- Open Access
Correlation functions for quantum dynamics of coupled quasinormal modes and quantum emitters interacting via finite-delay propagating photons
Phys. Rev. A 114, 023725 – Published 25 August, 2026
DOI: https://doi.org/10.1103/xlv4-1bst
Abstract
A time-dependent theory for the interactions between spatially separated lossy cavities in a homogeneous background medium using quantized quasinormal modes (QNMs) is presented. The cavities interact via a bath of traveling photons, described by nonbosonic operators that are orthogonal to the open-cavity QNMs. The retarded (i.e., time-delayed) intercavity dynamics are fully described by system-bath correlation functions, in which the emission from one cavity appears as the input field for another. Coupling between quantum emitters (described as two-level systems), placed inside a cavity or embedded in an external medium, and the electromagnetic field (cavity modes and bath photons) is included in the theory, which gives rise to both bath-mediated and QNM-mediated interactions between the emitters.
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References (85)
- H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. 69, 1325 (2006).
- H. J. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields (Springer Science & Business Media, New York, 2009).
- E. M. Purcell, H. C. Torrey, and R. V. Pound, Resonance absorption by nuclear magnetic moments in a solid, Phys. Rev. 69, 37 (1946).
- D. Meschede, H. Walther, and G. Müller, One-atom maser, Phys. Rev. Lett. 54, 551 (1985).
- M. O. Scully, K. Wodkiewicz, M. S. Zubairy, J. Bergou, N. Lu, and J. M. T. Vehn, Two-photon correlated–spontaneous-emission laser: Quantum noise quenching and squeezing, Phys. Rev. Lett. 60, 1832 (1988).
- K. An, J. J. Childs, R. R. Dasari, and M. S. Feld, Microlaser: A laser with one atom in an optical resonator, Phys. Rev. Lett. 73, 3375 (1994).
- A. Pscherer, M. Meierhofer, D. Wang, H. Kelkar, D. Martín-Cano, T. Utikal, S. Götzinger, and V. Sandoghdar, Single-molecule vacuum Rabi splitting: Four-wave mixing and optical switching at the single-photon level, Phys. Rev. Lett. 127, 133603 (2021).
- Y. Wang, X. Li, V. Nalla, H. Zeng, and H. Sun, Solution-processed low threshold vertical cavity surface emitting lasers from all-inorganic perovskite nanocrystals, Adv. Funct. Mater. 27, 1605088 (2017).
- J. D. Rivero, M. Pan, K. G. Makris, L. Feng, and L. Ge, Non-Hermiticity-governed active photonic resonances, Phys. Rev. Lett. 126, 163901 (2021).
- D. J. Bergman and M. I. Stockman, Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems, Phys. Rev. Lett. 90, 027402 (2003).
- M. Noginov, G. Zhu, A. Belgrave, R. Bakker, V. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, Demonstration of a spaser-based nanolaser, Nature (London) 460, 1110 (2009).
- J. Y. Suh, C. H. Kim, W. Zhou, M. D. Huntington, D. T. Co, M. R. Wasielewski, and T. W. Odom, Plasmonic bowtie nanolaser arrays, Nano Lett. 12, 5769 (2012).
- P. Song, J.-H. Wang, M. Zhang, F. Yang, H.-J. Lu, B. Kang, J.-J. Xu, and H.-Y. Chen, Three-level spaser for next-generation luminescent nanoprobe, Sci. Adv. 4, eaat0292 (2018).
- Y.-H. Hsieh, B.-W. Hsu, K.-N. Peng, K.-W. Lee, C. W. Chu, S.-W. Chang, H.-W. Lin, T.-J. Yen, and Y.-J. Lu, Perovskite quantum dot lasing in a gap-plasmon nanocavity with ultralow threshold, ACS Nano 14, 11670 (2020).
- I. Epstein, D. Alcaraz, Z. Huang, V.-V. Pusapati, J.-P. Hugonin, A. Kumar, X. M. Deputy, T. Khodkov, T. G. Rappoport, J.-Y. Hong, et al., Far-field excitation of single graphene plasmon cavities with ultracompressed mode volumes, Science 368, 1219 (2020).
- T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, Decoherence, continuous observation, and quantum computing: A cavity QED model, Phys. Rev. Lett. 75, 3788 (1995).
- J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Quantum state transfer and entanglement distribution among distant nodes in a quantum network, Phys. Rev. Lett. 78, 3221 (1997).
- T. Pellizzari, Quantum networking with optical fibres, Phys. Rev. Lett. 79, 5242 (1997).
- A. Imamoğlu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Quantum information processing using quantum dot spins and cavity QED, Phys. Rev. Lett. 83, 4204 (1999).
- L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Long-distance quantum communication with atomic ensembles and linear optics, Nature (London) 414, 413 (2001).
- A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
- M. Benito, J. R. Petta, and G. Burkard, Optimized cavity-mediated dispersive two-qubit gates between spin qubits, Phys. Rev. B 100, 081412(R) (2019).
- F. Borjans, X. Croot, X. Mi, M. Gullans, and J. Petta, Resonant microwave-mediated interactions between distant electron spins, Nature (London) 577, 195 (2020).
- J. Li, Y.-P. Wang, W.-J. Wu, S.-Y. Zhu, and J. Q. You, Quantum network with magnonic and mechanical nodes, PRX Quantum 2, 040344 (2021).
- H. Pichler and P. Zoller, Photonic circuits with time delays and quantum feedback, Phys. Rev. Lett. 116, 093601 (2016).
- S. A. Regidor, G. Crowder, H. Carmichael, and S. Hughes, Modeling quantum light-matter interactions in waveguide QED with retardation, nonlinear interactions, and a time-delayed feedback: Matrix product states versus a space-discretized waveguide model, Phys. Rev. Res. 3, 023030 (2021).
- M. Richter and S. Hughes, Enhanced TEMPO algorithm for quantum path integrals with off-diagonal system-bath coupling: Applications to photonic quantum networks, Phys. Rev. Lett. 128, 167403 (2022).
- M. Bundgaard-Nielsen, D. Englund, M. Heuck, and S. Krastanov, WaveguideQED.jl: An efficient framework for simulating non-Markovian waveguide quantum electrodynamics, Quantum 9, 1710 (2025).
- S. A. Regidor, M. Kozma, and S. Hughes, QwaveMPS: An efficient open-source Python package for simulating non-Markovian waveguide-QED using matrix product states, arXiv:2602.15826.
- E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
- J. I. Cirac, H. Ritsch, and P. Zoller, Two-level system interacting with a finite-bandwidth thermal cavity mode, Phys. Rev. A 44, 4541 (1991).
- S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics, Phys. Rev. Lett. 122, 213901 (2019).
- J. Ren, S. Franke, and S. Hughes, Connecting classical and quantum mode theories for coupled lossy cavity resonators using quasinormal modes, ACS Photon. 9, 138 (2022).
- P. T. Leung, S. Y. Liu, and K. Young, Completeness and orthogonality of quasinormal modes in leaky optical cavities, Phys. Rev. A 49, 3057 (1994).
- E. A. Muljarov, W. Langbein, and R. Zimmermann, Brillouin-Wigner perturbation theory in open electromagnetic systems, Europhys. Lett. 92, 50010 (2010).
- P. T. Kristensen, C. Van Vlack, and S. Hughes, Generalized effective mode volume for leaky optical cavities, Opt. Lett. 37, 1649 (2012).
- C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators, Phys. Rev. Lett. 110, 237401 (2013).
- P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Light interaction with photonic and plasmonic resonances, Laser Photon. Rev. 12, 1700113 (2018).
- P. T. Kristensen, K. Herrmann, F. Intravaia, and K. Busch, Modeling electromagnetic resonators using quasinormal modes, Adv. Opt. Photon. 12, 612 (2020).
- A.-W. El-Sayed and S. Hughes, Quasinormal-mode theory of elastic Purcell factors and Fano resonances of optomechanical beams, Phys. Rev. Res. 2, 043290 (2020).
- J. Ren, S. Franke, and S. Hughes, Quasinormal modes, local density of states, and classical Purcell factors for coupled loss-gain resonators, Phys. Rev. X 11, 041020 (2021).
- C. Gustin, S. Franke, and S. Hughes, Gauge-invariant theory of truncated quantum light-matter interactions in arbitrary media, Phys. Rev. A 107, 013722 (2023).
- C. Gustin, J. Ren, and S. Hughes, What is the spectral density of the reservoir for a lossy quantized cavity? Phys. Rev. Lett. 134, 123601 (2025).
- K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, Second quantization of open systems using quasinormal modes, Phys. Rev. E 58, 2965 (1998).
- R. Fuchs and M. Richter, Hierarchical equations of motion analog for systems with delay: Application to intercavity photon propagation, Phys. Rev. B 107, 205301 (2023).
- R. Fuchs, J. Ren, S. Franke, S. Hughes, and M. Richter, Quantization of optical quasinormal modes for spatially separated cavity systems with finite retardation, Phys. Rev. A 110, 043718 (2024).
- S. Franke, M. Richter, J. Ren, A. Knorr, and S. Hughes, Quantized quasinormal-mode description of nonlinear cavity-QED effects from coupled resonators with a Fano-like resonance, Phys. Rev. Res. 2, 033456 (2020).
- H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002).
- Y. Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM), J. Chem. Phys. 153, 020901 (2020).
- C. Gustin, J. Ren, S. Franke, and S. Hughes, Dissipation in the broadband and ultrastrong coupling regimes of cavity quantum electrodynamics: An ab initio quantized quasinormal mode approach, arXiv:2507.21408.
- J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, et al., Satellite-to-ground entanglement-based quantum key distribution, Phys. Rev. Lett. 119, 200501 (2017).
- J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, et al., Satellite-based entanglement distribution over 1200 kilometers, Science 356, 1140 (2017).
- R. M. Fuchs and M. Richter, Quasinormal mode quantization of bound and propagating photons in complex lightguiding nanostructures for integrated devices, arXiv:2605.22153.
- H. T. Dung, L. Knöll, and D.-G. Welsch, Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics, Phys. Rev. A 57, 3931 (1998).
- L. G. Suttorp and M. Wubs, Field quantization in inhomogeneous absorptive dielectrics, Phys. Rev. A 70, 013816 (2004).
- T. G. Philbin, Canonical quantization of macroscopic electromagnetism, New J. Phys. 12, 123008 (2010).
- S. Scheel, L. Knöll, and D.-G. Welsch, QED commutation relations for inhomogeneous Kramers-Kronig dielectrics, Phys. Rev. A 58, 700 (1998).
- R. Matloob, Electromagnetic field quantization in an absorbing medium, Phys. Rev. A 60, 50 (1999).
- T. Gruner and D.-G. Welsch, Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous Kramers-Kronig dielectrics, Phys. Rev. A 53, 1818 (1996).
- S. Franke, J. Ren, and S. Hughes, Quantized quasinormal-mode theory of coupled lossy and amplifying resonators, Phys. Rev. A 105, 023702 (2022).
- B. VanDrunen, J. Ren, S. Franke, and S. Hughes, Gain-modified emission dynamics between two quantum emitters in a plasmonic gain cavity system, Phys. Rev. A 112, 013532 (2025).
- P.-O. Guimond, A. Roulet, H. N. Le, and V. Scarani, Rabi oscillation in a quantum cavity: Markovian and non-Markovian dynamics, Phys. Rev. A 93, 023808 (2016).
- D. Zueco and J. García-Ripoll, Ultrastrongly dissipative quantum Rabi model, Phys. Rev. A 99, 013807 (2019).
- K. Lee, P. Leung, and K. Pang, Dyadic formulation of morphology-dependent resonances. I. Completeness relation, J. Opt. Soc. Am. B 16, 1409 (1999).
- C. Müller, Foundations of the mathematical theory of electromagnetic waves, Die Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, Heidelberg, 1969), Vol. 155.
- S. Silver, in Microwave Antenna Theory and Design, Electromagnetic Waves Series (IET, London, 1984), Vol. 19.
- N. A. Gumerov and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions (Elsevier, San Diego, CA, 2005).
- S. Franke, J. Ren, and S. Hughes, Impact of mode regularization for quasinormal-mode perturbation theories, Phys. Rev. A 108, 043502 (2023).
- P. Leung and K. Pang, Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres, J. Opt. Soc. Am. B 13, 805 (1996).
- R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics, New J. Phys. 16, 113048 (2014).
- M. I. Abdelrahman and B. Gralak, Completeness and divergence-free behavior of the quasi-normal modes using causality principle, OSA Continuum 1, 340 (2018).
- R. M. Fuchs, J. Ren, S. Hughes, and M. Richter, Green's-function expansion for multiple coupled optical resonators with finite retardation using quasinormal modes, Phys. Rev. A 113, L041503 (2026).
- J. Ren, S. Franke, A. Knorr, M. Richter, and S. Hughes, Near-field to far-field transformations of optical quasinormal modes and efficient calculation of quantized quasinormal modes for open cavities and plasmonic resonators, Phys. Rev. B 101, 205402 (2020).
- C. A. Balanis, Antenna Theory: Analysis and Design (John Wiley & Sons, New York, 2016).
- A. Vagov, M. D. Croitoru, M. Glässl, V. M. Axt, and T. Kuhn, Real-time path integrals for quantum dots: Quantum dissipative dynamics with superohmic environment coupling, Phys. Rev. B 83, 094303 (2011).
- A. Strathearn, B. W. Lovett, and P. Kirton, Efficient real-time path integrals for non-Markovian spin-boson models, New J. Phys. 19, 093009 (2017).
- A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587 (1983).
- H. Grabert, P. Schramm, and G.-L. Ingold, Quantum Brownian motion: The functional integral approach, Phys. Rep. 168, 115 (1988).
- V. Chernyak and S. Mukamel, Collective coordinates for nuclear spectral densities in energy transfer and femtosecond spectroscopy of molecular aggregates, J. Chem. Phys. 105, 4565 (1996).
- B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).
- H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. 70, 2273 (1993).
- H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, Exciton Rabi oscillation in a single quantum dot, Phys. Rev. Lett. 87, 246401 (2001).
- Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure, Opt. Express 21, 27371 (2013).
- P. T. Kristensen and S. Hughes, Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators, ACS Photon. 1, 2 (2014).
- M. Barth, R. McLeod, and R. Ziolkowski, A near and far-field projection algorithm for finite-difference time-domain codes, J. Electromagn. Waves Appl. 6, 5 (1992).