How to observe off-diagonal long range order in superfluid?

Question

off-diagonal long range order in superfluid is an effect that the matrix element of the single particle's density matrix remains finite in the long distance limit.

My question is: how to prove this experimentally?

Answer 1

It should first be noted that Off-Diagonal Long-Range Order (ODLRO) and superfluidity do no necessarily go hand in hand. ODLRO is associated with a Bose-Einstein condensed phase (BEC), which usually also behaves as a superfluid and the latter thus inherits the ODLRO property. However, you can have systems that are superfluid but where BEC is not possible (and hence lack true ODLRO, but can display some sort of quasi-ODLRO) such as the BKT phase.

So anyway. Following the above, let's rephrase your question to How to observe off-diagonal long range order in a Bose-Einstein condensate?

Let's look at the asymptotic behaviour of the off-diagonal one-body density matrix, used in the Penrose-Onsager criterion as a rigorous definition for a BEC. In the first quantisation formalism, this is defined as: \begin{equation} \begin{gathered} n^{(1)}(\mathbf{r}, \mathbf{r}') = \sum_i n_i\, \psi_i^\ast(\mathbf{r}) \psi_i(\mathbf{r}') \\ = n_0\,\phi_0(\mathbf{r})^\ast \phi_0(\mathbf{r}')+ \sum_{i\neq 0} n_i\, \psi_i(\mathbf{r})^\ast \psi_i(\mathbf{r}) \\ = n_0\, \phi_0(\mathbf{r})^\ast \phi_0(\mathbf{r}')+ \sum_{i\neq 0} n_i\, \mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}')}, \end{gathered} \end{equation} where in the last term a special case of free particles was assumed, to express them as plane waves. $n_0$ is the density of atoms in the $0$ state (ground state), which we have taken out of the sum $\sum_i$ for reasons below.
At increasing separation, this tends to a constant value because the contributions of $\mathbf{p} \neq 0$ average out: \begin{equation} \lim_{|\mathbf{r} - \mathbf{r}'| \rightarrow \infty} n^{(1)}(\mathbf{r}, \mathbf{r}')\rightarrow n_0 \neq 0. \end{equation} This is exactly the definition of Off-Diagonal (because of the $\mathbf{r}$ and $\mathbf{r}'$) Long-Range (because of the limit $|\mathbf{r} - \mathbf{r}'| \rightarrow \infty$) order, ODLRO.

To confirm this experimentally, then, you have to devise an experiment where you can see whether or not phase coherence is preserved over 'long distances'. One such experiment is reported here (plot shown below). What is plotted is the visibility of the fringes in a matter-wave interference pattern as a function of the spatial extent of the atomic cloud (in this sense, $z$ larger than the inteatomic spacing is considered 'large distances'). For a thermal state, this decays to zero within the thermal de Broglie wavelength, whereas for a BEC it stays constant owing to the presence of off-diagonal long-range order (the actual constant is related to $n_0$).