Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate

We study an ultracold Bose gas in the presence of 1D disorder for repulsive interatomic interactions varying from zero to the Thomas-Fermi regime. We show that for weak interactions the Bose gas populates a finite number of localized single-particle Lifshits states, while for strong interactions a delocalized disordered Bose-Einstein condensate is formed. We discuss the schematic quantum-state diagram and derive the equations of state for various regimes.

Figure 1

Schematic quantum-state diagram of an interacting ultracold Bose gas in 1D disorder. The dashed lines represent the boundaries (corresponding to crossovers) which are controlled by the parameter ${\alpha }_R={\hslash }^2/2m{\sigma }_R^2{V}_R$ (fixed in the figure, see text), where $V_R$ and ${\sigma }_R$ are the amplitude and correlation length of the random potential. The hatched part corresponds to a forbidden zone ($\mu <V_{\min }$).

Figure 2

(a) Cumulative density of states of single particles in a speckle potential with ${\sigma }_R=2\times {10}^{-3}L$ and $V_R={10}^4{E}_0$, where $E_0={\hslash }^2/2m{L}^2$ ($V_{\min }=-V_R$). Inset: Participation length [25]. (b) Low-energy Lifshits eigenstates. For the considered realization of disorder, ${ϵ}_0\simeq -5\times {10}^3{E}_0$.

Figure 3

Chemical potential of a Bose gas in a speckle potential with the same parameters as in Fig. 2, in the case of a tight radial confinement (${\mu }^{\prime }-{ϵ}_{\nu }\ll \hslash {\omega }_{\perp }$). The points are given by numerical calculations; the solid and dotted lines represent the analytical formulas Eqs. (13, 16) derived for the Lifshits and BEC regime, respectively.