Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models
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arXiv
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| Format: | Preprint |
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2024
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| author | Gough, John E. Ding, Haijin Amini, Nina H. |
| author_facet | Gough, John E. Ding, Haijin Amini, Nina H. |
| contents | We give a derivation of the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arises as a consequence of using the Bargmann-Segal (complex wave) representation of the bath. In particular, we construct a reproducing kernel Hilbert space for the bath auto-correlation and realize the space of complex trajectories as a Hilbert subspace. The reproducing kernel naturally arises from a feature space where the underlying feature space is the one-particle Hilbert space of the bath quanta. We exploit this to derive the unravelling of the open quantum system dynamics and show equivalence to the equation of Di{ó}si and Strunz. We also give an explicit expression for the reduced dynamics of a two-level system coupled to the bath via a Jaynes-Cummings interaction and show that this does indeed correspond to an exact solution of the Di{ó}si-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07231 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models Gough, John E. Ding, Haijin Amini, Nina H. Quantum Physics We give a derivation of the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arises as a consequence of using the Bargmann-Segal (complex wave) representation of the bath. In particular, we construct a reproducing kernel Hilbert space for the bath auto-correlation and realize the space of complex trajectories as a Hilbert subspace. The reproducing kernel naturally arises from a feature space where the underlying feature space is the one-particle Hilbert space of the bath quanta. We exploit this to derive the unravelling of the open quantum system dynamics and show equivalence to the equation of Di{ó}si and Strunz. We also give an explicit expression for the reduced dynamics of a two-level system coupled to the bath via a Jaynes-Cummings interaction and show that this does indeed correspond to an exact solution of the Di{ó}si-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable. |
| title | Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2407.07231 |