Maximally localized Wannier functions: Theory and applications

55
1995; Bradley et al., 1995; Davis et al., 1995) and the
award of Nobel prizes in 1997 and 2001. Since then, the
study of cold atoms trapped in optical lattices has flourished.
For a reviews see Bloch et al. (2008) and Morsch
and Oberthaler (2006).
Ultracold atoms trapped in optical lattices provide a
versatile alternative to electrons in crystal lattices for the
study of quantum phenomena. Indeed, they have a number
of advantages over the solid state in this respect, such
as the absence of lattice defects, the absence of a counterpart
to the electron-phonon interaction, and the ability
to control precisely both the nature of the inter-atomic
interactions and the depth and periodicity of the optical
lattice potential.
The second quantized Hamiltonian for a system of N
weakly interacting bosons of zero spin and mass m in a
(periodic) external potential V 0 (r) = V 0 (r + R) is given
by (Yukalov, 2009)
∫
Ĥ =
dr ˆΨ † (r)
[− ħ2
∫
+ g 2
]
2m ∇2 + V 0 (r) ˆΨ(r)
dr ˆΨ † (r) ˆΨ † (r) ˆΨ(r) ˆΨ(r), (153)
where g = 4πa s ħ 2 /m, it is assumed that the atoms
interact via a short-range pseudopotential with a s as
the s-wave scattering length, and ˆΨ(r) and ˆΨ † (r) are
bosonic field operators obeying canonical commutation
relations (Fetter and Walecka, 2003).
In a Bose-Einstein condensate, wherein the condensate
particle densities are typically of the order of 10 14 cm −3
or more, the mean-field limit of this Hamiltonian is usually
taken, which leads to the Gross-Pitaevskii equation,
also known as the non-linear Schrödinger equation
(NLSE),
iħ ∂ [
]
∂t φ(r, t) = − ħ2
2m + V 0(r) + g |φ(r, t)| 2 φ(r, t).
(154)
φ(r, t) is the condensate wavefunction, the squared norm
of which gives the condensate density. The Gross-
Pitaevskii equation has been used with remarkable success
in the study of BEC (Leggett, 2001).
As shown, for example, by Alfimov et al. (2002), a
basis of Wannier functions, localized to each site α of the
optical lattice, may be used to expand the condensate
wavefunction,
φ(x, t) = ∑ n,α
c nα (t)w nα (x). (155)
The Wannier functions are related to the Bloch eigenstates
ψ nk (x) of the eigenvalue equation
]
[− d2
dx 2 + V 0(x) ψ nk (x) = ϵ nk ψ nk (x) (156)
by the usual Wannier transformation
w nα (x) = L ∫
dk e iϕn(k) e −ikαL ψ nk (x). (157)
2π
BZ
Substituting Eq. (155) into Eq. (154), leads to a tightbinding
formulation known as the discrete non-linear
Schrödinger equation (DNLSE),
i d dt c nα = ∑ β
where
c nβ˜ϵ n,α−β + ∑ β,γ,η
˜ϵ nα = L 2π
∫
BZ
∑
i,j,k
c ∗ iβc jγ c kη U αβγη
nijk ,
(158)
dk e −ikαL ϵ nk , (159)
and the interaction matrix is given by
∫
= g dx w nα (x)w iβ (x)w jγ (x)w kη (x). (160)
U αβγη
nijk
Truncating the first term on the right-hand side of
Eq. (158) to nearest-neighbors only, and the second term
to on-site (α = β = γ = η) terms within a single band
(n = i = j = k) results in the usual tight-binding description
(Chiofalo et al., 2000; Trombettoni and Smerzi,
2001),
i d dt c nα = c nα˜ϵ n0 +˜ϵ n1 (c n,α−1 +c n,α+1 )+Unnnn αααα |c nα | 2 c nα .
(161)
As pointed out by Alfimov et al. (2002), using a WF
basis enables the range and type of interactions encapsulated
in the DNLSE to be systematically controlled and
improved. On the most part, however, WFs have been
used in the context of the NLSE in order to carry out
formal derivations and to justify the use of empirical or
semi-empirical tight-binding models.
An interesting analogy with electrons in atomic lattices
manifests itself when the filling of sites in the optical
lattice is low and hence particle correlations need to be
accounted for more rigorously. This is done via the Bose-
Hubbard model, developed by Fisher et al. (1989) in the
context of He-4, and first applied to cold atoms in optical
lattices by Jaksch et al. (1998). The Bose-Hubbard
Hamiltonian is derived from Eq. (153) by expanding the
boson field operator in terms of WFs of a single band,
localized at the lattice sites,
ˆΨ = ∑ α
ˆbα w α (r), (162)
where the bosonic particle creation and annihilation operators,
ˆb α and ˆb † α, respectively, satisfy the usual commutation
rules (Fetter and Walecka, 2003). This, on
approximation to nearest-neighbor coupling, and on-site
only interactions, results in the standard Bose-Hubbard
Hamiltonian (Jaksch et al., 1998)
Ĥ BH = −J
∑ ˆb† αˆbβ + U ∑
ˆn α (ˆn α − 1), (163)
2
α