Bose Einstein condensates

Bose-Einstein condensation is a phenomenon which occurs when bosonic, or pseudo-bosonic (e.g. Cooper pairs), particles forms a quantum liquid (including both liquid and gas phases) and are cooled below some critical temperature. Roughly speaking, Bose Einstein condensates (BEC) are those component of the quantum liquids that can be described by a macroscopically occupied quantum state (or often known as order parameter in the language of phase transition).

To give a rough-and-ready argument, we note that the quantum mechanical properties of a particles may be estimated by the de Broglie wave-length

\lambda =\frac{h}{p},

where \, p is the (characteristic) momentum of the particle and \, h is the Planck's constant.

In thermal equilibrium, the momentum is related to temperature by k_{B}T\approx p^{2}/m. On the other hand, to allow any quantum mechanical effect to occur (e.g. interference), the wavefunctions between the particle have to have a substantial overlap (so that they can exchange positions in space easily). We should expect \lambda \ge n^{-1/3}. Combining these relations, we conclude that to have BEC, it is necessary that temperature is sufficiently low:

k_{B}T<\frac{n^{2/3}\hbar ^{2}}{m},

which roughly gives the correct critical temperature for a three-dimensional non-interacting Bose gas.

BEC for interacting systems

Often, in undergraduate text books, the phenomena of BEC is demonstrated using Einstein's original argument for non-interacting Bose gases. It is not at all trivial to ask (even theoretically) if a BEC could still exist when interaction is introduced: This necessarily becomes a many-body problem and the exact solutions are not available. There are some qualitative arguments suggesting that BEC is energetically more favourable if when the interaction is "weakly" repulsive.

To illustrate one of the arguments, let us consider a very dilute gas of Bosons with no internal degrees of freedom. Suppose the interaction between the particles is dominated by the two-body contact-type interaction:

U\left( r_{1}-r_{2} \right)=U_{0}\delta \left( r_{1}-r_{2} \right),

where \, U_{0}>0 characterizes the interaction strength. This interaction is correct provided that the effective range of the interaction is short, compared with the average particle spacing.

We shall compare two different cases (note that both cases are properly symmetrized):

(a) Two particles in the same state: for \Psi \left( r_{1},r_{2} \right)=\chi \left( r_{1} \right)\chi \left( r_{2} \right), the interaction energy is
\left\langle E_{\operatorname{int}} \right\rangle _{same}=U_{0}\int{dr}\left| \chi \left( r \right) \right|^{4}.
(b) Two particles in different states: for \Psi \left( r_{1},r_{2} \right)=\tfrac{1}{\sqrt{2}}\left\{ \chi _{a}\left( r_{1} \right)\chi _{b}\left( r_{2} \right)+\chi _{a}\left( r_{2} \right)\chi _{b}\left( r_{1} \right) \right\}, the interaction energy is
\left\langle E_{\operatorname{int}} \right\rangle _{diff}=2U_{0}\int{dr}\left| \chi _{a}\left( r \right) \right|^{2}\left| \chi _{b}\left( r \right) \right|^{2},

which is twice of that for distinguishing particles. We may conclude that for homogeneous systems, or specifically \left| \chi \left( r \right) \right|^{4}=\left| \chi _{a}\left( r \right) \right|^{2}\left| \chi _{b}\left( r \right) \right|^{2}, the energy for the interacting particles in different states is higher\left\langle E_{\operatorname{int}} \right\rangle _{diff}>\left\langle E_{\operatorname{int}} \right\rangle _{same} and therefore not, relatively, energetically stable.