Anderson Localization of Bogolyubov Quasiparticles in Interacting Bose-Einstein Condensates

Anderson Localization of Bogolyubov Quasiparticles in Interacting Bose-Einstein Condensates

Abstract

We study the Anderson localization of Bogolyubov quasiparticles in an interacting Bose-Einstein condensate (with healing length ) subjected to a random potential (with finite correlation length ). We derive analytically the Lyapunov exponent as a function of the quasiparticle momentum and we study the localization maximum . For 1D speckle potentials, we find that when while when , and that the localization is strongest when . Numerical calculations support our analysis and our estimates indicate that the localization of the Bogolyubov quasiparticles is accessible in experiments with ultracold atoms.

pacs:
05.30.Jp,03.75.Hh,64.60.Cn,79.60.Ht

An important issue in mesoscopic physics concerns the effects of disorder in systems where both quantum interference and particle-particle interactions play crucial roles. Multiple scattering of non-interacting quantum particles from a random potential leads to strong Anderson localization (AL) (1), characterized by an exponential decay of the quantum states over a typical distance, the localization length. AL occurs for arbitrarily weak disorder in 1D and 2D, and for strong-enough disorder in 3D (2). The problem is more involved in the presence of interactions. Strong disorder in repulsively interacting Bose gases induces novel insulating quantum states, such as the Bose (3) and Lifshits (4) glasses. For moderate disorder and interactions, the system forms a Bose-Einstein condensate (BEC) (5); (6); (4), where the disorder induces a depletion of the condensed and superfluid fractions (7) and the shift and damping of sound waves (8).

These studies have direct applications to experiments on liquid He in porous media (9), in particular as regards the understanding of the absence of superfluidity. Moreover, the realization of disordered gaseous BECs (10); (11); (12); (13); (14) has renewed the issue due to an unprecedented control of the experimental parameters. Using optical speckle fields (15), for instance, one can control the amplitude and design the correlation function of the random potential almost at will (14), opening possibilities for experimental studies of AL (16); (17). Earlier studies related to localization in the context of ultracold atoms include dynamical localization in -kicked rotors (18) and spatial diffusion of laser-cooled atoms in speckle potentials (19).

Transport processes in repulsively interacting BECs can exhibit AL (17); (20). However, for BECs at equilibrium, interaction-induced delocalizing effects dominate disorder-induced localization, except for very weak interactions (5); (6). The ground state of an interacting BEC at equilibrium is thus extended. Beyond, one may wonder how the many-body (collective) excitations of the BEC behave in weak disorder. In dilute BECs, these excitations correspond to quasiparticles (particle-hole pairs) described by the Bogolyubov theory (21). In this case, the interplay of interactions and disorder is subtle and strong arguments indicate that the Bogolyubov quasiparticles (BQP) experience a random potential screened by the BEC density (5). This problem has been addressed in the idealized case of uncorrelated disorder (random potentials with a delta correlation function) in Ref. (22).

In this Letter, we present a general quantitative treatment of the localization of the BQPs in an interacting BEC with healing length in a weak random potential with arbitrary correlation length . For weak disorder, we introduce a transformation that maps rigorously the many-body Bogolyubov equations onto the Schrödinger equation for a non-interacting particle in a screened random potential, which we derive analytically. We calculate the Lyapunov exponent (inverse localization length) as a function of the BQP wavenumber for 1D speckle potentials. For a given ratio , we determine the wavenumber for which is maximum: We find for , and for . The absolute maximum appears for , so that the finite-range correlations of the disorder need to be taken into account. Numerical calculations support our analysis. Finally, possibilities to observe the AL of BQPs in BECs placed in speckle potentials are discussed.

We consider a -dimensional Bose gas in a potential with weak repulsive short-range atom-atom interactions, characterized by the coupling constant . Its physics is governed by the many-body Hamiltonian

 ^H=∫dr{(ℏ2/2m)[(∇^θ)2^n+(∇√^n)2] +V(r)^n+(g/2)^n2−μ^n} (1)

where is the atomic mass, is the chemical potential, and and are the phase and density operators, which obey the commutation relation . According to the Bogolyubov-Popov theory (21); (23); (24), for small phase gradients () and small density fluctuations (, where and ), Hamiltonian (1) can be diagonalized up to second order as , where is the annihilation operator of the excitation (BQP) of energy . The many-body ground state of the Bose gas is a BEC with a uniform phase and a density governed by the Gross-Pitaevskii equation (GPE):

 μ=−ℏ2∇2(√nc)/2m√nc+V(r)+gnc(r). (2)

Expanding the density and phase in the basis of the excitations, and , the Hamiltonian reduces to the above diagonal form provided that obey the Bogolyubov-de Gennes equations (BdGE) (25):

 [−(ℏ2/2m)∇2+V+1gn%c−μ]f+ν=ϵνf−ν (3) [−(ℏ2/2m)∇2+V+3gnc−μ]f−ν=ϵνf+ν, (4)

with the normalization . Equations (2)-(4) form a complete set to calculate the ground state (BEC) and excitations (BQPs) of the Bose gas, from which one can compute all properties of finite temperature or time-dependent BECs.

Here, we analyze the properties of the BQPs in the presence of weak disorder. According to Eqs. (3) and (4), they are determined by the interplay of the disorder and the BEC density background . Let be a weak random potential (, see below) with a vanishing average and a finite-range correlation function, , where is the standard deviation, and the correlation length of . As shown in Refs. (4); (5); (6), the BEC density profile is extended for strong-enough repulsive interactions (i.e. for , where is the healing length and the size of the BEC). More precisely, up to first order in , the GPE (2) yields

 nc(r)=[μ−˜V(r)]/g (5)

where and , the Green function of the linearized GPE (6); (4), reads in Fourier space (26). Then,

 ˜V(q)=V(q)/[1+(|q|ξ)2]. (6)

Thus, is a threshold in the response of the density to the potential , as for , while for . The potential is a smoothed potential (6). If is a homogeneous random potential, so is , and according to Eq. (5), the BEC density profile is random but extended (6); (4).

Solving the BdGEs (3) and (4) is difficult in general because they are strongly coupled. Yet, we show that for a weak (possibly random) potential this hurdle can be overcome by using appropriate linear combinations of the functions, namely , with and . For , the equations for are uncoupled [see Eqs. (7) and (8)] and we recover the usual plane-wave solutions of wave vector and energy (21). For weak but finite , inserting Eq. (5) into Eqs. (3) and (4), we find

 ℏ2k22mg+k = −ℏ22m∇2g+k−2ρk˜V1+ρ2kg−k (7) Missing or unrecognized delimiter for \right −ρ2kℏ2k22mg−k = −ℏ22m∇2g−k−2ρk˜V1+ρ2kg+k (8) +[V−1+3ρ2k1+ρ2k˜V]g−k.

Equations (7) and (8), which are coupled at most by a term of the order of [since and ], allow for perturbative approaches. Note that the functions and have very different behaviors owing to the signs in the left-hand-side terms in Eqs. (7) and (8). Equation (8) can be solved to the lowest order in with a Green kernel defined in Ref. (6): we find , where . Equation (7) cannot be solved using the same method because the perturbation series diverges. Nevertheless, from the solution for , we find that . The coupling term in Eq. (7) can thus be neglected to first order in and we are left with the closed equation

 −(ℏ2/2m)∇2g+k+Vk(r)g+k≃(ℏ2k2/2m)g+k, (9)

where

 Vk(r)=V(r)−1+4(kξ)21+2(kξ)2˜V(r). (10)

Equation (9) is formally equivalent to a Schrödinger equation for non-interacting bare particles with energy , in a random potential . This mapping allows us to find the localization properties of the BQPs using standard methods for bare particles in 1D, 2D or 3D (27). However, since depends on the wave vector itself, the localization of the BQPs is dramatically different from that of bare particles as discussed below.

In the remainder of the Letter, we restrict ourselves to the 1D case, for simplicity, but also because AL is expected to be stronger in lower dimensions (2). The Lyapunov exponent is a self-averaging quantity in infinite 1D systems, which can be computed in the Born approximation using the phase formalism (27) (see also Ref. (17)). We get , where is the Fourier transform of the correlation function of , provided that (16); (17); (27). Since , the component of relevant for the calculation of is . From Eqs. (6) and (10), we find

 Vk(2k)=S(kξ)V(2k);\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak S(kξ)=2(kξ)21+2(kξ)2, (11)

and the Lyapunov exponent of the BQP reads

 Γk=[S(kξ)]2γk (12)

where is the Lyapunov exponent for a bare particle with the same wavenumber (27); (17).

Let us summarize the validity conditions of the perturbative approach presented here. It requires: (i) the smoothing solution (5) to be valid (i.e. ); (ii) the coupling term proportional to in Eq. (7) to be negligible (which is valid if ); and (iii) the phase formalism to be applicable. The latter requires , i.e. , which is valid for any if .

Applying Eq. (12) to uncorrelated potentials [ with , and ], one recovers the formula for found in Ref. (22). Our approach generalizes this result to potentials with finite-range correlations, which proves useful since uncorrelated random potentials are usually crude approximations of realistic disorder, for which can be significantly large. We show below that if , as e.g. in the experiments of Refs. (10); (11); (12); (13); (14), the behavior of versus is dramatically affected by the finite-range correlations of the disorder.

Let us discuss the physical content of Eqs. (9) and (10). According to Eqs. (3) and (4), the properties of the BQPs are determined by both the bare random potential and the BEC density in a non-trivial way. Equation (9) makes their roles more transparent. As the occurrence of the smoothed potential in Eq. (10) is reminiscent of the presence of the meanfield interaction in the BdGEs (3) and (4), it appears that the random potential results from the screening of the random potential by the BEC density background (5). More precisely, the expression (11) for the Fourier component , relevant for the Lyapunov exponent of a BQP, shows that the screening strength depends on the wavenumber . In the free-particle regime (), we find that the Lyapunov exponent of a BQP equals that of a bare particle with the same wavenumber (), as expected. In the phonon regime (), the disorder is strongly screened and we find , as in models of elastic media (28). Here, the localization of a BQP is strongly suppressed by the repulsive atom-atom interactions, as compared to a bare particle in the same bare potential. These findings agree with and generalize the results obtained from the transfer matrix method, which applies to potentials made of a 1D random series of -scatterers (22).

Our approach applies to any weak random potential with a finite correlation length. We now further examine the case of 1D speckle potentials used in quantum gases (10); (11); (12); (13); (14). Inserting the corresponding reduced correlation function, where is the Heaviside function (17), into Eq. (12), we find

 Γk=π8(V\tiny Rμ)2σ\tiny Rk2(1−kσ\tiny R)[1+2(kξ)2]2Θ(1−kσ\tiny R) (13)

which is plotted in Fig. 1. To test our general approach on the basis of this example, we have performed numerical calculations using a direct integration of the BdGEs (3) and (4) in a finite but large box of size . The Lyapunov exponents are extracted from the asymptotic behavior of , where is the localization center and is the envelope of the function , obtained numerically. The numerical data, averaged over 40 realizations of the disorder, are in excellent agreement with formula (13) as shown in Fig. 2. These results validate our approach. It should be noted, however, that our numerical calculations return BQP wave functions that can be strongly localized for very small momenta . This will be discussed in more details in a future publication (29).

Of special interest are the maxima of , which denote a maximum localization of the BQPs. It is straightforward to show that, for a fixed set of parameters , is non-monotonic and has a single maximum, , in the range (see Fig. 2). This contrasts with the case of bare particles, for which the Lyapunov exponent decreases monotonically as a function of , provided that decreases versus (which is valid for a broad class of random potentials (27)). The existence of a localization maximum with respect to the wavenumber is thus specific to the BQPs and results from the strong screening of the disorder in the phonon regime. In general, the value of , plotted in the inset of Fig. 2 versus the correlation length of the disorder, depends on both and . For , we find , so that the localization is maximum near the crossover between the phonon and the free-particle regimes as for uncorrelated potentials (22). For however, we find , so that is no longer determined by the healing length but rather by the correlation length of the disorder, and lies deep in the phonon regime. For , vanishes. This defines an effective mobility edge due to long-range correlations in speckle potentials, as for bare particles (17); (30).

Finally, let us determine the absolute localization maximum. The Lyapunov exponent decreases monotonically versus and . However, for fixed values of and , has a maximum at and (see Fig. 1) and we find the corresponding localization length ():

 Lmax(ξ)=(512√6/9π)(μ/V\tiny R)2ξ. (14)

At the localization maximum, we have , so that the disorder cannot be modeled by an uncorrelated potential, and the long-range correlations must be accounted for as in our approach. For m (14) and , we find m, which can be smaller than the system size in disordered, ultracold gases (11); (14); (31).

In conclusion, we have presented a general treatment for the AL of BQPs in an interacting BEC subjected to a random potential with finite-range correlations. We have calculated the Lyapunov exponents for a 1D speckle potential and we have shown that the localization is strongest when . We have found that the localization length can be smaller than the size of the BEC for experimentally accessible parameters. We expect that the AL of BQPs could be observed directly, for instance as a broadening of the resonance lines in Bragg spectroscopy, a well mastered technique in gaseous BECs (32).

We thank M. Lewenstein, G. Shlyapnikov, S. Stringari, and W. Zwerger for stimulating discussions during the Workshop on Quantum Gases at the Institut Henri Poincaré - Centre Emile Borel. This work was supported by the French DGA, MENRT and ANR, and the ESF QUDEDIS program. The Atom Optics group at LCFIO is a member of the Institut Francilien de Recherche sur les Atomes Froids (IFRAF).

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