Bose-Hubbard model describes the hopping of bosonic particles in the presence of (repulsive, and usually on-site) interaction. Depending on the ratio between the tunneling term $\backslash ,\; t$ and the interaction term $\backslash ,\; U$, the macroscopic properties of the system may change from superfluid to Mott insulator.
The Bose-Hubbard model is described by the following Hamiltonian:
:$H\_\{BH\}=-\backslash frac\{t\}\{2\}\backslash sum\backslash limits\_\{\backslash left\backslash langle\; i,j\; \backslash right\backslash rangle\; \}\{\backslash left(\; a\_\{i\}^\{\backslash dagger\}a\_\{j\}+a\_\{j\}^\{\backslash dagger\}a\_\{i\}\; \backslash right)\}+\backslash frac\{U\}\{2\}\backslash sum\backslash limits\_\{i\}\{n\_\{i\}\backslash left(\; n\_\{i\}-1\; \backslash right)\}$,
where $\backslash left\backslash langle\; i,j\; \backslash right\backslash rangle$ denotes the sum over nearest-neighbor sites, and $n\_\{i\}\backslash equiv\; a\_\{i\}^\{\backslash dagger\}a\_\{i\}$ with bosonic commutation relations, e.g. $\backslash left[\; a\_\{i\},a\_\{j\}^\{\backslash dagger\}\; \backslash right]=\backslash delta\; \_\{ij\}$.
== Physical Realization of the Bose-Hubbard model ==
The Bose-Hubbard model is nowadays most commonly realized by neutral atomic gases trapped in a optical lattice.
=== Atom trapped in a Optical lattice ===
We shall consider a neutral atom interacting with a spatially varying laser field. Let us for the moment assume the frequency of the laser field is comparable but detuned from the transition between an electronic excited state and the ground state. Then, from second-order perturbation theory, the time-average energy shift $\backslash ,\; \backslash Delta\; E$ is proportional to the field intensity $\backslash ,\; I$, but inversely proportional to the detuning $\backslash Delta\; \backslash equiv\; \backslash omega\; \_\{laser\}-\backslash omega\; \_\{eg\}$,
:$\backslash Delta\; E\backslash propto\; \backslash frac\{I\}\{\backslash Delta\; \}$.
Therefore, an atom will see an effective potential which is proportional to the intensity $\backslash ,\; I$ and is attractive when $\backslash ,\; \backslash Delta\; <0$, repulsive when $\backslash ,\; \backslash Delta\; >0$.
A regular "lattice" can be constructed by the interference of pairs of (counter-propagating) polarized light beams. For example, $\backslash vec\{E\}\_\{1\}=\backslash vec\{E\}\_\{0\}\backslash cos\; \backslash left(\; \backslash vec\{k\}\backslash cdot\; \backslash vec\{r\}+\backslash omega\; t\; \backslash right)$ and $\backslash vec\{E\}\_\{2\}=\backslash vec\{E\}\_\{0\}\backslash cos\; \backslash left(\; -\backslash vec\{k\}\backslash cdot\; \backslash vec\{r\}+\backslash omega\; t\; \backslash right)$. The total electric field is simply $\backslash vec\{E\}\_\{total\}=\backslash vec\{E\}\_\{1\}+\backslash vec\{E\}\_\{2\}=2\backslash vec\{E\}\_\{0\}\backslash cos\; \backslash left(\; \backslash omega\; t\; \backslash right)\backslash cos\; \backslash left(\; \backslash vec\{k\}\backslash cdot\; \backslash vec\{r\}\; \backslash right)$. We therefore have a spatially varying potential of the form (apart from a constant and consider 1D):
:$V\backslash left(\; x\; \backslash right)=V\_\{0\}\backslash cos\; \backslash left(\; 2kx\; \backslash right)$,
where $V\_\{0\}\backslash propto\; I/\backslash Delta$ is a controllable constant. It is straight forward to generalize the analysis into 3D by including two extra pairs of counter propagating beams. Here we note that in real experiments, this potential is superposed by a much slowly varying trapping potential to avoid loss of atoms.
=== Tight-binding approximation ===
In principle the width (wavelength of the laser beams) of the wells, and the height (light intensity) of the wells are adjustable. The range of interest for constructing the Bose-Hubbard Hamiltonian is the range where the wavefunction of the system may be described by the localized ground-state wavefunction for each well, and the correction due to the overlap of the wavefunctions nearby. This regime is also known as the tight-binding approximation.
Under this approximation, the hopping (non-interacting) part of the Hamiltonian can be described by
:$H\_\{free\}=-\backslash frac\{t\}\{2\}\backslash sum\backslash limits\_\{\backslash left\backslash langle\; i,j\; \backslash right\backslash rangle\; \}\{\backslash left(\; a\_\{i\}^\{\backslash dagger\}a\_\{j\}+a\_\{j\}^\{\backslash dagger\}a\_\{i\}\; \backslash right)\}$,
where the tunneling term $\backslash ,\; t$ depends on the overlap of the localized wavefunctions $\backslash phi\; \_\{j\}\backslash left(\; x\; \backslash right)\backslash equiv\; \backslash left\backslash langle\; x\; \backslash right|a\_\{j\}^\{\backslash dagger\}\backslash left|\; 0\; \backslash right\backslash rangle$ between two wells, and is assumed to be the same throughout the lattice.
=== Interaction term ===
For a dilute gas, the effective two-body (s-wave) interaction is the dominate interaction, and is given by
:$V\_\{eff\}\backslash left(\; r\; \backslash right)=\backslash frac\{4\backslash pi\; \backslash hbar\; ^\{2\}a\_\{s\}\}\{m\}\backslash delta\; \backslash left(\; r\; \backslash right)$,
where $\backslash ,\; a\_\{s\}$ is the scattering length. The wavefunction overlap between different wells are exponentially small, compared with the on-site part. We therefore keep only the part where two bosons are in the same well. Provided that the interaction is "weak", compared with the excitation energy of the well (which we shall justify below), the second quantized form of the interaction term is
:$H\_\{\backslash operatorname\{int\}\}=\backslash frac\{U\}\{2\}\backslash sum\backslash limits\_\{i\}\{n\_\{i\}\backslash left(\; n\_\{i\}-1\; \backslash right)\}$.
==== Constraint for the scattering length ====
We shall now consider the condition where higher excitations of the localized states can be neglected. First, we assume each localized well can be approximated by a harmonic potential with energy $\backslash hbar\; \backslash omega\; \_\{0\}$, which depends on the curvature of the bottom part of the well, and the atomic mass. Let us define the length scale $\backslash ,\; a\_\{zp\}$ (called zero-point length) associated with this energy scale by
:$\backslash hbar\; \backslash omega\; \_\{0\}\backslash equiv\; \backslash frac\{\backslash hbar\; ^\{2\}\}\{ma\_\{zp\}^\{2\}\}$.
The ground state wavefunction of a harmonic oscillator is given by $\backslash psi\; \_\{0\}\backslash left(\; x\; \backslash right)=\backslash left(\; \backslash frac\{1\}\{\backslash pi\; \}\; \backslash right)^\{1/4\}\backslash frac\{1\}\{\backslash sqrt\{a\_\{zp\}\}\}e^\{-x^\{2\}/2a\_\{zp\}^\{2\}\}$. Consider two particles in the same state, with $\backslash psi\; \backslash left(\; x\_\{1\},x\_\{2\}\; \backslash right)=\backslash psi\; \_\{0\}\backslash left(\; x\_\{1\}\; \backslash right)\backslash psi\; \_\{0\}\backslash left(\; x\_\{2\}\; \backslash right)$,
:$U=\backslash frac\{4\backslash pi\; \backslash hbar\; ^\{2\}a\_\{s\}\}\{m\}\backslash int\{\backslash left|\; \backslash psi\; \backslash left(\; r\; \backslash right)\; \backslash right|^\{4\}dr\backslash approx\; \backslash frac\{\backslash hbar\; ^\{2\}a\_\{s\}\}\{ma\_\{zp\}^\{3\}\}\}$,
apart from some constant. To avoid the excited states, we impose a constraint $U\backslash ll\; \backslash hbar\; \backslash omega\; \_\{0\}$, which implies that
:$a\_\{s\}\backslash ll\; a\_\{zp\}$.
This is the result we want in this section.

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Monday, October 26, 2015 - 17:37