Stability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates
SStability and dispersion relations ofthree-dimensional solitary waves in trappedBose-Einstein condensates
A. Mu˜noz Mateo
E-mail: [email protected]
Departament d’Estructura i Constituents de la Mat`eria, Facultat de F´ısica,Universitat de Barcelona, E08028 Barcelona, SpainCentre for Theoretical Chemistry and Physics and New Zealand Institute forAdvanced Study, Massey University, Private Bag 102904 NSMC, Auckland 0745,New Zealand
J. Brand
E-mail:
Dodd-Walls Centre for Photonic and Quantum Technologies and Centre forTheoretical Chemistry and Physics, New Zealand Institute for Advanced Study,Massey University, Private Bag 102904 NSMC, Auckland 0745, New ZealandPACS numbers: 03.75.Lm,67.85.De,03.75.Kk,05.30.Jp
Abstract.
We analyse the dynamical properties of three-dimensional solitarywaves in cylindrically trapped Bose-Einstein condensates. Families of solitarywaves bifurcate from the planar dark soliton and include the solitonic vortex,the vortex ring and more complex structures of intersecting vortex-line knowncollectively as Chladni solitons. The particle-like dynamics of these guidedsolitary waves provides potentially profitable features for their implementationin atomtronic circuits, and play a key role in the generation of metastable loopcurrents. Based on the time-dependent Gross-Pitaevskii equation we calculatethe dispersion relations of moving solitary waves and their modes of dynamicalinstability. The dispersion relations reveal a complex crossing and bifurcationscenario. For stationary structures we find that for µ/ (cid:126) ω ⊥ > .
65 the solitonicvortex is the only stable solitary wave. More complex Chladni solitons stillhave weaker instabilities than planar dark solitons and may be seen as transientstructures in experiments. Fully time-dependent simulations illustrate typicaldecay scenarios, which may result in the generation of multiple separated solitonicvortices. a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates
1. Introduction
Nonlinear waves in superfluids are the subject of intense theoretical and experimentalresearch. The exquisite control achieved in manipulating ultracold atomic gases hasenabled the creation, manipulation, and detection of dark solitons [1, 2], vortex rings[3, 4] and solitonic vortices in Bose-Einstein condensates (BECs) [5] and Fermi gasesalong the BEC–BCS crossover [6]. All these structures are strongly influenced bythe confinement of the particle cloud and represent solitary waves in the sense thatthey are characterised by an excitation energy density above the condensate groundstate that is localised with respect to the long axis of the confining geometry. Thesemulti-dimensional solitary waves distinguish themselves by their non-trivial topologyassociated with the constituting superfluid currents.The combination of confinement and superfluid currents is also the mainconstituent in the development of atomtronic devices, and, to this end, an in-depthunderstanding of the nonlinear phenomena involved in such dynamics is required.In particular, the role played by nonlinear waves deserves special attention. On theone hand, the shape-preserving evolution of solitary waves, in both repulsively andattractively interacting systems, could be a useful feature to be implemented in futureapplications, in a similar way as optical solitons are being currently used in opticalfibers [7]. On the other hand, the necessity to better understand the role played bysolitary waves in the generation of superfluid currents has manifested itself in a series ofexperiments with superfluid rings at NIST [8, 9]. Therein, vortices of solitonic nature,due to the transverse trapping along the radius of the rings, have been found afterdriving the superfluid into motion. Additionally, the generation of such metastableloop currents has been demonstrated to be mediated by the existence of solitary wavesthat produce an energy barrier preventing phase slips [10].Dark solitons, or kinks , are density dips with an associated jump in the phase ofthe order parameter, and represent nonlinear excitations in Bose-Einstein condensateswith repulsive interparticle interactions [11]. In one dimensional rings, kink excitationsrepresent intermediate stages connecting states with different winding numbers [12]. Aone dimensional dark soliton can be understood as a vortex which is crossing the ring,and hence providing a characteristic density depletion and phase slip that depends onthe position of the vortex. In higher dimensions, the structure of a vortex line crossingthe ring, a solitonic vortex, can be more easily identified on the cross section of thesystem. Alternatively, other transverse states containing vortex lines can be excitedin order to produce a given phase jump along the ring circumference. In general,multidimensional stationary kinks are dynamically unstable [13], unless a tight trapcould keep the system in the quasi-one dimensional regime so that higher energytransverse excitations were excluded [14, 15].In trapped superfluids, Chladni solitons [16] emerge from the decay of threedimensional kinks, as a result of the excitation of standing waves on the nodal planeof the kink. Such waves produce patterns of vorticity along the nodal lines of thetransverse modes in analogy to the Chladni figures visualising the nodal lines of platevibration modes [17]. In traps with cylindrical symmetry, the different families ofsolitary waves can be described by the radial p and azimuthal l quantum numbers,indicating the number of transverse nodal lines along their respective directions.Solitonic vortices, belonging to the family ( p = 0 , l = 1), are the lowest energystates in elongated condensates [15, 18], while vortex rings [19, 20], presenting higherexcitation energies, belong to the family ( p = 1 , l = 0). Along with these previously tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates p and l quantum numbers, as was pointed out by the authors[16]. Very recently evidence for the observation of the Φ-shaped Chadni soliton withquantum numbers ( p = 1 , l = 1) in a superfluid Fermi gas at unitary was reported inRef. [21]. The relative strength of the decay modes that can produce Chladni solitonsfrom the decay of the kink, as well as the robustness of the Chladni solitons are keypoints that remain to be clarified.Motivated by the previous considerations, in the present work we study thedynamics of solitary wave excitations within the framework of the time-dependentGross-Pitaevskii equation. Section 2 is devoted to characteristic mass parametersrelevant for the Landau quasiparticle dynamics of solitary waves. Numerical data ispresented and compared to analytical approximations for energy, inertial, and physicalmasses of the dark soliton in subsection 2.1, and the solitonic vortex and vortex ring insubsection 2.2. The snaking instability of the kink state is analysed in detail in section3 by solving the Bogoliubov equations of linearised excitations for the trapped kinkstate numerically and by developing a semi-analytical theory of the unstable modes.Numerical results for stationary Chladni solitons are reported in section 4, while thedispersion relations and phase step of moving Chladni solitons are considered in section5. A stability analysis of Chladni solitons – stationary and moving – is performed insection 6, where also results from real-time evolution beyond the linear response regimeare reported. We identify two characteristic scenarios in the fate of Chladni solitons:either a chain of decay episodes into single vortex lines which are localized arounda transverse plane of the system, or the generation of secondary travelling waves.Finally, the dynamical features pointed out in our study are used to propose feasibleprotocols for the experimental realization of these solitary waves.
2. Energy and inertial and physical masses
Solitary waves often exhibit particle-like dynamics. One manifestation of such particle-like dynamics occurs when solitary waves move across a slowly varying backgroundwhere energy radiation is being suppressed. As a consequence of energy conservation,the solitary wave will then adapt adiabatically to the changing environmentalconditions, adjusting its internal parameters as to maintain its local energy constant,and acting as a Landau quasiparticle [22]. The nonlinear wave solutions considered inthis article are all solitary waves in this sense, because they are localised with respectto the long axis of a trapped geometry and thus may perform guided quasiparticlemotion, even though their properties are significantly influenced by the presence ofa transverse confining potential. In sections 3 and 6 we present evidence to showthat only two types of solitary waves are dynamically stable in cylindrically trappedBose-Einstein condensates: The solitonic vortex is stable when µ/ ( (cid:126) ω ⊥ ) > .
65 whilethe dark soliton is stable below this value, where µ is the chemical potential and ω ⊥ the frequency of the transverse harmonic trapping potential. Dynamically stablesolitary wave can be expected to perform near-hamiltonian quasiparticle dynamics fora long time and have been observed in experiments with trapped superfluid Fermigases for several seconds [23, 6]. Unstable solitary waves like vortex rings may stillexhibit quasiparticle like dynamics if the competing decay dynamics is slow enoughor suppressed by symmetry constraints [24, 25].In the framework of Landau quasiparticle dynamics, the equations of motion ofa solitary wave in a trapped quantum gas can be derived from knowing the excitation tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates E s ( µ, v s ) of the solitary wave as a function of the chemical potential µ and itsvelocity v s [22]. In a trapped gas, the chemical potential is then treated as a (slowlyvarying) function of the position Z of the solitary wave, while the velocity is the timederivative of position v s = ˙ Z . Requiring the energy to be a constant of motion thenleads to 0 = dE s dt = (cid:32) ∂E s ∂µ (cid:12)(cid:12)(cid:12)(cid:12) v s dµdZ + M in ¨ Z (cid:33) ˙ Z, (1)where M in = 1 v s ∂E s ∂v s (cid:12)(cid:12)(cid:12)(cid:12) µ , (2)is the inertial mass of the solitary wave, often also called the effective mass . Equation(1) already looks like Newton’s law. For weak harmonic trapping potential along the z axis where the Thomas Fermi approximation demands that µ ( Z ) = µ − mω z Z ,we arrive at Newton’s equation of motion in the form M in ¨ Z = − M ph ω z Z, (3)where the physical mass defined by M ph = m ∂E s ∂µ (cid:12)(cid:12)(cid:12)(cid:12) v s , (4)is to be interpreted characteristic parameter of the solitary wave that gives rise tothe buoyancy-like force on the right hand side of Eq. (3). The interpretation of thephysical mass to be related to a buoyancy phenomenon is further supported by itbeing closely related to number of missing particles N s , i.e. particles depleted fromthe background density due to the presence of the solitary wave, where M ph = mN s holds in many cases [26].Solitary waves in repulsively interacting quantum gases typically have negativeinertial and physical masses, which leads to oscillatory motion of the solitary waves ina trapped gas. This, e.g. is the case for the one-dimensional Gross-Pitaevskii equationdescribing Bose-Einstein condensates with tight transverse confinement [27, 22]. If thephysical and inertial masses are independent of position and velocity, or in the limit ofsmall-amplitude motion, we obtain simple harmonic oscillations Z ( t ) ∝ sin(Ω t ) with[26, 28] ω z Ω = M in M ph . (5)Such harmonic oscillations have already been observed in experiments and thefrequency ratio measured for dark solitons [29] and for solitonic vortices [30] in Bose-Einstein condensates and for solitonic vortices in the superfluid Fermi gas [23, 6]. Inthe remainder of this section we present numerical data of dispersion relations andmass parameters evaluated for the dark soliton, solitonic vortex and vortex rings, andcompare with approximate analytical expressions. tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates We will model solitary waves within the mean field theory given by the Gross-Pitaevskiiequation for the condensate order parameter Ψ( r , t ) i (cid:126) ∂ Ψ ∂t = (cid:18) − (cid:126) m ∇ + V ( r ) + g | Ψ | (cid:19) Ψ , (6)where g = 4 π (cid:126) a/m is the interaction strengh determined by the positive s -wavescattering length a and the bosonic mass m , V ( r ) = mω ⊥ r ⊥ / r ⊥ = x + y ,and the condensate particle number N follows from normalization N = (cid:82) d r | Ψ | .Our starting point is the search for stationary solutions to Eq. (6), Ψ( r , t ) = e − iµt/ (cid:126) ψ ( r ), with chemical potential µ , having the form of planar kinks across theaxial coordinate z . This task has been carried out numerically because no analyticalsolution is known for the 3D Gross-Pitaevskii equation (6). Nevertheless, we havebeen guided by the asymptotic analytical solution proposed in Ref. [16] ψ ( r ) = χ T F ( r ⊥ ) tanh( z/ξ ( r ⊥ )) , (7)which is valid in the Thomas-Fermi regime, where χ T F = (cid:112) µ loc ( r ⊥ ) /g is thetransverse ground state, µ loc ( r ⊥ ) = µ − V ( r ⊥ ) is the local chemical potential, anda local healing length is defined by ξ ( r ⊥ ) = (cid:126) / (cid:112) mµ loc ( r ⊥ ). Employing Eq. (7) asinitial ansatz, the numerical solution of Eq. (6) is obtained without difficulty eitherusing a Newton method or by imaginary time evolution.The ansatz (7) also provides an excellent description of relevant properties ofthe kink, such as excitation energy or missing number of particles. We define theexcitation energy of a soliton ψ , relative to the ground state ψ , by means of E s = E [ ψ ] − µN − ( E [ ψ ] − µN ) , (8)with the energy defined by the functional E [ ψ ] = (cid:90) d r (cid:18) (cid:126) m |∇ ψ | + V ( r ) | ψ | + g | ψ | (cid:19) , (9)Substituting the ansatz (7) into Eq. (8) and neglecting the derivatives of χ T F ( r ⊥ ) inEq. (9), according to the Thomas-Fermi approximation, we get the energy of a planardark soliton confined by a transverse harmonic trapping as E s (cid:126) ω ⊥ = 415 ˜ µ ˜ a , (10)where the quantities with tilde are measured in the characteristic units of the trap,˜ µ = µ/ (cid:126) ω ⊥ , ˜ a = a/a ⊥ , and a ⊥ = (cid:112) (cid:126) /mω ⊥ . Normalization of Eq. (7) gives themissing number of particles in the soliton N s = N − N : N s = −
23 ˜ µ ˜ a , (11)which can also be obtained from the relation ∂E s /∂µ = − N s . (12) tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates µ > a ˜ E s and ˜ aN s arefixed, as reflected in equations (10) and (11). µ / h _ ω ⊥ E xc i t a t i on ene r g y / a ⊥ n h _ ω ⊥ Kink theory Eq. (10)Kink theory Eq. (13)KinkSV theory Eq. (15)SVVR3SV2VR5SV µ / h _ ω ⊥ − N s a / a ⊥ Kink theory Eq. (11)Kink theory Eq. (14)SV theory Eq. (18)
Figure 1.
Excitation energy (upper panel) and number of missing particles N s (lower panel) for a three dimensional solitary waves in an infinite cylindrical BECwith transverse trapping frequency ω ⊥ as a function of chemical potential µ .Results from fully numerical solutions of Eq. (6) are shown as circles for the kink(dark soliton) and squares for the solitonic vortex (SV). Formulas (10) and (11)based on the Thomas Fermi approximation for the kink are shown in a full redline. The improved approach of equations (13) and (14) is also shown (dashed redline) for comparison. The analytical approximations of equations (15) and (18)for the solitonic vortex are shown as a full green line. In addition, grey lines shownumerical results for other stationary Chladni solitons: single vortex ring (VR),triple solitonic vortex (3SV), double vortex ring (2VR), and quintuple solitonicvortex (5SV). tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates
7A further improvement can easily be introduced in previous expressions foraverage quantities. Following Ref. [31], by properly incorporating the zero point energyof the harmonic oscillator in the calculations we obtain the improved expressions E s (cid:126) ω ⊥ = 415 (˜ µ − ˜ a + 23 (˜ µ − ˜ a , (13)and N s = −
23 (˜ µ − ˜ a − (˜ µ − ˜ a , (14)which have the correct limits both in the Thomas-Fermi and in the quasi-onedimensional regimes. They also interpolate in between with very good accuracy ascan be seen in Fig. 1. We have determined the inertial and physical masses for solitary waves by computingthe energy of the fully numerical solution of the 3D Gross-Pitaevskii equation (6)and taking numerical derivatives with respect to chemical potential and velocity nearthe stationary point. Figure 3 reports the resulting mass ratios for dark solitons (redcurve), solitonic vortices (green curve with open squares), and vortex rings (blue curvewith open circles) in 3D condensates confined by isotropic radial harmonic traps. Forthe kink, one can observe discontinuities arising at the bifurcation points of ( p, E SV = π (cid:126) mm B n ln R ⊥ ξ = √
23 ˜ µ ln(2˜ µ ) a ⊥ a (cid:126) ω ⊥ (15)= 4 √ √ ˜ µ ln(2˜ µ ) a ⊥ n (cid:126) ω ⊥ , (16)and the Thomas Fermi approximation has been used for the one- and two-dimensionaldensities, in particular 4 an = ˜ µ . The missing particle number is obtained bydifferentiation N SV = − mm B πn ξ (cid:18) γ + 1 γ ln R ⊥ ξ + 2 γ (cid:19) (17)= − √ a ⊥ a (cid:112) ˜ µ (cid:18) ln(2˜ µ ) + 23 (cid:19) , (18)where γ = µn ∂n∂µ is a polytropic index characterising the equation of state, whichevaluates to γ = 1 for the case of a BEC. For the inertial mass the following expression tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates µ / h _ ω ⊥ -2024681012 M i n / M ph DSSVVR
Figure 2.
The ratio of inertial to physical mass as a function of the chemicalpotential µ for kinks (DS, red line), solitonic vortices (SV, green line with opensquare), and vortex rings (VR, blue line with open circles) in a cylindrical BECwith isotropic transverse harmonic trapping with frequency ω ⊥ . was obtained M SV in = − π γ + 1 n R ⊥ ln R ⊥ ξ m (19)= − √
29 ˜ µ ln(2˜ µ ) a ⊥ a m. (20)
3. Snaking instability of the dark soliton
After finding stationary kink solutions to Gross-Pitaevskii equation (6), we look forelementary excitations { u ( r ) , v ( r ) } with angular frequency ω around every equilibriumstate ψ with chemical potential µ . The perturbed state can be written as Ψ( r , t ) = e − iµt/ (cid:126) (cid:2) ψ + (cid:80) ω ( u e − iωt + v ∗ e iωt ) (cid:3) , and the excitation modes are the solutions to thelinear Bogoliubov equations, (cid:0) H − µ + 2 g | ψ | (cid:1) u + gψ v = (cid:126) ωu , (21a) − gψ ∗ u − (cid:0) H − µ + 2 g | ψ | (cid:1) v = (cid:126) ωv , (21b)where H = − (cid:126) ∇ / m + V ( r ⊥ ). Dynamical instabilities are related to solutionsto Eq. (21) with complex frequencies ω , where the imaginary part of ω is a rate ofexponential growth of the corresponding unstable mode. The inverse of the imaginarypart of the unstable frequency can thus be interpreted as the lifetime of the particularmode. Below we will report fully numerical solutions of Eq. (21) for the dark solitonand Chladni solitons. Here we will proceed with finding analytical solutions to theBogoliubov equations for the dark soliton based on the Thomas Fermi approximation. tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates For a real-valued stationary state, as it is the case of the dark soliton or kink of Eq.(7), the Bogoliubov equations (21) can be transformed using f ± ( r ) = u ( r ) ± v ( r ) and g ± = (2 ± g into (cid:0) H − µ + g ± ψ (cid:1) f ± = (cid:126) ωf ∓ . (22)For ω = 0 the two equations decouple. This was the problem solved in Ref. [16] inorder to find the bifurcation points of Chladni solitons from the dark soliton. Here, weare interested in the more general problem of finding solutions to Eq. (22) with finiteimaginary or complex values of ω . Aiming at an approximate separation of transverseand longitudinal degrees of freedom we introduce the rescaled variable ¯ z = z/ξ ( r ⊥ ).For the dark soliton state of Eq. (7) we can write gψ = µ loc ( r ⊥ ) tanh ¯ z , where µ loc = µ − V ( r ⊥ ) and may thus rewrite the Bogoliubov equation (22) as (cid:18) − (cid:126) m ∇ ⊥ + µ loc A ± (cid:19) f ± = (cid:126) ωf ∓ , (23)with A − = − ∂ ∂ ¯ z − ¯ z, (24a) A + = − ∂ ∂ ¯ z − ¯ z. (24b)The operators A ± are both Hamilton operators of one-dimensional Schr¨odingerequations with a shifted Rosen-Morse potential, whose eigenfunctions are knownanalytically [32]. Here we use the localised ground states of the respective operatorsas a restricted basis for the z -dependence of f ± , since we expect the unstable modesto be localised near the dark soliton plane. The ground state eigenfunction of A + is ϕ (¯ z ) = √ sech ¯ z with eigenvalue 0. It corresponds to the well-known Goldstonemode of translation of the dark soliton in z direction. This does not constitutean instability in itself but the mode will be relevant for constructing the decayingmodes with imaginary omega. The operator A − has the ground state wave function ϕ − / − (¯ z ) = √ sech ¯ z with eigenvalue − . It is this mode that is responsible for theexistence of unstable Bogoliubov modes.The z dependence can now be removed from the Bogoliubov equation (22) bystarting from the ansatz ( f + , f − ) t = χ + ( x, t )( ϕ , t + χ − ( x, y )(0 , ϕ − / − ) t . Ignoringany x, y derivatives of the functions ϕ ± (¯ z ) and projecting onto the respective groundstates by multiplying from the left with ( ϕ ,
0) and (0 , ϕ − / − ), and integrating over¯ z , we obtain the matrix equation (cid:32) − (cid:126) m ∇ ⊥ − (cid:126) ωζ − (cid:126) ωζ − (cid:126) m ∇ ⊥ − µ loc (cid:33) (cid:18) χ + χ − (cid:19) = 0 , (25)where ζ = (cid:82) ϕ − / − ϕ d ¯ z = π (cid:113) ≈ .
962 is a numerical constant close to one.
The transverse Bogoliubov equation (25) is easily solved in the absence of a transversetrapping potential, where µ loc = µ = (cid:126) mξ = const, with plane wave solutions, e.g., tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates χ ± ∝ exp( iqx ). For the unstable eigenvalues we find ω hom = iq √ µ c √ m (cid:112) − q ξ . (26)For small q the growth rate Im( ω ) grows linearly with the slope √ µ ζ √ m ≈ . √ µ √ m as isdemanded by general hydrodynamic arguments [33]. Although being approximate dueto the restricted basis expansion of the z dependence, this result compares very wellwith the previously obtained ones in Refs. [34, 13, 33] , where the exact slope for theGross-Pitaevskii equation is √ µ √ m ≈ . √ µ √ m . The snaking instability is suppressedand eigenvalues become real-valued for wave numbers larger than q crit = 1 /ξ , which isthe exact value. For intermediate values 0 < q < /ξ the growth rate has previouslyonly been obtained numerically, and Eq. (26) reproduces the results of Refs. [34, 13]very closely. We now proceed with solving the transverse Bogoliubov equation (25) for an isotropictransverse trapping potential with µ loc ( r ⊥ ) = µ − mω ⊥ r ⊥ . Assuming an azimuthaldependence ∝ cos( lθ ) with the quantum number l = 0 , , , . . . reduces Eq. (25) to aset of ordinary differential equations in the radial coordinate r ⊥ . It is now convenientto move to harmonic oscillator units. Introducing the rescaled radial coordinate˜ r ⊥ = r ⊥ /a ⊥ and dividing the equation by (cid:126) ω ⊥ , we obtain (cid:18) H l − ˜ ωζ − ˜ ωζ H l − ˜ µ (cid:19) (cid:18) χ l + χ l − (cid:19) = 0 , (27)where H l = − (cid:18) ∂ ∂ ˜ r ⊥ + 1˜ r ⊥ ∂∂ ˜ r ⊥ − l ˜ r ⊥ (cid:19) , (28) H l = H l + 14 ˜ r ⊥ , (29)where H l represents a two-dimensional Laplacian and H l is the Hamiltonian of atwo-dimensional harmonic oscillator with weakened trap potential compared to theone experienced by the atoms. Equation (27) can also be rewritten in the form H l (cid:18) H l − ˜ µ (cid:19) χ l − = ˜ ω ζ χ l − . (30)Even though this represents a non-hermitian eigenvalue problem, we have only foundreal eigenvalues ˜ ω ζ in numerical investigations. Solutions of Eq. (27) with ˜ ω = 0 have special significance as they indicate the transitionof a specific mode from representing an instability ˜ ω < ω >
0. At the same time they indicate a bifurcation of the stationary tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates χ lp (˜ r ⊥ ) = 2 − R lp (2 − ˜ r ⊥ ) of the 2D harmonic oscillatorHamiltonian H l with eigenvalues (cid:15) lp = (2 p + l + 1) / √ R lp ( r ) = (cid:115) p !( p + l )! r l L lp (cid:0) r (cid:1) e − r , (31)where L lp ( x ) is the generalised Laguerre polynomial. It is easily seen that χ lp solvesthe Bogliubov equation (30) with ˜ ω = 0 when (cid:15) lp = ˜ µ/
2, which translates into thecondition µ (cid:126) ω ⊥ = √ p + l + 1) , (32)for the bifurcation points of Chladni soliton solutions from the dark soliton, as foundpreviously in Ref. [16]. For finite instability rates Im(˜ ω ) the eigenvalue equation (30) can be expanded in abasis of the normalised eigenfunctions | p, l ) ≡ χ lp of H l , which transforms it into atridiagonal matrix eigenvalue equation (cid:88) p (cid:48) B lp,p (cid:48) v p (cid:48) = ˜ ω ζ v p (33)with B lp,p (cid:48) = ( p, l | H l ( H l − ˜ µ ) | p (cid:48) , l ) = (cid:82) ∞ ˜ r ⊥ d ˜ r ⊥ χ lp H l ( H l − ˜ µ ) χ lp (cid:48) . For the matrixelements we find B lp,p = (2 p + l + 1) − ˜ µ p + l + 14 √ , (34) B lp,p − = 14 [2( p −
1) + l + 1] (cid:112) p + lp − ˜ µ (cid:112) p + lp √ , (35) B lp − ,p = 14 (2 p + l + 1) (cid:112) p + lp − ˜ µ (cid:112) p + lp √ , (36)and B lp,p (cid:48) = 0 for | p − p (cid:48) | >
1. The matrix is block-diagonal in the azimuthalquantum number l due to the azimuthal symmetry of the problem. We have solvedthe corresponding eigenvalue equations numerically and have found the approximateasymptotic behaviour ˜ ω ln ∼ ˜ µ − √ n + 1)4 ζ , (37)for large ˜ µ and values of the azimuthal quantum number l >
1. All these eigenvalueshave a zero crossing for a finite value of ˜ µ corresponding to Eq. (32), where n = p canbe identified. These results were obtained by diagonalising a truncated matrix B lp,p (cid:48) with p, p (cid:48) < p c . Changing the cutoff value p c affects the large-˜ µ regime but leavesthe zero crossings for n (cid:28) p c unaffected. The asymptotic behaviour reported aboverepresents the limit of p c → ∞ . tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates l = 0 sector a special case occurs, where the | ,
0) state needs to be excludedfrom the basis in order to eliminate an unphysical unstable eigenvalue with p = 0 thatotherwise occurs in solving Eq. (33). The mode with p = 0 , l = 0 corresponds to zeropoint motion of the kink, which is not captured correctly by the underlying ThomasFermi approximation. Indeed, no unstable mode with these quantum numbers is foundin the full numerical solution of the Bogoliubov equations. Diagonalising Eq. (33) ina truncated basis with 0 < p < p c produces the asymptotic behaviour˜ ω l =0 n ∼ ˜ µ − √ n − ζ , (38)where still n = p can be identified at the zero crossings of ˜ ω .The results of the truncated eigenvalue problem (33) with the asymptotic results(37) and (38) closely resemble the full numerical results. We have also solved the Bogoliubov equation (21) numerically in full three dimensionswithout making use of the approximations discussed in the previous paragraphs.The results for the unstable modes and associated frequencies are collected as afunction of the chemical potential of the kink in Fig. 3. The insets represent axialviews of phase-colored density isocontours of the excitation modes ( p, l ) just after theappearance of bifurcation points. These modes present a structure of nodal lines,derived from the radial p and azimuthal l quantum numbers, characteristic of thelinear excitations of the transverse trap. As can be seen in the lower left inset, theonly one displaying a longitudinal view, they are strongly localized around the planeof the kink. Their emergence follow in a very good approximation the analyticalprediction for bifurcations given by Eq. (32), which are indicated by red arrows belowthe horizontal axis. As the interaction energy increases from the quasi-onedimensionalconfiguration, where kink states are stable structures and generate only real frequencyexcitations, the first bifurcation point denoting the appearance of a complex frequencyfor the mode | p = 0 , l = 1 (cid:105) comes into existence at µ , = 2 . (cid:126) ω ⊥ , very close to the2 . (cid:126) ω ⊥ value predicted by (32). From this point on, kinks are unstable states, andfurther increase of the chemical potential is accompanied by the emergence of newbifurcation points grouped around the integer energy values (cid:15) pl = 2 p + l + 1, with thecharacteristic degeneracy of the two-dimensional harmonic oscillator.
4. Stationary Chladni solitons
Every unstable mode of the kink is associated with a stationary Chladni soliton. Thezero crossings of the unstable mode frequencies in Fig. 3 indicate bifurcation pointsof the Chladni solitons from the dark soliton. The sign patterns of the unstable modefunctions at the bifurcation points χ lp ( r ⊥ ) cos( θ ) determine the direction of flow alongor counter the z direction and the nodal lines translate into vortex lines. Numericallyobtained mode functions are also shown in Fig. 3. Increasing the nonlinearityparameter ˜ µ = µ/ ( (cid:126) ω ⊥ ) above the bifurcation point, we obtain a family of stationaryChladni solitons with the same structure and symmetries, as endowed by the unstablemode at the bifurcation point. A number of Chladni soliton solutions is shown inFig. 4. For instance, the first excited mode | p = 0 , l = 1 (cid:105) generates the family ofsolitonic vortex states, while the next two unstable modes, | , (cid:105) and | , (cid:105) , that tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates µ / h _ ω ⊥ I m ag ( ω ) / ω ⊥ (1,0)(0,1) Figure 3.
Upper panel: Unstable frequencies ω and density isocontours at 5% ofmaximum density (insets) of Bogoliubov modes, classified by their radial andangular quantum numbers ( p, l ), responsible for the decay of the kink. Theshaded background of the isocontours indicates the BEC density distribution. Theaxial view for the (0 ,
1) mode (bottom left inset) clearly demonstrates the axiallocalisation of the mode function. All the mode functions have been generatedclose to their respective bifurcation points. The arrows below the horizontal axisindicate the bifurcation points according to the analytical prediction (32). Lowerpanel: Following the unstable mode frequencies to larger interaction parameters µ/ ( (cid:126) ω ⊥ ) demonstrates the linear growth. branch off near µ , = 4 (cid:126) ω ⊥ in Fig. 3, generate the families of vortex rings and twocrossed vortices states, respectively.The linear solutions of a general, isotropic or anisotropic, twodimensionalharmonic oscillator can be written in terms of Hermite modes with cartesiansymmetries instead of the Laguerre modes of cylindrical symmetry. The Hermitemodes are characterised by the pair ( n x , n y ) of quantum numbers indicating thenodal lines along x and y directions. Corresponding Chladni solitons would bemade of vortex lines in such rectangular patterns. Although we have found the tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates Figure 4.
Density isocontours (at 5 % of maximum density) of static Chladnisolitons with µ = 10 (cid:126) ω ⊥ . The cross-sections are 9 a ⊥ in width. stationary Chladni solitons bifurcating from the kink to all conform to the Laguerretype, one may still expect to find Hermite-type structures to be relevant in thedecay process of the kink. In the linear regime, the Hermite modes | n x , n y (cid:105) canbe constructed as superposition of Laguerre modes | p, l (cid:105) with degenerate energies (cid:15) n x n y = (cid:15) pl . For example, | n x = 0 , n y = 2 (cid:105) ∝ | p = 0 , l = 2 (cid:105) − | p = 1 , l = 0 (cid:105) , and | n x = 1 , n y = 2 (cid:105) ∝ | p = 0 , l = 3 (cid:105) − | p = 1 , l = 1 (cid:105) . Because of the energy splittingof the bundle of linear degenerate states presented in Fig. 3, the linear superpositionmechanism is not acting in the nonlinear case. However, we have found that Hermite-like modes do emerge as nonlinear bifurcations at a second stage. In the first stage,families of stationary solitary waves bifurcate from the kink in close relation to itsunstable Laguerre-like modes. Except for the solitonic vortex, all these families aremade of unstable states, the unstable excitation modes of which can generate newsolitary waves at a second stage. It is then when the new solitonic families turn outto be composed of Hermite-like modes ( n x , n y ). The instabilities of Chladni solitonswill be discussed in more detail in section 6.
5. Moving Chladni solitons
Up to now we have been looking at static solitons. In order to understand how thedifferent families of solitonic states appear and connect, it is convenient to considera more general picture of moving solitary waves. Here, we extend upon the work ofKomineas and Papanicolao, who already computed energy dispersion relations andphase steps for axisymmetric solitary waves (kinks and nested vortex rings) [19, 20]and for the solitonic vortex [18]. tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates π π P ha s e s t ep ( ∆ φ ) Kink2SVVRSV1D -0.1 0 0.1 π (a) -1 -0.5 0 0.5 1 Speed / c E xc i t a t i on ene r g y / a ⊥ n h _ ω ⊥ -0.05 0 0.052.222.23 (b) π π Momentum / h _ n E xc i t a t i on ene r g y / a ⊥ n h _ ω ⊥ B A C (c)
Figure 5.
Characteristic quantities for solitary waves with chemical potential µ = 5 (cid:126) ω ⊥ : axial phase step ∆ φ (a), and excitation energy E s (b) as a functionof the axial velocity v z in units of the speed of sound c , and energy ( E s ) vs.momentum dispersion relations (c). In order to construct the full dispersion relations for moving solitary waves, wenumerically search for states Ψ( r , t ) = e − i ( µ (cid:48) t + mv z z ) / (cid:126) ψ ( r ), moving along the z -axiswith a constant velocity v z = (0 , , v z ), which are solutions of the stationary Gross-Pitaevskii equation for a co-moving reference frame:12 m ( − i (cid:126) ∇ − m v z ) ψ + V ( r ) ψ + g | ψ | ψ = µ (cid:48) ψ , (39) tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates µ (cid:48) = µ + mv z / π ) phase jumps, and their velocitiesare limited by the speed of sound c , at which solitonic states become linear soundexcitations. In order to calculate the speed of sound, which will be used as velocityunit in what follows, we will make use of the analytical expression given in Ref. [35]: c (cid:112) (cid:126) ω ⊥ /m = (cid:18) ˜ µ − µ (cid:19) , (40)where ˜ µ = µ/ (cid:126) ω ⊥ . This expression provides the speed of sound for elongated,harmonically trapped condensates with arbitrary values of the interaction, and givesthe exact limits both in the quasi-onedimensional and Thomas-Fermi regimes.Figure 5b shows the excitation energy E s as a function of the axial velocity v z for moving solitary waves with chemical potential µ = 5 (cid:126) ω ⊥ . Kinks, represented bythe solid red line at the top of the figure, have the highest excitation energy amongthe solitary waves, and exist in this case only for low velocities, | v z | < . c . Belowthem, at lower excitation energies, only solitary waves bifurcating at energies less orequal than ˜ µ = 5 can be found in the graph, as per Eq. (32). As can also be deducedfrom the unstable frequencies of Fig. 3, indeed, only vortex rings (blue solid line),double solitonic vortex (dashed black) and the single solitonic vortex (solid green) areavailable at ˜ µ = 5 and appear in Fig. 5. At small velocity, and very close in energyto the vortex ring and the static cross soliton, a new type of solution emerges (seethe inset of Fig. 5b). It is composed of a couple of almost parallel vortices, indeed avortex-antivortex pair or vortex dipole, which is not coming directly from a bifurcationof the kink, but from a secondary bifurcation of the cross soliton. Indeed, the solutioncan be connected to a decay instability of the cross soliton produced by the Hermitemodes ( n x = 2 , n y = 0) or ( n x = 0 , n y = 2). Figure 6a shows the density configurationof these states around zero velocity, which can be cross-checked with their featuresextracted from the three panels of Fig. 5, where clear differences in the associatedphase step arise for the three states A, B and C. At higher velocities, the picture isslighly different. The structure of the kink changes and so do its excitation modes.Since the cross soliton is a static state, it can not be found between the bifurcationsof moving kinks, and is substituted by the mentioned, moving Hermite modes. Forgrowing values of the chemical potential, new, higher energy Hermite modes emergeby equivalent mechanisms. In Fig. 6b we show density isocontours for some of suchmodes with ˜ µ = 5 , and 10.It is also interesting to look at the phase jump along the axial z -direction∆ φ = φ ( z → ∞ ) − φ ( z → −∞ ) created by the different solitary waves (Fig.5a). The phase shift of dark solitons of the one-dimensional nonlinear Schr¨odingerequation (dotted line) is shown as a reference. Its phase shows a π jump in the staticconfiguration, and grows up to 2 π as the velocity approaches + c , or alternativelyreduces to zero as v z → − c . Similar behaviour, but with different variation rates, isin general followed by the phases of 3D solitonic states. However, a particular featurecan be noted as characteristic of the 3D case. It is the existence of turning points withvertical phase slopes. Looking at the curves for vortex rings (blue lines with triangles),one can notice that there are two separated branches ending at respective turningpoints, and connected by kink states (red curve). A more striking manifestation ofthis phenomenon can be observed in the inset of Fig. 5b, on the curve corresponding tothe vortex-antivortex pair. Between turning points, labeled as B and C in Fig. 5c, there tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates Figure 6. (a) Broken symmetry in a static cross soliton (2SV) with chemicalpotential µ = 5 (cid:126) ω ⊥ for small velocity increments around v z = 0, correspondingto points A (at the center), B (left), and C (right) of Fig. 5. (b) Hermite-type( n x , n y ) Chladni solitons: for µ = 5 (cid:126) ω ⊥ , H(2,0) solitons have v z = 0 (left) and v z = 0 . c (right), whereas H(3,0) corresponds to a static soliton with µ = 10 (cid:126) ω ⊥ .The cross-section of H(2,0) is 6 . a ⊥ in width, whereas H(2,0) is 9 a ⊥ . exist a set of almost static and degenerate states of this type which produce differentphase jumps. The turning points here indicate a transition between the Hermite-likesymmetry of the vortex-antivortex pair and the Laguerre-like vortex cross. π/ π π/ π Momentum / h _ n E xc i t a t i on ene r g y / a ⊥ n h _ ω ⊥ DS(0,5)(2,0)(0,3) (1,1)(1,0)(0,1)
Figure 7.
Dispersion curves (non-exhaustive diagram) for Chladni solitons with µ = 10 (cid:126) ω ⊥ , some of which correspond to the isocontour diagrams in Fig. 4 forstatic solitons, and in Fig. 8 for moving solitons. The precedent analysis, in terms of phase and velocity, can now be completed byconstructing the dispersion relations of Chladni solitons. To this end, as usual, wedefine the axial canonical momentum P c of a solitonic state as the conjugate variable tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates z , that fulfils v z = ∂E s ∂P c . (41)Along with the axial physical momentum p z = − i (cid:126) (cid:82) d r ( ψ ∗ ∂ z ψ − ψ∂ z ψ ∗ ), carried bythe particles traversing the plane of the moving soliton, the canonical momentumincludes the contribution coming from the phase jump ∆ φ between the axialboundaries of the condensate P c = p z − (cid:126) n ∆ φ , (42)where n is the axial density of the ground state of the system [20, 26]. Figure 5cshows the dispersion relation, excitation energy versus axial canonical momentum, forsolitonic states with µ = 5 (cid:126) ω ⊥ moving along the axial coordinate. The turning pointsthat have been described on the phase graph, appear here as the vertexes of cuspsconnecting states with the twofold symmetry: vortex rings and vortex-antivortex pairs.It is also worth noticing that the lowest excitation energy levels for fixed momentumare occupied by solitonic vortices, wherever they exist. This last remark accounts forthe fact that there exist a small regime of soliton speeds, approaching the speed ofsound, where the only solitonic state is the continuation of the vortex ring family, herean axisymmetric solitary wave very much like a grey soliton with a vortex ring phasesingularity outside the Thomas-Fermi density of the trapped BEC, as is apparent inFigs. 5a,b. In this regime, vortex rings are dynamically stable states.When the chemical potential increases, and the number of bifurcations grows,the dispersion diagram of Chladni solitons becomes more complex, because of theemergence of new connections between solitary waves sharing symmetries. As aninstance of this complexity, Fig. 7 displays some of the curves making the dispersiondiagram for µ = 10 (cid:126) ω . For this value of the chemical potential, the family of doublevortex rings [( p = 2 , l = 0), violet lines] is available, and produces a new couple ofturning points compared to the case for µ = 5 (cid:126) ω . Following the different curves awayfrom their maximum (corresponding to zero velocity v z = ∂E s /∂P c = 0) the densitypatterns of Chladni solitons can change dramatically from their static configuration.To illustrate this point, Figure 8 shows the density isocontours of some of the movingChladni solitons that can be found with µ = 10 (cid:126) ω . It is apparent how their symmetrychanges when compared to their static counterparts in 4.
6. Stability of Chladni solitons
It remains to know how robust the Chladni solitons are. To this end, we havenumerically solved the Bogoliubov equations (21) for the linear excitation modes ofthe stationary solitonic solutions (as those represented in Fig. 3). In addition, we havechecked the stability of these nonlinear systems against perturbations by monitoringtheir evolution in real time through the full time dependent Gross-Pitaevskii equation(6).
As mentioned before, and aside from the stability of vortex rings moving near to thespeed of sound, our results indicate that the solitonic vortex (SV) branch is the only tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates Figure 8.
Steady state configurations of moving Chladni solitons (representedby density isocontours at 5% of maximum density) sampling the dispersion curvesin Fig. 7 for µ = 10 (cid:126) ω ⊥ . The labels indicate the quantum numbers ( p, l ) of theassociated solitonic family. Several isocontours from the same family correspondto different velocities, with increasing value from left to right. In this order, thecanonical momentum is in units of π (cid:126) n for (0,1): 1.97 and 1.99; for (1,0): 1.2and 1.6; for (1,1): 1.04, 1.3 and 1.7; for (2,1): 1.06 ; for (0,3): 1.2, 1.3 and 1.6;for (0,5): 1.3. The cross-sections are 9 a ⊥ in width. one containing dynamically stable states. Solitons corresponding to other families areunstable, and decay through the instability channels opened by lower energy branches.Specifically for the stationary solitary waves, the solitonic vortex as the lowest energysolitary wave has no channel of instability, since there is no other, lower energy solutionbifurcating from it (or from the kink). However, the second excited state, which isthe single vortex ring (VR), does present one instability channel associated to thebifurcation of solitonic vortices from the kink with lower energy. The next family isthat of cross solitons (2SV), and is unstable through two channels, and so on. Thisanalysis for static states is displayed in Fig. 9a. The red curve with open circlescorresponds to the unstable frequencies for vortex rings as a function of the chemicalpotential, and the two blue curves with open triangles indicate the two instabilitychannels for the cross soliton. The dotted vertical lines mark the bifurcation pointsfor the Chladni solitons considered (VR and 2SV), by intersecting the instabilitycurves of kinks (gray dashed lines). It is worth to mention that, as can be seen inFig. 9a, for intermediate values of the chemical potential, between 4 and 5 (cid:126) ω ⊥ , thecross soliton (2SV, dashed black curve) has smaller values of unstable frequenciesthan vortex rings. This fact suggests that cross solitons are good candidates for beingexperimentally realised in elongated BECs. Vortex rings have already been observed inexperiments [3, 37]. In this regard, we have noticed that in Ref. [38], where the decayof dark solitons in anisotropic cigar-shaped condensates was observed in experiments,travelling solitary waves composed of vortex-antivortex pairs (see below) were clearly tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates µ / h _ ω ⊥ . . I m ag ( ω ) / ω ⊥ (a) π π Momentum / h _ n . . . . M a x [ I m ag ( ω ) ] / ω ⊥ KinkVRCross 2SV2SV A (b) Figure 9. (a) Growth rates of unstable modes from numerical solution ofBogoliubov equations (21), for stationary vortex ring (VR) and double solitonicvortex (2SV) states. Dotted vertical lines indicate the bifurcation from DS (withunstable frequencies represented by dashed grey lines). In the inset, numericalresults for vortex rings (open symbols) are compared with the analytical prediction(43) (solid line) of Ref. [36]. (b) Unstable mode growth rates for moving solitarywaves with µ = 5 (cid:126) ω ⊥ , as a function of the canonical momentum, from a numericalsolution of the Bogoliubov equations (21). Point A refers to the cross-vortexsoliton labeled in Fig. 5 and represented in the centre of Fig. 6. identified, and a cross soliton structure was found at the turning points of motion.Additional remarks about vortex ring states are in order. As shown in Fig. 9a forthe static case, vortex rings are unstable against decay modes with quantum numbers( p = 0 , l = 1). Our numerical results (open circles in the inset of Fig. 9a) show thatthis instability decreases at slow rate with increasing chemical potential, in agreementwith the analytical prediction of Ref. [36] (solid red line in the inset): ω = − (cid:126) √ ξ ( R − ⊥ + κ / mR ⊥ , (43)where κ is the curvature of the vortex, ξ is the healing length, and R ⊥ is the Thomas-Fermi radius. This expression is valid for vortex rings in harmonically trappedelongated condensates within the Thomas-Fermi regime, and gives nonzero unstablefrequencies ω in the limit of very high chemical potential, thus providing an estimation tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates VR2SVDS
Figure 10.
Density isocontours (at 5% of maximum density) during realtime evolution showing the decay dynamics of a kink (DS), a vortex ring(VR) and a cross soliton (2SV), with µ/ (cid:126) ω ⊥ = 5 and v z = 0, obtained bysolving numerically the time-dependent Gross-Pitaevskii equation starting fromthe stationary configuration seeded with a small amount of numerical noise. Thetypical cross-section radius is 3 . a ⊥ . for the life time of vortex rings. For instance, for µ = 21 (cid:126) ω ⊥ both the numerical andanalytical methods predict an unstable frequency ω (cid:39) . ω ⊥ corresponding to a lifetime of about 20 ms for 50-Hz transverse trap.In the case of moving solitons, the linear stability analysis follows essentially thepreceding procedure for static states. Figure 9b, generated for µ = 5 (cid:126) ω ⊥ , shows ournumerical result for the unstable frequencies of moving Chladni solitons as a functionof the canonical momentum. The unstable frequencies decrease rapidly for vortexrings (blue lines with open triangles) of increasing speed (and thus their life timeincreases), and indeed they become stable past the bifurcation with solitonic vorticesclose to P c = 0 . π (cid:126) n , (and P c = 1 . π (cid:126) n ) where the speed approaches the soundspeed. As anticipated, there are also no unstable frequencies for solitonic vortices. Asit is the case for the cross soliton, moving vortex dipoles (black dashed curves) presentlower unstable frequencies than vortex rings. This may seem surprising consideringthe cylindrical symmetry of the system, and gives support to their possible detectionin experiments. In order to check the predictions given by the linear stability analysis, we have alsotested the nonlinear stability of Chladni solitons by the real time evolution of theirstationary configurations. For, on the initial states Ψ( r , t = 0), we have added arandom noise perturbation δ Ψ( r ), which typically amounts to 2% of the wave function tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates Figure 11.
Sling shot events in the real time evolution of a 2SV state (up)with chemical potential µ/ (cid:126) ω ⊥ = 8, and a 2VR state (down) with µ/ (cid:126) ω ⊥ = 15.Density isocontours at 5% of maximum density are shown in both cases, with atransparent external surface in the axial view of the 2VR. amplitude. Afterwards, we have allowed these wave functions to evolve in time,without dissipation, at constant chemical potential according to Eq. (6). For example,we have followed this procedure for the static Chladni solitons with µ = 5 (cid:126) ω ⊥ , namelydark soliton, vortex ring, cross soliton and solitonic vortex, which have been previouslycharacterized in Figs. 5 and Figs. 9b. As expected, we have observed the decay of allsolitonic states except the soltionic vortex, which, as a result, emerges at the finalstage of the time evolution in all cases. Figure 10 summarises the decay processes,showing snapshots of the evolution at intermediate times. In particular it showscomplex patterns localised in the plane of the initial stationary state at intermediatetimes and the emergence of a single solitionic vortex at late times, while some smallamplitude radiation moves away from the solitary wave at the speed of sound.For higher values of the interaction parameter ˜ µ , different scenarios can be foundin the decay of Chladni solitons. In particular more than one solitary wave can appearand move away from the location of the initial unstable soliton. The interaction energyreleased from the parent state can transform into translational kinetic energy of adescendant solitary wave, leaving a simpler structure at the initial position. This is tability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates ∼
7. Conclusions
We have analysed the dynamical properties of static and moving Chladni solitonsin cylindrically symmetric Bose-Einstein condensates, within the mean-field regimedescribed by Gross-Pitaevskii equation. These states, strongly influenced by thegeometry of the trap, emerge from the excitation of standing waves on planar kinkstates, and inherit particle-like features characterised by lower excitation energies andhigher inertial masses than the kink. We have calculated numerically such quantities,and presented analytical expressions for their evaluation. The unstable standing wavesproducing the decay of the kink have been object of a detailed analysis, and a formulafor the prediction of the unstable frequencies has been proposed.The linear excitations and the real time evolution of Chladni solitons have alsobeen addressed. Our results suggest that these states, even other than the recentlyrealised solitonic vortex, could be observed in current experiments with ultracoldgases, given that their lifetimes are expected to be of tens of milliseconds. In thisregard, several procedures could be followed. In particular, in Ref. [16], we haveproposed a feasible protocol for seeding a particular Chladni soliton on a planarkink. By means of a dark-bright soliton [39] in a two component condensate in theimmiscible regime, a proper density and phase pattern could be imprinted on thebright soliton of one of the components occupying the kink depletion in the othercomponent. The subsequent transfer of the selected pattern into the kink component,through a controlled Raman pulse, could serve the purpose of seeding the decay intothe corresponding Chladni soliton. Other procedures with scalar condensates relyingon an adequate trap geometry, have already been demonstrated. This is the case inRef. [38], where vortex dipoles and the cross soliton has been identified after the decayof kinks in anisotropic harmonic traps. Very recently also the Φ soliton ( p = 1 , l = 1)has been identified in what appears to be a seeded decay of a kink in a unitary Fermigas [21]. Acknowledgments
We acknowledge useful discussions with Martin Zwierlein, Shih-Chuan Guo, and P´eterJeszenszki.
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