© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

The Quantum Backflow Phenomenon

Harkan J. Kirk-Karakaya

Published Jan 1, 2026• 0 citations

Abstract

It has been known since at least 1969, that there exist quantum states of a free particle on the line with purely positive momentum that exhibit probability transfer towards the negative half-line. This phenomenon, the quantum backflow effect, is the subject of this thesis. The largest amount of backflow any state can exhibit over a single time interval, the Bracken–Melloy constant, was initially calculated to be cBM ≈ 0.04. In the first half of this thesis, we consider the question of how much quantum backflow states can exhibit over disjoint time intervals. This is done by formulating the question in terms of the spectra of bounded operators. When considering more than one disjoint time interval, we discuss a new phenomenon where a state exhibits more probability transfer in the same direction as its momentum than any classical state. We call this effect quantum overflow. Given a number M of disjoint time intervals, we give bounds on the maximum amount of backflow and overflow a state can exhibit over any M many intervals. The limiting cases in which two disjoint intervals merge into one is particularly studied. Finally, we show plots of the time t = 0 momentum space wavefunction of states exhibiting multiple quantum backflow and quantum overflow. The second half of the thesis contains a detailed numerical investigation into the single backflow problem. Based on previous conjectures, we construct the maximum backflow state from a given set of normalised states closely resembling the sinc function. The Bracken—Melloy constant is then bounded from below by the largest solution to a generalised eigenvalue problem. Using a multi-step numerical procedure, we solve a high precision large dense generalised eigenvalue problem. From this we obtain new numerical approximation to the backflow state and the best known rigorous lower bound of the Bracken—Melloy constant.

Related papers

Repeated quantum backflow and overflow

Christopher J. Fewster, Harkan J. Kirk-Karakaya

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.

arXiv:2505.13184•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 15, 2025•2 citations

A new approach to quantum backflow

Penz, M, Gr"ubl, G, Kreidl, S +1 more

Abstract. 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. PACS numbers: 03.65 1. Introduction and

•Jan 1, 2005•26 citations

Analytical examples, measurement models, and classical limit of quantum backflow

James M. Yearsley, J. J. Halliwell, R. Hartshorn +1 more

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.

arXiv:1202.1783•Physical Review A•Oct 15, 2012•42 citations

A new approach to quantum backflow

Markus Penz, Gebhard Grübl, Sabine Kreidl +1 more

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.

arXiv:quant-ph/0511109•Journal of Physics A Mathematical and General•Dec 14, 2005•51 citations

An introduction to the quantum backflow effect

James M. Yearsley, J. J. Halliwell

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.

arXiv:1301.4893•Journal of Physics Conference Series•Jun 10, 2013•30 citations

Probability backflow for correlated quantum states

Arseni Goussev

This paper addresses the effect of probability backflow in quantum states with position-momentum correlations. The author constructs correlated states that are accompanied by classically-forbidden probability transfer exceeding its largest possible value for uncorrelated states, the so-called Bracken-Melloy bound

arXiv:2002.03364•Physical Review Research•Aug 6, 2020•19 citations

It has been known since at least 1969, that there exist quantum states of a free particle on the line with purely positive momentum that exhibit probability transfer towards the negative half-line. This phenomenon, the quantum backflow effect, is the subject of this thesis. The largest amount of backflow any state can exhibit over a single time interval, the Bracken–Melloy constant, was initially calculated to be cBM ≈ 0.04. In the first half of this thesis, we consider the question of how much quantum backflow states can exhibit over disjoint time intervals. This is done by formulating the question in terms of the spectra of bounded operators. When considering more than one disjoint time interval, we discuss a new phenomenon where a state exhibits more probability transfer in the same direction as its momentum than any classical state. We call this effect quantum overflow. Given a number M of disjoint time intervals, we give bounds on the maximum amount of backflow and overflow a state can exhibit over any M many intervals. The limiting cases in which two disjoint intervals merge into one is particularly studied. Finally, we show plots of the time t = 0 momentum space wavefunction of states exhibiting multiple quantum backflow and quantum overflow. The second half of the thesis contains a detailed numerical investigation into the single backflow problem. Based on previous conjectures, we construct the maximum backflow state from a given set of normalised states closely resembling the sinc function. The Bracken—Melloy constant is then bounded from below by the largest solution to a generalised eigenvalue problem. Using a multi-step numerical procedure, we solve a high precision large dense generalised eigenvalue problem. From this we obtain new numerical approximation to the backflow state and the best known rigorous lower bound of the Bracken—Melloy constant.

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. 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. PACS numbers: 03.65 1. Introduction and

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 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 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.