# Thomas-Fermin approximation of cold atoms in a harmonic trap

In Bose-Hubbard model of cold atoms, one can use Thomas-Fermi approximation to get a rough number of total particle number. In Thomas-Fermi approximation, the site density is $n(R_i)=（\mu-\kappa R_i^2）/U$,where $\mu$ is the chemical potential and $\kappa$ is the trapping potential. The atomic cloud extends to the Thomas-Fermi radius $r_{TF}=\sqrt{\mu/\kappa}$. Why in Thomas-Fermi approximation, one can neglect the kinetic energy. Does it mean it only work for strong interaction regime?

When I use Gutzwiller mean field method to calculate the total particle number in a trap, I find the total particle number varies a lot as the hopping amplitude varies.

• I think you have forgotten the interaction strength in your formula for $n$. – Adam Jun 3 '14 at 23:58

Another comment: be careful when you change the hopping that you should also change the chemical potential if you want keep a constant density. In particular, at large hopping, the chemical potential should be negative (and $\mu\simeq -2dt+{\rm const})$.
• Now I am confused.Thomas-Fermi approximation is valid for Gross-Pitaevski regime in which the hopping is dominant, but why the Thomas-Fermi ignores the kinetic energy term? In Bose-Hubbard model, only given micro-parameters hopping t,interaction U and chemical potential $\mu$, one can calculate the total particle number,right? – Jeremy Jun 4 '14 at 14:51
• @Jeremy: These are different levels of approximation. You start from a many-body Hamiltonian on a lattice. The first approximation made is the a mean-field approximation (GP), which is valid at large hopping (or small interaction). (This is because at large interaction the correlation between atoms is important for the Mott physics.) If there were no trap, then the condensate/density would be homogeneous. If there's a trap, the Thomas-Fermi approximation is equivalent to the LDA, i.e. just replace $\mu$ by $\mu-V(r)$. – Adam Jun 4 '14 at 15:28
• @Jeremy: Considering your second question: yes, of course, if you solve the model exactly, you can compute the density as a function of $t$, $\mu$ and $U$ (assuming no trap). But what is your point? I was just saying that usually one likes to work at a given density, and in that case, one has to adjust the chemical potential when $t$ is varied. For example, with a trap, one usually want the same density at the center of the trap, which again implies fixing the chemical potential accordingly. – Adam Jun 4 '14 at 15:30
• I totally agree with all your points above. My question is: is there any quick method to roughly get the total density in a trap when given $t$,$U$,$\mu$ and trapping potential $\Omega$. For a 2D system in strong interacting regime with a Mott plateau of unitary filling in the trap center, one can get the roughly density $N_{tot}\simeq \pi*R_{Mott}^2=\pi*\frac{\mu}{\Omega}.$ But how to quickly get an approximation total particle number for weakly interacting regime without performing numeric calculation? – Jeremy Jun 4 '14 at 15:55