Thermodynamics of ultrastrongly coupled light-matter systems

Philipp Pilar; Daniele De Bernardis; Peter Rabl

doi:10.22331/q-2020-09-28-335

Thermodynamics of ultrastrongly coupled light-matter systems

Philipp Pilar, Daniele De Bernardis, and Peter Rabl

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria

Published:	2020-09-28, volume 4, page 335
Eprint:	arXiv:2003.11556v5
Doi:	https://doi.org/10.22331/q-2020-09-28-335
Citation:	Quantum 4, 335 (2020).

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

Abstract

We study the thermodynamic properties of a system of two-level dipoles that are coupled ultrastrongly to a single cavity mode. By using exact numerical and approximate analytical methods, we evaluate the free energy of this system at arbitrary interaction strengths and discuss strong-coupling modifications of derivative quantities such as the specific heat or the electric susceptibility. From this analysis we identify the lowest-order cavity-induced corrections to those quantities in the collective ultrastrong coupling regime and show that for even stronger interactions the presence of a single cavity mode can strongly modify extensive thermodynamic quantities of a large ensemble of dipoles. In this non-perturbative coupling regime we also observe a significant shift of the ferroelectric phase transition temperature and a characteristic broadening and collapse of the black-body spectrum of the cavity mode. Apart from a purely fundamental interest, these general insights will be important for identifying potential applications of ultrastrong-coupling effects, for example, in the field of quantum chemistry or for realizing quantum thermal machines.

► BibTeX data

@article{Pilar2020thermodynamicsof,
  doi = {10.22331/q-2020-09-28-335},
  url = {https://doi.org/10.22331/q-2020-09-28-335},
  title = {Thermodynamics of ultrastrongly coupled light-matter systems},
  author = {Pilar, Philipp and De Bernardis, Daniele and Rabl, Peter},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {335},
  month = sep,
  year = {2020}
}

► References

[1] C. Ciuti, G. Bastard, and I. Carusotto, Quantum vacuum properties of the intersubband cavity polariton field, Phys. Rev. B 72, 115303 (2005).
https://doi.org/10.1103/PhysRevB.72.115303

[2] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
https://doi.org/10.1103/RevModPhys.91.025005

[3] A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
https://doi.org/10.1038/s42254-018-0006-2

[4] Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Ultrastrong Light-Matter Coupling Regime with Polariton Dots, Phys. Rev. Lett. 105, 196402 (2010).
https://doi.org/10.1103/PhysRevLett.105.196402

[5] G. Scalari, C. Maissen, D. Turcinkova, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial, Science 335, 1323 (2012).
https://doi.org/10.1126/science.1216022

[6] D. Dietze, A. M. Andrews, P. Klang, G. Strasser, K. Unterrainer, and J. Darmo, Ultrastrong coupling of intersubband plasmons and terahertz metamaterials, Appl. Phys. Lett. 103, 201106 (2013).
https://doi.org/10.1063/1.4830092

[7] C. R. Gubbin, S. A. Maier, and S. Kéna-Cohen, Low-voltage polariton electroluminescence from an ultrastrongly coupled organic light-emitting diode, App. Phys. Lett. 104, 233302 (2014).
https://doi.org/10.1063/1.4871271

[8] Q. Zhang, M. Lou, X. Li, J. L. Reno, W. Pan, J. D. Watson, M. J. Manfra, and J. Kono, Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons, Nature Phys. 12, 1005 (2016).
https://doi.org/10.1038/nphys3850

[9] A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Huber, D. Bougeard, and C. Lange, Terahertz Light-Matter Interaction beyond Unity Coupling Strength, Nano Lett. 17, 6340 (2017).
https://doi.org/10.1021/acs.nanolett.7b03103

[10] B. Askenazi, A. Vasanelli, Y. Todorov, E. Sakat, J.-J. Greffet, G. Beaudoin, I. Sagnes, and C. Sirtori, Midinfrared Ultrastrong Light-Matter Coupling for THz Thermal Emission, ACS Photonics 4, 2550 (2017).
https://doi.org/10.1021/acsphotonics.7b00838

[11] T. Schwartz, J. A. Hutchison, C. Genet and T. W. Ebbesen, Reversible Switching of Ultrastrong Light-Molecule Coupling, Phys. Rev. Lett. 106, 196405 (2011).
https://doi.org/10.1103/PhysRevLett.106.196405

[12] J. George, T. Chervy, A. Shalabney, E. Devaux, H. Hiura, C. Genet, and T. W. Ebbesen, Multiple Rabi Splittings under Ultrastrong Vibrational Coupling, Phys. Rev. Lett. 117, 153601 (2016).
https://doi.org/10.1103/PhysRevLett.117.153601

[13] J. Flick, M. Ruggenthaler, H. Appel, and A. Rubio, Atoms and Molecules in Cavities: From Weak to Strong Coupling in QED Chemistry, Proc. Natl. Acad. Sci. 114, 3026 (2017).
https://doi.org/10.1073/pnas.1615509114

[14] R. F. Ribeiro, L. A. Martinez-Martinez, M. Du, J. Campos-Gonzalez-Anguloand, and J. Yuen-Zhou, Polariton chemistry: controlling molecular dynamics with optical cavities, Chem. Sci. 9, 6325 (2018).
https://doi.org/10.1039/C8SC01043A

[15] V. N. Peters, S. Prayakarao, S. R. Koutsares, C. E. Bonner, and M. A. Noginov, Control of Physical and Chemical Processes with Nonlocal Metal–Dielectric Environments, ACS Photonics 6, 3039 (2019).
https://doi.org/10.1021/acsphotonics.9b00734

[16] J. A. Hutchison, T. Schwartz, C. Genet, E. Devaux, and T. W. Ebbesen, Modifying Chemical Landscapes by Coupling to Vacuum Fields, Angew. Chem., Int. Ed. 51, 1592 (2012).
https://doi.org/10.1002/anie.201107033

[17] A. Thomas, J. George, A. Shalabney, M. Dryzhakov, S. J. Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Ground-State Chemical Reactivity under Vibrational Coupling to the Vacuum Electromagnetic Field, Angew. Chem., Int. Ed. 55, 11462 (2016).
https://doi.org/10.1002/anie.201605504

[18] S. Wang, A. Mika, J. A. Hutchison, C. Genet, A. Jouaiti, M. W. Hosseini, and T. W. Ebbesen, Phase Transition of a Perovskite Strongly Coupled to the Vacuum Field, Nanoscale 6, 7243 (2014).
https://doi.org/10.1039/c4nr01971g

[19] A. Canaguier-Durand, E. Devaux, J. George, Y. Pang, J. A. Hutchison, T. Schwartz, C. Genet, N. Wilhelms, J.-M. Lehn, and T. W. Ebbesen, Thermodynamics of Molecules Strongly Coupled to the Vacuum Field, Angew. Chem., Int. Ed. 52, 10533 (2013).
https://doi.org/10.1002/anie.201301861

[20] M. H. Devoret, S. Girvin, and R. Schoelkopf, Circuit-QED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be?, Ann. Phys. (NY) 16, 767 (2007).
https://doi.org/10.1002/andp.200710261

[21] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nature Phys. 6, 772 (2010).
https://doi.org/10.1038/nphys1730

[22] P. Forn-Diaz, J. Lisenfeld, D. Marcos, J. J. Garcia-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime, Phys. Rev. Lett. 105, 237001 (2010).
https://doi.org/10.1103/PhysRevLett.105.237001

[23] P. Forn-Díaz, J. J. García-Ripoll, B. Peropadre, J.-L. Orgiazzi, M. A. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu, Ultrastrong coupling of a single artificial atom to an electromagnetic continuum, Nature Phys. 13, 39 (2017).
https://doi.org/10.1038/nphys3905

[24] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime, Nature Phys. 13, 44 (2017).
https://doi.org/10.1038/nphys3906

[25] R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99 (1954).
https://doi.org/10.1103/PhysRev.93.99

[26] T. Brandes, Coherent and collective quantum optical effects in mesoscopic systems, Physics Reports 408, 315 (2005).
https://doi.org/10.1016/j.physrep.2004.12.002

[27] J. J. Hopfield, Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals, Phys. Rev. 112, 1555 (1958).
https://doi.org/10.1103/PhysRev.112.1555

[28] K. Rzazewski, K. Wodkiewicz, and W. Zakowicz, Phase Transitions, Two-Level Atoms, and the $A^2$ Term, Phys. Rev. Lett. 35, 432 (1975).
https://doi.org/10.1103/PhysRevLett.35.432

[29] O. Viehmann, J. von Delft, and F. Marquardt, Superradiant Phase Transitions and the Standard Description of Circuit QED, Phys. Rev. Lett. 107, 113602 (2011).
https://doi.org/10.1103/PhysRevLett.107.113602

[30] Y. Todorov and C. Sirtori, Intersubband polaritons in the electrical dipole gauge, Phys. Rev. B 85, 045304 (2012).
https://doi.org/10.1103/PhysRevB.85.045304

[31] M. Bamba, and T. Ogawa, Stability of polarizable materials against superradiant phase transition, Phys. Rev. A 90, 063825 (2014).
https://doi.org/10.1103/PhysRevA.90.063825

[32] T. Jaako, Z.-L. Xiang, J. J. Garcia-Ripoll, and P. Rabl, Ultrastrong coupling phenomena beyond the Dicke model, Phys. Rev. A 94, 033850 (2016).
https://doi.org/10.1103/PhysRevA.94.033850

[33] D. De Bernardis, T. Jaako, and P. Rabl, Cavity quantum electrodynamics in the non-perturbative regime, Phys. Rev. A 97, 043820 (2018).
https://doi.org/10.1103/PhysRevA.97.043820

[34] V. Rokaj, D. M. Welakuh, M. Ruggenthaler, and A. Rubio, Light-matter interaction in the long-wavelength limit: no ground-state without dipole self-energy, J. Phys. B: At. Mol. Opt. Phys. 51, 034005 (2018).
https://doi.org/10.1088/1361-6455/aa9c99

[35] G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Cavity quantum electrodynamics of strongly correlated electron systems: A no-go theorem for photon condensation, Phys. Rev. B 100, 121109(R) (2019).
https://doi.org/10.1103/PhysRevB.100.121109

[36] D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl, Breakdown of gauge invariance in ultrastrong-coupling cavity QED, Phys. Rev. A 98, 053819 (2018).
https://doi.org/10.1103/PhysRevA.98.053819

[37] A. Stokes and A. Nazir, Gauge ambiguities imply Jaynes-Cummings physics remains valid in ultrastrong coupling QED, Nat. Commun. 10, 499 (2019).
https://doi.org/10.1038/s41467-018-08101-0

[38] O. Di Stefano, A. Settineri, V. Macri, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Resolution of gauge ambiguities in ultrastrong-coupling cavity QED, Nature Phys. 15, 803 (2019).
https://doi.org/10.1038/s41567-019-0534-4

[39] M. Roth, F. Hassler, and D. P. DiVincenzo, Optimal gauge for the multimode Rabi model in circuit QED, Phys. Rev. Research 1, 033128 (2019).
https://doi.org/10.1103/PhysRevResearch.1.033128

[40] Y. A. Kudenko, A. P. Slivinsky, and G. M. Zaslavsky, Interatomic Coulomb interaction influence on the superradiance phase transition, Phys. Lett. A 50, 411 (1975).
https://doi.org/10.1016/0375-9601(75)90103-6

[41] J. Keeling, Coulomb interactions, gauge invariance, and phase transitions of the Dicke model,J. Phys: Cond. Mat. 19, 295213 (2007).
https://doi.org/10.1088/0953-8984/19/29/295213

[42] A. Vukics and P. Domokos, Adequacy of the Dicke model in cavity QED: A counter-no-go statement, Phys. Rev. A 86, 053807 (2012).
https://doi.org/10.1103/PhysRevA.86.053807

[43] T. Grießer, A. Vukics, and P. Domokos, Depolarization shift of the superradiant phase transition, Phys. Rev. A 94, 033815 (2016).
https://doi.org/10.1103/PhysRevA.94.033815

[44] A. Stokes and A. Nazir, Uniqueness of the phase transition in many-dipole cavity QED systems, arXiv:1905.10697 (2019).
arXiv:1905.10697

[45] K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model, Ann. Phys. 76, 360 (1973).
https://doi.org/10.1016/0003-4916(73)90039-0

[46] Y. K. Wang, and F. T. Hioe, Phase Transition in the Dicke Model of Superradiance, Phys. Rev. A 7, 831 (1973).
https://doi.org/10.1103/PhysRevA.7.831

[47] H. J. Carmichael, C. W. Gardiner, and D. F. Walls, Higher order corrections to the Dicke superradiant phase transition, Phys. Lett. A 46, 47 (1973).
https://doi.org/10.1016/0375-9601(73)90679-8

[48] J. Galego, F. J. Garcia-Vidal, and J. Feist, Cavity-Induced Modifications of Molecular Structure in the Strong-Coupling Regime, Phys. Rev. X 5, 041022 (2015).
https://doi.org/10.1103/PhysRevX.5.041022

[49] J. A. Cwik, P. Kirton, S. De Liberato, and J. Keeling, Excitonic spectral features in strongly coupled organic polaritons, Phys. Rev. A 93, 033840 (2016).
https://doi.org/10.1103/PhysRevA.93.033840

[50] L. A. Martinez-Martinez, R. F. Ribeiro, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, Can Ultrastrong Coupling Change Ground-State Chemical Reactions?, ACS Photonics 5, 167 (2018).
https://doi.org/10.1021/acsphotonics.7b00610

[51] Y. Todorov and C. Sirtori, Few-Electron Ultrastrong Light-Matter Coupling in a Quantum LC Circuit, Phys. Rev. X 4, 041031 (2014).
https://doi.org/10.1103/PhysRevX.4.041031

[52] H. J. Lipkin, N. Meshkov, and A. J. Glick, Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory, Nucl. Phys. 62, 188 (1965).
https://doi.org/10.1016/0029-5582(65)90862-X

[53] M. Schuler, D. De Bernardis, A. M. Läuchli, and P. Rabl, The Vacua of Dipolar Cavity Quantum Electrodynamics, arXiv:2004.13738 (2020).
arXiv:2004.13738

[54] M. Bamba, K. Inomata, and Y. Nakamura, Superradiant Phase Transition in a Superconducting Circuit in Thermal Equilibrium, Phys. Rev. Lett. 117, 173601 (2016).
https://doi.org/10.1103/PhysRevLett.117.173601

[55] A. Settineri, O. Di Stefano, D. Zueco, S. Hughes, S. Savasta, and F. Nori, Gauge freedom, quantum measurements, and time-dependent interactions in cavity and circuit QED, arXiv:1912.08548 (2019).
arXiv:1912.08548

[56] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley, New York, 1997).

[57] A. Das, K. Sengupta, D. Sen, and B. K. Chakrabarti, Infinite-range Ising ferromagnet in a time-dependent transverse magnetic field: Quench and ac dynamics near the quantum critical point, Phys. Rev. B 74, 144423 (2006).
https://doi.org/10.1103/PhysRevB.74.144423

[58] H. T. Quan and F. M. Cucchietti, Quantum fidelity and thermal phase transitions, Phys. Rev. E 79, 031101 (2009).
https://doi.org/10.1103/PhysRevE.79.031101

[59] J. Wilms, J. Vidal, F. Verstraete, and S. Dusuel, Finite-temperature mutual information in a simple phase transition, J. Stat. Mech. P01023 (2012).
https://doi.org/10.1088/1742-5468/2012/01/P01023

[60] E. Irish, Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling, Phys. Rev. Lett. 99, 173601 (2007).
https://doi.org/10.1103/PhysRevLett.99.173601

[61] Q.-H. Chen, Y.-Y. Zhang, T. Liu, and K.-L. Wang, Numerically exact solution to the finite-size Dicke model, Phys. Rev. A 78, 051801(R) (2008).
https://doi.org/10.1103/PhysRevA.78.051801

[62] A. Le Boite, Theoretical methods for ultrastrong light-matter interactions, Adv. Quantum Technol. 3, 1900140 (2020).
https://doi.org/10.1002/qute.201900140

[63] M. Aparicio Alcalde, M. Bucher, C. Emary, and T. Brandes, Thermal phase transitions for Dicke-type models in the ultrastrong-coupling limit, Phys. Rev. E 86, 012101 (2012).
https://doi.org/10.1103/PhysRevE.86.012101

[64] A. Ridolfo, S. Savasta, and M. J. Hartmann, Nonclassical Radiation from Thermal Cavities in the Ultrastrong Coupling Regime, Phys. Rev. Lett. 110, 163601 (2013).
https://doi.org/10.1103/PhysRevLett.110.163601

[65] A. Ridolfo, M. Leib, S. Savasta, and M. J. Hartmann, Thermal emission in the ultrastrong-coupling regime, Phys. Scr. 2013, 014053 (2013).
https://doi.org/10.1088/0031-8949/2013/T153/014053

[66] T. Chervy, A. Thomas, E. Akiki, R. M. A. Vergauwe, A. Shalabney, J. George, E. Devaux, J. A. Hutchison, C. Genet, and T. W. Ebbesen, Vibro-Polaritonic IR Emission in the Strong Coupling Regime, ACS Photonics 5, 217 (2018).
https://doi.org/10.1021/acsphotonics.7b00677

[67] F. Armata, G. Calajo, T. Jaako, M. S. Kim, and P. Rabl, Harvesting Multiqubit Entanglement from Ultrastrong Interactions in Circuit Quantum Electrodynamics, Phys. Rev. Lett. 119, 183602 (2017).
https://doi.org/10.1103/PhysRevLett.119.183602

[68] T. Holstein and H. Primakoff, Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet, Phys. Rev. 58, 1098 (1940).
https://doi.org/10.1103/PhysRev.58.1098

[69] A. M. Bratkovsky and A. P. Levanyuk, Continuous Theory of Ferroelectric States in Ultrathin Films with Real Electrodes, Journal of Computational and Theoretical Nanoscience 6, 10.1166/jctn.2009.1058 (2008).
arXiv:0801.1669

[70] T. Jaako, J. J. Garcia-Ripoll, and P. Rabl, Ultrastrong-coupling circuit QED in the radio-frequency regime, Phys. Rev. A 100, 043815 (2019).
https://doi.org/10.1103/PhysRevA.100.043815

[71] L. Fusco, M. Paternostro, and G. De Chiara, Work extraction and energy storage in the Dicke model, Phys. Rev. E 94, 052122 (2016).
https://doi.org/10.1103/PhysRevE.94.052122

[72] Y. Ma, S. Su, and C. Sun, Quantum thermodynamic cycle with quantum phase transition, Phys. Rev. E 96, 022143 (2017).
https://doi.org/10.1103/PhysRevE.96.022143

[73] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffeves, R. Azouit, P. Rouchon, and B. Huard, Observing a quantum Maxwell demon at work, Proc. Natl. Acad. Sci. 114, 7561 (2017).
https://doi.org/10.1073/pnas.1704827114

[74] M. Naghiloo, J. J. Alonso, A. Romito, E. Lutz, and K. W. Murch, Information Gain and Loss for a Quantum Maxwell's Demon, Phys. Rev. Lett. 121, 030604 (2018).
https://doi.org/10.1103/PhysRevLett.121.030604

[75] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, Information-to-work conversion by Maxwell’s demon in a superconducting circuit quantum electrodynamical system, Nat. Commun. 9, 1291 (2018).
https://doi.org/10.1038/s41467-018-03686-y

[76] M. A. Alcalde, E. Arias, Quantum Heat Engine and Quantum Phase Transition: through Anisotropic LMG and Full Dicke models, arXiv:1906.00292 (2019).
arXiv:1906.00292

Cited by

[1] Yuto Ashida, Takeru Yokota, Ataç İmamoğlu, and Eugene Demler, "Nonperturbative waveguide quantum electrodynamics", Physical Review Research 4 2, 023194 (2022).

[2] Kanta Masuki and Yuto Ashida, "Berry phase and topology in ultrastrongly coupled quantum light-matter systems", Physical Review B 107 19, 195104 (2023).

[3] M Salado-Mejía, R Román-Ancheyta, F Soto-Eguibar, and H M Moya-Cessa, "Spectroscopy and critical quantum thermometry in the ultrastrong coupling regime", Quantum Science and Technology 6 2, 025010 (2021).

[4] Katharina Lenk, Jiajun Li, Philipp Werner, and Martin Eckstein, "Dynamical mean-field study of a photon-mediated ferroelectric phase transition", Physical Review B 106 24, 245124 (2022).

[5] Yuto Ashida, Ataç İmamoğlu, and Eugene Demler, "Cavity Quantum Electrodynamics at Arbitrary Light-Matter Coupling Strengths", Physical Review Letters 126 15, 153603 (2021).

[6] Ricardo Herrera Romero, Miguel Angel Bastarrachea-Magnani, and Román Linares, "Critical Phenomena in Light–Matter Systems with Collective Matter Interactions", Entropy 24 9, 1198 (2022).

[7] Olesia Dmytruk and Marco Schirò, "Controlling topological phases of matter with quantum light", Communications Physics 5 1, 271 (2022).

[8] Jacopo Fregoni, Francisco J. Garcia-Vidal, and Johannes Feist, "Theoretical Challenges in Polaritonic Chemistry", ACS Photonics 9 4, 1096 (2022).

[9] Takeru Yokota, Kanta Masuki, and Yuto Ashida, "Functional-renormalization-group approach to circuit quantum electrodynamics", Physical Review A 107 4, 043709 (2023).

[10] I. V. Tokatly, D. R. Gulevich, and I. Iorsh, "Vacuum anomalous Hall effect in gyrotropic cavity", Physical Review B 104 8, L081408 (2021).

[11] Michael Schuler, Daniele De Bernardis, Andreas Läuchli, and Peter Rabl, "The vacua of dipolar cavity quantum electrodynamics", SciPost Physics 9 5, 066 (2020).

[12] Anton Trushechkin, "Quantum master equations and steady states for the ultrastrong-coupling limit and the strong-decoherence limit", Physical Review A 106 4, 042209 (2022).

[13] Juan Román-Roche, Fernando Luis, and David Zueco, "Photon Condensation and Enhanced Magnetism in Cavity QED", Physical Review Letters 127 16, 167201 (2021).

[14] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, "Open quantum system dynamics and the mean force Gibbs state", AVS Quantum Science 4 1, 012301 (2022).

[15] Vincenzo Macrì, Fabrizio Minganti, Anton Frisk Kockum, Alessandro Ridolfo, Salvatore Savasta, and Franco Nori, "Revealing higher-order light and matter energy exchanges using quantum trajectories in ultrastrong coupling", Physical Review A 105 2, 023720 (2022).

[16] Juan Román-Roche and David Zueco, "Effective theory for matter in non-perturbative cavity QED", SciPost Physics Lecture Notes 50 (2022).

[17] D. D. Sedov, V. Shirobokov, I. V. Iorsh, and I. V. Tokatly, "Cavity-induced chiral edge currents and spontaneous magnetization in two-dimensional electron systems", Physical Review B 106 20, 205114 (2022).

[18] Yuto Ashida, Ataç Imamoǧlu, Jérôme Faist, Dieter Jaksch, Andrea Cavalleri, and Eugene Demler, "Quantum Electrodynamic Control of Matter: Cavity-Enhanced Ferroelectric Phase Transition", Physical Review X 10 4, 041027 (2020).

[19] Giacomo Jarc, Shahla Yasmin Mathengattil, Angela Montanaro, Francesca Giusti, Enrico Maria Rigoni, Francesca Fassioli, Stephan Winnerl, Simone Dal Zilio, Dragan Mihailovic, Peter Prelovšek, Martin Eckstein, and Daniele Fausti, "Cavity control of the metal-to-insulator transition in 1T-TaS$_{2}$", arXiv:2210.02346, (2022).

[20] Tao E. Li, Abraham Nitzan, and Joseph E. Subotnik, "Collective Vibrational Strong Coupling Effects on Molecular Vibrational Relaxation and Energy Transfer: Numerical Insights via Cavity Molecular Dynamics Simulations**", Angewandte Chemie 133 28, 15661 (2021).

[21] Zeno Bacciconi, Gian Marcello Andolina, Titas Chanda, Giuliano Chiriacò, Marco Schiró, and Marcello Dalmonte, "First-order photon condensation in magnetic cavities: A two-leg ladder model", arXiv:2302.09901, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-06-08 06:52:33) and SAO/NASA ADS (last updated successfully 2023-06-08 06:52:33). 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.