Dipole moments quantum interference

The effect of quantum interference on spontaneous emission in atomic and molecular systems is the generation of superposition states that can be manipulated, to reduce the interaction with the environment, by adjusting the polarizations of the transition dipole moments, or the amplitudes and phases of the external driving fields. With a suitable choice of parameters, the superposition states can decay with controlled and significantly reduced rates. This modification can lead to subnatural linewidths in the fluorescence and absorption spectra [5,10]. Furthermore, as will be shown in this review, the superposition states can even be decoupled from the environment and the population can be trapped in these states without decaying to the lower levels. These states, known as dark or trapped states, were predicted in many configurations of multilevel systems [11], as well as in multiatom systems [12],... 

The presence of the additional damping terms F12 may suggest that quantum interference enhances spontaneous emission from two coupled systems. However, as we shall illustrate in the following sections, the presence of these terms in the master equation can, in fact, lead to a reduction or even suppression of spontaneous emission. According to Eq. (62), the reduction and suppression of spontaneous emission can be controlled by changing the mutual orientation of the dipole moments of the bare systems. 

Phase dependent effects in spontaneous emission have been predicted in atomic systems with nonorthogonal as well as with orthogonal dipole moments. In the first case the phase-dependent effects, which arise from quantum interference between two nonorthogonal dipole moments, can be observed with two driving fields [25-28]. In the latter case the observation of phase-dependent effects requires at least three driving fields [29,30], It is of particular interest to observe the phase-dependent effects, as they represent interference effects that can be induced by driving fields even in the absence of the vacumm-induced quantum interference. 

The discussion, presented in Section IV, has been concentrated on analysis of the effect of quantum interference on spontaneous emission in a V-type three-level atom. With the specific examples we have demonstrated that spontaneous emission can be controlled and even suppressed by quantum interference. In this section, we extend the analysis to the case of coherently driven systems. We will present simple models for quantum interference in which atomic systems are composed of two coupled dipole subsystems. In particular, we consider interference effects in coherently driven V and A-type three-level atoms. Each of the three systems is represented by two dipole moments, p, and p2, interacting through the vacuum field. 

Figure 4. Dressed states of two neighboring manifolds, N I I and N. Solid arrows indicate transitions at oiL 2 that are only slightly affected by quantum interference, while the dashed arrow indicates the transition at the laser frequency aL, which is strongly affected by quantum interference and vanishes for parallel dipole moments and p12 — p,21 ...

The dressed-atom predictions clearly explain the origin of the cancellation of the spectral line arising from the cancellation of the transition dipole moment due to quantum interference between the two atomic transitions. 

Another area of interest in quantum interference effects, which has been studied extensively, is the response of a V-type three-level atom to a coherent laser field directly coupled to the decaying transitions. This was studied by Cardimona et al. [36], who found that the system can be driven into a trapping state in which quantum interference prevents any fluorescence from the excited levels, regardless of the intensity of the driving laser. Similar predictions have been reported by Zhou and Swain [5], who have shown that ultrasharp spectral lines can be predicted in the fluorescence spectrum when the dipole moments of the atomic transitions are nearly parallel and the fluorescence can be completely quenched when the dipole moments are exactly parallel. 

The CPT effect and its dependence on quantum interference can be easily explained by examining the population dynamics in terms of the superposition states. v) and a). Assume that a three-level A-type atom is composed of a single upper state 3) and two ground states 1) and 2). The upper state is connected to the lower states by transition dipole moments p31 and p32. After introducing superposition operators 5+ = (S ) = 3)(.v and 5+ = (Sa) = 3)(a, where. v) and a) are the superposition states of the same form as Eqs. (107) and (108), the Hamiltonian (65) can be written as... 

Consider the Menon-Agarwal approach to the Autler-Townes spectrum of a V-type three-level atom. The atom is composed of two excited states, 1) and 3), and the ground state 2) coupled by transition dipole moments with matrix elements p12 and p32, but with no dipole coupling between the excited states. The excited states are separated in frequency by A. The spontaneous emission rates from 1) and 3) to the ground state 2) are Tj and T2, respectively. The atom is driven by a strong laser field of the Rabi frequency il, coupled solely to the 1) —> 2) transition. This is a crucial assumption, which would be difficult to realize in practice since quantum interference requires almost parallel dipole moments. However, the difficulty can be overcome in atomic systems with specific selection rules for the transition dipole moments, or by applying fields with specific polarization properties [26]. 

Figure 13, together with the transition dipole moments and transition rates, provides a simple interpretation of the absorption rate shown in Fig. 11. According to Eq. (139), the emissive peak in the absorption rate appears on an almost completely inverted transition ( a,iV) — —, N — 1)), whose dipole moment is significantly reduced by quantum interference. One could expect that the weaker field should not couple to an almost canceled dipole moment. However, we have assumed that the probe field couples only to the dipole moment p12. From Eq. (136), we find that the coupling strength of the probe field to the transition a. (V) —. /V 1) is proportional to j (Sp12 despite the...

Thus, in terms of the quantum dressed states, the gain features predicted by Menon and Agarwal [48] actually appear on completely inverted transitions whose dipole moments are canceled by quantum interference. Therefore, the gain features can be regarded as the amplification on dark transitions [51]. 

IY In their case the atom prefers to stay in the transition with the larger decay rate (strong transition) and there is a small probability of finding the system in the other (weak) transition. The extended dark periods, predicted for the V-type atom with almost parallel dipole moments, appear simultaneously on both transitions independent of the decay rates. This indicates that in the presence of quantum interference the atomic states 1) and 3) are not the preferred radiative states of the atom. 

This represents a formidable practical problem, as one is very unlikely to find isolated atoms with two nonorthogonal dipole moments and quantum states close in energy. Consider, for example, a V-type atom with the upper states 11), 3) and the ground state 2). The evaluation of the dipole matrix elements produces the following selection rules in terms of the angular momentum quantum numbers J — J2 = 1,0, J3 — J2 = 1,0, and Mi — M2 = M3 — M2 = 1,0. Since Mi / M3, in many atomic systems, p12 is perpendicular to p32 and the atomic transitions are independent. Xia et al. [62] have found transitions with parallel and antiparallel dipole moments in sodium molecules (dimers) and have demonstrated experimentally the effect of quantum interference on the fluorescence intensity. We discuss the experiment in more details in the next section. Here, we point out that the transitions with parallel and antiparallel dipole moments in the sodium dimers result from a mixing of the molecular states due to the spin-orbit coupling. 

We now can calculate the transition dipole moments (in n between the doubly dressed states, corresponding to the transitions at (Do, and find that the dipole moments are equal to zero. Thus, in the doubly driven atom the effective dipole moments at (Do are zero due to quantum interference between the two... 

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