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Effect of scattering lengths on the dynamics of a two-component Bose-Einstein condensate

We examine the effect of the intra- and interspecies scattering lengths on the dynamics of a two-component Bose-Einstein condensate, particularly focusing on the existence and stability of solitonic excitations. For each type of possible soliton pairs stability ranges are presented in tabulated form. We also compare the numerically established stability of bright-bright, bright-dark and dark-dark solitons with our analytical prediction and with that of Painlev\'e-analysis of the dynamical equation. We demonstrate that tuning the inter-species scattering length away from the predicted value (keeping the intra-species coupling fixed) breaks the stability of the soliton pairs.


I. INTRODUCTION
Since Bose-Einstein condensates (BECs) can be routinely prepared in laboratories, ultracold gases became a very important testbed for many predictions of condensed matter physics [1]. The experimental examination of binary condensates started nearly the same time as for single condensates by using two different quantum states of the same species, such as 87 Rb [2] or 23 Na [3]. With the development of sympathetic cooling ultracold mixtures have been assembled from two different alkalies, 41 K-87 Rb [4,5], 7 Li-133 Cs [6] and 87 Rb-133 Cs [7] or for different isotopes of the same alkali atom 85 Rb-87 Rb [8]. The tunability of the interand intra-species scattering lengths via driving the mixture through a Feshbach-resonance has also been experimentally demonstrated [5,7].
The ability to create Bose-Einstein condensate(s), a highly coherent form of matter, also facilitated the convergence of two fields of physics: condensed matter physics and quantum optics, and therefore BECs became favourable candidates for examining the effects of nonlinearity in matter waves, where this nonlinearity originates from the mean-field representation of the interatomic interaction. The similarity between electromagnetic waves in nonlinear medium and coherent matter waves is also expressed in the equations of motion which is the nonlinear Schrödinger equation (NLS) for the former and the Gross-Pitaevskii (GP) equation for the latter. Although the physical interpretation of these equations is different, their structures are the same apart from the external potential term. Furthermore, in some cases this extra term can even be removed [9] and the GP equation is transformed into a form coinciding with the NLS equation. Consequently all results for the NLS equation known in nonlinear optics can be readily adapted to Bose-Einstein condensates.
However, the question of existence of solitons needs more attention than simply recognising the similarity between the two governing equations. The existence of solitons is strongly related to the integrability of the given physical model. An usual test to determine whether or not an equation is integrable is the Painlevé-test (P-test) [20]. It was shown that without any potential term or inhomogeneity the one-dimensional NLS equation, iu t + u xx ± 2 |u| 2 u = 0, is completely integrable both in its one-component [20,21] or multi-component [22,23] form. However, the inclusion of a potential term, v(x, t)u, in the one-component NLS equation or different coupling strengths in the multi-component NLS/GP equations fundamentally changes their integrability [24,25]. It was shown that integrability is preserved provided the external potential, v(x, t), has a specific form [25].
The authors examined [26] the integrability of the two-component coupled Gross-Pitaevskii (CGP) equations and lead to similar conclusion: the scattering lengths and the external potentials cannot be arbitrary if the integrability of CGP is to be preserved. The inter-and intraspecies scattering lengths must satisfy the following equation where ξ 1 = a 11 /a 21 , ξ 2 = a 22 /a 12 , κ 1 = µ 11 /µ 21 , κ 2 = µ 22 /µ 12 and µ ij denotes the reduced mass of pair of particles composed by an atom from the ith and jth species. On the right hand side of Eq. (1) n is a non-negative integer number. One may call n a classification number , because it determines the form of the external potentials for which CGP equations remain integrable. For example, for n = 2, the external potential, apart from the quadratic trapping potential, may even contain an imaginary time-dependent term [26]. This term can mimic the loss or gain in the number of particles of the given species. We note here that usually dissipation works against long-living coherent matter waves, however, the importance of this imaginary potential term has been analysed in [27,28] and shown to permit exact soliton solution [29]. In the context of BECs at finite temperature, the interaction of the condensate with the thermal cloud could also be taken into account as an imaginary term in the governing GP equation. This interaction, due to its stochastic nature, can influence the dynamics of the solitons, via density-fluctuation.
In this paper we carry out an analysis on how the intra-and interspecies interaction influences the dynamics of a binary mixture of Bose-Einstein condensates. We select out of the many possible systems of two component BECs the pairs 87 Rb-87 Rb (prepared in two distinct hyperfine states), 23 Na-87 Rb, and 7 Li-39 K. In the former two systems there is a possibility to study stability of bright-dark and dark-dark soliton pairs, while the last pair is capable to sustain bright-bright and bright-dark excitations.
The organization of the paper is as follows. Section II defines a quasi one-dimensional model derived from the general three-dimensional, coupled Gross-Pitaevskii equations assuming cigar-like harmonic oscillator trap potential. In the first part of section III we shall perform a stability analysis based on coupled soliton excitations, and on Eq. (1) of the P-test. In the second part of section III, possible new interesting modes exhibited by the bright-bright solitons will be shown. Section IV is devoted to a conclusion and the summary.

II. THE MODEL
In ultracold gases the interaction between two particles can usually be well described by a scalar parameter, the scattering length. For a two-component Bose-Einstein condensate one has to introduce three, possibly different, scattering lengths, characterising the intra-species interactions (a 11 , a 22 ) and the inter-species coupling (a 12 = a 21 ).
In the mean-field approximation a two-component BEC is described by the coupled Gross-Pitaevskii equations [30,31] in the form: where m i denotes the individual mass of the ith atomic species, Ω ij = 2π 2 a ij /µ ij with a ij being the 3D scattering length, µ ij = m i m j /(m i + m j ) is the reduced mass, and V i denotes the external trapping potential. In the following, indices i and j label the components, therefore, take only two values, 1 and 2. In the case of real trap potentials the normalisation of the wave functions reads as N i = |Ψ i | 2 dV with N i denoting the number of atoms in the ith component. We exclude those cases from our analysis where the species can transform into each other, therefore the number of atoms in each component hereafter is conserved.

A. Transformed equations
If the three dimensional quadratic trapping potential is weak in one direction, i.e.
where ω ix ≪ ω i⊥ , one may replace the three-dimensional equations (2) with a coupled system of quasi one-dimensional GP equations. Although Eq.
(2) is a nonlinear equation, physically we may assume that the weak x-direction decouples from the strong [yz]-plane, therefore the macroscopic wave functions can be written as with χ i,⊥ represents the ground-state solution of the corresponding two-dimensional Schrödinger equation in the [yz]-plane. The external potential introduces suitable units of length and time as a ⊥ = /m 1 ω 1,⊥ and τ = 1/ω 1,⊥ , respectively. By rescaling the spatial and temporal variable with a ⊥ and τ one obtains two quasi one-dimensional GP equations The normalisation is such that |ψ 1 | 2 dx = 1 and |ψ 2 | 2 dx = N 2 /N 1 . Moreover, the relation γ 2 = κ must hold if both species experience the same harmonic potential. (Note the slight departure from Ref. [32] in

B. Thomas-Fermi background
If the kinetic energy term is negligible compared to the potential energy terms in (5a-b), then one may apply the Thomas-Fermi approximation to determine the density distribution of the ground state. Following [33] we write the corresponding wave-functions as resulting in TF densities and TF energies The parameters A i and x i represent the amplitude of the density and the extension of the condensates, respectively. All these quantities are determined by the system parameters b ij , N i , λ i according to the following relations: while the extensions are Although the Thomas-Fermi density distribution is not physical at x = x i , it still provides a good starting point for analytical calculations. In our numerical treatment we will not use this approximation, rather start our simulations from the appropriate ground state solution of the one-dimensional GP equation.

C. Coupled soliton excitations
Now we are seeking solutions of Eqs. (5a-b) which support soliton excitations. A static soliton excitation can be written as [33] By inserting this ansatz into the GP Eqs. (5a-b) and neglecting the small potential contributions one obtains the coupled soliton equations as follows The normalisation of the soliton solutions reads as follows where the integrations, in both cases, are over the spatial extension, L i , of the solitons.
The above coupled equations admit generic moving soliton solutions of the types: brightbright (BB), bright-dark (BD), dark-bright (DB) and dark-dark (DD). We shall investigate here a simple static bright-dark soliton pair solution by taking the first component to be a static bright soliton and the second component to be the static dark soliton with yet unknown complex amplitudes q i and wave-vector, k i . The latter one is related to the width of the soliton k i ∼ 1/L i . By inserting the above ansatz into the Eqs. (12a-b) and equating the coefficients of the constant and x-dependent terms, respectively, one may conclude that the wave-vectors of the dark and bright solitons must be equal, k 1 = k 2 ≡ k.
The amplitudes are expressed by the system parameters as The energy of these excitations read as Suppose now that the width parameter k is a real number. The modulus of the amplitudes of the bright-dark soliton pair superimposed on the Thomas-Fermi background must be real numbers, thus one obtains a the following set of conditions for the existence of this brightdark soliton excitations One may apply the same method to generate static bright-bright or dark-dark soliton excitations. In the bright-bright case one obtains the following solutions with the conditions while the energies are E BB 1 = E BB 2 /κ = −k 2 /2. The dark-dark coupled soliton solutions read as follows with the conditions and energies E DD Our interesting result shows that in the bright-bright and dark-dark cases the energies are uniquely determined by the wave-vector and the mass ratio.
Note that the existence conditions (17), (19), and (21) are just the same as obtained in Ref. [32] for the existence of moving soliton pairs, while the constraints for static excitations were published in [33].

III. STABILITY TESTS BY SIMULATION
Below we are going to numerically investigate the stability of soliton pairs. To solve the time dependent coupled Gross-Pitaevskii equations (5a-b), a third-order accurate splitstep Fourier transform method is used as described in Ref. [34] for a single-component condensate. Here we solve the time independent coupled Gross-Pitaevskii equations for their numerically exact ground states using imaginary time method [35] combined with the split-step operator technique. Choosing initial distributions is necessary to this method, and the Thomas-Fermi approximate solution proved to be an effective initial guess for this purpose.
The procedure explained in the previous section can be generalised for solitons moving with velocity v. Such a solution is given by At v = 0 we obtain the static soliton excitation solution as given by (11) in the preceding section.
The existence conditions (17) prescribe various relations between domains of the interand intra coupling strengths, b ij , where it is possible to create bright-dark soliton pairs.
These domains are listed in Table I. Although the creation of a bright soliton is generally associated with attractive interaction (b ii < 0) among the particles, Table I clearly shows that, due to the appropriate other couplings, it may also be possible to create a bright soliton in case of repulsive interaction (b ii > 0). Such a situation occurs in case 4a of Tab. I which we are going to analyse below.

A. Moving bright-dark soliton pairs
Let us now investigate the stability of a bright-dark soliton pair for the case 4a of Table I by considering two experimentally accessible two-component BECs. The first is composed of the two hyper-fine states of 87 Rb atoms, the second is obtained from 23 Na and 87 Rb atoms. In both cases the scattering lengths are well known and can be tuned over a broad limit by using the Feshbach resonance method. Our aim here is to explore the sensitivity of the temporal evolution of soliton pair when the inter-atomic coupling strength, b 12 , is varied around the value prescribed by the ratio formula (1) obtained by performing a Painlevé-analysis of the coupled GP equations [26]. In this respect we fix the intra-atomic strengths, b ii , to values easily accessible to the experiments and vary the inter-atomic interaction values, b 12 , within a small range allowed by the case 4a listed in Table I. In order to represent a more realistic situation, a small velocity of v = 0.04 is given the solitons, a weak harmonic trapping potential is added along the longitudinal direction and the solitons are superimposed on the ground state density distribution. This procedure spatially confines the solitons without affecting their essential stability properties [32].
In Fig. 1, we plotted a bright-dark soliton pair composed of two hyperfine states of 1 + − no bright-dark soliton pair  the dark soliton has to drag this extra mass as well. This effect has recently been observed [18] with a 87 Rb-87 Rb condensate, prepared in the |F = 2, m F = 0 (bright soliton) and |F = 1, m F = 0 (dark soliton) hyperfine states.
However, if a 12 (or equivalently b 12 ) is tuned away from this particular value, the stability is lost, and the initial forms of the solitons are destroyed by the destructive interference of In Fig. 2 the evolution of a bright-dark soliton pair in binary BEC composed of 23 Na and 87 Rb atoms is plotted with corresponding snapshots of the density distribution. Upper two panels exhibit the bright ( 23 Na) and dark ( 87 Rb) excitations, respectively, which are stable for a long period due to the fine tuning of scattering lengths a ij (i, j = 1, 2) which satisfy the P-test formula (1). On the lower two panels of Fig. 2 we see, however, that the initial soliton excitations do not remain stable but are rapidly dissolved due to detuning the interspecies scattering length a 12 from the value obeying the integrability condition expressed by Eq.
(1). The existence conditions (19) prescribe the relations between domains of the inter-and intra-species coupling strengths, b ij , which are listed in Table II. Moreover, the common wave-vector is k = 1/(2C 1 ).  Another interesting scenario here is the one listed under the case 3a in Table II showing that it is possible to create a bright-bright pair, within the range 0 < b 12 < √ b 11 b 22 , in spite of the repulsive inter-atomic interaction. In order to investigate the stability of the brightbright soliton pair in this domain we simulate the temporal evolution of the BEC system composed of 7 Li and 39 K atoms accessible for experiments. As Fig. 3 shows, depending on the value of the interspecies interaction, two types of instability may occur. At values of a 12 less than a critical value of about 0.275 nm, the two standing solitons begin to repel each other (first two panels) and depart from each other as they were particles, preserving the total zero momentum. Above the critical a 12 value the soliton with constituents of the smaller mass splits into two equal parts and the heavier component becomes a breather keeping its original place (third pair of panels). At the critical value of a 12 both types of instability can be observed (third and fourth panel) simultaneously.

Summarising, both examples exhibited in
Similar instability effects of a bright-bright and other soliton pairs have also been observed in Ref. [39]. The instability is explained by Kevrekidis at al. [40] as the appearance of a negative eigenvalue pair obtained by a linear stability analysis. It has also been shown that higher order bright-bright soliton pairs would exhibit instability irrespectively of the system parameters. Interestingly, it is possible for one of the higher-order bright solitons, i.e., for the one which has the stronger self-attraction, to recover its stability by collapsing into one bright soliton and expel the other component from its original position.
Finally, it is important to note here, that the phases of the divided solitons are equal, therefore these particle-like wave-packets remain coherent with each other. Similarly to an optical beam splitter where light is divided into two coherent beams, one could divide these matter waves and use one of them as a probe and the other one as a control packet. The probe packet could undergo transformations, while the control packet is left to evolve freely, thereby a phase-difference could build up between the two packets. If the two packets are brought together again, the phase-difference could cause interference pattern which allows one to quantify coherence. This may potentially be helpful for calibrational purposes as well.

C. Dark-dark soliton pairs
At the end we are examining the third possible combination of soliton pairs; a dark soliton is excited in each condensate. Interestingly this pairing can be stable even if the interaction inside each condensate is attractive (a 11 , a 22 < 0). The dark-dark solitons are described by the following formulas For our numerical investigations a 87 Rb-87 Rb system has been chosen, with repulsive intraspecies interactions, a 11 = 5.335 nm and a 22 = 5.665 nm taken from [41]. This choice of coupling corresponds to scenario 4 of Table III

IV. SUMMARY
We have considered the existence and stability of soliton excitations in a two-component Bose-Einstein condensate both analytically and numerically. Our model allows the components to represent different elements (m 1 = m 2 ), but we also included those cases when two hyperfine states of the same element constitute the condensates. We excluded the possibil-ity that these components can transmute into each other, i.e. the hyperfine states cannot be driven into each other. The dynamics of these condensates, within the mean-field zerotemperature approximation, are governed by the coupled Gross-Pitaevskii equations. We chose our presentation to be suitable for combining analytical results with earlier numerical investigations, e.g. [26,32,42].
The occurrence of particle-like excitations together with conserved quantities are associated with the integrability of a nonlinear evolution field-equation, such as the coupled Gross-Pitaevskii equations. Using the results of a recent Painlevé analysis of the coupled Gross-Pitaevskii equation [26], we showed how the system parameters determine the integrability of this system. However, for the CGP equations the studies so far restricted themselves for either equal coupling coefficients (b 11 = b 12 = b 22 ) and/or equal masses. We note here that excitations with long lifetime may exist for a nonlinear evolution equation even when the integrability conditions are violated [43], but these cases are possibly exceptional and do not represent the generic behaviour.
We examined those ranges of system parameters which permit coupled soliton solutions of the governing equations (5a-b). Here, we utilised the two well-known one-soliton solutions of the one-component nonlinear Schrödinger equation; the bright and dark solitons. These excitations then spatially modify the Thomas-Fermi ground state in our theoretical description or by the appropriate ground state in our numerical simulation. We found analytically that each possible pair of these solitons have a range of system parameters where they are stable, and presented these ranges in tabulated form in Tabs. I, II, and III. One can present these findings differently, if the intra-species coupling coefficients are held relatively fixed, and only the inter-species one is tuned. For example, if we assume b 11 and b 22 to be positive (effective repulsive interaction) then the three tables I, II, and III can be combined into one graph (see Fig. 5).
Moreover, we also examined how the stability of these soliton pairs is changing if the inter-species coupling coefficient is detuned from the value predicted by the Painleé-analysis via Eq. (1). It was shown, irrespectively of the pairing, that the stability is lost, although the pairs were not equally sensitive to the detuning, e.g. the motion of the bright-dark pair became erratic and preserved its periodicity only qualitatively after changing a 12 from 5.5 nm to 5.4 nm.
Two types of instability of static bright-bright soliton pairs composed of species with unequal masses have been observed. When the interspecies interaction lies below a critical value (0 < b 12 < b cr 12 ), the static bright-bright pair evolves into a repulsive, momentum conserving, moving soliton pair. When the value of b 12 > b cr 12 then one of the bright soliton (the constituent with smaller mass) splits into two equal portion of same phase while the other bright soliton becomes a breather.
Well below the critical temperature of the Bose-Einstein condensate our description is expected to be adequate and the results could help experimentalist to modify the scattering lengths via Feshbach resonance into a range where stable soliton pairs exist. As the temperature increases, however, the interaction with the thermal cloud becomes more and more important, and could not be neglected any more. We have not yet examined how the interplay between the condensates and the thermal clouds (for each component) would modify the dynamics. This needs further research beyond the mean field description.