Exact propagators for atom - laser interactions

A class of exact propagators describing the interaction of an N -level atom with a set of on-resonance δ-lasers is obtained by means of the Laplace transform method. State-selective mirrors are described in the limit of strong lasers. The ladder, V and Λ configurations for a three-level atom are discussed. For the two level case, the transient effects arising as result of the interaction between both a semi-infinite beam and a wavepacket with the on-resonance laser are examined. PACS numbers: 03.75.Be, 03.75.-b, 31.70.Hq The spacetime propagator can be considered as one of the most important tools in quantum physics for it governs any dynamical process. However, the knowledge of propagators corresponding to non-quadratic Hamiltonians is severely restricted. In this line, the spacetime propagator for a δ-potential relevant to tunnelling problems has excited much attention [1, 2, 3, 4, 5]. Such interactions turn out to be particularly useful to gain physical insight in systems where only integrated quantities are to be considered. A thorough discussion of point interactions as solvable models using a functional approach can be found in [6], and a formalism to incorporate general pointinteractions and dealing with different boundary conditions has been developed by Grosche [7, 8]. Even though the method is particularly suitable to calculate the energydependent Green function, a wide class of propagators was derived in such a fashion. The incorporation of time-dependent point-interactions has been possible through different approaches as Duru’s method [9] or the use of integrals of motion [10]. However, most of the effort has been focused on the dynamics of structureless particles and to the knowledge of the authors no attention has been paid to problems involving internal levels. Such state of affairs contrasts dramatically with the current surge of activity in atom optics. In this paper we use the method of Laplace transform [4, 5] to tackle particles with internal structure. In particular, we shall focus on exact propagators for atom-laser interactions, namely, those of an atom interacting with a set of δ-laser on-resonance with given interatomic transitions. The method is introduced in section 1 to obtain the exact propagator for a two-level atom. Details of the calculations relevant to the following sections are here provided. In section 2 the propagators for a ladder, V, and Λ configuration (see Fig. 1) of lasers interacting with a three-level atom are obtained. The Exact propagators for atom-laser interactions 2 general case in which a given state is coupled to an arbitrary number of levels is discussed in section 3 where the high intensity limit of the laser is related to state selective mirrors. Such kind of systems presents manifold applications in laser coherent control techniques such as cold atomic cloud compression [11], atom mirrors and beam splitters [12], and different schemes where fast transitions are required as in the implementation of logic gates for ion trap quantum computing [13]. Moreover, idealised time-of-arrival measurements [14] and recently proposed improvements in Ramsey-interferometry with ultracold atoms [15] rest in a full quantum mechanical treatment of the dynamics of such systems. 1. The two-level atom In this section, we use the method of Laplace transform to obtain the propagator for a two-level atom incident on a narrow perpendicular on-resonance laser beam. Spontaneous decay is assumed to be negligible throughout the paper and we shall consider effective one-dimensional systems in which the transverse momentum components can be neglected as it is the case for atoms in narrow waveguides [16]. In a laser adapted interaction picture, and using the rotating wave approximation, the Hamiltonian describing the system is Hc = p̂ 2m 12 +Vδ(x̂− ξ) = p̂ 2m 12 + ~Ω 2 δ(x̂− ξ) ( 0 1 1 0 ) , (1) where p̂ is the momentum operator conjugate to x̂, the ground state |1〉 is in vectorcomponent notation ( 1 0 ) , and the excited state |2〉 is ( 0 1 ) . The second term in the right hand side defines the potential strength matrix V and 12 the two-dimensional identity matrix. Equation (1) may be regarded as the ǫ → 0 limit of a laser of width ǫ and Rabi frequency ΩL keeping Ω = ΩLǫ constant. ΩL here and in the following is chosen to be real. We start then by considering the free propagator for a one-channel problem on a Hilbert space of square integrable functions H (see for instance [8]), K0(x, t|x, 0) = √ m 2πi~t e im(x−x′)2 2t~ . (2) In what follows we shall be interested in describing the dynamics of particles with two internal levels. The free propagator (Ω = 0) for states on the Hilbert space H⊗C2 is given by K0(x, t|x′, t) = K0(x, t|x′, t)12, in the same interaction picture than (1). Moreover, δ-type of perturbations can be generally taken into account using the method of Laplace transform [5] which assumes the unperturbed propagator to be known. More precisely, the full propagator can be related to the free one through the Lippmann-Schwinger equation [4, 5] K(x, t|x, t) = K0(x, t|x, t)− i ~ ∫ t