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Theory of three-step absorption of three light pulses

The theory of the absorption of three light pulses with an arbitrary duration in four-level systems is proposed. The pulses are in resonance with the corresponding transitions. By using the time-dependent perturbation theory the probability that the fourth level is excited at the moment t is found. The possible absorption lines and parameters on which their widths depend on are analysed.


Introduction
Multistep absorption/excitation has been investigated in many ways and also both monochromatic light and light pulses have been used [1][2][3][4][5][6][7]. In [7] the possible time dependent two-step absorption spectra of two light pulses are theoretically investigated by using the simplest model for matter and in the case of an arbitrary duration of absorbed light pulses from monochromatic light to ultrashort pulses.
Thereby, we note that if at the two-step absorption in the three-level system the monochromatic light with the frequency ω 1 is absorbed at the first step 0 → 1, then in the spectrum where the frequency ω 1 is fixed and the frequency ω 2 of the maximum of the second light pulse (which is absorbed at the second step 1 → 2) varies, there exists one line whose width is determined by the rate of energy relaxation of the second excited level 2 and by the spectral width of the second pulse. In our opinion, this is exactly the same situation as in [6]. We note that if at the first step of the absorption the light pulse is absorbed and/or the pure phase relaxation of the first excited level exists, the second line is added to the aforementioned spectrum. The width of this line depends on the rates of the energy relaxations of both excited levels 1 and 2, the rate of the pure phase relaxation and on the spectral width of the second pulse, but not on the spectral width of the first pulse. On the other hand, the width of the initial line, in addition to the aforementioned parameters, depends on the spectral width of the first pulse. This line corresponds to the coherent contribution of the spectrum [3]. We note that the linewidths depend on the timeframes between the pulses and the time t as well.
In this paper the three-step absorption of three light pulses in a four-level system (the pulses are in resonance with the corresponding transitions) is observed to analyse which absorption lines are possible at all and which parameters their widths depend on. The interaction with the environment (phonons) is taken into account phenomenologically via the relaxation constants of the electronic levels. The presented model holds for impurity centers at low temperatures with weak electron-phonon coupling.
By using the time-dependent perturbation theory [8,9] analogically with the theory of the absorption of two light pulses in three-level systems [7], the probability that at the time moment t all three pulses are absorbed and the fourth level 3 is excited is the following: where F is the correlation function of the four-level system and S 1 , S 2 and S 3 are the correlation functions of the first, second and third pulses. This probability W(t) decreases to zero with the increase of time t because of the energy relaxation of the fourth level.
To simplify the calculations, the light pulses are taken coherent and of a single-sided exponential shape. The corresponding correlation functions are where () x  is Heaviside step function, τ 1 , τ 2 and τ 3 are the time moments when the pulses begin to pass through the impurity centre, Δ 1 , Δ 2 and Δ 3 are the FWHM spectral widths of the pulses. The correlation function of the four-level system is [9] In (3) They describe the annihilation of the photons of the frequencies ω 1 , ω 2 and ω 3 . In (4) N is the initial state of the electromagnetic field. In our elementary model the phase relaxation and the phonon wings are not taken into account, the relaxation processes of the excited levels 1, 2 and 3 are described by the rates of energy relaxation γ 1 , γ 2 and γ 3 . Thus the interaction with phonons is taken into account phenomenologically via the relaxation constants of the electronic levels.
In figure 1 the dependence of the probability W(t) on the frequency of the maximum of the first pulse ω 1 is calculated, the other frequencies ω 2 , ω 3 are fixed and the quantities ω 2 -Ω 12 , ω 3 -Ω 23 are chosen different from zero to separate possible lines spectroscopically.
In figure 2 the dependence of the probability W(t) on the frequency of the maximum of the second pulse ω 2 is presented, the other frequencies ω 1 , ω 3 are fixed. In figure 3 the dependence of the probability W(t) on the frequency of the maximum of the third pulse ω 3 is calculated, the other frequencies ω 1 , ω 2 are fixed. At different values of the spectral width of the second pulse Δ 2 different lines exist. When the frequency ω 1 is variable, then the possible maximums of the lines are at x = 0, x = -y (in figure 1 x = 10γ 3 ) and x = -yz (in figure 1 x = 25γ 3 ). When the frequency ω 2 is variable, the possible maximums of the lines are at y = 0, y = -x (in figure 2 y = 15γ 3 ), y = -z and y = -xz (in figure 2 y = 25γ 3 ). Finally, when the frequency ω 3 is variable, the possible maximums of the lines are at z = 0, z = -y and z = -xy (in figure 3 z = 25γ 3 ).  The widths of the different lines depend on different parameters (see table). It can be seen from the Table that if z varies, the width of the line with the maximum at z = 0 depends on the parameters γ 2 , γ 3 and Δ 3 ; the width of the line with the maximum at z = -y depends on the parameters γ 1 , γ 3 , Δ 2 and Δ 3 ; the width of the line with the maximum at z = -xy depends on the parameters γ 3 , Δ 1 , Δ 2 and Δ 3 .
In the case, where the monochromatic light (Δ 1 = 0) is used at the first step of the absorption (0 → 1) in the spectrum where x varies, all three lines remain, in the spectrum where y varies the lines with the maximums at y = -x and y = -xz remain, and in the spectrum where z varies the lines with the maximums at z = 0 and z = -xy remain.
If, additionally, the monochromatic light is used at the second step of the absorption (1 → 2) in the spectrum where z is a variable, the line with the maximum at z = 0 disappears as well.
Note, that the lines for which the equation x + y + z = 0 applies are the coherent parts of the spectrums. Furthermore, the widths of all lines depend on the timespans T 1 , T 2 and the time t. In general, with the increase of T 1 , T 2 and t the lines become narrower.