Dressed molecular states

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Using the dressed atomic states we construct the dressed molecular states as direct products. Explicitly, we construct the two states tpA(t) and tpB t) given by3... 

It is convenient to treat intense electromagnetic field problems in the dressed molecular states picture (see review by Giusti-Suzor, et al, (1995)). This picture allows one to think about intense field problems in a framework that closely resembles the weak field, diabatic or adiabatic states picture that is the primary focus of this book. In the dressed states picture the photon energy is added to, or subtracted from, the field-free potential energy curves. One obtains field-dressed potential curves. 

As the excitation process in an external field can be regarded as being a nonadiabatic transition between dressed adiabatic states [32], effective laser control can be achieved by manipulating the parameters of these nonadiabatic transitions directly. Based on this idea, two control schemes have been proposed. The first one is a control scheme for the branching ratio during the molecular photodissociation, achieved by utilizing the phenomenon of complete reflection [24,43,44], The second is to control the population transfer by using a laser pulse with periodically swept parameters [24-29], In both cases the best parameters of the laser pulse can be easily estimated from the ZN theory of nonadiabatic transitions. 

Consider the generic two-level model, Eq. (2.13), with the levels now denoted g and. S with energies Eg > Eg. An extended system that includes also the environment may be represented by states that will be denoted l, e ), g, e ) where e defines states of the environment. A common phrase is to say that these molecular states are dressed by the environment. Now consider this generic molecule in state s and assume that the environment is at zero temperature. In this case e = e g is the ground state of the enviromnent. Obviously the initial state lx, e g) is energetically embedded in a continuum of states g, e x) where e x are excited states of the environment. This is exactly the situation represented in Fig. 9.1, where level 11) represents the state 5, e g while levels /) are the states g, e x) with different excited state of the environment. An important aspect common to all models of this type is that the continuous manifold of states Z> is bound from below State Ig, e g is obviously its lowest energy state. 

The generality of this picture is emphasized by the observation that even for a single atom or molecule in vacuum the ever present radiation field constitutes such an environment (see Section 9.2.3 below). Any excited molecular state is coupled to lower molecular states dressed by photons. 

Note that in Fig. 9.2 5) represents a molecular state, while in Fig. 9.3 it stands for the dressed state 5, vac). Note also that the physical nature of the continuum ]/) and the coupling F, / depends on the physical process under consideration. In the dressed state picture of Fig. 9.3 this continuum may represent the radiative channel g, k) or a nonradiative channel, for example, g, v vac) of vibrational levels V associated with the electronic ground state g. In the former case the coupling... 

Fig. 9.3 Same as Fig. 9.2, now cast in the dressed states (eigenstates oF/Zm +HR)form. 0> = g k) corresponds to the molecule in the ground state with a single photon of mode k.. v) describes the molecule in an excited state and the radiation field in its vacuum state. The coupling between 0) and x> is proportional to the dipole matrix element /j.obetween the corresponding molecular states. 

The model (9.73)—(9.75) was presented as an initial value problem We were interested in the rate at which a system in state 0) decays into the continua L and R and have used the steady-state analysis as a trick. The same approach can be more directly applied to genuine steady state processes such as energy resolved (also referred to as continuous wave ) absorption and scattering. Consider, for example, the absorption lineshape problem defined by Fig. 9.4. We may identify state 0) as the photon-dressed ground state, state 1) as a zero-photon excited state and the continua R and L with the radiative and nonradiative decay channels, respectively. The interactions Fyo and correspond to radiative (e.g. dipole) coupling elements between the zero photon excited state 11 and the ground state (or other lower molecular states) dressed by one photon. The radiative quantum yield is given by the flux ratio Yr = Jq r/(Jq r Jq l) = Tis/(Fijj -F F1/,). 

It should be noted that instead of dressing the molecular states g and s by 1 and 0 photons, respectively, we could use any photon numbers p and p — 1. The corresponding matrix elements are than proportional to p. In processes pertaining to linear spectroscopy it is convenient to stick with photon populations 1 and 0, keeping in mind that all observed fluxes should be proportional to the incident photon numberp or, more physically, to the incident field intensity fo P- With this in mind we will henceforth use the notation g, k) (or g, m if the incident direction is not important for the discussion) as a substitute for g, Ik)-... 

We therefore envision the molecule as an entity characterized by its vibronic spectrum, interacting with a dissipative environment. As indicated above, a useful characteristic of the truncated dressed state approach is the simplification provided by considering only states that pertain to the process considered. In experiments involving a weak incident field of frequency co these states are found in the energy range about hoo above the molecular ground state. For simplicity we assume that in this range there is only one excited electronic state. We therefore focus in what follows on a model characterized by two electronic states (See Fig. 18.1) and include also their associated vibrational manifolds. The lower molecular state g)... 

Fig. 18.1 A dressed-state model that is used in the text to describe absorption, emission, and elastic (Rayleigh) and inelastic (Raman) light scattering. g) and. v> represent particular vibronic levels associated with the lower (1) and upper (2) electronic states, respectively. These are levels associated with the nuclear potential surfaces of electronic states 1 and 2 (schematically represented hy the parabolas). Rj are radiative continua— 1 -photon-dressed vibronic levels of the lower electronic states. The quasi-continuum L represents a nonradiative channel—the high-energy regime of the vibronic manifold of electronic state 1. Note that the molecular dipole operator /t couples ground (g) and excited (s) molecular states, but the ensuing process occurs between quasi-degenerate dressed states g,k and 5,0).

Fig. 18.7 A schematic display of light scattering/excitation-fluorescence process. Shown are the relevant molecular states and the dressed states in) and out) used in the calculation. The arrows denote thermal population transfer within the intermediate state manifold. The shading on levels p and 5 corresponds to energy level fluctuations that leads to pure dephasing.

The electromagnetic radiation field is taken into account by adding the energy of the photons to the various molecular potential curves, Vi(R). If the photon number is initially N, when n photons are absorbed, the remaining number of photons is N — n. The resultant field-dressed diabatic state has potential energy... 

The ARPA process in a thermal ensemble of atoms as well as in a Bose-Einstein condensate of atoms, has been studied theoretically in great detail [1,7,8,11-18]. Its first experimental demonsfiations [21-23] have verified the existence of the field-dressed dark state, predicted in Ref. [1] to arise from a strongly coupled A-type system made up of an initial-continuum and two (intermediate and final) bound states. In addition, the closely related process of adiabatic passage from a loosely bound excited molecular state, produced by the Feshbach resonance switching process, to a deeply bound low-lying state, was observed [49-51]. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

E )/h = CO — co2 correspond to an implementation of the dressed-state picture given the molecular states 1) and 2) with energy spacing E2 = E2 — E, Eqs (18.43) are written for the density matrix elements in the representation of the dressed states l,co) (the molecular state 1 plus a photon of frequency co) and 2,0) (molecular state 2 with no photons) whose spacing is rj. 

Once the mechanisms of dynamic processes are understood, it becomes possible to think about controlling them so that we can make desirable processes to occur more efficiently. Especially when we use a laser field, nonadiabatic transitions are induced among the so-called dressed states and we can control the transitions among them by appropriately designing the laser parameters [33 1]. The dressed states mean molecular potential energy curves shifted up or down by the amount of photon energy. Even the ordinary type of photoexcitation can be... 

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