Tomography, control, and characterization of entanglement in a three-level atomic system

Abstract

We study the quantum correlations of the radiation emitted by three level atoms (cascade type) interacting with two driving fields. In the linear regime, and in the Weisskopf-Wigner approximation, we show that the atomic and the two-photon density matrix are equivalent to each other. This facilitates the tomography of the two mode state to be realized by measurements on either the atomic system or the emitted fields. While, in general, one needs 4 N measurements for the tomography of a N photon state, we show that one needs (N + 1) 2 − 1 observables for the tomography of photons emitted by an atomic system. Thus there is an exponential reduction in the number of observables for the reconstruction of the class of N photon states emitted by atoms. We show that the driving field strengths and detunings provide the control parameters for the preparation of a specific target state. Finally, we study the characterization of entanglement of the two photon state. We observe that a characterization of entanglement in terms of a single parameter is not possible when the system is in a mixed state; therefore, we provide a description in terms of the newly introduced probability distribution for entanglement, in various regimes of interest.

Geometric phase: an indicator of entanglement

“…At two photon resonance ∆ = 0. Some of the non-classical properties of this two photon field have been reported in the literature [20,19]. In particular, a detailed numerical study of the entanglement distribution as a function of the these control parameters has been reported in [19].…”

“…Some of the non-classical properties of this two photon field have been reported in the literature [20,19]. In particular, a detailed numerical study of the entanglement distribution as a function of the these control parameters has been reported in [19]. In order to establish the dependence of entanglement and purity on these control parameters we make a small change in the scheme used above, namely, we replace the metastable state (Γ3 = 1M Hz) by an infinitely long lived state i.e, Γ3 = 0 which we call scheme II.…”

“…The two photon state may now be obtained by taking the partial trace over the atomic indices. It has been further shown [19], that the two photon density matrix at a time t is equivalent to the atomic density matrix at a retarded time t − r/c [21]. Utillizing this equivalence we write the two-photon normalized density matrix in the basis {|0 , |1 , |3 } ≡ {|00 , |01 , |11 } as …”

“…The two photon density matrix for scheme I (Fig.1) may be determined using the quantum tomographic method described in [19]. We briefly recapitulate the procedure here.…”

“…Atom photon interaction provides common ground for exploring the relationship between geometric phase and entanglement with the added advantage of the existence of control parameters for manipulating the two photon states. In this letter we study the non-adiabatic, non-cyclic, non-unitary evolution of the geometric phase of the two-photon state corresponding to the two modes emitted by a three level cascade system interacting with two driving fields [19]. The two photon state is in general a mixed state, however it may also be prepared in a pure state by proper control.…”