Four-level system

A dye molecule has one or more absorption bands in the visible region of the electromagnetic spectrum (approximately 350-700 nm). After absorbing photons, the electronically excited molecules transfer to a more stable (triplet) state, which eventually emits photons (fluoresces) at a longer wavelength (composing three-level system.) The delay allows an inverted population to build up. Sometimes there are more than three levels. For example, the europium complex (Figure 18.15) has a four-level system. 

Commonly, a three-level or four-level system, illustrated in Figures 9.2(b) and 9.2(c), is necessary for population inversion to be obtained between two of the levels. 

The four-level system in Figure 9.2(c) is even more efficient in the creation of a population inversion, in this case between levels 3 and 2. The reason for the greater efficiency is that, not only is level 3 populated through the fast 4-3 process, but the population of level 2 is rapidly depleted by the fast 2-1 process. 

In more complex systems there may be more levels between 4 and 3 and between 2 and 1, all involved in fast processes to lower levels, but they are still referred to as four-level systems. 

Laser action involves mainly the 3/2 hi/i transition at about 1.06 pm. Since is not the ground state, the laser operates on a four-level system (see Figure 9.2c) and consequently is much more efficient than the ruby laser. 

Fig. 1. Pumping methods for lasers where is the pump light frequency and is the laser frequency, wavy lines represent radiationless transitions, and the dashed line collisions (a) optical pumping in three-level systems (b) optical pumping in four-level systems (c) pumping by electron impact and...

Figure lb shows a four-level system. The terminal level, level 2, is ordinarily empty. Atoms are optically pumped to level 4. From level 4, the atoms make a rapid radiationless transition to level 3. The first few atoms to arrive begin to contribute to the population inversion. Therefore, laser operation can begin with much less intense pumping light. After the laser transition, the atoms return to the ground state (level 1) by a radiationless transition. 

The situation is more complicated in the adiabatic limit when this inequality is reversed. According to Eq. (4.36) and Eq. (4.4) the off-diagonal parts of T and y are different. To elucidate this difference and explore its consequences we shall examine the spectra of the four-level system passing from non-adiabatic to adiabatic broadening. 

Let us reconsider the four-level system shown in Fig. 4.1(6), which has two doublets in the spectrum split by 2A and 2e (Fig 4.4(a)). Since diagonal elements of G,kjm are the same in impact and Markovian theories we assume that F , = 0 without any restriction of generality. This is actually the case for any electric multipolar interaction and hence Ac0 = 0. The non-zero elements of the perturbation... 

The condition for observing induced emission is that the population of the first singlet state Si is larger than that of So, which is far from the case at room temperature because of the Boltzmann distribution (see above). An inversion of population (i.e. NSi > Nso) is thus required. For a four-level system inversion can be achieved using optical pumping by an intense light source (flash lamps or lasers) dye lasers work in this way. Alternatively, electrical discharge in a gas (gas lasers, copper vapor lasers) can be used. 

Fig. 1. Transition probabilities in a two-spin, four-level system. Reprinted from Ann. Rep. NMR Spectrosc., vol. 22, Kowalewski, J., Nuclear Spin Relaxation in Diamagnetic Fluids. Part 1. General Aspects and Inorganic Applications , pp. 307-414, Copyright 1990, with permission from Elsevier.

Figure 1. Jablonski diagram for a four-level system depicting absorption, non-radiative (wavy arrows) and radiative processes between singlet (total spin S = 0) and triplet (total spin S = 1) states. Emissions respect Kasha s rule. IC internal conversion. ISC intersystem crossing.

The Stokes shift between the absorption and emission wavelengths creates a four-level system. Consequently, population inversion is obtained with very low excitation densities. 

Paspalakis and Knight [28] have considered a V-type three-level system driven from an auxiliary level by two laser fields of the same frequencies. They have predicted linewidth narrowing and cancellation of the fluorescenc, which can be controlled via the phase difference between the two laser fields used for the excitation. Ghafoor et al. [29] have considered a four-level system in which quantum interference can be generated by three driving fields and have shown that the linewidths and intensities of the spectral lines can be controlled by the phases and amplitudes of the driving fields. 

The four-level system considered by Zhu and Scully is shown in Fig. 3. The laser held is coupled to nondecaying 1) — b) and 3) — b) transitions, whereas spontaneous emission occurs from the levels 1), 3) to the ground level 2). 

Agarwal [67] has proposed a totally different mechanism to produce atomic transitions with parallel dipole moments. In this method the interference between two perpendicular dipole moments can be induced by an anisotropic vacuum field. Using the second-order perturbation theory, it can be shown that transition probability from the ground state g) of a four-level system to the final state /) through two intermediate states ) and y) is given by... 

The collective states are shown in Fig. 2. It is seen that in the collective states representation, the two-atom system behaves as a single four-level system with the ground state g), the upper state e), and two intermediate states the symmetric state s) and the antisymmetric state a). The energies of the intermediate states depend on the dipole-dipole interaction and these states suffer a large shift when the interatomic separation is small. 

However, it is difficult experimentally to fulfil the second requirement that interatomic separations should be much smaller than the resonant wavelength. In fact, present atom trapping and cooling techniques can trap two atoms only within distances of the order of a resonant wavelength [11-13]. It is therefore of interest to examine the effect of increasing the interatomic separation so that the simple three-state representation of two atoms, presented in the preceding section, eventually ceases to be valid. With a finite interatomic separation, the two-atom system is represented by the full four-level system of (35). 

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