Microscopic scattering theory for interacting bosons in weak random potentials
MMicroscopic scattering theory for interacting bosonsin weak random potentials
Tobias Geiger, Andreas Buchleitner, and Thomas Wellens
Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg, D-79104 Freiburg,GermanyE-mail:
Abstract.
We develop a diagrammatic scattering theory for interacting bosons ina three-dimensional, weakly disordered potential. Based on a microscopic N -bodyscattering theory, we identify the relevant diagrams including elastic and inelasticcollision processes that are sufficient to describe diffusive quantum transport. Bytaking advantage of the statistical properties of the weak disorder potential, wedemonstrate how the N -body dynamics can be reduced to a nonlinear integral equationof Boltzmann type for the single-particle diffusive flux. Our theory reduces to theGross-Pitaevskii mean field description in the limit where only elastic collisions aretaken into account. However, even at weak interaction strength, inelastic collisionslead to energy redistribution between the bosons – initially prepared all at the samesingle-particle energy – and thereby induce thermalization of the single-particle current.In addition, we include also weak localization effects and determine the coherentcorrections to the incoherent transport in terms of the coherent backscattering signal.We find that inelastic collisions lead to an enhancement of the backscattered cone ina narrow spectral window for increasing interaction strength. a r X i v : . [ qu a n t - ph ] J u l icroscopic scattering theory for interacting bosons in weak random potentials
1. Introduction
In recent years, increasing interest has been devoted to the behaviour of ultracoldatoms in disordered potentials. Whereas the first experiments [1, 2, 3] concentratedon the realization of Anderson localization [4] in one dimension, this intriguing disordereffect – which leads to complete suppression of diffusive transport due to destructiveinterference – has now also been observed in three dimensions [5, 6]. The 3D case isespecially interesting since it exhibits a transition from extended to localized single-particle eigenstates: In the absence of interactions, particles with low energy arelocalized, whereas those with higher energy (in comparison with the strength of thedisorder potential) propagate diffusively in the random potential. Also in the lattercase – on which we concentrate in the present paper – wave interference effects arerelevant, though less pronounced: They lead to weak localization [7] (i.e. reduction ofthe diffusion constant instead of complete suppression of diffusion) and, associated withthat, coherent backscattering [8, 9, 10] (i.e. enhancement of backscattering), which hasrecently been observed also with atomic matter waves [11, 12, 13]. Beyond the scope of[11, 12] lies the investigation of the interplay between disorder and interactions, whereit is not well understood, especially in the higher-dimensional case, to what extentinteraction leads to a loss of coherence, i.e. to a breakdown of localization effects [14, 15].Most theoretical works, e.g. [16, 17, 18, 19, 20], focus on the regime of – or close to –thermal equilibrium and examine, e.g., the effect of disorder on the condensate fraction,superfluid fraction or the sound velocity [17, 18, 20]. In this case, weak interactions canusually be treated perturbatively, e.g., by introducing Bogoliubov quasiparticles [21].In contrast, the present paper investigates a stationary scattering setup far fromthermal equilibrium. Here, bosonic atoms are continuously emitted from a coherentsource (‘atom laser’ [22, 23]) and guided into the random potential until a stationaryscattering state is reached. Theoretical studies of this scattering scenario so fareither neglect the interparticle interaction [24, 25], treat it on the mean-field level[26, 27, 28, 29], or apply a Hartree-Fock-Bogoliubov approach [30], which is appropriatein the case of a large condensate fraction. If all atoms enter the scattering regionat fixed initial energy, the Gross-Pitaevski equation obtained within the mean-fieldapproach predicts either a stationary regime with the same final energy for all scatteredatoms, or a non-stationary, time-dependent behavior [26, 30]. In contrast, accordingto the microscopic scattering theory developed in the present paper, atoms exchangeenergy with each other due to mutual collision events, leading to strong depletion ofthe condensate already for small interactions. As shown in [31], this finally leads to astationary state with thermal Maxwell-Boltzmann distribution for those atoms whichpropagate deeply into the scattering region.The present paper is devoted to a detailed presentation of the underlying bosonicmany-particle scattering theory. Starting from the N -particle Hamiltonian, we derivea nonlinear transport equation for the average particle density. Since this transportequation amounts to a stationary version of the Boltzmann equation [32], our approach icroscopic scattering theory for interacting bosons in weak random potentials k(cid:96) dis (cid:29) k and disorder scattering mean free path (cid:96) dis ) is crucial,since it allows to reduce the – in principle infinitely complicated [42] – hierarchy ofmany-particle diagrams to a tractable subclass of diagrams, i.e. ladder and crosseddiagrams [43], which are composed out of a small number of building blocks. As shownin Sec. 4, the sum of all ladder diagrams amounts to a Boltzmann-like equation fordiffusive transport eventually leading to complete thermalization due to inelastic atom-atom collisions in case of an infinitely large scattering region. Sec. 5 is devoted to thederivation of transport equations describing coherent backscattering based on crosseddiagrams. Finally, in Sec. 6 we present the results of numerical solutions of the ladderand crossed transport equations exemplifying the behaviour of diffusive transport for afinitely large scattering region, and the effect of elastic and inelastic atom-atom collisionson coherent backscattering, respectively. Sec. 7 concludes the paper. Several technicalaspects are relegated to Appendices A-E.
2. Scattering theory for a single particle
We write the Hamiltonian for a single particle in the following form:ˆ H = ˆ H + ˆ V , (1) icroscopic scattering theory for interacting bosons in weak random potentials H denotes free propagation and ˆ V the disorder potential. The eigenstates | k (cid:105) ofˆ H are plane waves with wave vector k :ˆ H = (cid:90) d k (2 π ) E k | k (cid:105)(cid:104) k | , (2)and energy E k = k , (3)where we set ¯ h / (2 m ) ≡
1. The matrix elements of ˆ V are given by the Fourier transformof the disorder potential V ( r ): (cid:104) k | ˆ V | k (cid:105) = (cid:90) d r V ( r ) e i ( k − k ) r . (4)In order to obtain a properly defined scattering scenario, we assume that V ( r ) is non-zero only inside a finite scattering region V . This allows us to define an asymptoticallyfree initial state: | i (cid:105) = (cid:90) d k (2 π ) w ( k ) | k (cid:105) , (5)with normalized wavepacket w ( k ), i.e. (cid:82) d k | w ( k ) | = (2 π ) , which we assume to bea quasi-monochromatic wavepacket, i.e., sharply peaked around the initial wavevector k i with energy E i = k i , see Eq. (3). Therefore, the spatial density resulting from theFourier transform of w ( k ): | (cid:101) w ( r ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d k (2 π ) e i k · r w ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d k (2 π ) w ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6)is approximately constant inside the scattering region, i.e. for r ∈ V . If the stateexp( − i ˆ H T ) | i (cid:105) is prepared at time T → −∞ , the wavepacket arrives at the scatteringregion at time t = 0, and a quasi-stationary scattering state | f + , (cid:105) = ˆΩ ( V )+ ( E i ) | i (cid:105) (7)is reached at that time. Here, the operator ˆΩ ( V )+ ( E ) is defined byˆΩ ( V )+ ( E ) = + ˆ G V ( E ) ˆ V , (8)where ˆ G V ( E ) = 1 E − ˆ H − ˆ V + i(cid:15) , (9)with infinitesimally small (cid:15) >
0, denotes the (retarded) Green’s operator associated tothe Hamiltonian ˆ H = ˆ H + ˆ V . The operators ˆΩ ( V )+ ( E ) and ˆ G V ( E ) fulfill the followingversions of the Lippmann-Schwinger equation:ˆΩ ( V )+ ( E ) = + ˆ G ( E ) ˆ V ˆΩ ( V )+ ( E ) , (10)ˆ G V ( E ) = ˆ G ( E ) + ˆ G ( E ) ˆ V ˆ G V ( E ) , (11)where ˆ G ( E ) denotes the vacuum Green’s operator:ˆ G ( E ) = 1 E − ˆ H + i(cid:15) . (12) icroscopic scattering theory for interacting bosons in weak random potentials ( V )+ ( E ) is closely related to the Møller operator ˆΩ ( V )+ =lim T →−∞ exp[ i ( ˆ H + ˆ V ) T ) exp( − i ˆ H T ), as their action on an eigenstate | ψ (cid:105) of ˆ H withenergy E is identical, i.e. ˆΩ ( V )+ | ψ (cid:105) = ˆΩ ( V )+ ( E ) | ψ (cid:105) if ˆ H | ψ (cid:105) = E | ψ (cid:105) . Since, in the fol-lowing, we will apply ˆΩ ( V )+ ( E ) only to such eigenstates – or quasi-eigenstates, as | i (cid:105) in Eq. (7) – we will henceforth refer also to ˆΩ ( V )+ ( E ) as ‘Møller operator’. Finally, theexpectation value of an arbitrary observable ˆ A in the (quasi-)stationary scattering stateresults as (cid:104) ˆ A (cid:105) = (cid:104) f + , | ˆ A | f + , (cid:105) .Let us note that, instead of using the Møller operator, a scattering process can alsobe characterized by the S -matrix, ˆ S = (cid:16) ˆΩ ( V ) − (cid:17) † ˆΩ ( V )+ (where ˆΩ ( V ) − is defined in the sameway as ˆΩ ( V )+ , but with T → + ∞ instead of −∞ ). We could formulate the following N -particle scattering theory equally well in terms of the S -matrix. However, since the S -matrix maps incoming onto outgoing asymptotically free states, it does not allow –in contrast to the Møller operator – to evaluate what is happening inside the scatteringregion, e.g. to calculate the (quasi-)stationary density or flux of particles inside V . Forthis reason, we prefer using the (quasi-)stationary scattering state | f + , (cid:105) , see Eq. (7) (orits N -particle counterpart | f + (cid:105) , see Eq. (22) below) in the following.
3. Scattering theory for many bosonic particles
We add a term ˆ U to the Hamiltonian, Eq. (1), denoting the interaction between particles:ˆ H = ˆ H + ˆ V + ˆ U . (13)As compared to Eqs. (2,4), the operators ˆ H and ˆ V are generalized as follows to themany-particle Hilbert space:ˆ H = (cid:90) d k (2 π ) E k ˆ a † k ˆ a k , (14)ˆ V = (cid:90) d r V ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) , (15)with creation and annihilation operators ˆ a † k and ˆ a k for particles with wave vector k ,whereas the operators ˆ ψ ( r ) = (cid:82) d k exp( i k · r )ˆ a k / (2 π ) and ˆ ψ † ( r ) = (cid:82) d k exp( − i k · r )ˆ a † k / (2 π ) annihilate and create, respectively, a particle at position r .In contrast to ˆ H and ˆ V , the interaction ˆ U acts on two particles:ˆ U = 12 (cid:90) d r d r U ( r − r ) ˆ ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) ˆ ψ ( r ) , (16)with atom-atom interaction potential U ( r ). In the following, a collision event betweentwo particles will be described by the T -matrix [44]:ˆ T U ( E ) = ˆ U + ˆ U ˆ G ( E ) ˆ U + ˆ U ˆ G ( E ) ˆ U ˆ G ( E ) ˆ U + . . . . (17)According to Eq. (17), the matrix elements of ˆ T U ( E ) with respect to two-particlestates describe repeated application of the interaction ˆ U on the same pair of particles, icroscopic scattering theory for interacting bosons in weak random potentials G ( E ). Separating the center-of-mass from the relativecoordinates, the two-body T matrix fulfills momentum conservation: (cid:104) k , k | ˆ T U ( E ) | k , k (cid:105) = (2 π ) δ ( k + k − k − k ) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) , (18)where ˆ T (1) U ( E ) is the T -matrix for a single particle (with reduced mass m/
2) scatteredby the potential U ( r ) at energy E = E − ( k + k ) / | k (cid:105) = (cid:16) | ( k − k ) / (cid:105) + | ( k − k ) / (cid:105) (cid:17) / √
2, and | k (cid:105) = (cid:16) | ( k − k ) / (cid:105) + | ( k − k ) / (cid:105) (cid:17) / √
2. The single-particle T -matrix, in turn, fulfills the optical theorem [44]: (cid:16) ˆ T (1) U ( E ) (cid:17) † (cid:16) ˆ G † ,m/ ( E ) − ˆ G ,m/ ( E ) (cid:17) ˆ T (1) U ( E ) = (cid:16) ˆ T (1) U ( E ) (cid:17) † − ˆ T (1) U ( E ) , (19)expressing conservation of the particle and the energy flux (where ˆ G ,m/ denotes thevacuum Green’s operator for a particle with mass m/ E = 2 k ).Our many-particle scattering theory presented below, and in particular thetransport equations in Secs. 4 and 5, are valid for an arbitrary interaction potential U ( r )– as long as it is sufficiently weak in the sense specified below (mean distance betweencollision events larger than between disorder scattering events). Only for the numericalresults presented in Sec. 6, we will assume a short-range potential with corresponding s -wave scattering approximation, see Eq. (E.1).Finally, we note that, in principle, the vacuum T -matrix as defined in Eq. (17)is modified by the presence of the disorder potential. To take this into account, thevacuum Green’s operator ˆ G ( E ) must be replaced by the disorder Green’s operatorˆ G V ( E ), see Eq. (9), in Eq. (17). However, since the present paper assumes the case ofa very weak disorder potential, we will neglect the disorder during each collision eventin the following, and therefore use the vacuum T -matrix as introduced above. Thisapproximation is valid if the range of the interaction potential U ( r ) is much smallerthan the disorder mean free path (cid:96) dis introduced in Sec. 4. We now generalize the scattering scenario outlined in Sec. 2 to the case of many particles.For this purpose, we assume that, both, the disorder and the particle-particle interactionare non-zero only inside a finite region V (which, for simplicity, we assume to be thesame for ˆ V and ˆ U ). Note that the introduction of a finite interaction region in principlebreaks translational invariance, and therefore the δ -function expressing momentumconservation in Eq. (18) turns into an approximate δ -function. Since, however, weassume the size L of the scattering region V to be much larger than the disorder meanfree path, i.e. L (cid:29) (cid:96) dis (cid:29) k − (see below), we can safely neglect the associated smallwidth ( ∝ /L ) of this δ -function, and still work with the T -matrix as given by Eq. (18).Our initial state for N particles reads: | i (cid:105) = 1 √ N ! (cid:90) d k . . . d k N (2 π ) N w ( k ) . . . w ( k N ) | k , . . . , k N (cid:105) , (20) icroscopic scattering theory for interacting bosons in weak random potentials N atoms are described by the same quasi-monochromatic single-atomwavepacket w ( k ) as given in Eq. (5). The factor 1 / √ N ! arises from theindistinguishability of bosonic particles. The corresponding density of particles reads: ρ = (cid:104) i | ˆ ψ † ( r ) ˆ ψ ( r ) | i (cid:105) (cid:39) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d k (2 π ) w ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (21)As mentioned above, this density is approximately uniform within the whole scatteringregion V for a wavepacket sharply peaked around the initial wavevector k i . Since, inthis quasi-monochromatic limit, the density, Eq. (6), for N = 1 approaches zero (sincethe wave packet is spread over an increasingly large region of space), the number N ofparticles correspondingly must tend to infinity in order to obtain a finite density ρ .The Møller operator, which yields the quasi-stationary N -particle scattering state | f + (cid:105) = ˆΩ + ( N E i ) | i (cid:105) , (22)is defined in the same way as above, see Eqs. (8,9) but with ˆ V + ˆ U instead of ˆ V . Ittherefore fulfills the Lippmann-Schwinger equation:ˆΩ + ( E ) = + ˆ G ( E ) (cid:16) ˆ V + ˆ U (cid:17) ˆΩ + ( E ) , (23)which, using Eqs. (8,11), can be rewritten as:ˆΩ + ( E ) = ˆΩ ( V )+ ( E ) + ˆ G V ( E ) ˆ U ˆΩ + ( E ) . (24)Iteration of Eq. (24) yields an expansion in powers of ˆ U :ˆΩ + ( E ) = ˆΩ ( V )+ ( E )+ ˆ G V ( E ) ˆ U ˆΩ ( V )+ ( E )+ ˆ G V ( E ) ˆ U ˆ G V ( E ) ˆ U ˆΩ ( V )+ ( E )+ . . . . (25)Remember that, according to Eq. (16), each operator ˆ U annihilates and creates twoparticles. In contrast, the Green’s operator ˆ G V and the Møller operator ˆΩ ( V )+ act on all N particles. However, since these operators describe non-interacting particles, they canbe factorized into single-particle operators. As an example, we give here the factorizationformulas for the case N = 2: (cid:104) k , k | ˆΩ ( V )+ ( E k + E k ) | k , k (cid:105) = (cid:104) k | ˆΩ ( V )+ ( E k ) | k (cid:105)(cid:104) k | ˆΩ ( V )+ ( E k ) | k (cid:105) + (cid:104) k | ˆΩ ( V )+ ( E k ) | k (cid:105)(cid:104) k | ˆΩ ( V )+ ( E k ) | k (cid:105) , (26)and (cid:104) k , k | ˆ G V ( E ) | k , k (cid:105) = 1( − πi ) (cid:90) ∞−∞ d E (cid:48) (cid:104) (cid:104) k | ˆ G V ( E (cid:48) ) | k (cid:105)×(cid:104) k | ˆ G V ( E − E (cid:48) ) | k (cid:105) + (cid:104) k | ˆ G V ( E (cid:48) ) | k (cid:105)(cid:104) k | ˆ G V ( E − E (cid:48) ) | k (cid:105) (cid:105) . (27)As mentioned above, the energy argument of our Møller operator, Eq. (8), is alwaysfixed to the energy of the state it acts on. In contrast, Green’s operators also act onstates with different energies. Hence, the energy E of a two-particle Green’s operatorhas to be distributed among two one-particle Green’s operators according to Eq. (27).Using the above factorization formulas – and analogous ones for N > U . Repeated interaction between the same pair of particles icroscopic scattering theory for interacting bosons in weak random potentials k k k p p p p k k k p p p p Figure 1.
Example of a three-particle scattering process with initial state | k , k , k (cid:105) and final state | k , k , k (cid:105) . The three arrows associated with the initial state representthe Møller operator ˆΩ ( V )+ ( E i ) of the disorder potential, see Eq. (8), whereas theremaining arrows refer to the disorder Green’s operator ˆ G V , Eq. (9). Squarescorrespond to the two-body T -matrix of the particle-particle interaction, Eq. (18).The transition amplitude corresponding to this scattering process is given in Eq. (28). is included in the T -matrix, see Eq. (17) (and the discussion at the end of Sec. 3.1).We hence replace two-particle matrix elements of ˆ U by matrix elements of ˆ T U ( E ) (withappropriately defined two-particle energy E , see below) in Eq. (25), and thereby obtaina sequence of collision events between different pairs of particles. An example of athree-particle scattering process is demonstrated in Fig. 1. As shown in Appendix A,this diagram gives rise to the following contribution to the transition amplitude: (cid:104) k , k , k | ˆΩ (fig . (3 E i ) | k , k , k (cid:105) = (cid:90) ∞−∞ d E d E ( − πi ) (cid:90) d p . . . d p (2 π ) × (cid:104) k | ˆ G V (3 E i − E − E ) | p (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105)× (cid:104) p , p | ˆ T U (3 E i − E ) | p , p (cid:105)(cid:104) p | ˆ G V (2 E i − E ) | p (cid:105)(cid:104) p , p | ˆ T U (2 E i ) | p , p (cid:105)× (cid:104) p | ˆΩ ( V )+ ( E i ) | k (cid:105)(cid:104) p | ˆΩ ( V )+ ( E i ) | k (cid:105)(cid:104) p | ˆΩ ( V )+ ( E i ) | k (cid:105) , (28)with E k (cid:39) E k (cid:39) E k (cid:39) E i , according to our above assumption of a quasi-monochromatic wavepacket.In general, the rules for constructing an arbitrary N -particle scattering amplitudefor a given diagram are as follows: (i) Apply the disorder Møller operator ˆΩ ( V )+ ( E i ),see Eq. (8), to each initial single-particle state | k (cid:105) , . . . , | k N (cid:105) . The energy associatedto each initial particle is given by E i . (ii) Integrate over all intermediate particles( p , . . . , p in Fig. 1). (iii) Write down the corresponding two-body T -matrix element,see Eq. (18), for any collision between two particles. The energy argument of ˆ T U isgiven by the sum of the two incoming single-particle energies. (iv) For each ˆ T U ( E ),write down an integral (cid:82) ∞−∞ d E (cid:48) / ( − πi ) which determines the energy arguments of theGreen’s operators ˆ G V ( E (cid:48) ) and ˆ G V ( E − E (cid:48) ), see Eq. (27), for the two particles after thecollision. (v) These two particles may then collide with other particles, and so on ... .The total transition amplitude defining the stationary scattering state | f + (cid:105) , seeEq. (22), is then obtained by summing the contributions from all possible differentdiagrams. For example, in addition to the diagram shown in Fig. 1, eight more diagrams icroscopic scattering theory for interacting bosons in weak random potentials k , k , k ) and ( k , k , k )also contribute to | f + (cid:105) . As the finally measured quantity, we determine the expectation value of the flux densityoperator ˆ J ( r ) = 2Im (cid:16) ˆ ψ † ( r ) ∇ ˆ ψ ( r ) (cid:17) = (cid:90) d k d k (cid:48) (2 π ) (cid:32) k + k (cid:48) (cid:33) e − i ( k − k (cid:48) ) · r ˆ a † k ˆ a k (cid:48) , (29)with respect to the stationary scattering state | f + (cid:105) . Since ˆ J ( r ) is a one-particle operator,this implies a partial trace of the density matrix | f + (cid:105)(cid:104) f + | over N − J ( r ) = (cid:104) f + | ˆ J ( r ) | f + (cid:105) = NN ! (cid:90) d k d k (cid:48) (2 π ) (cid:32) k + k (cid:48) (cid:33) e − i ( k − k (cid:48) ) · r × (cid:90) d k . . . d k N − (2 π ) N − (cid:104) k , . . . , k N − , k (cid:48) | f + (cid:105)(cid:104) f + | k , . . . , k N − , k (cid:105) . (30)Placing the detector at position R in the far field of the scattering region (i.e. | R | (cid:29) | r | for r ∈ V ), the scattered flux is finally expressed as a dimensionless quantity (the so-called ‘bistatic coefficient’ [45]): γ (ˆ k d ) = lim R →∞ (cid:32) R · J ( R ) 4 πR A ρ √ E i (cid:33) , (31)normalized with respect to the incident flux A ρ √ E i , where A denotes the transversearea (with respect to the incident wave) of the scattering volume V , and ˆ k d = R / | R | is the direction of the detected particle’s wavevector. The limit R → ∞ is to be taken after the quasi-stationary limit N → ∞ , see the discussion after Eq. (21). Apart fromthe total flux density γ (ˆ k d ), we will also be interested in the spectral density γ E (ˆ k d ), i.e.the flux of particles scattered into direction ˆ k d with energy E , which is given by: γ E (ˆ k d ) = lim R →∞ (cid:90) d k d k (cid:48) π ( R · K ) e − i ( k − k (cid:48) ) · R (cid:104) f + | ˆ a † k ˆ a k (cid:48) | f + (cid:105)A ρ √ E i /R δ ( E − K ) , (32)where K = ( k + k (cid:48) ) /
2, such that (cid:82) ∞ d E γ E (ˆ k d ) = γ (ˆ k d ).The factor 1 /N ! in Eq. (30) arises from the indistinguishability of the bosonicparticles. It turns out, however, that this factor – together with the factors 1 / √ N ! inEq. (20) – is exactly counterbalanced once we sum the amplitudes of all processes wherethe initial and/or final particles are exchanged. In total, we get the same result as ifthe particles were distinguishable. This equivalence is generally valid if all particles areprepared in the same initial state, and if the Hamiltonian is symmetric under exchangeof particles [46].Remember that the number N of particles tends to infinity in the quasi-stationarylimit, whereas, in case of a finite scattering region, only a finite number of particles willeventually interact with the finally detected particle. The evolution of the remaining icroscopic scattering theory for interacting bosons in weak random potentials E E E i − E E i − E E E i − E E i − E = a)b) = E i E E i − E E i E i Figure 2.
Graphical equations exemplifying the trace over the undetected particles.Arrows and squares refer to single-particle propagators and two-body T -matrices, asdefined in Fig. 1. Dashed arrows correspond to adjoint propagators ˆ G † V and (cid:16) ˆΩ ( V )+ (cid:17) † .The half circle symbol denotes the detector, whereas the dots on the left-hand sideof both equations represent the trace over the undetected particle. a) Inelasticscattering of two particles. On the right-hand side, the trace has been performedusing Eq. (33). This results in the dashed-solid double arrow representing the spectralfunction (cid:104) ˆ G † V ( E ) − ˆ G V ( E ) (cid:105) / (2 πi ). The energy of the detected particle is 2 E i − E . b) Elastic scattering of two particles. The trace is performed using Eq. (34). Theresulting diagram on the right-hand side is equivalent to a diagram obtained from theGross-Pitaevski equation. Since, in contrast to a), the conjugate particles (dashedlines) do not undergo a collision, the energy of the detected particle is unchanged ( E i ). particles (which do not interact with the detected particle) does not influence the resultof the partial trace, Eq. (30). This follows from the factorization property, Eq. (26),and the left-unitarity, (cid:16) ˆΩ + (cid:17) † ˆΩ + = of the Møller operator. Consequently, in orderto calculate the detection signal, we may disregard all scattering processes concerningthose particles which do not interact (neither in | f + (cid:105) nor in (cid:104) f + | ) with the detectedparticle. (The presence of these particles only leads to a prefactor giving rise to thecorrect dependence of a given scattering diagram on the density ρ , see the discussionat the end of Appendix B.) According to the recipe given above, the flux density for an arbitrary N -particlescattering process is obtained as follows: take a diagram contributing to | f + (cid:105) , a conjugatediagram contributing to (cid:104) f + | , apply the observable ˆ J ( r ) to one of the final particles ofboth diagrams, and trace over the undetected particles. An example for two particlesis shown in Fig. 2a) (left-hand side). Since both conjugate diagrams (solid and dashedlines, respectively) exhibit a collision event, which redistributes the energy among thetwo particles according to the factorization formula, Eq. (27), the energy of the detected icroscopic scattering theory for interacting bosons in weak random potentials =+ Figure 3.
Elastic scattering diagram where the conjugate undetected amplitudeoriginates from a previous elastic scattering event. The sum of the two processesshown on the left-hand side reproduces the Gross-Pitaevskii diagram (cf. [29]) on theright-hand side. particle is different from the initial energy E i . For this reason, we call this scatteringprocess ‘inelastic’. This means that the energies of the single particles change – althoughtheir sum remains conserved. In contrast, Fig. 2b) shows an elastic scattering process.Here, the conjugate diagram (dashed lines) on the left-hand side does not exhibit acollision event. As shown below, this implies that the energies of both particles remainunchanged.We will now demonstrate how to perform the trace over the undetected particle forinelastic and elastic collisions, respectively. The result is represented on the right-handside of Fig. 2. Inelastic collisions.
The complete expression for the inelastic scattering diagram,Fig. 2a), is given in Eq. (B.1). Focusing on those terms which are relevant for thetrace over the undetected particle, this trace can be written in the following generalform: (cid:90) ∞−∞ d E d E (cid:48) | πi | (cid:90) d k (2 π ) ( . . . ) ( l )( − E (cid:48) ) ˆ G † V ( E (cid:48) ) | k (cid:105)(cid:104) k | ˆ G V ( E )( . . . ) ( r )( − E ) = (cid:90) ∞−∞ d E πi ( . . . ) ( l )( − E ) (cid:16) ˆ G † V ( E ) − ˆ G V ( E ) (cid:17) ( . . . ) ( r )( − E ) . (33)On the left-hand side of Eq. (33), k corresponds to the final state of the undetectedparticle, whereas ˆ G V ( E ) and ˆ G † V ( E (cid:48) ) refer to the (single-particle) Green’s operatorsexpressing propagation from the collision event to the final state. According to therules given in Sec. 3.2, the collision events are associated with integrals (cid:82) d E/ ( − πi )and (cid:82) d E (cid:48) / (2 πi ) which determine the energies of the undetected ( E and E (cid:48) ) and thedetected particle (2 E i − E and 2 E i − E (cid:48) ). The brackets ( . . . ) ( l )( − E (cid:48) ) and ( . . . ) ( r )( − E ) denoteall the remaining parts of the scattering diagram where the energy argument enters witha negative sign, see Eq. (B.1). Their precise form is irrelevant for Eq. (33) – except forthe fact that ( . . . ) ( l )( − E (cid:48) ) is a complex analytic function with poles only in the lower halfof the complex plane, and ( . . . ) ( r )( − E ) in the upper half. Due to the negative sign, this isin contrast to the respective contributions ˆ G † V ( E (cid:48) ) and ˆ T † U ( E (cid:48) ), as well as ˆ G V ( E ) andˆ T U ( E ), which exhibit poles only in the upper (or lower) half plane.Under these conditions – which do not only hold for the example shown in Fig. 2, icroscopic scattering theory for interacting bosons in weak random potentials E and E (cid:48) are set equal to each other, and thetwo conjugate Green’s functions ˆ G † V ( E ) and ˆ G V ( E ) are replaced by their difference (cid:104) ˆ G † V ( E ) − ˆ G V ( E ) (cid:105) / (2 πi ) (which is also known as the ‘spectral function’, since theimaginary part of the Green’s function determines the density of states [47]). Elastic collisions.
In a similar way, the trace in the elastic scattering diagram, seeEq. (B.2), is performed as follows: (cid:90) ∞−∞ d E πi (cid:90) d k (2 π ) (cid:104) k i | (cid:16) ˆΩ ( V )+ ( E i ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E )( . . . ) E = (cid:104) k i | (cid:16) ˆΩ ( V )+ ( E i ) (cid:17) † ( . . . ) E i , (34)see Appendix C. According to Eq. (34) – which is graphically depicted in Fig. 2b) –the outgoing solid arrow emitted from the two-body collision event is replaced by anincoming dashed arrow with energy E i . We note that precisely this diagram is theonly interaction contribution generated by the Gross-Pitaevskii equation [29]. Thereby,we have shown that our N -particle scattering theory reproduces the Gross-Piatevskiiequation if only elastic scattering is taken into account.In Eq. (34), the conjugate undetected particle originates directly from the initialstate (cid:104) k i | propagated in the disorder potential through the Møller operator (cid:16) ˆΩ ( V )+ ( E i ) (cid:17) † .The formula can be generalized, however, to the case where the undetected particleundergoes previous collisions with other particles before colliding with the detectedparticle. An example is depicted in Fig. 3. Also in this case, the correspondingGross-Pitaevskii diagram is reproduced (i.e. the outgoing solid arrow is replaced byan incoming dashed arrow). In a similar way, also the inelastic trace formula, Eq. (33),is valid in the case where the undetected particle undergoes further collisions with otherparticles before the trace is taken – provided that none of these other particles, in turn,collides with the detected particle which, as discussed in Sec. 4, is the case for a weakdisorder potential. This allows us to take the trace over the undetected particles directlyafter their last collision with the detected particle – without being obliged to follow theirfurther evolution before finally leaving the scattering region.
4. Incoherent transport
The N -particle scattering formalism outlined above is valid for an arbitrary potential V ( r ). Now, we consider V ( r ) as a random potential, and calculate the correspondingaverage density matrix | f + (cid:105)(cid:104) f + | . For this purpose, we assume a Gaussian white noise icroscopic scattering theory for interacting bosons in weak random potentials r r r r r r r E i E i E i E i E d E = 2 E i − E d r r Figure 4.
Example of a ladder diagram describing the propagation of three interactingparticles in a slab with a random scattering potential. Pairs of conjugate amplitudes(solid and dashed arrows, respectively) undergo the same sequence of scattering events(encircled crosses) induced by the disorder potential, see Eq. (35), at r , . . . , r . Dueto particle-particle collision events (squares), the particles redistribute their energies.Here, solid and dashed arrows correspond to disorder averaged single-particle Green’sfunctions, Eq. (36), and their complex conjugates, respectively. Upon flux detection,one particle is annihilated, while the undetected particles are traced over (dots). potential, specified by the mean value (cid:104) V ( r ) (cid:105) = 0 and the two-point correlation function: V ( r ) V ( r ) = 4 π(cid:96) dis δ ( r − r ) . (35)Furthermore, the disorder potential is assumed to be weak, i.e. √ E(cid:96) dis (cid:29) E , see Eq. (3). Initially, this is the case if √ E i (cid:96) dis (cid:29) E i , with only a negligible fraction of particlesthat reach single-particle energies E (cid:39) k(cid:96) dis (cid:29) G , see Eq. (12), is replaced by theaverage single-particle Green’s function: (cid:104) k (cid:48) | ˆ G ( E ) | k (cid:105) = (2 π ) δ ( k − k (cid:48) ) G E ( k ) , (36)with G E ( k ) = 1˜ k E − k , (37)where ˜ k E = √ E + i/ (2 (cid:96) dis ). In position representation, this leads to an exponentialdecay of the average density with 2 Im˜ k E = 1 /(cid:96) dis as the decay constant, see Eq. (39)below. This establishes (cid:96) dis as the mean free path, i.e. the average distance betweensubsequent disorder scattering events. Second, when calculating the average densitymatrix | f + (cid:105)(cid:104) f + | , and representing both | f + (cid:105) and (cid:104) f + | as a sum of diagrams, only thosecombination of diagrams survive where both | f + (cid:105) and (cid:104) f + | undergo the same sequence ofdisorder scattering events. Here, a disorder scattering event is induced by the correlationfunction, Eq. (35), where V ( r ) acts in | f + (cid:105) and V ( r ) in (cid:104) f + | (or vice versa – whereas icroscopic scattering theory for interacting bosons in weak random potentials == a)b) Figure 5.
Trace over the undetected particle for disorder-averaged diagrams (seeFig. 4), in case of a) inelastic or b) elastic collisions. In contrast to Fig. 2, arrows referto the disorder-averaged Green’s function, Eq. (37). The solid-dashed double arrow in a) denotes the average spectral function [ G ∗ E ( k ) − G E ( k )] / (2 πi ). correlators with V ( r ) and V ( r ) both acting in | f + (cid:105) or both in (cid:104) f + | are accounted forby the average Green’s function (37) [48]). These combinations of diagrams give rise toso-called ladder diagrams for the average density [49].We now apply the same procedure to the N -particle scattering processes presentedin Sec. 3.2. First, we take a diagram contributing to | f + (cid:105) and another one (called‘conjugate diagram’ in the following) contributing to (cid:104) f + | . Then, we replace all vacuumGreen’s functions by average Green’s functions and correlate, using Eq. (35), eachdisorder scattering event with another one in the conjugate diagram such that bothconjugate diagrams undergo the same sequence of disorder scattering events. Finally,we choose one of the final particles as detected particle, and trace over the remaining N − N -particle ladder diagrams thus constructed, we neglect all those wheretwo particles which interacted once meet again. This approximation is equivalent to theneglect of recurrent scattering [50] for a single particle, which, alike the neglect of non-ladder diagrams, is valid for k(cid:96) dis (cid:29)
1. It allows us to trace away the undetectedparticles directly after their interaction with the detected particle. Finally, we assumethat at least one disorder scattering event occurs between two collision events. This isjustified if (cid:96) int (cid:29) (cid:96) dis where (cid:96) int = 1 σρ , (38)with σ denoting the scattering cross section of the atom-atom interaction potential U ( r ), defines the average distance between two inelastic collision events. For s -wave icroscopic scattering theory for interacting bosons in weak random potentials + r r E E r r r E E r r r E E E r r r E E E E a) b) c) Figure 6.
The three building blocks from which all ladder diagrams (see Fig. 4) areconstructed. a) Single-particle propagation in the disorder potential, see Eq. (39). b) Elastic two-particle collision g E ,E ( r , r , r ), see Eq. (40). c) Inelastic two-particlecollision f E ,E ,E ( r , r , r ), see Eq. (41). scattering, σ = 8 πa s , see Eq. (71). The trace over the undetected particle allows us to decompose every ladder diagram(like the one shown in Fig. 4) into independent building blocks. These building blocksare shown in Fig. 6. The first one, Fig. 6a), represents a single average propagation stepof a single particle with energy E and corresponding wave vector k in the disorderedpotential from r to r : P E ( r , r ) = 4 π(cid:96) dis (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d k (2 π ) e − i k · ( r − r ) G E ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e −| r − r | /(cid:96) dis π(cid:96) dis | r − r | , (39)for E ≥
0. Note that, for a white noise potential as defined in Eq. (35), themean free path (cid:96) dis is independent of E [53]. For E <
0, the propagation isexponentially suppressed; in this case, Eq. (39) is multiplied by an additional factorexp (cid:16) − | r − r | (cid:113) | E | (cid:17) ). Since the typical distance between two scattering events isgiven by the mean free path (cid:96) dis , we can neglect the occurrence of negative energies if (cid:113) | E | (cid:96) dis (cid:29) g E ,E ( r , r , r ) = 2 (cid:18) π(cid:96) dis (cid:19) (cid:40) (cid:90) d k . . . d k (2 π ) × e − i [( k − k ) · r +( k − k ) · r +( k − k ) · r ] (cid:104) k , k | ˆ T U ( E + E ) | k , k (cid:105)× G E ( k ) G E ( k ) G E ( k ) G ∗ E ( k ) G ∗ E ( k ) (cid:41) . (40)The trace over the undetected particle was performed according to Fig. 5b), givingrise to an average Green’s function G ∗ E ( k ). For reasons of clarity, the wave vectors k , . . . , k are not explicitly shown in Fig. 6. They can, however, be easily deduced fromthe phase factors exp( ± i k · r ) describing annihilation or creation of a particle k due todisorder scattering at r , see Eqs. (4,15), with the help of following rule: outgoing solid(dashed) arrows always contribute with negative (positive) sign, the opposite holds for icroscopic scattering theory for interacting bosons in weak random potentials k – with phase factor exp[ i k · ( r − r )] in Eq. (40) –is associated with the dashed arrow pointing from r to r in Fig. 6b).The first factor 2 in Eq. (40) originates from the fact that the solid and dashedincoming amplitudes can be grouped together in two different ways. It can be shownthat this accounts for fluctuations of the atomic density inside the disordered slab [28].The factor 1 / E and E change to E and E = E + E − E : f E ,E ,E ( r , r , r ) = 2 (cid:18) π(cid:96) dis (cid:19) (cid:90) d k (2 π ) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d k d k d k (2 π ) e − i ( k · r + k · r − k · r ) (cid:104) k , k | ˆ T U ( E + E ) | k , k (cid:105)× G E ( k ) G E ( k ) G E ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (41)where Fig. 5a) was used for the trace over the undetected particle. The outgoing arrows of each building block may now be attached to the incoming arrowsof the next building block, and so on. The sum of ladder diagrams resulting from allcombinations of these building blocks is expressed by the following nonlinear integralequation: I E ( r ) = I ( r ) δ ( E − E i ) + (cid:90) V d r (cid:48) P E ( r , r (cid:48) ) I E ( r (cid:48) )+ (cid:90) ∞ d E (cid:48) (cid:90) V d r (cid:48) (cid:90) V d r (cid:48)(cid:48) (cid:20) g E (cid:48) ,E ( r (cid:48) , r (cid:48)(cid:48) , r ) I E ( r (cid:48)(cid:48) )+ (cid:90) ∞ d E (cid:48)(cid:48) f E (cid:48) ,E (cid:48)(cid:48) ,E ( r (cid:48) , r (cid:48)(cid:48) , r ) I E (cid:48)(cid:48) ( r (cid:48)(cid:48) ) (cid:21) I E (cid:48) ( r (cid:48) ) , (42)where I ( r ) = ρ e − z r , − ˆ k i /(cid:96) dis (43)represents the incoming wave propagating to r without being scattered. Correspond-ingly, z r , − ˆ k i denotes the distance from the surface of the scattering region V to r along a straight line parallel to the direction k i of the incident wavepacket. Thequantity I E ( r ) can be interpreted as the average density of particles with energy E at position r (at least in the case (cid:96) dis √ E (cid:29) G ∗ E ( k ) − G E ( k )] / (2 πi ) → δ ( E − k ) approaches a δ -function, suchthat a particle with wavevector k possesses a well defined energy). In particular, I ( r ) = (cid:82) d EI E ( r ) = (cid:104) f + | ˆ ρ ( r ) | f + (cid:105) gives the disorder-averaged expectation value of thesingle-particle density operator ˆ ρ ( r ) = ˆ ψ † ( r ) ˆ ψ ( r ) with respect to the quasi-stationaryscattering state | f + (cid:105) . From I E ( r ), the diffuse flux γ ( L ) (ˆ k d ) – i.e. the disorder average of icroscopic scattering theory for interacting bosons in weak random potentials γ (ˆ k d ), see Eq. (31), in ladder approximation – of particles scattered into direction ˆ k d isfinally obtained as: γ ( L ) (ˆ k d ) = (cid:90) ∞ d E γ ( L ) E (ˆ k d ) , (44)where γ ( L ) E (ˆ k d ) = (cid:90) V d r A (cid:96) dis e − z r , ˆ k d /(cid:96) dis (cid:115) EE i I E ( r ) ρ (45)denotes the ladder component of the average spectral flux density, i.e. the flux ofparticles scattered into direction ˆ k d with energy E , see Eq. (32). In Eq. (45), z r , ˆ k d denotes the distance from r to the surface of the scattering region in direction ˆ k d .Note that, in the far-field limit, only positive energies contribute to the scattered flux,Eq. (44). Within the scattering medium, negative energies are neglected in the transportequation (42) due to the exponential suppression mentioned after Eq. (39).Since we assume that disorder scattering events – represented by the term P E inEq. (42) – are much more frequent than collision events, we may neglect the spatialdependence of the collision terms in Eq. (42) and approximate them by δ -functions: g E (cid:48) ; E ( r (cid:48) , r (cid:48)(cid:48) , r ) (cid:39) δ ( r (cid:48) − r ) δ ( r (cid:48)(cid:48) − r ) g E (cid:48) ; E and f E (cid:48) ,E (cid:48)(cid:48) ,E ( r (cid:48) , r (cid:48)(cid:48) , r ) (cid:39) δ ( r (cid:48) − r ) δ ( r (cid:48)(cid:48) − r ) f E (cid:48) ,E (cid:48)(cid:48) ,E ,where g E (cid:48) ,E = (cid:90) d r (cid:48) d r (cid:48)(cid:48) g E (cid:48) ,E ( r (cid:48) , r (cid:48)(cid:48) , r ) , (46) f E (cid:48) ,E (cid:48)(cid:48) ,E = (cid:90) d r (cid:48) d r (cid:48)(cid:48) f E (cid:48) ,E (cid:48)(cid:48) ,E ( r (cid:48) , r (cid:48)(cid:48) , r ) . (47)The transport equation (42) then reduces to: I E ( r ) = I ( r ) δ ( E − E i ) + (cid:90) V d r (cid:48) P E ( r , r (cid:48) ) I E ( r (cid:48) )+ (cid:90) ∞ d E (cid:48) (cid:20) g E (cid:48) ,E I E ( r ) + (cid:90) ∞ d E (cid:48)(cid:48) f E (cid:48) ,E (cid:48)(cid:48) ,E I E (cid:48)(cid:48) ( r ) (cid:21) I E (cid:48) ( r ) . (48)As shown in [51], this equation can also be derived from the nonlinear Boltzmanntransport equation. Due to the above collision approximation, the spatial transport ofparticles in Eq. (48) is solely governed by the propagation P in the disorder potential,whereas the collision terms g and f lead to a redistribution of energies. As comparedto Eqs. (40,41), these terms simplify as follows: g E ,E = (cid:18) π(cid:96) dis (cid:19) (cid:90) d k d k (2 π ) | G E ( k ) | (cid:110) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105)× | G E ( k ) | G E ( k ) (cid:111) , (49) f E ,E ,E = 2 (cid:18) π(cid:96) dis (cid:19) (cid:90) d k d k d k (2 π ) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi × (cid:12)(cid:12)(cid:12) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) (cid:12)(cid:12)(cid:12) | G E ( k ) | | G E ( k ) | | G E ( k ) | , (50)with k = k + k − k , E = E + E − E k + k /
2, and | k (cid:105) , | k (cid:105) as defined afterEq. (18). icroscopic scattering theory for interacting bosons in weak random potentials For a given form of the two-body T -matrix (e.g. s-wave scattering, see below), we cannow calculate the collision terms g and f according to Eqs. (49,50), and then numericallysolve the transport equation (48) by iteration. Before presenting the correspondingnumerical results in Sec. 6, however, we will discuss, in the remainder of this section,some general properties of g and f , which, as shown below, lead to thermalization ofthe single-particle energies for an infinite system.As shown in Appendix D, the collision terms fulfill the following relations: (cid:113) E g E ; E = − (cid:90) ∞ d E √ Ef E ,E ,E , (51)( E + E ) (cid:113) E g E ; E = − (cid:90) ∞ d E E √ Ef E ,E ,E . (52)Both relations follow from the fact that the T -matrix associated to the atom-atominteraction potential U ( r ) fulfills the optical theorem, Eq. (19), and express conservationof the particle and the energy flux, respectively. Moreover, from Eq. (41), one can showthat: f E ,E ,E √ E + E − E = f E ,E + E − E ,E √ E . (53)This equation expresses microscopic reversibility of the collision dynamics: given twoparticles with energy E and E , the collision process E , E → E , E occurs with thesame probability as the reverse process E , E → E , E given two particles with energy E and E . The square roots in the denomimators of (53) result from the traces over theundetected particle with energy E = E + E − E (left-hand side) or E (right-handside), respectively.Using Eqs. (51,52) – and the fact that the linear propagator P E ( r , r (cid:48) ) = P ( r , r (cid:48) )is independent of E – it follows that the quantities J ( r ) = (cid:82) ∞ d E J E ( r ) and K ( r ) = (cid:82) ∞ d E K E ( r ), with J E ( r ) = √ EI E ( r ) , K E ( r ) = E √ EI E ( r ) , (54)corresponding to the particle and energy flux, respectively, both fulfill the same lineartransport equation: J ( r ) = J ( r ) + (cid:90) V d r (cid:48) P ( r , r (cid:48) ) J ( r (cid:48) ) , (55) K ( r ) = K ( r ) + (cid:90) V d r (cid:48) P ( r , r (cid:48) ) K ( r (cid:48) ) , (56)where the source terms J ( r ) = √ E i I ( r ) and K ( r ) = E i √ E i I ( r ) differ only by theconstant factor E i . Due to Eqs. (51,52), the collision terms drop out from Eq. (48)when integrating over E . Since the linear transport equation fulfills flux conservation,this, in turn, implies that, both, particle and energy flux are conserved. Furthermore,since K ( r ) = E i J ( r ), the same relation holds for the solutions of the linear equations(55,56): K ( r ) = E i J ( r ) . (57) icroscopic scattering theory for interacting bosons in weak random potentials r r r r r r r E i E i E i E d E = 2 E i − E d r r E d E i E i θ θ Figure 7.
Example of a crossed diagram contributing to coherent backscattering inthe case of three interacting particles. It is obtained from the ladder diagram shownin Fig. 4 by reversing the direction of propagation of the dashed propagators along thepath r → r → r → r → r → r → r . After these preparatory steps, we can now look for a solution of the transport equation(48) in case of a semi-infinite medium. Far away from its boundary, I E ( r ) = I E shouldbecome independent of r , and I ( r ), Eq. (43), tends to zero. Hence, the constant solution I E must fulfill: (cid:90) ∞ d E (cid:48) (cid:20) g E (cid:48) ,E I E + (cid:90) ∞ d E (cid:48)(cid:48) f E (cid:48) ,E (cid:48)(cid:48) ,E I E (cid:48)(cid:48) (cid:21) I E (cid:48) = 0 . (58)Using Eqs. (51,53), one can show that I E = √ Ee − γE fulfills Eq. (58) for γ >
0. Theconstant γ , in turn, is determined by Eq. (57) as γ = 2 /E i . Hence the normalizedparticle flux distribution is given by: J E ( r ) J ( r ) = 4 EE i e − E/E i . (59)This corresponds to a Maxwell-Boltzmann distribution the temperature of which isdetermined by the initial energy ( E i = k B T /
5. Coherent transport
Before turning to the numerical results, however, we will extend the general formalismof Sec. 4 in order to calculate the leading interference correction (in the weak disorderparameter 1 / ( k(cid:96) dis )) to the average scattered flux density. This correction is describedby crossed diagrams [52], which are obtained from the ladder diagrams by reversing thedirection of propagation of a single amplitude. Starting from the ladder diagram shownin Fig. 4, we can construct, for example, the crossed diagram shown in Fig. 7. It amountsto an interference between two amplitudes where the detected atom is emitted from r icroscopic scattering theory for interacting bosons in weak random potentials r , respectively. For a given wavevector k d of the detected atom, the backscatteringangle θ is defined by cos θ = − ˆ k i · ˆ k d . Since annihilation and creation of atoms withwavevector k by the disorder potential at position r are associated with factors e ± i k · r ,respectively, see Eqs. (4,15), this leads to a phase factor e i q · ( r − r ) with respect to theladder diagram, Fig. 4, where q = k i + k d . (60)Since r and r refer to randomly chosen positions of scattering events, this phase factorvanishes on average unless q (cid:39) ⇔ k i (cid:39) − k d , corresponding to exact backscattering( θ = 0). Therefore, this effect of interference between reversed amplitudes is called‘coherent backscattering’ [8, 9, 10]. More precisely, one can show that the angularwidth of the coherent backscattering interference peak is approximately given by∆ θ (cid:39) / ( k(cid:96) dis ) [53]. For a single particle, the height of this peak at θ = 0 equals theincoherent background as described by the ladder diagrams (except for single scatteringwhich only contributes to the background), what amounts to an enhancement of thebackscattered flux by a factor 2. We will show below how this enhancement factorchanges as a consequence of elastic and inelastic atom-atom collisions.For this purpose, we will derive a transport equation for the ‘crossed density’which describes a pair of amplitudes propagating in opposite directions. In Fig. 7, thecorresponding crossed scattering path is given by r → r → r → r → r → r → r (where we define the direction of the path to be fixed by the solid arrows, whereasthe dashed arrows propagate in the opposite sense). The remaining parts ( r and r )correspond to ladder diagrams already treated in Sec. 4. Due to energy conservation,the energies E and (cid:101) E associated with the two counterpropagating conjugate amplitudesalways fulfill the following relation: (cid:101) E = E i + E d − E . (61)
This leads us to the first crossed building block: P ( C ) E ( r , r ) = 4 π(cid:96) dis (cid:90) d k d k (2 π ) e − i ( k − k ) · ( r − r ) G E ( k ) G ∗ (cid:101) E ( k )= e | r − r | (cid:16) i √ E − i √ (cid:101) E − /(cid:96) dis (cid:17) π(cid:96) dis | r − r | , (62)describing single-particle propagation with different energies E (wave vector k ) and (cid:101) E (wave vector k ) for the conjugate amplitudes, see Fig. 8a). For E = (cid:101) E , i.e. E = ( E i + E d ) / P E , see Eq. (39).The following building block, Fig. 8b) , g ( C ) E ,E = (cid:18) π(cid:96) dis (cid:19) (cid:90) d k d k (2 π ) | G E ( k ) | G E ( k ) G ∗ (cid:101) E ( k ) (63) × (cid:34) G E ( k ) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) + G ∗ (cid:101) E ( k ) (cid:104) k | (cid:16) ˆ T (1) U ( E (cid:48) ) (cid:17) † | k (cid:105) (cid:35) , icroscopic scattering theory for interacting bosons in weak random potentials r r E r r r E E + r r r E E E r r r E E r r r E E r r r E E E E E E a) d)b) c)e) E E E E E E E E E E E E E Figure 8.
Building blocks for crossed diagrams. a) Single-particle propagation P ( C ) E ( r , r ), see Eq. (62). Note that the energies E and (cid:101) E associated tocounterpropagating amplitudes (solid and dashed line, respectively) fulfill E + (cid:101) E = E i + E d , see Eq. (61). b) Elastic collision g ( C ) E ,E , see Eq. (63), obtained by reversingthe lowermost line of the ladder building block g E ,E , Fig. 6b). c) Inelastic collision f ( C ) E ,E ,E , see Eq. (64). d) Crossed collision h ( C ) E ,E , see Eq. (65), obtained byreversing the solid arrow between r and r for g E ,E , Fig. 6b). e) Conjugate crossedcollision (cid:0) h ( C ) (cid:1) ∗ (cid:101) E , (cid:101) E . Note that, due to different possibilities of reversing single-particleamplitudes, there are more crossed than ladder building blocks (see Fig. 6). where E = E + E − E k + k / E (cid:48) = E + (cid:101) E − E k + k /
2, and | k (cid:105) as definedafter Eq. (18), represents the crossed counterpart of the elastic collision g E ,E , seeFig. 6b). Again, it reduces to the corresponding ladder term, Eq. (49), for (cid:101) E = E .The wavevector k in Eq. (63) is associated with Green’s functions emitted from position r , and k with those propagating between r and r .Similarly, Fig. 8c), f ( C ) E ,E ,E = 4 (cid:18) π(cid:96) dis (cid:19) (cid:90) d k d k d k (2 π ) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi × (cid:104) k | ˆ T (1) U ( E ) | k (cid:105)(cid:104) k , + | (cid:16) ˆ T (1) U ( E (cid:48) ) (cid:17) † | k , + (cid:105)× | G E ( k ) | G E ( k ) G ∗ (cid:101) E ( k ) G E ( k ) G ∗ (cid:101) E ( k ) , (64)where k = k + k − k , E = E + E − E k + k / E (cid:48) = E + (cid:101) E − E k − k / | k , + (cid:105) = (cid:16) | ( k + k ) / (cid:105) + | ( − k − k ) / (cid:105) (cid:17) / √ | k , + (cid:105) = (cid:16) | ( k + k ) / (cid:105) + | ( − k − k ) / (cid:105) (cid:17) / √ | k (cid:105) , | k (cid:105) as defined after Eq. (18), represents the crossed counterpart of inelasticcollision f E ,E ,E , see Fig. 6c). It reduces to two times the corresponding ladder term,Eq. (50), for (cid:101) E = (cid:101) E = E = E . The factor 2 originates from the fact that we canreverse the single-particle amplitudes of the ladder building block, Fig. 6c), also in adifferent way (with the outgoing dashed arrow pointing to r instead of r ) giving rise toan identical term. The wavevectors k , k and k in Eq. (64) are associated with Green’sfunctions emitted from (or pointing towards) positions r , r and r , respectively.Similarly, there also exist two different possibilities for reversing the ladder buildingblock g E ,E , Fig. 6b). Apart from g ( C ) E ,E , see Eq. (63) and Fig. 8b), this gives rise to a icroscopic scattering theory for interacting bosons in weak random potentials h ( C ) E ,E = (cid:18) π(cid:96) dis (cid:19) (cid:90) d k d k (2 π ) G E ( k ) G ∗ (cid:101) E ( k ) (cid:104) k | ˆ T (1) U ( E i + E d ) | k (cid:105)× (cid:12)(cid:12)(cid:12) G (cid:101) E ( k ) (cid:12)(cid:12)(cid:12) G E ( k ) , (65)where | k (cid:105) = (cid:16) | k (cid:105) + |− k (cid:105) (cid:17) / √
2. The corresponding conjugate diagram, see Fig. 8e),is given by (cid:18) h ( C ) (cid:101) E , (cid:101) E (cid:19) ∗ . Note that the two colliding particles exhibit opposite wavevectors( k and − k ), and therefore the energy of the collision event is fixed to ˜ E + E = E i + E d due to Eq. (61).Each of the above crossed building blocks exhibits an incoming and an outgoingcrossed density (defined by the direction of the solid arrow, as mentioned above).Additionally, the two-particle building blocks, Figs. 8b-e), exhibit an incoming ladderdensity. The latter is given by the solution I E ( r ) of the ladder transport equation (48). Propagation of the crossed density can now be described by an integral equationaccounting for all possible combinations of the above crossed building blocks (see Fig. 8).An example is displayed in Fig. 9a). Here, the outgoing crossed density of the buildingblock shown in Fig. 8d) serves as the incoming crossed density for the building blockshown in Fig. 8e). The resulting combination, Fig. 9a), exhibits the following remarkableproperty: if we look at the outgoing arrows (solid arrow pointing to r , dashed arrowpointing to r ) corresponding to the detected particle, we see that the detected particleexhibits no collision with the other particles involved in Fig. 9a). The evolution of theseundetected particles therefore has no impact on the detected particle and, consequently,as discussed at the end of Sec. 3.2, the process shown in Fig. 9a) may be disregardedwhen calculating the detection signal. The same remains true if – instead of attachingFig. 8e) directly to Fig. 8d) – an arbitrary sequence of the remaining crossed buildingblocks, Figs. 8a), b) or c), is inserted in between. In contrast, for any other combinationof building blocks, e.g. Fig. 9b), all involved particles turn out to be connected to eachother (through collision events or partial traces), thus contributing to the propagationof the crossed density.In order to exclude combinations of the former type from the transport equation, wesplit the crossed density into two parts, i.e. C E ( r ) = C (1) E ( r ) + C (2) E ( r ). All combinationsof the building blocks Figs. 8a-c) and e) are contained in C (1) E ( r ), and the remainingones, i.e. those involving Fig. 8d), in C (2) E ( r ). According to the rules mentioned above,the building block Fig. 8e) is excluded from the transport equation for C (2) E ( r ). In total,the transport equations therefore read as follows: C (1) E ( r ) = I ( C )0 ( r ) δ ( E − E i ) + (cid:90) V d r (cid:48) P ( C ) E ( r , r (cid:48) ) C (1) E ( r (cid:48) )+ (cid:90) ∞ d E (cid:48) (cid:34) g ( C ) E (cid:48) ,E C (1) E ( r ) + (cid:90) E i + E d d E (cid:48)(cid:48) f ( C ) E (cid:48) ,E (cid:48)(cid:48) ,E C (1) E (cid:48)(cid:48) ( r ) (cid:35) I E (cid:48) ( r ) icroscopic scattering theory for interacting bosons in weak random potentials r r r r r r r r a) b) Figure 9.
Combinations of crossed building blocks. a) When attaching Fig. 8d) toFig. 8e), the detected atom (solid arrow pointing to r , dashed arrow pointing to r )exhibits no collision with the undetected atoms. Combinations of this type therefore donot contribute to the detection signal, and must be excluded from the crossed transportequation. b) In contrast, the inverse combination, i.e. attaching Fig. 8e) to Fig. 8d),contributes to the propagation of the crossed density, and must be taken into accountin the transport equation. + (cid:90) E i + E d d E (cid:48) (cid:18) h ( C ) (cid:101) E, (cid:101) E (cid:48) (cid:19) ∗ I E ( r ) C (1) E (cid:48) ( r ) , (66)and C (2) E ( r ) = (cid:90) V d r (cid:48) P ( C ) E ( r , r (cid:48) ) C (2) E ( r (cid:48) )+ (cid:90) ∞ d E (cid:48) (cid:34) g ( C ) E (cid:48) ,E C (2) E ( r ) + (cid:90) E i + E d d E (cid:48)(cid:48) f ( C ) E (cid:48) ,E (cid:48)(cid:48) ,E C (2) E (cid:48)(cid:48) ( r ) (cid:35) I E (cid:48) ( r )+ (cid:90) E i + E d d E (cid:48) h ( C ) E (cid:48) ,E I (cid:101) E (cid:48) ( r ) (cid:16) C (1) E (cid:48) ( r (cid:48)(cid:48) ) + C (2) E (cid:48) ( r ) (cid:17) . (67)In Eq. (66), the incoming crossed density is given by: I ( C )0 ( r ) = ρ e i q · r − (cid:16) z r , − ˆ k i + z r , ˆ k d (cid:17) / (2 (cid:96) dis ) , (68)where q = k i + k d , see Eq. (60), where the wave vector of the detected particle isdetermined by the energy E d and the position R of the detector (in the far field)as k d = √ E d R /R , and z r , − ˆ k i (or z r , ˆ k d ) corresponds to the distance an incoming (oroutgoing) particle travels inside the scattering region, as defined in Eqs. (43,45). Finally,the coherently backscattered flux density results as γ ( C ) (ˆ k d ) = (cid:90) ∞ d E d γ ( C ) E d (ˆ k d ) , (69)with associated spectral density γ ( C ) E d (ˆ k d ) = (cid:90) V d r A (cid:96) dis ρ (cid:115) E d E i e − i q · r − (cid:16) z r , − ˆ k i + z r , ˆ k d (cid:17) / (2 (cid:96) dis ) × (cid:104)(cid:16) C (1) E d ( r ) + C (2) E d ( r ) (cid:17) − δ ( E d − E i ) I ( C )0 ( r ) (cid:105) , (70)where the last term accounts for single scattering. The total average flux measured by adetector placed in direction ˆ k d , see Eq. (31), then corresponds to the sum of the ladderand the crossed component, γ (ˆ k d ) = γ ( L ) (ˆ k d ) + γ ( C ) (ˆ k d ). icroscopic scattering theory for interacting bosons in weak random potentials
6. Numerical solutions of the transport equations
After having developed the general scattering formalism valid for arbitrary shapes ofthe interaction potential U ( r ) and the scattering region V , we will now focus on the caseof a short-range potential U ( r ) and a slab geometry for V . As explained in AppendixE, the T -matrix is then described by a single parameter – the s -wave scattering length a s . The corresponding average distance between (inelastic) collision events is given by: (cid:96) int = 18 πa s ρ . (71)In the following, we measure the interaction strength in terms of the ratio between (cid:96) dis and (cid:96) int : α = (cid:96) dis (cid:96) int = 8 πa s (cid:96) dis ρ , (72)which, as explained in Sec. 4.1, should fulfill the condition α (cid:28)
1, and, due to Eq. (71),is proportional to a s . Indeed, we see from Eqs. (E.2,E.3) that the ladder collision terms g and f both depend on α . The terms proportional to a s in g drop out as a consequenceof flux conservation, see Eq. (51). The same is not true for the crossed collision terms g ( C ) and h ( C ) , see Eqs. (E.4,E.5), which depend on a second parameter proportional to a s : β = 8 πa s (cid:96) dis ρ √ E i = α √ E i a s . (73)Since √ E i a s (cid:28) s -wave scattering, it follows that β (cid:29) α . The parameter β can alsobe expressed in terms of the healing length ξ = (8 πρ a s ) − / [55], i.e. β = (cid:96) dis / ( √ E i ξ ),or in terms of the interaction parameter g = 8 πa s appearing in the Gross-Pitaevskiiequation, i.e. β = gρ (cid:96) dis / √ E i . The Gross-Pitaevskii equation is valid in the limit a s → ρ → ∞ [56] such that a s ρ = const. Since α → α = 0, as it is indeed the case if we insert the s -wave expressions, Eqs. (E.2-E.6),evaluated at α = 0.Concerning the geometry of the scattering medium, we choose a slab confinedbetween two planes, z = 0 and z = L , respectively, with perpendicular incidentwavevector, i.e. k i = (0 , , k i ). The thickness of the slab in units of the disordermean free path defines its optical thickness b = L/(cid:96) dis . The slab geometry is veryconvenient from a numerical point of view, since the integration over x and y can beperformed analytically in Eqs. (48,66,67), such that the resulting transport equationsonly depend on z [54]. Moreover, due to rotational symmetry around the z -axis, thebackscattered flux g (ˆ k d ) = g ( θ ) depends only on the backscattering angle θ defined byˆ k i · ˆ k d = − cos θ , and the distances appearing in Eqs. (43,45,68,70) simplify to z r , − ˆ k i = z and z r , ˆ k d = z/ cos θ , respectively. Finally, the integration over the scattering volume V in Eqs. (45,70) reduces to (cid:82) V d r / A → (cid:82) L d z . The one-dimensional versions of thetransport equations (48,66,67) can now be solved numerically, e.g. by iteration. icroscopic scattering theory for interacting bosons in weak random potentials b)a) E d /E i z/ dis J ( z ) / J J ( i n e l ) ( E ) Figure 10. (Color online) a) Different components of the average flux density J ( z ),plotted as a function of position z in the slab, for weak interaction α = 1 / b = 40. The linear flux density (red solid line) coincides with the total fluxdensity for the case of many particles (black long-dashed). The latter splits into anelastic (green dashed) and inelastic component (blue dotted). In spite of the weakinteraction, the transport is dominated by inelastically scattered particles, especiallydeep inside the slab. b) Normalized energy distribution J (inel) E ( z ) of inelasticallyscattered atoms for different positions z = 0 (blue dotted), z = L/ z = L (black long-dashed) in the slab, and otherwise the same parameters as in a). Thethin black line displays √ Ef E i ,E i ,E / ( −√ E i g E i ,E i ) (see Eq. (51) for the normalization)according to Eq. (E.3), i.e. the distribution after a single inelastic scattering event.The kink of this distribution is recovered at the beginning of the slab (i.e., for z = 0).Deep inside the slab (i.e., for z = L/ z = L ), the spectrum collapses onto athermal Maxwell-Boltzmann distribution with average energy E av = E i (red solid). Fig. 10a) shows the resulting flux density J ( z ) = (cid:82) ∞ d E √ EI E ( z ), see Eq. (54), asa function of the position z inside the slab, for weak interactions α = 1 / b = 40. As already proven after Eq. (55), the totalflux J ( z ) (black long-dashed line) equals the linear flux (red solid) as obtained fromEq. (48) with α = 0. In contrast to the linear case, however, the flux J ( z ) splitsinto an elastic (green dashed) and an inelastic component (blue dotted), defined by J E ( z ) = J (el) E ( z ) δ ( E − E i ) + J (inel) E ( z ). We see that, in spite of the weakness of theinteraction ( α = 1 / ≈ b ) of scattering events requiredto traverse a slab with thickness b . The expected number of two-body collision eventsthus results as approximately αb = 16. Let us note that the inelastic component ofthe flux is associated with a non-condensed fraction of atoms, since an N -fold productof a single-particle state (as required from the formal definition of a condensate via the icroscopic scattering theory for interacting bosons in weak random potentials J (inel) E ( z ) of the inelastic component is shown inFig. 10b), for different positions z inside the slab. We see that, far inside the slab, i.e.at z = 10 (cid:96) dis (green dashed) and z = 40 (cid:96) dis (black long-dashed), the energy distributionapproaches a Maxwell-Boltzmann distribution J ( MB ) E /J = 4 E exp( − E/E i ) /E i (redsolid), see Eq. (59). Thereby, we confirm the analytical result derived in Sec. 4.4 for aninfinite slab. In contrast, at the beginning of the slab (blue dotted), the distributionis not yet thermalized, and lies between the Maxwell-Boltzmann distribution and thedistribution √ Ef E i ,E i ,E / ( −√ E i g E i ,E i ) obtained after a single inelastic collision event(thin black line) and normalized according to Eq. (51). Fig. 11a) shows the background and interference contributions to the backscatteredflux in exact backscattering direction θ = 0 for (cid:96) dis √ E i = 10. Since, in general, thebackscattered flux is dominated by scattering paths which do not penetrate very deeplyinto the slab, we restrict ourselves to the case of a moderate slab thickness b = 10 (ascompared to b = 40 in Fig. 10). As already known from previous work on nonlinearcoherent backscattering in the purely elastic Gross-Pitaevskii limit [27, 28, 29], thebackscattered flux γ ( C ) (0) for α = 0 (red line) exhibits a transition from constructive todestructive interference for increasing β . This transition can be explained by the factthat the nonlinearity effectively introduces dephasing between two reversed scatteringpaths [27, 28, 29]. For larger β , the phase difference accumulated along the shortestscattering paths, i.e. those exhibiting only a few, but at least two scattering events(since, as mentioned above, single scattering does not contribute), may exceed π/ α = β/
10 corresponding to a s √ E i = 1 /
10, see Eq. (73), is taken intoaccount (blue line): At first, the interference drops faster than for α = 0, since inelasticscattering changes the frequency and thus leads to an additional dephasing mechanism.At larger values β , however, the decrease of the backscattered flux is slowed down ascompared to the purely elastic case, such that a transition to destructive interference isnot observed in Fig. 11a).This behaviour can be explained by examining the spectral distribution γ ( C, inel) E d of the inelastic interference component as a function of the detected frequency E d in Fig. 11b). Since the amount of inelastic scattering is governed by α = β/ γ ( L, inel) E d (0) (dashed lines) and γ ( C, inel) E d (0) (solid lines) increase as a function of β (green, blue and red lines corresponding to β = 0 .
02, 0 .
08 and 0 .
2, respectively.)The interference spectra γ ( C, inel) E d (0) exhibit narrow peaks close to the initial energy, E d (cid:39) E i . The relative width of these peaks approximately equals ∆ E d /E i (cid:39) / ( (cid:96) dis √ E d ) = 1 /
10, which can be understood as a consequence of frequency-induceddephasing given by Eq. (62) with E (cid:54) = (cid:101) E = E i + E d − E . Less intuitive is the icroscopic scattering theory for interacting bosons in weak random potentials b)a) β γ ( ) E d /E i γ ( i n e l ) E d ( ) / E − i Figure 11. (Color online) a) Background and interference contributions γ ( L ) (0) and γ ( C ) (0) to the scattered flux in exact backscattering direction ( θ = 0) for a slab withthickness b = 10 and (cid:96) dis √ E i = 10 as a function of the crossed collision strength β ,see Eq. (73). The Gross-Pitaevskii equation ( α = 0, red line) predicts a crossover fromconstructive ( γ ( C ) (0) >
0) to destructive interference ( γ ( C ) (0) <
0) at β (cid:39) .
13. Inpresence of inelastic collisions ( α = β/
10, blue solid line), the decrease of γ ( C ) (0) isinitially faster, but for β > .
04 slower than in the case α = 0, due to the steadilyrising inelastic component γ ( C, inel) (0) (blue dotted line). For comparison, the totalbackground contribution γ ( L ) (0) (green solid), which is independent of α and β , andits inelastic component γ ( L, inel) (0) (for α = β/
10, green dashed) are also shown. b) Spectral distributions γ ( C, inel) E d (0) (solid lines) and γ ( L, inel) E d (0) (dashed lines) for thesame parameters as in a) and β = 0 .
02 (red), 0.08 (green) and 0.2 (blue). In anarrow spectral region around E d (cid:39) E i (but slightly shifted with respect to E i ), theinterference contribution γ ( C, inel) E (0) exceeds the background γ ( L, inel) E (0), correspondingto a coherent backscattering enhancement factor larger than two. fact that the maxima of the peaks are slightly shifted with respect to E i , for which,at present, we are lacking a simple explanation. Please note, however, that theinterference peaks exceed the background (most remarkably for larger values of β ),corresponding to a coherent backscattering enhancement factor of the inelastic fluxcontribution η (inel) E d = (cid:16) γ ( C, inel) E d (0) + γ ( L, inel) E d (0) (cid:17) /γ ( L, inel) E d (0) > E d where the crossed contribution is maximal. Thisenhancement is a consequence of the many-wave interference character of nonlinearcoherent backscattering [57, 58] resulting from the fact that, as discussed in Sec. 5.2,there are several ways of reversing the scattering paths when constructing crossed fromladder diagrams. The number of these possibilities increases with increasing number ofinelastic scattering events. Although, due to the small width of these peaks, the totalinelastic component γ ( C, inel) (0) (integrated over E d ) turns out to be smaller than the icroscopic scattering theory for interacting bosons in weak random potentials γ ( L, inel) (0) [cf. the dashed blue and dotted green lines in Fig. 11a)], thismany-wave interference effect contributes to the above observed slowing down of thedecrease of the backscattered flux.
7. Conclusion
Within this paper, we have derived a microscopic N -body scattering theory forinteracting particles in a weak disorder potential in three dimensions. We have appliedthis diagrammatic theory to a stationary scattering scenario for an asymptotically non-interacting quasi-plane matter wave incident on a three-dimensional slab, with thedisorder potential and inter-particle collisions confined to the slab region, and herebyverified the viability of our theory to address, on the one hand, very fundamental but,on the other hand, very timely questions of quantum transport for interacting particlesin random environments. In a clear and precise manner, we demonstrated how one canbridge the gap between strictly unitary many-body evolution and its implications on themesoscopic level governed by a transport equation similar to the nonlinear Boltzmanntransport equation. Furthermore, we have determined the coherent corrections due tothe wave nature of the particles leading to the effect of coherent backscattering. Wehave demonstrated that inelastic scattering slows down the decrease of the coherentbackscattering peak as compared to the purely elastic case described by the Gross-Pitaevskii equation.Let us briefly summarize the basic assumptions of our theory: first, we assume anoptically thick scattering medium ( b = L/(cid:96) dis (cid:29)
1) allowing for multiple scattering in aweak disorder potential, where the mean free path is much larger than the wavelength ofthe incoming particles ( (cid:96) dis √ E i (cid:29) α and β for ladder and crossed collisions, see Eqs. (72,73)for the case of s -wave scattering, should fulfill the condition α, β (cid:28)
1, such that atom-atom collisions occur less frequently than disorder scattering events. With view atfuture work, we expect that the condition β (cid:28) β (cid:39) (cid:96) dis √ E i , another type of collision process – corresponding to scattering inducedby the fluctuating background density – sets in which is described by diagrams similarto our ladder diagrams [59]. Furthermore, relaxing the contact approximation for thecollision terms, Eqs. (46,47), will allow to determine the effect of attractive or repulsiveinteractions onto the spatial atomic density profile. It will hence be a worthwhile taskto extend our theory to stronger interactions, although we surely expect to encountercertain limits (e.g., the regime of superfluidity) where other methods will be required.Concerning an experimental verification of our results, the application to astationary scattering setup with matter waves constitutes, on the one hand, a very timelyscenario, as, e.g., the developments of atom lasers and matter wave interferometers onatom chips [22, 23, 60] rapidly progress. On the other hand, many years of expertise havebeen gathered within the field of wave-packet spreading upon releasing the condensate icroscopic scattering theory for interacting bosons in weak random potentials k p p p p E E i − E (i) (ii) (iv)(iii) k k p p p k k k p p Figure A1.
The dashed lines split the diagram of Fig. 1 into 4 subdiagrams (i), (ii),(iii), and (iv). Note that the subdiagrams (i) and (ii) – and likewise (iii) and (iv) – arenot connected to each other. This allows us to factorize the 3-particle diagram into 1-and 2-particle diagrams. from a trap into a new environment, where, e.g., the first experiments on coherentbackscattering of (non-interacting) matter waves have been reported recently [11, 12].Consequently, an extension of our theory to time-dependent scenarios based on recentprogress in this field [51, 61, 62] presents a significant and feasible task. Similarly, also afinite correlation length of the disorder potential can be taken into account, see [24, 25]for the non-interacting case.In conclusion, we are confident, that our present theory and the ratherstraightforward extensions discussed above will substantially foster a more completeunderstanding of quantum transport under the interplay of disorder and inter-particleinteraction and can contribute to a unifying picture from microscopic to macroscopicscales.
Acknowledgements
We thank Nicolas Cherroret, Pierre Lugan, Cord A. M¨uller and Peter Schlagheck forfruitful discussions. We acknowledge partial support by DFG research unit FG760. T.G. acknowledges funding through DFG Grant No. BU1337/8-1.
Appendix A. Factorization of the transition amplitude
In this appendix, we show how an arbitrary N -particle scattering diagram can befactorized into single-particle propagators and two-body collisions. We first look atthe example diagram shown in Fig. 1. As shown in Fig. A1, this diagram can be splitinto four independent subdiagrams. The two subdiagrams connected to the initial state– (i) and (ii) in Fig. A1 – correspond to Møller operators, and the remaining ones – (iii)and (iv) – to Green’s operators. This gives rise to the following matrix elements:Ω (i)+ ( E ) = (cid:104) p | ˆΩ ( V )+ ( E ) | k (cid:105) , (A.1)Ω (ii)+ ( E ) = 12 (cid:90) d p d p (2 π ) (cid:104) p , p | ˆ T U ( E ) | p , p (cid:105)(cid:104) p , p | ˆΩ ( V ) ( E ) | k , k (cid:105) , (A.2) icroscopic scattering theory for interacting bosons in weak random potentials G (iii) ( E ) = 14 (cid:90) d p d p d p d p (2 π ) (cid:104) k , k | ˆ G V ( E ) | p , p (cid:105) (A.3) × (cid:104) p , p | ˆ T U ( E ) | p , p (cid:105)(cid:104) p , p | ˆ G V ( E ) | p , p (cid:105) ,G (iv) ( E ) = (cid:104) k | ˆ G V ( E ) | p (cid:105) . (A.4)Note that the diagrams (i) and (ii) are not connected to each other in Fig. A1. Thecorresponding Møller operators can therefore be factorized as in Eq. (26). Likewise, theGreen’s functions corresponding to (iii) and (iv) are factorized according to Eq. (27).The prefactors 1 / / | p , p (cid:105) and | p , p (cid:105) are identical and therefore mustnot be summed over twice). It turns out that these factors are compensated, however,by the two possibilities to associate the initial and final single-particle states with eachother in Eqs. (26,27).The total transition amplitude results as: (cid:104) k , k , k | ˆΩ (fig . (3 E i ) | k , k , k (cid:105) = 12 (cid:90) d p d p d p (2 π ) Ω (i)+ ( E i )Ω (ii)+ (2 E i ) × (cid:90) ∞−∞ d E ( − πi ) G (iii) (3 E i − E ) G (iv) ( E ) . (A.5)Now, we again apply Eqs. (26,27) to factorize the two-particle Møller and Green’soperators on the right-hand side of Eqs. (A.2,A.3) into single-particle operators. Inthis way, we recover most of the terms in Eq. (28). The only ones which appear to differfrom Eq. (28) are those associated to k , p , p and p , which we reformulate as follows:12 (cid:90) d p (2 π ) (cid:104) p , p | ˆ G V (3 E i − E ) | p , p (cid:105)(cid:104) p | ˆΩ ( V )+ ( E i ) | k (cid:105) = (cid:90) ∞−∞ d E ( − πi ) (cid:104) p | ˆ G V ( E ) ˆΩ ( V )+ ( E i ) | k (cid:105) G ( − E ) , (A.6)where we again applied Eq. (27), used the completeness relation (cid:82) d p | p (cid:105)(cid:104) p | = (2 π ) ,and defined: G ( − E ) = (cid:104) p | ˆ G V (3 E i − E − E ) | p (cid:105) . (A.7)Note that G ( − E ) is a complex analytic function of E with poles only in the upperhalf of the complex plane. This, again, is due to the fact that ˆ G V ( E ) as a function of E exhibits poles only in the lower half, whereas E enters with negative sign in the righthand side of Eq. (A.7). We now reformulate some terms in Eq. (A.6) as follows:ˆ G V ( E ) ˆΩ ( V )+ ( E i ) | k (cid:105) = (cid:16) + ˆ G V ( E ) ˆ V (cid:17) ˆ G ( E ) | k (cid:105) + ˆ G V ( E ) ˆ G V ( E i ) ˆ V | k (cid:105) = 1 E − E i + i(cid:15) (cid:104) + (cid:16) ˆ G V ( E ) + ˆ G V ( E i ) − ˆ G V ( E ) (cid:17) ˆ V (cid:105) | k (cid:105) = 1 E − E i + i(cid:15) ˆΩ ( V )+ ( E i ) | k (cid:105) , (A.8)where we used Eq. (8), the alternative but equivalent expression ˆ G V ( E ) = ˆ G ( E ) +ˆ G V ( E ) ˆ V ˆ G ( E ) with respect to Eq. (11), and the identityˆ G V ( E ) ˆ G V ( E i ) = 1 E − E i + i(cid:15) (cid:16) ˆ G V ( E i ) − ˆ G V ( E ) (cid:17) , (A.9) icroscopic scattering theory for interacting bosons in weak random potentials ab = b − a (cid:16) a − b (cid:17) (where we set the imaginary part in the denominatorof ˆ G V ( E ), see Eq. (9), equal to 2 (cid:15) instead of (cid:15) , and used the fact that ˆ G V ( E ) andˆ G V ( E i ) have the same set of eigenvectors). After inserting Eq. (A.8) into Eq. (A.6), weperform the integral over E by closing the integration contour in the lower half of thecomplex plane. (Remember that G ( − E ) has no poles in the lower half!) Thereby, theenergy E is set to E i , and we finally recover the missing terms in Eq. (28): (cid:90) ∞−∞ d E ( − πi ) (cid:104) p | ˆ G V ( E ) ˆΩ ( V )+ ( E i ) | k (cid:105) G ( − E )= (cid:104) p | ˆΩ ( V )+ ( E i ) | k (cid:105) G ( − E i ) . (A.10)The above procedure can be generalized to an arbitrary many-particle scatteringdiagram: We first divide the whole diagram into independent subdiagrams. Then,some of these subdiagrams turn out to be connected to each other by single-particlepropagators. In the above example, Fig. A1, this is the case for the subdiagram (i) and(iii), which are connected by the single-atom propagators from k to p with energy E i and from p to p with energy E . We have to show that these propagators mergeinto a single propagator (from k to p with energy E i ). For the case that one ofthe propagators is connected to the initial state (and thus corresponds to a Mølleroperator), the corresponding general identity is given by Eq. (A.10). If both propagatorscorrespond to Green’s operators, e.g. ˆ G V ( E ) and ˆ G V ( E ) below, the required identityis proven as follows: (cid:90) ∞−∞ d E d E ( − πi ) ˆ G V ( E ) ˆ G V ( E ) G (1) ( − E ) G (2) ( − E )= (cid:90) ∞−∞ d E d E ( − πi ) E − E + i(cid:15) (cid:16) ˆ G V ( E ) − ˆ G V ( E ) (cid:17) G (1) ( − E ) G (2) ( − E )= (cid:90) ∞−∞ d E ( − πi ) ˆ G V ( E ) G (1) ( − E ) G (2) ( − E ) , (A.11)where we again used Eq. (A.9) and the fact that G (1) ( − E ) and G (2) ( − E ) (whichcorrespond to arbitrary other subdiagrams where the energy E or E enters withnegative sign) exhibit no pole in the lower half of the complex plane. Note that theterm with ˆ G V ( E ) in the second line of Eq. (A.11) vanishes after integrating over E ,since no pole remains in the lower half. In total, the concatenation of two Green’soperators, i.e. ˆ G V ( E ) ˆ G V ( E ) on the left-hand-side of Eq. (A.11), reduces to a singleGreen’s operator, i.e. ˆ G V ( E ) on the left-hand-side, whereas the energy E is set equalto E . Appendix B. Inelastic and elastic example diagrams
The inelastic diagram shown in Fig. 2a) gives the following contribution to the fluxdensity: J (fig . ( r ) = (cid:18) (cid:19) (cid:90) d k d k d k d k d k (cid:48) d k (cid:48) d k (cid:48) d p d p d p d p d p (cid:48) d p (cid:48) d p (cid:48) d p (cid:48) (2 π ) icroscopic scattering theory for interacting bosons in weak random potentials × (cid:90) ∞−∞ d E d E (cid:48) | πi | w ∗ ( k (cid:48) ) w ∗ ( k (cid:48) ) w ( k ) w ( k ) (cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | p (cid:48) (cid:105)(cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | p (cid:48) (cid:105)× (cid:104) p (cid:48) , p (cid:48) | (cid:16) ˆ T U (2 E i ) (cid:17) † | p (cid:48) , p (cid:48) (cid:105)(cid:104) p (cid:48) | (cid:16) ˆ G V (2 E i − E (cid:48) ) (cid:17) † | k (cid:48) (cid:105)(cid:104) k (cid:48) | ˆ J ( r ) | k (cid:105)× (cid:104) p (cid:48) | (cid:16) ˆ G V ( E (cid:48) ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105)(cid:104) k | ˆ G V (2 E i − E ) | p (cid:105)× (cid:104) p , p | ˆ T U (2 E i ) | p , p (cid:105)(cid:104) p | ˆΩ ( V ) ( E i ) | k (cid:105)(cid:104) p | ˆΩ ( V ) ( E i ) | k (cid:105) . (B.1)Here, we used the following labels for the wave vectors: the incoming solid arrows arecalled k and k , whereas the detected and the traced out solid arrows are given by k and k , respectively. The intermediate solid arrows before and after the interaction eventare labeled by p and p and by p and p , respectively. The same notation holds forthe dashed arrows, which are, however, denoted by an additional prime.The trace formula, Eq. (33), can now be applied as follows: (i) re-place (cid:104) p (cid:48) | (cid:16) ˆ G V ( E (cid:48) ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105) by (cid:104) p (cid:48) | (cid:20)(cid:16) ˆ G V ( E ) (cid:17) † − ˆ G V ( E ) (cid:21) | p (cid:105) , (ii) deletethe integrals (cid:82) d k / (2 π ) and (cid:82) d E (cid:48) / ( − πi ), and (iii) replace E (cid:48) by E in (cid:104) p (cid:48) | (cid:16) ˆ G V (2 E i − E (cid:48) ) (cid:17) † | k (cid:48) (cid:105) .For the elastic diagram, Fig. 2b), we obtain: J (fig . ( r ) = (cid:18) (cid:19) (cid:90) d k d k d k d k d k (cid:48) d k (cid:48) d k (cid:48) d p d p d p d p (2 π ) × (cid:90) ∞−∞ d E πi w ∗ ( k (cid:48) ) w ∗ ( k (cid:48) ) w ( k ) w ( k ) (cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | k (cid:48) (cid:105)(cid:104) k (cid:48) | ˆ J ( r ) | k (cid:105)× (cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105)(cid:104) k | ˆ G V (2 E i − E ) | p (cid:105)× (cid:104) p , p | ˆ T U (2 E i ) | p , p (cid:105)(cid:104) p | ˆΩ ( V ) ( E i ) | k (cid:105)(cid:104) p | ˆΩ ( V ) ( E i ) | k (cid:105) . (B.2)Here, the labels of the wave vectors are identical to Eq. (B.1), with the only difference,that the intermediate wave vectors p (cid:48) , . . . , p (cid:48) are not needed due to the missinginteraction event for the dashed amplitudes. The trace formula, Eq. (34), is applied asfollows: (i) replace (cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E ) | p (cid:105) by (cid:104) k (cid:48) | (cid:16) ˆΩ ( V ) ( E i ) (cid:17) † | p (cid:105) , (ii) deletethe integrals (cid:82) d k / (2 π ) and (cid:82) d E/ (2 πi ), and (iii) replace E by E i in (cid:104) k | ˆ G V (2 E i − E ) | p (cid:105) .It is also insightful to take a look at the prefactors: the first factor 1 / / √ in the initial states | i (cid:105) and (cid:104) i | , see Eq. (20). The integration overthe final states | k , k (cid:105) and (cid:104) k (cid:48) , k (cid:48) | (with k = k (cid:48) = k due to the trace) goes along withtwo more factors 1 / | k (cid:105) and (cid:104) k (cid:48) | . Therefore, we have to include a factor 2 to take into account the otherpossibilities. In Eq. (B.2), we obtain an additional factor 2 due to the two possibilitiesin the factorization formula, Eq. (26), for the dashed amplitudes: k (cid:48) can be associatedwith k (cid:48) and k (cid:48) with k (cid:48) = k – or vice versa.Finally, the diagrams shown in Fig. 2 can be generalized to N > N − icroscopic scattering theory for interacting bosons in weak random potentials / → N ( N − / → N ( N − / N ( N − (cid:39) N (for N (cid:29)
1, since N → ∞ inthe quasi-stationary limit) are accounted for by the source term ρ in Eq. (43), whichis proportional to N , see Eq. (21), and occurs two times for a two-particle processproportional to the density squared. What remains is a factor 1 / T -matrix for indistinguishable particles, seeEq. (18), differs by a factor 2 from the one for distinguishable particles, this must becounterbalanced by the above factor 1 / Appendix C. Trace formulas
Here, we prove the trace formulas, Eqs. (33,34), for the trace over the undetected particleoriginating from an inelastic or an elastic collision. In both cases, we apply first thecompleteness relation (cid:82) d k | k (cid:105)(cid:104) k | = (2 π ) , and then the following identity for the productof two Green’s operators:ˆ G † V ( E (cid:48) ) ˆ G V ( E ) = 1 E − E (cid:48) + i(cid:15) (cid:16) ˆ G † V ( E (cid:48) ) − ˆ G V ( E ) (cid:17) , (C.1)which is similar to Eq. (A.9). Thereby, Eq. (33) is proven as follows: (cid:90) ∞−∞ d E d E (cid:48) | πi | (cid:90) d k (2 π ) ( . . . ) ( l )( − E (cid:48) ) ˆ G † V ( E (cid:48) ) | k (cid:105)(cid:104) k | ˆ G V ( E )( . . . ) ( r )( − E ) = (cid:90) ∞−∞ d E d E (cid:48) | πi | ( . . . ) ( l )( − E (cid:48) ) ˆ G † V ( E (cid:48) ) ˆ G V ( E )( . . . ) ( r )( − E ) = (cid:90) ∞−∞ d E d E (cid:48) | πi | E − E (cid:48) + i(cid:15) ( . . . ) ( l )( − E (cid:48) ) (cid:16) ˆ G † V ( E (cid:48) ) − ˆ G V ( E ) (cid:17) ( . . . ) ( r )( − E ) = (cid:90) ∞−∞ d E πi ( . . . ) ( l )( − E ) (cid:16) ˆ G † V ( E ) − ˆ G V ( E ) (cid:17) ( . . . ) ( r )( − E ) . (C.2)In the last step, we have used the fact that ( . . . ) ( r )( − E ) is a complex analytic functionwithout poles in the lower half of the complex plane. Similarly, ( . . . ) ( l )( − E (cid:48) ) exhibitsno poles in the upper half. Thereby, considering the two terms ˆ G † V ( E (cid:48) ) or ˆ G V ( E ),respectively, we can perform the integral either over E or over E (cid:48) , closing the integrationcontour in the lower or upper half, respectively. In both cases, the term 1 / ( E − E (cid:48) + i(cid:15) )gives the only pole. This fixes E (cid:48) = E , and we arrive at the final result, Eq. (C.2).Concerning the trace formula for elastic collisions, Eq. (34), we proceed in a similarway as in Eq. (A.10). We use the definition of ˆΩ ( V )+ ( E i ), Eq. (8), and the Lippmann- icroscopic scattering theory for interacting bosons in weak random potentials G V ( E ) as follows: (cid:90) ∞−∞ d E πi (cid:90) d k (2 π ) (cid:104) k i | (cid:16) ˆΩ ( V )+ ( E i ) (cid:17) † | k (cid:105)(cid:104) k | ˆ G V ( E )( . . . ) ( − E ) = (cid:90) ∞−∞ d E πi (cid:104) k i | (cid:104) ˆ G V ( E ) + ˆ V ˆ G † V ( E i ) ˆ G V ( E ) (cid:105) ( . . . ) ( − E ) = (cid:90) ∞−∞ d E πi E − E i + i(cid:15) (cid:104) k i | (cid:104) + ˆ V (cid:16) ˆ G V ( E ) + ˆ G † V ( E i ) − ˆ G V ( E ) (cid:17)(cid:105) ( . . . ) ( − E ) = (cid:104) k i | (cid:104) + ˆ V ˆ G † V ( E i ) (cid:105) ( . . . ) ( − E i ) = (cid:104) k i | (cid:16) ˆΩ ( V )+ ( E i ) (cid:17) † ( . . . ) − ( E i ) . (C.3)This proves Eq. (34). Appendix D. Particle and energy flux conservation
In this appendix, we prove Eqs. (51) and (52). Starting from Eq. (50) for f E ,E ,E , wecalculate (cid:82) ∞ d E √ E f E ,E ,E . For this purpose, we first note that: (cid:90) ∞ d E (cid:113) E (cid:32) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi (cid:33) | G E ( k ) | = (cid:90) ∞−∞ d E (cid:32) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi (cid:33) (cid:32) G ∗ E ( k ) − G E ( k )2 i/(cid:96) dis (cid:33) (cid:39) (cid:96) dis i (cid:32) E + E − k − k − iε − E + E − k − k + 2 iε (cid:33) (cid:39) (cid:96) dis i (cid:16)(cid:104) G (0 ,m/ E (( k − k ) / (cid:105) ∗ − G (0 ,m/ E (( k − k ) / (cid:17) , (D.1)with E = E + E − E k + k / k + k = k + k and | k (cid:105) as defined after Eq. (18).Here, we have first used the identity √ E | G E ( k ) | = (cid:96) dis [ G ∗ E ( k ) − G E ( k )] / (2 i ) forthe average Green’s function, then replaced the average Green’s functions by vacuumGreen’s functions (which is appropriate in the weak disorder limit), and evaluatedthe integral over E using residual calculus. By setting the imaginary part 2 ε in thedenominator to ε and using momentum conservation, i.e. ( k + k ) / → E k + k / − k k we arrived at Eq. (D.1). Again, G (0 ,m/ E ( k ) = 1 / ( E − k + i(cid:15) ) denotes the vacuumGreen’s function for a particle with mass m/
2, cf. Eq. (19).Inserting Eq. (D.1) into Eq. (50), and substituting the variable k → k =( k − k ) /
2, the integration over k reduces to:2 (cid:90) d k (2 π ) (cid:104) k | (cid:16) ˆ G † ,m/ ( E ) − ˆ G ,m/ ( E ) (cid:17) | k (cid:105) (cid:12)(cid:12)(cid:12) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) (cid:12)(cid:12)(cid:12) = 2 (cid:104) k | (cid:16) ˆ T (1) U ( E ) (cid:17) † (cid:16) ˆ G † ,m/ ( E ) − ˆ G ,m/ ( E ) (cid:17) ˆ T (1) U ( E ) | k (cid:105) . (D.2)Applying the optical theorem, Eq. (19), yields in total: (cid:90) ∞ d E (cid:113) E f E ,E ,E = − (4 π ) (cid:96) dis (cid:90) d k d k (2 π ) Im (cid:110) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) (cid:111) × | G E ( k ) | | G E ( k ) | . (D.3) icroscopic scattering theory for interacting bosons in weak random potentials r and r in Eqs. (40,46), see Eq. (49), togetherwith the formula √ E | G E ( k ) | = (cid:96) dis [ G ∗ E ( k ) − G E ( k )] / (2 i ), yields: − (cid:113) E g E ,E = − (4 π ) (cid:96) dis (cid:90) d k d k (2 π ) | G E ( k ) | Im (cid:110) (cid:104) k | ˆ T (1) U ( E ) | k (cid:105)× (cid:16) G ∗ E ( k ) − G E ( k ) (cid:17) G E ( k ) (cid:111) . (D.4)The term | G E ( k ) | in the second line of Eq. (D.4) exactly reproduces Eq. (D.3), whereasthe remaining term [ G E ( k )] gives a negligible contribution in the limit √ E (cid:96) dis (cid:29) (cid:82) ∞ d E √ E E f E ,E ,E , Eq. (D.1) is replaced by: (cid:90) ∞ d E (cid:113) E E (cid:32) G ∗ E + E − E ( k ) − G E + E − E ( k )2 πi (cid:33) | G E ( k ) | (cid:39) (cid:96) dis i (cid:16)(cid:104) G (0 ,m/ E ( k ) (cid:105) ∗ − G (0 ,m/ E ( k ) (cid:17) (cid:16) E + E + k − k (cid:17) . (D.5)When integrating over k as in Eq. (D.2), the term ( k − k ) vanishes due to symmetry,but the factor ( E + E ) remains. This proves Eq. (52). Appendix E. Collision terms for s -wave scattering For a short-range interaction potential U ( r ), the T -matrix (for a particle with mass m/
2) has the following form [49]: (cid:104) k (cid:48) | ˆ T (1) U ( E ) | k (cid:105) = 8 πa s − i (cid:115) E a s + O (cid:16) √ Ea s (cid:17) , (E.1)valid for arbitrary plane wave states | k (cid:105) , | k (cid:48) (cid:105) in the limit √ Ea s (cid:28)
1, where a s is the s -wave scattering length associated to the potential U ( r ). The prefactor 8 πa s ≡ πa s ¯ h /m for ¯ h / (2 m ) ≡ m/
2. Note that, due to thesymmetrization of the states | k (cid:105) and | k (cid:105) , an additional factor 2 appears whenevaluating (cid:104) k | ˆ T (1) U ( E ) | k (cid:105) in Eq. (18), cf. the remark at the end of AppendixB. Inserting Eq. (E.1) into the general expressions, Eqs. (40,41,63,64,65), yields thefollowing results for the collision terms (in the limit (cid:96) dis (cid:113) E , , (cid:29) g E ,E = − α ρ √ E E (cid:34)(cid:18)(cid:113) E + (cid:113) E (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:113) E − (cid:113) E (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) , (E.2) f E ,E ,E = αρ √ E E E min (cid:18)(cid:113) E , (cid:113) E , (cid:113) E , (cid:113) E + E − E (cid:19) , (E.3) g ( C ) E ,E = − ρ √ E Re (cid:20) − i(cid:96) dis (cid:18) √ E − (cid:113) (cid:101) E (cid:19)(cid:21) icroscopic scattering theory for interacting bosons in weak random potentials × iβ √ E i + α (cid:16) √ E + √ E (cid:17) − (cid:12)(cid:12)(cid:12) √ E − √ E (cid:12)(cid:12)(cid:12) √ E E , (E.4) h ( C ) E ,E = − iβ √ E i − α √ E i + 2 E d ρ (cid:20) − i(cid:96) dis (cid:18) √ E − (cid:113) (cid:101) E (cid:19)(cid:21) (cid:20) √ E + (cid:113) (cid:101) E − i(cid:96) dis ( E − (cid:101) E ) (cid:21) , (E.5) f ( C ) E ,E ,E = αρ √ E (cid:20) √ E + (cid:113) (cid:101) E − i(cid:96) dis ( E − (cid:101) E ) (cid:21) (cid:20) √ E + (cid:113) ˜ E − i(cid:96) dis ( E − (cid:101) E ) (cid:21) × (cid:88) s i ∈{ , } ( − s + s + s + s +1 (cid:32) | k s | + 2 ik s π ln | k s | (cid:33) , (E.6)where k s = ( − s (cid:113) E + ( − s (cid:20) s (cid:113) E + (1 − s ) (cid:113) (cid:101) E (cid:21) + ( − s (cid:20) s (cid:113) E + (1 − s ) (cid:113) (cid:101) E (cid:21) + ( − s (cid:113) E + E − E . (E.7)In the above expressions, Eqs. (E.2-E.7), all energies appearing under a square root mustbe positive – otherwise, the corresponding expression is set to zero (e.g. f E ,E ,E = 0 if E > E + E ). Furthermore, note that the density ρ appearing in the denominatorsof Eqs. (E.2-E.6) drops out when expressing the densities I E ( r ), C (1) E ( r ) and C (2) E ( r ) inEqs. (48,66,67) in units of the incoming density ρ , see also Eqs. (45,70). Therefore,the effective strength of the collision terms, Eqs. (E.2-E.6), is solely governed by theparameters α and β introduced in Eqs. (72,73). References [1] Cl´ement D, Var´on A F, Hugbart M, Retter J A, Bouyer P, Sanchez-Palencia L, Gangardt D M,Shlyapnikov G V, and Aspect A 2005
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