The Quantum Stochastic Limit for Non-Markovian Driving Processes

Abstract

We express the stochastic limit (weak-coupling limit) of a quantum system interacting with its environment as the limit of solutions to a parametrized family of quantum stochastic differential equations (QSDEs). We prove the convergence of such solutions in the case where the driving processes may be non-Markovian and the system may be infinite-dimensional with unbounded Hamiltonian. We achieve this by imposing a particular condition on the n-point functions of the driving process which we verify for annihilation, creation, and gauge operators on a bosonic Fock space with sufficient mixing. Additionally, we include not only the usual fast-evolving system Hamiltonian which is averaged out in the limit, but also a slow-evolving system Hamiltonian which survives the limit. We also relax the requirement for the fast-evolving Hamiltonian to have a purely discrete spectrum allowing accumulation points under some circumstances. Moreover, our convergence mode for the processes is convergence of all time-centered n-point functions which is stronger than the typical semigroup convergence for the stochastic limit.

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BibTeX RIS

BibTeX

@misc{reese2026quantum,
  title = {{The Quantum Stochastic Limit for Non-Markovian Driving Processes}},
  author = {Tanner J. Reese},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06202},
  eprint = {2611.06202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06202},
}

RIS

TY  - UNPB
TI  - The Quantum Stochastic Limit for Non-Markovian Driving Processes
AU  - Tanner J. Reese
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06202
UR  - https://arxiv.org/abs/2611.06202
AB  - We express the stochastic limit (weak-coupling limit) of a quantum system interacting with its environment as the limit of solutions to a parametrized family of quantum stochastic differential equations (QSDEs). We prove the convergence of such solutions in the case where the driving processes may be non-Markovian and the system may be infinite-dimensional with unbounded Hamiltonian. We achieve this by imposing a particular condition on the n-point functions of the driving process which we verify for annihilation, creation, and gauge operators on a bosonic Fock space with sufficient mixing. Additionally, we include not only the usual fast-evolving system Hamiltonian which is averaged out in the limit, but also a slow-evolving system Hamiltonian which survives the limit. We also relax the requirement for the fast-evolving Hamiltonian to have a purely discrete spectrum allowing accumulation points under some circumstances. Moreover, our convergence mode for the processes is convergence of all time-centered n-point functions which is stronger than the typical semigroup convergence for the stochastic limit.
ER  -

Abstract

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