- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermodynamic Uncertainty Relations for Coherent Transport

Kay Brandner1,2 and Keiji Saito3

PDF

Phys. Rev. Lett. 135, 046302 – Published 23 July, 2025

DOI: https://doi.org/10.1103/6nww-8wcp

Abstract

We derive a universal thermodynamic uncertainty relation for fermionic coherent transport, which bounds the total rate of entropy production in terms of the mean and fluctuations of a single particle current. This bound holds for any multiterminal geometry and arbitrary chemical and thermal biases, as long as no external magnetic fields are applied. It can further be saturated in two-terminal settings with boxcar-shaped transmission functions and reduces to its classical counterpart in linear response. Upon insertion of a numerical factor, our bound also extends to systems with broken time-reversal symmetry. As an application, we derive trade-off relations between the figures of merit of coherent thermoelectric heat engines and refrigerators, which show that such devices can attain ideal efficiency only at vanishing mean power or diverging power fluctuations. To illustrate our results, we work out a model of a coherent conductor consisting of a chain of quantum dots.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (83)

  1. U. Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica (Amsterdam) 504A, 176 (2018).
  2. U. Seifert, From stochastic thermodynamics to thermodynamic inference, Annu. Rev. Condens. Matter Phys. 10, 171 (2019).
  3. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  4. P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Phys. Rev. Lett. 120, 190602 (2018).
  5. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  6. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  7. T. Koyuk and U. Seifert, Operationally accessible bounds on fluctuations and entropy production in periodically driven systems, Phys. Rev. Lett. 122, 230601 (2019).
  8. T. Koyuk and U. Seifert, Thermodynamic uncertainty relation for time-dependent driving, Phys. Rev. Lett. 125, 260604 (2020).
  9. J. P. Garrahan, Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables, Phys. Rev. E 95, 032134 (2017).
  10. I. Di Terlizzi and M. Baiesi, Kinetic uncertainty relation, J. Phys. A 52, 02LT03 (2019).
  11. J. Yan, A. Hilfinger, G. Vinnicombe, and J. Paulsson, Kinetic uncertainty relations for the control of stochastic reaction networks, Phys. Rev. Lett. 123, 108101 (2019).
  12. K. Hiura and S.-i. Sasa, Kinetic uncertainty relation on first-passage time for accumulated current, Phys. Rev. E 103, L050103 (2021).
  13. A. Dechant and S.-i. Sasa, Fluctuation–response inequality out of equilibrium, Proc. Natl. Acad. Sci. U.S.A. 117, 6430 (2020).
  14. Y. Wang, Sub-Gaussian and subexponential fluctuation-response inequalities, Phys. Rev. E 102, 052105 (2020).
  15. H.-M. Chun and J. M. Horowitz, Trade-offs between number fluctuations and response in nonequilibrium chemical reaction networks, J. Chem. Phys. 158, 174115 (2023).
  16. E. Kwon, H.-M. Chun, H. Park, and J. S. Lee, Fluctuation-response inequalities for kinetic and entropic perturbations, arXiv:2411.18108.
  17. K. Macieszczak, K. Brandner, and J. P. Garrahan, Unified thermodynamic uncertainty relations in linear response, Phys. Rev. Lett. 121, 130601 (2018).
  18. F. Carollo, R. L. Jack, and J. P. Garrahan, Unraveling the large deviation statistics of Markovian open quantum systems, Phys. Rev. Lett. 122, 130605 (2019).
  19. G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Thermodynamics of precision in quantum nonequilibrium steady states, Phys. Rev. Res. 1, 033021 (2019).
  20. Y. Hasegawa, Quantum thermodynamic uncertainty relation for continuous measurement, Phys. Rev. Lett. 125, 050601 (2020).
  21. E. Potanina, C. Flindt, M. Moskalets, and K. Brandner, Thermodynamic bounds on coherent transport in periodically driven conductors, Phys. Rev. X 11, 021013 (2021).
  22. Y. Hasegawa, Thermodynamic uncertainty relation for general open quantum systems, Phys. Rev. Lett. 126, 010602 (2021).
  23. A. Rignon-Bret, G. Guarnieri, J. Goold, and M. T. Mitchison, Thermodynamics of precision in quantum nanomachines, Phys. Rev. E 103, 012133 (2021).
  24. P. Menczel, E. Loisa, K. Brandner, and C. Flindt, Thermodynamic uncertainty relations for coherently driven open quantum systems, J. Phys. A 54, 314002 (2021).
  25. T. Van Vu and K. Saito, Thermodynamics of precision in Markovian open quantum dynamics, Phys. Rev. Lett. 128, 140602 (2022).
  26. K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Verdozzi, P. Samuelsson, and P. P. Potts, Entanglement and thermokinetic uncertainty relations in coherent mesoscopic transport, Phys. Rev. Res. 5, 023155 (2023).
  27. T. V. Vu, Fundamental bounds on precision and response for quantum trajectory observables, PRX Quantum 6, 010343 (2025).
  28. S. V. Moreira, M. Radaelli, A. Candeloro, F. C. Binder, and M. T. Mitchison, Precision bounds for multiple currents in open quantum systems, Phys. Rev. E 111, 064107 (2025).
  29. E. Kwon and J. S. Lee, A unified framework for classical and quantum uncertainty relations using stochastic representations, arXiv:2412.04988.
  30. B. J. Van Wees, H. Van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. Van Der Marel, and C. T. Foxon, Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett. 60, 848 (1988).
  31. D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, One-dimensional transport and the quantisation of the ballistic resistance, J. Phys. C 21, L209 (1988).
  32. K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L. Roukes, Measurement of the quantum of thermal conductance, Nature (London) 404, 974 (2000).
  33. J. Matthews, F. Battista, D. Sánchez, P. Samuelsson, and H. Linke, Experimental verification of reciprocity relations in quantum thermoelectric transport, Phys. Rev. B 90, 165428 (2014).
  34. J. M. Krans, J. M. Van Ruitenbeek, V. V. Fisun, I. K. Yanson, and L. J. De Jongh, The signature of conductance quantization in metallic point contacts, Nature (London) 375, 767 (1995).
  35. H. E. Van Den Brom and J. M. Van Ruitenbeek, Quantum suppression of shot noise in atom-size metallic contacts, Phys. Rev. Lett. 82, 1526 (1999).
  36. L. Cui, W. Jeong, S. Hur, M. Matt, J. C. Klöckner, F. Pauly, P. Nielaba, J. C. Cuevas, E. Meyhofer, and P. Reddy, Quantized thermal transport in single-atom junctions, Science 355, 1192 (2017).
  37. O. S. Lumbroso, L. Simine, A. Nitzan, D. Segal, and O. Tal, Electronic noise due to temperature differences in atomic-scale junctions, Nature (London) 562, 240 (2018).
  38. J.-P. Brantut, J. Meineke, D. Stadler, S. Krinner, and T. Esslinger, Conduction of ultracold fermions through a mesoscopic channel, Science 337, 1069 (2012).
  39. J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, and A. Georges, A thermoelectric heat engine with ultracold atoms, Science 342, 713 (2013).
  40. S. Krinner, D. Stadler, D. Husmann, J.-P. Brantut, and T. Esslinger, Observation of quantized conductance in neutral matter, Nature (London) 517, 64 (2015).
  41. M. Lebrat, P. Grišins, D. Husmann, S. Häusler, L. Corman, T. Giamarchi, J.-P. Brantut, and T. Esslinger, Band and correlated insulators of cold fermions in a mesoscopic lattice, Phys. Rev. X 8, 011053 (2018).
  42. U. Sivan and Y. Imry, Multichannel Landauer formula for thermoelectric transport with application to thermopower near the mobility edge, Phys. Rev. B 33, 551 (1986).
  43. M. Büttiker, Scattering theory of current and intensity noise correlations in conductors and wave guides, Phys. Rev. B 46, 12485 (1992).
  44. G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Fundamental aspects of steady-state conversion of heat to work at the nanoscale, Phys. Rep. 694, 1 (2017).
  45. S. Saryal, H. M. Friedman, D. Segal, and B. K. Agarwalla, Thermodynamic uncertainty relation in thermal transport, Phys. Rev. E 100, 042101 (2019).
  46. K. Brandner, T. Hanazato, and K. Saito, Thermodynamic bounds on precision in ballistic multiterminal transport, Phys. Rev. Lett. 120, 090601 (2018).
  47. B. K. Agarwalla and D. Segal, Assessing the validity of the thermodynamic uncertainty relation in quantum systems, Phys. Rev. B 98, 155438 (2018).
  48. T. Ehrlich and G. Schaller, Broadband frequency filters with quantum dot chains, Phys. Rev. B 104, 045424 (2021).
  49. M. Gerry and D. Segal, Absence and recovery of cost-precision tradeoff relations in quantum transport, Phys. Rev. B 105, 155401 (2022).
  50. A. M. Timpanaro, G. Guarnieri, and G. T. Landi, Quantum thermoelectric transmission functions with minimal current fluctuations, Phys. Rev. B 111, 014301 (2025).
  51. J. Eriksson, M. Acciai, L. Tesser, and J. Splettstoesser, General bounds on electronic shot noise in the absence of currents, Phys. Rev. Lett. 127, 136801 (2021).
  52. L. Tesser, M. Acciai, C. Spånslätt, J. Monsel, and J. Splettstoesser, Charge, spin, and heat shot noises in the absence of average currents: Conditions on bounds at zero and finite frequencies, Phys. Rev. B 107, 075409 (2023).
  53. L. Tesser and J. Splettstoesser, Out-of-equilibrium fluctuation-dissipation bounds, Phys. Rev. Lett. 132, 186304 (2024).
  54. M. Acciai, L. Tesser, J. Eriksson, R. Sánchez, R. S. Whitney, and J. Splettstoesser, Constraints between entropy production and its fluctuations in nonthermal engines, Phys. Rev. B 109, 075405 (2024).
  55. G. Blasi, R.  R. Rodríguez, M. Moskalets, R. López, and G. Haack, Quantum kinetic uncertainty relations in mesoscopic conductors at strong coupling, arXiv:2505.13200.
  56. D. Palmqvist, L. Tesser, and J. Splettstoesser, Kinetic uncertainty relations for quantum transport, arXiv:2410.10793.
  57. Y. Hasegawa and T. Van Vu, Fluctuation theorem uncertainty relation, Phys. Rev. Lett. 123, 110602 (2019).
  58. A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from exchange fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019).
  59. P. P. Potts and P. Samuelsson, Thermodynamic uncertainty relations including measurement and feedback, Phys. Rev. E 100, 052137 (2019).
  60. G. Falasco, M. Esposito, and J.-C. Delvenne, Unifying thermodynamic uncertainty relations, New J. Phys. 22, 053046 (2020).
  61. M. Büttiker, Coherent and sequential tunneling in series barriers, IBM J. Res. Dev. 32, 63 (1988).
  62. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6nww-8wcp for further details on the derivations of the bounds (2) and (16).
  63. G. Nenciu, Independent electron model for open quantum systems: Landauer-Büttiker formula and strict positivity of the entropy production, J. Math. Phys. (N.Y.) 48, 033302 (2007).
  64. Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H. Shtrikman, An electronic Mach–Zehnder interferometer, Nature (London) 422, 415 (2003).
  65. G. Granger, J. P. Eisenstein, and J. L. Reno, Observation of chiral heat transport in the quantum Hall regime, Phys. Rev. Lett. 102, 086803 (2009).
  66. S.-G. Nam, E. H. Hwang, and H.-J. Lee, Thermoelectric detection of chiral heat transport in graphene in the quantum Hall regime, Phys. Rev. Lett. 110, 226801 (2013).
  67. S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, and F. Pierre, Quantum limit of heat flow across a single electronic channel, Science 342, 601 (2013).
  68. C. Bäuerle, D. Christian Glattli, T. Meunier, F. Portier, P. Roche, P. Roulleau, S. Takada, and X. Waintal, Coherent control of single electrons: A review of current progress, Rep. Prog. Phys. 81, 056503 (2018).
  69. E. Sivre, H. Duprez, A. Anthore, A. Aassime, F. D. Parmentier, A. Cavanna, A. Ouerghi, U. Gennser, and F. Pierre, Electronic heat flow and thermal shot noise in quantum circuits, Nat. Commun. 10, 5638 (2019).
  70. R. S. Whitney, Most Efficient quantum thermoelectric at finite power output, Phys. Rev. Lett. 112, 130601 (2014).
  71. R. S. Whitney, Finding the quantum thermoelectric with maximal efficiency and minimal entropy production at given power output, Phys. Rev. B 91, 115425 (2015).
  72. L. Tesser, R. S. Whitney, and J. Splettstoesser, Thermodynamic performance of hot-carrier solar cells: A quantum transport model, Phys. Rev. Appl. 19, 044038 (2023).
  73. S. Mishra and C. Benjamin, Reaching the Van den Broeck limit in linear response and the Whitney limit in nonlinear response in edge mode quantum thermoelectrics and refrigeration, Phys. Rev. B 110, 235430 (2024).
  74. C. Volk, A. M. J. Zwerver, U. Mukhopadhyay, P. T. Eendebak, C. J. Van Diepen, J. P. Dehollain, T. Hensgens, T. Fujita, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Loading a quantum-dot based “Qubyte” register, npj Quantum Inf. 5, 29 (2019).
  75. A. R. Mills, D. M. Zajac, M. J. Gullans, F. J. Schupp, T. M. Hazard, and J. R. Petta, Shuttling a single charge across a one-dimensional array of silicon quantum dots, Nat. Commun. 10, 1063 (2019).
  76. A. R. Mills, M. M. Feldman, C. Monical, P. J. Lewis, K. W. Larson, A. M. Mounce, and J. R. Petta, Computer-automated tuning procedures for semiconductor quantum dot arrays, Appl. Phys. Lett. 115, 113501 (2019).
  77. H. Qiao, Y. P. Kandel, K. Deng, S. Fallahi, G. C. Gardner, M. J. Manfra, E. Barnes, and J. M. Nichol, Coherent multispin exchange coupling in a quantum-dot spin chain, Phys. Rev. X 10, 031006 (2020).
  78. T.-K. Hsiao, C. Van Diepen, U. Mukhopadhyay, C. Reichl, W. Wegscheider, and L. Vandersypen, Efficient orthogonal control of tunnel couplings in a quantum dot array, Phys. Rev. Appl. 13, 054018 (2020).
  79. Y. P. Kandel, H. Qiao, S. Fallahi, G. C. Gardner, M. J. Manfra, and J. M. Nichol, Adiabatic quantum state transfer in a semiconductor quantum-dot spin chain, Nat. Commun. 12, 2156 (2021).
  80. C. J. Van Diepen, T.-K. Hsiao, U. Mukhopadhyay, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Quantum simulation of antiferromagnetic Heisenberg chain with gate-defined quantum dots, Phys. Rev. X 11, 041025 (2021).

  81. We have verified this result using computer algebra for N=1,…,10 and numerically for N=11,…,100.

  82. K. Brandner, Coherent transport in periodically driven mesoscopic conductors: From scattering amplitudes to quantum thermodynamics, Z. Naturforsch. 75, 483 (2020).
  83. D. C. Ohnmacht, J. C. Cuevas, W. Belzig, R. López, J. S. Lim, and K. W. Kim, Thermodynamic uncertainty relations in superconducting junctions, Phys. Rev. Res. 7, L012075 (2025).

Outline

Information

Phys. Rev. Lett. 135, 046302– Published 23 July, 2025
  • Received 18 February 2025
  • Accepted 27 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation