function (and therefore, theHusimi function) bifurcates into two separated lobes inthe phase space. Figures 7(e–h) demonstrate the quan-tum OD state in the phase space using Wigner function(upper row) and Husimi function (bottom row) for tworepresentative values of q : Figs. 7(e, f) are for q = 1 . q = 1 .
5. One can observe thatthe probability density is concentrated in the two lobes. The separation between the two lobes increases with in-creasing q . The three dimensional plot of Fig. 7(g) asshown in Fig. 7(i) adds more clarity to the occurrence ofsymmetry-breaking bifurcation and creation of the quan-tum OD state. We observed that the Wigner function inthe OD state is not just two lobes separated in the phasespace, which is nothing but classical representation ofprobability of two possible outcomes [38], however, in thiscase quantum interference terms appear in the middleand exhibits symmetry-breaking inhomogeneous steadystates. This fact is also verified by the non zero coher-ence terms in the density matrix of this state: the inset ofFig. 7(i) shows the histogram of the real part of the ele-ments of the density matrix that exhibits the presence ofoff-diagonal terms in the density matrix. Since the quan-tum OD state occurs in the deep quantum regime, wecan not draw a one to one correspondence with the clas-sical OD state as now the classical amplitude equation(6) and the quantum master equation (15) are no longerexactly equivalent. It is noteworthy that in Ref. [11] asymmetry-breaking bifurcation in the Wigner distribu-tion function occurs in a squeezing driven van der Poloscillator. However, that does not resemble an OD stateas it occurs in a single driven oscillator. In our case thesymmetry-breaking bifurcation occurs due to the cou-pled interaction of two oscillators, therefore, the notionof emergent dynamics is applicable here.Finally, in the deep quantum regime we tried to mapthe zone of occurrence of the quantum OD state in the ε/k − q space. Figure 8 shows the mean phonon numberin the ε/k − q space ( k = 1 , k = 3): quantum ODemerges above the dashed line which is plotted by visualinspection of the bifurcation of Wigner function. In thequantum OD state one observes a drastic increase in themean phonon number, which resembles the fact that inclassical OD, the inhomogeneous fixed points have nonzero values. However, an exact demarcation of the quan-tum OD in the parameter space is difficult in the absenceof any quantitative measure of this state.At this point it is important to raise the issue of quan-tum mechanical analog of classical Turing-type bifurca-tion. As discussed earlier, in Ref. [17] the transitionfrom AD to OD through a symmetry-breaking bifurca-tion was established as equivalent to the Turing-type bi-furcation of spatially extended system. In the presentstudy also, in the deep quantum regime we get quantumAD and OD in the same system. Moreover, we noticea symmetry-breaking transition from the squeezed ADstate to the quantum OD state with increasing q (cf.Fig. 7): this may be thought of as the “quantum” analogof the Turing-type bifurcation. However, since quantumOD occurs in the deep quantum regime only, therefore,the presence of strong quantum noise makes it difficult todistinguish quantum AD from oscillations and to identifythe exact route of transition from quantum AD to quan-tum OD. Therefore, more quantitative measures are re-quired to draw any strong conclusions regarding this. (b) I m ( (cid:1) ) R e ( (cid:0) ) W ( (cid:2) ) -44 -4 4 (i) (a) (c) (e) (g)(d) (f) (h) Re( (cid:3) ) Re( (cid:3) ) Re( (cid:3) ) Re( (cid:3) ) I m ( (cid:4) ) I m ( (cid:4) ) (cid:5) /k =0 q=0.6, (cid:6) /k =20 q=1.25, (cid:6) /k =20 q=1.5, (cid:6) /k =20 -4 -4 -4 -4
00 9 90.180
FIG. 7.
Quantum manifestation of oscillation death (OD):
Deep quantum region ( k = 1 and k = 3). (a, c, e, g) Wignerfunction (b, d, f, h) Husimi function. (a, b) Limit cycle oscillation in uncoupled oscillators ( ε/k = 0) (c, d) Squeezed qantumAD for ε/k = 20, q = 0 .
6. Quantum OD state for (e, f) q = 1 .
25 and (g, h) q = 1 .
5: note the emergence of inhomogeneoussteady states. (i) Three dimensional representation of (g). It clearly shows two separated lobes in the phase space. Inset showsthe histogram of the real part of elements of the density matrix: note the presence of coherence terms. Other parameter: ω = 2.FIG. 8. Deep quantum region k = 1 and k = 3 : Meanphonon number (cid:104) a † a (cid:105) in the two parameter space. Quan-tum OD occurs above the dashed line. Solid horizontal lineindicates q = 1. ω = 2. VI. CONCLUSIONS
In this paper we have studied the quantum mechanicalmanifestations of oscillation suppression states, namelythe amplitude death and the oscillation death states intwo mean-field diffusively coupled identical quantum vander Pol oscillators. Our study has unraveled two ques-tions that we asked in the beginning of this paper. First, identical quantum oscillators can exhibit two types ofquantum amplitude death states, namely squeezed andnon squeezed quantum AD. Second, oscillation deathstate indeed appears in coupled quantum oscillators; itis manifested in the deep quantum regime as the cre- ation of inhomogeneous steady states due to symmetry-breaking bifurcation in Wigner function and Husimi func-tion. Moreover, our results hint at the occurrence of thequantum analog of the Turing-type bifurcation.First we have shown that under nonscalar couplingquantum AD state appears in its non squeezed form.With the scalar coupling we observed a squeezed quantumAD state, which is unlike nonscalar or diffusive couplinginduced quantum AD state [15, 16]. In the higher exci-tation regime (i.e., outside the deep quantum zone) thequantum AD has a one to one correspondence with theclassical and semiclassical results. However, in the deepquantum regime the notion of quantum AD is not ob-vious because in this regime only a few Fock levels arepopulated around the quantum mechanical ground state.In the deep quantum regime with high mean-field den-sity we have discovered a quantum OD state that emergesas the consequence of symmetry-breaking bifurcation inthe Wigner distribution function. To the best of ourknowledge this is the first instance where the quantumequivalence of the OD state has been observed. Since thisstate is exhibited in the deep quantum regime, therefore,a one to one correspondence with the classical OD stateis not possible. However, both quantum and classicalOD share two common features. First, they appear un-der the scalar coupling, and second, their manifestationin the phase space is equivalent, viz., the appearance ofinhomogeneous steady states in the phase space. Sinceone of our main goals in this paper is the observation ofOD in quantum regime, therefore, we restrict our studyto two coupled oscillators. In the case of more than twooscillators, multicluster OD may appear [3] and the iden-tification of the same in the quantum regime may becomeillusive.With the advancement of experimental techniques webelieve that the present coupling schemes can be re-alized experimentally, e.g., using the ion trap [4, 39]and “membrane-in-the-middle” experimental set up [40].Quantum amplitude death is thought to be an efficientmean of cavity cooling [15]; since in the present couplingscheme no parameter mismatch is required and one hastwo control parameters — coupling strength and den-sity of mean-field, therefore, we believe that the presentscheme offers a more flexible option for cooling. Fur-ther, since a strong squeezing appears in the quantumAD state under the scalar coupling, therefore, generationof the squeezed state in coupled oscillators and its possi-ble real life applications can be explored further [41, 42].On the other hand, OD is generally thought of as the un-derlying mechanism of cellular differentiation and othersymmetry breaking phenomena in biological systems [3],however, we have to figure out the exact implication ofthe quantum OD state in real quantum systems. Onlythen we will be able to identify the application potential-ity of quantum OD in quantum technology.The observation of quantum OD state opens up a myri-ads of scopes in the study of symmetry-breaking dynam-ics in the quantum regime. The shape of the Wignerfunction in the quantum OD state has a striking resem-blance with that of the single-photon-subtracted two-mode states with vortex structure in quadrature space[43] (see Chapter 4 of [44]); also, it shares some of thevisual features of the squeezed Schr¨odinger cat (like)state [45, 46]. However, unlike the vortex state andSchr¨odinger cat state in our system the Wigner functionis always positive. Nevertheless, this visual resemblancecalls for the further investigation. Our observation ofquantum Turing-type bifurcation hints at the possibilityof Turing pattern in quantum domain. However, a deepunderstanding of this scenario demands much more in-depth investigations. Recently, the symmetry-breakingpartially synchronized states, namely the chimeras havebeen reported in quantum regime by Bastidas et al. [10].The connection between the “quantum” chimera statesand the quantum OD state will be an interesting prob-lem to study in a network of coupled quantum oscillators[47, 48].
Appendix A: Derivation of amplitude equation(Eq. 2)
We consider the complex amplitude of the oscillator(1) as α = x + iy . Therefore,˙ α = ˙ x + i ˙ y, = ωy − iωx + ik y − ik x y, = − iωα + k α − α ∗ ) − k ( α + α ∗ ) ( α − α ∗ ) . (A1) Using polar coordinate α = ηe − iφ we get,˙ η − iη ˙ φ = − iωη + k η (1 − e iφ ) − k η ( e − iφ + e iφ ) (1 − e iφ ) . (A2)From Eq.(A2) we can extract equations for ˙ η and ˙ φ ,˙ η = k η (1 − cos 2 φ ) − k η cos φ (1 − cos 2 φ ) , ˙ φ = ω + k φ − k η cos φ sin 2 φ. (A3)Now at this point we apply the method of averaging.It can be done by directly averaging the equations of ˙ η and ˙ φ over one time period T = πω (for details see [2],Chapter 7 and references therein). We get,˙ η = ( k − k η ) η, ˙ φ = ω. (A4)Putting these averaged values of ˙ η and ˙ φ in the equation˙ α = e − iφ ( ˙ η − iη ˙ φ ) we get the following equation,˙ α = − iωα + ( k − k | α | ) α. (A5)This is the amplitude equation as given in Eq.(2). Appendix B: Correspondence between masterequation (Eq. 3) and amplitude equation (Eq. 2)
In quantum optics the average annihilation operator( (cid:104) a (cid:105) ) and the complex amplitude ( α ) are equivalent [45],i.e., (cid:104) a (cid:105) ≡ α . This property bridges the master equa-tion and amplitude equation. Let us consider the masterequation in the Lindblad form as ˙ ρ = − i [ H, ρ ] + D [ L ]( ρ ).Now the average of any operator ˆ O is given by (cid:104) ˆ O (cid:105) =Tr( ρ ˆ O ). So the dynamical equation of (cid:104) ˆ O (cid:105) is given by, d (cid:104) ˆ O (cid:105) dt = ddt Tr( ρ ˆ O ) , = i (cid:104) [ H, ˆ O ] (cid:105) + Tr( ˆ O D [ L ]( ρ )) , = i (cid:104) [ H, ˆ O ] (cid:105) + (cid:104) ˜ D [ L ]( ˆ O ) (cid:105) , (B1)where ˜ D [ L ]( ˆ O ) is called the ‘adjoint operator’, having thefollowing form.˜ D [ L ]( ˆ O ) = L † ˆ OL − { L † L, ˆ O } , = 12 (cid:16) L † [ ˆ O, L ] + [ L † , ˆ O ] L (cid:17) . (B2)Following Eq. (B1) and using the mater equation Eq. (3),we evaluate the dynamical equation of expectation value0of the annihilation operator ( (cid:104) a (cid:105) ) as˙ (cid:104) a (cid:105) = i (cid:104) [ ωa † a, a ] (cid:105) + k (cid:104) ˜ D [ a † ]( a ) (cid:105) + k (cid:104) ˜ D [ a ]( a ) (cid:105) , = iω (cid:104) [ a † a, a ] (cid:105) + k (cid:104) ( a [ a, a † ] + [ a, a ] a † ) (cid:105) + k (cid:104) ( a † [ a, a ] + [ a † , a ] a ) (cid:105) , = − iω (cid:104) a (cid:105) + k (cid:104) a (cid:105) − k (cid:104) a † a (cid:105) , (B3)which is similar to Eq. (2) as (cid:104) a (cid:105) ≡ α and (cid:104) a † a (cid:105) ≈| (cid:104) a (cid:105) | (cid:104) a (cid:105) [6]. ACKNOWLEDGMENTS
B.B. and T.K. acknowledge the University GrantsCommission (UGC), India for providing Junior Re-search Fellowship. T. B. acknowledges the financial sup-port from the Science and Engineering Research Board(SERB), Govt. of India, in the form of a Core ResearchGrant [CRG/2019/002632]. [1] S. Strogatz,
Sync: The emerging science of spontaneousorder (Penguin UK, 2004).[2] A. Pikovsky, M. Rosenblum, and J. Kurths,
Synchroniza-tion: A Universal Concept in Nonlinear Sciences (Cam-bridge University Press, England, 2003).[3] A. Koseska, E. Volkov, and J. Kurths, Oscillation quench-ing mechanisms: Amplitude vs oscillation death, Phys.Reports , 5109 (2014).[4] T. E. Lee and H. R. Sadeghpour, Quantum synchroniza-tion of quantum van der pol oscillators with trapped ions,Phys. Rev. Lett. , 234101 (2013).[5] S. Walter, A. Nunnenkamp, and C. Bruder, Quan-tum synchronization of a driven self-sustained oscillator,Phys. Rev. Lett. , 094102 (2014).[6] T. E. Lee, C.-K. Chan, and S. Wang, Entanglementtongue and quantum synchronization of disordered os-cillators, Phys. Rev. E , 022913 (2014).[7] S. Walter, A. Nunnenkamp, and C. Bruder, Quantumsynchronization of two van der pol oscillators, Ann. der.Phys. , 131 (2015).[8] N. L¨orch, S. E. Nigg, A. Nunnenkamp, R. P. Tiwari, andC. Bruder, Quantum synchronization blockade: Energyquantization hinders synchronization of identical oscilla-tors, Phys. Rev. Lett. , 243602 (2017).[9] L. Morgan and H. Hinrichsen, Oscillation and synchro-nization of two quantum self-sustained oscillators, J.Stat. Mech. , P09009 (2015).[10] V. M. Bastidas, I. Omelchenko, A. Zakharova, E. Sch¨oll,and T. Brandes, Quantum signatures of chimera states,Phys. Rev. E , 062924 (2015).[11] S. Sonar, M. Hajduˇsek, M. Mukherjee, R. Fazio, V. Ve-dral, S. Vinjanampathy, and L. Kwek, Squeezing en-hances quantum synchronization, Phys. Rev. Lett. ,163601 (2018).[12] W.-K. Mok, L.-C. Kwek, and H. Heimonen, Synchro-nization boost with single-photon dissipation in the deepquantum regime, Phys. Rev. Res. , 033422 (2020).[13] A. W. Laskar, P. Adhikary, S. Mondal, P. Katiyar,S. Vinjanampathy, and S. Ghosh, Observation of quan-tum phase synchronization in spin-1 atoms, Phys. Rev.Lett. , 013601 (2020).[14] M. Koppenh¨ofer, C. Bruder, and A. Roulet, Quantumsynchronization on the IBM Q system, Phys. Rev. Re-search , 023026 (2020).[15] K. Ishibashi and R. Kanamoto, Oscillation collapse incoupled quantum van der pol oscillators, Phys. Rev. E , 052210 (2017).[16] E. Amitai, M. Koppenh¨ofer, N. L¨orch, and C. Bruder,Quantum effects in amplitude death of coupled anhar-monic self-oscillators, Phys. Rev. E , 052203 (2018).[17] A. Koseska, E. Volkov, and J. Kurths, Transition fromamplitude to oscillation death via turing bifurcation,Phys. Rev. Lett , 024103 (2013).[18] A. Turing, The chemical basis of morphogenesis, Philos.Trans. R. Soc. Lond. , 37 (1952).[19] B. van der Pol, On oscillation hysteresis in a triode gen-erator with two degrees of freedom, Philos. Mag. , 700(1922).[20] T. Banerjee and D. Ghosh, Transition from amplitudeto oscillation death under mean-field diffusive coupling,Phys. Rev. E , 052912 (2014).[21] T. Banerjee and D. Ghosh, Experimental observation of atransition from amplitude to oscillation death in coupledoscillators, Phys. Rev. E , 062902 (2014).[22] D. V. Ramana Reddy, A. Sen, and G. L. Johnston, Timedelay induced death in coupled limit cycle oscillators,Phys. Rev. Lett , 5109 (1998).[23] G. Saxena, A. Prasad, and R. Ramaswamy, Amplitudedeath: The emergence of stationarity in coupled nonlin-ear systems, Physics Reports , 205 (2012).[24] D. G. Aronson, G. B. Ermentrout, and N. Kopell, Ampli-tude response of coupled oscillators, Physica D , 403(1990).[25] W. Zou, D. V. Senthilkumar, J. Duan, and J. Kurths,Emergence of amplitude and oscillation death in identicalcoupled oscillators, Phys. Rev. E , 032906 (2014).[26] J. Garc´ıa-Ojalvo, M. B. Elowitz, and S. H. Strogatz,Modeling a synthetic multicellular clock: Repressilatorscoupled by quorum sensing, Proc. Natl. Acad. Sci. USA , 10955 (2004).[27] E. Ullner, A. Zaikin, E. I. Volkov, and J. Garc´ıa-Ojalvo,Multistability and clustering in a population of syntheticgenetic oscillators via phase-repulsive cell-to-cell commu-nication, Phys. Rev. Lett , 148103 (2007).[28] D. Ghosh, T. Banerjee, and J. Kurths, Revival of oscilla-tion from mean-field-induced death: Theory and experi-ment, Phys. Rev. E , 052908 (2015).[29] T. Banerjee, P. S. Dutta, and A. Gupta, Mean-fielddispersion-induced spatial synchrony, oscillation and am-plitude death, and temporal stability in an ecologicalmodel, Phys. Rev. E , 052919 (2015). [30] E. H.Hellen and E. Volkov, How to couple identical ringoscillators to get quasiperiodicity, extended chaos, mul-tistability, and the loss of symmetry, Comm. in NonlinSci. Num. Simulat. , 462 (2018).[31] A. Zakharova, I. Schneider, Y. N. Kyrychko, K. B.Blyuss, A. Koseska, B. Fiedler, and E. Sch¨oll, Time de-lay control of symmetry-breaking primary and secondaryoscillation death, EPL , 50004 (2013).[32] B. Ermentrout, Simulating, Analyzing, and AnimatingDynamical Systems: A Guide to Xppaut for Researchersand Students (Software, Environments, Tools) (SIAMPress, 2002).[33] J. Johansson, P. Nation, and F. Nori, Qutip 2: A pythonframework for the dynamics of open quantum systems.,Comput. Phys. Commun. , 1234 (2013).[34] J. Weinbub and D. K. Ferry, Recent advances in wignerfunction approaches, Appl. Phys. Rev. , 041104 (2018).[35] H. J. Carmichael, Statistical Methods in Quantum Optics1 (Springer, 1999).[36] G. Ansmann, Efficiently and easily integrating differen-tial equations with JiTCODE, JiTCDDE, and JiTCSDE,Chaos , 043116 (2018).[37] C. K. Zachos, D. B. Fairlie, and T. L. Curtright, QuantumMechanics in Phase Space (World Scientific, Singapore,2005).[38] R. P. Rundle, P. W. Mills, T. Tilma, J. H. Samson, andM. J. Everitt, Simple procedure for phase-space mea-surement and entanglement validation, Phys. Rev. A ,022117 (2017).[39] M. R. Hush, W. Li, S. Genway, I. Lesanovsky, and A. D.Armour, Spin correlations as a probe of quantum syn- chronization in trapped-ion phonon lasers, Phys. Rev. A , 061401(R) (2015).[40] A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson,S. Girvin, A. Clerk, F. Marquardt, and J. Harris, Disper-sive optomechanics: a membrane inside a cavity, New J.Phys. , 095008 (2008).[41] S. Lloyd and S. L. Braunstein, Quantum computationover continuous variables, Phys. Rev. Lett. , 1784(1999).[42] K. Goda, O. M. E. E. Mikhailov, S. Saraf, R. Ad-hikari, K. McKenzie, R. Ward, S. Vass, A. J. Wein-stein, and N. Mavalvala, A quantum-enhanced prototypegravitational-wave detector, Nat. Phys. , 472 (2008).[43] G. S. Agarwal, Engineering non-gaussian entangledstates with vortices by photon subtraction, New J. Phys. , 073008 (2011).[44] G. S. Agarwal, Quantum Optics (Cambridge UniversityPress, England, 2012).[45] C. Gerry and P. Knight,
Introductory Quantum Op-tics (Cambridge University Press, Cambridge, England,2005).[46] B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li,and G. Rempe, Deterministic creation of entangled atom-light schr¨odinger-cat states, Nature Photonics , 110(2019).[47] A. Zakharova, M. Kapeller, and E. Sch¨oll, Chimeradeath: Symmetry breaking in dynamical networks, Phys.Rev. Lett , 154101 (2014).[48] T. Banerjee, Mean-field-diffusion–induced chimera deathstate, EPL110