Non-Hermitian quantum dynamics and entanglement of coupled nonlinear resonators
Evren Karakaya, Ferdi Altintas, Kaan Güven, Özgür E. Müstecaplıoğlu
aa r X i v : . [ qu a n t - ph ] F e b epl draft Non-Hermitian quantum dynamics and entanglement of couplednonlinear resonators
Evren Karakaya , Ferdi Altintas , Kaan G¨uven and ¨Ozg¨ur E. M¨ustecaplıo˘glu Department of Physics, Ko¸c University, Sarıyer, ˙Istanbul, 34450, Turkey Department of Physics, Abant Izzet Baysal University, Bolu, 14280, Turkey
PACS – Charge conjugation, parity, time reversal, and other discrete symmetries
PACS – Cavity quantum electrodynamics; micromasers
PACS – Collective excitations (including excitons, polarons, plasmons and other charge-density excitations)
Abstract –We consider a generalization of recently proposed non-Hermitian model for resonantcavities coupled by a chiral mirror by taking into account number non-conservation and nonlinearinteractions. We analyze non-Hermitian quantum dynamics of populations and entanglement ofthe cavity modes. We find that an interplay of initial coherence and non-Hermitian coupling leadsto a counterintuitive population transfer. While an initially coherent cavity mode is depleted, theother empty cavity can be populated more or less than the initially filled one. Moreover, presenceof nonlinearity yields population collapse and revival as well as bipartite entanglement of thecavity modes. In addition to coupled cavities, we point out that similar models can be found in PT symmetric Bose-Hubbard dimers of Bose-Einstein condensates or in coupled soliton-plasmonwaveguides. We specifically illustrate quantum dynamics of populations and entanglement in aheuristic model that we propose for a soliton-plasmon system with soliton amplitude dependentasymmetric interaction. Degree of asymmetry, nonlinearity and coherence are examined to controlplasmon excitations and soliton-plasmon entanglement. Relations to PT symmetric lasers andJahn-Teller systems are pointed out. Introduction. –
Recently, an intriguing quantum op-tical model for resonant cavities coupled by a chiral mirrorhas been proposed [1]. The chiral mirror is a planar meta-material array of asymmetric split rings, through whichtransmission of circularly polarized electromagnetic wavesbecomes different in the opposite direction without violat-ing Lorentz reciprocity principle [2]. The transmission ma-trix of such a mirror is a two dimensional non-Hermitianmatrix. The proposed quantum optical system is thendescribed by a non-Hermitian Hamiltonian. As a specialclass of non-Hermitian systems, parity-time ( PT ) sym-metric systems have been attracted much attention [3–5].More recently, it is shown that hybridized metamaterialscan simulate effectively spontaneous symmetry breakingin PT - symmetric non-Hermitian quantum systems [6].Nonreciprocal light transmission in PT - symmetry bro-ken phase using whispering-gallery microcavities has beenobserved very recently [7].Non-Hermitian interactions are also reported for Bose-Einstein condensates [8, 9], optical lattices [10], waveg- uides [11–13] and soliton-plasmon systems [14, 15].Soliton-plasmon system has unique properties such as non-linear coupling in addition to local nonlinearity due to soli-ton and can be examined from a pure quantum perspec-tive in terms of generalized models of coupled cavities [1]or Bose-Einstein condensates [8]. Effects of local nonlin-earities on systems with PT symmetry have been subjectto recent attention [16, 17]. Moreover, chiral mirror cou-pled cavities or soliton-plasmon systems we examine herehave non-local non-Hermiticity in the coupling in contrastto typical systems considered in literature with local non-Hermiticity associted with balanced gain and loss.The non-Hermitian quantum model for resonant cavi-ties coupled by a chiral mirror uses a number conservingHeisenberg approach to determine the quantum dynam-ics [1]. In the present letter, we use a more general vonNeumann approach [18–21] which allows for violation ofnumber conservation due to the non-Hermitian nature ofthe system. These two approaches are not equivalent toeach other and von Neumann approach is proper treat-p-1. Karakaya et al. ment of non-conservative systems with the non-Hermitianmodels. We determine the population dynamics of theresonator modes when one resonator is in a coherent stateand the other is empty. Strong excitations of an initiallyempty resonator mode is found to be possible by a weaklypopulated coherent one depending on the asymmetry oftheir coupling. Curious role of coherence on particle non-conservation is revealed. In addition we characterize bi-partite quantum entanglement of the resonators by usingvon Neumann entropy [22] and as expected initially unen-tagled state remains factorized with zero entropy. This istypical as one cannot create entanglement by mixing co-herent states [23, 24]. We then propose generalized mod-els where resonator and their coupling could be nonlinear.In this case, quantum entanglement between the modesemerges. We emphasize that the proposed nonlinear mod-els are physically different from the coupled linear res-onator model.The specific purpose of the present letter is to proposemodels which are experimentally achievable in soliton-plasmon and in Bose-Hubbard dimers as well as in nonlin-ear coupled cavities, and to reveal the interplay of coher-ence, local and non-local nonlinearities, and asymmetricnon-local non-Hermitian coupling on quantum entangle-ment and on number non-conserving nonlinear quantumdynamics. Our models are of broad interest to the sub-jects of cavity QED, Bose-Einstein condensates, coupledwaveguides, optical solitons and quantum plasmonics. Wepoint out a close relation to Jahn-Teller physics [25] and PT symmetric lasers and anti-lasers as well [26–29].From more fundamental perspective, one can learn moreabout the PT symmetry in such systems by further stud-ies of the spectral properties. In contrast to typical classi-cal electrodynamical PT symmetric systems we have purequantum models which could be used to examine quan-tum correlations and PT symmetry relations. Our vonNeumann approach can be extended to open system con-ditions, which can be implemented by the coupled cavityand soliton models, to examine PT symmetry effects inopen system. In contrast to typical local PT symmetricterms, our models have non-local non-reciprocal couplingas a different class of PT symmetry and hence they canbring new perspectives to nonlinearity, non-locality andbroken PT symmetry relations. Non-Hermitian dynamics and Entanglement. –
Before investigating the mean-field dynamics and theentanglement between coupled field modes in two exper-imentally feasible quantum models which are representedby non-Hermitian Hamiltonians, we give a brief introduc-tion to the non-Hermitian dynamics and the entanglementmeasure.The general formulation for the time evolution of quan-tum systems under non-Hermitian Hamiltonians can befound in [18–21]. A non-Hermitian Hamiltonian opera-tor can be partitioned into Hermitian and anti-Hermitian parts: H = H + + H − , (1)where H ± = (cid:0) H ± H † (cid:1) , H ± = ± H †± , and A † de-notes the Hermitian conjugate of A . The non-HermitianSchr¨odinger equation can be written as (Here and in therest of the paper we take ¯ h = 1.) ∂∂t | Ψ( t ) i = − i ( H + + H − ) | Ψ( t ) i . (2)Introducing the density matrix ρ ( t ) = | Ψ( t ) i h Ψ( t ) | leadsto the von Neumann type master equation in the followingform [18]: ∂∂t ρ ( t ) = − i [ H + , ρ ( t )] − i { H − , ρ ( t ) } , (3)where [ . . . ] and { . . . } stand for commutator and anti-commutator, respectively. Here we assure that Eq. (3) alsoholds for mixed states [18]. Since the dynamics governedby (2) is not unitary, the trace of the density operator maynot be conserved. Therefore, we introduce a normalizeddensity operator, ρ ′ ( t ) = ρ ( t ) /T r ( ρ ( t )) , (4)then the time evolution of the quantum averages of theobservables can be calculated through the formula h A i t = T r ( Aρ ( t )) /T r ( ρ ( t )) . (5)Entangled states are essential resource for many appli-cations of quantum information and computation proto-cols. It would be very desirable to study the possibility ofdetecting entanglement between the coupled modes [22].In our study, we consider an initial pure product state ofthe field modes and have carefully checked that the purityof the considered initial state is conserved during the dy-namics (3) in accordance with the fact that an initial purestate remains pure under non-Hermitian dynamics [21].The entanglement between coupled modes in a pure statecan be quantified by the von Neumann entropy S = − T r (cid:16) ρ ′ i log ρ ′ i (cid:17) , (6)where ρ ′ i = T r j ρ ′ is the reduced density matrix of themode i obtained by taking a partial trace over the re-maining mode j . It is zero for separable states and reachesthe maximum value of log N for a N dimensional Hilbertspace. Non-zero values of the entropy indicate that themodes are in an entangled state.We apply the formalism described above to systemati-cally examine the entanglement and population dynamicsin a generic resonant two-mode model H = ω ( a † a + b † b ) + ua † a † aa + g AB ab † + g BA f ( a † a ) a † b, (7)p-2on-Hermitian quantum dynamics and entanglement of coupled resonatorswhere a ( a † ) and b ( b † ) are the lowering (raising) op-erators of the bosonic cavity field modes A and B, re-spectively, which obey the Weyl-Heisenberg h algebra[ a, a † ] = [ b, b † ] = 1; ω is the resonance frequency; u isthe parameter of local nonlinearity, while f ( a † a ) is an in-tensity dependent deformation function describing non-local nonlinearity. Non-reciprocal coupling of the modesis described by the coefficients g AB and g BA , which areassumed to be real numbers for simplicity. More local andnon-local sources for nonlinearity could be introduced butthis model has optimal number of parameters to capturethe essential physics and relevant to experimental systems.We first consider linear model with u = 0 and f ( a † a ) = 1which corresponds to chiral mirror coupled optical cavi-ties [1]. After that we discuss the case of local nonlinear-ity per se with f ( a † a ) = 1 and point out that the modeltranslates to a PT Bose-Hubbard dimer [11]. Finally weinclude the nonlocal nonlinearity by taking f ( a † a ) = √ a † a to make the model relevant to asymmetric amplitude de-pendent coupled soliton-plasmon system [14]. Resonant cavities coupled by a chiral mirror. –
We first reconsider the non-Hermitian quantum modelthat describes the behavior of a cavity mode in two res-onant cavities coupled through a 2-D chiral mirror [1].The mode in each cavity can be described by independentquantum oscillators. The Hamiltonian of the system inthe weak coupling scenario can be written as H = H + H I , (8)where H = ω a † a + ω b † b (9)is the free Hamiltonian of the oscillators having the tran-sition frequencies ω , and H I = g AB ab † + g BA a † b (10)is the interaction Hamiltonian which describes a non-reciprocal coupling between the oscillators if g AB = g BA .The analytical non-Hermitian dynamics of the meanexcitation number in cavity A and B for the consideredmodel have been investigated in Ref. [1] by solving thestandard Heisenberg equation of motion in an ad hoc man-ner. The exchange of a photon from one cavity to the an-other through the 2-D chiral mirror has been found. Also,this formalism has been found to preserve the total exci-tation number, since the total number operator a † a + b † b commutes with the Hamiltonian (8). On the other hand,the standard Heisenberg equation is based on the Hamil-tonian being Hermitian, so it is not expected to capturemore general and subtle features of non-Hermitian dynam-ics. Therefore, our aim in this section is to reexaminethe mean-field dynamics by using the formalism describedabove.We first derive the Heisenberg equation for the non-Hermitian Hamiltonian, and show that the expectation H a L r = Ω t H b L r = Ω t H c L r = Ω t Fig. 1: (cid:10) a † a (cid:11) t (solid line), (cid:10) b † b (cid:11) t (dashed line) and (cid:10) a † a + b † b (cid:11) t (dot-dashed line) versus ω t for the initial state | Ψ(0) i = | α A i | B i , α A = 1, g AB = g.r , g BA = g , g = 0 . ω , r = 1 (a), r = 0 . r = 2 (c). value of the total number operator may not be constant.The time evolution of the expectation value of an ob-servable through the normalized state can be written as: h A i t = h Ψ( t ) | A | Ψ( t ) i / h Ψ( t ) | Ψ( t ) i . By using Eq. (2), thegeneralized Heisenberg equation for the expectation valueof an operator can be written as [19]: ∂∂t h A i t = − i h [ A, H + ] i t − i h{ A, H − }i t + 2 i h A i t h H − i t . (11)It is simple to show that the expectation value of the totalexcitation number evolves as: ∂∂t (cid:10) a † a + b † b (cid:11) t = i [ g AB − g BA ][ (cid:10) ( a † ) ab − a † a b † (cid:11) t + (cid:10) a † b − ab † (cid:11) t + (cid:10) b † b a † − ( b † ) ba (cid:11) t + (cid:10) a † a + b † b (cid:11) t (cid:10) ab † − a † b (cid:11) t ] , (12)which can be different than zero if g AB = g BA . Strictly,the time evolution in Eq. (12) depends on the initial state,as well.p-3. Karakaya et al. In Fig. 1, we have plotted the time evolution of the meanexcitation number in cavity A, (cid:10) a † a (cid:11) t , in cavity B, (cid:10) b † b (cid:11) t ,and the total excitation number, (cid:10) a † a + b † b (cid:11) t , by numeri-cally solving Eq. (3) for ρ ( t ) and by employing Eq. (5), foran initial state where the mode A in the coherent state andthe mode B in its vacuum, | Ψ(0) i = | α A i | B i , for the pa-rameters g AB = g.r , g BA = g , r = 1 (Hermitian case) and r = 0 . , g AB < g BA ,the creation rate of initially empty mode B by initially co-herent mode A is smaller than its destruction, then modeB is weakly excited. Even if the mode A is completelydepleted, its population can only be partially transferredto mode B as can be seen in Fig. 1(b). In this casenon-Hermiticity and initial coherence act as a decoher-ence and population loss channel. On the other hand,if g AB > g BA , an amplification of the total excitation isfound as in Fig. 1(c). In this case, weak amplitude Focknumber states in the coherent state of mode A can con-tribute to excitation of the mode B due to asymmetricinteraction in favor of mode B. As a result mode B can beexcited with higher amplitudes then the exciting field.Our model can be related to the Jahn-Teller systemin classical limit [25]. The states of that system is al-ways a coherent state and there is always number non-conservation. In our quantum system, Fock number stateslead to number conservation and the significance of initialcoherence is revealed. We emphasize that the photons arenot generated or destroyed by the mirror but they arealready present in the initial coherent state. Number non-conservation is just a shift of the mean excitation numberdue to non-reciprocal photon exchange between the cavi-ties which allows for unbalanced growth of less fortunateexcitation manifolds relative to the initial average.Number non-conserving dynamics is related to PT sym-metric lasers and anti-lasers as well [26–29]. To see this ex-plicitly we can introduce the operators of non-local modes c = ( a + i b ) / √ d = ( a − i b ) / √ H − = ( g AB − g BA )( ab † − ba † ) / H − = iΓ( c † c − d † d ) with Γ =( g AB − g BA ) /
2. The Hermitian part remains Hermitianas H + = ω N + i G ( c † d − d † c ) with N = c † c + d † d and G = ( g AB + g BA ) /
2. Non-local non-reciprocal coupling oflocal modes then effectively describes local gain and losson the non-local modes. In the single excitation mani-folds, the Hamiltonian blocks of the local and nonlocalmodes H (1)l and H (1)nl are respectively given by H (1)l = (cid:18) ω g AB g BA ω (cid:19) , (13) H (1)nl = (cid:18) ω + iΓ i G − i G ω − iΓ (cid:19) . (14)Effective model with ω ± i Γ energies of non-local modesare similar to the scenarios proposed to generate PT lasersusing a medium with spatially alternating gain and lossregions. Using chiral mirrors could be an efficient and ef-fective way to design PT symmetric lasers and anti-lasers.We have also analyzed whether or not the modes becomeentangled in the processes shown in Fig. 1 via the vonNeumann entropy (6) and by using the reduced densitymatrices of the modes A and B . Our results demonstratethat the entanglement entropy is always zero and hencethe states of the modes remain factorized. This is ex-pected as a beam splitter type bilinear mixing interactioncannot be used to create entanglement out of initial coher-ent states [23, 24]. If at least one of the cavities containsa local nonlinearity (for example of Kerr type ua † a † aa ),then the cavity modes is found to be entangled. Further-more the population oscillations exhibit collapse and re-vival dynamics. These effects are due to non-Hermitianand nonlinear interactions and in principle could be real-ized not only in coupled cavities but in other systems aswell. We first show similarity to Bose-Hubbard dimer ofa Bose-Einstein condensate model and then propose an-other system of coupled soliton-plasmon waveguides belowto illustrate these effects. Bose-Hubbard dimer of a Bose-Einstein conden-sate. –
Atomic Bose-Einstein condensate trapped in adouble well potential possesses local Kerr type nonlinear-ity due to atomic collisions at the trap sites. Generaliza-tion of the two-mode condensate model to a PT symmet-ric non-Hermitian model H BEC has been discussed in theliterature [8] in the form H BEC = − i2 γL z + 2 vL x + 2 cL z , (15)where γ, v, c are constants, and L + = L x +i L y = a † b, L − = L † + and L z = ( a † a − b † b ) / su (2) spin algebra.The case of asymmetric coupling of the resonators canbe translated to such a model and becomes H = ω N + g x L x − i g y L y , (16)where g x = g AB + g BA , g y = g AB − g BA and N = a † a + b † b .Similar model has been discussed very recently [11]. Non-linear spin term 2 cL z can be generated by considering Kerrtype nonlinear cavities with additional N, N dependentterms. Such terms can be significant due to possible viola-tion of number conservation. For initial conditions whichare number conserving, or with small number fluctations,the similarity of the two systems can be improved. Weconclude that nonlinear cavities coupled by chiral mirrorscan effectively simulate the non-Hermitian Bose-Hubbarddimer of Bose-Condensate in a double well. Below we en-vision a more general scenario that allows us to includenon-local nonlineraties by proposing a heuristic model ofsoliton-plasmon system.p-4on-Hermitian quantum dynamics and entanglement of coupled resonators Non-linear soliton-plasmon interaction. –
Thesecond non-Hermitian model that we propose in thepresent study describes a non-linear quantum interactionbetween optical soliton photons in a Kerr medium and sur-face plasmons in a metal through a dielectric layer [14,15].One particular advantage of this model is that the non-linear coupling can be larger and more tunable than thelocal nonlinearity. The Hamiltonian of the system can bewritten as H = ω a † a + ua † a † aa + ω b † b + g AB ab † + g BA √ n A a † b, (17)where a ( a † ) and b ( b † ) are the field operators for the soli-ton photons and for the surface plasmons, respectively,and u is the Kerr interaction strength. Here g BA √ n A with n A = a † a is the nonlinear operator-type interactionstrength and describes the weak soliton amplitude. With-out such a nonlinear interaction, the model describes lin-ear and nonlinear cavities coupled through a chiral mirror.One should note that the soliton-plasmon Hamiltonian isalways non-Hermitian, even in the case g AB = g BA .The time dependence of the average soliton photons andsurface plasmons number have been plotted in Fig. 2 forthe initial state | Ψ(0) i = | α A i | B i , where | α i is the co-herent state, u = − . ω , g AB = g.r , g BA = g , g = 0 . ω and r = 0 . , ,
2. Here we have introduced a small Kerrnonlinearity, so that the system remains almost in soli-plasmon resonance. Due to the nature of initial coherentstate and the nonlinear interactions, the dynamics of theaverage excitation numbers exhibit collapse and revivalphenomena. Nonlinearity in either local ( u ) or non-local( g BA ) interaction is sufficient for the collapse-revival ef-fect; though local nonlinearity per se yields longer timefor collapse and revivals.The total excitation number in none of the cases is con-served due to the non-Hermitian dynamics. For the cases r = 1 (Fig. 2(a)) and r = 0 . r = 2 (Fig. 2(c)), the total excitation number exceedsits initial value. There is no complete population trans-fer in soliton photons and surface plasmons, except the r = 2 case, where time average value of the mean numberof plasmon excitations is larger than the time average ofthe soliton population. In other words, a weak coherentsoliton excites strong plasmon, populated more than theinitial soliton mode. This happens when g AB > g BA orwhen transfer rate from soliton to plasmon is greater thanthe one from plasmon to soliton and the soliton is in thecoherent state initially. In the coherent state, weak ampli-tude Fock states with large photon numbers find chanceto contribute to plasmon excitation due to the higher un-balanced transfer rate from soliton to plasmon mode. Ifwe consider initially Fock state for the soliton mode thiseffect disappears and the number conservation cannot beviolated. These conclusions are consistent with the caseof coupled cavities by a chiral mirror.In Fig. 3, we have investigated the dynamics of entangle-ment between coupled modes by means of entropy, S ( ρ A ), H a L r = Ω t H b L r = Ω t H c L r = Ω t Fig. 2: (cid:10) a † a (cid:11) t (solid line) and (cid:10) b † b (cid:11) t (dashed line) versus ω t for the initial state | Ψ(0) i = | α A i | B i , α A = 1, g AB = g.r , g BA = g , g = 0 . ω , u = − . ω , r = 1 (a), r = 0 . r = 2 (c). Ω t S Fig. 3: Entropy versus ω t for the initial state | Ψ(0) i = | α A = 1 i | B i , g AB = g.r , g BA = g , g = 0 . ω , u = − . ω , r = 1 (black, solid), r = 0 . r = 2 (blue,dotdashed). for the same parameters and initial state in Fig. 2. Theinitial separable soliton photons and surface plasmons be-p-5. Karakaya et al. come entangled in time. They never become separablein the considered time domain. Contrary to the previousdiscussed non-Hermitian model (8), the non-linearities inEq. (17) are found to lead to the bipartite entanglementof the modes. The entropy increases in the collapse re-gion of the populations and takes its largest values aroundthe collapse time. The largest entropy value is foundfor the case of strong plasmon excitation ( r = 2). Themaximum entanglement corresponds to maximum entropy S max = log M for a Hilbert space dimension of M . Wetake M = 10 for each soliton and plasmon Fock space. Weconclude that the generated bipartite entanglement is notmaximal. As in the case of collapse-revival effect, thoughany source of nonlinearity suffices for entanglement, localnonlinearity per se would yield longer time to generatelargest entanglement. Conclusions. –
We considered effects of local non-linearity and amplitude dependent, asymmetric interac-tions on the non-Hermitian quantum dynamics and en-tanglement of a two resonator system. Specifically were-examined recently proposed chiral mirror coupled two-resonator system from the point of view of a number non-conserving approach. In addition we proposed anotherheuristic model which could be realized in coupled soliton-plasmon systems. We find that as a result of the interplaybetween the asymmetry of the non-Hermitian couplingand initial coherence of cavity fields, a counter-intuitivepopulation transfer can occur between the resonators. Thepopulation of an initially empty resonator can exceed thatof the initially available population in the other resonator.Moreover, the excitations can exhibit collapse and revivaldynamics in the presence of nonlinearity. Collapse and re-vival times as well as the average populations depend onthe asymmetry of the non-Hermitian interaction. Finallywe investigated the bipartite entanglement between themodes of the resonators using von Neumann entropy. Wefound that while nonlinearity induces entanglement, itsdynamics and amount can be controlled by the asymmetryof the non-Hermitian interaction. In addition, we pointedout that using Kerr-type nonlinear cavities and chiral mir-rors one can simulate the double-well non-Hermitian Bose-Condensate models. Relations to PT symmetric lasersand Jahn-Teller systems are pointed out. ∗ ∗ ∗ We acknowledge illuminating discussions byR. B. B. Santos, U. G¨unther and C. Mili´an. Thiswork is supported by T ¨UB˙ITAK (Grant No. 111T285).F. A. acknowledges the support and the hospitality ofthe Office of Vice President for Research and Develop-ment (VPRD) and Department of Physics of the Ko¸cUniversity.
REFERENCES[1]
Santos R. B. B. , EPL , (2012) 24005.[2] Fedotov V. A., Mladyonov P. L., Prosvirnin S. L.,Rogacheva A. V., Chen Y. and
Zheludev N. I. , Phys.Rev. Lett. , (2006) 167401.[3] Boettcher S. and
Bender C. M. , Phys. Rev. Lett , (1998) 5243.[4] Mostafazadeh A. , J. Math. Phys. , (2002) 205.[5] Bender C. M. , Rep. Prog. Phys. , (2007) 947.[6] Kang M., Liu F. and
Li J. , Phys. Rev. A , (2013)053824.[7] Peng B., ¨Ozdemir S¸. K., Lei F., Monifi F., GianfredaM., Long G. L., Fan S., Nori F., Bender C. M. and
Yang L. , arXiv:1308.4564.[8]
Graefe E. M, G¨unther U., Korsch H. J. and
NiderleA. E. , E-print, ArXiv , (2008) 255206.[9] Dast D., Haag D., Cartarius H., Main J. and
Wun-ner, G. , J. Phys. A: Math. Theor. , (2013) 375301.[10] Regensburger A., Miri M.-A., Bersch C., N¨ager J.,Onishchukov G., Christodoulides D. M. and
PeschelU. , Phys. Rev. Lett. , (2013) 223902.[11] Hong-Hua Z., Qiong-Tao X. and
Jun X. , Chin. Phys. B , (2014) 020314.[12] Klaiman S., G¨unther U. and
Moiseyev N. , Phys. Rev.Lett. , (2008) 080402.[13] Bludov Y. V., Konotop V. V. and
Malomed B. A. , Phys. Rev. A , (2013) 013816.[14] Mili´an C., Ceballos-Herrera D. E., Skryabin D. V. and
Ferrando A. , Opt. Lett. , (2012) 4221.[15] Ferrando A., Mili´an C. and
Skryabin D. V. , J. Opt. Soc. Am. B , (2013) 2507.[16] Lumer Y., Plotnik Y., Rechtsman M. C. and
SegevM. , Phys. Rev. Lett , (2013) 263901.[17] Miroshnichenko A., Malomed B. A. and
KivsharY. S. , Phys. Rev. A , (2011) 012123.[18] Sergi A. and
Zloshchastiev K. G. , Int. J. Mod. Phys.B , (2013) 1350163.[19] Graefe E. M., Korsch H. J. and
Niederle A. E. , Phys. Rev. Lett. , (2008) 150408.[20] Dattoli G., Torre A. and
Mignani R. , Phys. Rev. A , (1990) 1497.[21] Brody D. C. and
Graefe E.-M. , Phys. Rev. Lett. , (2012) 230405.[22] Sivakumar S. , J. Phys. B: At. Mol. Opt. Phys. , (2009)095502.[23] Kim M. S., Son W., Buˇzek V., and
Knight P. L , Phys.Rev. A , (2002) 032323.[24] Xiang-bin W. , Phys. Rev. A , (2002) 024303.[25] Wu J. and
Xie X.-T. , Phys. Rev. A , (2012) 032112.[26] Chong Y. D., Ge L., Cao H., and
Stone A. D. , Phys.Rev. Lett. , (2010) 053901.[27] Longhi S. , Phys. Rev. A , (2010) 031801.[28] Liertzer M., Ge L., Cerjan A., Stone A. D., T¨ureciH. E. and
Rotter S. , Phys. Rev. Lett. , (2012) 173901.[29] Yoo G., Sim H.-S. and
Schomerus H. , Phys. Rev. A , (2011) 063833.(2011) 063833.