Modulational instability and soliton generation in chiral Bose-Einstein condensates with zero-energy nonlinearity
MModulational instability and soliton generation in chiral Bose-Einstein condensateswith zero-energy nonlinearity
Ishfaq Ahmad Bhat, S. Sivaprakasam, and Boris A. Malomed
2, 3 Department of Physics, Pondicherry University, Pondicherry 605014, India Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile
By means of analytical and numerical methods, we address the modulational instability (MI) inchiral condensates governed by the Gross-Pitaevskii equation including the current nonlinearity. Theanalysis shows that this nonlinearity partly suppresses off the MI driven by the cubic self-focusing,although the current nonlinearity is not represented in the system’s energy (although it modifies themomentum), hence it may be considered as zero-energy nonlinearity . Direct simulations demonstrategeneration of trains of stochastically interacting chiral solitons by MI. In the ring-shaped setup, theMI creates a single traveling solitary wave. The sign of the current nonlinearity determines thedirection of propagation of the emerging solitons.
PACS numbers: 42.65.Sf, 03.75.Kk, 03.75.Lm
I. INTRODUCTION
Since the creation of the Bose-Einstein condensates(BECs) in 1995, they became versatile testbeds for thestudy of various physical phenomena in quantum statesof matter [1, 2]. One of striking properties of the conden-sates is their ability to emulate physics of charged parti-cles under the action of magnetic fields [3–5] through en-gineering of synthetic gauge fields in such charge-neutralultracold atomic gases.Synthetic gauge fields can be introduced by means ofrapid rotation of the condensate [6, 7], optical couplingbetween internal states of atoms [8–10], laser-assistedtunneling [11, 12], and Floquet engineering [13]. Thenature of these gauge fields is essentially static, as param-eters of field-inducing laser beams, including their inten-sity and phase gradients, cannot reproduce the time de-pendence of the Maxwell’s equations. Dynamical gaugefields, affected by nonlinear feedback from matter, towhich the fields are coupled, are required to emulate afull time-dependent field theory. A number of schemes[14–19] have been proposed for creating dynamical gaugefields with ultra-cold atoms, including the ones whichgive rise to density-dependent gauge potentials and cur-rent nonlinearities [20, 21]. Following these proposals,such dynamical gauge fields have been experimentally re-alized recently [22–24].The chiral condensates, interacting with gauge poten-tials, enrich physics of atomic and nonlinear systems,thus drawing much interest in experimental and theo-retical studies. The presence of density-dependent gaugefields makes it possible to create anyonic structures [25]and chiral solitons [20, 26], which opens new perspec-tives for quantum simulations. The chiral solitons moveunidirectionally, the selected direction being determinedby the current nonlinearity. The chiral solitons wereconsidered for emulation of quantum time crystals (firstenvisaged by Wilczek in 2012 [27]) in circumferentially confined condensates with density-dependent gauge po-tentials [28–30]. However, this proposal was disputedsince, in the thermodynamic limit of the latter setting,the lowest-energy ground state is realized by a static soli-ton, hence a genuine time crystal is not feasible [31, 32].The chiral condensates have also been studied in thepresence of persistent currents [20] and collective excita-tions [33] in them. The evolution of the excitations isaffected by the current nonlinearity, leading to irregulardynamics in a strongly nonlinear regime and related vio-lation of the Kohn’s theorem. The current nonlinearitiesin chiral condensates, in addition to being responsiblefor nonintegrable collision dynamics in soliton pairs [34],help to maintain rich dynamics in trapped condensates[35].In this paper, we study effects of the current nonlinear-ity on modulational instability (MI) (alias Benjamin-Feirinstability [36]) of the chiral BEC. It is well known thatMI is a natural precursor to the formation of solitoniccoherent structures, as a result of the interplay betweenintrinsic self-interaction of the medium and diffractionor dispersion, and has been studied in diverse physicalsettings theoretically [36–38] and experimentally [39–41].The MI strongly depends on the nature of two-body in-teractions in single-component condensates, where it oc-curs only in the case of self-attraction [42, 43]. However,the intercomponent interactions make MI scenarios morediverse in multicomponent condensates. In particular,binary condensates with repulsive interactions are mod-ulationally unstable under the condition of immiscibil-ity [44–46]. Static gauge fields which impose spin-orbitcoupling make condensates still more vulnerable to MI[47, 48]. Further, helicoidal gauge potentials break theMI symmetry and thus strongly modify patterns of in-stability regions and gain in the underlying parameterspace [49]. In this connection, we investigate the effect ofdensity-dependent gauge potentials on the MI and sub-sequent generation of solitons. Additionally, we considerthe solitons in a circumferentially confined condensate in a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b a moving reference frame. A specific peculiarity of thesystem is that the gauge potential is represented in therespective Gross-Pitaevskii (GP) equation by a currentnonlinearity, which, however, is not represented by anyterm in the system’s energy. Thus, this term may beidentified as an example of zero-energy nonlinearity . Tothe best of our knowledge, effects of such terms on theMI were not studied before.The subsequent material is organized as follows. Sec.II introduces the model and the corresponding GP equa-tion including the density-dependent gauge potential. InSec. III the dispersion relation produced by the linearMI analysis is derived and discussed. Sec. IV reportsresults of numerical simulations of the system under theconsideration. The work is concluded by Sec. V. II. THE MODEL
We consider a condensate of two-level atoms with theRabi coupling imposed by an incident laser beam, as de-scribed by the following mean-field Hamiltonian [5, 20]:ˆ H = (cid:18) ˆp m + V ( r ) (cid:19) ˇI + (cid:18) g | Ψ | + g | Ψ | (cid:126) Ω r e − iφ ( r ) (cid:126) Ω r e iφ ( r ) g | Ψ | + g | Ψ | (cid:19) (1)where ˆp is the momentum operator, V ( r ) is the trappingpotential, ˇI is the unity matrix, Ω r is the Rabi-couplingstrength, φ ( r ) is a spatially varying phase of the couplingbeam, and g µν = (cid:0) π (cid:126) /m (cid:1) a µν are the mean-field in-teraction strengths, proportional to respective scatteringlengths, a µν , of collisions between atoms in internal states µ and ν ( µ, ν = 1 , A = A + n g − g r ∇ φ ( r ) (2)where A = − ( (cid:126) / ∇ φ ( r ) represents the single-particlevector potential, and n is the density of the condensate.By projecting the coupled two-level atom onto a singledressed state, one arrives at the following mean-field GPequation governing the dynamics of chiral condensates:[20, 26, 33, 35]: i (cid:126) ∂ Ψ ∂t = (cid:20) ( ˆp − A ) m + W ( r )+ V ( r )+ g | Ψ | + a · J (Ψ , Ψ ∗ ) (cid:21) Ψ(3)where Ψ is the wave function of the dressed state.Further, W ( r ) = | A | / (2 m ), together with the trap-ping potential, V ( r ), defines the scalar potential, a = ( g − g ) / (8Ω r ) ∇ φ ( r ) is the strength of the density-dependent vectorial gauge potential, and g = (1 / g + g + 2 g ) determines the effective interaction in thedressed-state picture. The unconventional nonlinearity inthe chiral condensates manifests itself in Eq. (3) throughthe current, J (Ψ , Ψ ∗ ) = (cid:126) im (cid:20) Ψ (cid:18) ∇ + i (cid:126) A (cid:19) Ψ ∗ − Ψ ∗ (cid:18) ∇ − i (cid:126) A (cid:19) Ψ (cid:21) (4)Defining φ ( x ) = k l x , where k l is the wavenumber of theincident laser beam, following the usual procedure of thedimensional reduction [34], and applying the nonlinearphase transformation,Ψ( x, t ) = ψ ( x, t ) exp (cid:20) − i φ ( x )2 + i a (cid:126) (cid:90) x −∞ dx (cid:48) n ( x (cid:48) , t ) − i W t (cid:126) (cid:21) ,(5)Eq. (3) is cast in the form of the following one-dimensional (1D) GP equation: i (cid:126) ∂∂t ψ = (cid:18) − (cid:126) m ∂ ∂x + V ( x ) + g | ψ | − αJ ( x ) (cid:19) ψ (6)where V ( x ) = (1 / mω x x is the 1D trapping poten-tial with axial trapping frequency ω x , g ≡ g / πa ⊥ is the usual coefficient of the cubic nonlinearity, and a ⊥ = (cid:112) (cid:126) / ( mω ⊥ ) is the harmonic-oscillator length ofthe transverse confining potential. In Eq. (6) α = (cid:0) πa ⊥ (cid:1) − [ k l ( g − g ) / (8Ω r )] is the strength of the un-conventional current nonlinearity, which involves the cur-rent density, J ( x ) = ( i (cid:126) / m ) ( ψ∂ x ψ ∗ − ψ ∗ ∂ x ψ ). Spa-tiotemporal rescaling, t (cid:48) = ω ⊥ t, x (cid:48) = xa ⊥ , ψ (cid:48) = √ a ⊥ ψ, (7)casts Eq. (6) in the normalized form: i∂ t ψ = (cid:18) − ∂ xx + σ | ψ | − ρJ ( x ) (cid:19) ψ. (8)In Eq. (8), the current density is now defined as J ( x ) = ( i/
2) ( ψ∂ x ψ ∗ − ψ ∗ ∂ x ψ ) , (9)and the axial potential, which is V ( x ) in Eq. (6),is dropped, as the following analysis deals with thedynamics in free space. The generalized GP equa-tion (8), with scaled interaction parameters σ =( a + a + 2 a ) / (2 a ⊥ ) and ρ = ( (cid:126) k l / m Ω r ) ( a − a ) /a ⊥ , is used below for the consideration of MI andformation of chiral solitons. Note that solutions of Eq.(8) with σ < µ <
0, are precisely the same asin the case of ρ = 0: ψ sol = e − iµt (cid:112) µ/σ sech (cid:16)(cid:112) − µx (cid:17) . (10)Equations (8) and (9) conserve the integral norm(proportional to the total number of particles), N = (cid:82) + ∞−∞ dx | ψ | , energy, E = − (cid:90) + ∞−∞ dx (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂x (cid:12)(cid:12)(cid:12)(cid:12) + σ | ψ | (cid:33) , (11)and momentum, P = − (cid:90) + ∞−∞ dx (cid:0) J ( x ) + ρ | ψ | (cid:1) (12)[34, 50]. Unlike the underlying chiral GP equation (3),the transformed equation (8) cannot be written in theLagrangian form. It cannot be written in the Hamilto-nian form either, but, nevertheless, energy E , defined byEq. (11), is its dynamical invariant. It is worthy to notethat the energy conservation holds in spite of the fact E does not include any term ∼ ρ (in fact, E is the sameas for the usual non-chiral GP equation), i.e., the cur-rent term may be identified as zero-energy nonlinearity [although it affects expression (12) for the conserved mo-mentum]. The conservation of E can be readily verifiedby the straightforward calculation of dE/dt , substituting ∂ψ/∂t and ∂ψ ∗ /∂t in the integrand by what is producedby Eq. (8). As a result, the terms ∼ ρ amount to ex-pressions in the form of full x -derivatives, hence theircontribution to the integral expression for dE/dt iden-tically vanishes for all localized states, the result being dE/dt = 0.Qualitatively, a term in conservative equations whichdoes not affect the energy may be compared to theLorentz force acting on a charged particle in magneticfield, or the Coriolis force for a body moving on a rotat-ing sphere. However, such an analogy is not an accurateone, because the Lorentz and Coriolis forces, althoughthey are not represented in the particle’s Hamiltonian,appear in the Lagrangian, while, as mentioned above,the term ∼ ρ in Eq. (8) cannot be derived from a La-grangian. In terms of models governed by GP-like equa-tions, somewhat similar effects are produced by the stim-ulated Raman scattering (SRS) in fiber optics [51] andits pseudo-SRS counterpart in the system of interact-ing high- and low-frequency waves (the Zakharov system)[52], as well as by the nonlinear Landau damping in plas-mas [53]. However, in those cases the non-Lagrangianterms conserve only the norm of the wave fields, whilecausing the dissipation of the energy and momentum (inparticular, self-decelaration of solitons [51]), therefore thelatter analogy is not accurate either.On the other hand, the current nonlinearity in Eq. (8)makes the total momentum, as given by Eq. (12), dif-ferent from the standard expression for the non-chiralGP equation. In this connection, it is also relevant tomention that, unlike the usual GP equation, Eq. (8) isnot invariant with respect to the Galilean transform: thesubstitution of˜ x = x − ut, ψ = exp (cid:0) iux − iu t/ (cid:1) ˜ ψ (˜ x, t ) , (13) transforms Eq. (8) not into the equation of the sameform for wave function ˜ ψ (˜ x, t ), written in the referenceframe moving with arbitrary velocity u , but into one with σ replaced by ˜ σ = σ − ρu. (14)Note that taking ρu > σ/ σ > σ < ρu <σ/
2, in the case of σ < u is obtained fromthe quiescent one (10) [34]: ψ sol = exp (cid:0) iux − i (cid:0) µ + u / (cid:1) t (cid:1) (cid:112) µ/ ( σ − ρu ) × sech (cid:16)(cid:112) − µ ( x − ut ) (cid:17) . (15)Obviously, this solution exists for µ < ρu − σ > III. THE LINEAR-STABILITY ANALYSIS (LSA)
We examine the MI via linear instability analysis(LSA) of the spatially uniform (alias continuous-wave,CW) state of the chiral condensate in the absence ofthe external potential, V ( x ) = 0. The respective lin-earized Bogoliubov-de Gennes equations for small per-turbations about the CW wave function were derived inRefs. [33, 34], though the MI spectrum has not been de-rived, and is addressed here. The usual CW solution toEq. (8) is ψ ( x, t ) = √ n exp ( iKx − iµ CW t ) , (16)with arbitrary density, n = | ψ | , real wavenumber K ,and chemical potential µ CW = σn − ρK . It is moreconvenient to develop the MI analysis in the referenceframe moving at velocity u = K , applying the Galileantransformation provided by Eq. (13). It removes term Kx from the CW phase, replacing σ by˜ σ = σ − ρK, (17)as per Eq. (14).We perturb the CW state (16), replacing it by (in themoving reference frame, if K (cid:54) = 0) ψ = ( √ n + δψ ( x, t )) exp ( − iµ CW t ) , (18)which results in the following linearized equation for thesmall perturbation, δψ ( x, t ): i∂ t ( δψ ) = − ∂ xx ( δψ )+ n ˜ σ ( δψ + δψ ∗ )+ inρ ( ∂ x δψ − ∂ x δψ ∗ )(19)where ∗ stands for the complex conjugate, and x is writ-ten instead of ˜ x , to simplify the notation, if the mov-ing reference frame is used. We look for eigenmodes ofthe perturbations in the usual plane-wave form, δψ = a cos( kx − Ω t ) + ib sin( kx − Ω t ), with real wavenumber k and complex eigenfrequency, Ω, assuming that the ac-tual system’s size is much greater than the healing lengthwhich determines a characteristic MI length scale. Thesubstitution of this in Eqs. (19) and (9) results in thefollowing dispersion equation:Ω + 2 nρk Ω − n ˜ σk − k , (20)which yields two branches of the Ω( k ) dependence:Ω ± = − nρk ± (cid:114) k k n ( nρ + ˜ σ ) (21)Equation (21) is the Bogoliubov dispersion relation [55]for the propagation of small perturbations on top of theCW background. The expression on the right-hand sideof Eq. (21) may be positive, negative or complex, de-pending on the signs and magnitudes of the interactionparameters. The CW solutions are stable if Ω ± ( k ) arereal for all real k ; otherwise, the instability gain is definedas ξ ≡ | Im(Ω ± ) | . For ρ = 0 (no current nonlinearity), theabove consideration reproduces the well-known resultsfor BEC with the cubic self-attraction ( σ < < k < (cid:112) n | σ | ,with the maximum MI gain, ξ max = n | σ | , attained at k max = (cid:112) n | σ | [36, 42].The effect of the current nonlinearity, quantified bycoefficient ρ , on the MI can be understood in the frame-work of Eq. (21). It is evident that the MI gain is essen-tially the same for both signs of the current nonlinear-ity, ρ . Further, such chiral condensates may be modula-tionally unstable only for the attractive sign of the two-body interactions, i.e. , ˜ σ <
0. The current-nonlinearity’sstrength, ρ , controls the MI region and gain for a fixedvalue of ˜ σ , as shown in Fig. 1. It is found from Eq. (21)that the system is subject to the MI in the wavenumberrange of 0 < k < (cid:112) n | ˜ σ | − n ρ , provided that ρ < (cid:112) | ˜ σ | /n. (22)For ρ ≥ (cid:112) | ˜ σ | /n , the CW state is modulationally stableeven for attractive two-body interactions, thereby reveal-ing the stabilizing effect of the current nonlinearity in thechiral condensates. Further, substituting definition (17)in Eq. (22), the parameter region in which the MI takesplace for ρK (cid:54) = 0 amounts to the following interval of thecurrent-nonlinearity’s strength: K − (cid:112) K − σn < nρ < K + (cid:112) K − σn, (23) which exists provided that the cubic-nonlinearity coeffi-cient satisfies condition σn < K . (24)In particular, this condition holds for all σ <
0, while K (cid:54) = 0 may impose the MI even in the case when thecubic term in Eq. (8) is self-defocusing, with σ > StableUnstable k = 0.5 1.0 1.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ρ -1-0.8-0.6-0.4-0.200.20.4 σ ~ Figure 1: (Color online) The MI stability diagram, deter-mined by Eq. (21) for n = 10 and different values of k in the(˜ σ, ρ ) plane: as it follows Eq. (21), the CW state is modula-tionally stable for ρ ≥ (cid:0) k − σn (cid:1) / (cid:0) n (cid:1) . Note the symme-try of the curves about ρ = 0. In combination with the Feshbach-resonance tech-niques [56–58], which makes it possible to adjust thevalue of σ , the chiral-interaction strength, ρ, can be usedto tune the MI gain, including suppression of the insta-bility. This possibility can be adequately described interms of the change of the maximum MI gain ξ max at k = k max , following the variation of ρ . It follows fromEq. (21) that, for ρ < (cid:112) | ˜ σ | /n , the largest MI gain, ξ max = n ( | ˜ σ | − nρ ) (25)is attained at k max = ± (cid:112) n ( | ˜ σ | − nρ ) . (26)It is seen from Eqs. (25) and (26) that both ξ max and k max decrease monotonically with the increase of ρ , upto ρ = (cid:112) | ˜ σ | /n , as shown in Fig. 2. The value ξ max vanishes at ρ = | ˜ σ | /n , which is the above-mentioned MIboundary, given by Eq. (22).A plot of the variation of k max with the change of thestrength of the nonlinearity coefficients, ρ and ˜ σ , is pre-sented, in the form of a heatmap, in Fig. 3, as per Eq.(26). This plot also shows that the addition of the currentnonlinearity results in the stabilization of the condensateagainst the modulational perturbations.Next we study the variation of MI gain with respectto the strength of the two-body attraction, ˜ σ , at a fixed Ρ Ξ m ax ; H k m ax L k max Ξ max Figure 2: (Color online) Variation of ξ max and k max as afunction of the current-nonlinearity’s strength, ρ , for n =10 and ˜ σ = − .
1, as per Eqs. (25) and Eq. (26). The MIboundary, ρ = 0 .
1, is given by Eq. (22).Figure 3: (Color online) The perturbation wavenumber, k max ,at which the MI gain attains its maximum, as a function ofthe strengths of the two-body (cubic) attraction, ˜ σ , and thecurrent nonlinearity, ρ , for n = 10. value of the current-nonlinearity’s strength, ρ . As men-tioned above, the system is modulationally unstable ifand only if the two-body interaction is attractive andcondition (22) holds. The maximum MI gain, ξ max , asgiven by Eq. (25), is plotted in Fig. 4. Accordingly, thedependence of ξ max on ˜ σ is linear, with the slope deter-mined by the CW density n . IV. NUMERICAL RESULTS
Proceeding to the numerical analysis, we employed asixth-order Runge-Kutta scheme for simulations of the È ΣŽ È Ξ m ax Ρ Figure 4: (Color online) Variation of ξ max as a function ofthe strength of the two-body interaction, | ˜ σ | , for three fixedvalues of the current-nonlinearity’s strength, ρ and n = 10. GP equation (8), cf. Refs. [59, 60]. The kinetic-energy term was dealt with by dint of the fast Fouriertransform, while the spatial derivative in the current-nonlinearity part we approximated by the fourth-ordercentral-difference formula. In the simulations, we fixedthe timestep, ∆ t = 0 . L = 100,and the spatial mesh size, ∆ x = L/N with N = 2048,unless mentioned otherwise.The initial condition was taken as the CW backgroundwith uniform density, n = 10, and K = 0 [see Eq. (16)],seeded by weak random perturbations. It is clearly seenfrom Fig. 2 that the system is modulationally unstablefor σ = − .
1, provided that ρ < .
1. Figure 5(a) showsthe spatiotemporal evolution of the initially perturbedCW density for parameters σ = − . , ρ = 0 .
08. It showsgeneration of a train of chiral solitons at t ≥
50, prop-agating in the negative x -direction. The unidirectionalpropagation is a signature of the chiral solitons, see Eq.(15). Further, solitons belonging to the train collide in-elastically due to the nonintegrability of the model [34].The direction of motion of the solitons is determined bythe sign of ρ , while the occurrence of the MI is indepen-dent of this sign, as shown in the previous section. Toconfirm this expectation, in Fig. 5(b) we set ρ = − . σ = − .
1. This simulation gives rise to asoliton train, with the same structure, but propagating inthe positive x -direction. The choice of the propagationdirection by the generated solitons can be understoodfrom the k -space density, n ( k x ), shown in Fig. 6, wheremodes with k x < n ( k x ) in the spontaneously excited mode is k x ≈ . k max = 0 .
84 predicted by theLSA. Further, values k max = 0 .
84 and ξ max = 0 . λ max ≡ π/k max ≈ . τ = 2 π/ξ max ≈
17. In turn, λ max determines thenumber of solitons in the train observed in Fig. 5, asn sol (cid:39) L/λ ≈
14, where L = 100 is the above-mentioned Figure 5: (Color online) The evolution of the condensate den-sity due to the development of the MI for the cubic-attractionstrength σ = − . ρ = 0 .
08, and (b) ρ = − . size of the simulation domain.We extend the numerical analysis by varying ρ at fixed σ . Figure 7 shows the temporal evolution of the midpointdensity, n ( x = 0), for ρ = (0 . , . , . ρ increases to-wards value (cid:112) | σ | /n , above which the MI is suppressed,instability-growth time increases, due to the decrease inthe MI gain, in agreement with Fig. 2. Numerical dataproduced by these simulations are summarized in TableI. It is clearly seen that the findings agree with the pre-dictions of the LSA analysis.Lastly, we address the 1D setting corresponding to thering geometry with radius R and periodic boundary con-ditions, the accordingly rescaled form of Eqs. (8) and (9) Figure 6: (Color online) The evolution of the condensatedensity in the k -space for the cubic-attraction and current-nonlinearity’s strengths σ = − . ρ = 0 .
08. The circleat t = 62 indicates the wavenumber, k max ≈ .
81, at whichthe MI-induced mode features the maximum of n ( k x ), see thetext. n ( x = ) t ρ =0.08 ρ =0.09 ρ =0.1 Figure 7: (Color online) The evolution of MI at different val-ues of the current-nonlinearity’s strengths, ρ , and fixed self-attraction coefficient, σ = − .
1, is displayed in terms of themidpoint density, n ( x = 0). For σ <
0, the condensate ismodulationally stable at ρ ≥ (cid:112) | σ | /n , see Eq. (22). being i ∂ϕ ( θ, τ ) ∂τ = (cid:18) − ∂ ∂θ + ˜ g | ϕ | − a J ( θ ) (cid:19) ϕ ( θ, τ ) , (27)with J ( θ ) = ( i/ ϕ∂ θ ϕ ∗ − ϕ ∗ ∂ θ ϕ ) and wavefunction ϕ subject to the normalization condition (cid:82) π dθ | ϕ ( θ, τ ) | = 1, where θ is the angular coordi-nate on the ring. The accordingly scaled parametersare ˜ g = ( g − αu ) (cid:0) mR / (cid:126) (cid:1) and a = αR/ (cid:126) , with Table I: The summary of the MI characteristics producedby the LSA and numerical simulations for the fixed cubic-attraction coefficient, σ = 0 .
1, fixed CW density, n = 10,and different values of the current-nonlinearity’s strength, ρ . Wavenumber k max and the respective wavelength, λ =2 π/k max , corresponding to the largest MI gain, ξ max , are givenby Eq. (26). The MI growth time, τ = 1 /ξ max , is determinedby Eq. (25). The number of solitons in the MI-generatedtrain is accurately approximated by estimate n sol (cid:39) L/λ , seethe text. σ ρ k max λ τ numberofsolitons − . .
05 1 .
22 5 .
13 8 . ∼ .
08 0 .
84 7 .
40 17 . ∼ .
09 0 .
61 10 .
19 33 . ∼ . ∞ ∞ α ≡ q ( g − g ) / (8Ω r R ) determined by integer windingnumber q of the laser beam used to induce the circumfer-ential gauge potentials. In this case, the CW state (18),including term Kθ in the phase, can be constructed too,with integer values of K allowed by the circular boundarycondition.As typical parameter values, we take a = π/ u = − / g = −
4, the respective cubic-term coefficientin Eq. (27) being ˜ g = − .
6. Note that this value satisfiescondition g < − π , which is necessary to the transitionfrom the uniform state in the ring to a soliton-like pat-tern [61]. For this case, Fig. 8 shows the nonlinear evolu-tion of the weak perturbations, initiated by the MI, thatultimately coalesce into a single chiral soliton, which per-forms circular motion on the ring. Note that the systemdoes not converge to an equilibrium state that would beappropriate for the realization of the quantum time crys-tal. In addition to the spatial distribution of the density,displayed in 8(a), panel 8(b) shows that the wavenumbercorresponding to the initially excited mode is | k θ | = 1.It matches well to the analytical estimate given by theEq. (26), even if it was derived for the infinite system,rather than for the circumference. As mentioned above,owing to the ring geometry of the condensate, only inte-ger values are allowed for wavenumbers k . Consequently,the condensate undergoes oscillation in the k − space, asobserved in Fig. 8(b). V. CONCLUSION
We have studied the MI (modulational instability) ofthe chiral condensate by means of LSA (linear stabil-ity analysis) and direct simulations. The chirality is in-troduced by the action of the current nonlinearity. Apeculiar property of this nonlinearity is its zero-energy character, as, being accounted for by term ∼ ρ in theunderlying GP equation (8), it is not represented in therespective energy, given by Eq. (11). Nevertheless, the Figure 8: (Color online) Numerically simulated evolution ofMI in a circumferentially confined condensate with ˜ g = − . a = π and (cid:82) π dθ | ϕ ( θ, τ ) | = 1. The white circle representsthe wave vector corresponding to the initially excited mode. current nonlinearity strongly affects the MI, tending tosuppress it. Direct simulations demonstrate that, as longas the MI is present, it generates a train of stochasticallyinteracting chiral solitons in the extended system, or asingle soliton in the ring-shaped one. The MI gain doesnot depend on the sign of the current nonlinearity, al-though the sign determines the direction of the motionof the generated solitons.An interesting extension of the work may be the con-sideration of GP (Gross-Pitaevskii) equations includingthe current nonlinearity in a combination with nonlinearterms that give rise to the onset of the critical collapse. Inthe 1D setting, the collapse is induced by the self-focusingquintic term, which may represent attractive three-bodyinteractions in the atomic condensate [62, 63]. A crudeestimate of the virial type [64] suggests that the currentnonlinearity, although it is not represented in the sys-tem’s energy, may compete on a par with the quinticself-attraction, hence it may essentially affect the col-lapse dynamics. Further, the same current term can beadded, as a quasi-1D one, to the two-dimensional (2D)GP equation, with the usual cubic self-attraction, whichdrives the development of the critical collapse in 2D [64].An estimate suggests that, in the 2D setup, the currentnonlinearity will be the strongest term in the collapsingregime, hence its effect may be crucially important. VI. ACKNOWLEDGEMENTS
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