= Solution
Let the uniform background <number density> be $n_\infty>0$ and set $\mu=Un_\infty$. The subtraction of this <chemical potential> requires a phase rotation as well as a rescaling:
$$
\psi_{\rm phys}(\mathbf x,t)=\sqrt{n_\infty}\,e^{-i\mu t/\hbar}\Psi(\widetilde{\mathbf x},\widetilde t),
\qquad \widetilde{\mathbf x}=\frac{\mathbf x}{\ell_0},\quad \widetilde t=\frac t{t_0}.
$$
Substitution into the physical <Gross–Pitaevskii equation> gives
$$
i\frac{\hbar}{\mu t_0}\Psi_{\widetilde t}
=-\frac{\hbar^2}{2m\mu\ell_0^2}\widetilde\nabla^2\Psi+(|\Psi|^2-1)\Psi.
$$
Consequently the requested coefficients follow from
$$
\boxed{\ell_0=\frac{\hbar}{\sqrt{2m\mu}},\qquad t_0=\frac{\hbar}{2\mu},\qquad \psi\text{ unit}=\sqrt{\mu/U}.}
$$
The length unit is the specified <healing length>. Removing tildes produces the paper's dimensionless equation; in particular the coefficient of its time derivative fixes $t_0$, not its reciprocal. Without the phase rotation the linear $+\Psi$ term would not appear.
The dimensionless <ground state> is \b[$\Psi=e^{i\chi}$ for any constant phase $\chi$], conventionally $\Psi=1$. Its subtracted energy is proportional to $\int[|\nabla\Psi|^2+(1-|\Psi|^2)^2/2]d^3x$, which is nonnegative and vanishes for this state. In physical variables it has density $n_\infty$ and phase rotating at $\mu/\hbar$.
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