Adiabatic Transitions in a Two-Level System Coupled to a Free Boson Reservoir
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Annales Henri Poincar´ e
Adiabatic Transitions in a Two-Level System Coupled to a Free Boson Reservoir Alain Joye , Marco Merkli
and Dominique Spehner
Abstract. We consider a time-dependent two-level quantum system interacting with a free Boson reservoir. The coupling is energy conserving and depends slowly on time, as does the system Hamiltonian, with a common adiabatic parameter ε. Assuming that the system and reservoir are initially decoupled, with the reservoir in equilibrium at temperature T ≥ 0, we compute the transition probability from one eigenstate of the twolevel system to the other eigenstate as a function of time, in the regime of small ε and small coupling constant λ. We analyse the deviation from the adiabatic transition probability obtained in the absence of the reservoir.
1. Introduction In this paper, we study the transition probability between the energy eigenstates of a driven two-level system in contact with an environment, a bosonic reservoir at zero or at positive temperatures. The Hamiltonian of the two-level system and the coupling with the reservoir both depend on time, varying on a slow timescale 1/ε; that is, they are functions of the rescaled time t = εtp , where tp is the physical time. We consider interaction Hamiltonians which are linear in the bosonic field operators and for which the system and reservoir do not exchange energy instantaneously, meaning that the system Hamiltonian commutes with the interaction at any given time. The initial system-reservoir state is taken to be disentangled, with the two-level system in an eigenstate of its Hamiltonian and the reservoir in equilibrium at temperature T ≥ 0. Such an initial state is very natural from an open quantum system perspective; it corresponds to the situation in which the system is put in contact with the reservoir at t = 0. Our main goal is to (λ,ε) determine the probability, denoted p1→2 (t), to find the system in the other eigenstate at some fixed rescaled time t > 0. We do this in the adiabatic and weak coupling regime, meaning that ε and the system-reservoir coupling constant λ are both small.
A. Joye et al.
Ann. Henri Poincar´e
The adiabatic regime yields rather detailed and precise approximations of the true quantum dynamics in a variety of physically relevant situations, and its study has a long history. The adiabatic theorem of quantum mechanics was first stated for self-adjoint time-dependent Hamiltonians with isolated eigenvalues in [7,23] and then extended to accommodate isolated parts of the spectrum, see [6,29]. This version applies to the two-level system we consider, in the absence of coupling. Adiabatic approximations for gapless Hamiltonians, where the eigenvalues are not isolated from the rest of the spectrum, were later established in [3,33]. This is in particular the situation for the total Hamiltonian of the two-level system coupled to a free boson reservoir. Then, adiabatic theorems were formulated in [1,4,19] for dynamics generated by nonself-adjoint operators, leading to extensions of the gapless, no
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