When does a particle arrive?

Simone Roncallo1,2, Krzysztof Sacha3, and Lorenzo Maccone1,2

1Dipartimento di Fisica, Università degli Studi di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy
2INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy
3Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland

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We compare the proposals that have appeared in the literature to describe a measurement of the time of arrival of a quantum particle at a detector. We show that there are multiple regimes where different proposals give inequivalent, experimentally discriminable, predictions. This analysis paves the way for future experimental tests.

Measurements of time are problematic in quantum mechanics since, unlike position and momentum, time is not described by an observable. Simple questions like "When does a particle arrive at a detector?" are difficult to treat. In the literature, this is the time of arrival problem. Several solutions have been considered, mostly grouped into three main approaches: Kijowski's axiomatic construction, the quantum flux and the quantum clock proposals. However, they all lead to different predictions!

We identify realizable regimes for experimentally discriminating these approaches. Our results show that discrepancies appear in strongly quantum regimes, namely when the particle displays quantum interference in the time of arrival: destructive interference at times when it is less likely to detect the particle, constructive interference when detection is more likely to happen.

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► References

[1] W. Pauli, General Principles of Quantum Mechanics (Springer, 1980).

[2] N. Vona and D. Dürr, The role of the probability current for time measurements, in The Message of Quantum Science: Attempts Towards a Synthesis, edited by P. Blanchard and J. Fröhlich (Springer, 2015) Chap. 5.

[3] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

[4] S. Das and W. Struyve, Questioning the adequacy of certain quantum arrival-time distributions, Phys. Rev. A 104, 042214 (2021).

[5] Y. Aharonov and D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy, Phys. Rev. 122, 1649 (1961).

[6] N. Grot, C. Rovelli, and R. S. Tate, Time of arrival in quantum mechanics, Phys. Rev. A 54, 4676 (1996).

[7] E. A. Galapon, F. Delgado, J. G. Muga, and I. L. Egusquiza, Transition from discrete to continuous time-of-arrival distribution for a quantum particle, Phys. Rev. A 72, 042107 (2005).

[8] J. Kijowski, On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time, Rep. Math. Phys. 6, 361 (1974).

[9] V. Delgado and J. G. Muga, Arrival time in quantum mechanics, Phys. Rev. A 56, 3425 (1997).

[10] A. Ruschhaupt and R. F. Werner, Quantum mechanics of time, in The Message of Quantum Science: Attempts Towards a Synthesis, edited by P. Blanchard and J. Fröhlich (Springer, 2015) Chap. 14.

[11] R. Werner, Screen observables in relativistic and nonrelativistic quantum mechanics, J. Math. Phys. 27, 793 (1986).

[12] Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, Measurement of time of arrival in quantum mechanics, Phys. Rev. A 57, 4130 (1998).

[13] T. Jurić and H. Nikolić, Arrival time from the general theory of quantum time distributions, Eur. Phys. J. Plus 137, 631 (2022).

[14] Y. Aharonov and T. Kaufherr, Quantum frames of reference, Phys. Rev. D 30, 368 (1984).

[15] Y. Aharonov, S. Popescu, and J. Tollaksen, Each instant of time a new universe, in Quantum Theory: A Two-Time Success Story (Springer, 2014) pp. 21–36.

[16] C. Rovelli, Relational quantum mechanics, Int. J. Theor. Phys. 35, 1637 (1996).

[17] M. Reisenberger and C. Rovelli, Spacetime states and covariant quantum theory, Phys. Rev. D 65, 125016 (2002).

[18] D. N. Page and W. K. Wootters, Evolution without evolution: Dynamics described by stationary observables, Phys. Rev. D 27, 2885 (1983).

[19] L. Maccone and K. Sacha, Quantum measurements of time, Phys. Rev. Lett. 124, 110402 (2020).

[20] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum time, Phys. Rev. D 92, 045033 (2015).

[21] R. Brunetti, K. Fredenhagen, and M. Hoge, Time in quantum physics: From an external parameter to an intrinsic observable, Found. Phys. 40, 1368–1378 (2009).

[22] S. Das and D. Dürr, Arrival time distributions of spin-1/​2 particles, Sci. Rep. 9, 2242 (2019).

[23] C. R. Leavens, Time of arrival in quantum and Bohmian mechanics, Phys. Rev. A 58, 840 (1998).

[24] A. Ananthaswamy, Can we gauge quantum time of flight?, Sci. Am. 326, 1 (2022).

[25] J. G. Muga, R. S. Mayato, and I. L. Egusquiza, Time in Quantum Mechanics, Vol. 1 (Springer, 2008).

[26] G. Muga, A. Ruschhaupt, and A. Campo, Time in Quantum Mechanics, Vol. 2 (Springer, 2009).

[27] M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, Coherent splitting of Bose-Einstein condensed atoms with optically induced bragg diffraction, Phys. Rev. Lett. 82, 871 (1999).

[28] S. Pandey, H. Mas, G. Drougakis, P. Thekkeppatt, V. Bolpasi, G. Vasilakis, K. Poulios, and W. von Klitzing, Hypersonic Bose–Einstein condensates in accelerator rings, Nature 570, 205 (2019).

[29] C. R. Leavens, Spatial nonlocality of the “standard” arrival-time distribution, Phys. Lett. A 338, 19 (2005a).

[30] C. R. Leavens, On the “standard” quantum mechanical approach to times of arrival, Phys. Lett. A 303, 154 (2002).

[31] S. Das and M. Nöth, Times of arrival and gauge invariance, Proc. R. Soc. A: Math. Phys. Eng. Sci. 477, 2250 (2021).

[32] I. L. Egusquiza, J. G. Muga, B. Navarro, and A. Ruschhaupt, Comment on: “On the standard quantum-mechanical approach to times of arrival”, Phys. Lett. A 313, 498 (2003).

[33] C. R. Leavens, Reply to Comment on: “On the ‘standard’ quantum-mechanical approach to times of arrival” [Phys. Lett. A 313 (2003) 498], Phys. Lett. A 345, 251 (2005b).

[34] A. J. Bracken and G. F. Melloy, Probability backflow and a new dimensionless quantum number, J. Phys. A: Math. Theor. 27, 2197 (1994).

[35] K. V. Kuchar, Time and interpretations of quantum gravity, Int. J. Mod. Phys. D 20, 3 (2011).

[36] J. Leon and L. Maccone, The Pauli objection, Found. Phys. 47, 1597–1608 (2017).

[37] B. S. DeWitt, Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160, 1113 (1967).

[38] M. Porrmann, Particle weights and their disintegration I, Commun. Math. Phys. 248, 269–304 (2004).

[39] R. Gambini and J. Pullin, The solution to the problem of time in quantum gravity also solves the time of arrival problem in quantum mechanics, New J. Phys. 24, 053011 (2022).

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[1] Ali Ayatollah Rafsanjani, MohammadJavad Kazemi, Alireza Bahrampour, and Mehdi Golshani, "Can the double-slit experiment distinguish between quantum interpretations?", Communications Physics 6 1, 195 (2023).

[2] Tajron Jurić and Hrvoje Nikolić, "Arrival Time from Hamiltonian with Non-Hermitian Boundary Term", Universe 10 1, 35 (2024).

[3] Mathieu Beau and Lionel Martellini, "Quantum delay in the time of arrival of free-falling atoms", Physical Review A 109 1, 012216 (2024).

[4] Ruben E. Araújo, Ricardo Ximenes, and Eduardo O. Dias, "Space-time-symmetric extension of quantum mechanics: Interpretation and arrival-time predictions", Physical Review A 109 1, 012221 (2024).

[5] Ali Ayatollah Rafsanjani, MohammadJavad Kazemi, Vahid Hosseinzadeh, and Mehdi Golshani, "Non-local temporal interference", Scientific Reports 14 1, 3615 (2024).

[6] Ranjan Modak and S. Aravinda, "Non-Hermitian description of sharp quantum resetting", arXiv:2303.03790, (2023).

[7] Tajron Jurić and Hrvoje Nikolić, "Passive Quantum Measurement: Arrival Time, Quantum Zeno Effect and Gambler's Fallacy", Fortschritte der Physik 71 10-11, 2300014 (2023).

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