The non-dissipative damping of the Rabi oscillations as a “which-path” information

Abstract

Rabi oscillations may be viewed as an interference phenomenon due to a coherent superposition of different quantum paths, like in the Young's two-slit experiment. The inclusion of the atomic external variables causes a non dissipative damping of the Rabi oscillations. More generally, the atomic translational dynamics induces damping in the correlation functions which describe non classical behaviors of the field and internal atomic variables, leading to the separability of these two subsystems. We discuss on the possibility of interpreting this intrinsic decoherence as a "which-way" information effect and we apply to this case a quantitative analysis of the complementarity relation as introduced by Englert [Phys. Rev. Lett. 77, 2154]. Young's double-slit experiment may be considered as an emblematic example to introduce the wave-particle duality, which, to quote Feynman [1], is the basic mystery of Quantum Mechanics (QM). This experiment is frequently discussed in the introductory part of QM textbooks [2] to explain how the interference pattern and the which-way information on the particle trajectory are mutually exclusive behaviors in QM. Moreover, this wave-particle dualism belongs to the enlarged contest of complementarity [3,4], which has its root in the superposition principle. Coherent superposition of two states or, more generally, of two quantum paths leads to interference effects which may manifest itself as probability oscillations of some "populations". For example, when a two-level atom interacts with the field of an optical cavity, the atomic level populations undergo to the well known Rabi oscillations. But, what is the interference effect underlying this phenomenon? In other words, which paths do interfere? When the atomic external degrees of freedom are included in the model, the internal dynamics of the atom correlates with the translational variables. This information transfer towards the atomic external variables is at the origin of an intrinsic decoherence effect, that is, it causes a non dissipative damping of the Rabi oscillations [5,6]. Non dissipative damping of Rabi oscillations has been observed experimentally [7,8] and it has also been analyzed by Bonifacio et. al. who consider the evolution time as random variable [9]. In this letter we will show that a damping also affects all the correlations functions which are involved in the Bell inequality [10] and in the separability criterion [11,12] for the field and atomic internal variables. Can this decoherence be interpreted as a "which-way" information effect? In this Letter we will try to throw light on these questions. Let us consider a two-level atom interacting with the field of an optical cavity. In the rotating wave approximation (RWA) the atom-field interaction is described by the Hamiltonian ǫ sin kx(â †Ŝ − +âŜ + ) , where the usual spin 1/2 operators refer to the internal dynamics of the two-level atom, whileâ andâ † are the annihilation and creation operators for the photons of k-mode of the cavity standing wave and ǫ is the at...

On the observability of Bell's inequality violation in the two-atoms optical Stern-Gerlach model

“…The damping factor shown by this last approximated expression, which is at the origin of the non dissipative damping of the Rabi oscillations [7,22,23], is due to the increasing distance in the phase space [24] of the two deflected components of the translational wave packet [25]. Similar behaviors hold for the other coefficients of eq.…”

“…The visibility V of the interference pattern and the distinguishability D of the quantum paths can in some extent coexist, and as shown by Englert in its quantitative analysis of complementarity [5,6], they satisfy the inequality D 2 + V 2 ≤ 1. According to this analysis and in the ambit of the optical Stern-Gerlach (SG) model, we have recently shown [7] that the visibility of the Rabi oscillations and the distinguishability of the two atomic translational paths satisfy the equality relation D 2 +V 2 = 1 when pure initial states are considered.When applied to a composite system, the superposition principle leads to quantum correlations (entanglement), which may hide the individuality of the subsystems. Differently from the classical case, and in idealized configurations, two quantum systems that have interacted for a time, generally do not recover their individuality, even if the subsystems become spatially separated.…”

Irreversible decay of nonlocal entanglement via a reservoir of a single degree of freedom

“…It has already been shown that this non dissipative coupling leads to a decoherence in the dynamics of a single atom [11,12,13] and also affects the entanglement in the internal dynamics of two atoms that successively interact with the same cavity [14]. As will shown in what follows, the same interaction gives rise to a disentanglement of the bipartite system which exhibits strong analogies with the disentanglement generated by the vacuum noise [4,8].…”

“…As a consequence, the density operator |ψ(t) ψ(t)| must be traced on the cavity fields and on the atomic translation variables. Taking into account the normalization of kets (12) and (13), and using relation (9) we obtain ρ AB (t) = T r transl,f ield (|ψ(t) ψ(t)|)…”

“…[13] for the one atom case. The same Γ(t) can be also interpreted as the visibility of the Rabi oscillations [12] in the ambit of the complementarity relation as introduced by Englert [16]. Considering the ordered basis |++ , |+− , |−+ , |−− , the reduced density operator (18) where (a 1 + a 2 + a 4 ) = 1. a 3 describes the coherence between the two atoms which assumes its maximum value at t = 0 and, contrarily to what happens in the JC case, goes irreversibly to zero because of the coupling of each qubit with its own external dynamics.…”