Finite-time Landauer principle beyond weak coupling

Alberto Rolandi; Martí Perarnau-Llobet

doi:10.22331/q-2023-11-03-1161

Finite-time Landauer principle beyond weak coupling

Alberto Rolandi and Martí Perarnau-Llobet

Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland

Published:	2023-11-03, volume 7, page 1161
Eprint:	arXiv:2211.02065v3
Doi:	https://doi.org/10.22331/q-2023-11-03-1161
Citation:	Quantum 7, 1161 (2023).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Landauer's principle gives a fundamental limit to the thermodynamic cost of erasing information. Its saturation requires a reversible isothermal process, and hence infinite time. We develop a finite-time version of Landauer's principle for a bit encoded in the occupation of a single fermionic mode, which can be strongly coupled to a reservoir. By solving the exact non-equilibrium dynamics, we optimize erasure processes (taking both the fermion's energy and system-bath coupling as control parameters) in the slow driving regime through a geometric approach to thermodynamics. We find analytic expressions for the thermodynamic metric and geodesic equations, which can be solved numerically. Their solution yields optimal processes that allow us to characterize a finite-time correction to Landauer's bound, fully taking into account non-markovian and strong coupling effects.

Featured image: Dissipation of optimal erasure protocols as a function of the chosen coupling to the thermal bath in the weak coupling approximation and in the exact case. These cases are compared to the fundamental bound obtained by optimizing the value of the coupling during the erasure protocol.

Popular summary

Landauer's principle gives a fundamental limit to the thermodynamic cost of erasing information. However, reaching this limit requires an infinite amount of time. In this work, we combine finite-time quantum thermodynamics and geometric thermodynamics to investigate finite-time corrections to Landauer's principle. Previous works derive general bounds that are obtained from the weakly-coupled Markovian regime, but the strong coupling regime remained unexplored so far. Our work investigates this regime and shows that strong coupling is needed for optimal finite-time erasure. Furthermore, our results suggest the appearance of the Planckian dissipation time as the shortest timescale for information erasure.

► BibTeX data

@article{Rolandi2023finitetimelandauer,
  doi = {10.22331/q-2023-11-03-1161},
  url = {https://doi.org/10.22331/q-2023-11-03-1161},
  title = {Finite-time {L}andauer principle beyond weak coupling},
  author = {Rolandi, Alberto and Perarnau-Llobet, Mart{\'{i}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1161},
  month = nov,
  year = {2023}
}

► References

[1] R. Landauer, IBM Journal of Research and Development 5, 183 (1961).
https://doi.org/10.1147/rd.53.0183

[2] T. Sagawa and M. Ueda, Phys. Rev. Lett. 102, 250602 (2009).
https://doi.org/10.1103/PhysRevLett.102.250602

[3] M. Esposito and C. V. den Broeck, EPL (Europhysics Letters) 95, 40004 (2011).
https://doi.org/10.1209/0295-5075/95/40004

[4] S. Hilt, S. Shabbir, J. Anders, and E. Lutz, Phys. Rev. E 83, 030102 (2011).
https://doi.org/10.1103/PhysRevE.83.030102

[5] S. Deffner and C. Jarzynski, Phys. Rev. X 3, 041003 (2013).
https://doi.org/10.1103/PhysRevX.3.041003

[6] D. Reeb and M. M. Wolf, New Journal of Physics 16, 103011 (2014).
https://doi.org/10.1088/1367-2630/16/10/103011

[7] P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, Nature Communications 6 (2015), 10.1038/ncomms8669.
https://doi.org/10.1038/ncomms8669

[8] S. Lorenzo, R. McCloskey, F. Ciccarello, M. Paternostro, and G. M. Palma, Phys. Rev. Lett. 115, 120403 (2015).
https://doi.org/10.1103/PhysRevLett.115.120403

[9] J. Goold, M. Paternostro, and K. Modi, Phys. Rev. Lett. 114, 060602 (2015).
https://doi.org/10.1103/PhysRevLett.114.060602

[10] A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, Phys. Rev. X 6, 041017 (2016).
https://doi.org/10.1103/PhysRevX.6.041017

[11] G. Guarnieri, S. Campbell, J. Goold, S. Pigeon, B. Vacchini, and M. Paternostro, New Journal of Physics 19, 103038 (2017).
https://doi.org/10.1088/1367-2630/aa8cf1

[12] H. J. Miller, G. Guarnieri, M. T. Mitchison, and J. Goold, Physical Review Letters 125 (2020), 10.1103/physrevlett.125.160602.
https://doi.org/10.1103/physrevlett.125.160602

[13] A. M. Timpanaro, J. P. Santos, and G. T. Landi, Phys. Rev. Lett. 124, 240601 (2020).
https://doi.org/10.1103/PhysRevLett.124.240601

[14] P. M. Riechers and M. Gu, Phys. Rev. A 104, 012214 (2021).
https://doi.org/10.1103/PhysRevA.104.012214

[15] L. Buffoni and M. Campisi, Journal of Statistical Physics 186 (2022), 10.1007/s10955-022-02877-8.
https://doi.org/10.1007/s10955-022-02877-8

[16] P. Taranto, F. Bakhshinezhad, A. Bluhm, R. Silva, N. Friis, M. P. Lock, G. Vitagliano, F. C. Binder, T. Debarba, E. Schwarzhans, F. Clivaz, and M. Huber, PRX Quantum 4, 010332 (2023).
https://doi.org/10.1103/PRXQuantum.4.010332

[17] C. Browne, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Phys. Rev. Lett. 113, 100603 (2014).
https://doi.org/10.1103/PhysRevLett.113.100603

[18] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature 483, 187 (2012).
https://doi.org/10.1088/1742-5468/2015/06/P06015

[19] Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. 113, 190601 (2014).
https://doi.org/10.1103/PhysRevLett.113.190601

[20] A. Bérut, A. Petrosyan, and S. Ciliberto, J. Stat. Mech. 2015, P06015 (2015).
https://doi.org/10.1088/1742-5468/2015/06/p06015

[21] M. Gavrilov and J. Bechhoefer, Phys. Rev. Lett. 117, 200601 (2016).
https://doi.org/10.1103/PhysRevLett.117.200601

[22] J. Hong, B. Lambson, S. Dhuey, and J. Bokor, Sci. Adv. 2 (2016), 10.1126/sciadv.1501492.
https://doi.org/10.1126/sciadv.1501492

[23] L. Martini, M. Pancaldi, M. Madami, P. Vavassori, G. Gubbiotti, S. Tacchi, F. Hartmann, M. Emmerling, S. Höfling, L. Worschech, and G. Carlotti, Nano Energy 19, 108 (2016).
https://doi.org/10.1016/j.nanoen.2015.10.028

[24] R. Gaudenzi, E. Burzurí, S. Maegawa, H. S. J. van der Zant, and F. Luis, Nat. Phys. 14, 565 (2018).
https://doi.org/10.1038/s41567-018-0070-7

[25] O. Saira, M. H. Matheny, R. Katti, W. Fon, G. Wimsatt, J. P. Crutchfield, S. Han, and M. L. Roukes, Phys. Rev. Res. 2, 013249 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013249

[26] S. Dago, J. Pereda, N. Barros, S. Ciliberto, and L. Bellon, Phys. Rev. Lett. 126, 170601 (2021).
https://doi.org/10.1103/PhysRevLett.126.170601

[27] S. Dago and L. Bellon, Phys. Rev. Lett. 128, 070604 (2022).
https://doi.org/10.1103/PhysRevLett.128.070604

[28] M. A. Ciampini, T. Wenzl, M. Konopik, G. Thalhammer, M. Aspelmeyer, E. Lutz, and N. Kiesel, arXiv:2107.04429 (2021).
https://doi.org/10.48550/arXiv.2107.04429
arXiv:2107.04429

[29] M. Scandi, D. Barker, S. Lehmann, K. A. Dick, V. F. Maisi, and M. Perarnau-Llobet, Phys. Rev. Lett. 129, 270601 (2022).
https://doi.org/10.1103/PhysRevLett.129.270601

[30] Nernst, W. Sitzber. Kgl. Preuss. Akad. Wiss. Physik-Math. Kl. 134 (1912), missing.

[31] L. Masanes and J. Oppenheim, Nature Communications 8 (2017), 10.1038/ncomms14538.
https://doi.org/10.1038/ncomms14538

[32] N. Freitas, R. Gallego, L. Masanes, and J. P. Paz, in Fundamental Theories of Physics (Springer International Publishing, 2018) pp. 597–622.
https://doi.org/10.1007/978-3-319-99046-0_25

[33] E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Phys. Rev. Lett. 106, 250601 (2011).
https://doi.org/10.1103/PhysRevLett.106.250601

[34] E. Aurell, K. Gawȩdzki, C. Mejía-Monasterio, R. Mohayaee, and P. Muratore-Ginanneschi, Journal of Statistical Physics 147, 487 (2012).
https://doi.org/10.1007/s10955-012-0478-x

[35] T. Van Vu and K. Saito, Phys. Rev. X 13, 011013 (2023).
https://doi.org/10.1103/PhysRevX.13.011013

[36] P. Salamon, B. Andresen, P. D. Gait, and R. S. Berry, J. Chem. Phys. 73, 1001 (1980).
https://doi.org/10.1063/1.440217

[37] P. Salamon and R. S. Berry, Phys. Rev. Lett. 51, 1127 (1983).
https://doi.org/10.1103/PhysRevLett.51.1127

[38] J. Nulton, P. Salamon, B. Andresen, and A. Qi, J. Chem. Phys. 83, 334 (1985).
https://doi.org/10.1063/1.449774

[39] B. Andresen, R. S. Berry, R. Gilmore, E. Ihrig, and P. Salamon, Phys. Rev. A 37, 845 (1988).
https://doi.org/10.1103/PhysRevA.37.845

[40] D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012).
https://doi.org/10.1103/PhysRevLett.108.190602

[41] S. Deffner and M. V. S. Bonança, EPL (Europhysics Letters) 131, 20001 (2020).
https://doi.org/10.1209/0295-5075/131/20001

[42] P. Abiuso, H. J. D. Miller, M. Perarnau-Llobet, and M. Scandi, Entropy 22, 1076 (2020).
https://doi.org/10.3390/e22101076

[43] T. V. Vu and Y. Hasegawa, Physical Review Letters 126 (2021), 10.1103/physrevlett.126.010601.
https://doi.org/10.1103/physrevlett.126.010601

[44] P. R. Zulkowski and M. R. DeWeese, Phys. Rev. E 89, 052140 (2014).
https://doi.org/10.1103/PhysRevE.89.052140

[45] K. Proesmans, J. Ehrich, and J. Bechhoefer, Phys. Rev. Lett. 125, 100602 (2020a).
https://doi.org/10.1103/PhysRevLett.125.100602

[46] K. Proesmans, J. Ehrich, and J. Bechhoefer, Phys. Rev. E 102, 032105 (2020b).
https://doi.org/10.1103/PhysRevE.102.032105

[47] A. B. Boyd, A. Patra, C. Jarzynski, and J. P. Crutchfield, Journal of Statistical Physics 187, 1 (2022).
https://doi.org/10.1007/s10955-022-02871-0

[48] J. S. Lee, S. Lee, H. Kwon, and H. Park, Phys. Rev. Lett. 129, 120603 (2022).
https://doi.org/10.1103/PhysRevLett.129.120603

[49] G. Diana, G. B. Bagci, and M. Esposito, Phys. Rev. E 87, 012111 (2013).
https://doi.org/10.1103/PhysRevE.87.012111

[50] M. Scandi and M. Perarnau-Llobet, Quantum 3, 197 (2019).
https://doi.org/10.22331/q-2019-10-24-197

[51] Y.-Z. Zhen, D. Egloff, K. Modi, and O. Dahlsten, Phys. Rev. Lett. 127, 190602 (2021).
https://doi.org/10.1103/PhysRevLett.127.190602

[52] T. Van Vu and K. Saito, Phys. Rev. Lett. 128, 010602 (2022).
https://doi.org/10.1103/PhysRevLett.128.010602

[53] Y.-Z. Zhen, D. Egloff, K. Modi, and O. Dahlsten, Phys. Rev. E 105, 044147 (2022).
https://doi.org/10.1103/PhysRevE.105.044147

[54] Y.-H. Ma, J.-F. Chen, C. P. Sun, and H. Dong, Phys. Rev. E 106, 034112 (2022).
https://doi.org/10.1103/PhysRevE.106.034112

[55] P. Strasberg, G. Schaller, N. Lambert, and T. Brandes, New Journal of Physics 18, 073007 (2016).
https://doi.org/10.1088/1367-2630/18/7/073007

[56] C. Jarzynski, Phys. Rev. X 7, 011008 (2017).
https://doi.org/10.1103/PhysRevX.7.011008

[57] H. J. D. Miller, in Fundamental Theories of Physics (Springer International Publishing, 2018) pp. 531–549.
https://doi.org/10.1007/978-3-319-99046-0_22

[58] A. Nazir and G. Schaller, in Fundamental Theories of Physics (Springer International Publishing, 2018) pp. 551–577.
https://doi.org/10.1007/978-3-319-99046-0_23

[59] P. Talkner and P. Hänggi, Rev. Mod. Phys. 92, 041002 (2020).
https://doi.org/10.1103/RevModPhys.92.041002

[60] A. Rivas, Phys. Rev. Lett. 124, 160601 (2020).
https://doi.org/10.1103/PhysRevLett.124.160601

[61] M. Brenes, J. J. Mendoza-Arenas, A. Purkayastha, M. T. Mitchison, S. R. Clark, and J. Goold, Phys. Rev. X 10, 031040 (2020).
https://doi.org/10.1103/PhysRevX.10.031040

[62] N. Pancotti, M. Scandi, M. T. Mitchison, and M. Perarnau-Llobet, Phys. Rev. X 10, 031015 (2020).
https://doi.org/10.1103/PhysRevX.10.031015

[63] S. Alipour, A. Chenu, A. T. Rezakhani, and A. del Campo, Quantum 4, 336 (2020).
https://doi.org/10.22331/q-2020-09-28-336

[64] K. Ptaszyński, Physical Review E 106 (2022), 10.1103/physreve.106.014114.
https://doi.org/10.1103/physreve.106.014114

[65] M. Carrega, L. M. Cangemi, G. De Filippis, V. Cataudella, G. Benenti, and M. Sassetti, PRX Quantum 3, 010323 (2022).
https://doi.org/10.1103/PRXQuantum.3.010323

[66] F. Cavaliere, M. Carrega, G. D. Filippis, V. Cataudella, G. Benenti, and M. Sassetti, Physical Review Research 4 (2022), 10.1103/physrevresearch.4.033233.
https://doi.org/10.1103/physrevresearch.4.033233

[67] F. Ivander, N. Anto-Sztrikacs, and D. Segal, Phys. Rev. E 105, 034112 (2022).
https://doi.org/10.1103/PhysRevE.105.034112

[68] D. Newman, F. Mintert, and A. Nazir, Phys. Rev. E 95, 032139 (2017).
https://doi.org/10.1103/PhysRevE.95.032139

[69] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert, Physical Review Letters 120 (2018), 10.1103/physrevlett.120.120602.
https://doi.org/10.1103/physrevlett.120.120602

[70] P. Strasberg, G. Schaller, T. L. Schmidt, and M. Esposito, Physical Review B 97 (2018), 10.1103/physrevb.97.205405.
https://doi.org/10.1103/physrevb.97.205405

[71] M. Wiedmann, J. T. Stockburger, and J. Ankerhold, New Journal of Physics 22, 033007 (2020).
https://doi.org/10.1088/1367-2630/ab725a

[72] J. Liu, K. A. Jung, and D. Segal, Phys. Rev. Lett. 127, 200602 (2021).
https://doi.org/10.1103/PhysRevLett.127.200602

[73] Y. Shirai, K. Hashimoto, R. Tezuka, C. Uchiyama, and N. Hatano, Phys. Rev. Research 3, 023078 (2021).
https://doi.org/10.1103/PhysRevResearch.3.023078

[74] S. Koyanagi and Y. Tanimura, The Journal of Chemical Physics 157, 084110 (2022).
https://doi.org/10.1063/5.0107305

[75] J. Liu and K. A. Jung, Phys. Rev. E 106, L022105 (2022).
https://doi.org/10.1103/PhysRevE.106.L022105

[76] G. Schaller, Open quantum systems far from equilibrium, Vol. 881 (Springer, 2014).
https://doi.org/10.1007/978-3-319-03877-3

[77] M. F. Ludovico, J. S. Lim, M. Moskalets, L. Arrachea, and D. Sánchez, Phys. Rev. B 89, 161306 (2014).
https://doi.org/10.1103/PhysRevB.89.161306

[78] M. Esposito, M. A. Ochoa, and M. Galperin, Phys. Rev. Lett. 114, 080602 (2015a).
https://doi.org/10.1103/PhysRevLett.114.080602

[79] M. Esposito, M. A. Ochoa, and M. Galperin, Phys. Rev. B 92, 235440 (2015b).
https://doi.org/10.1103/PhysRevB.92.235440

[80] A. Bruch, M. Thomas, S. Viola Kusminskiy, F. von Oppen, and A. Nitzan, Phys. Rev. B 93, 115318 (2016).
https://doi.org/10.1103/PhysRevB.93.115318

[81] P. Haughian, M. Esposito, and T. L. Schmidt, Phys. Rev. B 97, 085435 (2018).
https://doi.org/10.1103/PhysRevB.97.085435

[82] M. T. Mitchison and M. B. Plenio, New Journal of Physics 20, 033005 (2018).
https://doi.org/10.1088/1367-2630/aa9f70

[83] K. Tong and W. Dou, Journal of Physics: Condensed Matter 34, 495703 (2022).
https://doi.org/10.1088/1361-648x/ac99c8

[84] S. A. Hartnoll and A. P. Mackenzie, Rev. Mod. Phys. 94, 041002 (2022).
https://doi.org/10.1103/RevModPhys.94.041002

[85] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).
https://doi.org/10.1017/CBO9780511973765

[86] J. Maldacena, S. H. Shenker, and D. Stanford, Journal of High Energy Physics 2016 (2016), 10.1007/jhep08(2016)106.
https://doi.org/10.1007/jhep08(2016)106

[87] S. Pappalardi and J. Kurchan, SciPost Phys. 13, 006 (2022).
https://doi.org/10.21468/SciPostPhys.13.1.006

[88] P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese, Phys. Rev. E 86, 041148 (2012).
https://doi.org/10.1103/PhysRevE.86.041148

[89] M. V. S. Bonança and S. Deffner, J. Chem. Phys. 140, 244119 (2014).
https://doi.org/10.1063/1.4885277

[90] G. M. Rotskoff, G. E. Crooks, and E. Vanden-Eijnden, Phys. Rev. E 95, 012148 (2017).
https://doi.org/10.1103/PhysRevE.95.012148

[91] G. Li, J.-F. Chen, C. P. Sun, and H. Dong, Phys. Rev. Lett. 128, 230603 (2022).
https://doi.org/10.1103/PhysRevLett.128.230603

[92] J. Eglinton and K. Brandner, Phys. Rev. E 105, L052102 (2022).
https://doi.org/10.1103/PhysRevE.105.L052102

[93] A. G. Frim and M. R. DeWeese, Phys. Rev. Lett. 128, 230601 (2022).
https://doi.org/10.1103/PhysRevLett.128.230601

[94] J.-F. Chen, R.-X. Zhai, C. Sun, and H. Dong, arXiv preprint arXiv:2209.07269 (2022), 10.48550/arXiv.2209.07269.
https://doi.org/10.48550/arXiv.2209.07269
arXiv:2209.07269

[95] H. J. D. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, Phys. Rev. Lett. 123, 230603 (2019).
https://doi.org/10.1103/PhysRevLett.123.230603

[96] P. Abiuso and M. Perarnau-Llobet, Phys. Rev. Lett. 124, 110606 (2020).
https://doi.org/10.1103/PhysRevLett.124.110606

[97] K. Brandner and K. Saito, Phys. Rev. Lett. 124, 040602 (2020).
https://doi.org/10.1103/PhysRevLett.124.040602

[98] P. Terrén Alonso, P. Abiuso, M. Perarnau-Llobet, and L. Arrachea, PRX Quantum 3, 010326 (2022).
https://doi.org/10.1103/PRXQuantum.3.010326

[99] M. Mehboudi and H. J. D. Miller, Phys. Rev. A 105, 062434 (2022).
https://doi.org/10.1103/PhysRevA.105.062434

[100] S. Deffner and E. Lutz, Phys. Rev. Lett. 105, 170402 (2010).
https://doi.org/10.1103/PhysRevLett.105.170402

[101] J. Eisert, M. Friesdorf, and C. Gogolin, Nature Physics 11, 124 (2015).
https://doi.org/10.1038/nphys3215

[102] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics 65, 239 (2016).
https://doi.org/10.1080/00018732.2016.1198134

[103] Y. Subaşı, C. H. Fleming, J. M. Taylor, and B. L. Hu, Physical Review E 86 (2012), 10.1103/physreve.86.061132.
https://doi.org/10.1103/physreve.86.061132

[104] M. Merkli, Annals of Physics 412, 167996 (2020).
https://doi.org/10.1016/j.aop.2019.167996

[105] J. D. Cresser and J. Anders, Phys. Rev. Lett. 127, 250601 (2021).
https://doi.org/10.1103/PhysRevLett.127.250601

[106] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, AVS Quantum Science 4, 012301 (2022).
https://doi.org/10.1116/5.0073853

[107] V. Cavina, A. Mari, and V. Giovannetti, Phys. Rev. Lett. 119, 050601 (2017).
https://doi.org/10.1103/PhysRevLett.119.050601

[108] T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007).
https://doi.org/10.1103/PhysRevLett.98.108301

[109] M. Esposito, R. Kawai, K. Lindenberg, and C. Van Den Broeck, EPL 89, 20003 (2010).
https://doi.org/10.1209/0295-5075/89/20003

[110] S. Rochette, M. Rudolph, A.-M. Roy, M. J. Curry, G. A. T. Eyck, R. P. Manginell, J. R. Wendt, T. Pluym, S. M. Carr, D. R. Ward, M. P. Lilly, M. S. Carroll, and M. Pioro-Ladrière, Applied Physics Letters 114, 083101 (2019).
https://doi.org/10.1063/1.5091111

[111] F. Evers, R. Korytár, S. Tewari, and J. M. van Ruitenbeek, Rev. Mod. Phys. 92, 035001 (2020).
https://doi.org/10.1103/RevModPhys.92.035001

[112] F. Covito, F. G. Eich, R. Tuovinen, M. A. Sentef, and A. Rubio, Journal of Chemical Theory and Computation 14, 2495 (2018).
https://doi.org/10.1021/acs.jctc.8b00077

[113] L. W. Tu, Differential Geometry (Springer International Publishing, 2017).
https://doi.org/10.1007/978-3-319-55084-8

[114] L. Fox and D. F. Mayers, Numerical Solution of Ordinary Differential Equations (Springer Netherlands, 1987).
https://doi.org/10.1007/978-94-009-3129-9

[115] A. Soriani, E. Miranda, and M. V. S. Bonança, New Journal of Physics 24, 113037 (2022).
https://doi.org/10.1088/1367-2630/aca177

[116] M. Ciorga, A. S. Sachrajda, P. Hawrylak, C. Gould, P. Zawadzki, S. Jullian, Y. Feng, and Z. Wasilewski, Phys. Rev. B 61, R16315 (2000).
https://doi.org/10.1103/PhysRevB.61.R16315

[117] J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H. Willems van Beveren, S. De Franceschi, L. M. K. Vandersypen, S. Tarucha, and L. P. Kouwenhoven, Phys. Rev. B 67, 161308 (2003).
https://doi.org/10.1103/PhysRevB.67.161308

[118] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, Proceedings of the National Academy of Sciences 111, 13786 (2014a).
https://doi.org/10.1073/pnas.1406966111

[119] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola, Phys. Rev. Lett. 113, 030601 (2014b).
https://doi.org/10.1103/PhysRevLett.113.030601

[120] S. Ismail-Beigi, Yale notes (2013).
https://volga.eng.yale.edu/sites/default/files/files/general-lorentzian-integrals.pdf

[121] P. A. Erdman, A. Rolandi, P. Abiuso, M. Perarnau-Llobet, and F. Noé, Phys. Rev. Res. 5, L022017 (2023).
https://doi.org/10.1103/PhysRevResearch.5.L022017

Cited by

[1] Alberto Rolandi, Paolo Abiuso, and Martí Perarnau-Llobet, "Collective Advantages in Finite-Time Thermodynamics", Physical Review Letters 131 21, 210401 (2023).

[2] Hong-Bo Huang, Geng Li, and Hui Dong, "Qubit Reset with a Shortcut-to-Isothermal Scheme", arXiv:2310.18997, (2023).

[3] Sayan Mondal, Aparajita Bhattacharyya, Ahana Ghoshal, and Ujjwal Sen, "Modified Landauer's principle: How much can the Maxwell's demon gain by using general system-environment quantum state?", arXiv:2309.09678, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-12-06 22:30:35) and SAO/NASA ADS (last updated successfully 2023-12-06 22:30:36). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.