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Repeated quantum backflow and overflow

Christopher J. Fewster, Harkan J. Kirk-Karakaya

TL;DR

This work extends quantum backflow to multiple disjoint time intervals by formulating an M-fold backflow problem via bounded self-adjoint operators and analyzing their spectra. The authors prove that, unlike the classical case, quantum backflow and overflow can grow unbounded with the number of intervals, providing rigorous bounds and an asymptotic growth of at least O(M^{1/4}); they also establish conditions under which the total overflow Δ^{(M)} can fall below -1 for M≥2. The paper combines analytic operator-theoretic results with detailed numerical computations, delivering a refined estimate of the Bracken–Melloy constant c_{BM} ≈ 0.0384506 and numerical evidence that for M=2–4 the extremal backflow/overflow values exceed the single-interval bounds. These findings highlight a robust quantum advantage in repeated backflow phenomena and illuminate the spectral structure of the backflow operators, with potential implications for experimental verification and foundational understanding of quantum transport.

Abstract

Quantum backflow is a surprising phenomenon in which a quantum particle, moving in one dimension and with a state of rightwards momentum, can exhibit a net probability transfer to the left-hand half-line over a finite time interval. We generalise the setting of quantum backflow to allow for $M$ disjoint time intervals, considering the sum of probability differences for each interval. In classical statistical particle mechanics, the total backflow lies in the interval $[-1,0]$ for all $M$, indicating rightwards probability transfer. By contrast, we show that, in quantum mechanics, the maximum $M$-fold backflow is positive and unbounded from above as $M$ increases, demonstrating that there are states that exhibit repeated periods of backflow. Moreover, for $M\ge 2$, we discover a new phenomenon; namely, that there are states whose total backflow is below $-1$, giving a probability transfer to the right-hand half-line beyond that possible in classical statistical particle mechanics. We call this effect "quantum overflow". The maximum extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, where the $M$'th backflow (overflow) constant is the supremum (infimum) of the $M$'th backflow (overflow) function. The $M=1$ backflow constant was first identified by Bracken and Melloy. Our results are obtained by formulating the $M$-fold backflow problem in terms of the spectra of suitable bounded operators. Using this formulation, we also study limiting cases of the backflow and overflow functions, including cases in which two disjoint intervals merge. Our analytical results are supported by detailed numerical investigations. Among other things, by applying numerical acceleration methods, we obtain a new estimate of the Bracken-Melloy constant of $0.0384506$ which is slightly lower than the previously accepted value of $0.038452$.

Repeated quantum backflow and overflow

TL;DR

This work extends quantum backflow to multiple disjoint time intervals by formulating an M-fold backflow problem via bounded self-adjoint operators and analyzing their spectra. The authors prove that, unlike the classical case, quantum backflow and overflow can grow unbounded with the number of intervals, providing rigorous bounds and an asymptotic growth of at least O(M^{1/4}); they also establish conditions under which the total overflow Δ^{(M)} can fall below -1 for M≥2. The paper combines analytic operator-theoretic results with detailed numerical computations, delivering a refined estimate of the Bracken–Melloy constant c_{BM} ≈ 0.0384506 and numerical evidence that for M=2–4 the extremal backflow/overflow values exceed the single-interval bounds. These findings highlight a robust quantum advantage in repeated backflow phenomena and illuminate the spectral structure of the backflow operators, with potential implications for experimental verification and foundational understanding of quantum transport.

Abstract

Quantum backflow is a surprising phenomenon in which a quantum particle, moving in one dimension and with a state of rightwards momentum, can exhibit a net probability transfer to the left-hand half-line over a finite time interval. We generalise the setting of quantum backflow to allow for disjoint time intervals, considering the sum of probability differences for each interval. In classical statistical particle mechanics, the total backflow lies in the interval for all , indicating rightwards probability transfer. By contrast, we show that, in quantum mechanics, the maximum -fold backflow is positive and unbounded from above as increases, demonstrating that there are states that exhibit repeated periods of backflow. Moreover, for , we discover a new phenomenon; namely, that there are states whose total backflow is below , giving a probability transfer to the right-hand half-line beyond that possible in classical statistical particle mechanics. We call this effect "quantum overflow". The maximum extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, where the 'th backflow (overflow) constant is the supremum (infimum) of the 'th backflow (overflow) function. The backflow constant was first identified by Bracken and Melloy. Our results are obtained by formulating the -fold backflow problem in terms of the spectra of suitable bounded operators. Using this formulation, we also study limiting cases of the backflow and overflow functions, including cases in which two disjoint intervals merge. Our analytical results are supported by detailed numerical investigations. Among other things, by applying numerical acceleration methods, we obtain a new estimate of the Bracken-Melloy constant of which is slightly lower than the previously accepted value of .
Paper Structure (24 sections, 16 theorems, 223 equations, 27 figures, 5 tables)

This paper contains 24 sections, 16 theorems, 223 equations, 27 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $M$-fold backflow
  3. Classical statistical mechanics
  4. Quantum mechanics
  5. The spectrum of $M$-fold backflow operators
  6. Bounds on the spectrum
  7. Unbounded $M$-fold backflow and overflow as $M\to\infty$
  8. Limiting cases and monotonicity of the backflow constants
  9. Numerical calculation
  10. Background theory
  11. Numerical methodology and results
  12. Numerical acceleration
  13. Conclusion
  14. Acknowledgments
  15. The Bracken--Melloy operator
  16. ...and 9 more sections

Key Result

Theorem 2.1

For any $t_1<t_2$, $B_{\langle t_1,t_2\rangle}$ is a bounded self-adjoint operator with $\|B_{\langle t_1,t_2\rangle}\|=1$. There is a unitary equivalence between $B_{\langle t_1,t_2\rangle}$ and the operator $C \in\mathcal{B}(L^2(\mathbb{R}^+,dq))$ with action on $\varphi\in L^2(\mathbb{R}^+,dq)$, where eq:J1def holds pointwise almost everywhere on $\mathbb{R}^+$. The map $(t_1,t_2)\mapsto B_{\l

Figures (27)

  • Figure 1.1: Probability flux at $x=0$ of the $M$-fold backflow states $\chi_{\textnormal{back}}^{(M)}$ computed in Section \ref{['sec:numerical_methodology_results']}.
  • Figure 1.2: Time evolution of the position probability density for the $M=2$ backflow state $\chi^{(2)}_{\textnormal{back}}$.
  • Figure 1.3: Time evolution of the position probability density for the $M=2$ overflow state $\chi^{(2)}_{\textnormal{over}}$.
  • Figure 5.1: Plot of $\lambda^{(M)}_\textnormal{back}(N)$ for $100 \leq N \leq 500$.
  • Figure 5.2: Plot of $\lambda_\textnormal{over}^{(M)}$ for $100 \leq N \leq 500$.
  • ...and 22 more figures

Theorems & Definitions (31)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 21 more