Single-particle localization in dynamical potentials

Single-particle localization of an ultracold atom is studied in one dimension when the atom is confined by an optical lattice and by the incommensurate potential of a high-finesse optical cavity. In the strong-coupling regime the atom is a dynamical refractive medium, the cavity resonance depends on the atomic position within the standing-wave mode, and nonlinearly determines the depth and form of the incommensurate potential. We show that the particular form of the quasirandom cavity potential leads to the appearance of mobility edges, even in presence of nearest-neighbor hopping. We provide a detailed characterization of the system as a function of its parameters and, in particular, of the strength of the atom-cavity coupling, which controls the functional form of the cavity potential. For strong atom-photon coupling the properties of the mobility edges significantly depend on the ratio between the periodicities of the confining optical lattice and of the cavity field.

Figure 1

Eigenenergies (in units of $t$) as a function of the dimensionless potential depth $V=V_0/t$ for $\beta =\varphi$, $C=-2$, and $\delta =0$. The dots correspond to the energy values, the color (shade of gray in print) represents their $\mathrm{IPR}$ as visualized by the bar code. The horizontal black lines indicate the bandwidth for $V=0$. The gray vertical lines indicate the parameters used in Fig. 5.

Figure 2

Contour plot of the ratio of localized states $R$, Eq. (8), as a function of the dimensionless potential depth $V$ and of the cooperativity $C$ for $\beta =\varphi =(1+\sqrt{5})/2$ and offsets (a) $\delta =0$ and (b) $\delta =-2$ [see Eq. (6)].

Figure 3

Contour plots of (a) the average ratio of localized states $\langle R\rangle$, Eq. (10), and (b) the standard deviation $\sigma \left(R\right)$ of the distribution, Eq. (11), as a function of $C$ and $V$ for offset $\delta =0$ in the potential of Eq. (6). The average is taken over 122 different values of diophantic numbers $\beta ={\varphi }_{cd}^{ab}$ (see text for details).

Figure 4

Same as Fig. 3 but for the offset parameter $\delta =-2$ in the potential of Eq. (6).

Figure 5

Distribution ${\left|\psi \left(k\right)\right|}^2$ of momenta of waves, leaving the central quasidisordered region of the system, as a function of $k$. The distribution is evaluated numerically after evolving an initially localized wave function for a time $T=400/t$. The parameters are $\delta =0$, $C=-2$, while $V=2$ and $V=3$ for black and green lines, respectively. The atom is initially at the center of the quasidisordered potential. See text for further details.