Quantum backflow in biased tight-binding systems

TL;DR

This work analyzes quantum backflow in biased tight-binding systems with complex hopping, addressing how backflow manifests in both infinite and periodic lattices. By deriving the momentum operator and continuity equation on the lattice, the authors implement variational and Bracken–Melloy-type approaches to obtain maximal backward fluxes and universal-like bounds, highlighting how the drift term from the imaginary coupling expands the positive-momentum sector and enhances backflow. The results show that discrete systems can exceed the continuous $c_{ ext{BM}}$ bound, especially in small periodic lattices, and that the bounds converge to their continuous counterparts as system size grows. These findings offer theoretical guidance for potential experimental observation and illuminate transport phenomena in biased lattice structures.

Abstract

We study the phenomenon of quantum backflow in tight-binding systems with complex couplings, considering different boundary conditions and lattice sizes. Backflow is an intrinsically non-classical effect where the density flux associated with a particle described by the superposition of wave functions with, say, positive momentum, acquires negative values. We calculate the superposition of positive momentum states that gives rise to the strongest backflow in the systems. We also evaluate the bounds on the total amount of probability that flows in the opposite direction of the particle's momentum.

Figures (7)

• Figure 1: Segment of the chain, the lattice elements are separated by a distance $(\Delta x)$ and the nearest neighbor hopping is characterized by a complex number with imaginary part $\pm\epsilon$.
• Figure 2: The extreme values of the probability density flux as a at $j' = 3$, and the nearest sites are shown as a function of $t$ taking $t' = 3$ (dashed red line), for the infinite chain (a) and the periodic chain (b) with $\epsilon=1$ and $N=9$. We compute the flux using the function $\phi_{\pm}(k)$ (sequence $c_m^{\pm}$) and the bound associated : $\lambda_{+}$ (blue dashed line) and $\lambda_{-}$ (magenta dashed line).
• Figure 3: The extreme values of the probability density flux for different $\epsilon$ values at $j' = 0$, and $t' = 3$ (dashed red line) as a time function, for the infinite chain (a) and the periodic chain with $N=9$ (b). The dashed lines are the respective bounds, $\lambda_{+}$ and $\lambda_{-}$ for the corresponding values of $\epsilon$.
• Figure 4: Supremum of the minus time-integrated probability flux, $\min(\lambda_p)$, as a function of $\nu$ for different values of $\epsilon$. It can be observed that $\lambda_p$ appears to become bounded below by $c_{\text{BM}}$ as $\nu \rightarrow \infty$. The maximum of $\lambda_p$ tends to be larger for values of $\nu$ close to zero, and the effect of $\epsilon$ does not significantly change the qualitative behavior of the curves. We found that the three curves reach a maximum value of $0.0764734$.
• Figure 5: Maximum of the time negative integrated probability density flux as a function of $\nu$, for $N=10, 10^2$ and $10^3$ sites. The solid gray line corresponds to the result for the continuous version of the system, while the magenta dotted line denotes $c_{\text{BM}}$. We note that as the number of sites increases, the peak value of the curves converges toward $c^{\text{cont}}_{\text{ring}}$ from above, which is in accordance with the scaling law of eq. (\ref{['eqn: er']}).