Anderson localization of a Tonks-Girardeau gas in potentials with controlled disorder

We theoretically demonstrate features of Anderson localization in a Tonks-Girardeau gas confined in one-dimensional potentials with controlled disorder. That is, we investigate the evolution of the single-particle density and correlations of a Tonks-Girardeau wave packet in such disordered potentials. The wave packet is initially trapped, the trap is suddenly turned off, and after some time the system evolves into a localized steady state due to Anderson localization. The density tails of the steady state decay exponentially, while the coherence in these tails increases. The latter phenomenon corresponds to the same effect found in incoherent optical solitons.

Figure 1

Anderson localization in a Tonks-Girardeau gas dependent on the initial trap parameter $\nu$ (i.e., $\omega$). The parameters of the disordered potential are $\sigma =0.13$ and $V_0=0.465$. (a) Averaged density of the Tonks-Girardeau wave packet at $t=0$ and after $t=1450$ ($=4$ s) of propagation. The initial state corresponds to $\nu =8.67\times {10}^{-2}$ ($\omega =10$ Hz). (b) Density of a Tonks-Girardeau gas (in the localized steady state) after $t=1450$ ($=4$ s) of propagation in a disordered potential. The dot-dashed (blue) line corresponds to $\nu =4.34\times {10}^{-2}$ ($\omega =5$ Hz), whereas the solid (red) line corresponds to $\nu =8.67\times {10}^{-2}$ ($\omega =10$ Hz). (c) Same as b, on a logarithmic scale. The inner (blue) line corresponds to ($\omega =5$ Hz), and the outer (red) line corresponds to $\omega =10$ Hz; the arrow indicates the increase in $\omega$. For $\left|x\right|$ larger than some value (call it $L_t$), the density decays exponentially, which characterizes Anderson localization. The density tails decay more slowly for a larger initial trap parameter $\omega$ (see text for details).

Figure 2

Anderson localization in a Tonks-Girardeau gas dependent on the disorder parameter $\sigma$. Averaged density of a Tonks-Girardeau gas after $t=1450$ ($=4$ s) of propagation in the disordered potential. Plots correspond to $\sigma =0.13$ [outer (blue) line], $\sigma =0.25$ [middle (green) line), and $\sigma =0.40$ [inner (red) line]; the arrow indicates the increase in $\sigma$. The initial state corresponds to $\omega =10$ Hz, while the amplitude of the disordered potential is $V_0\approx 0.47$ (see text for details).

Figure 3

Exponential decay rates ${\gamma }^{(2)}$ [dotted (blue) line] and ${\gamma }^{(2)}+{\gamma }^{(3)}$ [solid (red) line], obtained with perturbation theory [42], dependent on the correlation length of the potential $\sigma$. Circles represent decay rates obtained numerically. The initial trap parameter is $\omega =10$ Hz (see text for details).

Figure 4

First-order correlations in the initial-state decay algebraically. (a) The single-particle density matrix $|{\rho }_B(0,x,0)|$ and (b) the degree of first-order coherence $|{\mu }_B(0,x,0)|$ at $t=0$ for the initial state corresponding to $\omega =10$ Hz. Graphs are plotted for three values of $\sigma$ as indicated in the legend. Dotted (black) lines depict the fitted curves $|{\rho }_B(0,x,0)|~\left|x\right|{}^{-0.54}$ and $|{\mu }_B(0,x,0)|~\left|x\right|{}^{-0.51}$.

Figure 5

Correlations in the steady (Anderson localized) state. Single-particle density matrix $|{\rho }_B(0,x,t)|$ [lower (blue) line in (a), indicated by the arrow] and degree of first-order coherence $|{\mu }_B(0,x,t)|$ [more horizontal (blue) line in (b), indicated by the arrow] at time $4$ s. Parameters used are $\sigma =0.13$ and $\omega =10$ Hz. The upper (red) line (a) and downward-trending (red) line (b) depict the single-particle density ${\rho }_B(x,x,t)$. Insets enlarge the region where $\left|x\right|$ is small and where correlations decay approximately exponentially. For $\left|x\right|$ in the region of the density tails ($\left|x\right|>L_t$), $|{\mu }_B(0,x,t)|$ reaches a plateau. See text for details.

Figure 6

Single-particle states $|{\psi }_j(x,t)|{}^2$ for $j=5$, 9, and 13 in the (Anderson localized) steady state. The arrow indicates the increase in $j$. Parameters used are $\sigma =0.13$ and $\omega =10$ Hz. Single-particle states for larger $j$ (higher in energy) decay more slowly with and increase in $\left|x\right|$. See text for details.

Figure 7

Absolute value of the real and imaginary part of $A_{\mathit{ij}}(0,x,t)$ for $i=1$ and $j=13$, and five realizations of the disordered potential. Parameters used in the simulation are $\sigma =0.13$ and $\omega =10$ Hz. For sufficiently large $\left|x\right|$, $A_{\mathit{ij}}(0,x,t)$ reaches a constant value. See text for details.

Figure 8

Correlations in the steady (Anderson localized) state dependent on the disorder parameter $\sigma$. Correlations at $t=1450$ ($=$4 s) for $\sigma =0.13$, $0.25,$ and 0.40 are shown. Arrows indicate the increase in $\sigma$, whereas vertical bars indicate the uncertainty in correlations at the plateau values. For a given set of parameters $\sigma$ and $\omega$, $90\%$ of the simulations (one simulation is made for a single realization of $V_D$) fall within the vertical bars. The initial trap parameter is $\omega =10$ Hz. See text for details.

Figure 9

Correlations in the steady state dependent on the initial trap parameter $\omega$. Correlations at $t=1450$ ($4$ s) for two initial conditions, corresponding to $\omega =5$ and $10$ Hz, are shown. Arrows indicate the increase in $\omega$, whereas vertical bars indicate the uncertainty in correlations at the plateau values. Other parameters are exactly as in Fig. 1.