Abstract
Starting from a microscopic system–environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system–environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory.
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24 April 2019
Correction. The bound (2.23) in Theorem 2.1.
24 April 2019
Correction. The bound (2.23) in Theorem 2.1.
Notes
We recall that a map on bounded operators on a Hilbert space, \(V:{\mathcal B}({\mathcal H})\rightarrow {\mathcal B}({\mathcal H})\), is called completely positive if \(V\otimes {\mathbf 1}:{\mathcal B}({\mathcal H}\otimes {\mathbb C}^n)\rightarrow {\mathcal B}({\mathcal H}\otimes {\mathbb C}^n)\) is positive (maps positive operators into positive ones) for all \(n\in \mathbb N\).
If the initial state is of product form \(\omega _\mathrm{S}\otimes \omega _{\mathrm{R},\beta }\), then the term \( C |\lambda |\mathrm{e}^{-\theta _0 t}\) in (2.13) can be replaced by \( C \lambda ^2\mathrm{e}^{-\theta _0 t}\), see Theorem 3.1 of [20], Resonance theory of decoherence and thermalization.
\(\Omega _{\mathrm{S},\beta }\) is cyclic, meaning that \(({\mathcal B}({\mathcal H}_\mathrm{S})\otimes {\mathbf 1}_\mathrm{S})\Omega _{\mathrm{S},\beta }={\mathcal H}_\mathrm{S}\otimes {\mathcal H}_\mathrm{S}\) and \(\Omega _{\mathrm{S},\beta }\) is separating, meaning that if \((X\otimes {\mathbf 1}_\mathrm{S})\Omega _{\mathrm{S},\beta }=0\) then \(X=0\). Due to the cyclic and separating property, (2.14) defines the map \(\delta ^t_\lambda \) uniquely, and it shows that \(t\mapsto \delta ^t_\lambda \) is a group.
One may also follow an extended Mourre theory approach, recently developed in [15]. This method is more powerful in that it requires less restrictive assumptions, but it is technically somewhat more demanding.
Use the estimate \(|\mathrm{e}^{-\lambda ^2 t\mathrm{Im }a_{e,j}} - \mathrm{e}^{-\lambda ^2 t\mathrm{Im }\widetilde{\lambda }_{e,j}}| = \mathrm{e}^{-\lambda ^2 t\mathrm{Im }a_{e,j}}|1-\mathrm{e}^{\lambda ^2 t\mathrm{Im }(a_{e,j}-\widetilde{\lambda }_{\widetilde{e},j})}| \leqslant \mathrm{const.} |\lambda |^3 t \mathrm{e}^{-\lambda ^2(1+O(\lambda ))\gamma t}\).
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Könenberg, M., Merkli, M. Completely positive dynamical semigroups and quantum resonance theory. Lett Math Phys 107, 1215–1233 (2017). https://doi.org/10.1007/s11005-017-0937-z
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DOI: https://doi.org/10.1007/s11005-017-0937-z