Superfluid to Mott Insulator Transition

A condenstate in a shallow standing wave potential is a BEC, well described by a Bogoliubov approximate solution. As the potential gets deeper, though, eventually the system transitions into a state of localized atoms, with no long-range coherence, known as a Mott insulator. These physics are important in a wide range of condensed matter systems, and can be explored deeply with BECs. The ultracold atoms are trapped in a periodic potential,

$V(x,y,z)=V_{0}\left({\sin ^{2}kx+\sin ^{2}ky+\sin ^{2}kz}\right)\,,$

and we would like to know what happens when neutral atoms are in this potential. Traditionally, this is studied in condensed matter physics in the context of atoms in a periodic potential, but many anologies exist for the neutral atom system. The Hamiltonian is

$H={\frac {\hbar ^{2}}{2m}}\nabla ^{2}-V_{0}\sin ^{2}kz\,,$

in one dimension, giving the Bloch ansatz wavefunction solution

$\psi _{q,n}(x)=e^{iqx/\hbar }U_{q,n}(x)\,,$

where $q$ is a quasi-momentum labeling the eigenstate, and $U_{q}(x)$ is a periodic function. For a given $q$ there are several solutions, labeled by $n$, the band index. To solve this equaiton, a Fourier expansion of $U_{q}(x)$ is used, with elements $c_{l}^{a,n}e^{i2lkx}$. The potential is also expanded as $V_{r}e^{i2lkx}$. The solutions to the Schrodinger equation,

$H\psi _{q,n}=E_{a,n}\psi _{a,n}\,,$

give

$\sum _{l'}e^{i(q+2l'kx)}\left[{\left({{\frac {(q+2l'\hbar k)^{2}}{2m}}-E_{q,n}}\right)c_{l}^{q,n}+\sum _{v}c_{l',r}^{q,n}}\right]=0\,.$

Graphically, these solutions are as follows. If energies are plotted as a function of $\hbar k$, we get:

File:Superfluid to Mott insulator transition-bec3-smott-dispersion

With no interations, we get a free particle dispersion diagram. But with the periodic potential, a Brilloiuin zone appears, forbidding momentum beyond $\pm \hbar k$, giving bands. The potential couples the bands; for example, at $V\sim 3E_{r}$ (where $E_{r}=k^{2}/2m$), the band degnericies are lifted to become:

File:Superfluid to Mott insulator transition-bec3-3er

\noindent At higher potentials, the bands become even flatter. We will be interestd in the limit in which $V_{r}/E_{r}$ is large, and we shall focus on the physics of the lowest, $n=1$ band, in which in the limit of tight binding, the energy is

$E_{1}(q)={\frac {3}{2}}\hbar \omega _{0}\,,$

becoming constant with respect to quasi-momentum. The derivation of the energy as a function of quasi-momentum gives

$-2J(\cos q_{x}a+\cos q_{y}a+\cos q_{z}a)\,,$

where $a={\frac {\lambda }{2}}=\pi /k$, and the paramter $J$ tells us how wide the band is and how large the dispersion region is.

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