Chaos-Assisted Long-Range Tunneling for Quantum Simulation

We present an extension of the chaos-assisted tunneling mechanism to spatially periodic lattice systems. We demonstrate that driving such lattice systems in an intermediate regime of modulation maps them onto tight-binding Hamiltonians with chaos-induced long-range hoppings $t_n\propto 1/n$ between sites at a distance $n$. We provide a numerical demonstration of the robustness of the results and derive an analytical prediction for the hopping term law. Such systems can thus be used to enlarge the scope of quantum simulations to experimentally realize long-range models of condensed matter.

Figure 1

Three representations of CAT in a driven lattice. (a) In situ description: a wave function tunnels between potential wells. (b) Phase space description: the wave function escapes from a stable island (blue) by regular tunneling, spreads in the chaotic sea (red), and tunnels in another island. (c) Tight-binding description: the system contains $N$ sites with coupling between the $i$th and $j$th sites proportional to $1/|i-j|$.

Figure 2

In a mixed lattice as in Fig. 1, a CAT resonance between a regular Bloch wave $|{\beta }_{\mathrm{reg}}⟩$ and a chaotic one $|{\beta }_{\mathrm{ch}}⟩$ leads to a discontinuity in the energy band of the associated tight-binding model (solid black line). The main panel shows the avoided crossing characterized by $|W|$ (the strength of the coupling between $|{\beta }_{\mathrm{reg}}⟩$ and $|{\beta }_{\mathrm{ch}}⟩$), $\alpha$ (the slope of the energy of $|{\beta }_{\mathrm{ch}}⟩$), ${\beta }_0$ (the point of equal mixing), and $\mathrm{\Delta }\beta$ (the crossing width). Near $\beta ={\beta }_0$, the eigenstates $|{\beta }_{\pm }⟩$ become a mixture of $|{\beta }_{\mathrm{reg}}⟩$ and $|{\beta }_{\mathrm{ch}}⟩$. The color code gives the intensity of the mixing (projection on $|{\beta }_{\mathrm{reg}}⟩$). Husimi representations [56, 57, 58] of $|{\beta }_{\pm }⟩$ are on top of the classical dynamics phase portrait. Inset: black solid line is the effective regular band and red dashed line a nearest-neighbor approximation with parameters extracted from the effective regular band at $\beta =0$ and $\beta =\pi /\lambda$ (${\hslash }_{\mathrm{eff}}=0.4,\gamma =0.20,\epsilon =0.15$).

Figure 3

Dynamics of a wave packet initially located on a single regular island or site $n_0$. Color plot of the time evolution of the spatial probability distribution, with $\gamma =0.2$, ${\hslash }_{\mathrm{eff}}=0.4$, and $ϵ=0.15$ for the modulated lattice (a) or for the unmodulated lattice, i.e., $ϵ=0$ (regular case) (b). The exact dynamics (left) is compared to the corresponding effective description (right). Note that the system is symmetric through $n-n_0\to n_0-n$.

Figure 4

Characterization of the dynamics of a wave packet initially located on a single regular island or site $n_0$ (corresponding to Fig. 3). (a) Spatial probability distribution of the wave packet after $t=1500\mathrm{T}$. (b) Overlap of the wave function with the chaotic sea vs time (see text). (c) Standard deviation of the spatial distribution vs time. Symbols are for the exact dynamics and solid lines for the effective dynamics. Red data correspond to modulated lattices with different sizes, and blue data correspond to the unmodulated lattice (regular case).

Figure 5

(a) Effective hopping amplitude $|t_n^{\mathrm{eff}}|$ vs distance between sites $n$ for $\gamma =0.2$, $ϵ=0.15$, and ${\hslash }_{\mathrm{eff}}=0.4$. Red data were extracted from numerical Fourier series of the effective band structure. Green data correspond to Eq. (3) with parameters extracted from the band structure. Black solid line is the typical value of Eq. (3) (without the phase term). Inset: small-distance behavior and additional blue data for unmodulated case $ϵ=0$. (b) Distribution of fluctuations around the $1/n$ law for five parameter sets: histogram corresponds to cumulative values for $1500<n<10000$, dots are partial datasets of 500 consecutive values of $n$, and black curve is analytical prediction (see text).