Nonlinear transport of Bose-Einstein condensates through waveguides with disorder Tobias Paul,

Nonlinear transport of Bose-Einstein condensates through waveguides with disorder

Tobias Paul,¹Patricio Leboeuf,²Nicolas Pavloff,² Klaus Richter,¹and Peter Schlagheck¹

1Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany

2Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris Sud, Bâtiment 100, F-91405 Orsay Cedex, France 共Received 19 September 2005; published 28 December 2005兲

We study the coherent flow of a guided Bose-Einstein condensate incident over a disordered region of length L. We introduce a model of disordered potential that originates from magnetic fluctuations inherent to micro- fabricated guides. This model allows for analytical and numerical studies of realistic transport experiments.

The repulsive interaction among the condensate atoms in the beam induces different transport regimes. Below some critical interaction共or for sufficiently smallL兲a stationary flow is observed. In this regime, the trans- mission decreases exponentially with increasingL. For strong interaction共or largeL兲, the system displays a transition toward a time-dependent flow with an algebraic decay of the time-averaged transmission.

DOI:10.1103/PhysRevA.72.063621 PACS number共s兲: 03.75.Kk, 72.15.Rn, 42.25.Dd

I. INTRODUCTION

The extraordinary experimental control achieved over atomic Bose-Einstein condensates 共BECs兲 provides new testgrounds for phenomena coming from many different fields. On the one hand these systems allow extensive study of nonlinear phenomena such as four-wave mixing 关1兴, propagation of bright关2兴and dark关3兴solitons, or the dynam- ics of Bloch oscillations in the presence of atom-atom inter- actions 关4,5兴. On the other hand the rapid progress in this field has led to a number of fascinating experiments probing complex condensed matter phenomena, such as the Mott transition in optical lattices关6兴, the creation of vortices 关7兴, the Josephson effect 关8兴 or the BEC-BCS crossover 关9兴.

Bose-Einstein condensates link these two prominent fields of current research in an exciting and unique way.

Wave mechanical transport in atomic vapors appears as a new direction for these trans-disciplinary studies that provide deeper insights into transport phenomena in the presence of interaction. Indeed, BEC systems are intrinsically phase co- herent, as are the clean two-dimensional electronic structures studied in mesoscopic physics at low temperatures. In addi- tion, interaction is much more simply modeled in BEC sys- tems than the electrostatic electron-electron potential and its sign 共repulsive or attractive兲 and strength can be tuned al- most at will. The link between matter-wave physics and elec- tronic transport phenomena became ultimately apparent with the advent of microscopic traps and waveguides for atoms, known as atom chips 关10–12兴. Related studies include the attempt to generalize Landauer’s theory of conductance to cold atoms 关13兴, the atom blockade phenomenon in quantum-dot-like potentials关14兴, as well as nonlinear reso- nant transport of Bose-Einstein condensates关15兴, to mention just a few examples.

A new direction in this context is the transport of Bose- Einstein condensates through disordered potentials. A rel- evant question is to what extent a Bose-Einstein condensate is subject to Anderson localization关16,17兴in the presence of disorder, as well as how this scenario is affected by the atom- atom interaction. There is a growing interest in the BEC community in issues related to the behavior of matter waves in disordered potentials. It started with the observation of

“fragmentation of the condensate” over a microchip 关18兴.

Nowadays a random potential is routinely engineered using an optical speckle pattern and its effects on the expansion of the condensate have been explored in Refs.关19,20兴.

In contrast to studies where the condensate is initially at rest, we focus in the present paper on the effect of disorder on apropagating Bose-Einstein condensate. In an adiabatic approximation, the dynamics reduces to an effective one- dimensional共1D兲transport problem; this is the so-called 1D mean-field regime 关21兴. We furthermore assume that the mean kinetic energy of the atoms in the condensate is larger than the typical height of the barriers induced by the disorder potential, i.e., perfect transmission is expected by classical mechanics. For the sake of concreteness, we restrict our- selves to one specific type of disorder: the one experienced by a condensate that is magnetically trapped above a corru- gated microchip. To this end we introduce a model that could be characterized as a “dirty-wire model” where the current in the microfabricated wire has white-noise fluctuations. This simple model captures most of the characteristics of the ran- dom potentials observed over corrugated microchips. We point out, however, that our results are not expected to be sensitive to the particular type of disorder, as long as the latter is sufficiently smooth and can be characterized by a well-defined correlation length.

Previous theoretical studies of the effect of disorder on the transmission of nonlinear waves mainly focused on attractive interaction and looked for stationary solutions of the problem 共see the review关22兴兲. Then, one has to choose between fixed input and fixed output boundary conditions. The latter case is less realistic, but simpler to discuss: It leads to algebraic decay of the transmission关23兴. The former case is compli- cated by the advent of multistability. However, the results of Knappet al.关24兴show that, for short sample size, the mean transmission is poorly affected by a weak nonlinearity 共as compared to the linear case兲, whereas for larger samples and stronger nonlinearity, evidence of delocalization is found.

Realistic transport processes of Bose-Einstein conden- sates are different from the above-mentioned studies because they typically involve particles experiencingrepulsive inter- actions. We will see below that in this case the assumption of stationarity is not appropriate because for large disordered 1050-2947/2005/72共6兲/063621共14兲/$23.00 063621-1 ©2005 The American Physical Society

regions or strong nonlinearity stationary solutions are dy- namically unstable. In typical experiments the population of a given final state can only be achieved through a time- dependent process共such as the gradual filling of an initially empty waveguide with matter waves兲. As a result, if a sta- tionary scattering state is unstable, the transport properties of the condensate may be unrelated to the transmission coeffi- cient associated with that state, whereas a study of stationary flows might misleadingly give some weight to this state 共if the transmission is averaged over all possible stationary so- lutions for instance兲.

We thus consider a setup that is relevant to experimental realizations and adapted to this specific transport scenario explicitly taking into account the possibility of time- dependent scattering: a coherent source of atoms emits mat- ter waves that propagate in the magnetic waveguide and encounter on their path a disorder region of length L. We show that the presence of a repulsive atom-atom interaction has dramatic effects on the transport properties of the con- densate. As is the case for attractive nonlinearity, Anderson localization is observed only in the regime of small interac- tion strengths and sample lengths. In this regime the transmission decreases exponentially with increasing L 关⬀exp共−L/Lloc兲兴, with a localization length Lloc modified by the interaction. For large sample lengths or strong inter- action, time-dependent scattering processes occur. In contrast to the previous regime, one observes an Ohmic decrease of the time-averaged transmission共⬀L⁻¹兲.

The paper is organized as follows. In Sec. II we set up the theoretical framework that is necessary to study transport through mesoscopic waveguides, introduce an effective one- dimensional Gross-Pitaevskii equation, and present a nu- merical method that is particularly suited to study transport processes of Bose-Einstein condensates in waveguides. In Sec. III we introduce a one-dimensional model for the ran- dom magnetic potential along the center of the waveguide.

We will show that a microscopic meandering of the current in the wire on the atom chip leads to a Lorentzian-correlated random potential. In Sec. IV we investigate the regime of weak disorder potentials and give a simple analytic expres- sion for the condensate wave function in the guide. In Sec. V we discuss numerical results for transport through moderate- and strong-disorder regions. We consider in particular the scaling of the transmission with the length of the disorder region. The paper closes with some concluding remarks.

Some technical points are given in the appendixes. In Appen- dix A we derive a relation between the mean transmission and the correlation function of the disorder potential. In Ap- pendix B we rederive, using standard WKB techniques, a result that is obtained heuristically in the main text.

II. TRANSMISSION THROUGH WAVEGUIDES We consider a coherent beam of Bose-Einstein-condensed atoms at zero temperature, propagating through a cylindrical magnetic waveguide of axisx. The condensate is formed by atoms of mass m which interact via a two-body potential characterized by its 3Ds-wave scattering lengthasc. We con- sider the case of repulsive effective interaction, i.e.,a_sc⬎0.

The condensate is confined in the transverse direction by a harmonic potential of pulsation␻_⬜. This transverse confine- ment is characterized by the harmonic oscillator length a_⬜

=共ប/m␻⬜兲^1/2.

In the following we restrict ourself to the 1D mean-field regime 关21兴 corresponding to a density range such that 共a_sc/a_⬜兲²Ⰶn_1Da_scⰆ1, where n_1D denotes the typical order of magnitude of the 1D densityn共x,t兲of the system. The first of these inequalities ensures that the system does not get into the Tonks-Girardeau limit and the second that the transverse wave function is the ground state of the linear transverse Hamiltonian; see, e.g., the discussion in Refs.关21,25兴. In this regime the system is described by a 1D order parameter

␺共x,t兲 关such that n共x,t兲=兩␺共x,t兲兩²兴 depending only on the spatial variable x along the guide. ␺共x,t兲 obeys the 1D Gross-Pitaevskii equation

iប⳵␺

⳵^{t =}

冉

⁻2m^ប2

⳵²

⳵^{x2+V共x兲+gn共x,t兲}

冊

^{␺, 共1兲}

with g= 2ប␻_⬜^asc 关26–28兴. V共x兲 is an effective one- dimensional potential along the waveguide, to which the condensate is exposed during the propagation process. We will see in Sec. III how it may originate from irregularities of a wire used for creating the magnetic confinement.

In the absence of a potential关V共x兲⬅0兴the plane wave

␺共x,t兲=

冑

n₀exp共ikx−i␮t/ប兲 共2兲 is obviously a solution of the Gross-Pitaevskii equation共1兲.

It satisfies the dispersion relation

␮^=m 2

J²

n₀²+gn₀, 共3兲 where the particle current is given byJ=n₀បk/m. Therefore, the chemical potential␮and the equilibrium constant density n₀ of a freely propagating condensate beam are determined by the currentJ, the wave vectork, and the effective inter- action strengthg. At this point, we mention that it was dem- onstrated in关28兴 that Eq. 共3兲 exhibits two constant-density solutions: a low-density共supersonic兲one and a high density 共subsonic兲one, where the transport is respectively dominated by the kinetic energy or by mutual interaction of the atoms.

Both solutions are plane waves of the form␺共x兲=Ae^ikx, but with different wave vectorskand particle densitiesA². As we are considering rather small condensate densities and large velocities in the waveguide, the supersonic solution will be the relevant one in the context of this paper.

We now assume the presence of a disorder potentialV共x兲 in the waveguide which is finite between x= 0 and L and vanishes elsewhere. In this case, a BEC that is injected into the initially condensate-free disorder region from the up- stream side共i.e., at x⬍0兲in general does not freely propa- gate to the downstream region 共at x⬎L兲, but undergoes a scattering process. In this paper we shall compute transport properties of a system where a monochromatic beam of con- densate with well-defined currentJiis injected into the dis- order region共see Fig. 1兲. This means, we consider the propa- gation process in terms of a so-called fixed input problem 关22,24兴.

Our purpose is now to compute transmission coefficients for the condensate transport through the disordered region.

Furthermore, we shall investigate to what extent it is possible to populate stationary scattering states, i.e., stationary solu- tions ␺共x,t兲=␺共x兲exp共−i␮t/ប兲 satisfying the outgoing boundary condition ␺共x兲=

冑

n₀e^ikx 共with k⬎0兲 for x→+⬁, wheren₀ is the density associated with the supersonic solu- tion关29兴. This question can be addressed by integrating the time-dependent Gross-Pitaevskii equation共1兲in the presence of a source term that is localized in the upstream region and emits monochromatic matter waves. Such a source models the coupling of the waveguide to a reservoir of condensate from which matter waves are injected into the guide. It has been demonstrated in关15兴that this approach is particularly well suited to compute transmissions for fixed input prob- lems. Additionally, it allows one to determine for a given potential V共x兲 whether an incident monochromatic beam populates a stationary scattering state or not.

Hence, we consider now the modified Gross-Pitaevskii equation with a source that is localized at the positionx₀in the upstream region,

iប⳵␺共x,t兲

⳵^{t =}

冉

^−2mប2⳵^{⳵x22+V共x兲+g兩}␺共x,t兲兩²

冊

^␺共x,t兲

+S₀exp共−i␮t/ប兲␦共x−x₀兲. 共4兲 S₀is the source amplitude which determines the emitted cur- rent. To understand the functionality of the source term, it is instructive to consider first solutions of Eq.共4兲in the absence of the potentialV共x兲. In this case there exist plane-wave so- lutions with constant density n. To demonstrate this, we switch to the Fourier space, where Eq.共4兲takes the form共for constantn兲

冉

បⁱ

⳵

⳵^t− ប²q²

2m −gn

冊

^{␺˜共q,t兲=S0e−iqx0e−i␮t/ប. 共5兲}

This equation admits a solution of the form

␺˜共q,t兲= S₀e^−iqx0e^−i␮t/ប

␮^−gn−ប²q²/共2m兲. 共6兲 By transforming back to real space, we find that the source emits in both directions the monochromatic wave

␺共x,t兲= S₀m

ikប²e^{ik兩x−x0兩}e^−i␮t/ប. 共7兲 In Eq. 共7兲 k is self-consistently defined by 共បk兲²= 2m关␮

−g兩S₀兩²m²/共ប⁴k²兲兴. The current emitted by the source can be calculated by evaluating the quantum mechanical current operator. We find Ji= ±兩S0兩²m/共ប³k兲 共⫹ for x⬎x₀;

⫺for x⬍x₀兲.

We now return to the general caseV共x兲⫽0. In order to perform the numerical integration, the wave function␺共x,t兲 is expanded on a finite lattice and is propagated in the real time domain. As we are dealing with an open system, artifi- cial backscattering at the boundaries of the lattice has to be avoided. For that purpose we impose absorbing boundary conditions that are well suited for transport problems 关30兴 and can be generalized to account for weak or moderate non- linearities关31兴.

As in real experiments we choose as initial condition

␺共x,t= 0兲⬅0. In order to compute the condensate wave function we numerically integrate ␺共x,t兲 in Eq. 共4兲 while adiabatically tuning the source amplitudeS₀ from 0 up to a given maximum value that corresponds to a desired incident current J_i. This approach simulates a realistic propagation process, where a coherent Bose-Einstein condensate beam with chemical potential␮is injected into the initially empty waveguide from a reservoir. For comparatively weak nonlin- earities a stationary scattering state of the form ␺共x,t兲

=␺共x兲e^−i␮t/ប, which corresponds to a supersonic solution in the downstream region, is generally obtained from the nu- merical propagation. This stationary wave function satisfies the time-independent Gross-Pitaevskii equation

␮␺共x兲=

冉

^−2mប2 ⳵^{⳵x22+V共x兲+g兩}␺共x兲兩²

冊

^{␺共x兲. 共8兲}

In contrast to the case of the linear Schrödinger equation the transmission coefficient cannot be computed by simply decomposing the upstream wave function into an incident and a reflected part because the superposition principle is not valid in the presence of the nonlinear term. Such a decom- position is only possible in the limit of small interaction strengths or small back reflections 关32兴. However, our nu- merical approach permits nevertheless a straightforward ac- cess to the transmission coefficient also in the nonlinear case.

The latter is evaluated by the ratio of the current Jt in the presence of the potentialV共x兲 共i.e., the transmitted current兲to the current J_i obtained in the absence of V共x兲 共the incident current that is emitted by the source兲. This approach provides a natural extension of the usual definition of transmission coefficients in quantum mechanics to the nonlinear case关15兴.

In the nonlinear regime, due to dynamical instabilities the wave function ␺共x,t兲 does not always converge toward a stationary state but can remain time dependent共cf. Sec. V兲. In that case, the downstream current is no longer constant FIG. 1.共Color online兲A condensed beam with incident current

J_ican populate a stationary scattering state. The solid line shows its longitudinal densityn共x兲 共in units of the equilibrium densityn₀兲. In the downstream region,␺ tends to a plane wave with transmitted currentJ_t. The dashed line displays the scattering potentialV共x兲in units of the chemical potential␮.

and therefore the transmission becomes a function of time. In this case we simulate the propagation process over a long period␶共ideally␶→⬁兲and characterize the transport prop- erties of the guide by means of the time-averaged transmis- sion

T¯= lim

␶→⬁

1

␶

冕

t t+␶

T共t

⬘

^兲dt

⬘

^{共t⬎0兲. 共9兲}

This choice of working with the mean value¯Tis inspired by common experimental setups: the number of condensed at- omsNAreaching the downstream region during the time␶^is NA=␶T¯ J_i. This number of atoms can be determined experi- mentally, e.g., by use of absorption spectroscopy.

III. A SIMPLE MODEL OF DISORDER

In order to compute transport properties through disor- dered regions in magnetic waveguides, it is necessary to in- troduce an appropriate model for the static random magnetic potential along the center of the waveguide. We first briefly recall the basic principle to generate elongated magnetic waveguides for cold atoms or condensates. A typical setup that is commonly implemented on atomic chips is the so- called side wire guide关33兴. As sketched in Fig. 2 a circular magnetic fieldB₀, created by an electric currentIthat flows along a straight microfabricated quasi-two-dimensional wire, and a homogeneous bias field B_⬜ form a minimum of the magnetic field parallel to the wire at distanceh. An offset fieldB储applied parallel to the wire reduces losses induced by spin-flip processes near the magnetic field minimum.

For a spatially homogeneous current density in an ideal- ized wire, the magnetic waveguide is perfectly uniform along its longitudinal axis. In reality, however, inhomogeneities in the current density inside the wire have to be taken into account. Such deviations from a homogeneous current flow can be induced by shape fluctuations of the wire or impuri- ties inside the metal. These imperfections cause a magnetic

field roughness along the center of the waveguide that acts as an additional potential and prevents perfect transmission of condensate beam through the guide 关34兴. This additional magnetic-field component increases as the distance to the chip surface diminishes and is expected to reduce the trans- mission noticeably.

In the following we consider a steady state current density j共r兲 flowing in a thin quasi-two-dimensional metallic wire.

Due to the wire imperfections the current density varies with the positionr. We decomposej共r兲into a large constant com- ponentj₀flowing parallel to the wire and a small component

␦j共r兲

j共r兲=j₀ex+␦j共r兲. 共10兲 At the center of the waveguide the circular magnetic fieldB₀ that is generated byj₀cancels with the bias fieldB_⬜. Hence, the total magnetic field along the center of the waveguide is given by

B共x,0,h兲=B_储ex+␦B共x,0,h兲, 共11兲 where␦^B=␦^Bxex+␦^Byey+␦^Bzezis computed from the Biot- Savart law

␦^B= ␮0

4␲

冕

^d3r

^⬘

^␦j共r

^⬘

兩r^兲⫻−r^共r

⬘

兩^3−r

^⬘

^兲. 共12兲 The effective potential for the atoms is proportional to the modulus of the magnetic field

兩B兩=

冑

共B_储+␦^Bx兲²+␦^By 2+␦^Bz

2. 共13兲

As␦^j is supposed to be small, we keep only terms of first order in␦j. This yields the simple result

兩B兩=B储+␦^Bx. 共14兲

Hence, within the approximation of small current fluctua- tions, the disorder potential along the center of the wave- guide is given by

V共x兲=␮B␦^Bx共x,0,h兲. 共15兲

We now consider a quasi-two-dimensional wire of length Lin thexdirection and widthwin theydirection. A proper description of the current density j共r兲 in the metallic wire would require an accurate microscopic model for structural dislocations of the wire as well as its impurities关34,35兴. In the present work we adopt a more simple and phenomeno- logical approach, which is valid if the length scalelon which j共r兲typically fluctuates is much smaller than the heighthof the waveguide. To this end, we divide the wire intoNequal blocks of lengthlwidthwand thickness⌬z共see Figs. 2 and 3兲. For each block of volume V we compute the average current density

j^␯= 1 V

冕

−⌬z/2

⌬z/2

dz

冕

共␯−1兲l

␯l

dx

冕

−w/2 w/2

dyj共r兲. 共16兲 共The index␯= 1 , . . . ,Nlabels the blocks and the correspond- ing mean current densities j^␯.兲 The total electric current along the wire is given by

FIG. 2.共Color online兲Main building block to create a magnetic waveguide on a chip. A current flowing in a microfabricated wire and a perpendicular bias field form an elongated microtrap. Imper- fections in the wire force the current to follow a weakly meandering path and generate therefore a magnetic disorder potential along the center of the guide.

I=

冕

−⌬z/2

⌬z/2

dz

冕

−w/2 w/2

dyj共r兲·ex=w⌬zj^␯·ex. 共17兲

Hence, in the usual case of a stationary electric currentI, the xcomponent of j^␯ is given by the constant value j₀ of Eq.

共10兲for all␯, and we have

j^␯=j₀ex+␦^jy␯ey+␦^jz␯ez. 共18兲 The thickness⌬z of the wire is assumed to be much smaller than all other relevant length scales. We therefore assume 兩␦^jz␯兩Ⰶ兩␦^jy␯兩and neglect the contribution of␦^jz␯to the disorder potential in the following. This yields

V共x兲=␮B_␯=1

兺

^{N ␦Bx␯共x,0,h兲 共19兲}

where the magnetic-field contribution of the␯th block at the center of the waveguide is computed from the Biot-Savart law according to

␦^Bx␯= ␮0

4␲

冕

_共␯−1兲l

␯l

dx

⬘ 冕

−w/2 w/2

dy

⬘

_关共x ^{⌬zh␦jy␯}

−x

⬘

兲²+h²+y

⬘

²兴^3/2

=␮0⌬z

2␲ ␦^jy␯

冋

^arctan

^{冉 冑}

^{u2wu/共2h兲+h2+w2/4}

^冊 册

^x−␯l x−共␯−1兲l

. 共20兲 Within the discretization procedure just described, we can introduce disorder by assuming ␦^jy␯ as a random variable, uniformly distributed in the interval 关−共3␴^2/ 2l兲^1/2,共3␴²/ 2l兲^1/2兴. This assumption corresponds to a zero average transverse current density共具␦^jy␯典= 0兲with a␦ ^corre- lation

具␦^jy␯␦^jy␯⬘典=␴²

l ␦␯,␯⬘, 共21兲 and allows us to reach a well-defined regime in the limitl

→0. In that limit␦^jy␯is replaced by anx-dependent quantity

␦^jy共x兲verifying关36兴

具␦^jy共x兲␦^jy共x

⬘

^兲典=␴²␦共x−x

⬘

^{兲. 共22兲} Here, the parameter␴fixes a scale for the typical deviation of the current density from a homogeneous current flow.

Since the fluctuations in␦^jy are certainly proportional to j₀

=I/共w⌬z兲, we can write␴^=j0

冑

_ᐉ^*. Hereᐉ^*is a characteristic length depending on the properties of the metallic wire, which can in principle be found from experimental investi- gations.

Due to the convolution procedure in Eq.共20兲, the short- range disorder in the electric wire induces a smoothly vary- ing potentialV共x兲 along the guide. This is clearly visible in Fig. 4, which shows the disorder potentials that result from three numerically generated sets of current densities j_y^␯, at three different heightsh of the waveguide. The disorder po- tential is smoother for large distancesh, and becomes more rough共and its typical intensity increases兲as h diminishes.

One has具V共x兲典= 0 and it is appropriate to characterize the random potential by studying the correlation function

C共x−x

⬘

兲=具V共x兲V共x

⬘

兲典. 共23兲 In Fig. 5 we show results forC共x−x

⬘

兲at different heightsh.

The correlation function is computed numerically by averag- ing over a large number of different disorder realizations. We find that it can be fitted with good accuracy by a Lorentzian curve

C共x−x

⬘

^{兲 ⯝ ␥lc}

l_c²+共x−x

⬘

兲². 共24兲 This allows extraction of the correlation lengthlcand estab- lishment of an empirical relation between the heighthandl_c. In the regime where the widthwof the wire and the discreti- zation lengthl are of the same order, we find that the corre- lation length depends linearly on the distance between wire and waveguide,l_c⯝sh, with a proportionality constantsthat varies between 1 and 2. For the experimentally relevant case ofw= 4␮m共a wire of this size has been realized by Ottet al.关11兴兲we finds= 1.2.

FIG. 3.共Color online兲Partitioning of the wire in equal blocks of lengthl, widthw, and thickness⌬z. For each block we compute an average current densityj^␯. The current component parallel to they direction is at the origin of the magnetic disorder potential along the center of the waveguide.

FIG. 4. Numerically computed representative examples for dis- order realizations at different distanceshbetween the center of the guide and the atomic chip surface. The panels show the transition from weak to strong disorder, with decreasing distanceh. The dis- order potential is given in units of␮0␮Bw⌬z␴/共2␲兲.

Theoretically, this result may be understood as follows. In the continuous limit l→0 共and in the idealized case of an infinitely long wire兲, Eq.共20兲takes, in the regimewⰇh, the particularly simple form

V共x兲=␮B␮0

2␲

冕

_−⬁^+⬁dx

^⬘

共x^h⌬−^zx^␦

⬘

^j兲^y2共x+

^⬘

h^兲2. 共25兲 In this case the disorder potential is exactly Lorentz corre- lated with l_c= 2h and␥⁼␲⁻¹共␮0␮B⌬z␴^{/ 2}兲². In the opposite regime wⰆh, the correlation function C共x−x

⬘

^{兲 cannot be} computed analytically, but its Fourier transform Cq

=兰₋^+⬁_⬁exp共−iqx兲C共x兲dx can be calculated. One obtains Cq=

冉

^␮0␮2^Bw⌬␲ ^z␴

冊

^{2关qK1共qh兲兴2, 共26兲}

where K₁ is the modified Bessel function of the first kind 关37兴.Cq as given in Eq. 共26兲 is not very different from the Fourier transform of a Lorentzian共a decreasing exponential兲, and this is the reason whyC共x−x

⬘

^兲can be fitted reasonably well by a Lorentzian also in the regime hⰇw. To find a sensible Lorentzian fit, one can for instance try to reproduce C共0兲andC

⬙

共0兲obtained from Eq.共26兲with the parameters␥ andlcof Eq.共24兲:C共0兲=␥/lcandC

⬙

^{共0兲= −2␥/l}c

3. This leads to

lc

h =

冢

²

^{冕 冕}

^{0+⬁0+⬁tt42KK1212共共tt兲兲dtdt}

冣

^{1/2⯝1.46. 共27兲}

Thus, again in this limit, we find that lc is of the type lc

⯝sh. The important outcome of this discussion is that, for the continuous model 共l→0兲—in both limits wⰆh and w Ⰷh—and also in the numerical realizations of the disorder with a finite grid l, we obtain a random potential which is Lorentz-like correlated, with a correlation length lc that is proportional to the heighth of the trap above the chip, and

with a proportionality constant of the order 1–2. This is con- firmed experimentally by the detailed studies presented in Ref.关34兴.

The model introduced in this section, where the disorder potential originates from white-noise-correlated fluctuating currents关see Eqs.共21兲and共22兲兴corresponds physically to a dirty-wire model, in the sense that the very erratic random current density 共22兲 can be considered as originating from the presence of impurities in the wire. We note that the white-noise current correlations lead to a disorder potential whose typical amplitude 共for the continuous limit l→0, when hⰇw兲 varies as 具V²共x兲典^1/2⬀Ih^−3/2, different from the experimental finding Ih^−2.2 of Kraft et al. 关38兴. In contrast, the model of a “clean wire with corrugated boundaries” in- troduced in关35兴and developed in关34兴yields in the case of a white-noise correlated boundary roughness a dependence of the form Ih^−5/2, in closer agreement with the experimental findings of Ref.关38兴. Note, however, that the experimental results of Estève et al. 关34兴 point to a boundary roughness which is not white-noise correlated, and a typical disorder potential which decreases less rapidly thanh^5/2, as found in the present study. Also the correlation function coming from the dirty-wire model is in better agreement with the experi- mental one determined in Ref.关34兴, which differs from the one resulting from a wire with a white-noise-disordered boundary 共which has a correlation function verifying C_q=0

= 0 关35兴兲. It thus appears that the simple dirty-wire model introduced in the present section allows one to construct a disordered potentialV共x兲that captures most of the character- istics of microfabricated magnetic guides.

IV. WEAK DISORDER

In this section we investigate the regime of weak disorder potentials and derive simple relations between the conden- sate density and the disorder potentialV共x兲. Weak disorder means in this context that the propagation of the condensate is only marginally affected by the scattering region. This implies that the kinetic energy per particle must be much larger than the typical intensity of the disorder potential 关which can be estimated, for instance, by the standard devia- tion具V²共x兲典^1/2兴. We shall argue below that a secondary crite- rion is necessary to characterize this regime, namely, that the length of the disordered region is small compared to the characteristic length scale Ld 共to be defined below兲 typical for the decrease of the transmission.

First, we rewrite the Gross-Pitaevskii equation共1兲 in the well-known form of the hydrodynamic equations

⳵

⳵^{tn= −}

⳵

⳵^{x共nu兲 共28兲}

and

m⳵^u

⳵^{t =}

⳵

⳵^x

冉

^{2mnប21/2⳵2}⳵^{nx1/22 −} mu²

2 −V共x兲−gn

冊

^{, 共29兲}

whereuis the condensate velocity. In the case of a stationary state we have⳵tn= 0 and ⳵tu= 0, from which it follows that the current J=nu is constant. Integration of Eq. 共29兲 then yields

FIG. 5. 共Color online兲Numerically computed correlation func- tions for different distanceshbetween the center of the guide and the atomic chip surface共solid lines兲. Dashed lines: Fit to a Lorent- zian curve.

␮⁼V共x兲+gn+ J²

2mn²− ប² 2mn^1/2

⳵^2n1/2

⳵^{x2 . 共30兲} This is the time-independent Gross-Pitaevskii equation for a current-carrying scattering state. In the downstream region an outgoing plane wave␺共x兲=

冑

n₀e^ikxis expected. The equi- librium densityn₀ coincides with the supersonic solution of the dispersion relation 共3兲. Defining ␳共x兲⬅n共x兲/n₀ and v共x兲⬅2mV共x兲/共ប²k²兲 one may rewrite Eq.共30兲in a dimen- sionless form:

− 1

␳^1/2

⳵²␳^1/2

⳵^{x2 +}

␳^{− 1}

␰^{2 +k2}

冉

␳^{12− 1 +v共x兲}

冊

^{= 0. 共31兲}

In this expression we made use of the dispersion relation Eq.

共3兲 and expressed J=n₀បk/m in terms of the downstream densityn₀ and of the outgoing wave vectork. The quantity ប²k²/共2m兲=␮−gn₀ is the kinetic energy of the outgoing plane wave with equilibrium densityn₀.␰⁼ប/

冑

2mn₀g is the condensate healing length.

In order to find perturbative solutions of Eq. 共31兲 for v共x兲Ⰶ1, we insert the ansatz ␳共x兲= 1 +␦␳共x兲 into Eq. 共31兲 and keep only terms that are linear in␦␳共x兲:

⳵²

⳵^x2␦␳共x兲+ 4␬²␦␳共x兲= 2k²v共x兲, 共32兲 where

␬=k

冑

^{1 −}2␰^{12k2. 共33兲} The solution of Eq.共32兲in the presence of the downstream boundary conditions ␦␳共L兲= 0, ␦␳

⬘

共L兲= 0 共flat downstream density兲is关40兴

␦␳共x兲=k²

␬

冕

x L

sin关2␬共x

⬘

^−x兲兴v共x

⬘

兲dx

⬘

^,

␦␳

⬘

共x兲= − 2k²

冕

x L

cos关2␬共x

⬘

⁻x兲兴v共x

⬘

兲dx

⬘

^. 共34兲

This implies that the density profile in the upstream region 共x⬍0兲 deduced from the linearized Eq. 共32兲is of the form n共x兲=n₀关1 +␦␳共x兲兴with

␦␳共x兲=␦␳^¯cos共2␬^x+␪兲. 共35兲 The amplitude ␦␳^¯ and the phase factor ␪ in Eq. 共35兲 are determined by the disorder potentialV共x兲 via Eq. 共34兲. The modified wave number␬ fixes the period of the density os- cillations.

As we are obviously in the regime of small back reflec- tions we adopt the method of Ref. 关32兴 to determine the transmission coefficient in an approximative way. To this end we make the ansatzn共x兲=兩␺inc共x兲+␺ref共x兲兩²with

␺inc共x兲=aexp兵i␬x其,

␺ref共x兲=bexp兵i共␬^x+␪兲其. 共36兲 Comparing the corresponding density profile with Eq.共35兲, one obtains the following expressions for the amplitudes a andb 关39兴:

a² n₀= 1 −1

4␦␳^¯2+O共␦␳^¯4兲, b²

n₀=1

4␦␳^¯2+O共␦␳^¯4兲. 共37兲 It was pointed out in Ref. 关32兴, and numerically confirmed for single- and double-barrier potentials关31兴, that␺refcan be approximately identified with the reflected component of the condensate in the case of almost perfect transmission. This corresponds to a reflexion coefficient R=b²/a²=¹₄␦␳^¯2 +O共␦␳^¯4兲 and to a transmission coefficient which can be ex- pressed关using Eq.共35兲兴as

T= 1 −1

4␦␳^{¯2= 1 −1}

4

冉

^{关␦␳共0兲兴2+41}␬^2关␦␳

⬘

共0兲兴²

冊

^{. 共38兲}

In this final expression␦␳共0兲 and ␦␳

⬘

^共0兲 are related to the disordered potential by means of Eq.共34兲. Therefore, deter- mining the transmissionTfor a given potentialV共x兲amounts to compute the integrals Eq.共34兲.

As shown in Appendix A the above procedure allows us to determine the disorder average具T典from knowledge of the correlation function of the disorder potential. For the relevant case of a Lorentzian correlation关of the form共24兲兴we obtain

具T典= 1 − L Ld

, 共39兲

where

Ld= ប⁴␬²

␲␥^m2e2␬

l_c 共40兲

is the characteristic length scale for the decay of the trans- mission. We recall here that the above analysis is valid only in the regime ␦␳^¯Ⰶ1, i.e., the linear decrease of 具T典 in Eq.

共39兲 is valid only for LⰆLd. Thus, we have to refine our definition of weak disorder: not only should the intensity of the potential be small, but also the length of the disordered region should not exceed the valueLd.

As we see from expression 共40兲, the effect of the atom- atom interaction is entirely contained within the modified wave number␬关Eq.共33兲兴which describes the period of the upstream density oscillations. For repulsive atom-atom inter- actions, we have␬⬍k, which implies that the mean trans- mission is reduced compared to the noninteracting case. This behavior is indeed well confirmed by numerical computa- tions based on the approach presented in Sec. V. This interaction-induced decrease of the transmission was already observed in Ref. 关32兴 and interpreted as a lack of kinetic energy compared to the interaction-free case.

In the limit of very small correlation lengths, i.e., ␬^lc

Ⰶ1, the disorder potential can be approximated by a white- noise potential with correlation function C共x−x

⬘

^兲⯝␥␲␦共x

−x

⬘

兲. Considering the noninteracting case共␬⁼k兲we recover in this regime the well-known expression Ld=Lloc

⬅共ប⁴k²兲/共␲^m2␥兲 for the localization length of ␦-correlated disorder potentials共see, e.g., Ref.关41兴兲.

The opposite limit␬lcⰇ1 can be considered as thesemi- classical regime, where the de Broglie wavelength ␭

⬅2␲^/kof the condensate is much smaller than the correla- tion length l_c of the disorder potential. In this regime, the length scale Ld is dominated by the exponential prefactor exp共2␬lc兲, and the deviations from perfect transmission 具T典

⬅1 vanish exponentially fast with increasing ratio ␬lc. The semiclassical condition␬lcⰇ1 furthermore allows us to de- rive a simple analytical expression for the density n共x兲 throughout the scattering region. We start from the zeroth- order solution n共x兲⬅n₀ valid for V⬅0. Then, for given ␮ andJ, the densityn₀ can be obtained by iteratively solving the self-consistent equation关strictly equivalent to Eq.共3兲兴

n₀=

冑

^m2J关␮^−gn0兴^−1/2, 共41兲 starting, e.g., withn₀=J

冑

^m/共2␮兲. This procedure guarantees convergence toward the supersonic solution of Eq.共3兲.

The natural generalization of Eq. 共41兲 to the case of a small but nonvanishing potentialV共x兲 is obtained by using Eq.共30兲instead of Eq.共3兲. This yields

n=n₀

冉

⁻␮^{V共x兲−gn}0

+␮^−gn+共ប²/2m

冑

n兲共⳵^2/⳵^x2兲

冑

n

␮^−gn0

冊

^−1/2,

共42兲 where the currentJwas substituted by means of the disper- sion relation共3兲. We shall now find approximate solutions of this self-consistent equation in the case of weak disorder, i.e., 兩v共x兲兩Ⰶ1, where the typical value ofVis much smaller than

␮−gn₀, which is the kinetic energy per particle. We empha- size that this does not imply that the nonlinear term gn₀ should be small.

The zeroth-order solution of Eq.共42兲 is simply the con- stant equilibrium densityn₀. Resubstituting this constant so- lution into the recursive equations yields the first-order solu- tion for the condensate density

n^共1兲共x兲= n₀

冑

1 −v共x兲. 共43兲 Corrections to this first-order expression particularly arise from the quantum pressure term 共ប²/ 2m

冑

n兲共⳵^2/⳵^x2兲

冑

n. It can be shown, however, that the latter is suppressed by a factor ⬃1 /共kl_c兲² as compared to the kinetic energy ប²k²/共2m兲 when n^共1兲共x兲 is resubstituted in Eq. 共42兲. In the semiclassical regimeklcⰇ1, the quantum pressure term be- comes negligible, and the expression共43兲represents a very good approximation to the actual density of the condensate in the scattering region. We show in Appendix B that the result共43兲can be derived in a way that is directly analogous to the semiclassical WKB approach.

This result is illustrated in Fig. 6, whose lower panel shows a random potential generated with the method pre- sented in the previous section. In the upper panel we com-

pare the result of Eq. 共43兲 with an exact, i.e., numerically computed solution of the Gross-Pitaevskii equation. Excel- lent agreement between the first-order solution and the exact solution is found. We note here that it is quite natural to find that the density profile mirrors the potential because we are dealing with current-carrying states: the relation共30兲 共in the absence of quantum correction兲 ␮^{=mu2/ 2 +}V共x兲+gn pre- dicts that the condensate velocity becomes minimal close to the maxima of the disorder potential. It follows then from the continuity equationJ=nu= const that the densityn assumes its maxima when the velocity becomes minimal.

It is instructive to realize that classically forbidden back reflections can be taken into account by inserting the ansatz n共x兲=n^共1兲共x兲+␦ⁿ共x兲 into Eq.共30兲and linearizing the result- ing equation for small␦n共x兲/n₀. To the lowest nonvanishing order in v, we again obtain the result 共39兲 for the mean transmission.

Finally we consider experimental realizations of waveguides on atom chips. Typical distancesh between the chip surface and the guide are in the range 20– 100␮m.

Typical disorder correlation lengths are of the same order as h. In recent transport experiments关43兴the velocity of propa- gating⁸⁷Rb condensates is of the order of a few millimeters per second, resulting in a mean wavelength of a few mi- crometers. This corresponds to the regime ␬lcⰇ1 with—

from Eq.共40兲—a very large value ofLd. Hence the regime of weak disorder is presently the most relevant one; the kinetic energy is much larger than the typical intensity of the disor- dered potential andLdis large compared to the typical length of the disordered region; one thus expects almost perfect transmission.

V. MODERATE AND STRONG DISORDER

In Sec. IV we focused on weak-disorder potentials, in the limit of small reflection. The analysis was done in the regime

␮Ⰷ具V²共x兲典^1/2andLⰆLd. In the present section we still par- FIG. 6.共Color online兲The upper panel displays a comparison of the first-order solution关WKB, Eq. 共43兲兴with a numerically com- puted solution共QM兲of the Gross-Pitaevskii equation for a weak- disorder potentialV共x兲 共shown in the lower panel兲. The correlation length isl_c= 30␮m; the wavelength is␭= 3␮m. The ratio between interaction and kinetic energy in the incident beam is E_int/E_kin

= 1 / 10.

tially satisfy the first of these inequalities, but drop the sec- ond one. We will see that the behavior of the system is quite different, ranging from a regime of localization共in the limit of weak interaction兲to a time-dependent behavior for larger interaction, with a power-law decay of the time-averaged transmission.

First, we shall discuss some elementary differences be- tween the scattering problem in linear quantum mechanics and the nonlinear Gross-Pitaevskii equation. In linear quan- tum mechanics共g= 0兲one finds for any scattering potential a unique stationary scattering state that is dynamically stable, and the associated transmission coefficientTrelates the con- stant incident currentJione to one with the transmitted cur- rentJt. For the nonlinear Gross-Pitaevskii equation the trans- mission T depends on the density of the propagated condensate and thereby on the current. Additionally, the phe- nomenon of multistability may arise. This means that for a given incident currentJi two or more scattering states with different transmissions can coexist.

In principle all stationary scattering states that are associ- ated with a given incident currentJ_i can be found by inte- grating the time-independent Gross-Pitaevskii equation 共8兲 from the downstream to the upstream region. A systematic variation of the downstream currentJt allows one to select the desired states. This procedure, however, does not reveal any information about their dynamical stability properties, which are crucial for answering the question whether an in- cident condensate beam populates a stationary scattering state or not. For instance, in the case of coherent condensate transport through a double-barrier potential, three possible scattering states are expected close to the resonances, but only one of them is dynamically stable关15兴. Here the advan- tage of integrating the time-dependent Gross-Pitaevskii equation becomes apparent: If this integration converges to a stationary scattering state we know automatically that this state is dynamically stable共otherwise small numerical devia- tions would exponentially increase with propagation time兲.

We consider an ensemble ofN disorder realizations with randomly varying sample lengthsL that are uniformly dis- tributed between 0 and a maximal sample length. For each realization 共labeled with index ␣兲 we numerically compute the time evolution of the wave function and extract either the time-independent transmission T_␣ 关if ␺共x,t兲 converges to a stationary state兴 or the time-averaged transmission ¯T_␣ 关if

␺共x,t兲remains time dependent兴. For the sake of definiteness and due to its experimental relevance we consider the propa- gation of condensed⁸⁷Rb atoms 共whose scattering length is asc= 5.77 nm兲. Our numerical computations were performed for a guide with radial trapping frequency ␻_⬜^{= 2}␲

⫻100 s⁻¹共oscillator lengtha_⬜= 1␮m兲. The disorder is gen- erated as in the previous section. The regime of strong dis- order is reached by choosing a rather short distance h

= 5␮m between the center of the guide and the chip surface, which corresponds to a correlation lengthl_c= 6␮m. In order to avoid excitations of the condensate into higher transversal modes we adjust the standard deviation of the potential 共which is a measure of the mean potential height兲 to 具V²共x兲典^1/2⯝0.12ប␻_⬜^{. Inall}the following numerical calcula- tions we consider an incident monochromatic beam with cur-

rent Ji= 10³ atoms per second and wavelength ␭= 10␮^m.

Then the chemical potential is ␮^{= 0.25}ប␻_⬜ 共in the linear case the chemical potential takes the slightly different value

␮^{= 0.23}ប␻_⬜兲.

It is instructive to focus first on the linear case 共g= 0兲 which has already been extensively investigated in the con- text of localization theory 关22,41兴. In the localized regime the transmission decays exponentially with increasing sys- tem length L, i.e., 具T典= exp共−L/Lloc兲 where Lloc is the so- called localization length. The points in the upper panel of Fig. 7 mark for each disorder realization the associated trans- missionT_␣共L兲 as a function of the sample lengthL. To ex- tract from these data a characteristic scaling law for the L dependence of the transmission we divideLinto equal inter- vals of length⌬LⰆL. We then compute the mean transmis- sion at sample length L by summing up all the values T_␣ corresponding to a sample length lying in the interval of width⌬L centered atL:

具T典a共L兲= 1 NL

兺

␣

T_␣共L

⬘

^{兲, L−⌬L}

2 ⬍L

⬘

^⬍L+⌬L 2 .

共44兲 NLis the number of samples in the interval under consider- ation.

FIG. 7. 共Color online兲 Transmission through a disordered sample as a function of sample lengthLfor the noninteracting case.

Each point corresponds to a different realization of the disordered potential. Upper panel: arithmetically averaged transmission共blue staircase function兲. Lower panel: The geometric averaged transmis- sion共blue staircase function兲decreases exponentially with increas- ingL, as revealed by the fit withL_loc= 586␮m共straight black line兲. The arrows mark the logarithmic standard deviation.

The step function in Fig. 7 shows the decrease of具T典afor 30 000 disorder realizations and⌬L= 50共for the sake of clar- ity we show only 2000 points in the plot兲. In the context of localization theory it is convenient to investigate scaling laws by means of the geometrically averaged transmission

具T典g=e^{具ln共T兲典}, 具ln共T兲典= 1

NL

兺

_␣ ^{ln关T␣共L}

^⬘

^{兲兴, 共45兲}

because, contrary to 具T典a, the average 具ln共T兲典 is a self- averaging quantity of the system关41,42兴. The lower panel of Fig. 7 shows具T典g, which follows clearly an exponential law.

This is clear evidence for the appearance of localization. We can extract the localization length, which here is L_loc

= 586␮m. We note the wide spread of the data points around their average. This spread is quantified by the logarithmic standard deviation

⌬ln共T兲=

冉 ^冑

^N1L

^兺

␣ 关ln共T_␣兲−具ln共T兲典兴²

冊

^{, 共46兲}

which is shown as arrows in the lower panel of in Fig. 7. We find an almost linear increase of ⌬ln共T兲 with the sample length.

Is the conventional localization scenario, with the charac- teristic exponential decrease of the transmission关16,17兴, still valid in the case of interacting atoms? To address this ques- tion we now calculate the transport in presence of a moderate nonlinearity where the ratio of interaction and kinetic energy in the incident beam is E_int/E_kin⯝1 / 10. Contrarily to the linear case, time-dependent behavior becomes now a domi- nant feature as shown in Fig. 8. We find that dynamical stable scattering states 共black dots in Fig. 8兲 are populated only for sample lengths that are smaller than a critical length L^*, which is here of order of 125␮m. For samples with

length LⲏL^* we find a crossover region where time- dependent behavior sets in and convergency to a stationary state is achieved for only a certain fraction of disorder samples.␺共x,t兲remains time dependent共orange crosses兲for all samples when we reach the regime where L is notably larger than L^*. In the time-dependent case the data points display the time-averaged transmissionsT¯

␣共9兲.

In order to extract a scaling law from our data, we com- pute the ensemble-averaged transmission 关in the time- dependent casesT_␣in Eqs.共44兲–共46兲is replaced byT¯

␣兴. We find that the geometrically averaged transmission具T典g 共step function in Fig. 8兲 decreases inversely with the sample lengthL and is well approximated by the algebraic function 共smooth line in Fig. 8兲

具T典g= L₀

L+L₀ 共47兲

with the decay lengthL₀. Such a scaling law is characteristic for transport in systems with loss of phase coherence be- tween the single-scattering events. Indeed, if one considers a series of successive scatterers and calculates the transmission by neglecting all interference effects, one derives exactly the scaling law of Eq.共47兲 关44,45兴. Such an Ohmic behavior is observed for electron transport through mesoscopic metal structures in the limit of small dephasing lengths关44,46兴.

Another striking feature is the distribution of the data points in Fig. 8. Contrarily to the linear case, this distribution is now clearly restricted to the neighborhood of the average transmission, and the standard deviation⌬T共L兲decreases for long sample lengthsL. Hence, in the regime of large lengths one expects to find the ¯T

␣’s in a narrow interval centered around the averaged transmission. Loosely speaking,¯T

␣ be- comes more or less sample independent. For the sake of completeness we mention that ideally the time averages¯T

␣

should be computed for an infinitely long period. Of course this cannot be done numerically, but we verified that the averaged transmission and the standard deviation do not change if we increase in Eq.共9兲the averaging time window from␶^{to 2}␶^{and 3}␶^.

The above presented computations demonstrate that even a moderate nonlinearity leads to a dramatic change of the transmission properties. In particular, the usual interpretation of the transmission behavior in terms of localization is put in question in the case of interacting particles. In order to obtain deeper insight into that matter, we redo the above computa- tion with a very weak nonlinearity, such that Eint/Ekin

= 1 / 100. Figure 9 shows that for this case the crossover from time-independent to time-dependent behavior is shifted to larger sample lengths共Lⲏ600␮^m兲. This indicates the exis- tence of a critical nonlinearity beyond which the system ex- hibits time dependence. Indeed, preliminary studies show that for each disorder sample lengthLthere is a critical value g^* above which no stationary scattering state can be popu- lated, or, equivalently, for each strength of interaction g, there is a critical disorder lengthL^* above which the flow is time dependent. We find that L^* decreases with increasing nonlinearity. This is the reason why stationary states can be FIG. 8. 共Color online兲 Transmissions through an ensemble of

disorder realizations for a moderate nonlinearity E_int/E_kin= 1 / 10;

the characteristics of the incident beam are given in the main text. A transition from a time-independent to a time-dependent regime is observed at critical lengthL^*⯝125␮m. The black dots represent the transmissions in the time-independent regime and the共orange online兲 crosses are the time-averaged transmissions in the time- dependent regime. The staircase function is the geometrically aver- aged transmission. It is well approximated by the algebraic scaling lawL₀/共L+L₀兲 共smooth solid line兲withL₀= 287␮m.