Description
In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system-bath couplings, or to special cases where specific numerical techniques become effective.
I will present a general and yet exact numerical method to calculate the dynamics of an open quantum system strongly coupled to an environment. The method involves expressing the equations of motion for such an open quantum system as a tensor network [1]. In fact, the structure of this network is such that the environment parts can be pre-contracted efficiently, independent of the system Hamiltonian [2]. This so-called process tensor matrix product operator (PT-MPO) approach enables the calculation of dynamics for different time-dependent system Hamiltonians and allows the efficient calculation of any system multi-time correlation function.
I will show that such an approach enables us to optimize a control pulse for creating excitons in quantum dots [3], before going on to show how to combine it with standard tensor network methods to treat a chain of coupled systems interacting with independent environments [4]. I will also discuss how the non-additive properties of multiple environments are exposed using this technique [5].
I will go on to present work on using PT-MPO to model two-dimensional electronic spectroscopy, which is a technique used to measure coherent energy transport in molecular systems [6.7]. Finally, I will discuss recent work on extending our methods to systems with larger Hilbert spaces [8].
Our codes are publicly available [9] and we would be happy to support colleagues who would like to try them out on other problems.
[1] 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).
[2] Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-Markovian path integrals, M. R. Jørgensen and F. A. Pollock, Phys. Rev. Lett. 123 240602 (2019).
[3] G. E. Fux, E. Butler, P. R. Eastham, B. W. Lovett and J. Keeling, Efficient exploration of Hamiltonian parameter space for optimal control of non-Markovian open quantum systems Phys. Rev. Lett. 126 200401 (2021).
[4] Thermalization of a spin chain strongly coupled to its environment G. E. Fux, D. Kilda, B. W. Lovett and J. Keeling, Phys. Rev. Research 5 033078 (2023).
[5] Exact dynamics of non-additive environments in non-Markovian open quantum systems, D. Gribben, D. M. Rouse, J. Iles-Smith, A. Strathearn, H. Maguire, P. Kirton, A. Nazir, E. M. Gauger and B. W. Lovett, Physical Review X Quantum 3 010321 (2022).
[6] Process tensor approaches to modeling two-dimensional spectroscopy, R. de Wit, J. Keeling, B. W. Lovett and A. W. Chin, Phys. Rev. Research 7 013209 (2025).
[7] Extracting Coupling-Mode Spectral Densities with Two-Dimensional Electronic Spectroscopy, R. de Wit, J. Keeling, B. W. Lovett and A. W. Chin, arXiv:2503:21685, J. Phys. Chem. Lett. 16 6594 (2025).
[8] Efficient construction of time-invariant process tensors for simulating high-dimensional non-Markovian open quantum systems, É. Cochin, J. Keeling, B. W. Lovett and A. W. Chin arXiv:2603.06840 (2026).
[9] OQuPy - Open Quantum Systems in Python, oqupy.readthedocs.io