Solitary wave complexes in two-component mixture condensates

a r X i v :c o n d -m a t /0412099v 1 [c o n d -m a t .s o f t ] 4 D e c 2004

Solitary wave complexes in two-component mixture condensates

Natalia G.Berlo?

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge,Cambridge,CB30WA

(Dated:4December 2004)

Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intra-component but varying intercomponent interaction strengths.Several families of solitary wave complexes are found:(1)vortex rings of various radii in each of the components,(2)a vortex ring in one component coupled to a rarefaction solitary wave of the other component,(3)two coupled rarefaction waves,(4)either a vortex ring or a rarefaction pulse coupled to a localised disturbance of a very low momentum.The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative di?erences between the chemical potentials of two components.Solitary wave formation,their stability and solitary wave complexes in two-dimensions are discussed.

PACS numbers:03.75.Lm,05.45.-a,67.40.Vs,67.57.De

Solitons and solitary waves represent the essence of many nonlinear dynamical processes from motions in ?uids to energy transfer along biomolecules,as they de?ne possible states that can be excited in the sys-tem.These are localised disturbances of the uniform ?eld that are form-preserving and move with a con-stant velocity.The nonlinear Schr¨o dinger (NLS)equa-tion iψt +?2ψ+γ|ψ|2

ψ=0is canonical and universal equation which is of major importance in continuum me-chanics,plasma physics,nonlinear optics and condensed matter (where it describes the behaviour of a weakly in-teracting Bose gas and known as the Gross-Pitaevskii (GP)equation).The reason for its importance and ubiq-uity is that it describes the evolution of the envelope ψof an almost monochromatic wave in a conservative system of weakly nonlinear dispersive waves.Similarly,systems of the coupled NLS equations have been used to describe motions and interactions of more than one wave envelopes in cases when more than one order pa-rameter is needed to specify the system.The coupled NLS equations have been receiving a lot of attention with recent experimental advances in multi-component Bose-Einstein condensates (BECs).BECs can excite various exotic topological defects and provide a perfect testing ground to investigate their physics,because almost all parameters of the system can be controlled experimen-tally.Topological defects in two-component BECs have been predicted theoretically,but there is still no under-standing of what the complete families of these defects and solitary waves are,nor of their properties,formation mechanisms and dynamics.It has also been suggested [1]that multi-component BECs o?er the simplest tractable microscopic models in the proper universality class of cos-mological systems and solitary waves in multi-component BECs may have their analogs among cosmic strings.The goal of this Letter is to ?nd and characterise the fami-lies of solitary waves that exist in systems of the coupled

FIG.1:(colour online)The dispersion curves of four families of the axisymmetric solitary wave solutions of (3)(A)(red line)α=0.7and Λ2=0.1(c ?=0.2738);(B)(black line)α=0.1and Λ2=0.1(c ?=0.6169);(C)(green line)α=0.5and Λ2=0.25(c ?=0.3317);(D)(blue line)α=0.5and Λ2=0.1(c ?=0.3905).The numbers next to the dots give the velocity of the corresponding solitary wave.For (D)(blue line)these are 0.3,0.32,0.34,0.36,0.38.All these solutions are VR-VR complexes except for U =0.58on (B)branch which is VR-RP.The top inset shows the density isosurface at |ψ1|2=110ψ22∞

for a half of the VR-VR complex for α=0.5,Λ2=0.25that is moving with U =0.3.The radii are b 1=5.194and b 2=4.796.The density contour plots of this solution are shown in the bottom inset.

NLS equation in three dimensions.The implications of these solitary waves are wide-ranging,but they will be discussed in the context of two-component BECs.

The simplest example of a multi-component system is a mixture of two di?erent species of bosons,for in-stance,41K-87Rb [2].Since alkali atoms have spin,it is also possible to make mixtures of the same isotope,but in di?erent internal spin states,for instance,for 87Rb

2 [3].The multi-component BECs are far from being a

trivial extension of a one-component BEC and present

novel and fundamentally di?erent scenarios for their ex-

citations and ground state[4].The theory for a mixture

of two di?erent bosonic atoms can be developed similar to

that for a one-component condensate whose equilibrium

and dynamical properties can be accurately described by

the GP equation[5]for the wave functionψof the con-

densate

iˉh ?ψ(r,t)

2m?2ψ(r,t)+V0|ψ(r,t)|2ψ(r,t),(1)

where m is the mass of the atom,V0=4πˉh2a/m is the e?ective interaction between two particles,and a is the scattering length.The GP model has been remarkably successful in predicting the condensate shape in an exter-nal potential,the dynamics of the expanding condensate cloud and the motion of quantised vortices.The family of the solitary waves for(1)was numerically obtained in [6].In a momentum-energy(p E)plot,the sequence of solitary waves has two branches meeting at a cusp where p and E simultaneously assume their minimum values. For each p in excess of the minimum p c,two values of E are possible,and E→∞as p→∞on each branch.On the lower(energy)branch the solutions are asymptotic to large circular vortex rings.As p and E decrease from in?nity on this branch,the solutions begin to lose their similarity to vortex rings.Eventually,for a momentum p0slightly in excess of p c,they lose their vorticity,and thereafter the solutions may better be described as“rar-efaction waves”.The upper branch solutions consist en-tirely of these waves and,as p→∞,they asymptotically approach the rational soliton solution of the Kadomtsev-Petviashvili Type I equation.

For two components,described by the wave functions ψ1andψ2,with N1and N2particles respectively,the GP equations become

iˉh ?ψ1

2m1?2+V11|ψ1|2+V12|ψ2|2

ψ1,

iˉh ?ψ2

2m2?2+V12|ψ1|2+V22|ψ2|2

ψ2,(2)

where m i is the mass of the atom of the i th conden-sate,and the coupling constants V ij are proportional to scattering lengths a ij via V ij=2πˉh2a ij/m ij,where m ij=m i m j/(m i+m j)is the reduced mass.

Two-component one-dimensional BECs have recently been considered and various structures have been iden-ti?ed[7]such as bound dark-dark,dark-bright,dark-antidark,dark-grey,https://www.sodocs.net/doc/818906748.html,plexes.In higher dimen-sions,domain walls[8]and skyrmions(vortons)[9]have been identi?ed by numerical simulations.Numerical sim-ulations of two-dimensional rotating two-component con-densates were performed[10]and the structure of vor-tex states were investigated.A phase diagram in the intercomponent-coupling versus rotation-frequency plane revealed rich equilibrium structures such as triangular, square and double-core lattices and vortex sheets.These simulations give a taste of a rich variety of static and dy-namic phenomena in multi-component condensates.One would expect the existence of various other families of solutions many of which have not yet been detected.

In what follows I determine the families of three-dimensional axisymmetric solitary wave solutions that move with a constant velocity U in uniform two-component mixture BECs.The trap geometry,relevant to experiments,introduces a harmonic-oscillator poten-tial in(2)together with additional parameters,places restrictions on studies of solitary waves and their stabil-ity and is irrelevant in the view of our interest to e?ects that occur in large systems.Also,I believe that addi-tional physical mechanisms should be introduced only after simpler models are well understood.Nevertheless, the results I obtain will be relevant to experiments with a su?ciently shallow trap,so the linear dimensions of the trap are much larger than the healing length.To reduce the number of parameters in the system,I will assume that the intracomponent scattering lengths and masses of individual components in the mixture are equal,so that m i=m and a ii=a,but the intercomponent scattering lengths di?er from a.

To?nd axisymmetric solitary wave solutions moving with velocity U in positive z?direction,I solve

2iU

?ψ1

?z

=?2ψ2+(1?α|ψ1|2?|ψ2|2?Λ2)ψ2,(3)

ψ1→ψ1∞,ψ2→φ2∞,as|x|→∞, where a dimensionless form of(2)is used such that the distances are measured in units of the correlation(heal-ing)lengthξ=ˉh/

√

3

FIG.2:(colour online)The dispersion curves of three fam-ilies of the axisymmetric solitary wave solutions of (3)with α=0.1and Λ2=0.1.The numbers next to the dots give the velocity of the solitary wave solution.The top (black)branch corresponds to VR-VR (VR-RP for U =0.58)complexes.The middle (green)branch shows p vs E for VR-SW com-plexes and the bottom (red)branch is the dispersion curve of SW-VR (SW-RP for U =0.58)complexes.The radii of the vortex ring solutions are shown on the upper inset as a function of U :the two top (red and green)lines give b 1and b 2correspondingly for VR-VR complexes.The two bottom (black and blue)lines represent b 1in VR-SW complex and b 2in SW-VR complex correspondingly.The bottom inset shows 3D plots of |ψi (s,z )|2of the SW-RP complex moving with the velocity U =0.

58.

and ψ21∞=1?αψ2

2∞.In the long-wave limit (k →0),(4)gives two acoustic branches ω±≈c ±k with the corresponding sound velocities c ±=1

(ψ21∞?ψ22∞)2+4α2ψ21∞ψ22∞

)1/2.The solitary waves I seek below are all subsonic,i.e.U

Each solitary wave complex that belongs to a family

of the solitary wave solutions for a chosen set of (α,Λ2)will be characterised by its velocity,U ,vortex radii b i ,momenta p i =(0,0,p i ),and energy E .The momentum (or impulse)of the i ?th component is p i =1

2i

ψ?i ?ψi ?

ψi ?ψ?

i dV with the convergent integrals p i were spelled out in [6]for a one component GP equation.Also,similar to [6],we form the energy,E ,by subtracting the energy of an undisturbed system of the same mass for which ψi =const everywhere,from the energy of the system with a solitary wave,so that the energy of the system

becomes

E =

14

(ψ21∞?|ψ1|2)2+(ψ22∞?|ψ2|2)2dV

(5)

+

α

?z

2+

?ψ2

2

(1?|ψ1|2?α|ψ2|2)(2ψ21∞?ψ1∞(ψ1+ψ?

1))dV

+1

4

(1?|ψ1|2?α|ψ2|2)

×(3ψ21∞?ψ1∞(ψ1+ψ?1)?|ψ1|2

)dV

+

1

1?2U 2were

introduced and the in?nite domain was mapped onto

the box (0,π2,

π

example of various transitions from one complex to an-

other in the system withα=0.05andΛ2=0.1.Notice that the radii of the vortex rings in VR-VR complexes

di?er with b2→b1as U→0.Also notice,that there is a cusp in the dispersion curve E vs p1+p2,since the

solitary wave complex moving with U=0.63belongs to

the upper branch.

Table1.The velocity,U,energy,E,momenta,p i and radii,b i, of the solitary wave solutions of(3)withα=0.05andΛ2=0.1. The sequence terminates at U=c?≈0.646.

0.5511892.380.81.831.50VR-VR

0.5810781.072.61.470.57VR-VR

0.6010174.969.61.12–VR-RP

0.6310266.279.0––RP-RP

αE p1p2b1b2complex 0.21401.12330– 4.787SW-VR 0.496.74.89237– 4.354SW-VR 0.663.017.5174– 2.986SW-VR

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