Scattering of atomic dark-bright solitons from narrow impurities

Scattering of atomic dark-bright solitons from narrow impurities A. ´ Alvarez1, J. Cuevas2, F.R. Romero1, C. Hamner3, J.J. Chang3, P. Engels3, P.G. Kevrekidis4, and D.J. Frantzeskakis5 1Grupo de F´ısica No Lineal, Universidad de Sevilla. ´ Area de F´ısica Te´orica. Facultad de F´ısica, Avda. Reina Mercedes, s/n, E-41012 Sevilla, Spain 2Grupo de F´ısica No Lineal, Universidad de Sevilla. Departamento de F´ısica Aplicada I. Escuela Polit´ecnica Superior, C/ Virgen de ´ Africa, 7, E-41011 Sevilla, Spain 3Washington State University, Department of Physics and Astronomy, Pullman, Washington 99164, USA 4Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA 5Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece In this work, we examine the collision of an atomic dark-bright soliton, in a two-component Bose-Einstein condensate, with a Gaussian barrier or well. The study has both an experimental component and a theoretical/computational one. First, we present the results of an experiment, illustrating the classical particle phenomenology (transmission or reflection) in the case of an equal barrier or well in both components. Then, motivated by the experimental observations, we perform systematic simulations considering not only the case of equal heights, but also the considerably more complex setting, where the potential affects only one of the two components. We systematically classify the ensuing cases within a two-parameter diagram of barrier amplitudes in the two components, and provide intuitive explanations for the resulting observations, as well as of their variations as the size of the barrier changes. I. INTRODUCTION Atomic Bose-Einstein condensates (BECs) [1, 2] provide an ideal platform for the study of nonlinear phenomena at the mesoscopic scale (see, e.g., the reviews [3–6]). In this context, of particular interest are multi-component BECs, which – in their simplest version – are composed of two different hyperfine atomic levels of the same atom (e.g., 87Rb [7, 8]). In such systems, a rich variety of structures can be observed, which can not arise in single-component BECs. Regarding macroscopic nonlinear excitations of multi-component BECs, a distinctive feature of interest involves the potential formation of dark-bright (DB) solitons. This type of vector soliton consists of a dark soliton in one component coupled to a bright soliton in the second component. These solitons in repulsive homogeneous BECs are usually referred to as “symbiotic” solitons, with this characterization stemming from the fact that the bright component cannot be supported in a stand-alone fashion (it is only supported as such in attractive BECs [9, 10]; see also the review [5]), and is only sustained because of the presence of its dark-counterpart, which acts as an external trapping potential. We highlight here the fact that we refer to homogeneous BECs, as in the presence of external potentials (such as optical lattices [11]), bright localized structures can be sustained but are critically shaped by the form of the potential. Dark-bright solitons have been studied extensively in different settings in a large number of theoretical works (see, e.g., Refs. [12–21]), while they have also been observed in experiments, both in two-component 87Rb BECs [22–27] and in nonlinear optics [29–31]. In the BEC context, these experimental studies have chiefly involved the dynamics of a single DB soliton in a trap [22, 24], the generation of multiple DB solitons in a counterflow experiment [23], the study of their interactions [25], as well as the creation of SU(2)-rotated DB solitons, in the form of beating dark-dark solitons [26, 27]. On the other hand, the fundamental problem of the interaction of solitons with localized impurities has been considered both in nonlinear wave theory [32] and solid state physics [33]. The interaction of either bright or dark solitons with δ-like impurities has been investigated in the framework of the nonlinear Schr¨odinger (NLS) equation (see, e.g., Refs. [34–38]). Relevant studies in the physics of atomic BECs have also appeared some time ago (see, e.g., Refs. [39–41]). Among these works [39] considered a dark soliton as perturbed by a defect (both the steady state, and the associated dynamical problem), while [40] examined the collision of a moving dark soliton with a defect and captured also the leading order effect of produced radiation. Recent studies that we are aware of have chiefly been concerned with bright solitons both in the setting of potential wells [42, 43] (examining resonances that can arise therein), and in that of barriers [44, 45]. Most of the above works have been restricted to the mean-field description of the NLS equation. Nevertheless, the potential role of quantum fluctuations in such interactions (especially for smaller atom numbers or narrower barriers) has been illustrated; see e.g. [45]. The examination of such quantum effects in soliton-barrier [46, 47] and even soliton-soliton [48] interactions is a subject of increasing interest over the past few years. It has been argued, in particular, that the scattering of a bright solitary wave in an attractive BEC from a barrier potential leads to the 2 formation of a quantum superposition state of two distinct wave packets traveling through real space [46]. The full quantum mechanical scattering problem for short range potentials and at low energy was considered in [47] and error bounds on the scattering coefficients were derived. Finally, in [48] it was argued that initially uncorrelated quantum solitons lead to entanglement, upon their pair-wise interaction. Finally, the work of [49] has also mostly focused on bright soliton-barrier interactions at the NLS level, yet is the first one to touch upon two-component dark-bright soliton interactions with a barrier acting on the bright soliton only. In this context, localized impurities can be created as focused far-detuned laser beams and have already been used in experiments with dark solitons [50, 51]; we also note very recent experiments with matter-wave bright solitons of 7Li [52] and 85Rb [53] atoms and localized barriers. However, such soliton-defect interactions are far less well explored in the case of the multi-component setting (see, e.g., Ref. [20] where the statics of DB solitons was studied in the presence of δ-like impurities). It is the aim of this work to address this problem and study, in particular, the scattering of atomic DB solitons at narrow impurities. Nevertheless, it should be noted that here we will restrict our considerations to the mean-field level considerations. It would be an interesting problem for further study to examine how quantum mechanical effects modify the phenomenology presented herein. Our presentation will be structured as follows. In section II, we present the relevant prototypical model setup in the form of two coupled Gross-Pitaevskii (GP) equations describing the dynamics of a binary BEC with repulsive interactions; DB solitons for this model are presented as well. We also present results of a prototypical experiment where the scattering of DB solitons at a barrier – which is present (and equal) in both atomic components – is studied; this experiment provides the motivation for a more systematic theoretical study which is presented in the next section. In particular, in sections III, we numerically explore the dynamics of single DB solitons in a harmonic trap. The simpler scenario that is studied refers to the case where the impurity is equal in the two components – as in the case of the experiment; we find that a particle-based phenomenology is sufficient to capture the main characteristics here. On the other hand, we identify a far more significant wealth of possibilities in the setting where the barrier (or well) is applicable only in one of the two components. We present a systematic study within the plane of the barrier or well amplitudes for the two components, providing intuitive explanations (on the basis of effective potentials), wherever possible, for the observed phenomenology. Finally, in section IV, we summarize our findings and present our conclusions, as well as a number of directions for potential future studies. II. MODEL AND EXPERIMENTAL MOTIVATION A. Gross-Pitaevskii equations and dark-bright solitons We consider a two-component BEC composed of two different hyperfine states of the same alkali isotope. If this binary condensate is confined in a highly anisotropic trap (with longitudinal and transverse trapping frequencies ωx≪ω⊥), then the mean-field dynamics of the BEC can be described by the following system of two coupled GP equations [1, 2]: i~∂tψj= −~2 2m∂2 xψj+Vj(x)−µj+ 2 X k=1 gjk|ψk|2!ψj,(1) where ψj(x, t) (j= 1,2) are the macroscopic wave functions of the two components normalized to the numbers of atoms Nj=R+∞ −∞ |ψj|2dx,mis the atomic mass, µjare the chemical potentials, gjk = 2~ω⊥ajk are the effective 1D coupling constants (ajk are the s-wave scattering lengths), while Vj(x) denote the external trapping potentials for each species. In our considerations below, we will assume that the component 1 (2) supports a dark (bright) soliton; additionally, we will assume that both components are confined by the usual harmonic trap, namely VH(x) = (1/2)mω2 xx2, while – for each component – an additional localized “impurity” potential, which can be generated by off-resonant Gaussian laser beams, is also present. Thus, the external potentials Vj(x) for each of the two components are described as: V1(x) = VH+Edexp −2x2 ǫ2 d, V2(x) = VH+Ebexp −2x2 ǫ2 b,(2) where the parameters Ed,Eband ǫd,ǫbset, respectively, the amplitudes and widths of the impurities in each component. Notice that for a blue- or red-detuned laser beam, the impurity potentials can either repel (Ed,b >0) or attract (Ed,b <0) the atoms of the respective component of the condensate. We now cast Eqs. (1) into a dimensionless form as follows: measuring the densities |ψj|2, length, time and energy 3 in units of 2a11,a⊥=p~/(mω⊥), ω−1 ⊥and ~ω⊥, respectively, Eqs. (1) are reduced to the form: i∂tu=−1 2∂2 xu+V1(x)u+ (|u|2+ ˜g12|v|2−µ1)u, (3) i∂tv=−1 2∂2 xv+V2(x)v+ (˜g12|u|2+ ˜g22|v|2−µ2)v. (4) In Eqs. (3)-(4), the wave functions uand vcorrespond to ψ1and ψ2respectively, the normalized nonlinearity coefficients are given by ˜gj2=gj2/g11, while the normalized harmonic trap potential (incorporated in V1(x) and V2(x) as discussed above) is now given by VH(x) = (1/2)Ω2x2, where Ω = ωx/ω⊥. Notice that in the GP Eqs. (3)-(4) the number of atoms ND,B of each component is conserved; in fact, ND,B are given by ND,B = (a⊥/2a11)˜ ND,B, where ˜ ND=R∞ −∞ |u(x)|2dxand ˜ NB=R∞ −∞ |v(x)|2dxare the respective integrals of motion of the normalized GP Eqs. (3)-(4). As mentioned above, in the physically relevant setting of 87Rb, the scattering lengths characterizing the intra- and inter-component atomic collisions are almost equal; thus, to a first approximation, one may assume that ˜g12 = ˜g22 ≈1, which means that the system of Eqs. (3)-(4) is of the Manakov type [54]; in this case, the system is integrable in the absence of the external potentials V1,2(x) and admits exact analytical dark-bright soliton solutions. Particularly, considering the boundary conditions |u|2→µ1and |v|2→0 as |x| → ∞, Eqs. (3)-(4) possess an exact analytical one-DB-soliton solution of the following form (see, e.g., Ref. [12]): uDB(x, t) = √µ1{cos φtanhξ+isin φ},(5) vDB(x, t) = ηsechξexp[ikx +iθ(t)],(6) where ξ=D(x−x0(t)), φis the dark soliton’s phase angle, cos φand ηrepresent the amplitudes of the dark and bright solitons, and Dand x0(t) are associated with the inverse width and the center position of the DB soliton. Furthermore, k=Dtanφ= const and θ(t) are the wavenumber and phase of the bright soliton, respectively. The above parameters of the DB-soliton are connected through the equations: D2=µ1cos2φ−η2, ˙x0=Dtanφ, and θ(t) = (1/2)(D2−k2+µ2−µ1)t, where ˙x0is the DB soliton velocity. B. Experimental results Having introduced our setup, we now proceed by presenting results of an experiment dealing with scattering of atomic DB solitons at barriers. In fact, the results that will be presented below, motivate the more detailed theoretical investigation of this paper, but also illustrate the experimental tractability of this direction, and showcase prototypical results along this vein. Our experimental results are summarized in Fig. 1. There, it is shown that, depending on the barrier height, DB solitons are either reflected by or transmitted through a repulsive barrier. The experiment is conducted with a BEC of 4.5·105atoms of 87Rb confined in an optical dipole trap with trapping frequencies {ωaxial, ωvertical, ωhorizontal}= 2π× {1.4,120,174}Hz. The solitons are generated by transferring a small fraction of the atoms from the initial |F, mFi=|1,−1ihyperfine state to the |2,−2istate and exploiting a counter-flow induced modulational instability [23] generated by a magnetic gradient along the axial direction. The number of solitons, as well as their initial positions, can be controlled by adjusting experimental parameters such as the number of atoms transferred into the second state and the strength and duration of the magnetic gradient used to induce the counter-flow. For the data presented in Fig. 1, solitons traveling towards the right are generated in the left part of the BEC, the magnetic gradient is subsequently turned off, and the solitons continue to move towards the trap center. The oscillations of individual solitons in a trap have been investigated in detail in [24]. For the present data, we additionally ramp on a repulsive barrier at the center of the trap. The barrier is generated from a 660 nm laser beam with a narrow waist of approximately 18 µm in the direction of the BEC axis and has an aspect ratio of 4. For imaging, the two components of the BEC are vertically separated during a short free expansion time of 7 ms for the upper cloud and 8 ms for the lower cloud [23, 24]. For barrier depths larger than the chemical potential [cf. Fig. 1(e-h)] we observe confinement of the dark-bright solitons to the left half of the BEC. This is consistent with having two isolated BECs. For a barrier depth of approximately half the chemical potential [cf. Fig. 1(a-d)] we observe solitons penetrating through the barrier; see, e.g., especially the panel (c) in this setting. The dynamics observed here is a subset of the rich behavior expected for soliton-barrier interactions. These dynamics can be extended to more exotic regimes, e.g., by the addition of a species selective barrier. The latter will be examined in more detail in our theoretical investigation below. 4 a) b) c) d) e) f) g) h) |2, -2> |1, -1> 500 mm a) b) c) d) e) f) g) h) |2, -2> |1, -1> 500 mm FIG. 1: Experimental images of soliton-barrier interaction. (a-d) Peak barrier potential is 0.56 ·µBEC . The in-trap evolution times after generation of the solitons are a) 250 ms, b) 500 ms, c) 750 ms, d) 1000 ms. (e-h) Similar to (a-d) but for a peak barrier potential of 1.11 ·µBEC and evolution times e) 300 ms, f) 600 ms, g) 700 ms, h) 900 ms. In all cases the chemical potential of the BEC, µBEC , is approximately 36 nK (about 750 Hz). III. NUMERICAL INVESTIGATIONS In our numerical simulations below, we will assume that the two-component BEC under consideration consists of two different hyperfine states of 87Rb, namely the states |1,−1iand |2,−2iused in the experiment presented in the previous section (see also Refs. [22–27]). In this case, the scattering lengths take the values a11 = 100.4a0, a12 = 98.98a0and a22 = 98.98a0(where a0is the Bohr radius); accordingly, the normalized nonlinearity coefficients in Eqs. (3)-(4) take equal values: ˜g12 = ˜g22 ≈0.986. Furthermore, we will assume that the trap frequencies are ω⊥= 2π×116 Hz and ωz= 2π×1.3 Hz, i.e., Ω ≈0.0112, and the numbers of the atoms in each component are ND= 70,000 and NB= 1,000 resulting in a chemical potential of approx. 305 Hz for the total atom number. These values are similar to the respective ones used in experiments [22–27]. Concerning the parameters of the Gaussian impurity potential, the values of Eb,d are taken in the interval [−200,200] Hz, and we fix the value ǫd=ǫb=ǫ= 3 µm. Notice that we focus here on relatively narrow impurities, as for those we have explored the steady state problem [20] and, as will be seen below, they already present a rich phenomenology. An examination of the effect of the width of the impurity will be deferred to a future study. Our principal aim in what follows is to study the scattering of solitons at the impurity potential. To do so, we displace the solitons from the trap center, using the initial position value x0=−40 µm, which is sufficiently far from x= 0, so that the solitons do not overlap with the impurity. We then “release” the solitons and observe their subsequent interaction with the Gaussian barrier, measuring the fraction and observing also the shape of the coherent atomic structures that are transmitted, reflected, and trapped at the potential. We will study, at first, the case Eb=Ed, as per our experimental results (cf. subsection A below) and then the case where the impurity acts only on one component, i.e., either Ed= 0, Eb6= 0 or Ed6= 0, Eb= 0 (cf. subsection B below). At this point, it is relevant to present a sketch of a state diagram in the parameter space (Eb, Ed), as depicted in Fig. 2. The different regimes that appear after the collisions are illustrated by colors and they will be discussed below. The cases I (Ed=Eb=E), II (Ed= 0, Eb>0) and III (Ed= 0, Eb<0) correspond to the principal cases that we will examine in what follows; the capital letters Aand Bcorrespond, respectively, to a small and large value of the parameters in each region. In particular, we have taken E= 10 Hz (E= 50 Hz) in the IA (IB) case, Eb= 10 Hz (Eb= 90 Hz) in the IIA (IIB) case, and Eb=−10 Hz (Eb=−90 Hz) for the IIIA (IIIB) case. For each Aand Bwe will illustrate the contour plots of the densities of both components. Furthermore, in the case where Ed=Eb=Ewhere as we will see below the “particle-like” picture is most relevant, we will also display an effective 5 potential energy landscape encountered by the DB solitons. This amounts to computing the turning point, say x1, of each of our initializations of the DB soliton at position x0(whose potential energy VH(x0) in a harmonic oscillator trap we can evaluate). Then, (x1, VH(x0)) is identified as a point in the effective potential energy surface. Ed Eb IB IA IIA IIB IIIA IIIB FIG. 2: (Color online) A theoretical state diagram in the parameter space (Eb, Ed). The four different regimes that appear after the collisions are illustrated by colors / line styles: (Blue / dotted line) Transmission regime. (Red / full line) Reflection regime. (Green / dash-dotted line) Transmission-reflection regime. (Brown / dashed line) Trapping-transmission-reflection regime. Points Aand Bcorrespond, respectively, to a small and large value of the parameters in each region: I (Ed=Eb=E), II (Ed= 0, Eb>0) and III (Ed= 0, Eb<0). A. Scattering of DB solitons from identical impurities We start with the case where both impurities are identical, i.e., Ed=Eb=E, either repulsive (E > 0) or attractive (E < 0). In the repulsive case of Ed=Eb=E > 0, the simulations reveal the existence of two different regimes. For sufficiently small values of the repulsive barrier E, the DB solitons are transmitted (transmission regime, blue color in region I of Fig. 2). For values of Ebigger than a critical value, i.e., E&20 Hz (for solitons launched from x0= 40 µm) the DB solitons are reflected (reflection regime, red color in the same region I of Fig. 2). Thus, in this case, the solitons behave as classical particles: if they have potential energy (recall that the solitons start with zero kinetic energy) larger than the height of the barrier then they are transmitted through it, while they are reflected in the opposite case. This particle-like behavior can be quantitatively described as follows. First, we consider the increment of the DB-soliton potential energy caused by the defect, which can be defined as: ∆U=Udef −U0,(7) with Udef (U0) being the normalized potential energy of the DB soliton at the trap center with(out) the defect: Udef =R∞ −∞ dx V1(x)|u′(x)|2 R∞ −∞ dx|u′(x)|2+R∞ −∞ dx V2(x)|v′(x)|2 R∞ −∞ dx|v′(x)|2,(8) U0=R∞ −∞ dx VH(x)|u0(x)|2 R∞ −∞ dx|u0(x)|2+R∞ −∞ dx VH(x)|v0(x)|2 R∞ −∞ dx|v0(x)|2.(9) 6 0 20 40 60 80 100 0 10 20 30 40 50 60 70 Ed=Eb (Hz) ∆ U (Hz) FIG. 3: (Color online) A theoretical comparison of the the increment of the DB-soliton potential ∆Uenergy caused by the defect (solid line) with the maximum of the effective barrier height that the DB-soliton particle encounters at the defect, for different values of Ed=Eb=E(red circles). The very good agreement (especially for smaller values E) lends support to the consideration of this case as an example of a classical (solitonic) particle scattering from a barrier. In the above expressions, {u′(x), v′(x)}and {u0(x), v0(x)}denote, respectively, the DB solitons at the trap center with and without the defect, as found numerically by means of a fixed point algorithm, using as an initial guess Eqs. (5)-(6). Notice that the solutions with the prime are found by keeping fixed the chemical potentials µ1,2of each component to those of the solution without defect. Figure 3 shows the dependence of ∆Uwith respect to Ed=Eb(solid line in the figure) and this value is compared with the maximum of the effective barrier height that the DB-soliton particle encounters at the defect (see red circles in the figure). The last value is found by means of numerical simulation, fixing the initial soliton location x0or, equivalently, of the potential energy of the soliton, i.e., (1/2)Ω2x2 0, and varying Ed=Ebin order to determine the critical value which separates the reflection and transmission regimes for this value of x0. We find good agreement between the two for smaller values of Ed=Eb; for large values of the defect strength, the perturbative nature of the calculation of Eqs. (8)-(9) may be responsible for the somewhat decreased accuracy in capturing the effective barrier height. The contour plots of the densities of the dark and bright components corresponding to relatively small and large values of E > 0 (corresponding to IA and IB in Fig. 2) are illustrated in Fig. 4 (columns IA and IB). The top row shows the initial density profiles of the corresponding dark components. It is important to highlight here (as it will become also relevant for other cases) that the dark component sustains an increasing density dip as Eincreases in positive values, while it will correspondingly feature a density bump in the case of increasing negative such values. This is a feature of the ground state in the presence of the defect, as the latter repels for E > 0 and attracts for E < 0 the atoms in the neighborhood of x= 0. It is clear also by plotting the effective potentials (in the bottom panel of the figure) that the DB faces a weak barrier in the former case and its potential energy is sufficient to overcome it. On the other hand, the barrier is considerably higher in the right panel, inducing the reflection of the solitary wave. We complete this section by noting that in the case where both impurities are attractive, i.e., Ed=Eb=E < 0, the DB solitons are always transmitted after the collision, for every value of E, hence there is only a transmission regime depicted by blue color in Fig. 2. 7 B. Scattering of DB solitons from an impurity in one component While the dynamical evolution of the case where the defect acts on both components was found to be fairly straightforward, the case where the barrier is imposed selectively on only one of the components was found to be considerably more complex. We considered both the case where Ed= 0 and Eb6= 0 and that where Ed6= 0 and Eb= 0. The results show that the following two subcases are equivalent: a) (Ed= 0, Eb>0) ≡(Ed<0, Eb= 0). b) (Ed= 0, Eb<0) ≡(Ed>0, Eb= 0). The case where Ed= 0 and Eb<0 represents the existence of an attractive barrier in the bright component and absence of impurity in the dark component. The above equivalence can be most easily qualitatively appreciated in that case b), hence we present it for that setting. In particular, when an impurity attractively affects the atoms in the bright component, then it favors the “collection” of atoms near the origin. This, in turn, builds a population of atoms in that neighborhood which, through the term proportional to g12 in the equations of motion, provides a repulsive barrier for the dark component. Hence, the existence of an attractive well solely in the bright component becomes tantamount to having a repulsive barrier in the dark component. An analogous argument can be used to showcase that a repulsive barrier in the bright component, through favoring the absence of atoms in its vicinity, creates an effective well for the dark component atoms. The above feature is directly evident in the diagram of Fig. 2, hence we only focus on each of the representatives of the cases a) and b) above. After the collisions, for Ed= 0, Eb>0, part of the energy is transmitted and part of it is reflected. We denote this as a transmission-reflection regime (green color in Fig. 2). For small values of Ebthe DB solitons are mainly transmitted, and when Ebis high enough they are mostly reflected. Equivalent results, when the well depth |Ed| increases, are obtained for the case where Ed<0 and Eb= 0, which represents the existence of an attractive well in the dark component and the absence of impurity in the bright one. For small Eb>0, this dynamics, as shown in the left panel of Fig. 5, can be understood as follows. As discussed above, the repulsion of bright atoms produces an effective attraction of dark atoms, hence inducing an effective potential well, rather than what was anticipated as a potential barrier. It should be noted here that this counter-intuitive effect was quantified in the case of a δfunction potential in Ref. [20]. This effect leads to the acceleration of the soliton (with a small back-scatter due to the inelasticity of collision with the defect) visible in the left panels of the figure. To complete the discussion of Fig. 5, let us briefly touch upon the right panels of the figure. This concerns the case of Eb<0 (while Ed= 0, namely case b) above). The corresponding situation here, when Ebis small presents a repulsive effect for the dark atoms and as such results in an effective barrier. This prediction is also corroborated by the analytical considerations for the δ-function case of Ref. [20]. This, in turn, leads to the reflection dynamics observed in the right panels of Fig. 5. We now turn to the case of large barrier strength in Fig. 6. In this setting, there is a fundamental difference in comparison to the case of weak barrier presented previously. This consists of the fact that for small Eb, the defining characteristic in the DB-soliton and defect interaction is the nature of the potential for the dark component (which, as we saw, was somewhat counter-intuitively the opposite than the one for the bright component). However, for large Eb, the nature of the potential for the bright component becomes important and hence in this case, large positive Eb also induces a locally strong repulsive potential for the bright atoms. On the other hand, large negative Ebcreates a large attractive potential for the bright atoms. However, the latter tends to favor the trapping of the atoms of the bright component together with those of the dark component, leading essentially to the formation of a defect mode, alongside a partial reflection of the soliton. These characteristics, namely reflection for Eblarge and positive and the possibility of trapping, along with reflection for Eblarge and negative, are illustrated in the panels of Fig. 6. An example of intermediate values of Eband their associated dynamics can be found in Fig. 7. In these examples, for positive Eb, the impurity leads to partial transmission and partial reflection, but does not enable the possibility of trapping at the defect. In this case, the transmitted fraction of the soliton (which is also present in the cases of Figs. 5 and 6) is so energetic and localized that it can be directly observed; the transmitted soliton has a velocity different from that of the reflected soliton, and consequently, its oscillation frequency differs as well. The possibility of trapping at the defect is explored for Eb<0, whereby there is a fraction of atoms which is trapped at the defect, while also a considerable fraction appears to be reflected; see e.g. the right panel of Fig. 7. We notice that in the case where there is no impurity affecting the dark component and where there exists an attractive well in the bright one, as the well depth |Eb|increases, two regimes appear alternatively: a reflection regime and a trapping-transmission-reflection regime (red and brown colors, respectively, in region III of Fig. 2). Indeed, this alternation may be quite complex and a characteristic example thereof is presented in Fig. 8, where the fractions of atoms transmitted through the defect, reflected from it and trapped in the immediate vicinity of the barrier are measured. Those quantities are 8 quantitatively described by the transmission, reflection and trapping coefficients. Notice that the time-dependent coefficients are defined as: T(t) = R∞ ǫb|v(x, t)|2dx R∞ −∞ |v(x, t)|2dx, R(t) = R−ǫb −∞ |v(x, t)|2dx R∞ −∞ |v(x, t)|2dx , B(t) = Rǫb −ǫb|v(x, t)|2dx R∞ −∞ |v(x, t)|2dx.(10) Figure 8 depicts these coefficients at time t′, namely R′,T′and B′, with t′being the time where the bright component of the DB soliton reaches its maximum excursion after the first interaction with the defect. The figure illustrates that the full dynamics features more complicated resonant type transmission events, as well as alternating windows of predominantly reflection or predominantly trapping. These complex scenarios are beyond the scope of the particle analysis provided herein. We do note, however, the apparent similarity of these results with the ones obtained in the case of a single component bright soliton which scatters off of a quantum well [42]. In the latter case, the variational analysis was already fairly cumbersome even for a δ-function potential, while here it is rendered more elaborate by the presence of two-components and the Gaussian form of the barrier. A more detailed analysis, perhaps in the simpler δ-function attractive setting based on a two-mode variational ansatz would constitute an interesting problem for further studies. Nevertheless, we should point out that this behavior is similar to that obtained for the case where there exists a repulsive barrier in the dark component and there is no impurity affecting the bright component. IV. CONCLUSIONS AND FUTURE CHALLENGES In the present work, we have presented a survey of the collisions of dark-bright (DB) solitons with defects. Initially motivated by the potential of experimental studies (a prototypical example of which was shown herein), we considered the setting of a dark-bright soliton impinging on a defect potential. In the case of a bright soliton hitting a well [42, 43] or a barrier [44, 45], this theme has been of intense theoretical and even experimental [52, 53] interest recently. However, far less has been done in the realm of dark-bright solitons. We have shown that in the case of two equal repulsive barriers acting on both components, the DB solitons demonstrate a clear classical particle behavior, which involves transmission for weak potentials and reflection for strong ones. Similarly, predominantly transmission type events were observed for equal attractive potentials acting in both components. On the other hand, we illustrated that more complex scenarios can develop in the case where the impurity acts only on one of the two components. We categorized these cases, illustrating the analogies of a repulsive barrier in the first component with an attractive one in the second component (and vice-versa). We explained the low barrier amplitude cases on the basis of somewhat counter-intuitive, cross-component effective potentials and argued that the large amplitude cases may be significantly different due to the role of the defect in both components. We showcased the complexity of the latter by means of cases containing transmission and reflection, or trapping, transmission and reflection together and by monitoring the dependence of the different fractions (of trapping, transmission or reflection), as a function of the barrier amplitude. It would certainly be interesting to extend this chiefly numerical (but also experimental) study further. On the experimental side, it would be extremely interesting, although more challenging, to engineer potentials that are selective to particular hyperfine states, so that some of the predictions proposed herein could be tested. From a theoretical perspective, it would be very relevant to attempt to distill a simple setting (e.g. a δ-function potential) where a theoretical study of the above reported phenomenology could be appreciated in more quantitative terms. Numerically, it may also be quite significant to appreciate the effect of the width of the barrier, as here we have concentrated on the sign and magnitude (and inter-component interplay of the) barrier. Natural extensions may also concern the possibility of scattering in higher-dimensional settings and evaluation of the role of transverse degrees of freedom therein. Acknowledgments. P.E. acknowledges financial support from NSF and ARO. P.G.K. gratefully acknowledges support from NSF-DMS-0806762 and the Alexander von Humboldt Foundation, as well as the Binational Science Foundation. A.A., F.R.R. and J.C. acknowledge financial support from the MICINN project FIS2008-04848. The work of D.J.F. was partially supported by the Special Account for Research Grants of the University of Athens. [1] C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases (Cambridge University Press, Cambridge, 2002). [2] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003). [3] P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonz´alez, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment (Springer-Verlag, Heidelberg, 2008). 9 [4] R. Carretero-Gonz´alez, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinearity 21, R139 (2008). [5] F. Kh. Abdullaev, A. Gammal, A. M. Kamchatnov, and L. Tomio, Int. J. Mod. Phys. B 19, 3415 (2005). [6] D. J. Frantzeskakis, J. Phys. A: Math. Theor. 43, 213001 (2010). [7] C. J. Myatt, E. A.Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 78, 586 (1997). [8] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 81, 1539 (1998). [9] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, Science 296, 1290 (2002). [10] K.E. Strecker, G.B. Partridge, A.G. Truscott and R.G. Hulet, Nature 417, 150 (2002). [11] B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, M.K. Oberthaler, Phys. Rev. Lett. 92, 230401 (2004). [12] Th. Busch and J.R. Anglin, Phys. Rev. Lett. 87, 010401 (2001). [13] H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-Gonz´alez, Phys. Rev. A 77, 033612 (2008). [14] M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, Phys. Rev. A 77, 013820 (2008). [15] S. Rajendran, P. Muruganandam, M. Lakshmanan, J. Phys. B 42, 145307 (2009). [16] V. A. Brazhnyi and V. M. P´erez-Garc´ıa, Chaos, Solitons & Fractals 44, 381 (2011). [17] C. Yin, N. G. Berloff, V. M. Perez-Garcia, V. A. Brazhnyi, and H. Michinel, Phys. Rev. A 83, 051605 (2011). [18] K. J. H. Law, P. G. Kevrekidis, and L. S. Tuckerman, Phys. Rev. Lett. 105, 160405 (2010). [19] A. ´ Alvarez, J. Cuevas, F. R. Romero, and P. G. Kevrekidis, Physica D 240, 767 (2011). [20] V. Achilleos, P. G. Kevrekidis, V. M. Rothos, and D. J. Frantzeskakis, Phys. Rev. A 84, 053626 (2011). [21] V. Achilleos, D. Yan, P. G. Kevrekidis, and D. J. Frantzeskakis, New J. Phys. 14, 055006 (2012). [22] C. Becker, S. Stellmer, P. Soltan-Panahi, S. D¨orscher, M. Baumert, E.-M. Richter, J. Kronj¨ager, K. Bongs, and K. Sengstock, Nature Phys. 4, 496 (2008). [23] C. Hamner, J.J. Chang, P. Engels, M. A. Hoefer, Phys. Rev. Lett. 106, 065302 (2011). [24] S. Middelkamp, J. J. Chang, C. Hamner, R. Carretero-Gonz´alez, P. G. Kevrekidis, V. Achilleos, D. J. Frantzeskakis, P. Schmelcher, and P. Engels, Phys. Lett. A 375, 642 (2011). [25] D. Yan, J. J. Chang, C. Hamner, P. G. Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonz´alez, and P. Schmelcher, Phys. Rev. A 84, 053630 (2011). [26] M. A. Hoefer, J. J. Chang, C. Hamner, and P. Engels, Phys. Rev. A 84, 041605 (2011). [27] D. Yan, J. J. Chang, C. Hamner, M. A. Hoefer, P. G. Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, and J. Cuevas, J. Phys. B 45, 115301 (2012). [28] Yu.S. Kivshar and G.P. Agrawal, Optical solitons: from fibers to photonic crystals, Academic Press (San Diego, 2003). [29] Z. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, Yu. S. Kivshar, and V. V. Afanasjev, Opt. Lett. 21, 1821 (1996). [30] E. A. Ostrovskaya, Yu. S. Kivshar, Z. Chen, and M. Segev, Opt. Lett. 24, 327 (1999). [31] Z. Chen, M. Segev, T.H. Coskun, D.N. Christodoulides and Yu.S. Kivshar, J. Opt. Soc. Am. B 14, 3066-3077 (1997). [32] Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989). [33] I. M. Lifshitz and A. M. Kosevich, Rep. Prog. Phys. 29, 217 (1966). [34] A. M. Kosevich, Physica D 41, 253 (1990). [35] X. D. Cao and B. A. Malomed, Phys. Lett. A 206, 177 (1995). [36] R. H. Goodman, P. J. Holmes, and M. I. Weinstein, Physica D 192, 215 (2004). [37] J. Holmer, J. Marzuola, and M. Zworski, Comm. Math. Phys. 274, 187 (2007); J. Holmer, J. Marzuola, and M. Zworski, J. Nonlin. Sci. 17, 349 (2007). [38] V. V. Konotop, V. M. P´erez-Garc´ıa, Y.-F. Tang and L. V´azquez, Phys. Lett. A 236, 314 (1997). [39] D. J. Frantzeskakis, G. Theocharis, F. K. Diakonos, P. Schmelcher, and Yu. S. Kivshar, Phys. Rev. A 66, 053608 (2002). [40] N. Bilas and N. Pavloff, Phys. Rev. A 72, 033618 (2005). [41] G. Herring, P. G. Kevrekidis, R. Carretero-Gonz´alez, B. A. Malomed, D. J. Frantzeskakis, and A. R. Bishop, Phys. Lett. A345, 144 (2005). [42] T. Ernst and J. Brand, Phys. Rev. A 81, 033614 (2010). [43] C. Lee and J. Brand, Europhys. Lett. 73, 321 (2006). [44] J. L. Helm, T. P. Billam, and S. A. Gardiner, Phys. Rev. A 85, 053621 (2012). [45] A. D. Martin and J. Ruostekoski, New J. Phys. 14, 043040 (2012). [46] A.I. Streltsov, O.E. Alon, and L.S. Cederbaum, Phys. Rev. A 80, 043616 (2009). [47] C. Weiss and Y. Castin, J. Phys. A Math. Theor. 45, 455306 (2012). [48] M. Lewenstein and B.A. Malomed, New J. Phys. 11, 113014 (2009). [49] S.D. Hansen, N. Nygaard and K. Mølmer, arXiv:1210.1681. [50] P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007). [51] D. Dries, S. E. Pollack, J. M. Hitchcock, and R. G. Hulet, Phys. Rev. A 82, 033603 (2010). [52] P. Dyke, S. Lei, and R. G. Hulet, poster presentation in the 23rd International Conference on Atomic Physics (Palaiseau, France, 23-27 July, 2012). [53] A. L. Marchant, T. P. Wiles, S.H¨andel, M. M. H. Yu, S. A. Hopkins, and S. L.Cornish, poster presentation in the 23rd International Conference on Atomic Physics (Palaiseau, France, 23-27 July, 2012). [54] S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 505 (1973) [Sov. Phys. JETP 38, 248 (1974)].