Weakly Interacting Homogeneous Bose Gas

Typical introductory statistical mechanics courses examine BECs by assuming that they consist of many non-interacting atoms. That model shows that BECs can form at low temperatures, however it quantitatively and even qualitatively misses some of the properties of real BECs. The issue is that real atoms interact with each other and these interactions can dominate in many properties of a BEC. Fortunately, a simple mean-field treatment of the interactions can create an excellent model that captures much of the behavior seen in real BECs.

Sound propagation in Bose-Einstein condensates

We've seen two general cooling methods so far: Doppler cooling and, on trapped ions, sideband cooling. Last time: Bogolubov transform to diagonalize interacting Bose Einstein condensate.

$E_{k}={\sqrt {\left({\frac {\hbar ^{2}k^{2}}{2m}}\right)^{2}+(\hbar ck)^{2}}}\,.$

This dispersion relation shows us that the low lying excitaitons are phonons. At $k\rightarrow 0$, $E_{k}=\hbar ck$ that of sound, while at $k\rightarrow \infty$, $E_{k}=\hbar ^{2}k^{2}/2m$, a free particle. Free particles start with a quadratic dispersion relation, while phonons and other Bose systems start with a linear dispersion relation.

File:Superfluid to Mott insulator transition-bec3-sound1

File:Superfluid to Mott insulator transition-bec3-sound2

The Bogolubov solution has a great deal of physics in it. It gives the elementary excitation, and the ground state energy. In the simple model that we have a mean field, the ground state energy is

$E_{0}={\frac {U_{0}h}{2}}\left(1+{\frac {128}{15}}{\sqrt {na^{3}/\pi }}\right)\,.$

The extra correction term on the right is a small term, recently observed by the Innsbruck group, due to collective effects. The Bogolubov solution also gives the ground state wavefunction,

$|\psi _{0}\rangle =|0{\rangle }^{\otimes N}+\epsilon \,,$

where $\epsilon$ is the quantum depletion term, which makes the wavefunction satisfy

$\langle n_{k}\rangle ={\frac {v_{k}^{2}}{1-v_{k}^{2}}}$

and

$\langle n_{0}\rangle =N-\sum \langle n_{k}\rangle =N\left[{1-{\frac {8}{3}}{\sqrt {na^{3}/\pi }}}\right]\,.$

The quantum depletion term, which arises from the fact that the gas is weakly interacting, has now been experimentally observed. Recall that in the Bogolubov approximation, the original interaction

$H'=U_{0}\sum a_{p}^{\dagger }a_{q}^{\dagger }a_{r}a_{s}$

is approximated by

$H'=U_{0}a_{0}a_{0}\sum a_{p}^{\dagger }a_{-p}^{\dagger }$

The quantum depletion this leads to is very small. The effect can be more readily experimentally observed by increasing the mass of the particle, and this can be done by placing the particles in a lattice. Plotting the quantum depletion which can be obtained as a function of lattice depth, in such an experiment, one gets:

File:Superfluid to Mott insulator transition-bec3-qdepletion

Beyond a quantum depletion fraction of $0.6$, the Bogolubov approximation breaks down, as the condensate goes through a superfluid to Mott-insulator transition.

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