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PHYSICAL REVIEW A 73, 033608 共2006兲 Analytical study of resonant transport of Bose-Einstein condensates K. Rapedius, D. Witthaut, and H. J. Korsch* Technische Universität Kaiserslautern, FB Physik, D-67653 Kaiserslautern, Germany 共Received 19 December 2005; published 13 March 2006兲 We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a one-dimensional finite square-well potential. By neglecting the mean-field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behavior. Novel crossing scenarios of eigenstates similar to beak-to-beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The boundstate wave functions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed. DOI: 10.1103/PhysRevA.73.033608 PACS number共s兲: 03.75.Dg, 03.75.Kk, 42.65.Pc I. INTRODUCTION Due to the experimental progress in the field of BoseEinstein condensates 共BECs兲, e.g., atom-chip experiments achieved in the past few years 关1–3兴, it is now possible to study the influence of an interatomic interaction on transport 关4,5兴. At low temperatures, BECs can be described by the nonlinear Schrödinger equation 共NLSE兲 or Gross-Pitaevskii equation 共GPE兲 iប 冉 冊 共r,t兲 ប2 2 = − ⵜ + V共r兲 + g兩共r,t兲兩2 共r,t兲 t 2m 共1兲 in a mean-field approach 关6–9兴. Another important application of the NLSE is the propagation of electromagnetic waves in nonlinear media 共see, e.g., Ref. 关10兴, Chap. 8兲. In the case of vanishing interaction 共g = 0兲 it reduces to the linear Schrödinger equation of single particle quantum mechanics. The treatment of transport within this mean-field theory reveals new interesting phenomena which arise from the nonlinearity of the equation. For example a bistable behavior of the flux transmitted through a double barrier potential has been found 关5兴. For the same system transitions from resonances to bound states can occur due to the mean-field interaction 关11,12兴. However, previous studies of nonlinear transport and nonlinear resonances usually relied strongly on numerical solutions, such as complex scaling. There are only few model systems of the NLSE which have been studied analytically and most of these treatments concentrate on bound states. Among these few examples are the infinite square-well 关13–15兴, the finite square-well 关16–18兴, the potential step 关19兴, ␦ potentials 关20,21兴 and the ␦ comb 关22,23兴 as an example of a periodic potential. Therefore analytical solutions of the NLSE for other simple model potentials are of fundamental interest. In the present paper we solve the time-independent NLSE for a simple one-dimensional square-well potential in order *Electronic address: [email protected] 1050-2947/2006/73共3兲/033608共12兲/$23.00 to analytically indicate and discuss the nonlinear phenomena described above. To unambiguously define ingoing and outgoing waves and thus a transmission coefficient 兩T兩2, we neglect the mean-field-interaction g兩共x兲兩2 outside the potential well. Since the one-dimensional mean-field coupling constant g is inversely proportional to the square of the system’s radial extension a⬜, this can be achieved in an atom-chip experiment by a weaker radial confinement of the condensate outside the potential well 关6,17,24兴. Especially for bound states our approximation is well justified since in this case the condensate density 兩共x兲兩2 is much higher inside the potential well than outside. The paper is organized as follows. In Sec. II we analyze stationary scattering states of the NLSE with a finite squarewell potential. Then we investigate the transition from resonances to bound states for an attractive 共Sec. III兲 and a repulsive mean-field interaction 共Sec. IV兲. In addition to our discussion in terms of stationary states we have a brief look at the dynamics of the system in Sec. V, where we solve the time-dependent NLSE numerically. In Appendix A we briefly review some basic results for the linear Schrödinger equation with a square-well potential which has been treated in various textbooks on quantum mechanics 关25–27兴, whereas the Appendixes B and C contain details of our calculations of resonances and bound states. II. SCATTERING STATES We consider the one-dimensional NLSE with a squarewell potential, where we set the mean-field term g兩共x兲兩2 to zero outside the potential well. This means that for 兩x兩 ⬎ a the wave function must satisfy the free linear Schrödinger equation 冉 − 冊 ប2 d2 − 共x兲 = 0 2m dx2 共2兲 and for 兩x兩 艋 a the time-independent NLSE 033608-1 ©2006 The American Physical Society PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH 冉 − 冊 ប2 d2 + g兩共x兲兩2 + V0 − 共x兲 = 0 2m dx2 with a constant potential V0 ⬍ 0, where ⬎ 0 is the chemical potential. For an incoming flux from x = −⬁ we make the ansatz 冦 Aeikx + Be−ikx , x ⬍ − a, 兩x兩 艋 a, 共x兲 = f共x兲, Ceikx , x⬎a 冧 f共x兲 = 冑S共x兲ei⌽共x兲 , ␣ and S共x兲 = + dn2共x + ␦兩p兲 S共x兲 ប2 2 , gm 册 S⬘共− a兲 i⌽共−a兲 e . 2k 关S共− a兲 − S共a兲兴2 + S⬘共− a兲2/共4k2兲 关S共− a兲 + S共a兲兴2 + S⬘共− a兲2/共4k2兲 共15兲 and the transmission probability 兩T兩2 = 4S共a兲S共− a兲 S共a兲 . 2 2 2 = 关S共− a兲 + S共a兲兴 + S⬘共− a兲 /共4k 兲 兩A兩2 共16兲 It can easily be verified that 兩R兩2 + 兩T兩2 = 1. The scattering phase 共兲, defined by T共兲 = 兩T共兲兩exp关i共兲 − 2ika兴, can be expressed as 共兲 = arg共C兲 − arg共A兲 + 2ka 冕 +a −a S⬘共− a兲 kS共a兲 dx + arctan . 共17兲 S共x兲 2k关S共− a兲 + S共a兲兴 The condition 共10兲 can be further exploited by rewriting it as −2p dn共u 兩 p兲sn共u 兩 p兲cn共u 兩 p兲 = 0 with the abbreviation u = a + ␦. As the dn function is always nonzero, the solutions of this equation are given by the zeros of either of the two functions cn共u 兩 p兲 or sn共u 兩 p兲. Case 1. sn共a + ␦兩p兲 = 0 Û a + ␦ = 2jK共p兲 Þ ␦ = 2jK共p兲 − a. 3 ប2 共2 − p兲2 , = V0 + g − 2 2m − 兩R兩2 = = 共7兲 S共− a兲 − S共a兲 + i As in the linear case 共see Appendix A兲, we define the amplitudes of reflection and transmission as R ª B / A and T ª C / A. The above equations then lead to the reflection probability 共6兲 in terms of the Jacobi elliptic function dn with the real period 2K共p兲, where K共p兲 is the complete elliptic integral of the first kind and p is the elliptic parameter 关28–30兴. The real parameters ␣, p, , , , and ␦ have to satisfy the relations 冋 共14兲 共5兲 where S共x兲 and ⌽共x兲 are real functions. As already shown by Carr, Clark, and Reinhard 关13,14兴 a complex solution of the NLSE with a constant potential is given by the equations =− 2 共4兲 with the wave number k = 冑2m / ប. Inside the potential we assume an intrinsically complex solution ⌽⬘共x兲 = 冑S共− a兲Be+ika = 1 共3兲 共8兲 ប2 2 关 共p − 1兲 − ␣2兴 + g3 − 共 − V0兲2 = 0. 2m 共9兲 The variable j denotes an integer number. This means dn2共a + ␦ 兩 p兲 = dn2共2jK共p兲 兩 p兲 = 1 or S共a兲 = + . 共18兲 Case 2. cn共a + ␦ 兩 p兲 = 0. In analogy to case 1 we get ␦ = 共2j + 1兲K共p兲 − a and The wave function and its derivative must be continuous at x = ± a. At x = + a we obtain the equations f共a兲 = Ceika and f ⬘共a兲 = ikCeika. By inserting Eq. 共5兲 we arrive at 21 S⬘共a兲 + i␣ = ikS共a兲. Since we are considering positive chemical potentials ⬎ 0, the wave number k is real so that we obtain the conditions For given values of the chemical potential and the incoming amplitude 兩A兩, the parameters p, , and = −ប22 / gm are completely determined by the equations 共8兲 and 共9兲 and the absolute square of Eq. 共13兲, S⬘共a兲 = 0 and ␣ = kS共a兲. 共10兲 16k2兩A兩2S共− a兲 = 4k2关S共− a兲 + S共a兲兴2 + S⬘共− a兲2 . Ae−ika + Beika = f共− a兲, 共11兲 ik共Ae−ika − Beika兲 = f ⬘共− a兲. 共12兲 The positions of the resonances are defined by 兩T兩2 = 1 or 兩R兩 = 0. With Eq. 共15兲 this is equivalent to the conditions S⬘共−a兲 = 0 and S共−a兲 = S共a兲. Now we consider these conditions for the two cases mentioned above. Case 1. sn共a + ␦ 兩 p兲 = 0 or ␦ = 2jK共p兲 − a. The condition S⬘共−a兲 = 0 now reads At x = −a we obtain S共a兲 = + 共1 − p兲. 共19兲 共20兲 2 S⬘共− a兲 = − 2p cn共u兩p兲sn共u兩p兲dn共u兩p兲 = 0 With Eqs. 共5兲 and 共10兲 these two conditions can be written as 冑S共− a兲Ae−ika = 1 2 冋 S共− a兲 + S共a兲 − i 册 S⬘共− a兲 i⌽共−a兲 e , 2k 共13兲 with u = −a + ␦. The solutions of this equation are again given by the zeros of either the sn function or the cn function. One can show easily that only the zeros of the sn function are compatible with the condition S共−a兲 = S共a兲. Thus we 033608-2 PHYSICAL REVIEW A 73, 033608 共2006兲 ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼ obtain −2a + 2jK共p兲 = 2lK共p兲 where l and j are integer numbers. The resonance positions are then given by = K共p兲 n, a 共21兲 where the integer number n is defined as n = j − l. Case 2. cn共a + ␦ 兩 p兲 = 0 or ␦ = 共2j + 1兲K共p兲 − a. In analogy to case 1 one can show that the resonances are also determined by Eq. 共21兲. In the case of a resonance there is no reflected wave 共B = 0兲 so that Eqs. 共11兲, 共18兲, and 共19兲 lead to 兩A兩2 = S共− a兲 = S共a兲 = or 共p兲 = 再 再 + , case 1, + 共1 − p兲, case 2 兩A兩2 − , case 1, 兩A兩 − 共1 − p兲, case 2. 2 冎 冎 共22兲 共23兲 With the help of Eqs. 共23兲, 共7兲, and 共8兲 we obtain a formula for the chemical potential of a resonance 再 3 ប2K2共p兲 2 共1 + p兲, case 1, R共p兲 = V0 + g兩A兩2 + n 2 共1 − 2p兲, case 2. 2ma2 冎 共24兲 The chemical potential depends on the parameter p and the integer number n which must be sufficiently large to make R 艌 0. As in the linear case 关see Eq. 共A5兲兴 there is a term which grows quadratically in n. This term can be viewed as an effective increase of the chemical potential in comparison to the energy of the linear system in case 1 and, respectively, as a decrease in case 2. However, it turns out 共see below兲 that the energy shift due to nonlinear interaction is mainly determined by the term 23 g兩A兩2, which is a direct consequence of the effective increase or decrease of the potential well by the mean-field term g兩共x兲兩2. As we will show below, we have p → 0 if g → 0 so that K共p兲 → / 2 and Eq. 共24兲 reduces to the equivalent equation 共A5兲 for the resonance energies of the linear system. The Jacobi elliptic parameter p can be calculated analytically in terms of n and g兩A兩2 共Appendix B兲. Case 2 is valid for 0 艋 g 艋 兩V0兩 / 兩A兩2 and case 1 is valid for any other value of g 共see Appendix B兲. Since the parameter p and the chemical potential R only depend on the product g兩A兩2 we henceforth assume 兩A兩2 = 1 in all figures and numerical calculations. Using Eq. 共23兲 in Eqs. 共5兲 and 共6兲 we obtain the squared modulus of the resonance wave functions inside the potential well 兩x兩 艋 a. In dependence of p and n it is given by 兩R共x兲兩2 = 兩A兩2 − ប 2 2 2 关dn „x − nK共p兲兩p… − 1兴 gm 共25兲 in case 1 and 兩R共x兲兩2 = 兩A兩2 − ប 2 2 2 关dn „x + 共1 − n兲K共p兲兩p… − 1 + p兴 gm 共26兲 in case 2 with = nK共p兲 / a. FIG. 1. 共Color online兲 Squared modulus of the wave functions of the two most stable resonances of the potential V0 = −10, a = 2 for the attractive nonlinearity g = −2. The normalization 兩A兩 = 1 has been used. Figure 1 shows the squared modulus of resonance wave functions for attractive nonlinearities in the region of the potential well 兩x兩 艋 a where scaled units with ប = m = 1 are used, as for all figures and numerical calculations in this paper. The functions are symmetric with respect to x = 0 and have no nodes. The quantum number n indicates the number of the minima. The smaller the chemical potential R共n兲 is, the smaller is the value of 兩共x兲兩2 at its minima. In the case of repulsive nonlinearity the wave functions show exactly the same behavior. Figure 2 shows the transmission coefficient 兩T共兲兩2 in dependence of the chemical potential for the potential V0 = −10 and a = 2 for different attractive nonlinearities g ⬍ 0. For small values of 兩g兩, the behavior of the transmission coefficient is very similar to the linear system. With increasing attractive interaction, the curves bend more and more to the left so that the resonances are shifted towards smaller values of . This shift is mainly determined by the term 23 g兩A兩2 in Eq. 共24兲. For g = −2, the lowest resonance so far has disappeared into the bound state regime ⬍ 0. Between g = −6 and g = −8 the next resonance disappears. It can be shown that these resonances become bound states of the system 共see Sec. III兲. For higher values of 兩g兩 the bending of the curve in the vicinity of the resonances leads to a bistable behavior. 033608-3 PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH FIG. 2. 共Color online兲 Transmission coefficients for different attractive nonlinearities for the square-well potential V0 = −10, a = 2. The positions of the resonances, calculated with the formula 共24兲, are marked by asterisks. For a fixed value of , there are now different wave functions which lead to different transmission coefficients 兩T共兲兩2. Due to the nonlinear mean field interaction the state adopted by the system for an incoming flux from x = −⬁ is not only determined by the chemical potential and the amplitude A but also by the density distribution 兩共x , t兲兩2 of the condensate inside the potential well so that the system has a memory 共hysteresis兲. This will be further investigated in Sec. V. Now we consider repulsive nonlinearities g ⬎ 0. Figure 3 shows the transmission coefficient 兩T兩2 in dependence of for a potential V0 = −12, a = 4 for different repulsive nonlinearities. Here we observe a shift of the resonance positions towards higher values of , since the term 23 g兩A兩2 is now positive. New resonances appear for increasing 兩g兩 which originate from the bound states of the linear system 共see Sec. IV and Appendix C兲. Now the curves bend to the right and, as in the attractive case, the transmission coefficient shows a bistable behavior. Between g = 4.5 and g = 5, we observe a bifurcation phenomenon at ⬇ 1 which resembles an avoided crossing. This turns out to be a beak-to-beak scenario which also occurs in two-level systems. These phenomena will be discussed in detail in a subsequent paper 关31兴. In all transmission curves we observe that, for a fixed value of g兩A兩2, the effect of the nonlinearity on the shape of FIG. 3. 共Color online兲 Transmission coefficients for different repulsive nonlinearities for the square-well potential V0 = −12, a = 4. The positions of the resonances, calculated with the formula 共24兲, are marked by asterisks. the resonances is stronger for small values of than for large ones, because the kinetic energy is higher for larger and the mean-field energy has a comparatively smaller influence. This also manifests itself in formula 共24兲 for R, where the term proportional to n2 counteracts the dominant term 3 2 2 g兩A兩 . Figures 4 and 5 show scattering phases in dependence of the chemical potential which have been calculated by numerical integration of Eq. 共18兲 for the parameters of Figs. 2 and 3. For weak nonlinearities the scattering phase behaves 033608-4 PHYSICAL REVIEW A 73, 033608 共2006兲 ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼ FIG. 4. 共Color online兲 Scattering phases for different attractive nonlinearities for the potential V0 = −10, a = 2. The positions of the resonances, calculated with the formula 共24兲, are marked by dotted vertical lines. similar to the scattering phase of the linear system which is zero at the positions = R of the resonances. For stronger nonlinearities the resonance scattering phase increasingly deviates from zero. Since the squared modulus of the wave function enters Eq. 共18兲, the scattering phase 共兲 inherits the bistable behavior of the transmission coefficient 兩T共兲兩2. Wherever beak-to-beak structures occur in the transmission coefficient the scattering phase shows loops. III. BOUND STATES IN THE CASE OF ATTRACTIVE NONLINEARITY In this section, we consider bound states of the nonlinear system described by Eqs. 共3兲 and 共2兲 for attractive nonlinearities g ⬍ 0. In analogy to the linear case, bound states occur only for chemical potentials ⬍ 0. As in the linear case, we look for solutions of even and odd parity. We make the ansatz 冦 ␥ ±e x , x ⬍ − a, 共x兲± = I±cn共±x + K±兩p±兲, 兩x兩 艋 a, ± ␥ ±e −x , x⬎a 冧 FIG. 5. 共Color online兲 Scattering phases for different repulsive nonlinearities for the potential V0 = −12, a = 4. The positions of the resonances, calculated with the formula 共24兲, are marked by dotted vertical lines. The solutions inside the potential well 兩x兩 艋 a are given by Jacobi elliptic functions cn with amplitudes I± = 冑 ប2±2 p± gm 共28兲 2m共 − V兲 . ប2共1 − 2p±兲 共29兲 − and wave numbers 共27兲 for even 共⫹ sign, K+ = 0兲 and odd parity solutions 关⫺ sign, K− = K共p−兲兴 with = 冑−2m / ប. ± = 冑 共1兲 Symmetric solutions. The continuity of the wave function and its derivative at x = ± a yield 033608-5 PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH ␥+e−a = I+cn共u兩p+兲 共30兲 and ␥+e−a = I++sn共u 兩 p+兲dn共u 兩 p+兲, with the abbreviation u = +a. Thus we arrive at the condition cn共u兩p+兲 − +sn共u兩p+兲dn共u兩p+兲 = 0 共31兲 for symmetric bound states. In the limit p → 0, the Jacobi elliptic functions converge to trigonometric functions and we regain the condition 共A9兲 of the linear system. 共2兲 Antisymmetric solutions. In analogy to the case of even parity we obtain the condition cn共u兩p−兲 − −sn共u兩p−兲dn共u兩p−兲 = 0 共32兲 for antisymmetric bound states with u = −a + K共p−兲, which also converges to Eq. 共A10兲 in the limit p → 0. Now we show explicitly that the resonances discussed in Sec. II become bound states for a sufficiently strong attractive mean-field interaction. Due to the behavior of the wave function outside the potential well, transitions from resonances to bound states occur at = 0. For = 0 the condition 共31兲 for symmetric bound states becomes sn共+a 兩 p+兲 = 0 which is solved by + = 2j+K共p+兲/a 共33兲 FIG. 6. 共Color online兲 Squared modulus of the wave function of the most stable resonance of the potential V0 = −10, a = 2 exactly at the critical point for the transition from resonances to bound states 共see Table I兲. Here, the minima are also zeros of the wave function 共cf. Fig. 1兲. The normalization 兩A兩 = 1 has been used. 共1 − 2pT−兲K2共pT−兲 = 共1 − 2p+兲K 共p+兲 = 2 2ប2 j+2 pT− = − 共34兲 . 24 + 23 冑冉 冊 24 23 冉 + + + O共p+3兲 共35兲 冊 冉 冊 2m兩V0兩a2 32 32 共R0/兲2 . 1− 2 2 2 = 1− 23 23 ប j+ j+2 共36兲 The quantity R0 / in + determines the number of bound states N+ = 关R0 / 兴⬎ for g = 0 共see Appendix A兲. Since p+ must be positive, + must be positive as well and we obtain the condition j+ 艌 N+. One can prove 共see Appendix C兲 that for a critical nonlinearity gT+ the wave function of a symmetric bound state continuously merges into a resonance wave function with even quantum number n. In a similar way one can show that resonances of odd quantum number n become antisymmetric bound states for a sufficiently strong attractive nonlinearity gT− ⬍ 0. We obtain 关see Eq. 共33兲兴 − = 共2j− + 1兲K共p−兲 / a with integer j− and n = 2j− + 1. If we choose the normalization 兩␥−兩 = 兩A兩, the transition occurs at ប n K共pT−兲 pT− . ma2兩A兩2 2 2 gT− = − − = 2 as an approximate solution for the parameter p+ where the transition occurs. Here we have introduced the abbreviation + = 2 The parameter pT− ª p− = pR is determined by 24 + 23 冑冉 冊 24 23 2 + − + O共p−3兲, 共39兲 where we have introduced the abbreviation For small values of p+ we can make a Taylor approximation and obtain p+ = − 共38兲 or with integer j+. Inserting Eq. 共33兲 into 共29兲 yields m兩V0兩a2 2m兩V0兩a2 ប2共2j− + 1兲2 共37兲 冉 冊 8m兩V0兩a2 32 1− 2 2 . 23 ប 共2j− + 1兲2 共40兲 The condition pT− ⬎ 0 implies − ⬎ 0 and thus j− 艌 N− where N− = 关R0 / − 1 / 2兴⬎ is the number of antisymmetric bound states for g = 0 as defined in Eq. 共A11兲. Figure 6 shows the squared modulus of wave functions at the transition from resonances to bound states. As in the case of resonances the functions are symmetric and have n minima 共cf. Fig. 1兲. The quantum number n counts all the resonances and bound states defined above and is equal to the number of minima of the squared modulus of the wave function. For bound states, these minima are also nodes of the wave function. Thus, the famous oscillation theorem of linear quantum mechanics 共see Ref. 关32兴兲 is also valid in the nonlinear case considered here. TABLE I. Critical parameters for the transition of the two most stable resonances of the potential V0 = −10, a = 2 to bound states. The approximations 共35兲 for p+ and 共39兲 for p− have been used. pT gT gTeff 033608-6 symmetric 共n = 6兲 antisymmetric 共n = 7兲 numerically exact approx. numerically exact approx. 0.0644 −1.4772 −0.7325 0.0643 −1.4749 −0.7331 0.2044 −6.9159 −3.3592 0.205 −6.969 −3.346 PHYSICAL REVIEW A 73, 033608 共2006兲 ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼ FIG. 7. Dependence of the effective nonlinear interaction strength g±eff on the parameter p± for the symmetric bound state n = 0 共solid line兲 and the antisymmetric bound state n = 1 共dashed line兲. In order to study the shift of the bound state energies due to mean-field interaction it is useful to introduce an effective nonlinear interaction strength g±eff = g 2a 冕 a 兩±共x兲兩2dx 共41兲 −a 共see Ref. 关20兴兲 in the interaction region 兩x兩 艋 a. The integration can be carried out analytically. We obtain FIG. 8. 共Color online兲 Wave functions of the two lowest bound states of the potential V0 = −10, a = 2 for geff = −0.15 共solid lines兲 in comparison with the respective wavefunctions of the linear system a 共dashed lines兲. The normalization 兰−a 兩共x兲兩2dx = 1 has been used. from its linear counterpart than the wave function of the state n = 1. Figure 9 shows the shift of bound states in dependence of 兩geff兩. The appearance of new bound states can be observed. For large values of 兩geff兩, which correspond to elliptic parameters p ⬎ 0.5, the states all lie below the potential well depth V0 = −10. For small values of p± the chemical potential 共g±eff兲 can be approximated linearly in terms of g±eff, g−eff = − ប − 关E„−a + K共p−兲兩p−… − E共p−兲 − 共1 − p−兲−a兴 − ma 2 g−eff = ប 2 + 关E共+a兩p+兲 − 共1 − p+兲+a兴 g+eff = − ma 冋 gT±兩A兩2 E共pT±兲 p T± K共pT±兲 册 − 共1 − pT±兲 . a −a cn2„−x + K共p−兲…dx 冉 冊 sin共2qa兲 + O共p−2兲, 2qa 共45兲 where 共43兲 for symmetric bound states where E共p兲 and E共u 兩 p兲 are the complete and incomplete elliptic integral of the second kind, respectively. The critical effective nonlinear interaction strength for the transition from resonances to bound states is given by gTeff = ± 冕 = − p−共E0 − V0兲 1 − 共42兲 for antisymmetric and ប2−2 p− 2ma E0 = V0 + ប2q2/共2m兲 共46兲 is the energy of the respective bound state of the linear system. For symmetric states we get 共44兲 As seen from Eqs. 共42兲 and 共43兲, 兩geff共p±兲兩 increases faster for bigger wave numbers ± 共respectively, bigger chemical potentials , see Fig. 7兲. This means that for a given value of geff, states which are strongly bound are more affected by the nonlinear interaction than weakly bound states. As in the case of resonances this is due to the fact that for weakly bound states the energy of the mean-field interaction is much smaller than the kinetic energy and thus has a comparatively weak influence. This can also be observed in Fig. 8 which compares the wave functions of the two lowest bound states of the potential V0 = −10, a = 2 for geff = −0.15 with the respective wave functions of the linear system 共geff = 0兲. The wave function of the state n = 0 differs much more clearly FIG. 9. 共Color online兲 Shift of the bound states towards lower chemical potentials in dependence of 兩geff兩 for p± 艋 0.99 for the potential V0 = −10, a = 2, for symmetric 共⫻兲 and antisymmetric 共䊊兲 bound states 共attractive nonlinearity兲. 033608-7 PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH 冉 g+eff = − p+共E0 − V0兲 1 + 冊 sin共2qa兲 + O共p+2兲. 2qa Then, for small values of p±, we have p± ⬇ − 冋 g±eff 冉 sin共2qa兲 共E0 − V0兲 1 ± 2qa 冊册 共47兲 sn共−a兩p−兲 + −cn共−a兩p−兲dn共−a兩p−兲 = 0 共48兲 ± = V0 + ប2±2共1 − 2p±兲/2m, 共49兲 where ± is the wave number of a Jacobi elliptic function cn with quarter period K共p±兲. In the limit p± → 0 the cn function becomes a trigonometric function with quarter period / 2 and wave number q. This motivates the approximation ± ⬇ 2qK共p±兲 / which leads to ប 2q 2 4 共1 − p±兲K2共p±兲. 2m 2 ± ⬇ E0 + 冉 3 sin共2qa兲 1± 2 2qa 冊 n= 3 ± ⬇ E0 + g±eff 2 共52兲 As in the previous section we look for solutions of odd or even parity. We make the ansatz ␥ ±e x , x ⬍ − a, 共x兲± = I±sn共±x + K±兩p±兲, 兩x兩 艋 a, ␥ ±e −x , x⬎a 冧 共53兲 for even 关⫹ sign, K+ = K共p+兲兴 and odd parity solutions 共- sign, K− = 0兲, where = 冑−2m / ប. For 兩x兩 艋 a the solutions are given by Jacobi elliptic sn functions with amplitude I± = 冑 冎 共58兲 ប2n2K共pT±兲2 pT± , ml2兩A兩2 共59兲 where the transition parameters pT± are given by 共1 + pT±兲K2共pT±兲 = 2m兩V0兩2a2 ប 2n 2 24 24 3 兲 − 共n兲 + O共pT± + 冑共 25 兲 with or pT± = − 25 共60兲 2 共n兲 = IV. BOUND STATES IN THE CASE OF REPULSIVE NONLINEARITY even parity, 2j− + 1, j− 苸 N0 , odd parity. gT± = for an infinitely deep square-well potential 关13–15兴. ប2±2 p± gm 2j+, j+ 苸 N0 , The transition occurs at 共51兲 A similar relation was derived for a delta-shell potential 关20兴. For small values of p± this approximation is in good agreement with the numerically exact calculations. In the limit V0 → −⬁ and thus q → ⬁ we obtain the formula 冦 再 −1 g±eff . 共57兲 for odd parity, which determine the bound states of the system. In the limit p → 0 the conditions 共56兲 and 共57兲 converge to Eqs. 共A9兲 and 共A10兲 of the linear system. For the case of an attractive nonlinearity, we have shown that resonances with even/odd quantum number n become bound states of even/odd parity due to the attractive meanfield interaction. In a similar way one can show for the case of repulsive nonlinearity that even/odd bound states become resonances with even/odd quantum number n. As before, we use the normalization 兩␥±兩 = 兩A兩. Thus we obtain the condition ±a = nK共p±兲, where 共50兲 Inserting Eqs. 共46兲 and 共48兲 and expanding K2共p兲共1 − 2p兲 = 共 / 2兲2共1 − 23 p兲 + O共p2兲 we obtain 共56兲 for even parity, with u = +a + K共p+兲, and −1 and the chemical potential is given by ± ⬇ sn共u兩p+兲 + +cn共u兩p+兲dn共u兩p+兲 = 0 冉 冊 8m兩V0兩a2 32 1− 2 2 2 . 25 បn 共61兲 Since pT± has to be positive this implies the conditions j± 艋 N±, where N+ and N− are the numbers of even and odd bound states of the linear system. This expresses the obvious fact that the number of newly created resonances cannot be greater than the number of bound states of the linear system. As in the case of an attractive nonlinearity, the quantum number n numbers the the resonances as well as the symmetric and antisymmetric bound states and is equal to the number of the minima of the squared modulus of the wavefunction. In the case of bound states, these minima are also zeros of the wavefunction, so that the oscillation theorem is valid. In terms of the effective nonlinearity defined by Eq. 共41兲, we obtain g−eff = 共54兲 ប 2 − 关−a − E共−a兩p−兲兴 ma 共62兲 for antisymmetric bound states and and wave number ± = 冑 2m共 − V兲 . ប2共1 + p±兲 g+eff = 共55兲 Again the wave functions and their derivatives must be continuous at x = ± a. Thus we obtain the equations ប 2 + 关+a + E共p+兲 − E„+a + K共p+兲兩p+…兴 ma 共63兲 for symmetric bound states. For the transition we get in both cases 033608-8 PHYSICAL REVIEW A 73, 033608 共2006兲 ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼ FIG. 10. 共Color online兲 Shift of the bound states towards higher chemical potentials in dependence of geff for p± 艋 0.995 for the potential V0 = −10, a = 2. ⫹: symmetric bound states, 䊊: antisymmetric bound states. The sudden disappearance of the curves at geff共p = 0.995兲 is due to the fact that in this figure the Jacobi elliptic parameter had to be “artificially” limited to p 艋 0.995 for numerical reasons. gTeff = ± gT±兩A兩2 p T± 冋 1− E共pT±兲 K共pT±兲 册 共64兲 . The magnitude of geff has the same qualitative dependence on p± as in the attractive case so that the nonlinearity has a stronger effect on lower bound states. Figure 10 shows how the bound states of the system are shifted towards higher chemical potentials if geff is increased, until they reach = 0 and develop into resonance states. Approximating the chemical potential linearly in terms of geff results in ± ⬇ E0 + 冉 3 sin共2qa兲 1± 2 2qa 冊 −1 geff , 共65兲 which is identical with the respective Eq. 共51兲 for attractive nonlinearities. FIG. 11. Transmission of the square-well potential V0 = −12, a = 4 with a repulsive nonlinearity g兩A兩2 = 4. Shown is the transmission coefficient of the stationary states calculated as described in the previous sections 共solid lines兲 in comparison to the transmission coefficients obtained from a numerical solution of the timedependent NLSE as described in the text 共crosses兲. initial state is an empty waveguide 共x , t = 0兲 = 0. The source strength S共t兲 is then ramped up adiabatically, so that finally a steady transmitting solution 共x , tend兲 is found. To determine the incident current and the transmission coefficient, a superposition of plane waves with momenta k = ± 冑2 is fitted to the wave function 共x , tend兲 for x0 ⬍ x ⬍ −a 共upstream region兲 and x ⬎ a 共downstream region兲. The NLSE is numerically integrated in real space using a predictor-corrector step 关33兴. To avoid reflections at the edges of the computational interval, one-way boundary conditions are used as described in Ref. 关34兴. An example of the wave function calculated in this way is shown in Fig. 12. The results of these simulations are shown in Fig. 11, where the transmission coefficient is plotted in dependence of the chemical potential for a fixed repulsive nonlinearity g兩A兩2 = 4. First, we notice that the final state 共x , tend兲 is indeed a stationary solution as discussed in the previous sections. The transmission coefficient 兩T共兲兩2 follows closely one of the steady state results which are computed as described in the previous sections. The small deviations are caused by numerical errors, in fact mainly by the 共still not V. DYNAMICS In this section we briefly discuss the implications of the results for the nonlinear eigenstates presented above on the dynamics. To this end we assume that a source outside of the square-well potential emits condensed atoms with a fixed chemical potential . In fact, we numerically solve the timedependent NLSE with an additional delta-localized source term 共see Ref. 关5兴兲 i 冉 冊 共x,t兲 1 2 = − + V共x兲 + g共x兲兩共x,t兲兩2 共x,t兲 2 x2 t + S共t兲exp共− it兲␦共x − x0兲, 共66兲 where units with ប = m = 1 are used. As above, V共x兲 = V0H共a − 兩x兩兲 is a square-well potential with depth V0 = −12 and halfwidth a = 4. The nonlinearity is nonzero only within the square-well g共x兲 = gH共a − 兩x兩兲. Here, H共x兲 denotes the Heaviside step function. The source is located at x0 = −15.5. The FIG. 12. Instability due to a beak-to-beak crossing scenario in the transmission through a square-well potential with V0 = −12 and. Shown is the squared modulus of the wave function 兩共x , t兲兩2 for = 2 in a grayscale plot 共lower panel兲 and the source strength S共t兲 共upper panel兲 for comparison 共repulsive nonlinearity兲. 033608-9 PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH perfectly absorbing兲 boundary conditions. Secondly, the final state 共x , tend兲 is the one with the smallest transmission. This behavior may suppress resonant transport, because a resonance state is not populated if different stationary states exist with a smaller transmission. This can be seen in Fig. 11 around ⬇ 5 and ⱗ 2. However, it is possible to access states with a higher transmission by a suitable preparation procedure as it has been shown in Ref. 关5兴. If there is no bistability in the vicinity of a resonance 共e.g., around ⬇ 7.5 in Fig. 11兲, resonant transport is found. A more complicated situation arises when a beak-to-beak bifurcation scenario occurs 共see Fig. 3兲. As an example, we consider the solution of the time-dependent NLSE with a source term as in Ref. 共66兲 for the same square-well potential as above 共V0 = −12, a = 4兲 and a fixed chemical potential = 2. Again the source strength is increased adiabatically 关35兴 to a level S共t兲 = S1 so that the system assumes a stationary transporting state, where g兩A兩2 = 3.5 is well below the critical value for the beak-to-beak bifurcation. Then the source strength is again slowly increased up to S共t兲 = S2, so that g兩A兩2 becomes larger than the critical value of the beak-to-beak bifurcation. The lowly transmitting stationary state ceases to exist and the time evolution cannot follow the stationary solutions adiabatically any longer. This is shown in Fig. 12. Indeed one observes that the system becomes unstable and finally does not assume a stationary state any more. However, the transmission through the square-well increases. The beak-to-beak crossing scenario and its implications on the dynamics will be discussed in detail in a subsequent paper 关31兴. VI. CONCLUSION The scattering of a BEC by a finite square-well potential has been discussed in terms of stationary states of the nonlinear Schrödinger equation. Neglecting the mean-field interaction outside the potential, ingoing and outgoing waves and thus amplitudes of reflection and transmission can be defined. The transmitted flux shows a bistable behavior in the vicinity of the resonances. For repulsive nonlinearities, the transmission coefficient also shows beak-to-beak bifurcations. Wherever these beak-to-beak scenarios occur, the scattering phase shows loops. An analytical expression for the position of the resonances is derived. The transitions from resonances to bound states and vice versa due to attractive or repulsive mean-field interaction are proven analytically and an explicit formula for the transition interaction strength gT is found. Furthermore, the positions of the bound states of the system are calculated and compared with the case of vanishing interaction g = 0. It is found that the oscillation theorem also holds for interaction parameters g ⫽ 0. Finally, we solve the time-dependent NLSE numerically in order to discuss the dynamics of the system for g ⬎ 0. If the amplitude 兩A兩 of the incoming flux is slowly ramped up for a fixed value of the chemical potential , the system populates the lowest branch of the stationary transmission coefficient, as it was first shown in Ref. 关5兴 for a double barrier potential, so that resonant transport is usually be sup- pressed. In the vicinity of a beak-to-beak bifurcation, the system can become unstable if g兩A兩2 becomes larger than the critical value of the beak-to-beak bifurcation so that the lowest branch of the stationary transmission coefficient suddenly ceases to exist. These beak-to-beak crossing scenarios will be further investigated in a subsequent paper 关31兴. ACKNOWLEDGMENTS Support from the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg “Nichtlineare Optik und Ultrakurzzeitphysik” is gratefully acknowledged. We thank Peter Schlagheck and Tobias Paul for inspiring discussions. APPENDIX A: REVIEW OF THE LINEAR SCHRÖDINGER EQUATION WITH A FINITE SQUARE-WELL POTENTIAL The scattering states of the linear time-independent Schrödinger equation 冉 − 冊 ប2 d2 + V共x兲 共x兲 = E共x兲 2m dx2 共A1兲 for a particle with mass m where the potential is given by V共x兲 = V0H共a − 兩x兩兲, where V0 ⬍ 0 and H共x兲 denotes the Heaviside step function, are obtained for positive particle energies E ⬎ 0. We assume an incoming flux from x = −⬁ and make the ansatz 冦 x ⬍ − a, Aeikx + Be−ikx , 共x兲 = Fe + Ge Ceikx , iqx −iqx , 兩x兩 艋 a, x ⬎ a, 冧 共A2兲 where the wave numbers inside 共兩x兩 艋 a兲 and outside 共兩x兩 ⬎ a兲 the potential well are given by q = 冑2m共E − V0兲 / ប and k = 冑2mE / ប, respectively. The amplitudes of transmission and reflexion are then determined by T ª C / A and R ª B / A. They satisfy the relation 兩R兩2 + 兩T兩2 = 1 as the Schrödinger equation is flux preserving. We have to make the wave function and its derivative continuous at x = ± a. Together with the normalisation condition A = 1 these boundary conditions determine the wavefunction of the system. By eliminating F and G we obtain the amplitude of transmission T共E兲 = exp共− 2ika兲 1 共A3兲 cos共2qa兲 1 − 共i/2兲关共q/k兲 + 共k/q兲兴tan2共qa兲 and thus the transmission probability 冋 冉 冊 兩T共E兲兩2 = 1 + 1 q k − 4 k q 2 sin2共2qa兲 册 −1 . 共A4兲 We define a resonance as a maximum of the transmission coefficient where the potential well is fully transparent, that is 兩T共E兲兩2 = 1. Then Eq. 共A4兲 implies the resonance condition 2qa = n. The resonance energies are then given by En = V0 + ប 2q 2 ប 2 2 2 = V0 + n , 2m 8ma2 共A5兲 where n is an integer number and sufficiently large to make En ⬎ 0. A Taylor expansion of Eq. 共A3兲 in terms of E − En yields 033608-10 PHYSICAL REVIEW A 73, 033608 共2006兲 ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼ 兩T共E兲兩2 ⬇ 共⌫n/2兲2 , 共E − En兲2 + 共⌫n/2兲2 共A6兲 where ⌫n is given by 2冑2ប 冑En共En − V0兲 ⌫n = 冑ma 2En − V0 . negative this means either g 艋 0 or g兩A兩2 艌 兩V0兩. For small values of p the left-hand side of Eq. 共B1兲 can be approxi4 mated by pK4共p兲 = 共 2 兲 共p + p2兲 + O共p3兲 and we obtain p⬇− 共A7兲 This means that the transmission coefficient can be approximated by a Lorentzian with width ⌫n in the vicinity of the resonances. We define the scattering phase 共E兲 by T共E兲 and obtain tan 共E兲 = 共q / k = 兩T共E兲兩exp关i共E兲 − 2ika兴 + k / q兲tan共2qa兲 / 2. In the vicinity of a resonance this reduces to 共E兲 ⬇ arctan关2共E − En兲 / ⌫n兴. Note that the scattering phase is zero for E = En. Bound states are found for energies V0 ⬍ E ⬍ 0. The area 兩x兩 ⬎ a outside the potential well is classically forbidden and the wavefunction decays exponentially with the coefficient = 冑−2mE / ប. Inside the well it still oscillates with the wavenumber q = 冑2m共E − V0兲 / ប. Due to the symmetry V共x兲 = V共−x兲 of the potential we can choose eigenfunctions which are either of odd or even parity. We make the ansatz 冦 ␥ ±e x , x ⬍ − a, ±共x兲 = ± cos共qx + ±兲, 兩x兩 艋 a, ␥ ±e −x , x⬎a 冧 共A8兲 for even 共⫹ sign, + = 0兲 and odd parity solutions 共⫺ sign, − = − / 2兲. As in the case of scattering states the wave function and its derivative must be continuous at x = ± a. For even parity we get the condition q tan共qa兲 = , 共A9兲 which determines the energy of the bound states. For odd parity we obtain q cot共qa兲 = − . 共A10兲 冋 册 R0 ⬎ and N− = 冋 册 R0 1 − 2 , ⬎ 共A11兲 冑 1 2g兩A兩2共V0 + g兩A兩2兲m2a4 + . 4 共/2兲4ប4n4 共B2兲 2m2a4 . ប 4n 4 共B3兲 Case 2. p共1 − p兲K4共p兲 = − g兩A兩2共V0 + g兩A兩2兲 For g = 0, this equation is solved by p = 0. The solution p = 1 is incompatible with Eq. 共24兲 because it leads to a negative chemical potential R. From p共1 − p兲K4共p兲 艋 0 we obtain the condition 0 艋 g兩A兩2 艋 兩V0兩. For small values of p, the lefthand side of Eq. 共B3兲 can be approximated by p共1 4 − p兲K4共p兲 = 共 2 兲 p + O共p3兲 and we get p⬇− 2g兩A兩2共V0 + g兩A兩2兲m2a4 . 共/2兲4ប4n4 共B4兲 APPENDIX C: PROOF THAT THE BOUND-STATE WAVE FUNCTIONS CONTINUOUSLY MERGE INTO RESONANCE WAVE FUNCTIONS Next we want to demonstrate that the wave function of a symmetric bound state continuously merges into a resonance wavefunction with an even quantum number n at = 0 if we use an appropriate normalization. To avoid confusion, all the parameters of the resonance wave function now carry an additional index R. For = 0 we have k = = 0 so that the squares of the wave functions outside the potential well agree if we use the normalization 兩␥+兩 = 兩A兩. Equations 共28兲, 共30兲, and 共33兲 then imply The transcendental equations 共A9兲 and 共A10兲 can be solved numerically. One can show 关27兴 that the number of symmetric or antisymmetric solutions is given by N+ = 1 + 2 兩A兩2 = 兩I+兩2 = − 4ប2 j+2K共p+兲2 p+ . gma2 共C1兲 Inside the potential well the squared modulus of the resonance wavefunction is given by respectively, where 关y兴⬎ denotes the smallest integer greater than y and R0 is given by R0 = 冑2ma2V0 / ប. 兩R共x兲兩2 = 兩A兩2 − 冋 冉 冊 册 nK共pR兲 ប2K2共pR兲n2 x兩pR − 1 dn2 2 gma a 共C2兲 APPENDIX B: DETERMINATION OF THE JACOBI ELLIPTIC PARAMETER The elliptic parameter p can be determined by inserting the expressions 共24兲 and 共23兲 into Eq. 共9兲. We obtain the following. Case 1. 2m2a4 pK 共p兲 = g兩A兩 共V0 + g兩A兩 兲 4 4 . បn 4 2 2 关see Eq. 共25兲兴 as g ⬍ 0 corresponds to case 1 in Sec. II. Because of k = 0, the phase of the resonance wave function is constant and the phases of R and + can be chosen identical. Using Eq. 共C1兲 and the relation between the squares of the Jacobi elliptic functions 关28–30兴 we arrive at 共B1兲 For g = 0 this equation can only be solved by p = 0. The inequality pK4共p兲 艌 0 implies g兩A兩2共V0 + g兩A兩2兲 艌 0. Since V0 is 033608-11 兩R共x兲兩2 = − ប24j+2K共p+兲2 p+ ប2K2共pR兲n2 − pR gma2 gma2 冋 冉 ⫻ cn2 冊 册 nK共pR兲 x兩pR − 1 . a 共C3兲 PHYSICAL REVIEW A 73, 033608 共2006兲 RAPEDIUS, WITTHAUT, AND KORSCH By comparing 兩R共x兲兩2 with the squared modulus of the bound-state wave function 冉 冊 gT+ = − ប2n2K共pT+兲2 pT+ . ma2兩A兩2 共C5兲 we see that 兩R共x兲兩2 is equal to 兩+共x兲兩2 if pR = p+¬pT+ and n = 2j+, where the index T denotes transition. Equation 共C1兲 implies that the transition occurs for the nonlinear interaction strength Inserting n = 2j+ and Eqs. 共C5兲 and 共34兲 into Eq. 共24兲 for R and Eq. 共B1兲, which connects g and pR, we see that R共p = pT+兲 = 0 and that Eq. 共B1兲 holds. Thus we have proved that pR = p+ = pT+ is valid so that resonances of even quantum number n indeed become symmetric bound states. In a similar way one can show that resonances of odd quantum number n become antisymmetric bound states. 关1兴 W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. 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