Josephson oscillations of chirality and identity in two-dimensional solitons in spin-orbit-coupled condensates
JJosephson oscillations of chirality and identity in two-dimensional solitons inspin-orbit-coupled condensates
Zhaopin Chen , Yongyao Li , and Boris A. Malomed , Department of Physical Electronics, School of Electrical Engineering,Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel School of Physics and Optoelectronic Engineering, Foshan University, Foshan 52800, China Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile
We investigate dynamics of 2D chiral solitons of semi-vortex (SV) and mixed-mode (MM) types inspin-orbit-coupled Bose-Einstein condensates with the Manakov nonlinearity, loaded in a dual-core(double-layer) trap. The system supports two novel manifestations of Josephson phenomenology:one in the form of persistent oscillations between SVs or MMs with opposite chiralities in the twocores, and another one demonstrating robust periodic switching ( identity oscillations ) between SVin one core and MM in the other, provided that the strength of the inter-core coupling exceeds athreshold value. Below the threshold, the system creates composite states, which are asymmetricwith respect to the two cores, or collapses. Robustness of the chirality and identity oscillationsagainst deviations from the Manakov nonlinearity is investigated too. These dynamical regimes arepossible only in the nonlinear system. In the linear one, exact stationary and dynamical solutionsfor SVs and MMs of the Bessel type are found. They sustain Josephson self-oscillations in differentmodes, with no interconversion between them.
I. INTRODUCTION
Josephson oscillations, induced by tunneling of wavefunctions between weakly coupled cores, is a ubiqui-tous effect in macroscopic quantum systems [1]. It hasbeen predicted and observed in superconductors sepa-rated by a thin insulating layer [2]-[12], superfluid He[13], atomic Bose-Einstein condensates (BECs) [14]-[21],optical couplers [22]-[25], and exciton-polariton waveg-uides [26]. Among other applications, the Josephson ef-fect in superconductors may be used for design of qubits[27, 28]. Josephson oscillations of angular momentumbetween annular BECs was investigated too [29]-[38]. Inthese contexts, chirality of the wave functions may playan important role [29], [39]-[44].In its linear form, the Josephson effect only gives riseto plasma waves in superconducting junctions. The mostsignificant modes in the junctions are topological solitons( fluxons ) supported by the nonlinear Josephson relationbetween the tunneling current and phase difference acrossthe junction. Both classical [7]-[10] and quantum [11]fluxons, as well as superfluxons , i.e., kink excitations influxon chains [12], were studied theoretically and exper-imentally.New possibilities for realization of dynamical effectsin BEC are offered by binary (pseudo-spinor) conden-sates with spin-orbit coupling (SOC) between their com-ponents [45]-[48]. In particular, SOC affects the Joseph-son tunneling between annular condensates [49]. Stable2D solitons, which include vortex components, were pre-dicted in the SOC system of the Rashba type [50] withattractive nonlinear interactions [53]. In terms of op-tics, these interactions are represented by self- and cross-phase-modulation (SPM and XPM) terms in the underly-ing system of coupled Gross-Pitaevskii equations. Whilein the absence of SOC, attractive cubic terms in 2D equa-tions always lead to collapse [52] (i.e., the system does not have a ground state), the interplay of the attractive SPMand XPM terms with SOC gives rise to ground states, inthe form of solitons of the semi-vortex (SV, alias half-vortex [54]) or mixed-mode (MM) types, provided that,respectively, the SPM nonlinearity is stronger or weakerthan its XPM counterpart. SVs are composed of a vortexin one component and zero-vorticity soliton in the other,while MMs combine vortical and zero-vorticity terms inboth components. In the experiment, such states can becreated by using helical laser beams that resonantly cou-ple to one component only, and thus transfer the angularmomentum onto that component, without exciting theother one [55–57]. Two or several solitons in the layers’plane can be created too, by means of an initially appliedcellular in-plane potential, which cuts the condensate intolateral fragments, and is lifted afterwards [58].If the SV soliton is the ground state, the MM is an un-stable excited one, and vice versa. Similar results were re-ported [59] for more general SOC systems, of the Rashba-Dresselhaus [60] type. Two-dimensional SV solitons ex-ist in the form of two different chiral isomers ( right- andleft-handed ones), with vorticity sets in their componentsbeing, respectively, ( S + , S − ) = (0 ,
1) or ( − ,
0) (mirrorimages of each other). Similarly, there exist two differentchiral forms of MMs, which are introduced below in Eq.(22).In the experimental realization of SOC in binary BEC,two components are represented by different hyperfinestates of the same atom, with nearly equal strengths ofthe SPM and XPM interactions, suggesting one to con-sider the
Manakov nonlinearity [61], with equal SPM andXPM coefficients. In such a case, the system is invariantwith respect to rotation of the pseudo-spinor wave func-tion in the plane of its two components. The 2D systemcombining SOC and the Manakov nonlinearity gives riseto an additional soliton family, which embeds the SV andMM solitons into a continuous set of intermediate states a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l [53]. The family is a degenerate one, in the sense that allsolitons with a fixed total norm share common values ofthe energy and chemical potential. The family is dynam-ically stable against small perturbations, but structurallyunstable, as a deviation from the SPM = XPM conditionbreaks the intermediate states, keeping only the SV andMM solitons as robust modes.The subject of this work is a junction formed by apair of Josephson-coupled 2D layers ( cores ), each onecarrying spin-orbit-coupled BEC. In the experiment, atwo-layer setup can be realized by loading the conden-sate in two adjacent valleys of a deep 1D optical lattice,which illuminates the setting in the perpendicular direc-tion (see, e.g., Ref. [62]). In a general form, a dual-core2D BEC system was introduced in Ref. [63]. In thatwork, 2D solitons, built of components in the two layers,were stabilized by a spatially-periodic in-plane potential,while SOC was not considered. The problem addressedin Ref. [63] was spontaneous breaking of the inter-coresymmetry in the solitons, but not Josephson oscillations.Here, we consider oscillations between Josephson-coupled2D SOC condensates, of both SV and MM types, in thedual-core system. In its linear version, Bessel-shaped ex-act solutions are found in an analytical form, for station-ary states and Josephson oscillatory ones alike. Thesesolutions demonstrate solely intrinsic oscillations in thetwo components of the condensate, which do not mixstates of the SV and MM types, nor different isomers ineach type. In the full nonlinear system, solutions for self-trapped states are obtained in a numerical form. Theyreadily demonstrate robust periodic oscillations betweencomponents of the 2D solitons with opposite chiralities inthe two layers. Furthermore, Josephson-type identity os-cillations , i.e., periodic mutual interconversion betweensolitons’ components of the SV and MM types in the twolayers, are reported too.The rest of the paper is structured as follows. Themodel is formulated in Section II, in which basic types ofthe considered states are introduced too, including newsolutions for the Bessel-shaped SV and MM modes inthe linear system. Results for the Josephson oscillationsare collected in Section III. It includes exact solutionsobtained in the linear system, and a summary of the sys-tematic numerical analysis of Josephson oscillations in2D solitons of different types in the full nonlinear system.The numerical results make it possible to identify robustregimes of oscillations of the chirality (right (cid:29) left) andidentity (SV (cid:29) MM) between the coupled layers. Thepaper is summarized in Section IV, where we also giveestimates of the predicted effects in physical units, anddiscuss directions for the further work; one of them maybe the use of beyond-mean-field effects [51] for the stabi-lization of the 2D system against the critical collapse [52],when its norm exceeds the respective threshold value.
II. THE MODEL: COUPLEDGROSS-PITAEVSKII EQUATIONSA. The single-layer spin-orbit-coupled (SOC)system
For the single effectively two-dimensional layer, thesystem of coupled Gross-Pitaevskii equations for compo-nents φ ± of the pseudo-spinor wave function, coupled bythe spin-orbit interaction of the Rashba type, is writtenas [53, 59] i∂ t φ + = − (cid:20) ∇ + (cid:0) | φ + | + γ | φ − | (cid:1)(cid:21) φ + + ( ∂ x − i∂ y ) φ − , (1) i∂ t φ − = − (cid:20) ∇ + (cid:0) | φ − | + γ | φ + | (cid:1)(cid:21) φ − − ( ∂ x + i∂ y ) φ + , (2)where t and x , y are scaled time and coordinates, theSPM and SOC coefficients (the latter one is the coeffi-cient in front of the first-order spatial derivatives) are setto be 1 by scaling, and γ is the relative XPM strength.We will chiefly address the Manakov nonlinearity, with γ = 1, which is close to the experimentally relevant sit-uation [45]-[48]; effects of deviation from the Manakov’scase are considered too. Equations (1) and (2) do not in-clude an external trapping potential, which is necessarilypresent in the experiment, but its effect on stable solitonsis negligible [53].Stationary self-trapped states of the pseudo-spinorcondensate (solitons) with chemical potential µ < φ ± ( x, y, t ) = Φ ± ( x, y ) e − iµt , (3)with the 2D norm N φ = N + + N − ≡ (cid:90) (cid:90) (cid:2) | Φ + ( x, y ) | + | Φ − ( x, y ) | (cid:3) dxdy. (4)Two isomers of solitons of the SV type, with right-and left-handed chiralities, are defined, respectively, byvorticity sets ( S + , S − ) = (0 , +1) and (cid:0) ¯ S + , ¯ S − (cid:1) = ( − , φ (0)+ = A exp (cid:0) − α r (cid:1) , φ (0) − = A r exp (cid:0) iθ − α r (cid:1) , (5) φ (0)+ = − A r exp (cid:0) − iθ − α r (cid:1) , φ (0) − = A exp (cid:0) − α r (cid:1) , (6)which are written in polar coordinates ( r, θ ), with real A , and α , >
0. These inputs are natural, as Eqs. (1)and (2) are fully compatible with the substitution of SV ans¨atze with opposite chiralities [53], { φ + , φ − } SV = e − iµt (cid:8) f ( r ) , e iθ g ( r ) (cid:9) , (7) (cid:8) φ + , φ − (cid:9) SV = e − iµt (cid:8) − e − iθ g ( r ) , f ( r ) (cid:9) , (8) (a)(b) (c) (d)(e)(f) b c d e f x x x FIG. 1. (Color online) (a) Angular momenta per particle( L z = L φ or L ± , as indicated in the panel), which are definedby Eqs. (25) and (26), are shown as functions of norm ratio R φ [see Eq. (24)], for the continuous family of stable SOCsolitons, numerically generated by input (20), with fixed norm N φ = 4, in the single-core system. Linear dependence L φ ( R φ )follows Eq. (27). The energy and chemical potential of allsolutions belonging to the degenerate family are E φ = − . µ = − .
87. Cross-sections of two components of the2D solitons, marked in (a), are displayed for the followingvalues of integral quantities (24) and (25), (26): R φ = − . L φ = − . R φ = − . L φ = − .
24 (c); R φ = 0, L φ = 0(d); R φ = 0 . L φ = 0 .
22 (e); R φ = 0 . L φ = 0 . where radial form-factors f ( r ) , g ( r ) obey equations (cid:20) µ + 12 (cid:18) d dr + 1 r ddr (cid:19) + (cid:0) f + g (cid:1)(cid:21) f − (cid:18) r g + dgdr (cid:19) = 0 , (9) (cid:20) µ + 12 (cid:18) d dr + 1 r ddr (cid:19) + (cid:0) f + g (cid:1)(cid:21) g − (cid:18) r g − dfdr (cid:19) = 0 . (10)More general soliton modes are introduced below. B. Exact Bessel modes of the linearizedsingle-layer system
Equations (9) and (10) give rise to solitons if µ belongsto the respective semi-infinite bandgap , µ < − / κ = 0]. At r → ∞ , the linearizationof Eqs. (9) and (10) readily predicts that exponentiallydecaying tails of solitons’ form-factors f ( g ) and g ( r ) arebuilt as combinations of terms( f, g ) tail ∼ r − / exp (cid:16) − (cid:112) − (2 µ + 1) r (cid:17) { cos r, sin r } . (11)Furthermore, the linearized equations make it possibleto find exact solutions for weakly localized Bessel-shapedmodes (with a diverging total norm) in the propagationband (at µ > − / µ >
0, thesolutions of the SV type are f lin ( r ) = A (0) J ( r/ρ ) , g lin ( r ) = A (0) sJ ( r/ρ ) (12) with arbitrary amplitude A (0) , Bessel functions J , , signfactor s = ± φ ± , ψ ± ), and ρ = ρ s ≡ (cid:16)(cid:112) µ + s (cid:17) / (2 µ ) . (13)In the remaining part of the propagation band, − / <µ <
0, solution (12) is relevant only for s = −
1, whileexpression (13) is replaced by ρ = ρ ± ≡ (cid:16) ± (cid:112) µ (cid:17) / ( − µ ) , (14)with another independent sign ± .At the edge of the semi-infinite gap, µ = − /
2, Eq.(14) gives a single solution with ρ = 1, f (1)lin (cid:18) r ; µ = − (cid:19) = A (0) J ( r ) ,g (1)lin (cid:18) r ; µ = − (cid:19) = − A (0) J ( r ) , (15)instead of two, produced inside the gap at µ > − / f (2)lin (cid:18) r ; µ = − (cid:19) ≡ lim µ +1 / → +0 f lin ( r ; ρ + ) − f lin ( r ; ρ − )2 √ µ = − A (0) rJ ( r ) ,g (2)lin (cid:18) r ; µ = − (cid:19) ≡ lim µ +1 / → +0 g lin ( r ; ρ + ) − g lin ( r ; ρ − )2 √ µ = A (0) [ rJ ( r ) − J ( r )] . (16)Exact linear SV states with the opposite chirality aregenerated from expressions (12)-(16) as per Eq. (8).Next, MMs states, as exact solutions of the linear SOCsystem, are constructed from the SVs pursuant to Eqs.(22) and (23). It is also relevant to mention that, forthe Rashba SOC replaced by the Dresselhaus form [60],the exact linear solutions for the SV and MM modes areessentially the same as found here for the Rashba system.In addition to solution (12), which represents the basicSV state in the linear system, it is possible to constructexact solutions for excited states, obtained by injection ofextra integer vorticity ∆ S = 1 , , ... in both componentsof the SV state. Such solutions are obtained replacingEqs. (12) and (15), (16), severally, by f lin ( r ) = A (0) J ∆ S ( r/ρ ) , g lin ( r ) = A (0) sJ S ( r/ρ ) , (17) f (1)lin (cid:18) r ; µ = − (cid:19) = A (0) J ∆ S ( r ) ,g (1)lin (cid:18) r ; µ = − (cid:19) = − A (0) J S ( r ) , (18) f (2)lin (cid:18) r ; µ = − (cid:19) = A (0) [∆ S · J ∆ S ( r ) − rJ S ( r )] ,g (2)lin (cid:18) r ; µ = − (cid:19) = A (0) [ rJ S ( r ) − (1 + ∆ S ) J S ( r )] , (19)while Eqs. (13) and (14) keep the same form as above,as well as the possibility to construct the SV with theopposite chirality and MMs, according to Eqs. (8) and(22), (23).It is relevant to mention that similarly defined excitedstates of SV and MM solitons are unstable in the non-linear system, unlike the ground states with ∆ S = 0[53]. On the other hand, the excited-state solitons maybe made stable in a model with self-trapping provided bythe local repulsive nonlinearity growing fast enough fromthe center to periphery [68]. C. Soliton families
The Manakov’s nonlinearity makes it possible to con-struct a general family of solitons, including SVs, MMs,and intermediate states. The creation of such states isinitiated by a combination of inputs (5) and (6) with theopposite chiralities, φ ± = M φ (0) ± + (cid:112) − M · φ (0) ± , (20)where the weight factor takes values − ≤ M ≤
1. Be-cause integrals of cross-products, generated by ans¨atze (5) and (6) vanish, (cid:90) (cid:90) (cid:16) φ (0)+ (cid:17) ∗ φ (0) − dxdy = 0 , (21)the norm of input (20), defined as per Eq. (4), does notdepend on M , hence this input gives rise to a family ofsolitons with the same norm.The choice of M = +1 / √ − / √ exact ansatz, which is compatible with theunderlying system of equations (1) and (2) [cf. Eq. (7),that produces the exact ansatz for SV modes]: { φ + , φ − } MM = 1 √ e − iµt (cid:8) f ( r ) − e − iθ g ( r ) , f ( r ) + e iθ g ( r ) (cid:9) ≡ √ (cid:0) { φ + , φ − } SV + (cid:8) φ + , φ − (cid:9) SV (cid:1) . (22)Here f and g are the same real functions which are intro-duced above as solutions to Eqs. (9) and (10). Another chiral MM isomer is produced by a different ansatz, (cid:8) φ + , φ − (cid:9) MM =1 √ e − iµt (cid:8) f ( r ) + e − iθ g ( r ) , − f ( r ) + e iθ g ( r ) (cid:9) ≡ √ (cid:0) { φ + , φ − } SV − (cid:8) φ + , φ − (cid:9) SV (cid:1) , (23)with radial form-factors f and g obeying the same equa-tions (9) and (10). The exact linear relations between theMM and SVs in Eqs. (22) and (23) are admitted by thesymmetry of the Manakov nonlinearity. These relationsentail an equality for the total norms (4): N MM ( µ ) = N SV ( µ ). To the best of our knowledge, the exact MM ans¨atze represented by Eqs. (22)-(23) were not reportedearlier.The solitons are characterized by the norm ratio be-tween the components, R φ = ( N φ + − N φ − ) / ( N φ + + N φ − ) , (24)and by the angular momentum per particle in each com-ponent and in the entire system, L ± = N − ± (cid:90) (cid:90) φ ∗± ˆ L z φ ± dxdy, (25) L φ = N − φ (cid:88) + , − (cid:90) (cid:90) φ ∗± ˆ L z φ ± dxdy, (26)where the angular-momentum operator is ˆ L z = − i ( x∂/∂y − y∂/∂x ) ≡ − i∂/∂θ , and the norms are takenas per Eq. (4). For the family generated by ansatz(20), a typical relation between L and R φ is displayedin Fig. 1(a). The linear L φ ( R φ ) dependence, observedin the figure, is explained by the fact that the inputin the form of Eqs. (5)-(20), leads, taking into regardthe orthogonality of φ (0)+ and φ (0) − [see Eq. (21)], to R φ ( M ) = (cid:0) M − (cid:1) R (SV) φ and L φ ( R φ ) = (cid:16) − R (SV) φ (cid:17) (cid:16) R (SV) φ (cid:17) − R φ , (27)where R (SV) φ is given by Eq. (24) for the SV soliton cor-responding to M = 1 in Eq. (20)) [in Fig. 1(a), a nu-merically computed value is R (SV) φ = 0 . R φ and L φ vanish for solitons of the MM type,due to the overall symmetry between the components inthis state. Terminal points of the curves in Fig. 1(a)correspond to | M | = 1 or M = 0 in Eq. (20), i.e., SVsolitons, in which either L + = ± L − = 0, or viceversa. Numerical data yields L φ = ± . < | M | < − . < L φ < .
4. Examples of solitons belonging to thefamily are presented in Figs. 1(b-f). In particular, SVsare shown in panels (b) and (f), while (d) displays anMM. The stability of the entire family was establishedin direct simulations. The solitons of all types exist inranges 0 < − µ < ∞ and 0 < N φ < N coll ≈ .
85, where N coll is the well-known threshold for the onset of thecritical collapse, predicted by the single Gross-Pitaevskiiequation [52]. In fact, the stability of the 2D spin-orbit-coupled solitons is predicated on the fact that their totalnorm takes values N < N coll , thus securing them againstthe onset of the collapse [53].
D. The Josephson-coupled dual-core system
The pair of Josephson-coupled layers, each carryingpseudo-spinor BEC, are modeled by the system of equa-tions for four components of wave functions, φ ± in onelayer and ψ ± in the other: i∂ t φ + = − (cid:20) ∇ + (cid:0) | φ + | + γ | φ − | (cid:1)(cid:21) φ + + ( ∂ x − i∂ y ) φ − − κψ + , (28) i∂ t φ − = − (cid:20) ∇ + (cid:0) | φ − | + γ | φ + | (cid:1)(cid:21) φ − − ( ∂ x + i∂ y ) φ + − κψ − , (29) i∂ t ψ + = − (cid:20) ∇ + (cid:0) | ψ + | + γ | ψ − | (cid:1)(cid:21) ψ + + ( ∂ x − i∂ y ) ψ − − κφ + , (30) i∂ t ψ − = − (cid:20) ∇ + (cid:0) | ψ − | + γ | ψ + | (cid:1)(cid:21) ψ − − ( ∂ x + i∂ y ) ψ + − κφ − , (31)cf. Eqs. (1) and (2), where κ is the Josephson-couplingstrength, which is defined to be positive. Note that theJosephson interaction does not linearly couple compo-nents φ ± to ψ ∓ , as ones with opposite subscripts rep-resent two distinct atomic states, and the tunneling be-tween the parallel layers does not include a mechanismwhich would lead to mutual transformation of differentatomic states.The system of Eqs. (28)-(31) conserves the total norm, N φ + N ψ , where N ψ is a counterpart of the φ norm forthe ψ -layer, see Eq. (4). Other dynamical invariants ofthe system are the total energy, E = (cid:90) (cid:90) (cid:88) χ = φ,ψ (cid:20) (cid:16) |∇ χ + | + |∇ χ − | − | χ + | − | χ − | (cid:17) − γ | χ + χ − | + (cid:0)(cid:0) χ ∗ + ∂ x χ − − iχ ∗ + ∂ y χ − (cid:1) + c . c . (cid:1)(cid:105) − κ (cid:0)(cid:0) φ ∗ + ψ + + φ ∗− ψ − (cid:1) + c . c . (cid:1)(cid:9) dxdy, (32)and the total angular momentum, N φ L φ + N ψ L ψ , cf. Eq.(26). III. ANALYTICAL AND NUMERICALRESULTS FOR JOSEPHSON OSCILLATIONSA. Exact solutions for Josephson oscillations in thelinear system
Linearization of Eqs. (28)-(31) for excitations( φ ± , ψ ± ) ∼ exp ( iq x x + iq y y − iµt ) yields four branchesof the dispersion relation: µ = 12 (cid:0) q x + q y (cid:1) + σ (cid:113) q x + q y − σ κ, (33)with two mutually independent sign parameters σ , = ± φ ± and ψ ± ).Solitons may exist in the semi-infinite gap, readily deter-mined as an interval of values of µ which is not coveredby Eq. (33) with −∞ < q x,y < + ∞ and all choices of σ , : µ sol < − (1 / κ ) (34)(remind we fix κ >
0, by definition). In particu-lar, this gap is obtained from the above-mentioned one, µ sol < − /
2, for the single-layer system, if one considerssymmetric solutions of Eqs. (28)-(31), with ψ ± = φ ± .On the other hand, for antisymmetric solutions, with ψ ± = − φ ± , the single-layer gap is replaced by an ex-panded one, µ sol < − (1 / − κ ). In the additional inter-val which does not belong to gap (34), viz ., − (1 / κ ) <µ sol < − (1 / − κ ), antisymmetric states may exist as embedded solitons [69], but they are unstable.Further, the linearized system gives rise to exact solu-tions for Josephson-oscillation states,( φ ± ( x, y, t )) two − layer = cos ( κt ) · ( φ ± ( x, y, t )) single − layer , ( ψ ± ( x, y, t )) two − layer = i sin ( κt ) · ( φ ± ( x, y, t )) single − layer , (35)where ( φ ± ( x, y, t )) single − layer are the component of anysolution of the linearized version of Eqs. (1) and (2). Infact, explicit single-layer solutions which may be substi-tuted in Eq. (35) are those given above by Eqs. (12)-(19). Note that the Josephson oscillations do not changethe chemical potential in Eq. (35), which is borrowedfrom the respective single-layered solutions, in the formof exp ( − iµt ). B. Josephson oscillations in 2D solitons
1. Chirality and identity oscillations
In the framework of the full nonlinear system, wefirst address the evolution initiated by an input com-posed of a stable single-layer SV soliton, with vorticities( S + , S − ) = (0 , S + , S − ) = ( − ,
0) in the other, which arecreated, at t = 0, as per ans¨atze presented in Eqs. (7) | ϕ + | | ϕ − | | ψ + | | ψ − | (a) (b) (c) (d) y x x x xt t t t (m) ty x x x x (e) (f) (g) (h) x x x x (i) (j) (k) (l) FIG. 2. (Color online) Numerically simulated Josephson oscil-lations of chirality (with random-noise perturbation at the 1%amplitude level, added for testing stability of the dynamicalregime), initiated by the input composed of a stable single-core SV in one layer, and its left-handed counterpart in theother, see Eqs. (7) and (8). Panels (a,b) and (c,d) display theevolution of the pseudo-spinor components in the first andsecond cores, respectively. The norm of the input in eachlayer is N = 4 (hence the total norm in the two-layer systemis 8), and the Josephson coupling constant is κ = 0 .
5. Panel(e)-(h) and (i)-(l) exhibit 2D density distributions of the com-ponents at, respectively, t = 0 and t = 10. (m) The respectiveoscillations of angular momenta per particle in both layers , L φ and L ψ , see Eq. (26). and (8). The respective stationary single-layer solutionswere produced by means of the accelerated imaginary-time-evolution method [64], and their stability was thentested by simulations of Eqs. (1) and (2) in real time.Simulations of Eqs. (28)-(31) with such an input pro-duce a robust regime of periodic transmutations dis-played in Fig. 2(a-d). Each quarter period, SVs in bothcores switch into their chiral counterparts, returning tothe initial state, but with the opposite sign, each half pe-riod. Thus, chiralities of the two SVs periodically flip, re-maining opposite in the two cores, while the correspond-ing density patterns oscillate. Distributions of densitiesof the four involved components, displayed at t = 0 inpanels (e-f), and at t = 10 in (i-l) (the latter time isclose to 0 . L φ,ψ = ± . ∓ . | ϕ + | | ϕ − | | ψ + | | ψ − | (a) (b) (c) (d) y x x x xt t t t (e) (f) (g) (h) y x x x xx x (i) (j) (k) (l) y FIG. 3. (Color online) The same as in Fig. 2 but for the inputin the form of MMs with opposite chiralities in the two cores,and equal norms in them, N = 4. The angular momentumper particle remains equal to zero in each core, due to thesymmetry between the components of the pseudo-spinor wavefunction in the MM states. In addition to the Josephsonoscillations of the MM chirality, the figure also demonstratesthe spatiotemporal helicity in the oscillating soliton, see thetext. Panel (e)-(h) and (i)-(l) exhibit 2D density distributionsof the components at, respectively, t = 0 and t = 10. solely oscillations that do not mix different chiralities.Indeed, the chirality flipping implies generation of newangular harmonics, which is not possible without nonlin-earity.Next, we consider Josephson oscillations between right-and left-handed chiral isomers of the MM type, which arecreated, at t = 0, in the two layers, as per ans¨atze (22)and (23). A typical numerical solution, displayed in Fig.3, features stable periodic switching between the MM iso-mers in each layer. It also demonstrates spatiotemporalhelicity , i.e., clockwise rotation of the density distribu-tions (unlike nonrotating SV patterns in Fig. 2). Theinput with the opposite relative sign between the MMisomers in two layers gives rise to a similar stable dy-namical state, but with rotation of the density profilesin the opposite, counter-clockwise, direction. Again, itis relevant to stress that flipping of the MM chirality inthe course of the Josephson oscillations is only possiblein the nonlinear system.Recall that the single-core SOC system with the Man-akov nonlinearity maintains the family of 2D solitonswhich are intermediate between the SV and MM states.Such states correspond to | M | (cid:54) = 0 , √ , | R φ | (cid:54) = 0 . L φ different from ± . κ the frequency of the oscillations androtation is exactly equal to κ , cf. the exact solution ofthe linear system given by Eqs. (28)-(31). Although, assaid above, the chirality oscillations represent a nonlin-ear effect, the Josephson frequency is not affected by thenonlinearity, due to its “isotopic” invariance with respectto the rotation of the pseudo-spinor wave function in theplane of its two components.The next step is to consider Josephson oscillations be-tween inputs corresponding to SV in one core and MM,with the same norm, in the other. In this case, the iden-tity of the 2D solitons periodically switches between theSV and MM types in each core [see Figs. 4(a-d) and (m)],therefore this stable dynamical state may be naturallycategorized as identity oscillations , coupled to the clock-wise helicity rotation, cf. Figs. 2 and 3. This regime,which is again possible solely in the nonlinear system, cf.exact solution (35) obtained in the linear limit (whichactually corresponds to the system strongly dominatedby the inter-core coupling), takes place if the strengthof the Josephson interaction exceeds a certain thresholdvalue, κ > κ thr . The dependence of the threshold on thetotal norm in one layer, N , is shown in Fig. 4(n); recallthat the establishment of the robust chirality oscillationsin the dual-core SV and MM solitons, considered above,does not require κ to be larger than any finite thresholdvalue.To explain the latter feature, we note that, as it fol-lows from Eqs. (28)-(31) and (32), the interplay of thekinetic energy, nonlinearity, SOC, and Josephson inter-action between the cores predicts scaling κ thr ∼ N atsmall N . On the other hand, the above-mentioned onsetof the collapse in the single-core system at N = N coll implies divergence of the system’s sensitivity to pertur-bations at N → N coll , hence κ thr also diverges in thislimit. These qualitative arguments are corroborated bythe numerically found dependence κ thr ( N ) in Fig. 4(n).At κ < κ thr , the inter-core coupling is too weak to pre-vent the occurrence of spontaneous symmetry breakingbetween the cores. Accordingly, additional numerical re-sults demonstrate, for values of N which are relativelysmall in comparison with N coll (namely, in an interval of N < N crit < N coll ) the existence of asymmetric compos-ite solitons, with different amplitudes in the two cores.In the adjacent interval of N crit < N < N coll , the col-lapse takes place at κ < κ thr . On the other hand, ifthe Josephson coupling is strong enough, with κ > κ thr ,it forestalls the onset of the collapse at all values of N smaller than N coll , by keeping the total norm equally dis-tributed between the two cores (for instance, the identity-oscillation dynamical regime, displayed in Fig. 4, is sta-ble at N = 4 < N coll ≈ . | ϕ + | | ϕ − | | ψ + | | ψ − | (a) (b) (c) (d) y x x x xt t t t (m) t (n) N (e) (f) (g) (h) x x x xyy (i) (j) (k) (l) FIG. 4. (Color online) Panels (a-m): the same as in Fig.2, but for identity oscillations , initiated by the input in theform of SV and MM solitons with equal norms, N = 4, inthe coupled cores. (n) The minimum (threshold) value of theinter-core coupling κ , which is necessary for the stability ofthe identity oscillations, vs. the norm of the 2D soliton ineach core, N .
2. Structural stability of the oscillation regimes
The fact that the Manakov’s form of the nonlinearity isapproximate in experimental settings makes it necessaryto test effects of deviation from it [ γ (cid:54) = 1 in Eqs. (1) and(2)], i.e., structural stability of the chirality- and identity-oscillation dynamical regimes. The result, illustrated byFigs. 5(a,b) for γ = 1 ± .
05, is that the chirality oscil-lations of the SV soliton persist in the course of dozensof periods (which is sufficient for the experimental ob-servation), and then gradually detune from the regularregime.Additional simulations (not shown here in detail) sug-gest similar conclusions concerning the effect of the devi-ation from the Manakov’s case on chirality and identityoscillations of MM solitons. As concerns the intermediatestates, defined above as per Eq. (36), the deviation fromthe Manakov’s case for them is more disruptive, as, evenin the single-core system, taking γ (cid:54) = 1 causes evolutionof such inputs towards SVs or MMs, for γ < γ > FIG. 5. (Color online) Simulations similar to those displayedfor the SV soliton in Fig. 2, but with γ = 1 .
05 in (a) and 0 . (cid:39)
40 and 20 oscillation periods, respectively. (c) The chiralityoscillations of the SV solitons, with different initial norms inthe two cores, N ψ = 0 . N φ = 3 .
8. The oscillations stayundisturbed in the course of (cid:39)
10 periods. ∼
10 periods.
IV. CONCLUSION
The objective of this work is to introduce and theo-retically elaborate a setting for the study of Josephsonoscillations of the chirality and species type ( identity )of 2D solitons in the double-layer trap, loaded with aspin-orbit-coupled condensate, including self- and cross-attractive nonlinearity (SPM and XPM) in its compo-nents. It is known that, in the Manakov’s case of SPM =XPM, the single-layer system supports a broad family ofcomposite 2D solitons, which includes dynamically andstructurally stable ones of the SV (semi-vortex) and MM(mixed-mode) types, each existing in two chiral isomers,left- and right-handed ones. In addition to that, the Man-akov’s nonlinearity admits the existence of the interme-diate family of solitons, between the SV and MM states,which are dynamically stable but structurally unstableagainst deviation from the condition SPM = XPM. Allthese states exist with the norm in each component tak-ing values smaller than the well-known threshold valuenecessary for the onset of the critical collapse,
N < N coll ,We consider Josephson oscillations initiated by inputsin the form of SV, MM, or intermediate-type solitonswith equal norms and opposite chiralities, originally cre-ated in the coupled cores (layers), or SV in one coreand MM in the other. In the former case, the systemfeatures robust chirality oscillations at all values of theinter-core coupling, κ . In the latter case, persistent peri-odic switching between the SV and MM species ( identityoscillations ) occur at κ > κ thr , where κ thr increases withthe growth of the condensate’s norm, while at κ < κ thr the system develops states with unequal amplitudes inthe cores at relatively small values of N , or collapses at larger N . In all cases, the oscillation frequency is equalto κ . Deviation of the nonlinearity from the Manakov’sform, as well as a difference in the norms of the inputsin the two cores, leads to slowly developing dephasingof the Josephson oscillations. These regimes are non-linear ones, while the exact solutions reported here forJosephson oscillations in the linear system admit solelyself-oscillations of SVs and MMs modes with the Besselspatial shape, without switching between different chi-ralities or mode types. In fact, the exact Bessel-shapedlinear modes are new stationary solutions for the single-layer system too.With typical values of the relevant physical parame-ters (see, e.g., Ref. [70]), such as the scattering lengthof the attractive inter-atomic interactions ∼ − . ∼
300 Hz, and the gap ofwidth ∼ µ m separating two layers, the expected num-ber of atoms in the stable solitons, which are capableto feature the Josephson oscillations, is in the range of N atom ∼ − , and the oscillation frequency ∼ N atom , thescaled norm will exceed value N coll , which will triggerthe onset of the collapse. On the other hand, it was re-cently predicted [51, 71], and demonstrated experimen-tally in various setups [72]-[76], that 3D and quasi-2Dself-trapped matter-wave states may be stabilized, in theform of soliton-like quantum droplets , by beyond-mean-field (Lee-Huang-Yang) effects of quantum fluctuationsaround the mean-field states. In the present context, thispossibility suggests to consider the dual-core SOC systemin the case of N > N coll , that may be stabilized by theLee-Huang-Yang effect. In this connection, it is relevantto mention that the stabilization of 2D spin-orbit-coupledsolitons by this effect in the single-layer system was ana-lyzed in Ref. [77].As another extension of the present work, it may beinteresting to consider Josephson oscillations in 2D spa-tiotemporal optical solitons carried by dual-core planarwaveguides and stabilized by a SOC-emulating mecha-nism, viz ., temporal dispersion of the inter-core couplingcoefficient [78, 79]. Another potentially interesting pointis the study of intrinsic vibrations of the composite soli-tons, initiated by a separation of their components in thetwo layers.
ACKNOWLEDGMENTS
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