Synchronisation in coupled quantum Hamiltonian superconducting oscillator via a control potential

Chaos, Solitons and Fractals 42 (2009) 1415–1421

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Synchronisation in coupled quantum Hamiltonian superconducting oscillator via a control potential Sameer Al-Khawaja * Department of Physics, Atomic Energy Commission of Syria (AECS), Damascus, P.O. Box 6091, Syria

a r t i c l e

i n f o

Article history: Accepted 11 March 2009

Communicated by Prof. M. Wadati

a b s t r a c t This paper presents chaos synchronisation in a SQUID device mutually coupled to a resonant LC classical circuit. Via the Hamiltonian of the coupled quantum-classical system and by means of a ‘‘control potential” in the form of a double-well, measure synchronisation has been found to exist. A transition from quasi-periodic to chaotically synchronised orbits in the phase space has been observed, as the strength of coupling is increased between both oscillators. The system reaches a non-synchronised state if the choice of the control potential were to render both oscillators non-identical. Ó 2009 Published by Elsevier Ltd.

1. Introduction There is an ever mounting interest in exploring the dynamics of coupled nonlinear oscillators such as SQUID devices, that enclose quantum mechanically operating Josephson junctions [1–3]. Networks or arrays, as well as stacked and annular configurations of these devices can be visualised as a number of sub-oscillators coupled to their neighbours [4]. Rich nonlinear phenomena stem from these systems such as bifurcation, multistability, multi-level states, chaotic diffusion, ratchet effect, relaxation oscillations and synchronisation [5–8]. Studying synchronisation phenomena in dynamical systems as such is of significance, particularly in those systems that exhibit chaotic behaviour [9]. The elaborate examination of synchronisation although qualitative, may be of importance, since one tends to understand coherent dynamical behaviour of different coupled classical or quantum systems, whether physical, chemical, ecological, or biological in order to explain diffusion, reaction and transport processes. Synchronisation by definition, is a phenomenon occurs in periodic dynamical systems exchanging weak interaction, so that frequency and phase locking are secured. Therefore, attention has recently been drawn towards chaotic systems, for investigating in particular the bifurcation from chaos to periodicity or adversely, in systems such as Josephson or SQUID devices [10]. Numerous types of synchronisation have also been reported and identified like, lag synchronisation, phase, generalised, anticipated, and complete synchronisation [11]. Hitherto the role of chaos in classical dynamics is very well established, in contrast to quantum systems, for which the quantum chaos is a phenomenon that is not clear-cut understood. The interplay between classical phase structures and quantum processes like tunnelling, has been investigated on the basis of a Hamiltonian describing a driven quantum oscillator in a bi-stable well. The results refer to a high-rate coherent tunnelling between certain regions, dominated by the effect of classical phase–space structures, which are bounded by KAM surfaces, given that a proper driving is present [12]. The arduous problem of how also to recover classical nonlinear dynamics and chaos from quantum–mechanical systems has found a great deal of interest [13], in view of coupling a quantum object to a classical environment. In contrast to dissipative systems, Hamiltonian systems conserve the phase volumes [14], and do not initially permit two non-identical trajectories be synchronised, i.e. reach asymptotically an identical trajectory. In addition, one can study quantum pro-

* Fax.: +963 11 6112289. E-mail address: scientifi[email protected] 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.03.059

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cesses via the Hamiltonian prospect, since the latter constitutes the foreground to apprehending the concept of synchronisation and exploiting it in applications within the quantum domain, as is the case with quantum computation. On the other hand, since synchronisation arises implicitly in at least two coupled oscillators, controlling it is an important key-issue [15,16]. The reduction of the extraneous effects of noise and parameter mismatch [17,18], the minimisation of both the synchronisation time and the threshold energy input necessary for the process [19], have emerged as leading topics in nonlinear dynamics. Handling these issues is also central to investigating the problems of optimising synchronisation. Thus, as a primary conception, one can generally consider that synchronisation is optimised when the synchronisation manifold is in the most stable formation in the state space of the coupled oscillators. The issue of defining the conditions and coupling parameters that lead to high-quality synchronisation has been and still an open problem. In addition to studies undertaken on the phenomena of spatiotemporal chaos [19], and complete synchronisation [11], there are reports on the so-called cluster synchronisation [20]. The latter characterises hybrid configurations as a ramification to symmetry breaking and spontaneous spatial reordering. Cluster patterns form whenever chaotic oscillators synchronise with one another in groups, whilst there is no synchronisation among the groups. The significant application of such hybrid phenomena lies in telecommunications within which cryptography and switching can greatly benefit from it [15]. Reference [14] outlined the types of synchronisation and laid emphasis on measure synchronisation (MS) in Hamiltonian coupled Duffing oscillators. Superconducting quantum devices as Hamiltonian systems have received little attention with respect to synchronisation processes, given that chaotic transport develops in such nonlinear systems. This article is devoted to investigating synchronisation in superconducting configurations, which currently receive interest as persistent-current qubits [21], and exploring the effect of initial conditions and coupling parameters on their collective behaviour. We demonstrate that synchronisation can be fulfilled via a ‘‘control potential” related to the classical oscillator, which is inductively coupled to the superconducting quantum device, and can be conformed in a way that alters the overall energy of the system. The initial conditions and the parameters of the system are identified accordingly. 2. The coupled quantum-classical Hamiltonian system We consider the Superconducting Quantum Interference Device with a single Josephson junction (RF SQUID). The SQUID as a quantum oscillator traditionally operates when coupled inductively to a LC classical oscillator (tank circuit), resonant at pffiffiffiffiffiffiffiffiffi RF ( a few MHz), with angular frequency xt ¼ Lt C t , as in Fig. 1. The mutual inductance is M = K(LtK)1/2, where K is the coupling coefficient. In addition, one may consider the main parameter b of a SQUID defined by

b ¼ 2pKIc =U0

ð1Þ

where Ic is the critical current of the Josephson junction, and U0 = h/2e is the flux quantum. According to b, the modes of operation of the device can be identified; for b > 1, the SQUID is operated in the hysteretic mode [2], whilst for b 6 1, the behaviour is reversible and the mode of operation is accordingly called inductive. The internal flux within the loop of the SQUID can be expressed in terms of the external flux Ue, and screening currents Is such as

U ¼ Ue  KIs

ð2Þ

In the limit of weakly superconducting loop, Eq. (2) may be declared as a current-phase relation

hs ¼ he  b sinðhÞ

ð3Þ

where hs = 2pU/U0 and he = 2pUe /U0 are the internal and external phases, respectively. Normally, the Hamiltonian of the system can be constructed from the Coulomb energy of the junction charge Q 2s =2C s , where Csis the capacitance, and from the potential energy EJ(hs  he)2/2b  EJ cos(hs), which includes the inductive diamagnetic energy of the loop given in terms of phases, and potential energy of Josephson junction, where EJ = IcU0/2p is the coupling energy. Now if we consider the SQUID in a bi-stable well of the type Vðhs Þ ¼ ah4s  bh2s ; a and b are constants, we can straightforwardly write down its Hamiltonian as

Hs ¼

" # Q 2s ðhs  he Þ2 þ EJ ah4s  bh2s þ 2C s 2b

ð4Þ

In the limit of weak damping b 6 1, the Hamiltonian h i (4) is quantum–mechanical [2], and the charge Qs and phase hs are operb s ¼ 2ei, with the associated Hamilton equations ators satisfying the commutation relation ^ hs ; Q

M Λ

EJ CS

θs

Lt

Ct

Fig. 1. The inductively coupled quantum-classical oscillators.

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@Hs @Hs and Q_ s ¼  h_ s ¼ @Q s @hs

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ð5Þ

where hs and Qs are playing the role of position and momentum of a particle, respectively. Analogously, the Hamiltonian Htof the resonant oscillator consists of two energy terms; capacitive Q 2t =2C t and inductive u2t =2Lt where ut is the magnetic flux stored within the classical circuit. Within the framework of a full quantum–mechanical picture, one may consider the SQUID being coupled to the electromagnetic filed (em) mode, which can be modelled via the classical oscillator [22]. Consequently, we can write down the total Hamiltonian of the ring-em oscillator system in terms of a p  q representation as a sum of Hs and Ht such as

H¼

  1 2 q2 q2 2 ðp1 þ p22 Þ þ a aq41  bq1 þ 1 þ Uðq2 Þ þ 2 þ lq1 q2 2 2b 2

ð6Þ

where a denotes the Josephson coupling, and the last term on the right hand side of the Hamiltonian (6) represents the coupling between the SQUID ring and em field defined via l(=M/Lt), which is a measure of the strength of coupling between both of them. The third term in (6), U(q2) represents the ‘‘control potential” related to ut and dictates the energy status of the em field, which as we shall demonstrate, permits synchronisation to occur between both mutually coupled oscillators when carefully chosen. 2.1. The canonical equations of motion The Hamiltonian (6) yields the canonical equations governing the motion of each oscillator, so that one obtains

  dp1 q ðtÞ  lq2 ðtÞ; ¼ a 4aq31 ðtÞ  2bq1 ðtÞ þ 1 b dt

Fig. 2. (a) The limit cycle periodic behaviour of the two oscillators in the (q,p) plane for l = 0, with no synchronisation between both oscillators. The initial conditions are: q1(0) = 0.3, q2(0) = 0, p1(0) = 0, and p2(0) = 0.2. (b) As in (a), but for l = 0.005; both oscillators are still desynchronised while oscillator (1) developing quasi-periodic motion. (c) and (d) as before, but with l = 4  102. Both oscillators share the same phase space and reach measure synchronisation with increasing quasi-periodicity.

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dp2 dU ¼  q2 ðtÞ  lq1 ðtÞ; dq2 dt dq1 dq2 ¼ p1 ðtÞ; ¼ p2 ðtÞ dt dt

ð7Þ

The constants a, b define the shape of the potential well, and have been taken 0.25 and 0.5, respectively. In order the coupled oscillators reach synchronisation, we choose the functional form of the control potential U so that it resembles the bi-stable potential well of the quantum oscillator, i.e. U(q2) = q2(t)/4  q2(t)/2. Further, we set the Josephson coupling a, which is comparable to the charging energy Q 2t =2C t equals to unity and leave the dynamics of the system drastically dependant on l, which has been varied for some given initial conditions, and ultimately played a role in setting the overall energy of the cou-

Fig. 3. The initial conditions are: q1(0) = 0, q2(0) = 0, p1 (0) = 1.42  102, and p2(0) = 0.632. (a) Non-synchronised periodic orbits between the two oscillators for l = 0. (b) and (c) both oscillators come chaotically to synchronisation for l = 0.19. (d) and (e) The coupled oscillators achieve strong chaotic measure synchronisation for higher coupling l = 0.6.

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pled oscillators. In this model, we also consider b = 1 for which the SQUID preserves a quantum–mechanical description, whilst it follows a minimum in its potential. As we shall demonstrate in the proceeding section, l has a decisive contribution to identifying the behaviour of the system, since both oscillators can be made identical, and any fraction of flux coupled to the quantum oscillator would alter its quantum state and dynamics. 2.2. Chaotic synchronisation in the coupled quantum oscillator We have solved the coupled Eq. (7) of the system to simulate the dynamical behaviour in the q  p plane. The initial conditions and the coupling strength parameter l have been considered the control parameters throughout, apart from our choice of U(q2). The results refer to different types of measure synchronisation; quasi-periodic and chaotic. The initial conditions were primarily set as follows: q1 (0) = 0.3, q2(0) = 0, p1(0) = 0, and p2(0) = 0.2. In Fig. 2, we plot the phase map in (qi,pi)(i = 1,2), where we observe in Fig. 2(a) a limit cycle for zero coupling strength. Both oscillators as apparent, have different energy surfaces governed by the given initial conditions. For a slight increase in l (e.g. l = 0.005), the situation is no different from what is monitored in Fig. 2(a) for the classical oscillator (oscillator 2). While for the quantum oscillator (oscillator 1), the motion is quasi-periodic and both oscillators are still desynchronised, as demonstrated in 2(b). As l is further raised to 4  102, one can clearly notice that the orbits of both oscillators (inner and external borders) become identical and occupy the same energy surfaces with increasing quasi-periodicity. Thus measure synchronisation is reached between the two mutually coupled oscillators, as shown in (c) and (d) of Fig. 2. In order to examine the system behaviour more closely, a different set of initial conditions has been chosen. These are q1(0) = 0, q2(0) = 0, p1(0) = 1.42  102, and p2 (0) = 0.632, for which the results are displayed in Fig. 3. For l = 0, the phase domain plots still indicate to a limit cycle periodic motion with non-synchronised states between the two oscillators, with oscillator (2) occupying a very small surface area in the phase domain as Fig. 3(a) shows. Sufficiently beyond a critical value (l > 0.05), the system starts to chaotically synchronise for l = 0.19, as in (b) and (c). The coupled oscillators eventually exhibit strong chaotic measure synchronisation for higher coupling values, as for instance in 3(d) and 3(e) for which l = 0.6. As a supplemental illustration, a three-dimensional phase domain plot for each case of which l = 0.19, and l = 0.6 is shown in Fig. 4(a and b), respectively. 2.3. Chaotic non-synchronised states In order to show the convenience of the control potential that has been used, we have solved the coupled autonomous set of Eq. (7) for U having the functional form U (q2) = cos(q2). In this case, the solutions describe the behaviour of both non-identical oscillators, one of which is in a bi-stable well while the other is in a periodically varying potential. The results are shown in Fig. 5, in which we select as an example, two coupling values of l, for the same initial conditions considered in Fig. 2. In Fig. 5(a), the coupled oscillators exhibit periodic orbits in the q  p plane for zero coupling parameter as before. However, for a substantially higher value of l (l = 0.6), both oscillators become desynchronised, developing chaotic dynamics on different q  p surfaces, as demonstrated in Fig. 5(b and c). It’s worthwhile to mention that, in order to bring the coupled oscillators back to synchronisation, one has to resort to extra techniques such as active control, recursive backstepping or active sliding mode control [23]. Optimising such techniques is still an open issue and will constitute a future work on the subject, which is currently in progress.

Fig. 4. A three-dimensional phase domain plot for the two coupling values taken in Fig. 3 showing the chaotically synchronised orbits. (a) for l = 0.19 and (b) for l = 0.6.

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Fig. 5. (a) The periodic orbits for the desynchronised oscillators for zero coupling strength and periodically varying potential characterising oscillator (2). The same initial conditions in Fig. 2 have been considered. (b) and (c) Chaotic dynamics on different q  p surfaces for l = 0.6 with no synchronisation achieved, for periodic time varying control potential.

3. Conclusion In this paper, we have demonstrated that a quantum superconducting oscillator coupled to another classical oscillator can be synchronised by means of a control potential rendering both oscillators identical. A transition from measure desynchronisation to quasiperiodic and chaotic synchronisation is revealed. The type and functional form of the control potential have been found important to preserve the mutually coupled system from desynchronising. Acknowledgement The author would like to thank the Director General of the Atomic Energy Commission of Syria for continual assistance and support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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