“Middle way” in the description of ultrashort electromagnetic interactions: between Fermi’s golden rule and the numerical solution of the Schrödinger equation

A simple efficient approach describing the interaction of various quantum targets with short electromagnetic (EM) pulses firstly proposed in [1] is reviewed. This approach is based on the expression for the photoprocess probability in terms of the cross section and the Fourier transform of the EM pulse. Our method makes it possible to take into account the internal dynamics of the target and the influence of the pulse parameters in a simple way. The proposed approach is used both for the probability for the entire pulse duration, and for its dependence on the current time. In addition, the developed method is applied to generalize the simulation of the population kinetics of a quantum system to take into account the dependence of the photoprocess rate on time. It is also shown that our expression for the probability of excitation of a micro-target corresponds to the formula describing the energy loss of a non-monochromatic field in a dispersive medium and can be generalized on other EM processes.


Introduction
There are two-well known approaches to describe electromagnetic interaction in the framework of quantum mechanics, (1) the simplest one is based on Fermi golden rule [2] and (2) cumbersome method using numerical solution of Schrodinger equation [3].The first approach is applicable for monochromatic radiation and fails in the case of short EM pulses.The second method requires significant resources, it is non-transparent and devoid of predictability.At the same time there is a simple "middle way" approach which is analytical and allows one to capture the main features of EM interaction without extensive calculations.
The purpose of the present paper is brief review of basic aspects of this "middle way" method and demonstration of its important applications.We also consider some generalizations of the initial version of the method and its correspondence with other descriptions of EM processes.

Main formula for the excitation probability
Our approach is based on simple formula for the excitation probability of micro-target by EM pulse [1].The perturbation operator is equal to . ( are the dipole moment operator and electric field strength in EM pulse.Standard quantummechanical procedure gives the following expression for the excitation probability during all time of pulse action (we suppose that ): ( ) ( ) (2) The principal moment of our treatment is that this equality can be rewrite in terms of photo-excitation cross section as follows (3) t is pulse duration, c is light velocity, ћ is reduced Planck constant.In the derivation of the expression (3) we use the following relation between the photo-excitation cross section and the Fourier transform of the correlator of dipole moments for stationary and spherically symmetric system: .
We would like to emphasis that expression (3) accounts both for the internal dynamics of the target and for the parameters of EM pulse.Therefore it is applicable for description of a wide range of photoinduced processes and can capture the main features of them.

Correspondence with Fermi equivalent photon method
It is instructive to trace the correspondence between the present approach and the method of equivalent photons proposed by E. Fermi for description of atom excitation during collisions with charged particles [4].To do this the expression (3) should be represented in the form , is the radiation energy at a frequency w that has passed through a unit area during the entire time of the pulse action.Similar formula was derived by E. Fermi in his paper [4] devoted to the atom excitation by charged particles in the framework of phenomenological approach.In cited paper means the Fourier transform of electric field strength of charged particle colliding with atom.It can be interpreted as equivalent photon of this field at frequency w.
Expressions ( 5)-( 6) are the basis for the generalization of formula (3) on other photo-induced processes, such as EM radiation scattering, photo-ionization, photo-dissociation etc.

Limit cases
Basic expression (3) allows one to derive simple formulas for excitation probability in the limits of long and short pulses.These limit cases explicitly demonstrate the connection between "middle way" approach and other methods of description of EM interactions including conventional one.
Further we use the following representation of photo process cross section: Here is the spectral profile of cross section with central frequency w 0 and spectral width g.Let consider two opposite limits of the excitation probability (3).The first one is monochromatic limit when (long pulses) and then the following relation is valid (8) wc is carrier frequency of the pulse.Substituting (8) in (3) we obtain One can see from this formula the linear probability dependence on pulse duration (t-dependence for short) as it follows also from conventional approach based on Fermi's golden rule.
The second limit is ultrashort one when pulses is sufficiently short and the following replacement in the right side of equality (3) is valid .(10) After substitution (10) in ( 3) we obtain Thus the excitation probability is quadratic function of pulse duration in ultrashort limit if in addition the inequality holds.This quadratic t-dependence is characteristic feature of ultrashort EM interactions.Note that the last inequality corresponds to the applicability of sudden perturbation approximation [5] which allows one to simplify essentially description of photo-processes in ultrashort EM field [6].

Similarity functions
To describe t-dependence for all pulse durations it is convenient to introduce similarity functions which also depend on carrier frequency and envelope of the pulse [7].This function ( ) can be extract from the formula (3) in the following way (we suggest for simplicity that ): , (13) Here we introduce dimensionless parameters y (dimensionless pulse duration) and r (dimensionless frequency detuning) according to the formulas: ; .(14) Function (13) depends solely on spectral line shape and pulse envelope and describe the excitation probability in universal manner for any values of t, wc, w0 if and pulse envelope are given.Let us write the explicit expression for similarity function for Gaussian pulse (15) and Gaussian spectral profile of cross section.After integration in (3) over all frequencies we obtain for multi-cycle pulse (wct>>1) . (16) Thus in this case the excitation probability has Gaussian profile with spectral width .
Note that y-dependence of FG,GP contains extrema (maximum and minimum) if r > 1.707 and corresponds to relations ( 9), (11) in the limit of long (y>>1) and short (y<<1) pulses.
In the case of Lorenzian line shape and exponential pulse we obtain the following similarity function (wct>>1) . (18) This function has Lorentz profile with spectral width: .Note that (18) in contrast with (16) describes monotonically increasing dependence on dimensionless pulse duration y.
Thus photoprocess spectra contain new theory parameter, namely, pulse duration which essentially determines the probability for short pulses if .In addition the t-dependence of probability depends significantly on carrier frequency and envelope of exciting pulse.

Photoprocess probability as a function of current time
Main formula (3) can be generalized to describe the dependence of the photoprocess probability on current time [8]: is the square modulus of incomplete Fourier transform of electric field strength in the pulse.
Further we consider the excitation of micro-target by exponential pulse (17).In this case the simple analytical expression for probability can be obtained.For multi-cycle exponential pulse in rotating wave approximation we have . ( Using formulas (19) and (21) one can obtain in the monochromatic limit There is linear dependence on current time for t<t and on pulse duration for t>t.
In the opposite ultra-short limit we have .
There are oscillations of photoprocess probability at difference frequency for t<t and wc¹w0.
In the resonance case wc=w0 for short time t<t probability increases quadratically with time.

Probability per unit time (rate) and generalized kinetics of populations
Probability per unit time (or the rate of the photoprocess) is expressed through dimensionless probability of the process at current time according to the obvious relation After substitution of (20) in the right side of this equation we obtain Here we introduce Rabi frequency , and are amplitude and dimensionless electric field strength in the pulse.
Rate of photoprocess ( 25) is the basis of so called generalized kinetics of quantum system populations which account for time dependence of the rate in consistent manner [9].
Generalized kinetics was tested by the comparison with the exact solution for two-level system populations with the help of Bloch equations in the case of Lorentzian spectral profile.It was shown that in the framework of the perturbation theory applicability the use of rate (25) leads to correct results for populations for any pulse durations and carrier frequencies.
Standard approaches for photoprocess rate use two following expressions , ( 26) It was shown [9] that formula (26) is correct in monochromatic limit and the expression ( 27) is suitable in ultra-short one while rate ( 25) is correct for all pulse durations.

From micro-to macro description
Basics formula (3) derived for the micro-target excitation has analogue in macroscopic electrodynamics of continuous media.To demonstrate this we rewrite (3) in terms of dynamical polarizability of the target : .
Here we use the following relation between excitation/absorption cross section and polarizability that follows from the optical theorem: .
Note that in formula (28) the singularity at zero frequency present in (3) disappears.
Using (28) one can obtain the expression for the energy transferred to micro-target from the pulse component at frequency w: .
To derive a macroscopic expression for the energy transfer we use the correspondence between polarizability of micro-target and dielectric function of the medium: (31) is concentration of medium atoms, is dielectric function (permittivity).After substitution (31) into (30) and integration over all frequencies of monochromatic components of the pulse we arrive at the formula for energy transfer from EM pulse to medium per atom . (32) This implies the following expression for the transferred energy per unit volume of the medium ( ) This formula coincides with the well-known expression describing the dissipation of the energy of a non-monochromatic field in a dispersive medium [10].We note that in the case under consideration, the non-monochromaticity of the EM field is due to the finite duration of the electromagnetic pulse.

Generalization of basic formula to other EM processes
As it is mentioned above formulas ( 5)-( 6) allows one to generalize "middle way" approach for the description of other EM processes induced be radiation pulse.To do this, it is necessary to replace the photoexcitation cross section with the cross section of the photo-process under consideration.
Consider, for example, the Rayleigh scattering of an electromagnetic pulse by an atom.Then the cross section integrated over the scattering angles and frequencies of the scattered radiation is given by the following expression . ( here w is the frequency of the monochromatic component of the EM pulse.Note that in the case of Rayleigh scattering the frequency of scattered radiation is equal to the frequency of monochromatic component of incident EM pulse .Thus we have for scattering cross section differential with respect to scattered frequency . ( Substituting ( 35) in (3) instead of the excitation cross section we obtain for Rayleigh scattering probability differential with respect to scattered frequency the following expression: .
It is interesting to note that formula (36) corresponds to the classical expression for intensity of dipole radiation .
To trace this correspondence one should write the energy of dipole radiation , decompose it into the Fourier integral and use the expression for Fourier transform of dipole moment of radiating system .Consistent quantum mechanical derivation of the formula (36) was done in [11] beyond the dipole approximation.
Formula (36) describes different kinds of Rayleigh scattering including low-frequency, highfrequency and resonance scattering.In each case it is necessary to use appropriate cross-sections corresponding to the frequency range under consideration.The same applies to other types of EM radiation scattering: Raman, Compton, scattering by a free electron, scattering on bound electrons in dense plasma [12] etc.

Conclusion
Present review shows wide applicability of "middle way" approach for description of photo-processes induced by EM pulse with various durations from long quasi-monochromatic till ultra-short pulse with frequencies higher than atomic frequency.
This approach combines simplicity and taking into account the main characteristics of the EM pulse and target when calculating the probability of a photoprocess in the framework of a consistent quantum mechanical treatment.
The similarity functions extracted from the main formula make it possible to universally describe the photo-process probability for various combinations of the spectral profile of the cross section and the envelope of the EM pulse for any values of pulse duration and carrier frequency.The rate of the photoprocess obtained with help of the excitation probability as function of current time can be used in the generalized kinetics of the populations of a quantum system in the field of EM pulses of various durations.
The basic expression for the excitation probability is easily generalized to the case of other photoinduced phenomena.