On the Interaction Between a Time-Dependent Field and a Two-Level Atom: Path Integral Treatment
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On the Interaction Between a Time-Dependent Field and a Two-Level Atom: Path Integral Treatment Hilal Benkhelil1
· Mekki Aouachria1
Received: 29 July 2020 / Accepted: 22 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We use the coherent state path integral and a Schwinger model for the spin to solve the generalized Jaynes-Cummings model with a time-dependent coupling parameter of the photons-field. The propagators are given explicitly as perturbation series. These are summed up exactly. The wave functions are deduced. Keywords Path integral · Propagator · Coherent state · Two level system · The Jaynes-Cummings model
1 Introduction Among the simplest and most significant systems in quantum optics, Jaynes–Cummings model (JCM). This model describes the interaction of a two-level atom with a single mode of the quantized electromagnetic field [1]. In the last decades, JCM has received a lot of attention; an important number of theoretical and experimental studies are done in this context [2–4]. For this reason and others, this model has many generalization versions, for example, time-dependent atom field coupling [5], intensity dependent coupling [6, 7], and the most recent one is done by introducing the Kerr non-linearity [8]. In this work, we use the bosonic and fermionic coherent states path integral via the Schwinger model of spin to explicitly solve the problem of the generalized JCM in the presence of a time-dependent photons-field parameter, governed by the Hamiltonian [9] H = H0 + Hint with
(1)
H0 = a † a + g(t)a † a, (2) ωa and † (3) σ z + λ aσ + + a σ − . Hint = 2 † Here a and a are the atomic flipping operators, is the field frequency, λ is some arbitrary real parameter, ωa is the transition frequency between the levels, σ ± = 12 (σ x ±iσ y ) are the spin projection operators, and we are considering the case of time-dependent field g(t). Mekki Aouachria
[email protected] 1
Laboratoire de Physique Energ´etique Appliqu´ee (LPEA), Facult´e des Sciences de la Mati`ere, D´epartement de Physique, Universit´e Batna1, Batna, Algeria
International Journal of Theoretical Physics
Our purpose in this paper is to present the adaptation manner of the given problem with the commonly used methods. The rest of this paper is organized as follows. In Section 2, we give some notation and the spin coherent state path integral for spin 1/2 system for our further computations. After setting up a path integral formalism for the propagator, we perform the direct calculations over the bosonic variables, in Section 3. Accordingly, the integration over the spin variables is easy to carry out and the result is given as a perturbation series, which is summed up exactly. The explicit result of the propagator is directly computed and the wave function is then deduced in Section 4. Finally, we draw our conclusion in Section 5.
2 Path Integral Formulation First of all we introduce some definitions, properties and notations needed in this paper. Describe briefly the coher
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