Exploring complex phenomena using ultracold atoms in bichromatic lattices

With an underlying common theme of competing length scales, we study the many-body Schrödinger equation in a quasiperiodic potential and discuss its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We propose a possible visualization of such connection in experimentally accessible many-body observables. Those observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson and band insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil’s staircases. Our numerical treatment is complemented with a perturbative analysis which provides insight on the underlying physics. The perturbation theory approach is particularly useful in illuminating the distinction between the Anderson insulator and the band insulator phases in terms of paired sets of dimerized states.

Figure 1

(Color online) Return maps in real space for (a) and (c) $\lambda =10$ (black square), 1 (red triangle), 0.5 (blue solid line) and the corresponding return map in momentum space (b) and (d) $\lambda =0.1$ (black square), 1 (red triangle), 2 (solid blue line). The filling factors are $\nu =0.25$ for the upper panels and $\nu =0.5$ for the lower panels. The insets are averaged over 50 random phases. The insets use the same symbol convention as the main plots.

Figure 2

(Color online) Return maps for the irrational filling factor $\nu =1-\sigma$. Panel (a) is in real space with $\lambda =10$ [black (outer) curve], 1 [red (middle) curve], 0.5 [blue (inner) curve] and panel (b) is in momentum space with the dual values $\lambda =0.1$ [black (outer) curve], 1 [red (middle) curve], 2 [blue (inner) curve]. The insets are averaged over 50 random phases. The insets use the same symbol convention as the main plots.

Figure 3

(Color online) Various boundaries of the fragmented Fermi sea.

Figure 4

(Color online) Momentum distribution for (a) $\lambda =0.1$, (b) $\lambda =1$, (c) $\lambda =2$. All are averaged over 50 random phases to mimic a realistic experimental case. We can see that the sharp features of these distributions survive phase averaging.

Figure 5

(Color online) Momentum distribution for fixed $Q\left(k=1045\right)$, $F_M=4181$ as a function of filling factors vs the disorder parameter.

Figure 6

(Color online) Momentum distribution vs the disorder parameter. (a) $\nu =0.5$, and (b) $\nu =F_{M-1}/F_M$.

Figure 7

Momentum distributions as a function of disorder parameter. (a) filling factor $\nu =0.5$. Here we see the formation of two tongues which merge at the critical value $\lambda =1$. (b) $\nu =\sigma$. When $\nu$ is irrational, the system is a band insulator and the Arnold tonguelike structure disappears. We have checked the existence of the similar structure in position space when converting $\lambda$ to $1/\lambda$.

Figure 8

(Color online) Bifurcation of local density at Fibonacci sites. The central curves (dotted red line) correspond to quarter filling while the lowest (solid black line) and the topmost (dashed blue line) curves, respectively, correspond to $\nu =1/5$ and $\nu =1/3$. Quarter filling is a special case where the local density at Fibonacci sites is 0 or 1 as $\lambda \to \infty$. For $\nu >0.25$, the Fibonacci sites are filled while for $\nu <0.25$ Fibonacci sites are empty as $\lambda \to \infty$. We have checked the existence of the same bifurcation phenomena in the quasimomentum distribution but with the weak and the strong coupling limits reversed $\left(\lambda \to 1/\lambda \right)$.

Figure 9

(Color online) Local density distribution as the disorder parameter and filling factor are varied for (a): Fibonacci site $j=377$ and (b): normal site $j=100$. We have checked the self-dual behavior in the quasimomentum distribution with the weak and the strong coupling limits reversed $\left(\lambda \to 1/\lambda \right)$.

Figure 10

(Color online) Noise correlation for (a) $\lambda =0.1$, (b) $\lambda =1$, (c) $\lambda =2$. All plots display averaged quantities over 50 random phases. We can see that structure in the noise correlations survives phase averaging, as for the momentum distributions of Fig. 4. $\Delta \left(Q=0\right)$ is not displayed in those plots.

Figure 11

(Color online) This figure shows the evolution of $\Delta \left(Q_n\right)$ $\left(n=0,1\right)$ as a function of the filling factor for different $\lambda$ values. In the limit $\lambda ⪡1$, it exhibits steps occurring at the filling factors ${\nu }_{ju}^{\left(m\right)}\left(m=1,2,\dots \right)$ (see text). The plot shows four different $\lambda$’s. From bottom to top: $\lambda =0.2$ (medium dashed red line), $\lambda =0.6$ (dotted blue line), $\lambda =1$ (solid black line) and $\lambda =2$ (long dashed gray line). In the limit $\lambda ⪢1$, $\Delta \left(Q_n\right)$ acquires a sinusoidal profile. The inset shows the self-similar nature of the steps at $\lambda =1$.

Figure 12

(Color online) This figure shows a measure of the gaps in $\Delta \left(0\right)$ vs $\lambda$ for different system sizes, $F_M$. We deem that a gap occurs if $\left|\Delta \left(0\right)\left(N_p+1\right)-\Delta \left(0\right)\left(N_p\right)\right|>ϵ$ (we choose $ϵ={10}^{-15}$ here). We can see that with increasing $M$, the measure of the gaps approaches a step function with the step position at $\lambda =1$.

Figure 13

(Color online) Mapping from $j$ to $k$, denoted by $K\left(j\right)$ used to determine momentum-space observables from position-space observables and vice versa. Larger points are Fibonacci sites. Here we choose $F_M=87$, and $K\left(j\right)$ is the relation that maps $j$ to $k$.

Figure 14

(Color online) Panel (a) shows numerically that at the two red points, $m_1=F_{M-1}$ and $m_2=F_{M-2}$, ${\Lambda }_m$ vanishes, so that degenerate perturbation theory is required. Panel (b) shows the relations of $L_m$, $B_m$ and $m$ (see text) for $\varphi =\frac{3\pi }{4}$. The relations between $L_m$ and $m$ are very sensitive to the value of $\varphi$ while the relations between $B_m$ and $m$ are not sensitive at all.