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# ST MATH 697

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PHYSICAL REVIEWA 67 063610 2003 Modulational instability of Gross Pitaevskii type equations in 1 1 dimensions G Theocharis1 Z Rapti2 P G Kevrekidis2 D I Frantzeskakis1 and V V Konotop3 1Department of Physics University of Athens Panepistimiopolis Zografos Athens 15784 Greece 2Department of Mathematics and Statistics University of Massachusetts Amherst Massachusetts 010034515 USA 3Ceritro de F isica Teoriea e Computaeional Universidade de Lisboa Avenida Professor Gama Pinto 2 Lisboa 1649003 Portugal Received 27 December 200239 published 26 June 2003 The modulational instability of the nonlinear Schrodinger NLS equation is examined in the case with a quadratic external potential This study is motivated by recent experimental results in the context of matter waves in BoseEinstein condensates BECs The theoretical analysis invokes a lenstype transformation that converts the GrossPitaevskii into a modi ed NLS equation without explicit spatial dependence This analysis suggests the particular interest of a speci c timevarying potential t t z We examine both this potential as well as the timeindependent one numerically and conclude by suggesting experiments for the production of solitonic wave trains in BECs DOI 101103PhysRevA67063610 I INTRODUCTION Intensive studies of BoseEinstein condensates BECs 1 have drawn much attention to nonlinear excitations in them Recent experiments have achieved to generate topological structures such as vortices 2 and vortex lattices 3 as well as solitons Especially as far as the latter are concerned two types of solitons have been created namely dark solitons 476 for condensates with repulsive interactions and bright ones 7 for condensates with attractive interactions Dark solitons are density dips characterized by a phase jump of the wave fmetion at the position of the dip and thus can be generated by means of phaseengineering techniques Bright solitons which were only recently created in BECs of 7Li are characterized by a localized maximum in the density pro le without any phase jump across it In the relevant experi ments this type of soliton was formed upon utilizing a Fesh bach resonance to change the sign of the scattering length from positive to negative An interesting question concerns how such solitary wave structures may arise ie which is the Lmderlying physical mechanism for their manifestation and how they may be generated in this context of matter waves in BECs It is well known that the dynamics of the BEC wave fme tion is described at the mean eld level which is an increas ingly accurate description as the zerotemperature limit is approached by the GrossPitaevskii GP equation a variant of the wellknown nonlinear Schrodinger NLS equation 8 which incorporates an external trapping potential term In the context of the traditional NLS equation without the external potential perhaps the most standard mechanism through which bright solitons and solitary wave structures appear is through the activation of the modulational instabil ity MI of plane waves In this case the continuous wave CW solution of the NLS equation becomes unstable to wards the generation of a chain of bright solitons It is the purpose of this work to demonstrate that Lmder certain con ditions this may also happen in the case of the GP equation as we MI is a general feature of continuum as well as of discrete nonlinear wave equations and its demonstrations 10502947200367606361082000 67 0636101 PACS numbers 0375Kk 0375Lm 4265Sf span a diverse set of disciplines ranging from uid dynamics 9 where it is usually referred to as the BenjaminFeir in stability and nonlinear optics 10 to plasma physics 11 Additionally the M has been examined recently in the con text of optical lattices in BECs both in onedimensional 1D and quasionedimensional systems as well as in multiple dimensions In such settings it has been predicted theoreti cally 1213 and veri ed experimentally 1415 to lead to destabilization of plane waves and in turn to delocalization in momentum space equivalent to localization in position space and the formation of solitary wave structures In this paper we discuss the MI conditions for the con tinuous NLS equation in 11 dimensions 1 spatial and 1 temporal iutumslulzuVxuO 1 in the presence of the external potential Vx This equation is actually a dimensionless effective GP equation which de scribes the evolution of the wave fmetion of a quasione dimensional cigarshaped BEC subject to de nite conditions see below In this context we will consider the harmonic potential Vx 7ktx2 2 which is relevant in particular to experimental setups in which the magnetic trap is strongly con ned in the two directions while it is much shallower in the third one 1 The prefactor kt is typically xed in current experiments but adiabatic changes in the strength and in fact even in the location of the center of the trap are experimentally feasible hence we examine the more general timedependent case A selfconsistent reduction of a 3D GP equation to a 1D NLS equation with external potential can be provided by means of the multiplescale expansion see eg Ref 12 for details In the case of a cigarshaped BEC such an ex pansion exploits a small parameter 52 N as a0klt 1 where N is the number of atoms as is the s wave scattering length 110 x m we is the longitudinal harmonicoscillator length we is the harmonic frequency and m is the atomic 2003 The American Physical Society THEOCHARIS et a mass The parameter 6 indicates the relative strength of the twobody interactions as compared to the kinetic energy of the atoms In the case at hand when the nitesize effects along the cigar axes are of primary interest the same small parameter de nes strong con nement cross section and to the cigar axis by the condition a a0 N 5 11 is the trans verse harmonicoscillator length It is worth pointing out that for example for a BBC of N104 of 23Na atoms 115275 nm having characteristic sizes 110 300 um and ati 10 um one obtains k011 and 52 001 It should be noted that herein different values of k were used as well as timedependent traps where kklisee below In this reduction the complex eld uxl in Eq 1 rep resents the rescaled mean eld wave function of the conden sate according to 6 12e Z quotWe rZlt2 ru Sxai ling2 Ia I rt 3 where I rt is the original order parameter ryz wi is the harmonic frequency corresponding to the cross section and physical space and time coordinates are used Respec tively in the reduced and rescaled GP equation 1 x is nor malized to the harmonicoscillator length a time is nor malized to the corresponding oscillation period the potential Vx is measured in units of zai8m and uxl is a func tion of order one uxl01 In the present paper 110 will be a varying quantity and then in the estimates 110 should be understood as an effective averaged quantity Notice also that in Eq 1 the subscripts denote partial derivatives with respect to the index while s isignas EI 1 illustrates the focusing 1 or de focusing 1 nature of the nonlinearity which represents the attractive or repulsive nature of the interatomic interac tions respectively 1 After brie y reviewing in Sec II the M1 for the N15 case we proceed to our main aim that is to study this insta bility in the context of the GP of Eq 1 with the potential of Eq 2 In Sec III we show that a lenstype transformation which transforms the GP equation into a relatively simpler form of the N18 equation provides insight in the latter case Two interesting cases are singled out the case where kI Ek ie for a xed trap and the case of kINII 2 which naturally arises in this setting In Sec IV we investi gate these cases numerically and nd a variety of interesting results including the generation of solitary wave trains This result indicates that the MI is indeed an underlying physical mechanism explaining the formation of matterwave soliton trains Finally we conclude with the discussion of Sec V which suggests this method as an experimental technique for the generation of soliton patterns in BBC II MODULATIONAL INSTABILITY FOR NLS We start by recalling the results for the modulational sta bility of the N15 1 without an external potential ie for Vx E 0 PHYSICAL REVIEWA 67 063610 2003 iutumslulzu0 4 We look for perturbed planewave solutions of the form x EbegtltPiqxwl EllINN 5 analyzing the 05 terms as bxlb0expiBxI l1061tlioegtltpiBxJ 6 Using Bxl Qxi t the dispersion relation connecting the wave number Q and frequency 9 of the perturbation see eg Ref 8 is found to be of the form 029Q2Q2Q22S 2 7 This implies that the instability region for the MS in the absence of an external potential appears for perturbation wave numbers Q2 lt2 152 and in particular only for focusing nonlinearil ies to which we will restrict our study from this point onwards A natural question is how this instability is manifested for wave numbers that satisfy the above condition An example is shown in Fig 1 There are two principal ways in which the instability can be detected see Fig 1 One of them is by probing the maxima of the original plane wave notice that to avoid problems with the boundaries the simulation shown in the gure was performed with periodic boundary conditions In the modulationally unstable case we have periodic recur rence of structures with very large amplitude while in the modulationally stable case of Q2 the perturbation only causes small amplitude oscillations In the Fourier picture the unstable perturbation generates sidebands of higher har monics as is well known 16 while similar structures are absent in the modulationally stable case III MODULATIONAL INSTABILITY FOR NLS WITH QUADRATIC POTENTIAL The quadratic potential of Eq 2 is clearly the most physically relevant example of an external potential in the BEC case given the harmonic con nement of the atoms by the experimentally used magnetic traps 1 To examine the MI related properties in this case we use a lenstype transformation 8 of the form HONeileXPIifUWIUMJ 8 where fl is a real function of time xKI and 739 71 To preserve the scaling we choose 817 Tl 1 6 2 9 The resulting equations can be satis ed by demanding that fz 4f2 klgt 10 efe4fez 11 Taking into account Eq 9 the last equation can be solved 0636102 MODULATIONAL INSTABILITY OF GROSSPITAEVSKII PHYSICAL REVIEW A 67 063610 2003 FIG 1 The evolution of the maximum amplitude left panels and the Fourier transform at the ending time of the simulation k right panels for a modulation ally unstable case Q1 top pan els and a modulationally unstable case Q 2 bottom panels An initial perturbation of 005 sinQx was added to the uniform solution of lt15 1 A 05 N 5 N A h A 04 2 4 g 2 CO 03 vx x 3 11 g 95 02 3 2 01 1 0 0 1o 20 30 40 0 t 112 01 11 N N 008 q 108 1 5 o 006 a 106 l Vx 004 gt5 104 5 2 D 102 002 1 0 0 10 20 30 o t Z t 0exp 4ffsds 12 0 This problem of nding the time dependence of the param eters is then reduced to the solution of Eq 10 Upon the above conditions the equation for v 7 be comes ivTv lvlzv2iv0 13 where f 2 14 and generically 1 is real and depends on time Thus we retrieve NLS with an additional term which represents either growth if 1gt0 or dissipation if 1lt0 A particularly interesting case is that of 1 constant Then from the system of equations 9 11 and 13 it follows that k must have a speci c form f and 7 can then be determined accordingly In fact the system of equations 9 11 and 13 with A constant has as its solution kttt 216 15 fttt 18 16 12Ktt 17 1 81 18 Notice that as per the assumption of an imaginary phase in the exponential of Eq 8 that our considerations are valid k only for AER In the above equations t is an arbitrary constant whose sign is related to the sign of 1 tgt0 and which essentially determines the width of the trap at time t0 according to Eq 15 Notice that tlt0 ie the case of dissipation in Eq 13 describes a BBC in a shrinking trap while the case tgt0 ie the case of growth in Eq 13 corresponds to a broadening condensate In this case the modulational condition remains un changed but now 0 satis es the dispersion relation wq2 7 221 so the growth if 1gt0 or dissipation if 1 lt 0 is inherent in Eq 5 Moreover all the terms are modu lated by the constant growth or decay rate exp27 and the instability when present will be developing according to the form 0 expiQ Qr7v2r with QQriv and Dr ZqQ If kkt is not given by Eq 15 then 1 must be time dependent eg REACH Here one cannot directly perform the MI analysis However still in this case we have con verted the explicit spatial dependence into an explicit tempo ral dependence An important example of this type the sim plest one in fact is the case of ktEkconst Then k mgtantzmmm 19 i cos2tt ti 0 Coszwzt 20 i i cos22 t ft 7t 0 k 21 0636103 THEOCHARIS et al Iuxt402 G Iuxt4ogtI2 MaXIuX2 Iuxt4o2 v70 80 90 100 110 120 130 140 150 160 170 180 X PHYSICAL REVIEW A 67 063610 2003 Iuxt4oI2 Z IUXt402 I 50 1 00 1 50 200 250 MaXIUXI2 Iuxt4oI2 FIG 2 The evolution of an initial condition of the form of Eq 22 for the timedependent potential of Eq 15 The left panels show the case on 1 while the right ones the case on 2 The panels show respectively the pro le of Iux2 at t40 in the top panel the same pro le is shown in a semilogarithmic plot in the second from the top panel clearly indicating the exponential nature of the localiza tion The time evolution of the maximum amplitude of the con guration is shown in the third from the top panel while the bottom panel shows a detail of the top one indicating the clearly solitary structure of the corresponding pulses This case imposes a timeperiodic driving term in Eq 13 with frequency 4 which is nothing but the oscillation frequency naturally following from the Ehrenfest theorem In this viewpoint the phase divergence at tn 77n 122E7t where n 0i1i 2 is understand able Indeed in the case at hand the chirp initial condition means existence of a current at the initial moment of time Due to the quadratic potential this current periodically changes the direction which is a straightforward conse quence of the Ehrenfest theorem The change of the current direction is accompanied by the phase singularity From the above it is clear that the most interesting cases in the setting with the harmonic potential are the ones with the inverse square dependence of the trap amplitude for a given x on time of Eq 15 as well as the one the regular harmonic trap of constant amplitude In both of these cases as well as more generally the lens transformation suggests the equivalence with a NLS equation with a gain In these special cases of interest the gain is constant or time periodic suggesting that similar phenomenology to the one of the regular NLS may be present A note of caution worth making here is that in reality in this case the evolution takes place in the setting of Eq 1 rather than that of Eq 13 The moti vation however is that upon suitable choice of the initial condition and for the types of traps discussed above the two equations the GP and the NLS with the gain term are equivalent at initial times hence one may expect that the instability that is present in the latter will manifest itself in some manner in the former However to examine the details of the time evolution of this instability we perform numerical simulations of the Eq 1 with appropriately chosen modulationally stable as well as modulationally unstable initial conditions IV NUMERICAL MANIFESTATIONS OF THE MODULATIONAL INSTABILITY FOR NLS WITH QUADRATIC POTENTIAL A Timedependent potential Perhaps the most interesting case due to the suggested analogy with an NLS with a constant coef cient gain where 0636104 MODULATIONAL INSTABILITY OF GROSSPITAEVSKII the modulational stability analysis can be performed is the case of kt t t 216 which we now examine numeri cally Notice that in our numerical investigations we will apply a loss term to Eq 1 close to the boundaries to emulate the loss of particles from the trap The rst case that we studied in this setting was the one of an initial condition 2 lecosQx 22 uxt exp 1 8t suggested by Eqs 8 and 15 l7 e was typically chosen in the range 0010l without signi cant variation in the qualitative nature of the results The parameter t was set to l The results are shown in Fig 2 for the case of Q 1 left panels and Q 2 right panels It is clear from the time evolution shown in the gure that in this setting we obtain and that is one of the main ndings of this work a soliton wave train formed as a result of the instability starting from such a modulated planewave initial condition An interesting feature of the obtained soliton train is that emerging solitons are of approximately equal shapes amplitudes in the presence of a broadening parabolic po tential in the case when the latter is static created solitons have essentially different shapes depending on their positions in space see eg Fig 4 One can argue that this outcome may not be a result of the modulational instability given that both modulationally stable and unstable Q s lead to such a manifestation How ever a careful inspection of the details of the evolution see also the shorttime runs reported below outrules that possi bility In particular the two features that happen for modula tionally unstable wave numbers are the following PHYSICAL REVIEW A 67 063610 2003 MaxtMaxxuxl2 FIG 3 The maximal amplitude over space and time for runs up to t40 is shown for different values of the wave number Q of the perturbation a The instability is manifested at earlier times see in particular the comparison of the third panels of Fig 2 b The instability leads to larger amplitudes in the modu lationally unstable regime see eg the comparison of the fourth panels of Fig 2 than in the modulationally stable one This is also clearly shown in Fig 3 where the cases with different Q in the perturbation have been examined for amplitude of the original plane wave 151 showing a clearly larger amplitude tendency for unstable wave num bers of Qlt 2 in this case The reason why in practice the instability occurs in both cases is that the dynamics of the potential in Eq 1 mix the wave numbers of the original perturbation and eventually A 6 A14 N N T Z Q13 35 4 5 a 312 X as gt15 11 2 o E 1 FIG 4 The cases of Ql 0 1 2 3 0 1 t 2 3 left panels and Q 2 right pan t 6 6 els are shown for the 1n1t1a1 con N dition of Eq 23 The top panel 7 N shows the evolution of the maxi oi 4 CR 4 mum amplitude as a function of x quot1 t1me for short t1mes the m1dd1e 5 2 5 2 anel shows the mod s d 3 p quare spa z I tial pro le for t3 the dashed 0 o 2 100 120 X 140 160 100 120 140 160 line here 11lustrates the trap at th1s t1me wh11e the bottom panel shows Fourier transform for the N 04 N 04 same time t3 t 5 has been oi 03 3 03 used here 4 1 5 02 quot 02 3 01 a 01 0 0 O 2 4 6 0 2 4 6 k k 0636105 THEOCHARIS et a result in the excitation of modulationally unstable ones However this only happens later because rst the modula tionally unstable Q s need to be excited and with a smaller amplitude in this case We also tried a different initial condition motivated by the experimental settings that led to the observation of bright matterwave solitons 7 In particular in these settings a Feshbach resonance is used to tune the sign of the nonlinear ity in the case of Eq 1 the sign of s starting from the repulsive case of slt0 and getting to the attractive case of sgt0 as time evolves in the experiment Given that in the case of slt0 the ground state of the system consists ap proximately of the socalled ThomasFermi TF cloud 1 we initialized the system in such a state emulating the time in the duration of the experiment in which the system is at slt0 and evolved the system from such an imtial condition In this case uxt0 is of the form uxt0uTFlecosQx 23 um lmaxx0u Vxt 0 l The chemical potential u is chosen as u l in this case A particular example of this type for 501 Q I left panels and Q2 right panels is shown in Fig 4 In this case we only show shorttime dynamics because at longer times the ThomasFermi which is not functionally close to the ground state of the case with s 1 will be destroyed leading to large amplitude localized excitations independent of the imtial value of Q In fact this short time experiment illustrates all the points that we made about modulationally stable and unstable shorttime evolution previously The modulationally unstable case of Ql rap idly develops the instability and deforms into a solitary wave train pattern On the contrary for the shorttimes reported in Fig 4 the modulationally stable case is limited to bemgn oscillations of small amplitude In the case of Ql the sidebands clearly form indicating the manifestation of the MI However notice also as highlighted previously that in the case of Q2 the dynamics of Eq I eventually tends to excite modulationally unstable wave numbers and hence will also result for longer times in localization B Timeindependent potential In the case in which the potential is time independent we rst once again tried an imtial condition with a modulation added to the plane wave in the form u 1 5 cosQx 24 Notice that in this case the chirp was not used in the imtial condition as it does not rid the equation of the explicit tem poral dependence In this case the ndings once again for Ql and Q 2 are shown in Fig 5 5005 was used in Eq 24 k 00001 In both cases for the GP equation due to the presence of the potential the condensate will become peaked towards the center gradually as time evolves However the development PHYSICAL REVIEW A 67 063610 2003 8 N 3 O V 2 39 7 O V 3 0398 I I I o 5 1o 15 20 1 FIG 5 The time evolution of the amplitude at x0 u0t2 is shown in the left panel for Ql and in the right one for Q 2 The comparison of the GP Equation solid line with the corresponding case for the NLS dotted line is also illustrated of the instability is clear from the comparison of the corre sponding amplitudes of the oscillation of the norm eld as a function of time The case with the ThomasFermi imtial condition of Eq 23 is shown in Fig 6 k00025 in this case and once again the cases of Q 1 and Q 2 are shown in the left and right panels respectively The development of the instability for short times is once again clear for the modulationally unstable case of Q 1 leading to the formation of a wave train while in the modu lationally stable case the perturbation is not ampli ed For longer times the destmction of the TF cloud will eventually lead in both cases to the generation of very strongly localized patterns However in essence here we take advantage of the separation of time scales for the appearance of the MI and for the destmction of the TF to illustrate in the shorttime evolution the development of the former instability 0636106 MODULATIONAL INSTABILITY OF GROSSPITAEVSKII 6 NE gtlt 4 3 6239 39 2 o I 39 I o 1 t 2 3 o a 4 6 k PHYSICAL REVIEW A 67 063610 2003 3gtI2 uxt quot o 00 IV k A 6 FIG 6 Same as Fig 4 but for the case of kt00025const The left panels correspond to Ql and the right ones to Q2 V CONCLUSIONS In this work we have examined the problem of modula tional instabilities of plane waves in the context of Gross Pitaevskii equations with an external in particular qua dratic potential The motivation for this study was its direct link to current experimental realizations of BoseEinstein condensates A lens transformation was used to cast the problem in a rescaled space and time frame in a way very reminiscent of the scaling in problems related to focusing 817 In this rescaled frame the external potential can be viewed as a form of external growth For speci c forms of temporal de pendence of the prefactor of the harmonic potential eg for kt t t 2 16 the resulting growth term is constant In such a context once again the MI analysis can be carried through completely producing similar conditions but now in the dynamically rescaled framevariables which can be ap propriately recast in the original variables This singles out the case of a temporally dependent potential of inverse square dependence with time Another case which was also examined due to its direct relevance to the experiment was the one of the constant amplitude trap Both of these cases were analyzed theoretically and then studied in detail numerically The theoretical predictions for modulational instability were veri ed by the numerical ex periments This was most clearly identi ed for shorttime dynamical evolution results that permit to clearly identify the instability through the formation of localized pulses and trains thereof For longer times trains are also formed for 0636107 THEOCHARIS et a modulationally stable cases due to the eventual excitation in the dynamics of unstable wave numbers However there are still stronger signatures of the instability in the unstable cases such as the larger amplitude of the resulting excita tions in such cases The main aim of this work is to advocate the use of the M as an experimental tool to generate solitonic trains in BoseEinstein condensates Our theoretical investigation and numerical ndings clearly support the formation of such trains in the context of the GP equation initialized with an appropriate modulation and possibly a chirp The latter is needed in the case of the timedependent trap that we have examined herein and which we argue may also be interesting to try to create in experimental settings Let us note in pass ing that traps with this type of time dependence of their amplitude have also been suggested as being of interest in quite different setups such as the study of explosion implosion dualities for the quintic critical GP 18 How ever our ndings should be observable even without the PHYSICAL REVIEWA 67 063610 2003 tim edependent trap as we have demonstrated The appropri ate modulation in the condensate initial condition can be generated by placing the condensate in an optical lattice 19 while the chirp can also be produced using appropriate phaseengineering techniques that are currently experimen tally available39 see eg Ref 5 We believe that such ex periments are within the realm of present experimental capa bilities and hope that these theoretical ndings may motivate their realization in the near future ACIGVOWLEDGMENTS PGK gratefully acknowledges support from a University of Massachusetts Faculty Research Grant from the Clay Mathematics Institute and from the NSF through Grant No DMS0204585 VVK gratefully acknowledges support from the European COSYC Grant No HPRNCTZOOO 00158 1 F Dalfovo S Giorgini LP Pitaevskii and S Stringari Rev Mod Phys 71 463 1999 AJ Leggett ibid 73 307 2001 2 MR Matthews et al Phys Rev Lett 83 2498 1999 KW Madison et al ibid 84 806 2000 S Inouye et al 87 080402 2001 3 JR Abo Shaeer et al Science 292 476 2001 JR Abo haeer C Raman and W Ketterle Phys Rev Lett 88 070409 2002 P Engels et 41 ibid 89 100403 2002 4 S Burger et al Phys Rev Lett 83 5198 1999 5 J Denschlag et al Science 287 97 2000 6 BP Anderson et al Phys Rev Lett 86 2926 2001 7 KE Strecker et al Nature London 417 150 200239 L Khaykovich et al Science 296 1290 2002 8 C Sulem and P L Sulern The Nonlinear Schrodinger Equa tion SpringerVerlag New York 1999 9 TB Benjamin and JE Feir J Fluid Mech 27 417 1967 10 LA Ostrovskii Sov Phys JETP 24 797 1969 11 T Taniuti and H Washimi Phys Rev Lett 21 209 1968 A Hasegawa ibid 24 1165 1970 12 VV Konotop and M Salerno Phys Rev A 65 021602R 2002 13 A Smerzi A Trombettoni PG Kevrekidis and AR Bishop Phys Rev Lett 89 170402 2002 14 ES Cataliotti et al eprint condmaU0207139 15 M Kasevich and A Tuchrnan private communication 16 A Hasegawa and WF Brinkman IEEE J Quantum Electron 16 694 1980 K Tai A Tomita and A Hasegawa Phys Rev Lett 56 135 1986 17 See eg C1 Siettos 1G Kevrekidis and PG Kevrekidis Nonlinearity 16 497 2003 and references therein 18 PK Ghosh eprint condma 0109073 19 See eg FS Cataliotti S Burger C Fort P Maddaloni F Minardi A Trombettoni A Smerzi and M Inguscio Science 293 843 2001 M Greiner O Mandel T Esslinger TW Hansch and 1 Bloch Nature London 415 39 2002 0636108

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