Corrections to the Hamiltonian induced by finite-strength coupling to the environment
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912203878170624 |
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| author | Łobejko, Marcin Winczewski, Marek Suárez, Gerardo Alicki, Robert Horodecki, Michał |
| author_facet | Łobejko, Marcin Winczewski, Marek Suárez, Gerardo Alicki, Robert Horodecki, Michał |
| contents | If a quantum system interacts with the environment, then the Hamiltonian acquires a correction known as the Lamb-shift term. There are two other corrections to the Hamiltonian, related to the stationary state. Namely, the stationary state is to first approximation a Gibbs state with respect to original Hamiltonian. However, if we have finite coupling, then the true stationary state will be different, and regarding it as a Gibbs state to some effective Hamiltonian, one can extract a correction, which is called "steady-state" correction. Alternatively, one can take a static point of view, and consider the reduced state of total equilibrium state, i.e., system plus bath Gibbs state. The extracted Hamiltonian correction is called the "mean-force" correction. This paper presents several analytical results on second-order corrections (in coupling strength) of the three types mentioned above. Instead of the steady state, we focus on a state annihilated by the Liouvillian of the master equation, labeling it as the "quasi-steady state." Specifically, we derive a general formula for the mean-force correction as well as the quasi-steady state and Lamb-shift correction for a general class of master equations. Furthermore, specific formulas for corrections are obtained for the Davies, Bloch-Redfield, and cumulant equation (refined weak coupling). In particular, the cumulant equation serves as a case study of the Liouvillian, featuring a nontrivial fourth-order generator. This generator forms the basis for calculating the diagonal quasi-steady-state correction. We consider spin-boson model as an example, and in addition to using our formulas for corrections, we consider mean-force correction from the reaction-coordinate approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_00643 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Corrections to the Hamiltonian induced by finite-strength coupling to the environment Łobejko, Marcin Winczewski, Marek Suárez, Gerardo Alicki, Robert Horodecki, Michał Quantum Physics Statistical Mechanics If a quantum system interacts with the environment, then the Hamiltonian acquires a correction known as the Lamb-shift term. There are two other corrections to the Hamiltonian, related to the stationary state. Namely, the stationary state is to first approximation a Gibbs state with respect to original Hamiltonian. However, if we have finite coupling, then the true stationary state will be different, and regarding it as a Gibbs state to some effective Hamiltonian, one can extract a correction, which is called "steady-state" correction. Alternatively, one can take a static point of view, and consider the reduced state of total equilibrium state, i.e., system plus bath Gibbs state. The extracted Hamiltonian correction is called the "mean-force" correction. This paper presents several analytical results on second-order corrections (in coupling strength) of the three types mentioned above. Instead of the steady state, we focus on a state annihilated by the Liouvillian of the master equation, labeling it as the "quasi-steady state." Specifically, we derive a general formula for the mean-force correction as well as the quasi-steady state and Lamb-shift correction for a general class of master equations. Furthermore, specific formulas for corrections are obtained for the Davies, Bloch-Redfield, and cumulant equation (refined weak coupling). In particular, the cumulant equation serves as a case study of the Liouvillian, featuring a nontrivial fourth-order generator. This generator forms the basis for calculating the diagonal quasi-steady-state correction. We consider spin-boson model as an example, and in addition to using our formulas for corrections, we consider mean-force correction from the reaction-coordinate approach. |
| title | Corrections to the Hamiltonian induced by finite-strength coupling to the environment |
| topic | Quantum Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2204.00643 |