Skip to main content
SpringerLink
Log in

Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

This work is a pedagogical survey about the hierarchical equations of motion and their implementation with the tensor-train format. These equations are a great standard in non-perturbative non-Markovian open quantum systems. They are exact for harmonic baths in the limit of relevant truncation of the hierarchy. We recall the link with the perturbative second-order time convolution equations also known as the Bloch–Redfield equations. Some theoretical tools characterizing non-Markovian dynamics such as the non-Markovianity measures or the dynamical map are also briefly discussed in the context of HEOM simulations. The main points of the tensor-train expansion are illustrated in an example with a qubit interacting with a bath described by a Lorentzian spectral density. Finally, we give three illustrative applications in which the system–bath coupling operator is similar to that of the analytical treatment. The first example revisits a model in which population-to-coherence transfer via the bath creates a long-lasting coherence between two states. The second one is devoted to the computation of stationary absorption and emission spectra. We illustrate the link between the spectral density and the Stokes shift in situations with and without nonadiabatic interaction. Finally, we simulate an excitation transfer when the spectral density is discretized by undamped modes to illustrate a situation in which the TT formulation is more efficient than the standard one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Availability of data and materials

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.]

References

  1. U. Manthe, A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. J. Chem. Phys. 128(16), 164116 (2008). https://doi.org/10.1063/1.2902982

    Article  ADS  Google Scholar 

  2. H. Wang, M. Thoss, Numerically exact quantum dynamics for indistinguishable particles: the multilayer multiconfiguration time-dependent Hartree theory in second quantization representation. J. Chem. Phys. 131(2), 024114 (2009). https://doi.org/10.1063/1.3173823

    Article  ADS  Google Scholar 

  3. O. Vendrell, H.D. Meyer, Multilayer multiconfiguration time-dependent Hartree method: implementation and applications to a Henon–Heiles Hamiltonian and to pyrazine. J. Chem. Phys. 134(4), 044135 (2011). https://doi.org/10.1063/1.3535541

    Article  ADS  Google Scholar 

  4. L.P. Lindoy, B. Kloss, D.R. Reichman, Time evolution of ML-MCTDH wavefunctions. II. Application of the projector splitting integrator. J. Chem. Phys. 155(17), 174109 (2021). https://doi.org/10.1063/5.0070043

    Article  ADS  Google Scholar 

  5. T.G. Kolda, B.W. Bader, Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009). https://doi.org/10.1137/07070111X

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. L. Grasedyck, D. Kressner, C. Tobler, A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36(1), 53–78 (2013). https://doi.org/10.1002/gamm.201310004

    Article  MathSciNet  MATH  Google Scholar 

  7. I.V. Oseledets, Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011). https://doi.org/10.1137/090752286

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Lubich, I.V. Oseledets, B. Vandereycken, Time integration of tensor trains. SIAM J. Numer. Anal. 53(2), 917–941 (2015). https://doi.org/10.1137/140976546

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, F. Verstraete, Unifying time evolution and optimization with matrix product states. Phys. Rev. B 94, 165116 (2016). https://doi.org/10.1103/PhysRevB.94.165116

    Article  ADS  Google Scholar 

  10. R. Orús, A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014). https://doi.org/10.1016/j.aop.2014.06.013

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. F.A.Y.N. Schröder, A.W. Chin, Simulating open quantum dynamics with time-dependent variational matrix product states: towards microscopic correlation of environment dynamics and reduced system evolution. Phys. Rev. B 93, 075105 (2016). https://doi.org/10.1103/PhysRevB.93.075105

    Article  ADS  Google Scholar 

  12. F.A.Y.N. Schröder, D.H.P. Turban, A.J. Musser, N.D.M. Hine, A.W. Chin, Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation. Nat. Commun. 10, 1062 (2019)

    Article  ADS  Google Scholar 

  13. A.M. Alvertis, F.A.Y.N. Schröder, A.W. Chin, Non-equilibrium relaxation of hot states in organic semiconductors: impact of mode-selective excitation on charge transfer. J. Chem. Phys. 151(8), 084104 (2019)

    Article  ADS  Google Scholar 

  14. T. Lacroix, A. Dunnett, D. Gribben, B.W. Lovett, A. Chin, Unveiling non-Markovian spacetime signaling in open quantum systems with long-range tensor network dynamics. Phys. Rev. A 104, 052204 (2021). https://doi.org/10.1103/PhysRevA.104.052204

    Article  ADS  MathSciNet  Google Scholar 

  15. A.J. Dunnett, D. Gowland, C.M. Isborn, A.W. Chin, T.J. Zuehlsdorff, Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: methylene blue. J. Chem. Phys. 155(14), 144112 (2021). https://doi.org/10.1063/5.0062950

    Article  ADS  Google Scholar 

  16. A. Baiardi, M. Reiher, Large-scale quantum dynamics with matrix product states. J. Chem. Theory Comput. 15(6), 3481–3498 (2019). https://doi.org/10.1021/acs.jctc.9b00301

    Article  Google Scholar 

  17. X. Xie, Y. Liu, Y. Yao, U. Schollwöck, C. Liu, H. Ma, Time-dependent density matrix renormalization group quantum dynamics for realistic chemical systems. J. Chem. Phys. 151(22), 224101 (2019). https://doi.org/10.1063/1.5125945

    Article  ADS  Google Scholar 

  18. A.D. Somoza, O. Marty, J. Lim, S.F. Huelga, M.B. Plenio, Dissipation-assisted matrix product factorization. Phys. Rev. Lett. 123, 100502 (2019). https://doi.org/10.1103/PhysRevLett.123.100502

    Article  ADS  MathSciNet  Google Scholar 

  19. N. Lyu, M.B. Soley, V.S. Batista, Tensor-train split-operator KSL (TT-SOKSL) method for quantum dynamics simulations. J. Chem. Theory Comput. 18(6), 3327–3346 (2022). https://doi.org/10.1021/acs.jctc.2c00209

    Article  Google Scholar 

  20. P. Gelß, R. Klein, S. Matera, B. Schmidt, Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains. J. Chem. Phys. 156(2), 024109 (2022). https://doi.org/10.1063/5.0074948

    Article  ADS  Google Scholar 

  21. R. Feynman, F. Vernon, The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys. 24, 118–173 (1963). https://doi.org/10.1016/0003-4916(63)90068-X

    Article  ADS  MathSciNet  Google Scholar 

  22. Y. Tanimura, R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian–Markoffian noise bath. J. Phys. Soc. Jpn. 58(1), 101–114 (1989). https://doi.org/10.1143/JPSJ.58.101

    Article  ADS  MathSciNet  Google Scholar 

  23. Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems. J. Phys. Soc. Jpn. 75(8), 082001 (2006). https://doi.org/10.1143/JPSJ.75.082001

    Article  ADS  Google Scholar 

  24. Y. Tanimura, Numerically “exact’’ approach to open quantum dynamics: the hierarchical equations of motion (HEOM). J. Chem. Phys. 153(2), 020901 (2020). https://doi.org/10.1063/5.0011599

    Article  Google Scholar 

  25. R.X. Xu, Y. Yan, Dynamics of quantum dissipation systems interacting with bosonic canonical bath: hierarchical equations of motion approach. Phys. Rev. E 75, 031107 (2007)

    Article  ADS  Google Scholar 

  26. Q. Shi, L. Chen, G. Nan, R.X. Xu, Y. Yan, Efficient hierarchical Liouville space propagator to quantum dissipative dynamics. J. Chem. Phys. 130(8), 084105 (2009). https://doi.org/10.1063/1.3077918

    Article  ADS  Google Scholar 

  27. S. Nakajima, On quantum theory of transport phenomena: steady diffusion. Prog. Theor. Phys. 20(6), 948–959 (1958). https://doi.org/10.1143/PTP.20.948

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. R. Zwanzig, Ensemble method in the theory of irreversibility. J. Chem. Phys. 33(5), 1338–1341 (1960). https://doi.org/10.1063/1.1731409

    Article  ADS  MathSciNet  Google Scholar 

  29. A. Ishizaki, G.R. Fleming, Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: reduced hierarchy equation approach. J. Chem. Phys. 130(23), 234111 (2009). https://doi.org/10.1063/1.3155372

    Article  ADS  Google Scholar 

  30. A. Bose, P.L. Walters, A multisite decomposition of the tensor network path integrals. J. Chem. Phys. 156(2), 024101 (2022). https://doi.org/10.1063/5.0073234

    Article  ADS  Google Scholar 

  31. Q. Shi, Y. Xu, Y. Yan, M. Xu, Efficient propagation of the hierarchical equations of motion using the matrix product state method. J. Chem. Phys. 148(17), 174102 (2018). https://doi.org/10.1063/1.5026753

    Article  ADS  Google Scholar 

  32. Y. Yan, T. Xing, Q. Shi, A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes. J. Chem. Phys. 153(20), 204109 (2020). https://doi.org/10.1063/5.0027962

    Article  ADS  Google Scholar 

  33. R. Borrelli, S. Dolgov, Expanding the range of hierarchical equations of motion by tensor-train implementation. J. Phys. Chem. B 125(20), 5397–5407 (2021). https://doi.org/10.1021/acs.jpcb.1c02724

    Article  Google Scholar 

  34. R. Borrelli, M.F. Gelin, Finite temperature quantum dynamics of complex systems: integrating thermo-field theories and tensor-train methods. WIREs Comput. Mol. Sci. 11(6), e1539 (2021). https://doi.org/10.1002/wcms.1539

    Article  Google Scholar 

  35. Y. Yan, M. Xu, T. Li, Q. Shi, Efficient propagation of the hierarchical equations of motion using the tucker and hierarchical tucker tensors. J. Chem. Phys. 154(19), 194104 (2021). https://doi.org/10.1063/5.0050720

    Article  ADS  Google Scholar 

  36. A. Ishizaki, T.R. Calhoun, G.S. Schlau-Cohen, G.R. Fleming, Quantum coherence and its interplay with protein environments in photosynthetic electronic energy transfer. Phys. Chem. Chem. Phys. 12, 7319–7337 (2010). https://doi.org/10.1039/C003389H

    Article  Google Scholar 

  37. C. Kreisbeck, T. Kramer, Long-lived electronic coherence in dissipative exciton dynamics of light-harvesting complexes. J. Phys. Chem. Lett. 3(19), 2828–2833 (2012). https://doi.org/10.1021/jz3012029. arXiv:1203.1485

    Article  Google Scholar 

  38. S. Saito, M. Higashi, G.R. Fleming, Site-dependent fluctuations optimize electronic energy transfer in the Fenna–Matthews–Olson protein. J. Phys. Chem. B 123(46), 9762–9772 (2019). https://doi.org/10.1021/acs.jpcb.9b07456. (PMID: 31657928)

    Article  Google Scholar 

  39. L. Chen, P. Shenai, F. Zheng, A. Somoza, Y. Zhao, Optimal energy transfer in light-harvesting systems. Molecules 20(8), 15224–15272 (2015). https://doi.org/10.3390/molecules200815224. https://www.mdpi.com/1420-3049/20/8/15224

  40. T.P. Fay, D.T. Limmer, Coupled charge and energy transfer dynamics in light harvesting complexes from a hybrid hierarchical equations of motion approach. J. Chem. Phys. 157(17), 174104 (2022). https://doi.org/10.1063/5.0117659

    Article  ADS  Google Scholar 

  41. M. Cainelli, Y. Tanimura, Exciton transfer in organic photovoltaic cells: a role of local and nonlocal electron–phonon interactions in a donor domain. J. Chem. Phys. 154(3), 034107 (2021). https://doi.org/10.1063/5.0036590

    Article  ADS  Google Scholar 

  42. M. Tanaka, Y. Tanimura, Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach. J. Chem. Phys. 132(21), 214502 (2010). https://doi.org/10.1063/1.3428674

    Article  ADS  Google Scholar 

  43. E. Mangaud, A. de la Lande, C. Meier, M. Desouter-Lecomte, Electron transfer within a reaction path model calibrated by constrained DFT calculations: application to mixed-valence organic compounds. Phys. Chem. Chem. Phys. 17, 30889–30903 (2015). https://doi.org/10.1039/C5CP01194A

    Article  Google Scholar 

  44. T. Firmino, E. Mangaud, F. Cailliez, A. Devolder, D. Mendive-Tapia, F. Gatti, C. Meier, M. Desouter-Lecomte, A. de la Lande, Quantum effects in ultrafast electron transfers within cryptochromes. Phys. Chem. Chem. Phys. 18, 21442–21457 (2016). https://doi.org/10.1039/C6CP02809H

    Article  Google Scholar 

  45. L. Chen, M.F. Gelin, V.Y. Chernyak, W. Domcke, Y. Zhao, Dissipative dynamics at conical intersections: simulations with the hierarchy equations of motion method. Faraday Discuss. 194, 61–80 (2016). https://doi.org/10.1039/C6FD00088F

    Article  ADS  Google Scholar 

  46. H.G. Duan, V.I. Prokhorenko, R.J. Cogdell, K. Ashraf, A.L. Stevens, M. Thorwart, R.D. Miller, Nature does not rely on long-lived electronic quantum coherence for photosynthetic energy transfer. Proc. Natl. Acad. Sci. 114(32), 8493–8498 (2017)

    Article  ADS  Google Scholar 

  47. A.G. Dijkstra, V.I. Prokhorenko, Simulation of photo-excited adenine in water with a hierarchy of equations of motion approach. J. Chem. Phys. 147(6), 064102 (2017). https://doi.org/10.1063/1.4997433

    Article  ADS  Google Scholar 

  48. E. Mangaud, B. Lasorne, O. Atabek, M. Desouter-Lecomte, Statistical distributions of the tuning and coupling collective modes at a conical intersection using the hierarchical equations of motion. J. Chem. Phys. 151(24), 244102 (2019). https://doi.org/10.1063/1.5128852

    Article  ADS  Google Scholar 

  49. G. Breuil, E. Mangaud, B. Lasorne, O. Atabek, M. Desouter-Lecomte, Funneling dynamics in a phenylacetylene trimer: coherent excitation of donor excitonic states and their superposition. J. Chem. Phys. 155(3), 034303 (2021). https://doi.org/10.1063/5.0056351

    Article  ADS  Google Scholar 

  50. A. Jaouadi, J. Galiana, E. Mangaud, B. Lasorne, O. Atabek, M. Desouter-Lecomte, Laser-controlled electronic symmetry breaking in a phenylene ethynylene dimer: simulation by the hierarchical equations of motion and optimal control. Phys. Rev. A 106, 043121 (2022). https://doi.org/10.1103/PhysRevA.106.043121

    Article  ADS  Google Scholar 

  51. J. Zhang, R. Borrelli, Y. Tanimura, Proton tunneling in a two-dimensional potential energy surface with a non-linear system-bath interaction: Thermal suppression of reaction rate. The Journal of Chemical Physics 152(21), 214–114 (2020). https://doi.org/10.1063/5.0010580

    Article  Google Scholar 

  52. E. Mangaud, R. Puthumpally-Joseph, D. Sugny, C. Meier, O. Atabek, M. Desouter-Lecomte, Non-Markovianity in the optimal control of an open quantum system described by hierarchical equations of motion. New J. Phys. 20, 043050 (2018)

    Article  ADS  Google Scholar 

  53. A. Kato, Y. Tanimura, Quantum heat current under non-perturbative and non-Markovian conditions: applications to heat machines. J. Chem. Phys. 145(22), 224105 (2016). https://doi.org/10.1063/1.4971370

    Article  ADS  Google Scholar 

  54. L. Song, Q. Shi, Hierarchical equations of motion method applied to nonequilibrium heat transport in model molecular junctions: transient heat current and high-order moments of the current operator. Phys. Rev. B 95(6), 064308 (2017). https://doi.org/10.1103/PhysRevB.95.064308

    Article  ADS  Google Scholar 

  55. J. Bätge, Y. Ke, C. Kaspar, M. Thoss, Nonequilibrium open quantum systems with multiple bosonic and fermionic environments: a hierarchical equations of motion approach. Phys. Rev. B 103, 235413 (2021). https://doi.org/10.1103/PhysRevB.103.235413

    Article  ADS  Google Scholar 

  56. E.C. Wu, Q. Ge, E.A. Arsenault, N.H.C. Lewis, N.L. Gruenke, M.J. Head-Gordon, G.R. Fleming, Two-dimensional electronic-vibrational spectroscopic study of conical intersection dynamics: an experimental and electronic structure study. Phys. Chem. Chem. Phys. 21(26), 14153–14163 (2019). https://doi.org/10.1039/C8CP05264F

    Article  Google Scholar 

  57. K.B. Zhu, R.X. Xu, H.Y. Zhang, J. Hu, Y.J. Yan, Hierarchical dynamics of correlated system-environment coherence and optical spectroscopy. J. Phys. Chem. B 115(18), 5678–5684 (2011). https://doi.org/10.1021/jp2002244

    Article  Google Scholar 

  58. Y. Tanimura, Reduced hierarchy equations of motion approach with Drude plus Brownian spectral distribution: probing electron transfer processes by means of two-dimensional correlation spectroscopy. J. Chem. Phys. 137(22), 22A550 (2012). https://doi.org/10.1063/1.4766931

    Article  Google Scholar 

  59. H. Liu, L. Zhu, S. Bai, Q. Shi, Reduced quantum dynamics with arbitrary bath spectral densities: hierarchical equations of motion based on several different bath decomposition schemes. J. Chem. Phys. 140(13), 134106 (2014). https://doi.org/10.1063/1.4870035

    Article  ADS  Google Scholar 

  60. A.J. Dunnett, A.W. Chin, Simulating quantum vibronic dynamics at finite temperatures with many body wave functions at 0 K. Front. Chem. 8 (2021). https://doi.org/10.3389/fchem.2020.600731. https://www.frontiersin.org/articles/10.3389/fchem.2020.600731

  61. D. Tamascelli, A. Smirne, J. Lim, S.F. Huelga, M.B. Plenio, Efficient simulation of finite-temperature open quantum systems. Phys. Rev. Lett. 123, 090402 (2019). https://doi.org/10.1103/PhysRevLett.123.090402

    Article  ADS  MathSciNet  Google Scholar 

  62. L. Zhu, H. Liu, W. Xie, Q. Shi, Explicit system-bath correlation calculated using the hierarchical equations of motion method. J. Chem. Phys. 137(19), 194106 (2012). https://doi.org/10.1063/1.4766358

    Article  ADS  Google Scholar 

  63. A. Chin, E. Mangaud, V. Chevet, O. Atabek, M. Desouter-Lecomte, Visualising the role of non-perturbative environment dynamics in the dissipative generation of coherent electronic motion. Chem. Phys. 525, 110392 (2019). https://doi.org/10.1016/j.chemphys.2019.110392

    Article  Google Scholar 

  64. H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

    MATH  Google Scholar 

  65. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2012)

    Book  MATH  Google Scholar 

  66. V. May, O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems (Wiley, Hoboken, 2011)

    Book  Google Scholar 

  67. A. Chenel, E. Mangaud, I. Burghardt, C. Meier, M. Desouter-Lecomte, Exciton dissociation at donor–acceptor heterojunctions: dynamics using the collective effective mode representation of the spin-boson model. J. Chem. Phys. 140(4), 044104 (2014). https://doi.org/10.1063/1.4861853

    Article  ADS  Google Scholar 

  68. J. Iles-Smith, A.G. Dijkstra, N. Lambert, A. Nazir, Energy transfer in structured and unstructured environments: master equations beyond the Born–Markov approximations. J. Chem. Phys. 144(4), 044110 (2016). https://doi.org/10.1063/1.4940218

    Article  ADS  Google Scholar 

  69. A. Pomyalov, C. Meier, D.J. Tannor, The importance of initial correlations in rate dynamics: a consistent non-Markovian master equation approach. Chem. Phys. 370(1–3), 98–108 (2010). https://doi.org/10.1016/j.chemphys.2010.02.017

    Article  Google Scholar 

  70. A.G. Dijkstra, Y. Tanimura, Non-Markovian entanglement dynamics in the presence of system-bath coherence. Phys. Rev. Lett. 104, 250401 (2010). https://doi.org/10.1103/PhysRevLett.104.250401

    Article  ADS  Google Scholar 

  71. Y. Tanimura, Reduced hierarchical equations of motion in real and imaginary time: correlated initial states and thermodynamic quantities. J. Chem. Phys. 141(4), 044114 (2014). https://doi.org/10.1063/1.4890441

    Article  ADS  Google Scholar 

  72. L. Song, Q. Shi, Calculation of correlated initial state in the hierarchical equations of motion method using an imaginary time path integral approach. J. Chem. Phys. 143(19), 194106 (2015). https://doi.org/10.1063/1.4935799

    Article  ADS  Google Scholar 

  73. C. Meier, D.J. Tannor, Non-Markovian evolution of the density operator in the presence of strong laser fields. J. Chem. Phys. 111(8), 3365–3376 (1999). https://doi.org/10.1063/1.479669

    Article  ADS  Google Scholar 

  74. S. Jang, J. Cao, R.J. Silbey, Fourth-order quantum master equation and its Markovian bath limit. J. Chem. Phys. 116(7), 2705–2717 (2002). https://doi.org/10.1063/1.1445105

    Article  ADS  Google Scholar 

  75. E. Mulvihill, A. Schubert, X. Sun, B.D. Dunietz, E. Geva, A modified approach for simulating electronically nonadiabatic dynamics via the generalized quantum master equation. J. Chem. Phys. 150(3), 034101 (2019). https://doi.org/10.1063/1.5055756

    Article  ADS  Google Scholar 

  76. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995)

    Google Scholar 

  77. H. Rahman, U. Kleinekathöfer, Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures. J. Chem. Phys. 150(24), 244104 (2019)

    Article  ADS  Google Scholar 

  78. L. Cui, H.D. Zhang, X. Zheng, R.X. Xu, Y. Yan, Highly efficient and accurate sum-over-poles expansion of Fermi and Bose functions at near zero temperatures: Fano spectrum decomposition scheme. J. Chem. Phys. 151(2), 024110 (2019)

    Article  ADS  Google Scholar 

  79. T. Ikeda, G.D. Scholes, Generalization of the hierarchical equations of motion theory for efficient calculations with arbitrary correlation functions. J. Chem. Phys. 152(20), 204101 (2020). https://doi.org/10.1063/5.0007327

    Article  ADS  Google Scholar 

  80. Z.H. Chen, Y. Wang, X. Zheng, R.X. Xu, Y. Yan, Universal time-domain Prony fitting decomposition for optimized hierarchical quantum master equations. J. Chem. Phys. 156(22), 221102 (2022). https://doi.org/10.1063/5.0095961

    Article  ADS  Google Scholar 

  81. N. Lambert, T. Raheja, S. Cross, P. Menczel, S. Ahmed, A. Pitchford, D. Burgarth, F. Nori, QuTiP-BoFiN: a bosonic and fermionic numerical hierarchical-equations-of-motion library with applications in light-harvesting, quantum control, and single-molecule electronics (2020). https://doi.org/10.48550/ARXIV.2010.10806

  82. U. Kleinekathöfer, Non-Markovian theories based on a decomposition of the spectral density. J. Chem. Phys. 121(6), 2505–2514 (2004). https://doi.org/10.1063/1.1770619

    Article  ADS  Google Scholar 

  83. S. Valleau, A. Eisfeld, A. Aspuru-Guzik, On the alternatives for bath correlators and spectral densities from mixed quantum-classical simulations. J. Chem. Phys. 137(22), 224103 (2012). https://doi.org/10.1063/1.4769079

    Article  ADS  Google Scholar 

  84. T. Renger, R.A. Marcus, On the relation of protein dynamics and exciton relaxation in pigment-protein complexes: an estimation of the spectral density and a theory for the calculation of optical spectra. J. Chem. Phys. 116(22), 9997–10019 (2002). https://doi.org/10.1063/1.1470200

    Article  ADS  Google Scholar 

  85. G. Ritschel, A. Eisfeld, Analytic representations of bath correlation functions for ohmic and superohmic spectral densities using simple poles. J. Chem. Phys. 141(9), 094101 (2014). https://doi.org/10.1063/1.4893931

    Article  ADS  Google Scholar 

  86. J. Hu, M. Luo, F. Jiang, R.X. Xu, Y. Yan, Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems. J. Chem. Phys. 134(24), 244106 (2011). https://doi.org/10.1063/1.3602466

    Article  ADS  Google Scholar 

  87. N. Lambert, S. Ahmed, M. Cirio, F. Nori, Modelling the ultra-strongly coupled spin-boson model with unphysical modes. Nat. Commun. 10(1), 3721 (2019). https://doi.org/10.1038/s41467-019-11656-1

    Article  ADS  Google Scholar 

  88. T.P. Fay, A simple improved low temperature correction for the hierarchical equations of motion. J. Chem. Phys. 157(5), 054108 (2022). https://doi.org/10.1063/5.0100365

    Article  ADS  Google Scholar 

  89. E. Mangaud, C. Meier, M. Desouter-Lecomte, Analysis of the non-Markovianity for electron transfer reactions in an oligothiophene-fullerene heterojunction. Chem. Phys. 494, 90–102 (2017). https://doi.org/10.1016/j.chemphys.2017.07.011

  90. R. Kubo, Stochastic Liouville equations. J. Math. Phys. 4(2), 174 (1963). https://doi.org/10.1063/1.1703941

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. A. Ishizaki, Y. Tanimura, Quantum dynamics of system strongly coupled to low-temperature colored noise bath: reduced hierarchy equations approach. J. Phys. Soc. Jpn. 74(12), 3131–3134 (2005). https://doi.org/10.1143/JPSJ.74.3131

    Article  ADS  MATH  Google Scholar 

  92. G.C. Wick, The evaluation of the collision matrix. Phys. Rev. 80, 268–272 (1950). https://doi.org/10.1103/PhysRev.80.268

  93. H. Tamura, R. Martinazzo, M. Ruckenbauer, I. Burghardt, Quantum dynamics of ultrafast charge transfer at an oligothiophene-fullerene heterojunction. J. Chem. Phys. 137(22), 22A540 (2012). https://doi.org/10.1063/1.4751486

    Article  Google Scholar 

  94. F. Di Maiolo, D. Brey, R. Binder, I. Burghardt, Quantum dynamical simulations of intra-chain exciton diffusion in an oligo (para-phenylene vinylene) chain at finite temperature. J. Chem. Phys. 153(18), 184107 (2020). https://doi.org/10.1063/5.0027588

    Article  ADS  Google Scholar 

  95. W. Popp, D. Brey, R. Binder, I. Burghardt, Quantum dynamics of exciton transport and dissociation in multichromophoric systems. Annu. Rev. Phys. Chem. 72(1), 591–616 (2021). https://doi.org/10.1146/annurev-physchem-090419-040306

    Article  Google Scholar 

  96. J. Schulze, M.F. Shibl, M.J. Al-Marri, O. Kühn, Multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) approach to the correlated exciton-vibrational dynamics in the FMO complex. J. Chem. Phys. 144(18), 185101 (2016). https://doi.org/10.1063/1.4948563

    Article  ADS  Google Scholar 

  97. C. Kreisbeck, T. Kramer, A. Aspuru-Guzik, Scalable high-performance algorithm for the simulation of exciton dynamics. application to the light-harvesting complex II in the presence of resonant vibrational modes. J. Chem. Theory Comput. 10(9), 4045–4054 (2014). https://doi.org/10.1021/ct500629s

  98. R. Martinazzo, K.H. Hughes, F. Martelli, I. Burghardt, Effective spectral densities for system-environment dynamics at conical intersections: S2–S1 conical intersection in pyrazine. Chem. Phys. 377(1), 21–29 (2010)

    Article  ADS  Google Scholar 

  99. J. Iles-Smith, N. Lambert, A. Nazir, Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems. Phys. Rev. A 90(3), 032114 (2014). https://doi.org/10.1103/PhysRevA.90.032114

    Article  ADS  Google Scholar 

  100. H.P. Breuer, E.M. Laine, J. Piilo, B. Vacchini, Colloquium: non-Markovian dynamics in open quantum systems. Rev. Mod. Phys. 88, 021002 (2016). https://doi.org/10.1103/RevModPhys.88.021002

  101. I. de Vega, D. Alonso, Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 89(1), 015–001 (2017). https://doi.org/10.1103/RevModPhys.89.015001

    Article  MathSciNet  Google Scholar 

  102. H.P. Breuer, E.M. Laine, J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009). https://doi.org/10.1103/PhysRevLett.103.210401

  103. A. Rivas, S.F. Huelga, M.B. Plenio, Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett. 105, 050403 (2010). https://doi.org/10.1103/PhysRevLett.105.050403

  104. M.J.W. Hall, J.D. Cresser, L. Li, E. Andersson, Canonical form of master equations and characterization of non-markovianity. Phys. Rev. A 89, 042,120 (2014). https://doi.org/10.1103/PhysRevA.89.042120. https://link.aps.org/doi/10.1103/PhysRevA.89.042120

  105. S. Lorenzo, F. Plastina, M. Paternostro, Geometrical characterization of non-Markovianity. Phys. Rev. A 88, 020102 (2013). https://doi.org/10.1103/PhysRevA.88.020102

  106. S.C. Hou, S.L. Liang, X.X. Yi, Non-Markovianity and memory effects in quantum open systems. Phys. Rev. A 91, 012109 (2015). https://doi.org/10.1103/PhysRevA.91.012109

  107. G. Kimura, The Bloch vector for N-level systems. Phys. Lett. A 314(5), 339–349 (2003). https://doi.org/10.1016/S0375-9601(03)00941-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  108. D. Aerts, M.S. de Bianchi, The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem. Ann. Phys. 351, 975–1025 (2014). https://doi.org/10.1016/j.aop.2014.09.020

    Article  ADS  MathSciNet  MATH  Google Scholar 

  109. R. Puthumpally-Joseph, O. Atabek, E. Mangaud, M. Desouter-Lecomte, D. Sugny, Towards laser control of open quantum systems: memory effects. Mol. Phys. 115(15–16), 1944–1954 (2017). https://doi.org/10.1080/00268976.2017.1319085

    Article  ADS  Google Scholar 

  110. R. Puthumpally-Joseph, E. Mangaud, V. Chevet, M. Desouter-Lecomte, D. Sugny, O. Atabek, Basic mechanisms in the laser control of non-Markovian dynamics. Phys. Rev. A 97, 033411 (2018). https://doi.org/10.1103/PhysRevA.97.033411

  111. C.A. Rodríguez-Rosario, K. Modi, A. Aspuru-Guzik, Linear assignment maps for correlated system-environment states. Phys. Rev. A 81, 012313 (2010). https://doi.org/10.1103/PhysRevA.81.012313

  112. A. Brodutch, A. Datta, K. Modi, A. Rivas, C.A. Rodríguez-Rosario, Vanishing quantum discord is not necessary for completely positive maps. Phys. Rev. A 87, 042301 (2013). https://doi.org/10.1103/PhysRevA.87.042301

  113. M.D. Choi, Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975). https://doi.org/10.1016/0024-3795(75)90075-0

  114. S.H. Kye, Choi matrices revisited. J. Math. Phys. 63(9), 092202 (2022). https://doi.org/10.1063/5.0107646

    Article  ADS  MathSciNet  MATH  Google Scholar 

  115. I.S. Dunn, R. Tempelaar, D.R. Reichman, Removing instabilities in the hierarchical equations of motion: exact and approximate projection approaches. J. Chem. Phys. 150(18), 184109 (2019). https://doi.org/10.1063/1.5092616

    Article  ADS  Google Scholar 

  116. C.P. Koch, Controlling open quantum systems: tools, achievements, and limitations. J. Phys.: Condens. Matter 28(21), 213001 (2016). https://doi.org/10.1088/0953-8984/28/21/213001

    Article  ADS  Google Scholar 

  117. B. Bylicka, D. Chruściński, S. Maniscalco, Non-Markovianity as a resource for quantum technologies (2013). https://doi.org/10.48550/ARXIV.1301.2585

  118. N. Mirkin, P. Poggi, D. Wisniacki, Information backflow as a resource for entanglement. Phys. Rev. A 99(6) (2019). https://doi.org/10.1103/physreva.99.062327

  119. C.P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S.J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, F.K. Wilhelm, Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe. EPJ Quantum Technol. 9(1), 19 (2022). https://doi.org/10.1140/epjqt/s40507-022-00138-x

  120. C. Lubich, I.V. Oseledets, A projector-splitting integrator for dynamical low-rank approximation. BIT Numer. Math. 54, 171–188 (2014). https://doi.org/10.1007/s10543-013-0454-0

    Article  MathSciNet  MATH  Google Scholar 

  121. C. Lubich, I.V. Oseledets, B. Vandereycken, Time integration of tensor trains. SIAM J. Numer. Anal. 53(2), 917–941 (2015). https://doi.org/10.1137/140976546

    Article  MathSciNet  MATH  Google Scholar 

  122. https://github.com/oseledets/ttpy. Accessed 2015

  123. A.J. Dunnett, A.W. Chin, Efficient bond-adaptive approach for finite-temperature open quantum dynamics using the one-site time-dependent variational principle for matrix product states. Phys. Rev. B 104, 214302 (2021). https://doi.org/10.1103/PhysRevB.104.214302

  124. S.V. Dolgov, A tensor decomposition algorithm for large odes with conservation laws. Comput. Methods Appl. Math. 19(1), 23–38 (2019). https://doi.org/10.1515/cmam-2018-0023

    Article  MathSciNet  MATH  Google Scholar 

  125. A. Potočnik, A. Bargerbos, F.A.Y.N. Schröder, S.A. Khan, M.C. Collodo, S. Gasparinetti, Y. Salathé, C. Creatore, C. Eichler, H.E. Türeci, A.W. Chin, A. Wallraff, Studying light-harvesting models with superconducting circuits. Nat. Commun. 9(1), 1–7 (2018). https://doi.org/10.1038/s41467-018-03312-x

    Article  Google Scholar 

  126. A.W. Chin, E. Mangaud, O. Atabek, M. Desouter-Lecomte, Coherent quantum dynamics launched by incoherent relaxation in a quantum circuit simulator of a light-harvesting complex. Phys. Rev. A 97, 063823 (2018). https://doi.org/10.1103/PhysRevA.97.063823

  127. P.R. Eastham, P. Kirton, H.M. Cammack, B.W. Lovett, J. Keeling, Bath-induced coherence and the secular approximation. Phys. Rev. A 94, 012110 (2016). https://doi.org/10.1103/PhysRevA.94.012110

  128. T. Brabec, F. Krausz, Intense few-cycle laser fields: frontiers of nonlinear optics. Rev. Mod. Phys. 72, 545–591 (2000). https://doi.org/10.1103/RevModPhys.72.545

  129. D.B. Milošević, G.G. Paulus, D. Bauer, W. Becker, Above-threshold ionization by few-cycle pulses. J. Phys. B: At. Mol. Opt. Phys. 39(14), R203–R262 (2006). https://doi.org/10.1088/0953-4075/39/14/r01

    Article  ADS  Google Scholar 

  130. G.F. Thomas, Validity of the Rosen–Zener conjecture for Gaussian-modulated pulses. Phys. Rev. A 27, 2744 (1983)

    Article  ADS  Google Scholar 

  131. M. Holthaus, B. Just, Generalized \(\pi\) pulse. Phys. Rev. A 49, 1950 (1994)

    Article  ADS  Google Scholar 

  132. Y. Jing, L. Chen, S. Bai, Q. Shi, Equilibrium excited state and emission spectra of molecular aggregates from the hierarchical equations of motion approach. J. Chem. Phys. 138(4), 045101 (2013). https://doi.org/10.1063/1.4775843

    Article  ADS  Google Scholar 

  133. J. Ma, J. Cao, Förster resonance energy transfer, absorption and emission spectra in multichromophoric systems. I. Full cumulant expansions and system-bath entanglement. J. Chem. Phys. 142(9), 094106 (2015). https://doi.org/10.1063/1.4908599

  134. H.D. Zhang, Q. Qiao, R.X. Xu, X. Zheng, Y. Yan, Efficient steady-state solver for hierarchical quantum master equations. J. Chem. Phys. 147(4), 044105 (2017). https://doi.org/10.1063/1.4995424

    Article  ADS  Google Scholar 

  135. T. Kramer, M. Noack, A. Reinefeld, M. Rodríguez, Y. Zelinskyy, Efficient calculation of open quantum system dynamics and time-resolved spectroscopy with distributed memory HEOM (DM-HEOM). J. Comput. Chem. 39(22), 1779–1794 (2018). https://doi.org/10.1002/jcc.25354. https://onlinelibrary.wiley.com/doi/abs/10.1002/jcc.25354

  136. H. Tamura, I. Burghardt, M. Tsukada, Exciton dissociation at thiophene/fullerene interfaces: the electronic structures and quantum dynamics. J. Phys. Chem. C 115(20), 10205–10210 (2011). https://doi.org/10.1021/jp203174e

    Article  Google Scholar 

  137. P.L. Walters, T.C. Allen, N. Makri, Direct determination of discrete harmonic bath parameters from molecular dynamics simulations. J. Comput. Chem. 38(2), 110–115 (2017). https://doi.org/10.1002/jcc.24527. https://onlinelibrary.wiley.com/doi/abs/10.1002/jcc.24527

  138. F. Häse, C. Kreisbeck, A. Aspuru-Guzik, Machine learning for quantum dynamics: deep learning of excitation energy transfer properties. Chem. Sci. 8, 8419–8426 (2017). https://doi.org/10.1039/C7SC03542J

    Article  Google Scholar 

  139. S. Bandyopadhyay, Z. Huang, K. Sun, Y. Zhao, Applications of neural networks to the simulation of dynamics of open quantum systems. Chem. Phys. 515, 272–278 (2018). https://doi.org/10.1016/j.chemphys.2018.05.019. Ultrafast Photoinduced Processes in Polyatomic Molecules: Electronic Structure, Dynamics and Spectroscopy (Dedicated to Wolfgang Domcke on the occasion of his 70th birthday)

  140. F.A.Y.N. Schröder, D.H.P. Turban, A.J. Musser, N.D.M. Hine, A.W. Chin, Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation. Nat. Commun. 10(1), 1–10 (2019). https://doi.org/10.1038/s41467-019-09039-7

    Article  Google Scholar 

Download references

Acknowledgements

This work was performed within the French GDR 686 3575 THEMS. We want to dedicate this work to the memory of Christoph Meier and Osman Atabek. Both authors made significant contributions to the research areas described above, such as the Meier–Tannor spectral density and the introduction of the auxiliary matrices. Both have also greatly contributed to many strategies of quantum control in a wide range of processes such as molecular orientation, dynamics of excited states, isomerization, molecular cooling to cite only few. Dominik Domin is warmly acknowledged for his efficient technical support.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. MSME, Université Gustave Eiffel, UPEC, CNRS, 77454, Marne-la-Vallée, France

    Etienne Mangaud

  2. LYRIDS, ECE-Paris, Graduate School of Engineering, 75015, Paris, France

    Amine Jaouadi

  3. Institut des Nanosciences de Paris, Sorbonne Université, CNRS, 75015, Paris, France

    Alex Chin

  4. Institut de Chimie Physique UMR8000, Université Paris-Saclay, CNRS, 91405, Orsay, France

    Michèle Desouter-Lecomte

Authors
  1. Etienne Mangaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amine Jaouadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alex Chin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michèle Desouter-Lecomte
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

Corresponding author

Correspondence to Michèle Desouter-Lecomte.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Ethics approval

Not applicable.

Appendix: Numerical implementation of HEOM with ttpy

Appendix: Numerical implementation of HEOM with ttpy

Most of the TT algebra is carried out by the ttpy package developed by Oseledets and coworkers [122]. In this appendix, we show a minimal code to build the system Liouvillian and time-integrate a given system density matrix with initial system–bath factorization in TT format with Python3, Numpy and ttpy packages.

The system Liouvillian is defined as \({{\mathcal {L}}_{S(\text{ADO})}}=-i(H\otimes {{I}_{n}}-{{I}_{n}}\otimes H)\) where H is the system Hamiltonian, \(I_n\) the identity matrix with n the number of system states. Thus, \({{\mathcal {L}}_{S(\text{ADO})}}\) is a \(n^2 \times n^2\) matrix which can be built from standard numpy functions (np.eye returns the identity matrix and np.kron the Kronecker product of both matrices):

figure a

where ids is the identity matrix of size \(n \times n\) and j the imaginary unit. To convert this numpy array to a TT format, tt.matrix routine performs an approximation of \({{\mathcal {L}}_{S(\text{ADO})}}\) for a maximal rank (rmax) and an accuracy eps:

figure b

where Ls is the system Liouvillian super-operator. At this point, we are still only dealing with a representation of an array of \(n^2 \times n^2\) dimensions. The HEOM super-operator \({\mathcal {L}}_{S}\) which spans over the whole Liouville space is defined as \({\mathcal {L}}_S = {\mathcal {L}}_{S(\text{ADO})} \otimes \prod _{k''=1}^K I_{n_{\textrm{HEOM}}} .\) To avoid memory issues due to the high dimensionality of the array, one must work with the Kronecker products (tt.kron) of ttpy packages instead of the one of numpy (np.kron). Indeed, numpy will build the full tensor which can be very large and thus might suffer from the dimensionality curse. To carry out this task, a single loop iterates over the number of artificial decay modes K with successive Kronecker products:

figure c

The total Liouvillian super-operator \({\mathcal {L}}\) is a sum of several other super-operators, i.e., \({\mathcal {L}}_{k'},\) \({\mathcal {L}}_{k'+},\) \({\mathcal {L}}_{k'-}\) (see Eq. (26)). Addition can be performed directly with the usual algebraic symbol (+) on TT objects. The ttpy library carries out automatically the correct tensor operations. However, tensor ranks increase at each iteration. Thus, rounding operations are regularly performed to reduce the rank for a given accuracy and maximum rank with the following command:

figure d

where L is the total HEOM Liouvillian super-operator.

The initial vectorized density matrix is defined directly from its cores. The first core is filled with the initial system density matrix. As all auxiliary density matrices are vectors of zeros when assuming system–bath factorization all the other cores are vectors of dimensions \(n_{\textrm{HEOM}}\) with only the first index equal to 1.

For a given system density matrix rho defined as a numpy array and r the initial core rank (equal to 1) in this example, we compute rho the initial density matrix in TT format with the following algorithm:

figure e

Time integration is performed with the KSL second-order splitting algorithm [120]. For each time step dt, the density matrix in TT format is updated using the function ksl implemented in tt.ksl routines:

figure f

By iterating over the desired number of timesteps, we compute the full density matrix at time t. To extract the system density matrix, projection techniques (by expressing projectors on the full Liouville space in the TT format) or core manipulations can be used.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mangaud, E., Jaouadi, A., Chin, A. et al. Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics. Eur. Phys. J. Spec. Top. 232, 1847–1869 (2023). https://doi.org/10.1140/epjs/s11734-023-00919-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-023-00919-0

Search

Navigation