A generalized kinetic model of atomic level populations in an optically dense plasma excited by laser pulses of arbitrary duration is formulated and studied. This model is based on a nonstationary expression for the probability of excitation of an atomic transition and takes into account the effects of laser pulse penetration into an optically dense medium. A universal formula for the excitation probability as a function of time and propagation length is derived and applied to the case of a Lorentzian spectral profile of an atomic transition excited by a laser pulse with a Gaussian envelope. The features of nonstationary excitation probabilities are presented for different optical depths of the plasma, laser pulse durations, and carrier frequencies. The formulas derived here will be useful for the description of atomic populations excited by laser pulses under realistic conditions of dense plasmas.

Topics
Electric fields, Optical properties, Electromagnetic interactions, Lasers, Plasma properties and parameters, Quantum mechanical systems and processes, Perturbation theory

The rapid development of the ultrashort laser pulse generation technique (2023 Nobel Prize for Physics1) has opened up new possibilities for investigation of plasma properties, including those of optically dense plasmas. The advantage of this approach lies in the ability of ultrashort pulses to penetrate optically dense media much more efficiently than standard long-duration laser pulses. This is due to the much wider laser spectrum, which allows radiation harmonics to be transferred far from resonant atomic frequencies without significant absorption (for details, see Refs. 2 and 3). In particular, ultrashort pulses of X-ray radiation generated by free-electron laser (FEL) systems provide unique possibilities for investigation of dense laser-driven plasmas.4–7 

In this context, the availability of a correct formulation of atomic kinetics in dense media, especially in optically dense plasmas, becomes relevant. Since the short laser pulses are highly time-dependent, it is necessary to deal with time-dependent atomic kinetics. Here, all radiation and relaxation constants associated with the interaction of an atom with a plasma are the same as in standard steady-state atomic kinetics,8 but a more detailed description of the interaction with laser pulses is required.

The purpose of this paper is to formulate and study a generalized kinetic model of dense plasma excited by a laser pulse with arbitrary duration, taking into account the specifics of ultrashort electromagnetic interaction and the influence of an optically dense medium on ultrashort pulse propagation.

Consideration of the laser pulse interaction with a plasma requires the introduction of an essential new parameter, namely, the pulse duration. The main problem to be solved here is correct determination of the probability per unit time for this type of interaction. There are two well-known standard approaches to this problem (see, e.g., Ref. 9).

The first approach to the description of population kinetics is applicable when the absorption line shape width is larger than the spectral width of the laser pulse. In this case, the transition probability per unit time (the photo-process rate) is given by the following expression:
(1)
where G(ω) is the spectral profile of the photoexcitation cross section, ωc is the carrier frequency, Ω0 = d0 E0/ℏ is the Rabi frequency (with d0 being the dipole moment operator matrix element), Ẽt=Et/E0 is the dimensionless electric field strength of the pulse (with E0 being the pulse amplitude), and τ is the pulse duration. It is important to note that (1) contains information on the spectral line widths determined by broadening mechanisms.
The second standard approach is valid for ultrashort laser pulses with spectral widths greater than the line-broadening widths. In this case, the probability per unit time is expressed in terms of the Einstein coefficient for induced transitions B (the broadband illumination approximation) as follows:
(2)
Note that this expression does not contain any information on broadening mechanisms in the plasma. Its use has frequently been proposed for the descriptions of laser-induced transitions in a variety of cases.10,11

According to (1) and (2), the transition rate as a function of time follows the time dependence of the laser pulse intensity. This is a consequence of the approximate nature of these expressions. A more accurate analysis predicts various types of time dependence of the probability of pulsed excitation of a quantum system and related quantities, as shown, for example, in Refs. 12–14.

Let us briefly describe a generalized approach to the calculation of probabilities per unit time that is applicable for an arbitrary ratio of atomic and laser pulse frequency widths.15 It can be derived by applying time-dependent perturbation theory to the total transition probability W(t), which leads to the following expression for the probability per unit time w(t):
(3)
with
(4)
where ω0 is the eigenfrequency of the excited transition and
(5)
Thus,
(6)
The kinetic equations for the populations of two levels of a quantum system between which a radiative transition occurs under the action of a laser pulse have the form
(7)
(8)
where N1,2e are the equilibrium populations of the energy levels, and T1l,u are the population relaxation times of the lower and upper levels. In the case of a two-level quantum system, the normalization condition N1(t) + N2(t) = 1 is satisfied, and instead of the system of Eqs. (7) and (8), we have the following equation for the population of the upper level under the assumption that N2e=0 and T1l = T1u = T1:
(9)
This equation has the following solution:
(10)
where W(t)=∫−∞tw(t′)dt′⁠. It should be noted that (10) includes only the total probability W, and not the photoexcitation rate w. In Ref. 15, excellent agreement was found between the population given by (10) and the exact solution of the corresponding Bloch equations for any laser pulse durations within the range of applicability of perturbation theory. In this approximation, the laser pulse does not change the level energy to the matrix element of the dipole moment operator of the quantum system.
For a long relaxation time T1 (10), simplifies to the following expression for the population of the upper energy level as a function of the current time:
(11)
In the perturbation theory limit (W ≪ 1), it follows from (11) that
(12)
i.e., the population of the upper energy level is equal to the excitation probability.

An important problem with regard to the interaction of atoms with laser pulses is associated with the conditions for the penetration of laser pulses into a dense medium.

To describe this effect, we use the complex amplitude coefficient T(ω, z) accounting for penetration of a monochromatic electromagnetic field into a homogeneous medium. It is given by
(13)
where ñ(ω) is the complex refractive index and z is the propagation length. The monochromatic component of the electric field inside the optically dense medium is connected with that at the boundary of the medium by the following relationship:
(14)
We can use these formulas to determine the total probability W(t, z) inside an optically dense medium:
(15)
Here, we have introduced the incomplete Fourier transform of the electric field strength inside the optically dense medium:
(16)
where Ẽ(ω′,z=0) is the Fourier transform at the initial point of laser pulse propagation. As a result, the expression for the total transition probability takes the form
(17)
In the long-time limit, we can use the relationship
(18)
Substituting (18) into (17), we obtain the following expression for the transition probability after pulse termination:
(19)
Let us consider the following approximation, which is valid for a homogeneous medium, neglecting refraction:
(20)
where N is the density of atoms in the medium and σ(ω) is the atomic absorption cross section. In this approximation, we only take into account the effect of absorption of the laser pulse.
Using (20), we have from (13) the following equality3 in the homogeneous approximation:
(21)
Let us rewrite (16) in terms of the optical depth Λ and penetration frequency ωpen, which are given by
(22)
We then have
(23)
Substituting (23) into (15), we arrive at the final expression for the excitation probability as a function of the current time and the optical depth of the medium:
(24)
This is a universal expression applicable to any spectral profile and pulse envelope.
Neglecting the “delay” [the term Λ/ωpen in the exponent in (24)], since the calculations show that it does not make a noticeable contribution to the probability for all reasonable values of the problem parameters, we obtain the following expression for the probability of excitation of atomic transition by a Gaussian multicycle pulse in terms of dimensionless variables:
(25)
where
(26)
is a function arising from the incomplete Fourier transform of a Gaussian pulse, taking into account its propagation in an optically dense medium. The following dimensionless variables have been introduced in (25) and (26):
(27)
where τ and ωc are the pulse duration and carrier frequency, and Δ is the spectral width of the excited transition. In view of (18), in the long-time limit (⁠t̃≫1⁠), (25)–(27) coincide with the corresponding expressions obtained in Ref. 3.

Let us use the formulas obtained above to describe the influence of the optical depth of the plasma and the pulse parameters on the time dependence of the excitation probability given by (25) and (26). As indicated by (12), this probability is equal to the population of the upper level in the framework of perturbation theory and for a long relaxation time T1.

For specificity, we assume that the shape of the spectral line is Lorentzian and that the exciting pulse has a Gaussian envelope. The figures that we now present show the normalized probability of excitation
(28)
as a function of normalized time t̃=t/τ for different optical depths and various values of the pulse duration and carrier frequency in terms of the dimensionless variables (27).

The time dependence of the excitation probability for different optical depths is shown in Figs. 1 and 2 in the cases of long and short pulses, respectively. It can be seen that as the pulse becomes shorter, the time dependence turns from monotonically increasing to oscillating for all optical depths, and the dependence of the probability on the optical thickness becomes less significant.

FIG. 1.
Time dependence of excitation probability in the case of a long pulse (α = 2) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 7; dashed line, Λ = 10.
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Time dependence of excitation probability in the case of a long pulse (α = 2) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 7; dashed line, Λ = 10.

FIG. 1.
Time dependence of excitation probability in the case of a long pulse (α = 2) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 7; dashed line, Λ = 10.
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Time dependence of excitation probability in the case of a long pulse (α = 2) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 7; dashed line, Λ = 10.

Close modal
FIG. 2.
Time dependence of excitation probability in the case of a short pulse (α = 0.1) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 10; dashed line, Λ = 20.
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Time dependence of excitation probability in the case of a short pulse (α = 0.1) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 10; dashed line, Λ = 20.

FIG. 2.
Time dependence of excitation probability in the case of a short pulse (α = 0.1) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 10; dashed line, Λ = 20.
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Time dependence of excitation probability in the case of a short pulse (α = 0.1) and δ = 0 for different optical depths: solid line, Λ = 5; dotted line, Λ = 10; dashed line, Λ = 20.

Close modal

Figures 3 and 4 show how the time evolution of the excitation probability for different pulse durations changes with increasing optical depth. It can be seen that for short pulses, an increase in the optical depth enhances the oscillations of the excitation probability in time, whereas for long pulses, the time dependence remains monotonically increasing. In addition, the difference between the values of the average probability for different pulse durations decreases with increasing optical depth. It can also be seen that the probability of excitation by a long pulse decreases more strongly with increasing value of the optical depth parameter Λ, such that it becomes less than the probability of excitation by short pulses. This behavior is associated with a change in the spectrum of the exciting pulse as it propagates through the medium (the penetration effect). In fact, when a resonant pulse propagates in an absorbing medium, a dip appears in its spectrum at the eigenfrequency of the medium. Thus, two spectral components arise, and the interference between their contributions leads to the oscillations. The separation into spectral components is more pronounced for a medium with greater optical thickness, and so the oscillations become more pronounced in this case.

FIG. 3.
Time dependence of excitation probability in the case of an optical depth of Λ = 5 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.
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Time dependence of excitation probability in the case of an optical depth of Λ = 5 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.

FIG. 3.
Time dependence of excitation probability in the case of an optical depth of Λ = 5 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.
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Time dependence of excitation probability in the case of an optical depth of Λ = 5 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.

Close modal
FIG. 4.
Time dependence of excitation probability in the case of an optical depth of Λ = 20 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.
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Time dependence of excitation probability in the case of an optical depth of Λ = 20 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.

FIG. 4.
Time dependence of excitation probability in the case of an optical depth of Λ = 20 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.
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Time dependence of excitation probability in the case of an optical depth of Λ = 20 and δ = 1 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.5; dashed line, α = 1.

Close modal

Figures 5 and 6 show the changes in the time dependence of the excitation probability with a change in the carrier frequency for different pulse durations at a given optical depth. It can be seen that the relative value of the probability of excitation by pulses with different durations changes significantly. Thus, in the case of a long pulse, the probability increases significantly with spectral detuning from resonance, whereas in the case of a short pulse, it remains practically constant. Calculations show that in the case of small optical depth Λ ≪ 1, all dependences depicted in these figures are monotonically increasing. Thus, the pulse penetration effect leads to the appearance of temporal oscillations in the excitation probability for sufficiently short pulses.

FIG. 5.
Time dependence of excitation probability in the resonant case (δ = 0) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.
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Time dependence of excitation probability in the resonant case (δ = 0) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.

FIG. 5.
Time dependence of excitation probability in the resonant case (δ = 0) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.
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Time dependence of excitation probability in the resonant case (δ = 0) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.

Close modal
FIG. 6.
Time dependence of excitation probability in the nonresonant case (δ = 1) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.
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Time dependence of excitation probability in the nonresonant case (δ = 1) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.

FIG. 6.
Time dependence of excitation probability in the nonresonant case (δ = 1) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.
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Time dependence of excitation probability in the nonresonant case (δ = 1) and Λ = 10 for different pulse durations: solid line, α = 0.1; dotted line, α = 0.2; dashed line, α = 1.

Close modal

Note that a qualitative difference in the time dependence of the excitation probability between the resonant and nonresonant cases arises in the case of a longer pulse (α = 1), since the its spectrum is narrower than that of a short pulse, and accordingly the detuning of the carrier frequency from the eigenfrequency of the quantum system is more pronounced, which has a stronger effect on the excitation probability.

We have introduced a generalized approach to the description of atomic population kinetics in optically dense plasmas driven by laser pulses of arbitrary duration, including ultrashort ones. This approach is based on the time-dependent excitation probability of atomic transitions, with account taken of the penetration effect during ultrashort pulse propagation in plasmas. A universal expression for the excitation probability as a function of time and propagation length has been derived in terms of dimensionless variables.

Our method has been applied to the calculation of the temporal evolution of the excitation probability of atomic transition for different optical depths, durations, and carrier frequencies of the laser pulse. Specific calculations have been performed for a Gaussian pulse and a Lorentzian spectral profile of the excited transition. In particular, we have shown that for short pulses, the excitation probability is an oscillating function of the current time, whereas in the case of long pulses it is monotonically increasing. The oscillations become more pronounced with increasing optical depth. We have also established that with increasing detuning of the carrier frequency from atomic resonance for Λ ≫ 1, the probability of excitation by short pulses remains practically constant, whereas in the case of long pulses, it increases significantly.

The formulas (24) and (26) derived for the time-dependent excitation probability determine the photo-process rate according to the definition,3 which is included in the kinetic equations. Therefore, the results of the present paper will be useful for the study of atomic kinetics in optically dense plasmas irradiated by laser pulses of arbitrary duration, with account taken of all radiative–collisional processes accompanying laser excitation of atomic transitions under realistic plasma conditions.

This research is supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye) No. 075-03-2024-107.

The authors have no conflicts to declare.

V. A. Astapenko: Conceptualization (equal); Investigation (equal); Methodology (equal). V. S. Lisitsa: Project administration (equal); Supervision (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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See www.nobelprize.org/prizes/physics/2023/summary for brief information about the Nobel Prize in Physics 2023.
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