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Efficient tensor-network simulation of multiemitter non-Markovian systems

Irene Papaefstathiou1,2, Daniel Malz3, J. Ignacio Cirac1,2, and Mari Carmen Bañuls1,2

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Phys. Rev. A 112, 013721 – Published 24 July, 2025

DOI: https://doi.org/10.1103/hmbj-lg7p

Abstract

We present a numerical method to simulate a system of multiple emitters coupled to a noninteracting bath in any parameter regime. Our method relies on a Block Lanczos transformation that maps the whole system onto a strip geometry whose width is given by the number of emitters. Utilizing the spatial symmetries of the problem and identifying the relevant range of energies of the bath, we achieve a more efficient simulation, which we perform using tensor network techniques. As a demonstration, we study the collective emission from multiple emitters coupled to a square lattice of bosons and observe how the departure from Markovianity as coupling strength and emitter number is increased prevents collective radiation. We also simulate the dynamic preparation of an excitation in a bound state from a multiexcitation initial state. Our work opens possibilities for the systematic exploration of non-Markovian effects in the dynamics and equilibrium properties of multiemitter systems. Furthermore, it can easily be extended to other setups, including finite bath temperature or impurities coupled to fermionic environments.

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References (81)

  1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2007).
  2. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  3. Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101 (1989).
  4. N. Makri and D. E. Makarov, Tensor propagator for iterative quantum time evolution of reduced density matrices. I. Theory, J. Chem. Phys. 102, 4600 (1995).
  5. N. Makri and D. E. Makarov, Tensor propagator for iterative quantum time evolution of reduced density matrices. II. Numerical methodology, J. Chem. Phys. 102, 4611 (1995).
  6. B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).
  7. P. Degenfeld-Schonburg and M. J. Hartmann, Self-consistent projection operator theory for quantum many-body systems, Phys. Rev. B 89, 245108 (2014).
  8. J. Iles-Smith, N. Lambert, and A. Nazir, Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems, Phys. Rev. A 90, 032114 (2014).
  9. F. A. Y. N. Schröder, D. H. P. Turban, A. J. Musser, N. D. M. Hine, and A. W. Chin, Multi-dimensional tensor network simulation of open quantum dynamics in singlet fission, Nat. Commun. 10, 1062 (2019).
  10. A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-Markovian quantum dynamics using time-evolving matrix product operators, Nat. Commun. 9, 3322 (2018).
  11. F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Non-Markovian quantum processes: Complete framework and efficient characterization, Phys. Rev. A 97, 012127 (2018).
  12. V. Link, H.-H. Tu, and W. T. Strunz, Open quantum system dynamics from infinite tensor network contraction, Phys. Rev. Lett. 132, 200403 (2024).
  13. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom—Photon Interactions: Basic Process and Appilcations (Wiley, New York, 1998).
  14. A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1993).
  15. P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124, 41 (1961).
  16. M. Thorwart, J. Eckel, J. Reina, P. Nalbach, and S. Weiss, Enhanced quantum entanglement in the non-Markovian dynamics of biomolecular excitons, Chem. Phys. Lett. 478, 234 (2009).
  17. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  18. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975).
  19. R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormalization group method for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008).
  20. F. Verstraete, V. Murg, and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  21. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
  22. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  23. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  24. A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech. (2004) P04005.
  25. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  26. A. Weichselbaum, F. Verstraete, U. Schollwöck, J. I. Cirac, and J. von Delft, Variational matrix-product-state approach to quantum impurity models, Phys. Rev. B 80, 165117 (2009).
  27. J. Prior, A. W. Chin, S. F. Huelga, and M. B. Plenio, Efficient simulation of strong system-environment interactions, Phys. Rev. Lett. 105, 050404 (2010).
  28. A. W. Chin, Á. Rivas, S. F. Huelga, and M. B. Plenio, Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials, J. Math. Phys. 51, 092109 (2010).
  29. A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigment–protein complexes, Nat. Phys. 9, 113 (2013).
  30. C. S. Muñoz, F. Nori, and S. De Liberato, Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics, Nat. Commun. 9, 1924 (2018).
  31. D. D. Noachtar, J. Knörzer, and R. H. Jonsson, Nonperturbative treatment of giant atoms using chain transformations, Phys. Rev. A 106, 013702 (2022).
  32. I. de Vega and M.-C. Bañuls, Thermofield-based chain-mapping approach for open quantum systems, Phys. Rev. A 92, 052116 (2015).
  33. F. Schwarz, I. Weymann, J. von Delft, and A. Weichselbaum, Nonequilibrium steady-state transport in quantum impurity models: A thermofield and quantum quench approach using matrix product states, Phys. Rev. Lett. 121, 137702 (2018).
  34. M. P. Woods, M. Cramer, and M. B. Plenio, Simulating bosonic baths with error bars, Phys. Rev. Lett. 115, 130401 (2015).
  35. I. de Vega, U. Schollwöck, and F. A. Wolf, How to discretize a quantum bath for real-time evolution, Phys. Rev. B 92, 155126 (2015).
  36. R. Trivedi, D. Malz, and J. I. Cirac, Convergence guarantees for discrete mode approximations to non-Markovian quantum baths, Phys. Rev. Lett. 127, 250404 (2021).
  37. C. Jayaprakash, H. R. Krishna-murthy, and J. W. Wilkins, Two-impurity Kondo problem, Phys. Rev. Lett. 47, 737 (1981).
  38. M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Phys. Rep. 93, 301 (1982).
  39. T. Shi, Y.-H. Wu, A. González-Tudela, and J. I. Cirac, Bound states in Boson impurity models, Phys. Rev. X 6, 021027 (2016).
  40. D. E. Chang, V. Vuletić, and M. D. Lukin, Quantum nonlinear optics: Photon by photon, Nat. Photon. 8, 685 (2014).
  41. A. I. Lvovsky, B. C. Sanders, and W. Tittel, Optical quantum memory, Nat. Photon. 3, 706 (2009).
  42. D. E. Chang, J. S. Douglas, A. González-Tudela, C.-L. Hung, and H. J. Kimble, Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons, Rev. Mod. Phys. 90, 031002 (2018).
  43. A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations, Rev. Mod. Phys. 95, 015002 (2023).
  44. A. González-Tudela, A. Reiserer, J. J. García-Ripoll, and F. J. García-Vidal, Light–matter interactions in quantum nanophotonic devices, Nat. Rev. Phys. 6, 166 (2024).
  45. T. Baba, Slow light in photonic crystals, Nat. Photon. 2, 465 (2008).
  46. 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).
  47. K. Sinha, A. González-Tudela, Y. Lu, and P. Solano, Collective radiation from distant emitters, Phys. Rev. A 102, 043718 (2020).
  48. C. Gonzalez-Ballestero, F. J. García-Vidal, and E. Moreno, Non-Markovian effects in waveguide-mediated entanglement, New J. Phys. 15, 073015 (2013).
  49. H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, Quantum optics of chiral spin networks, Phys. Rev. A 91, 042116 (2015).
  50. T. Ramos, B. Vermersch, P. Hauke, H. Pichler, and P. Zoller, Non-Markovian dynamics in chiral quantum networks with spins and photons, Phys. Rev. A 93, 062104 (2016).
  51. A. González-Tudela and J. I. Cirac, Quantum emitters in two-dimensional structured reservoirs in the nonperturbative regime, Phys. Rev. Lett. 119, 143602 (2017).
  52. A. González-Tudela and J. I. Cirac, Markovian and non-Markovian dynamics of quantum emitters coupled to two-dimensional structured reservoirs, Phys. Rev. A 96, 043811 (2017).
  53. Y. Liu and A. A. Houck, Quantum electrodynamics near a photonic bandgap, Nat. Phys. 13, 48 (2017).
  54. N. M. Sundaresan, R. Lundgren, G. Zhu, A. V. Gorshkov, and A. A. Houck, Interacting qubit-photon bound states with superconducting circuits, Phys. Rev. X 9, 011021 (2019).
  55. G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, Non-exponential decay of a giant artificial atom, Nat. Phys. 15, 1123 (2019).
  56. V. S. Ferreira, J. Banker, A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim, M. Mirhosseini, and O. Painter, Collapse and revival of an artificial atom coupled to a structured photonic reservoir, Phys. Rev. X 11, 041043 (2021).
  57. M. Stewart, J. Kwon, A. Lanuza, and D. Schneble, Dynamics of matter-wave quantum emitters in a structured vacuum, Phys. Rev. Res. 2, 043307 (2020).
  58. T. Shirakawa and S. Yunoki, Block Lanczos density-matrix renormalization group method for general Anderson impurity models: Application to magnetic impurity problems in graphene, Phys. Rev. B 90, 195109 (2014).
  59. A. Allerdt, C. A. Büsser, G. B. Martins, and A. E. Feiguin, Kondo versus indirect exchange: Role of lattice and actual range of RKKY interactions in real materials, Phys. Rev. B 91, 085101 (2015).
  60. G. Golub and R. Underwood, The Block Lanczos method for computing eigenvalues, in Mathematical Software, edited by J. R. Rice (Academic Press, Cambridge, 1977), pp. 361–377.
  61. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (NY) 349, 117 (2014).
  62. P. Silvi, F. Tschirsich, M. Gerster, J. Jünemann, D. Jaschke, M. Rizzi, and S. Montangero, The tensor networks anthology: Simulation techniques for many-body quantum lattice systems, SciPost Phys. Lect. Notes, 8 (2019).
  63. K. Okunishi, T. Nishino, and H. Ueda, Developments in the tensor network––from statistical mechanics to quantum entanglement, J. Phys. Soc. Jpn. 91, 062001 (2022).
  64. M. C. Bañuls, Tensor network algorithms: A route map, Annu. Rev. Condens. Matter Phys 14, 173 (2023).
  65. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand 45, 255 (1950).
  66. J. K. Cullum and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations (Society for Industrial and Applied Mathematics, Philadelphia, 2002).
  67. N. Schuch, S. K. Harrison, T. J. Osborne, and J. Eisert, Information propagation for interacting-particle systems, Phys. Rev. A 84, 032309 (2011).
  68. C. Yin and A. Lucas, Finite speed of quantum information in models of interacting bosons at finite density, Phys. Rev. X 12, 021039 (2022).
  69. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. (NY) 411, 167998 (2019).
  70. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  71. M. Zwolak and G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93, 207205 (2004).
  72. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  73. C. Zhang, E. Jeckelmann, and S. R. White, Density matrix approach to local Hilbert space reduction, Phys. Rev. Lett. 80, 2661 (1998).
  74. C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Critical and strong-coupling phases in one- and two-bath spin-boson models, Phys. Rev. Lett. 108, 160401 (2012).
  75. R. Langley, The response of two-dimensional periodic structures to point harmonic forcing, J. Sound Vib. 197, 447 (1996).
  76. A. Mekis, M. Meier, A. Dodabalapur, R. Slusher, and J. Joannopoulos, Lasing mechanism in two-dimensional photonic crystal lasers, Appl. Phys. A 69, 111 (1999).
  77. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  78. M. Ganahl, M. Aichhorn, H. G. Evertz, P. Thunström, K. Held, and F. Verstraete, Efficient DMFT impurity solver using real-time dynamics with matrix product states, Phys. Rev. B 92, 155132 (2015).
  79. F. A. Wolf, A. Go, I. P. McCulloch, A. J. Millis, and U. Schollwöck, Imaginary-time matrix product state impurity solver for dynamical mean-field theory, Phys. Rev. X 5, 041032 (2015).
  80. D. Bauernfeind, M. Zingl, R. Triebl, M. Aichhorn, and H. G. Evertz, Fork tensor-product states: Efficient multiorbital real-time DMFT solver, Phys. Rev. X 7, 031013 (2017).
  81. I. Papaefsthatiou, “Efficient tensor network simulation of multi-emitter non-Markovian systems (2024)”, Zenodo, https://doi.org/10.5281/zenodo.15764360.

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Phys. Rev. A 112, 013721– Published 24 July, 2025
  • Received 9 January 2025
  • Revised 28 June 2025
  • Accepted 7 July 2025
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