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arXiv:1903.07618 · Physica Scripta

Relativistic quantum backflow

Johar M. Ashfaque, J Lynch, P. Strange

Published Jun 3, 2019• 23 citations
arXivHTMLPDF

Abstract

Abstract In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is presented and it is shown that the backflow can be written as a function of a simple parameter ϵ, which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of ϵ, and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Backflow eigenvectors are then fitted with some standard functions, which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of ϵ . In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.

DOI
10.1088/1402-4896/ab265c

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Abstract In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is presented and it is shown that the backflow can be written as a function of a simple parameter ϵ, which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of ϵ, and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Backflow eigenvectors are then fitted with some standard functions, which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of ϵ . In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.
Abstract Quantum backflow is the unexpected effect that wavepackets consisting of only positive momentum components can apparently move in the negative direction. This is usually described in terms of the backflow constant, which is a dimensionless quantity describing least upper bound on the amount of probability that can flow backwards during a given time interval. Backflow is usually calculated for wavepackets that can be written as a sum of positive momentum plane waves. Here we present a calculation of the backflow constant using the localised free particle hermite wavefunctions where equal weights of positive and negative momentum eigenfunctions occur. The resulting backflow constant is substantially smaller than the accepted value. The reasons for this are discussed and finally we draw conclusions about the calculation of backflow more generally.
It is known that for a non-relativistic quantum particle traveling freely on the xxx-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value 0.0384517…0.0384517\dots0.0384517…, or about 4%4\%4% of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0
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.
It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.
Abstract Quantum particles moving in one dimension with rightwards momentum can exhibit the surprising phenomenon of quantum backflow (QB): a net probability transfer to the left-hand half-line over a finite time interval. We generalize this phenomenon by considering the sum of probability differences for M disjoint time intervals. In classical mechanics, the total backflow lies in the interval [−1,0] for all M, indicating rightwards probability transfer. By contrast, we show that the maximum M-fold QB is positive and unbounded from above as M increases, demonstrating the existence of repeated backflow. For M≥2, a new phenomenon of ‘quantum overflow’ is discovered: there are states whose total backflow is below −1, which is impossible for classical particles. The extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, of which the M=1 backflow constant was first studied by Bracken and Melloy. Limiting cases of the backflow and overflow functions are studied, including cases in which two disjoint intervals merge. The analytical results are supported by detailed numerical investigations. Using numerical acceleration methods, we obtain a new estimate of the Bracken–Melloy constant of 0.0384506, slightly lower than the previously accepted value of 0.038452.
Abstract New quantum probability flow bounds and associated eigenfunctions are determined numerically for a free non-relativistic particle, with momentum lying in the range ( p 0 , ∞), for each fixed p 0 chosen in the range (−∞ , ∞). It is found that as p 0 increases through positive values, the maximum possible probability backflow from right-to left (R to L), opposite to the direction of all contributing momenta, decreases monotonically from the well-known value ≈0.038452 at p 0 = 0 but never reaches 0. As p 0 decreases through negative values, the maximum R to L flow increases monotonically but never reaches 1. These new quantum effects are compared and contrasted with the corresponding classical behaviour. A surprising new effect is revealed: Even when p 0 is negative, L to R directed momenta contribute to the maximum R to L flow. The size of this contribution is indicated by comparing with the maximum R to L flow possible when only negative momenta with values between p 0 and 0 are allowed. The extended modeling admits a simple interpretation of the classical limit without the introduction of a mechanism external to the system, as an effective value of Planck’s constant approaches zero.