On the experiment-friendly formulation of quantum backflow

Maximilien Barbier1,2 and Arseni Goussev3

1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany
2Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles (ULB), Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
3School of Mathematics and Physics, University of Portsmouth, Portsmouth PO1 3HF, United Kingdom

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Abstract

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.

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[1] A. J. Brackenand G. F. Melloy ``Probability backflow and a new dimensionless quantum number'' J. Phys. A: Math. Gen. 27, 2197 (1994).
https:/​/​doi.org/​10.1088/​0305-4470/​27/​6/​040

[2] G. R. Allcock ``The time of arrival in quantum mechanics III. The measurement ensemble'' Ann. Phys. (N.Y.) 53, 311 (1969).
https:/​/​doi.org/​10.1016/​0003-4916(69)90253-X

[3] J. Kijowski ``On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time'' Reports Math. Phys. 6, 361 (1974).
https:/​/​doi.org/​10.1016/​S0034-4877(74)80004-2

[4] S. P. Eveson, C. J. Fewster, and R. Verch, ``Quantum Inequalities in Quantum Mechanics'' Ann. Henri Poincaré 6, 1 (2005).
https:/​/​doi.org/​10.1007/​s00023-005-0197-9

[5] J. M. Yearsley, J. J. Halliwell, R. Hartshorn, and A. Whitby, ``Analytical examples, measurement models, and classical limit of quantum backflow'' Phys. Rev. A 86, 042116 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.042116

[6] J. J. Halliwell, E. Gillman, O. Lennon, M. Patel, and I. Ramirez, ``Quantum backflow states from eigenstates of the regularized current operator'' J. Phys. A: Math. Theor. 46, 475303 (2013).
https:/​/​doi.org/​10.1088/​1751-8113/​46/​47/​475303

[7] A. J. Bracken ``Probability flow for a free particle: new quantum effects'' Phys. Scr. 96, 045201 (2021).
https:/​/​doi.org/​10.1088/​1402-4896/​abdd54

[8] G. F. Melloyand A. J. Bracken ``The velocity of probability transport in quantum mechanics'' Ann. Phys. (Leipzig) 7, 726 (1998).
https:/​/​doi.org/​10.1002/​(SICI)1521-3889(199812)7:7/​8<726::AID-ANDP726

[9] H. Bostelmann, D. Cadamuro, and G. Lechner, ``Quantum backflow and scattering'' Phys. Rev. A 96, 012112 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.012112

[10] P. Strange ``Large quantum probability backflow and the azimuthal angle-angular momentum uncertainty relation for an electron in a constant magnetic field'' Eur. J. Phys. 33, 1147 (2012).
https:/​/​doi.org/​10.1088/​0143-0807/​33/​5/​1147

[11] V. D. Paccoia, O. Panella, and P. Roy, ``Angular momentum quantum backflow in the noncommutative plane'' Phys. Rev. A 102, 062218 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.062218

[12] A. Goussev ``Quantum backflow in a ring'' Phys. Rev. A 103, 022217 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.022217

[13] G. F. Melloyand A. J. Bracken ``Probability Backflow for a Dirac Particle'' Found. Phys. 28, 505 (1998).
https:/​/​doi.org/​10.1023/​A:1018724313788

[14] H.-Y. Suand J.-L. Chen ``Quantum backflow in solutions to the Dirac equation of the spin-1/​2 free particle'' Mod. Phys. Lett. A 33, 1850186 (2018).
https:/​/​doi.org/​10.1142/​S0217732318501869

[15] J. Ashfaque, J. Lynch, and P. Strange, ``Relativistic quantum backflow'' Phys. Scr. 94, 125107 (2019).
https:/​/​doi.org/​10.1088/​1402-4896/​ab265c

[16] M. Barbier ``Quantum backflow for many-particle systems'' Phys. Rev. A 102, 023334 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.023334

[17] M. V. Berry ``Quantum backflow, negative kinetic energy, and optical retro-propagation'' J. Phys. A: Math. Theor. 43, 415302 (2010).
https:/​/​doi.org/​10.1088/​1751-8113/​43/​41/​415302

[18] J. G. Muga, C. R. Leavens, and J. P. Palao, ``Space-time properties of free-motion time-of-arrival eigenfunctions'' Phys. Rev. A 58, 4336 (1998).
https:/​/​doi.org/​10.1103/​PhysRevA.58.4336

[19] J. G. Muga, J. P. Palao, and C. R. Leavens, ``Arrival time distributions and perfect absorption in classical and quantum mechanics'' Phys. Lett. A 253, 21 (1999).
https:/​/​doi.org/​10.1016/​S0375-9601(99)00020-1

[20] J. G. Mugaand C. R. Leavens ``Arrival time in quantum mechanics'' Phys. Rep. 338, 353 (2000).
https:/​/​doi.org/​10.1016/​S0370-1573(00)00047-8

[21] G. Grübl, S. Kreidl, M. Penz, and M. Ruggenthaler, ``Arrival time and backflow effect'' AIP Conf. Proc. 844, 177 (2006).
https:/​/​doi.org/​10.1063/​1.2219361

[22] J. M. Yearsley ``Quantum arrival time for open systems'' Phys. Rev. A 82, 012116 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.82.012116

[23] J. J. Halliwell ``Leggett-Garg correlation functions from a noninvasive velocity measurement continuous in time'' Phys. Rev. A 94, 052114 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052114

[24] J. J. Halliwell, H. Beck, B. K. B. Lee, and S. O'Brien, ``Quasiprobability for the arrival-time problem with links to backflow and the Leggett-Garg inequalities'' Phys. Rev. A 99, 012124 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.012124

[25] S. Dasand M. Nöth ``Times of Arrival and Gauge Invariance'' Proc. R. Soc. A 477, 20210101 (2021).
https:/​/​doi.org/​10.1098/​rspa.2021.0101

[26] F. Albarelli, T. Guaita, and M. G. A. Paris, ``Quantum backflow effect and nonclassicality'' Int. J. Quantum Inf. 14, 1650032 (2016).
https:/​/​doi.org/​10.1142/​S0219749916500325

[27] A. Goussev ``Equivalence between quantum backflow and classically forbidden probability flow in a diffraction-in-time problem'' Phys. Rev. A 99, 043626 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.043626

[28] W. van Dijkand F. M. Toyama ``Decay of a quasistable quantum system and quantum backflow'' Phys. Rev. A 100, 052101 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.052101

[29] A. Goussev ``Probability backflow for correlated quantum states'' Phys. Rev. Research 2, 033206 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033206

[30] M. Palmero, E. Torrontegui, J. G. Muga, and M. Modugno, ``Detecting quantum backflow by the density of a Bose-Einstein condensate'' Phys. Rev. A 87, 053618 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.053618

[31] Sh. Mardonov, M. Palmero, M. Modugno, E. Ya. Sherman, and J. G. Muga, ``Interference of spin-orbit-coupled Bose-Einstein condensates'' EPL (Europhysics Lett.) 106, 60004 (2014).
https:/​/​doi.org/​10.1209/​0295-5075/​106/​60004

[32] Y. Eliezer, T. Zacharias, and A. Bahabad, ``Observation of optical backflow'' Optica 7, 72 (2020).
https:/​/​doi.org/​10.1364/​OPTICA.371494

[33] M. Miller, W. C. Yuan, R. Dumke, and T. Paterek, ``Experiment-friendly formulation of quantum backflow'' Quantum 5, 379 (2021).
https:/​/​doi.org/​10.22331/​q-2021-01-11-379

[34] H.-W. Lee ``Theory and application of the quantum phase-space distribution functions'' Phys. Rep. 259, 147 (1995).
https:/​/​doi.org/​10.1016/​0370-1573(95)00007-4

[35] L. Cohen ``The Weyl Operator and its Generalization'' Birkhäuser (2013).

[36] W. Appel ``Mathématiques pour la physique et les physiciens, 4è Ed.'' H & K Eds (2008).

[37] M. de Gosson ``Symplectic Geometry and Quantum Mechanics'' Birkhäuser (2006).

[38] M. Penz, G. Grübl, S. Kreidl, and P. Wagner, ``A new approach to quantum backflow'' J. Phys. A: Math. Gen. 39, 423 (2006).
https:/​/​doi.org/​10.1088/​0305-4470/​39/​2/​012

[39] K. Gottfriedand T.-M. Yan ``Quantum Mechanics: Fundamentals (2nd Edition)'' Springer (2003).

[40] S. Harocheand J.-M. Raimond ``Exploring the Quantum: Atoms, Cavities and Photons'' Oxford Univ. Press (2006).

[41] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, ``Direct measurement of the quantum wavefunction'' Nature 474, 188 (2011).
https:/​/​doi.org/​10.1038/​nature10120

[42] W.-W. Pan, X.-Y. Xu, Y. Kedem, Q.-Q. Wang, Z. Chen, M. Jan, K. Sun, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, ``Direct Measurement of a Nonlocal Entangled Quantum State'' Phys. Rev. Lett. 123, 150402 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.150402

[43] S. Zhang, Y. Zhou, Y. Mei, K. Liao, Y.-L. Wen, J. Li, X.-D. Zhang, S. Du, H. Yan, and S.-L. Zhu, ``$\delta$-Quench Measurement of a Pure Quantum-State Wave Function'' Phys. Rev. Lett. 123, 190402 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.190402

[44] S. N. Sahoo, S. Chakraborti, A. K. Pati, and U. Sinha, ``Quantum State Interferography'' Phys. Rev. Lett. 125, 123601 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.123601

Cited by

[1] Dripto Biswas and Subir Ghosh, "Quantum Backflow Across A Black Hole Horizon", arXiv:2105.03944.

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