https://doi.org/10.1140/epjb/s10051-026-01168-5
Research - Condensed Matter
Analytical insights into the spin–boson model and optimal control via the redfield master equation
1
, 26, rue Gervais Bussière 69100, Villeurbanne, France
2
School of Applied and Engineering Physics, Mohammed VI Polytechnic University (UM6P), 43150, Ben Guerir, Morocco
3
Department of Chemistry, Princeton University, 08544, Princeton, NJ, USA
a
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Received:
11
December
2025
Accepted:
8
April
2026
Published online:
12
May
2026
Abstract
In this work, we are interested in the dynamics of the driven spin–boson model. First, we derive a master equation describing the interaction of a two-level system (TLS) with an Ohmic heat bath. The system is subject to the action of two control fields, physically associated with the time-dependent bias energy and time-dependent tunneling. In the limit where tunneling is equal to zero, we derive an analytic solution of the master equation in the undriven case. Then, we apply quantum optimal control theory to the resolvent to compute the time-dependent bias energy and time-dependent tunneling which allows for creating a single-qubit quantum gate for the fully driven spin–boson model. The resolvent is the generalization of the evolution quantum propagator in dissipative dynamics. In order to solve the quantum optimal control problem, we use the technique of automatic differentiation. We are able to implement the X-gate and Z-gate with good fidelity. The advantage of applying optimal control theory to the resolvent is that the logic gate does not depend on the initial state of the qubit. Thus, the optimal solution is unique regardless of the initial state of the qubit.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
