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arXiv:2102.03621

Markovian approximation for Pauli Fierz operators

Laurent Amour, Jean Nourrigat

Published Feb 6, 2021• 0 citations
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Abstract

The purpose of this article is to derive a Markovian approximation of the reduced time dynamics of observables for the Pauli-Fierz Hamiltonian with a precise control of the error terms. In that aim, we define a Lindblad operator associated to the corresponding quantum master equation. In a particular case, this allows to study the transition probability matrix.

DOI
10.48550/arxiv.2102.03621

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The purpose of this article is to derive a Markovian approximation of the reduced time dynamics of observables for the Pauli-Fierz Hamiltonian with a precise control of the error terms. In that aim, we define a Lindblad operator associated to the corresponding quantum master equation. In a particular case, this allows to study the transition probability matrix.
Abstract. The Hamiltonian approach of the Pauli-Fierz model is studied by Derezinski and Jaksic (cf. [7]). In this paper, we give the Markovian description of this model. Using the weak coupling limit, we derive the quan-tum Markovian semigroup of the Pauli-Fierz system from its Hamiltonian description. Also, we give the explicit form of the associated Lindblad gener-ator. As a consequence, we study the properties of the associated quantum master equation: Quantum detailed balance condition and return to equilib-rium. Finally, we give the associated quantum Langevin equation which can be obtained by a repeated quantum interaction Hamiltonian. 1.
Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator [Formula: see text] is encoded into a quantum state [Formula: see text]. Then, given access to [Formula: see text] copies of the state [Formula: see text], the task is to simulate the corresponding Markovian dynamics for time [Formula: see text]. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses [Formula: see text] samples of [Formula: see text] to achieve the target dynamics, with an approximation error of [Formula: see text].
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.
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Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics.
Manipulation of a quantum system requires the knowledge of how it evolves. To impose that the dynamics of a system becomes a particular target operation (for any preparation of the system), it may be more useful to have an equation of motion for the dynamics itself, rather than the state. Here we develop a Markovian master equation for the process matrix of an open system interacting with a reservoir, which resembles the Lindblad Markovian master equation. We employ this equation to introduce a scheme for optimal local coherent process control at target times and extend the Krotov technique to obtain optimal control. We illustrate utility of this framework through several quantum coherent-control scenarios, such as optimal decoherence suppression, gate simulation, and passive and indirect control of the reservoir, in all of which we aim to simulate a given terminal process at a given final time.