Formation of solitons in atomic Bose-Einstein condensates by dark-state adiabatic passage
aa r X i v : . [ c ond - m a t . s o f t ] O c t FORMATION OF SOLITONS IN ATOMICBOSE-EINSTEIN CONDENSATES BYDARK-STATE ADIABATIC PASSAGE
G. Juzeli¯unas a , J. Ruseckas a , P. ¨Ohberg b , M. Fleischhauer c a Institute of Theoretical Physics and Astronomy of Vilnius University,A. Goˇstauto 12, 01108 Vilnius, Lithuania b Department of Physics, School of Engineering and Physical Sciences,Heriot-Watt University, Edinburgh EH14 4AS, UK c Fachbereich Physik, Technische Universit¨at Kaiserslautern,D-67663 Kaiserslautern, Germany
November 20, 2018
Abstract
We propose a new method of creating solitons in elongated Bose-Einstein Con-densates (BECs) by sweeping three laser beams through the BEC. If one of thebeams is in the first order (TEM10) Hermite-Gaussian mode, its amplitude has atransversal π phase slip which can be transferred to the atoms creating a soliton.Using this method it is possible to circumvent the restriction set by the diffractionlimit inherent to conventional methods such as phase imprinting. The method allowsone to create multicomponent (vector) solitons of the dark-bright form as well as thedark-dark combination. In addition it is possible to create in a controllable way twoor more dark solitons with very small velocity and close to each other for studyingtheir collisional properties. Keywords: cold atoms, atomic Bose-Einstein condensates, solitons
PACS:
Atomic Bose-Einstein condensates (BECs) have received a great deal of interest since theywere first produced a decade ago [1–3]. They can exhibit various topological excitations,such as vortices and solitons. The dynamics of solitons in elongated BECs [5] is theatom-optics version of the nonlinear propagation of light pulses in optical fibres [4]. The1EC offers a remarkable freedom in terms of controlling the physical parameters such asdimensionality and even the sign of the strength of the atom-atom interaction [5].Solitons in BECs can be of both dark and bright type. Dark solitons are formed inBECs with repulsive interaction between the atoms [5]. For completely dark solitons thecondensate wavefunction is zero at the centre and changes its sign then crossing the centralpoint, i.e. the condensate wave-function has an infinitely steep π phase slip at the centre [5].On the other hand, the bright solitons are formed in BECs with repulsive interaction be-tween the atoms. The wave-function of the BEC is then localised at the centre [5] andgoes to zero further away from this point. The dark solitons which manifests themselves asa density minimum moving with a constant speed against a uniform background density,as well as bright solitons which are shape preserving wave packets, have both been exper-imentally realised [6–9]. The dynamics of solitons in BECs has been extensively studied.This has included investigations of the stability properties [10], as well as soliton dynamicsin inhomogeneous clouds [11], in multicomponent BECs [12,13] and in supersonic flow [14].Solitons can be created in various ways with a variable degree of controllability, e.g., bycolliding clouds of BEC [15–17] or engineering the density [18, 19].Traditionally dark solitons in BECs are created using phase imprinting [6, 7, 20–22],where a part of the condensate cloud is illuminated by a far detuned laser pulse in or-der to induce a sharp π phase slip in the wave function. The subsequent dynamics canindeed develop solitons [6, 7]. There are, however, some rather severe drawbacks withsuch a method of phase engineering. The resolution of the required phase slip is naturallyrestricted by the diffraction limit, i.e. the width of the phase slip should be larger thanan optical wave-length. Furthermore the phase imprinting does not produce a densityminimum characteristic to the dark solitons in the region of the phase change. Hencecompletely dark stationary solitons are difficult to achieve, which consequently results inso called grey moving solitons with a shallow density dip.It is of a significant interest to be able to create slowly moving, or even completelystationary solitons in order to test for instance their scattering properties. The shapes ofthe colliding solitons are to be preserved. In addition, a relative spatial shift is expected.This spatial shift, however, can only be detected for extremely slow solitons due to theinherent logarithmic dependence of the spatial shift on the relative velocity between thesolitons [23, 24]. The standard phase imprinting also inevitably creates phonons in thetrapped cloud because the constructed initial state is not the exact soliton solution largelydue to the missing density notch [6, 7].In this paper we show how states which have the required phase slip and density profilefor solitons can be created by sweeping three laser beams through an elongated BEC asshown in Fig. 1. If one of the beams is in the first order (TEM10) Hermite-Gaussianmode, its amplitude has a transversal π phase slip which will be transferred to atoms thusproducing a soliton. More importantly, with a sequence of three laser beams it is possibleto circumvent the restriction set by the diffraction limit. The laser fields reshape an atomicwave-function so that it acquires a zero-point. This leads to a hole in the atomic density,the width of which is only limited by the intensity ratios between the incident laser beamsdue to the geometric nature [25] of the process. The formation of the hole is accompanied2 % Ω Ω Ω Ω Ω Ω Ω Ω | i | i| i| i D(cid:12) E(cid:12) Z HH S L QJ (cid:3) G L U HF WL RQ [ ] (cid:11)(cid:13) Figure 1: a) The level scheme for the three laser beams Ω i ( i = 1 , , → Ω anda final stage Ω → Ω which engineers the phase and density of the BEC to produce asoliton.by a step-like (infinitely sharp) π phase slip in the atomic wave-function when crossingthe zero-point. The method is particularly useful for creating multicomponent (vector)solitons of the dark-bright form as well as the dark-dark combination. In addition it ispossible to create in a controllable way two or more slowly moving dark solitons close toeach other for studying their collisional properties. Consider a cigar-shape atomic BEC elongated in the z -direction. To create solitons in theBEC, we propose to sweep three incident laser beams across the condensate. The laserbeams interact with the condensate atoms in a tripod configuration [25, 26], i.e. the atomsare characterized by three ground states | i , | i , | i and an excited state | i . The j -thlaser drives resonantly the atomic transition between the ground state | j i and the excitedstate | i , see Fig. 1a. Initially the atoms forming the BEC are prepared in the hyperfineground state | i . Subsequently the lasers are swept through the BEC in the x -direction,i.e. perpendicular to the longitudinal axis z of the condensate.The sweeping process is made of two stages depicted in Fig. 1b. In the first stage the3asers 1 and 2 are applied in a counter-intuitive sequence to transfer adiabatically the atomsfrom the ground state | i to another ground state | i . If an additional laser 3 is on duringthe first stage, a partial transfer of atoms from the ground states | i to | i is possible [26].In that case a coherent superposition of states | i and | i is created after completing thefirst stage. In the second stage, the lasers 1, 2 and 3 are applied once again to transferatoms from the state | i back to the state | i and from the state | i to the state | i . Ifthe amplitude of one of these lasers Ω or Ω changes the sign at z = z , the BEC picks upa π phase shift at this point after the sweeping, and a soliton can be formed. This is thecase e.g. if one of the beams is the first order Hermite-Gaussian beam centered at z = 0.It is important to realize that at least two laser fields are needed to complete theadiabatic transfer of population between the ground states. Therefore the adiabaticity canbe violated in the vicinity of the point z = z where one of the Rabi frequencies Ω or Ω goes to zero. Inclusion of the third (support) laser 3 helps to avoid such a violation of theadiabaticity. In fact the atoms would experience absorption in the vicinity of z = z if thesupport laser 3 was missing during the second stage.It should be mentioned that there are similar previous proposals for creating vortices ina BEC via the two-laser Raman processes involving the transfer of an optical vortex to theatoms [27–29]. In these schemes the lasers are far detuned from the single-photon resonanceto avoid the absorption at the vortex core. In our scheme the lasers are in an exact single-photon resonance, so the use of the third (support) laser is essential to avoid the losses.An advantage of the resonant scheme is that an efficient and complete population transferis possible between the hyperfine ground states, whereas in the non-resonant case only afraction of population can be transferred [29]. Let us now provide a quantitative description of our scheme. The j -th laser beam ischaracterised by the complex Rabi frequency ˜Ω j = Ω j exp( i k j · r + iS j ) , with j = 1 , , j is the real amplitude, the phase being comprised of the local phase k j · r as well asthe global (distance-independent) phase S j . In what follows, the Rabi frequencies Ω andΩ are considered to be positive: Ω >
0, Ω >
0. Yet, the Rabi frequency Ω is allowedto be negative. This makes it possible to include an additional π phase shift in the spatialprofile of the first beam when crossing the zero-point at z = z .The electronic Hamiltonian of a tripod atom reads in the interaction representation:ˆ H e = − ¯ h ( ˜Ω | ih | + ˜Ω | ih | + ˜Ω | ih | ) + H . c . (1)The tripod atoms have two degenerate dark states | D i and | D i of zero eigen-energy( ˆ H e | D n i = 0) containing no excited-state contribution [25, 26], | D i = 1 √ ζ ( | i ′ − ζ | i ′ ) (2) | D i = 1 √ ζ (cid:16) ξ ( ζ | i ′ + | i ′ ) − ξ (1 + ζ ) | i ′ (cid:17) , (3)4here | j i ′ = | j i exp( i ( k − k j ) · r + i ( S − S j )) (with j = 1 , ,
3) are the modified atomicstate-vectors accommodating the phases of the incident laser fields, ζ = Ω / Ω is the ratiobetween the Rabi frequencies of the first and second fields, and ξ j are the normalised Rabifrequencies ( j = 1 , , ξ j = Ω j Ω , Ω = q Ω + Ω + Ω (4)with ξ > −∞ < ζ < + ∞ . The atomic dark states | D i and | D i depend on thecentre of mass coordinate r through the spatial dependence of the Rabi frequencies Ω j andstate-vectors | j i ′ . The full atomic state-vector of a multicomponent BEC is | Φ( r , t ) i = P j =1 | j i Ψ j ( r , t ), wherethe constituent wave functions Ψ j ( r , t ) describe the translational motion of the BEC in theinternal state | j i of the tripod scheme. The wave functions Ψ j ( r , t ) obey a multicomponentGross-Pitaevski equation of the form i ¯ h ∂∂t | Φ( r , t ) i = (cid:20) M ∇ + ˆ H e + ˆ V (cid:21) | Φ( r , t ) i , (5)where ˆ H e from Eq. (1) describes the light-induced transitions between the different internalstates of atoms. The diagonal operatorˆ V = X l>j =0 ( V j + g jl | Ψ l | ) | j ih j | . (6)accommodates the trapping potential V j ( r ) for the j -th internal state, as well as the non-linear interaction between the components j and l characterised by the strength g jl =4 π ¯ h a jl /m , with a jl being the corresponding scattering length. We shall apply the adiabatic approximation [25, 30, 31] under which atoms evolve withintheir dark-state manifold during the sweeping. This is legitimate if the total Rabi fre-quency Ω is sufficiently large compared to the inverse sweeping duration τ − . The fullatomic state-vector can then be expanded as: | Φ( r , t ) i = P n =1 Ψ ( D ) n ( r , t ) | D n ( r , t ) i , where acomposite wavefunction Ψ ( D ) n ( r ) describes the translational motion of an atom in the darkstate | D n ( r , t ) i . The atomic centre of mass motion is thus represented by a two-componentwave-function Ψ = Ψ ( D )1 Ψ ( D )2 ! . (7)5beying the following equation of motion [25]: i ¯ h ∂∂t Ψ = (cid:20) M ( − i ¯ h ∇ − A ) + V ( r ) + φ − β (cid:21) Ψ , (8)where the effective vector potential A and the matrix β are the 2 × A n,m = i ¯ h h D n ( r , t ) |∇ D m ( r , t ) i and β n,m = i ¯ h h D n ( r , t ) | ∂/∂tD m ( r , t ) i . The former A is known as the Mead-Berry con-nection [32, 33], whereas the latter matrix β is responsible for the geometric phase [34].The 2 × φ is the effective trapping potential (explicitly presented in Ref. [25])appearing due to the spatial dependence of the dark states. Assuming that all three beamsco-propagate ( k ≈ k ≈ k ), the effective vector potential [25] reduces to A = ¯ h ξ ζ ∇ ζ i − i ! (9)Lastly, the 2 × V originating from the operator ˆ V , Eq. (6), accommodates thetrapping potential for the dark states [25] as well as the atom-atom coupling. Suppose the incident laser beams are swept through a trapped BEC along the x axis witha velocity v , as shown in Fig. 1b. This can be done either by shifting in the transversal( x ) direction the laser beams propagating along the y axis or by applying a set of laserpulses of the appropriate shape and sequence propagating in the x direction. In the lattercase, the sweeping velocity v will coincide with the speed of light. In both cases theadiabatic dark states depend on time in the following way: | D n ( r , t ) i ≡ | D n ( r ′ ) i , where r ′ = ( x ′ , y, z ) ≡ ( x − vt, y, z ) is the atomic coordinate in the frame of the moving laserfields. Let us assume that the time, τ sweep = d/v , it takes to sweep the laser beams througha BEC of the width d , is small compared to the time associated with the BEC chemicalpotential τ µ = ¯ h/µ which is typically of the order of 10 − s. In that case one can neglectthe dynamics of the atomic centre of mass during the sweeping. Consequently the timeevolution of the multicomponent wave-function during the sweeping is governed by thematrix-term β = − vA x featured in Eq. (8), giving i ¯ h∂ t Ψ = vA x Ψ , (10)where the A x the effective vector potential along the sweeping direction.In passing we note that the subsequent time evolution of the BEC after the two-stagesweeping will be described by the general Gross-Pitaevski equation (5) with the light fieldsoff ( ˆ H e = 0), as we shall do in Section 4.Returning to Eq. (10), since vA x commutes with itself at different times, one can relatethe wave-function Ψ( t ) at a final time t = t f to the one at the initial time t = t i asΨ( r , t f ) = exp ( − i Θ) Ψ( r , t i ) , (11)6here the exponent Θ is a 2 × h Z t f t i A x ( r − v t ) vdt = 1¯ h Z x i x f A x ( r ′ ) dx ′ . (12)and the integration is over the sweeping path r ′ = ( x − vt, y, z ) from x f = x − vt f to x i = x − vt i . In most cases of interest the initial and final times can be considered to besufficiently remote, so that the spatial integration can be from x f = −∞ to x i = + ∞ . Let us now analyze the proposed two-stage setup in more details. In the first stage bothRabi frequencies Ω and Ω are positive. The lasers 1 and 2 are applied in a counterintuitiveorder (see Fig. 1b), where the ratio ζ = Ω / Ω changes from ζ ( t ′ i ) = 0 to ζ ( t ′ f ) = + ∞ .On the other hand, the laser 3 is dominant for both the initial and final times where ξ = Ω / Ω = 1. Initially the BEC has the wave-function Ψ( r ) and is in the internalatomic ground state | i which coincides with the first dark state at the initial time t ′ i , i.e. | D ( r , t ′ i ) i = | i . The full initial atomic state-vector is therefore | Φ( r , t ′ i ) i = Ψ( r ) | D ( r , t ′ i ) i .This provides the following initial condition for the multicomponent wave-function:Ψ( r , t ′ i ) = Ψ( r )0 ! . (13)Equations (9) and (11)–(13) yield the multicomponent wave-function after the first stageΨ( r , t ′ f ) = Ψ( r ) cos β − sin β ! , (14)where β = Z + ∞−∞ ξ ∂ arctan ζ∂x ′ dx ′ (15)is the mixing angle between the dark states acquired in the first stage.Suppose we have the following laser beams. The second beam Ω is the Gaussian beamcharacterised by a waist σ z in the z direction. The beam is centered at x ′ = ¯ x + ∆ in thesweeping direction and at z = 0 in the z direction,Ω = Ae − z /σ z − ( x ′ − ¯ x − ∆) /σ x . (16)The first beam Ω is characterised by the same amplitude A , the same waist σ z and thesame width σ x . Yet it is centered at x ′ = ¯ x − ∆ in the sweeping direction, where 2∆is the separation between the two beams. The beam waists should be of the order of thecondensate length (or larger) in the z -direction, so that the whole condensate is illuminatedby the beams.The third beam is considered to change little along the sweeping direction x . Further-more it has the same width σ z in the z -direction as the first two beamsΩ = Aκe − z /σ z . (17)7 β / ( π / ) κ Figure 2: Dependence of the mixing angle β on the relative amplitude of the third beam κ .The spatial separation between the first and the second beams is taken to be 2∆ = 1 . σ x where σ x is the width of the beams in the sweeping direction.The first stage is aimed at creating a superposition of states | i and | i . Since we takeall the beams to be the Gaussian beams characterized by the same widths σ z , the Rabifrequency ratios Ω / Ω and Ω / Ω have no z -dependence. As a result the acquired mixingangle β has no z -dependence, i.e. it is uniform along the BEC. The magnitude of β dependson the relative intensity of the third laser. If the third laser is weak ( ξ = Ω / Ω → ζ = Ω / Ω = 1), the mixing between the states | i and | i is small: β ≪
1. On the other hand, if the Rabi frequency Ω is comparable with Ω and Ω at thecrossing point where ζ = Ω / Ω = 1, the mixing can be close to its maximum: β ≈ π/ In the second stage the Rabi frequency Ω can be both positive and negative depending onthe transversal coordinate z . The laser 1 is now applied first, so that the ratio ζ = Ω / Ω changes from ζ ( t i ) = ±∞ to ζ ( t f ) = 0 in the second stage. Again the third laser dominatesfor the initial and final times: Ω / Ω = 1. The second stage takes place immediately aftercompleting the first stage, so the multicomponent wave-function of the first stage (14)serves as an initial condition for the second stage.Equations (11), (12) and (14) together with (2) and (3) yield the total state vector afterthe second stage: | Φ( r , t f ) i = | i Ψ( r ) (cid:16) sin γ cos β − e iν cos γ sin β (cid:17) | i Ψ( r ) (cid:16) cos γ cos β + e iν sin γ sin β (cid:17) . (18)where ν = S − S + S ′ − S ′ is the phase mismatch between the Rabi frequencies Ω andΩ in the first and second stages. The resulting mixing angle acquired in the second stageis γ ≡ γ z = Z + ∞−∞ (1 − ξ ) ∂ arctan ζ∂x ′ dx ′ . (19)If the first and second lasers are weak (Ω / Ω → ζ = Ω / Ω =1), the mixing angle is small γ z ≪
1. On the other hand, if first and second lasers arestrong at this point, we have γ z → ∓ π/
2. The change in sign of γ z will introduce a phaseshift which is needed to create solitons.In the second stage the first beam Ω is a first-order (in the z direction) Hermite-Gaussian beam centered at z = 0 and x ′ = ˜ x + ˜∆Ω = A zB e − z /σ z − ( x ′ − ˜ x − ˜∆) /σ x , (20)where z = ± B represents a distance where Ω = ± Ω for x ′ = ˜ x . In most cases of interestthe distance B is much smaller than the waist of the beams: B ≪ σ z . The second beamΩ is the ordinary Gaussian beam centered at z = 0 along the BEC and x ′ = ˜ x − ˜∆ in thesweeping direction Ω = Ae − z /σ z − ( x ′ − ˜ x + ˜∆) /σ x , (21)where 2 ˜∆ is the separation between the two beams. The ratio between the Rabi frequenciesreads then ζ = Ω Ω = zB e x ′ − ˜ x ) /σ x , (22)Equation (22) provides the following limiting cases: ζ ≡ ζ ( z, x ′ ) = ( , for x ′ → + ∞ , ±∞ , for x ′ → −∞ . (23)Finally let us determine the crossing point where Ω = Ω . Using Eq. (22), the condition | ζ | = 1 yields the crossing point x ′ = x ′ cr for a fixed z coordinate: x ′ cr = ˜ x + σ zB . (24)Specifically, if z = B , the crossing point is: x ′ cr = ˜ x . Since B ≪ σ , the Rabi frequencies at z = B and x ′ = ˜ x are: Ω = Ω ≈ Ae − ˜∆ /σ x . (25)In the next subsection we shall analyse in more detail the multicomponent wave-functionafter completing the second stage. 9 Ψ z Figure 3: Multicomponent wave-function after completing the second stage in the casewhere the second component is populated after the first stage ( β = 0) and there is nophase mismatch between the lasers of the first and second stages ( ν = 0). The secondand third laser beams are taken to be the Gaussian beams with equal widths σ z . The firstlaser beam is the first order Hermite-Gaussian beam with same width σ z . The parametersused are 2 ˜∆ /σ x = 1 . B/σ z = 0 . κ = 0 .
1. The wave function of the first (second)component is plotted in a solid (dashed) line.
Suppose that there is no phase mismatch between the lasers of the first and second stages: ν = 0. In that case Eq. (18) yields | Φ( r , t f ) i = Ψ( r )[ − sin( γ z − β ) | i + cos( γ z − β ) | i ] , (26)If β = 0, the second component is populated after the first stage. After the whole sweepingthe state-vector then takes the form | Φ( r , t f ) i = Ψ( r )[ − sin γ z | i + cos γ z | i ] . (27)In this case the first component alters the sign at z = z where the Rabi frequency Ω or Ω (and hence γ z ) crosses the zero-point. On the other hand, the second componentis maximum at this point and symmetrically decays to zero away from this point. Sucha multicomponent wave-function has a shape close to that of a soliton of the dark-brightform (see Fig. 3). This will indeed lead to the formation of such a soliton, as we shall fromthe analysis of the subsequent time-evolution presented in the next Section.On the other hand, β = π/ Ψ z Figure 4: Multicomponent wave-function after completing the second stage in the casewhere where both components are initially populated after the first stage ( β = π/
4) andthere is no phase mismatch between the lasers of the first and second stages ( ν = 0). Thesecond and third laser beams are taken to be the Gaussian beams with equal widths σ z .The first laser beam is the first order Hermite-Gaussian beam with same width σ z . Theparameters used are 2 ˜∆ /σ x = 1 . B/σ z = 0 . κ = 0 .
1. The wave function of the first(second) component is plotted in a solid (dashed) line.populated with equal probabilities. Thus we have after the sweeping: | Φ( r , t f ) i = − Ψ( r )[ − sin( γ z − π/ | i + sin( γ z + π/ | i ] . (28)In that case both components of the wave-function acquire a π phase shift in a vicinity of z = z where Ω = 0, as one can see clearly in the Fig. 4 Note that the zero-points ofeach component are slightly shifted with respect to each other. This makes it possible toproduce two component dark-dark solitons oscillating around each other, as we shall seein the following Section.If β = π/ π/ ν = π/ | Φ( r , t f ) i = − Ψ( r ) e iγ z √ | i − i | i ] . (29)In that case both components are characterised by the same spatial modulation exp ( iγ z )and have a relative phase π/ ∇ γ z . Furthermore there is no hole inthe atomic density of neither component after the sweeping, similar the case in the phaseimprinting techniques. 11n this way, the creation of solitons can be controlled by changing the mixing angle β and the phase mismatch ν The optical preparation of the initial state of the two-component Bose-Einstein condensatedescribed in the previous section, is fast compared to any characteristic dynamics in theBose-Einstein condensate. This is the case if the time τ sweep = d/v it takes to sweep thelaser beams through a BEC of the width d , is small compared to the time associated withthe BEC chemical potential τ µ = ¯ h/µ which is typically of the order of 10 − s. With theprepared initial state and for sufficiently low temperatures we can therefore describe thesubsequent dynamics using a two-component Gross-Pitaevskii equation [12] i ¯ h ∂∂t Ψ = [ − ¯ h m ∇ + V ( z ) + g | Ψ | + g | Ψ | ]Ψ (30) i ¯ h ∂∂t Ψ = [ − ¯ h m ∇ + V ( z ) + g | Ψ | + g | Ψ | ]Ψ . (31)The external potential is here chosen to be quadratic in the z -direction, V ( z ) = 12 mω z , (32)where ω is the trap frequency and m the atomic mass. The two-body interactions aredescribed by g ij = 4 π ¯ h a ij mS , i, j = { , } (33)with the scattering lengths a ij which represents the intra and inter collisional interactionsbetween the atoms in the states 1 and 2. In Eq. (33) we have introduced the effectivecross-section S of the elongated cloud. Strictly speaking the elongated Bose-Einstein con-densate is three dimensional. If, however, the transversal trapping is sufficiently strong,the dynamics can be considered effectively one dimensional, as in Eqs. (30) and (31). Thisrequires that the corresponding transversal ground state energy is much larger than thechemical potential of the condensate. We choose the normalisation as R dz | Ψ i ( z ) | = N i ,where N i is the particle number in condensate i ( i = 1 , g : g : g = 1 . .
97 : 1 . g N = 286 and N = N . The unit of length is q ¯ hmω and time is in units of ω − .In figure 5 we show the dark-bright soliton dynamics whose initial state is prepared bychoosing β = 0 and ν = 0. The two-component system which has one dark soliton incomponent 1 and a bright soliton in component 2, is stable, i.e. the solitons are stationary.This shows that the initial state is indeed close to the exact soliton solution. If the initialstate is prepared with β = π/ ν = 0, on the other hand, the dynamics is strikingly12igure 5: The dark-bright soliton. The two figures show the one dimensional density as afunction of time for component 1 and 2. The lighter (darker) colours correspond to higher(lower) atomic densities.different, see figure 6. In this case we create two dark solitons with opposite phase gradients,hence there is an oscillatory motion, sometimes referred to as a soliton molecule. Such abound state is only stable if the soliton velocities are low [12] which is indeed the case here.Alternatively, with β = π/ ν = π/
2, the solitons move in unison as shown in figure7. The large oscillatory motion appearing in Fig. 7 stems from the fact that the condensatedensity is not homogeneous, hence the solitons experience an effective trap [11].
In summary, we have proposed a new method of creating solitons in elongated Bose-EinsteinCondensates (BECs) by sweeping three laser beams through the BEC. If one of the beamsis the first order (TEM10) Hermite-Gaussian mode, its amplitude has a transversal π phaseslip which will be transferred to the atoms thus creating a soliton. Using this method it ispossible to circumvent the restriction set by the diffraction limit. The method allows oneto create multicomponent (vector) solitons of the dark-bright form as well as the dark-darkcombination. In addition it is possible to create in a controllable way two or more slowlymoving dark solitons close to each other for studying the collisional properties. For thisthe first beam Ω should represent a superposition of the zero and second order Hermite-Gaussian modes in the second stage. The soliton collisions will be considered in moredetails elsewhere. Acknowledgements
This work was supported by the Alexander-von-Humboldt foundation through the insti-tutional collaborative grant between the University of Kaiserslautern and the Institute of13igure 6: The bound state dark-dark soliton. For sufficiently low initial soliton velocitiesthe two dark solitons perform an oscillatory motion around each other. The figures showthe one dimensional atomic density as a function of time for component 1 and 2. Thelighter (darker) colours correspond to higher (lower) densities.Figure 7: The co-propagating dark-dark solitons. If the initial phase gradients of the twosoliton solutions are chosen to be the same the dark solitons propagate in unison. The twofigures show the one dimensional density as a function of time for component 1 and 2. Thelighter (darker) colours correspond to higher (lower) atomic densities.14heoretical Physics and Astronomy of Vilnius University. P. ¨O. acknowledges support fromthe EPSRC and the Royal Society of Edinburgh.
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