Anderson localization in optical lattices with speckle disorder
We study the localization properties of non-interacting waves propagating in a speckle-like potential superposed on a one-dimensional lattice. Using a decimation/renormalization procedure, we estimate the localization length for a tight-binding Hamiltonian where site-energies are square-sinc-correlated random variables. By decreasing the width of the correlation function, the disorder patterns approaches a -correlated disorder, and the localization length becomes almost energy-independent in the strong disorder limit. We show that this regime can be reached for a size of the speckle grains of the order of (lower than) four lattice steps.
The Anderson model was proposed first to explain the absence of electronic diffusion in certain random lattices Anderson (1958) and later to explain the absence of diffusion of light in certain amorphous materials Anderson (1985). In the first case, the electron energy is lower than the maxima of the lattice potential and the particles diffuse by tunnelling. In the second case the particles (photons in Anderson (1985)) energy is higher than the potential and the particles are “free”. In both cases, in the absence of disorder, the eigenstates are delocalized states (Bloch waves in the first regime, plane waves in the second). The presence of a -correlated random potential freezes the wave propagation at a length of , the localization length, always in one (1D) and two dimension (2D), and depending on the disorder strength and the energy wave also in three dimension (3D) Abrahams et al. (1979). In the last few years the ultracold atom community has devoted a large effort to the experimental realization of Anderson localization. Anderson localization in the tight-binding regime was observed in momentum space with kicked-rotor set-ups, in 1D Moore et al. (1995), and 3D Chabé et al. (2008), and in real space by using a quasi-periodic potential (thus not strictly-speaking a random potential). The key-ingredient for the experimental study of Anderson localization of ultracold atoms in the “free”-particle regime has been the speckle potential Bouyer (2010). By using this handable optical potential, Anderson localization was observed in 1D Billy et al. (2008) and in 3D Kondov et al. (2011); Jendrzejewski et al. (2011), and anomalous diffusion was observed in 2D Robert-de Saint-Vincent et al. (2010). The auto-correlation function of the speckle potential is a square-sinc, thus it decays algebraically as a long-range correlated disorder, but is also characterized by a finite size correlation length , corresponding to the width of central bump of the square-sinc function.
In 1D, the presence of disorder patterns with auto-correlation functions decaying algebraically can mimic the presence of a mobility edge Tessieri (2002); Kuhl et al. (2008); Gurevich and Kenneth (2009); Lugan et al. (2009) and can even enhance localization, as shown in a microwave experiment Kuhl et al. (2008). Both phenomena are due to the fact that the disorder spectrum is non-zero in a finite momentum interval.
Very recently Semmler and coworkers Semmler et al. (2010) have studied the phase diagram of correlated fermions in 2D and 3D optical lattices and in the presence of a speckle potential. From the analysis of the local Density Of States (DOS) they identify an Anderson-Mott and a Mott localized phase as functions of the interaction strength and the strength of the speckle potential. In this article we analyze the possibility of observing Anderson localization of a non-interacting wave, for example a non-interacting Bose-Einstein condensate Roati et al. (2008), in a speckle potential superposed to a 1D lattice potential. By using a decimation/renormalization scheme Farchioni et al. (1992) we analyze how the DOS of a lattice is modified by the presence of the speckle, and we estimate the localization length as function of the disorder strength and of the width of the auto-correlation function. The speckle potential is introduced as an on-site disorder which has statistical properties which are the same as a genuine speckle potential. This is illustrated in Sec. II. In Sec. III we remind the reader of the decimation/renormalization procedure exploited to compute the DOS and . Our results show how the efficacy of the speckle potential to localize increases by increasing the disorder strength and by decreasing the correlation length . These results can be a guide to choosing the experimental parameters to observe Anderson localization in the tight binding regime with speckle disordered patterns.
Ii The model
To study the effect of a speckle potential in the presence of a 1D lattice on matter-wave transport we use the 1D Tight-Binding (TB) Hamiltonian,
where is the number of sites, the energy at the site . The hopping term is chosen site-independent. The effect of the speckle potential is introduced in the on-site energy distribution by setting
where is the fluctuation of with respect to the mean value , is the disorder strength and the width, in the units of the lattice step , of the correlation function in Eq. (2). The disorder spectrum is not uniform as in the Anderson model Anderson (1958), but is described by the triangular function Billy et al. (2008)
where , and is the Heaviside function.
ii.1 Generation of the disordered potential
We use the Fourier Filtering Method (FFM) Peng et al. (1991); Prakash et al. (1992); Makse et al. (1996) to generate the disorder pattern described by the correlation function (2). First we generate a sequence of -correlated random numbers , with from a uniform distribution centered in zero and of width 1. The second step is the generation of the desired distribution by “filtering”, in Fourier space, the uniform distribution . The filter being the spectral function , the ’s are evaluated directly from the expression
where et . By construction , namely , and verifies Eq. (2) in the limit . A different choice of would just shift the zero of the energy.
Iii Numerical results: the DOS and the localization length
In the continuous limit, the single-particle DOS for an optical speckle has been studied in Falco et al. (2010). In the presence of a lattice (and in the absence of the speckle potential), the low-energy single-particle DOS has a typical saddle shape with two horns that correspond respectively to the center and the edge of the first Brillouin zone. To evaluate how the speckle potential modifies the DOS of the lattice, we compute the DOS, , regarding the Hamiltonian (1) by using the Kirkman-Pendry relation Kirkman and Pendry (1984)
Here is the Green’s function related to the Hamiltonian at energy , and . With the aim of computing the matrix element , we reduce the dimensionality of the system by evaluating the effective Hamiltonian
where , and are functions of the energy and of the Hamiltonian elements of the decimated states (2, 3, …) Farchioni et al. (1992); Vignolo et al. (2003). The Green’s function of the effective Hamiltonian (6), , coincides with in the subspace by construction.
The numerical results for the DOS as a function of the energy in units of are shown in the first column of Fig. 1. One can observe that for large values of , the speckle disorder mainly affects the edge states of the DOS of the underlying perfect chain, while for smaller values of the disorder mainly influences the central part of the spectrum.
The presence of the disorder modifies not only the DOS, but also the nature of the states, from extended to localized. In the continuous limit, the presence of the correlations described by Eq. (2) does not destroy localization but deeply modifies the behaviour of the localization length as a function of the energy Tessieri (2002); Gurevich and Kenneth (2009); Lugan et al. (2009). To study the behaviour of the localization length in the tight-binding regime, we compute the Lyapunov coefficient , through the asymptotic relation
The results shown in the second column of Fig. 1 have been computed for the case , but we have checked that the values obtained do not change significantly by increasing the value of up to . Analogously to the continuous case, we observe that all states are localized. In the limit of weak disorder, the localization length at the center of the spectrum, , is quite large, of the order of 50 lattice sites. By increasing the strength of the disorder, decreases significantly only for small values of the correlation length ( and ), and becomes almost energy-independent in the whole band. Longer-range correlations ( and ) act instead more efficiently on the edge states. To better understand these reasults we can refer to the continuous case, where
in the Born approximation (see for instance Kuhl et al. (2008)). From Eq. (8) we can expect to observe (i) a decrease of the localization length for large values of in the limit , since , and (ii) an increase in the localization length for , namely where the Born approximation is no longer valid Gurevich and Kenneth (2009); Lugan et al. (2009). Since in the TB case there is a symmetry in the DOS due to the presence of the underlying lattice and correlations act symmetrically with respect to the center of the spectrum, the observation (ii) leads to the conclusion that, if one wants to experimentally observe Anderson localization in the whole low-energy band, should be greater than , thus should be lower than lattice steps. This finding, deduced from propagation in the continuous space, is in good agreement with the numerical results for the TB case.
In this article we have studied the effetcs of a speckle potential on the spectrum of a quantum particle (or a non-interacting wave) in a lattice potential. At fixed, large disorder strength (), the localization efficacy of the speckle potential depends strongly on the width of the auto-correlation function . Large values of enhance localization at very low energies and at the edge of the Brillouin zone. Shorter-range correlations ( lattice sites) act more efficiently on the center of the spectrum. More generally, our results show that a speckle superposed to an optical lattice is a suitable potential to study Anderson localization in the tight-binding regime: analogously to the continuous case, speckle correlations deeply modify the behaviour of the localization length as a function of the energy, but do not induce an insulator-metal transition.
This work was supported by the CNRS and the TUBITAK (exchange of researchers, grant No. 24543). P.V. aknowledges G. Modugno for fruitful discussions, and warmly thanks the Condensed Matter group of the Trakya University for the kind hospitality.
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