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arXiv:1309.2909 · Journal of Physics A Mathematical and Theoretical

Quantum backflow states from eigenstates of the regularized current operator

J. J. Halliwell, E Gillman, O Lennon, M Patel, I Ramirez

Published Nov 11, 2013• 28 citations
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Abstract

We present an exhaustive class of states with quantum backflow—the phenomenon in which a state consisting entirely of positive momenta has negative current and the probability flows in the opposite direction to the momentum. They are characterized by a general function of momenta subject to very weak conditions. Such a family of states is of interest in the light of a recent experimental proposal to measure backflow. We find one particularly simple state which has surprisingly large backflow—about 41% of the lower bound on flux derived by Bracken and Melloy. We study the eigenstates of a regularized current operator and we show how some of these states, in a certain limit, lead to our class of backflow states. This limit also clarifies the correspondence between the spectrum of the regularized current operator, which has just two non-zero eigenvalues in our chosen regularization, and the usual current operator.

DOI
10.1088/1751-8113/46/47/475303

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We present an exhaustive class of states with quantum backflow—the phenomenon in which a state consisting entirely of positive momenta has negative current and the probability flows in the opposite direction to the momentum. They are characterized by a general function of momenta subject to very weak conditions. Such a family of states is of interest in the light of a recent experimental proposal to measure backflow. We find one particularly simple state which has surprisingly large backflow—about 41% of the lower bound on flux derived by Bracken and Melloy. We study the eigenstates of a regularized current operator and we show how some of these states, in a certain limit, lead to our class of backflow states. This limit also clarifies the correspondence between the spectrum of the regularized current operator, which has just two non-zero eigenvalues in our chosen regularization, and the usual current operator.
We investigate the backflow effect in elementary quantum mechanics---the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We compute the current and flux for states consisting of superpositions of Gaussian wave packets. These are experimentally realizable but the amount of backflow is small. Inspired by the numerical results of Penz et al. [Penz, Gr\"ubl, Kreidl, and Wagner, J. Phys. A 39, 423 (2006)], we find two nontrivial wave functions whose current at any time may be computed analytically and which have periods of significant backflow, in one case with a backward flux equal to about 70% of the maximum possible backflow, a dimensionless number ${c}_{bm}\ensuremath{\approx}0.04$, discovered by Bracken and Melloy [Bracken and Melloy, J. Phys. A 27, 2197 (1994)]. This number has the unusual property of being independent of $\ensuremath{\hbar}$ (and also of all other parameters of the model), despite corresponding to an obviously quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We discuss some specific measurement models in which backflow may be identified in certain measurable probabilities.
We present an introduction to the backflow effect in quantum mechanics – the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We show that the effect is present even for simple states consisting of superpositions of gaussian wave packets, although the size of the effect is small. Inspired by the numerical results of Penz et al, we present a wave function whose current at any time may be computed analytically and which has periods of significant backflow, with a backwards flux equal to about 70 percent of the maximum possible backflow, a dimensionless number c bm ≈ 0.04, discovered by Bracken and Melloy. This number has the unusual property of being independent of ℏ (and also of all other parameters of the model), despite corresponding to a quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We conclude by discussing a specific measurement model in which backflow may be identified in certain measurable probabilities.
We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self adjoint, and not compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant. Finally, detailed position probability flow properties of a numerical approximation to the (perhaps improper) wave function of maximal backflow are displayed.
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 system. We also evaluate the bounds on the total amount of probability that flows in the opposite direction of the particle's momentum.
In its standard formulation, quantum backflow is a classically impossible phenomenon in which a free quantum particle in a positive-momentum state exhibits a negative probability current. Recently, Miller et al. [Quantum 5, 379 (2021)] have put forward a new, "experiment-friendly" formulation of quantum backflow that aims at extending the notion of quantum backflow to situations in which the particle's state may have both positive and negative momenta. Here, we investigate how the experiment-friendly formulation of quantum backflow compares to the standard one when applied to a free particle in a positive-momentum state. We show that the two formulations are not always compatible. We further identify a parametric regime in which the two formulations appear to be in qualitative agreement with one another.
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.