The two-photon transition rate

The excitation of an atomic transition simultaneously by two photons is not the only possibility for combined excitation by two particles it is also possible for a single transition to be excited simultaneously by one photon and one electron, neither of which have sufficient energy separately to excite the transition [458]. 

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To complete the two-photon transition, the two photons must arrive at the absorber within the virtual state lifetime r. For classical uncorrelated photons the probability of accidental overlap increases with photon flux density. Therefore excitation by short, tightly focused laser pulses is needed for the TPA. Probabilistic analysis gives the two-photon transition rate ... 

In this note, we discuss different approximation schemes for the evaluation of the two-photon transition rate between discrete states. Non relativistic atomic hydrogen is used as a test of the reliability of the methods. We consider a one particle system described by a Hamiltonian Ho, whose eigenstates and eigenvalues are denoted by n> and En, respectively. In the gauge with divA = 0, the interaction of the particle with the electromagnetic field has the usual form... 

Since we are interested in the two-photon transition rate, we will consider a vector potential... 

For the special case of non relativistic Hydrogen, the multiphoton transition rate can be obtained exactly using methods based on Green function techniques, which avoid summations over intermediate states. This approach was introduced in order to treat time independent problems, and later extended to time dependent ones [2]. In the Green function method, the evaluation of the infinite sums over intermediate states is reduced to the solution of a linear differential equation. For systems other than Hydrogen, this method can also be used, but the associated differential equation has to be integrated numerically. The two-photon transition rate can also be evaluated exactly by performing explicitly the summation over the intermediate states. 

We present exact calculations of the two-photon transition rates between discrete states of the non relativistic Hydrogen atom and we compare these results with those obtained with various approximation schemes. We plot in Fig. 1 the ls-3s resonant transition amplitude as a function of one of (he photon frequencies [3], where... 

Robinson [5] has shown how to obtain gauge-independent results for the two-photon transition rates in cases when only a limited number of intermediate states are used. Following Robinson, we first transform the transition amplitudes in a more compact form... 

The two-photon transition rate, JVgy, defined as the probability for... 

The two-photon spectrum is recorded by fixing the wave length of the first absorbed photon and scanning the wave length X2 of the second. The intensity of the two-photon line depends on the wave-length Xi The variation of the two-photon transition rate plotted against Xx... 

In these equations we can clearly see the two photon transition probabilities that directly connect levels I and 3 and are proj)ortional to the fourth power of Rabi frequency. These equations in the limit of weak excitation (small fl) lead to the phenomenologicar rate equations, which do not take into account two photon... 

We see that, taking into account assumptions made with respect to the relaxation rates and other parameters, we basically have arrived at the same absorption rate that we had for a two level system, compare to Eq. (12). It is interesting to note the ratio between two-photon transition rate and one-photon transition rate. This ratio is... 

Fig. 22 - Two-photon transition rate in sodium using two unequal photons as a function of the wavelength Xi of the fixed-frequency Laser. The points are experimental and the curves theoretical. The inset shows the behaviour of the central region with an expanded scale (from ref 37 ).

Fig. 2. Doppler-free spectra of the 15 — 2S two-photon transition (F = 1 —> F = 1) in atomic hydrogen, a) Spectra for three different nozzle temperatures and no delay time, b) Time resolved spectrum (nozzle temperature 6.5 K). This plot gives the 2S count rate as a function of the absolute optical frequency for different delay times. The inset shows the spectra with longer delay times on a magnified scale...

The transition amplitude exhibits dramatic structure over the entire frequency range. Besides the resonance enhancement for = 2it R [1 - l/ i2] with p = 2, 3, the plot clearly shows the two-photon transparency. We refer to [3, 4] for the behavior of the transition rate W to higher excited s and d states and for the dependence of W on the polarization of the incoming radiation. A detailed discussion of the transparencies for different transitions is also given in a recent paper by Florescu et al [4]. In Table 1, we reproduce some of their results, showing in particular that the position of the transparencies are almost independent of the final state. 

The exciton-exciton annihilation rate constant for singlet excitons in solid films of poly(n-propylmethylsilane) was determined by measuring the fluorescent intensity from the films as a function of incident-light intensity in two types of experiments one in which the excitons were created by single-photon transitions and one in which the excitons were created by two-photon transitions. The rate constant is cm ls. 

The laser-excited fluorescence from dibenzofuran in a biphenyl host at 4.2 K and three different sites in the lattice have been identified. The two-photon excitation spectrum of single crystals of carbazole at 4.2 K has been analysed in detail. Very extensive vibrationally unrelaxed fluorescence is observed following dye-laser excitation of single vibronic levels of naphthazanine and the rates and pathways of relaxation are examined by picosecond emission spectroscopy. The photophysics of rubrene peroxide have been measured for the first time by Bayrakceken. Picosecond fluorescence has been used to study the spectra and kinetics for Sj— So and Si transitions for... 

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