General quantum backflow in realistic wave packets

TL;DR

Quantum backflow is a small, counterintuitive effect where probability flow defies the particle's momentum, and this work generalizes backflow to arbitrary momentum distributions while linking it to quantum reentry. By formulating $\Delta$ as a bilinear form with a kernel $K(u,u')$ and solving the associated integral eigenproblem, the authors show that the maximal quantum violation is $\sup \Delta = 0.128100 \pm 0.000002$, well above the traditional Bracken–Melloy bound $c_{\text{BM}} \approx 0.0384506$, and they provide concrete state constructions (including two-Gaussian superpositions and piecewise Gaussian momentum-space states) that realize large backflow and reentry. The work also establishes a universal framework that applies to both backflow and reentry, demonstrates a fundamental trade-off with forwardflow, and outlines a practical route to experimental observation in realistic, noisy settings. These results deepen our understanding of nonclassical probability transport in quantum mechanics and have potential implications for quantum transport and foundational interpretations.

Abstract

Quantum backflow is a counterintuitive phenomenon in which the probability density of a quantum particle propagates opposite to its momentum. Experimental observation of backflow has remained elusive due to two main challenges: (i) the effect is intrinsically small, with less than 4% of the probability able to flow backward, and (ii) it requires wave packets with a well-defined momentum direction, which are difficult both to prepare and to verify under realistic, noisy conditions. Here, we overcome these challenges by introducing a general formulation of quantum backflow applicable to arbitrary momentum distributions. The framework recovers the standard backflow limit for unidirectional states and identifies general backflow as probability flow exceeding that predicted by the particle's momentum distribution alone. We show that this excess can reach nearly 13%, surpassing the standard backflow bound by more than a factor of three. Furthermore, we extend the framework to the closely related phenomenon of quantum reentry, provide explicit examples of quantum states exhibiting large general backflow and reentry, and discuss the foundational implications of these nonclassical effects. Our results open a pathway toward the experimental observation of quantum backflow in realistic settings.

Figures (11)

• Figure 1: Rescaled state maximizing general backflow and reentry compared with the standard (positive-momentum) backflow-maximizing state.
• Figure 2: Phase-space diagram used to derive inequalities characterizing the absence of backflow and reentry in classical mechanics.
• Figure 3: Approximation to probability current in standard backflow. The plot shows the probability current, label $J$, and the simple approximation given by Eq. (\ref{['EQ_SIM_J']}), label $J_s$, for the optimal parameters of Ref. Yearsley2012. The backflow interval is marked with light blue area. The two curves are practically indistinguishable up to $t \simeq 15$ units.
• Figure 4: $\max \{ \lambda_{10,N} \}$ versus $1/N$. The orange dots correspond to the values given in Table \ref{['alpha_for_L=10']}. The blue line represents the fit detailed in the text.
• Figure 5: $\max \{ \lambda_{15, N} \}$ versus $1/N$. The orange dots correspond to the values given in Table \ref{['alpha_for_L=15']}. The blue line represents the fit detailed in the text.