Abstract
The dynamics of open quantum systems are of great interest in many research fields, such as for the interaction of a quantum emitter with the electromagnetic modes of a nanophotonic structure. A powerful approach for treating such setups in the non-Markovian limit is given by the chain mapping where an arbitrary environment can be transformed to a chain of modes with only nearest-neighbor coupling. However, when long propagation times are desired, the required long chain lengths limit the utility of this approach. We study various approaches for truncating the chains at manageable lengths while still preserving an accurate description of the dynamics. We achieve this by introducing losses to the chain modes in such a way that the effective environment acting on the system remains unchanged, using a number of different strategies. Furthermore, we demonstrate that extending the chain mapping to allow next-nearest neighbor coupling permits the reproduction of an arbitrary environment, and adding longer-range interactions does not further increase the effective number of degrees of freedom in the environment.
Highlights
No quantum system is ever fully isolated
This implies that any excitation that propagates along the chain will not be reflected anymore after reaching a sufficient distance, and can not affect the system. This general structure implies that the chain mapping naturally provides a separation of any bath into (i) a non-Markovian part close to the system, where excitations can be reflected due to the position-dependent coupling and energy at each site, and interact again with the system, leading to memory effects, and (ii) a Markovian part far enough away from the system where excitations propagate along a featureless continuum, such that their ‘momentum’ along the chain is conserved and they are just transported away without ever affecting the system again
The approaches are based on adding dissipation to the chain modes in a way that minimizes reflections and reproduces the system dynamics optimally
Summary
No quantum system is ever fully isolated. Any system is coupled to its external environment with a large (essentially infinite) number of degrees of freedom, leading to the notion of open quantum systems [1,2,3]. For example, formed by electromagnetic modes [4], phonons [5], or ensembles of other quantum systems [6]. About the dynamics that the system underwent at earlier times. In this so-called Markovian limit, the dynamics of the system can be approximately described using a Lindblad master equation [7]. We note that lack of memory does not necessarily require or imply that the environment fully equilibrates to its original state—for example, a photon emitted from an atom in free space will never interact with the atom again even if it is not absorbed or otherwise affected by another object after being emitted
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