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One body density matrix

The one body density matrix is given by the Fourier transform of the momentum distribution (see (1.45)). There are three contributions to the one-body density matrix coming from condensate, particle-particle interaction effects and particle-impurity interaction effects:
$\displaystyle \rho(r) = \frac{N_0}{V} + \rho^{(1)}(r) + \rho^{(2)}(r)$     (2.23)

The result for $\rho^{(1)}(r)$ has been given in (1.47). Here we calculate the contribution due to disorder
$\displaystyle \rho^{(2)}(r)
= \frac{2\chi\left(\frac{b}{a}\right)^2}{\pi V x}
\...
...a^3}\,
\chi\left(\frac{b}{a}\right)^2
\exp\left(-\frac{r}{\sqrt{2}r_0}\right)n,$     (2.24)

where $r_0 = a/\sqrt{8\pi na^3}$ is the healing length. Notice that at $r = 0$ the value of the integral equals to the density of particles which are scattered out of the condensate due to the presence of the external field.