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Bosonic hard spheres in quasi-one dimensional bichromatic optical lattices

We calculated the phase diagram of a continuous system of hard spheres loaded in a quasi-one dimensional bichromatic optical lattice. The wavelengths of both lattice-defining lasers were chosen to model an incommensurate arrangement. Densities of one particle and half a particle per potential well were considered. Our results can be compared directly to those of the experimental system [Fallani et al. PRL, {\bf 98} 130404 (2007)] from which our initial parameters were taken. The phase diagrams for both densities are significatively different to those obtained by describing the same experimental setup with a Bose-Hubbard model.


INTRODUCTION
An optical lattice is the result of the interference of one or several pairs of laser beams to produce a stationary wave, felt by a neutral atom as an external potential of the form [1,2]: This expression, that depends on the laser wavelengths (λ x , λ y , λ z ), defines a three dimensional lattice. V x , V y ,V z are the depths of the potential barriers felt by the atoms and can be modified by changing the light intensity. In a very common arrangement, two of those depths are fixed to values high enough to produce a tight confinement in two directions. We have then quasi-one dimensional systems well described by the continuous Hamiltonian: Here, V (r ij ) represents the interaction between the atoms i and j, separated a distance r ij from each other, and m is the mass of those atoms. ω ⊥ is the harmonic frequency we obtain at developing the first two terms of the right-hand side of Eq. (1) around x = 0 and y = 0, a good approximation when V x and V y are deep enough. When the interaction between particles is modeled by a hard spheres one, a system described by this Hamiltonian at a density of one particle per potential well exhibits superfluid-Mott insulator transitions [3] for values of V z that depend on ω ⊥ [4]. Eq. (2) does not consider the system to be harmonically trapped in the longitudinal direction, z. That would mean to include an additional term of the form 1 2 mω 2 z z 2 i in the Hamiltonian [19]. Disorder can be introduced on this simple arrangement. Two are the most common experimental routes: the use of two superimposed lasers of different (and incommensurate) wavelengths on the z direction [5][6][7][8], or the introduction of a speckle field on the optical lattice [9,10]. In both cases, we end up having potential wells of unequal depths, with a quasi-periodic variation in the first case and a more randomly distribution of potential minima in the second. What the phase diagrams of those arrangements have in common is the presence of at least a superfluid (SF) and a Bose glass (BG) phase, depending on the particular experimental setup. If the average number of atoms per potential well is an integer, a Mott insulator (MI) can be also stable for some values of the lattice defining parameters [11].
In this work, we considered the first disorder option, and solved the Schrödinger equation corresponding to the continuous Hamiltonian in Eq. (2) with an optical lattice potential of the form: where λ 1 and λ 2 are the two wavelengths that define the bichromatic lattice. Here, E R 1 and E R 2 are the recoil energies corresponding to λ 1 and λ 2 (E Rx = h 2 /(2mλ 2 x )). Since we chose to reproduce as closely as possible the experimental setup used in Ref. 5, we took λ 1 = 830 nm, s ⊥ = 40 (in units of E R 1 ), and m as the mass of a 87 Rb atom. There is a difference, tough. We considered λ 2 = 1067 nm instead of the 1076 nm originally used. This makes the ratio λ 1 /λ 2 = 7/9 ∼ 0.7778 instead of the 0.77137 of the experimental data. s 1 and s 2 where taken to be in the same range as the experimental data (s 1 = 16; 0 ≤ s 2 ≤ 2.5, in units of their respective recoil energies). To use rational approximations to the real wavelength ratio is a common practice (see for instance Ref. 12, in which λ 1 /λ 2 = 5/7, or Ref. 13 in which λ 1 /λ 2 = 97/126), and allowed us to use relatively small simulation cells.
The interatomic potential was described by a hard-spheres (HS) interaction: V (r ij ) = +∞ for r ij < a and 0 otherwise, a being fixed to a single value, that of the scattering length of the 87 Rb atoms [14]. This potential was previously used to describe bosons loaded in optical lattices [4,[15][16][17][18][19].
In most previous literature, the continuous Hamiltonian of Eq.(2) is approximated by the discrete Bose-Hubbard (BH) model [20]: in which all the particles are allowed to stay only at the minima of their corresponding optical lattice potentials (i sites). Besides, only interactions of the type < ij > were considered, j being one of the two nearest neighbors (a quasi-one dimensional system translates into a pure one-dimensional Hamiltonian) to any particular i. b + i (b i ) is the bosonic creation (annihilation) operator at i, and n i represents the number of neutral atoms at that same location. J, U and ǫ i are parameters that can be related meaningfully to the experimental ones (V z , λ, and ω ⊥ ) only when the optical lattice potentials are deep enough [2,20]. For the kind of Hamiltonians we are dealing with in this work [2]: and J is the hopping matrix element between sites i and j, and U models the repulsive interaction between two atoms located at the same i site. To deduce the expression for U, it was supposed that the full interaction potential between atoms in the original continuous Hamiltonian was adequately described by a pseudopotential [2]. Eqs. (5) and (6) were deduced for monochromatic lattices, i.e., lattices defined by a single laser of wavelength λ, and recoil energy E R . If another laser is introduced, both expressions would have to be recalculated to include the effect of this second wavelength [21]. However, this is usually not done [12,13,[21][22][23][24][25], even though a variation in J could produce an appreciable change in the phase diagram. [26] The remaining parameter in Eq.(4), ∆ i , represents the energy offset at each potential well, and for a monochromatic lattice with no longitudinal confinement is the same for each site.
In this work, we calculate the phase diagram of the continuous Hamiltonian defined by Eq. (2) with the optical lattice potential of Eq. (3). This implies that the disorder arises naturally at the introduction of the second laser, with no ad hoc change in the depths of the potential wells. The only approximation was to use a hard spheres interparticle interaction, a very common approach [4, 15-17, 19, 19]. We considered two densities, one particle per optical lattice minimum, and half a particle per minimum, and compared our results to those calculated with a Bose-Hubbard model for the same systems [21]. However, our primary goal will be to compare our results to the experimental ones, when available. Since the lattice defined by λ 1 is in most cases deeper than the defined by λ 2 (s 1 E R 1 > s 2 E R 2 ), to define the average density we used the number of potential minima corresponding to the lattice defined by λ 1 . This convention will not affect the limits of the phase diagram, since the system is always a superfluid, as we will see below.

METHOD
The initial step to calculate the phase diagrams was to obtain the ground state of a set of atoms described by the Hamiltonian of Eq. (2). This means that we are operating a T=0, a reasonable approximation commonly used [12,13,21] to describe the very low temperature regimes in which the experiments are performed [5,8]. In any case, we expect our description should be valid for temperatures lower than the corresponding to the excitations of the system, which are in the range of the kHz [5]. To do so, we used the diffusion Monte Carlo (DMC) method [27]. That technique needs an initial approximation to the ground state wavefunction to work, the so-called trial function. That initial expression depends on the positions of the N atoms in the simulation cell (r 1 ,· · · ,r N ). In this work, we used: where x i , y i , z i are the coordinates of the atom i and r lm is the distance between atoms l and m. We took ψ(x i , y i ) to be a Gaussian of variance σ 2 ⊥ = /(mω ⊥ ), i.e., the exact solution to the harmonic potential that traps the particles transversally. φ(z i ) was taken to be the numerical solution to an one-dimensional Schrödinger equation including only the monochromatic lattice defined by λ 1 . The two body correlations were modeled by: [18,19,28]: a is here the scattering length of the pair of atoms. The particular way to obtain the five constants in this equation has been detailed elsewhere [18,19].
As indicated above, we chose the s 1 and s 2 parameters in the same range that the experimental ones of Ref. 5. The main difference is that instead of limiting ourselves to the 5 for a density of one atom per potential well and 0 ≤ s 2 /E R 2 ≤ 2 when the average atom occupation was 0.5. In most cases, a simulation cell including 36 potential minima (= 18 λ 1 ) was used. We checked that to increase that number to 54, 72 or even 90 did not vary the results presented in this work.

One particle per potential well
A Bose glass is usually defined as an insulating (non-superfluid) and finite-compressibility (gapless) phase [11,23]. On the other hand, a Mott insulator is a non-superfluid gaped state.
So, to determine whether a particular arrangement of atoms can be described as a glass, we should calculate the superfluid fraction (to know if we are dealing with an insulator) and the compressibility of the system (to see if we have a MI or a BG). We define the compressibility, κ, as: where µ is the chemical potential, Here, E is the total energy of the hard spheres arrangement loaded in the quasi-one dimensional optical lattice and n is the density of the system (n = N/L, N, number of atoms; L, length of the simulation cell). This means that a finite κ at a given n is associated with a continuous slope of the energy per atom versus the density at that n.
In Fig. 1 Fig. 2 were obtained in the same way that the already described for s 1 /E R 1 = 16 and indicate the upper limit of stability of the Mott insulator phase for each s 1 /E R 1 value.
We calculated the superfluid fraction (ρ s /ρ) using a variant of the winding number tech- nique [29,30]. The formal expression employed was: where D 0 = 2 /2m. τ indexes each Monte Carlo step of a given simulation, from the initial step after the Monte Carlo equilibration (τ = 0) to the end of the Markov chain (in our case, τ = 20000). D(τ ) is defined as: Here, r CM (τ ) is the position of the center of mass of all the particles in the simulation cell at a step labeled by τ . N is, as before, the number of particles. This definition means that if we represent D(τ )/D 0 as a function of τ , for large enough τ , we should get a straight line whose slope will give us the value of the superfluid fraction. An example of that procedure is displayed in Fig. 3. Each curve represents the average of 50 independent DMC simulations.
There, s 2 /E R 2 = 1. This allows us to obtain a range of most probable slopes, and from that, a superfluid fraction interval. What we observe is that, except for the s 1 /E R 1 = 5 case, there is a range of s 2 /E R 2 values above which the system is an insulator, and below which we have a superfluid. We consider an arrangement to be non-superfluid when the error bar for the superfluid fraction includes the value ρ s /ρ = 0. So, for instance, when s 1 /E R 1 = 12 (full squares in Fig. 4), the last insulator corresponds to s 2 /E R 2 = 1.6, since in that case ρ s /ρ= 0.01 ± 0.01. This translates into the point s 1 /E R 1 = 12, s 2 /E R 2 = 1.6 in Fig. 2. For larger s 2 /E R 2 's, we have a superfluid, and when 1.3 < s 2 /E R 2 < 1.6, the phase is a gapless insulator. i.e. a Bose glass. The remaining of the full symbols in Fig. 2 have been obtained in a similar way for other s 1 /E R 1 values. The error bars displayed there have the same origin as the ones for the MI-BG limit. The full symbols in Fig. 2 are related to the superfluid behavior of the system (see below).

Half a particle per potential well
Exactly the same calculations can be performed for an average density of half a particle per potential well. The only difference is that in the phase diagram we will have only superfluid and Bose glass phases, since a MI is only possible for a filling fraction of one. On the other hand, for s 1 /E R 1 = 10, an insulator (Bose glass) appears for s 2 /E R 2 ≥ 0.9.
This last value is represented in Fig. 6 as the leftmost full symbol. Additional calculations indicate that for this density no Bose glass appears when s 1 /E R 1 < 8, at least for s 2 /E R 2 's in the range displayed in Fig. 6.  < s 2 /E R 2 <1.5) [5].
However, the standard approximation to describe an optical lattice is the use of a Bose-Hubbard model. In a classic case of putting the cart before the horse, the fact that a set of atoms loaded in an monochromatic optical lattice is a very good experimental realization of a BH Hamiltonian for some values of the lattice defining parameters [3,20], has been used as an excuse to describe any optical lattice in any circumstances by this model. The main advantage of this approach is that the BH model is relatively easy to use and its behavior is well understood; the main problem being that it is only an approximation that breaks down even for monochromatic lattices with shallow potential wells, as it has been established both experimentally [31] and theoretically [4,18] for the description of the superfluid-Mott insulator transition.
In addition, the study of a bichromatic lattice implies corrections for both the J and U parameters that would include the effect of the second standing wave and that there are usually not considered. This means that even in those cases in which the primary lattice is deep enough to be well described by a BH Hamiltonian, the description of the lattice is not exact. That compounded error would be the cause of the differences between the phase diagram for half a particle per potential well calculated with a continous and a BH Hamiltonian shown in Fig. 6. Unfortunately, to our knowledge, there are no experimental data to compare those diagrams to. We can only say that the stability zone of the superfluid phase is reduced with respect to the case of a continuous Hamiltonian, the same general trend found for monochromatic lattices [4].
The situation is similar for a density of one particle per potential well. As indicated above, for s 1 /E R 1 = 16, there is a reasonable agreement between the position of the MI-BG transition predicted from our calculations and the experiment. Regretably, we cannot compare any BH predictions to that value, since s 1 = 16 translates (using Eqs. (5) and (6), with the scattering length taken from Ref. [14]) into U/J ∼ 130, a larger value than the ones considered in the literature [12,21,22]. On the other hand, the upper stability limit for the MI phase in Ref. 21 for U/J = 10 (larger U/J used) corresponds to s 2 E R 2 /J ∼ 10.
This means s 1 /E R 1 ∼ 6.1 and s 2 /E R 2 ∼ 1.1, to be compared to s 1 /E R 1 = 6, s 2 /E R 2 = 0.6 ± 0.2, from Fig. 6. Thus, we have that a BH description produces a shrinking of the SF stability zone with respect to the case of a continuous model, as in the case of the lower density described above. This is corroborated by the onset of the SF-MI transition for s 2