Correlations and superfluidity of a one-dimensional Bose gas in a quasiperiodic potential

We consider the correlations and superfluid properties of a Bose gas in an external potential. Using a Bogoliubov scheme, we obtain expressions for the correlation function and the superfluid density in an arbitrary external potential. These expressions are applied to a one-dimensional system at zero temperature subject to a quasiperiodic modulation. The critical parameters for the Bose glass transition are obtained using two different criteria and the results are compared. The Lifshitz glass is seen to be the limiting case for vanishing interactions.

Figure 1

Density profiles for some values of $\mathit{gn}$ for $V=5\hspace{0.5em}{E}_d$ (top) and $V=9\hspace{0.5em}{E}_d$ (bottom); $\mathit{gn}$ is expressed in units of $E_d=\hslash {}^2/{\mathit{md}}^2$ and $x$ in units of $d$. Upon decreasing the interaction strength, the condensate breaks up in sets of spikes with little overlap.

Figure 2

Exponent of the correlation function (35) for the same values of $\mathit{gn}$ as in Fig 1, for $V=5\hspace{0.5em}{E}_d$ (top) and $V=9\hspace{0.5em}{E}_d$ (bottom); $\mathit{gn}$ is expressed in units of $E_d=\hslash {}^2/m\hspace{0.5em}{d}^2$ and $x$ in units of $d$. The insets show the same plot with a log scale on the horizontal axis. In the main panels it is seen that for lower $\mathit{gn}$ the values of $\ln \hspace{0.5em}{g}_1$ (35) follow a linear behavior. For higher $\mathit{gn}$ the plot appears to depend logarithmically on $x$.

Figure 3

Phase diagram of the Bose glass transition. The open circles correspond to a state with nonvanishing superfluid component, while for the red dots the system has only a normal part. The solid line indicates when the linear interpolation of $\ln \hspace{0.5em}{g}_1$ [Eq. (35)] gives a smaller error than a logarithmic fit; that is, at the left side of the solid line the correlation function decays exponentially. The dashed line indicates the point at which the smallest excitation energies cannot be resolved numerically. The inset shows the sum of the residuals of the fits (in arbitrary units) as a function of $\mathit{gn}$, for $V=9\hspace{0.5em}{E}_d$: the triangles refer to a linear regression, the diamonds to a logarithmic interpolation. $V$ and $\mathit{gn}$ are expressed in units of $E_d=\hslash {}^2/m\hspace{0.5em}{d}^2$.

Figure 4

Surface plot of the superfluid density (29) (normalized to the total density $n$) as a function of $V$ and $\mathit{gn}$, for $L=89\hspace{0.5em}d$. The dashed line is the boundary that we estimate for the validity of our numerical calculations (see text). The inset shows the inverse participation ratio $P$ (36) of the quasicondensate wave function. Both $V$ and $\mathit{gn}$ are measured in units of $E_d=\hslash {}^2/m\hspace{0.5em}{d}^2$.

Figure 5

Inverse participation ratio $P$ (36) for $L=377\hspace{0.5em}d$ as a function of $\mathit{gn}$, for some values of the external potential strength $V$. The inset shows a surface plot of $P$ as a function of $V$ and $\mathit{gn}$. All the energies are measured in units of $E_d=\hslash {}^2/m\hspace{0.5em}{d}^2$.