Time distribution of stimulated transition probabilities

In recent years, an increasing number of experimental results have been published for a wide variety of systems in which high-precision, ultra-fast, and ultra-short laser pulses techniques have been used to study time-resolved dynamical processes. In the absence of a theoretical prediction of the time distribution of the transition probabilities, the lifetimes τ are generally extracted by fitting the decay time with an exponential, bi-exponential or three-exponential function. In fitting the data, the short-time behavior is generally neglected. The purpose of this study is to show that an explicit formula for the time distribution of the transition probability can be determined rigorously using time-dependent perturbation theory. A formula that perfectly fits an important subset of the experimental results from t = 0 to t=∞ . We show that by following a route different from the usual procedure for deriving transition amplitudes and Einstein coefficients in time-dependent perturbation theory, one ends up with a time-dependent factor, that is, a temporal distribution function of the transition probabilities, and for the Einstein coefficients. The time distribution reported here looks in the time domain similar to the Planck distribution in the frequency domain. In fact, the behavior of the time distribution, with a maximum at 2τln⁡2 , and with τ playing the role of temperature, resembles the behavior of the Planck frequency distribution. We present several examples to demonstrate that the theoretical formula allows for easier fitting of the experimental results reported in the literature. We also show that the time distribution explains the difference in resonance heights between the experimental and theoretical blue laser spectra.


Introduction
The development of high-precision ultrafast measurement techniques has boosted experimental research on timeresolved ultrafast dynamical processes in a variety of systems with different relaxation mechanisms and time scales.
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The research and development of ultra-short laser pulses as tools for excitation and stop-action measurements has reached a high level of understanding [1][2][3] and has been applied to obtain high-resolution time-resolved spectroscopy of ultrafast processes occurring in different ways in a variety of systems .Among them are time-resolved studies of stimulated emission, time-resolved spectroscopy of charge trapping, ultra-fast photoluminescence decay time in quantum wells or multi quantum wells, ultra-fast hole/electron transfer dynamics in semiconductor nano-crystals, and charge carrier or exciton dynamics in CdSe, CdTe, and CdS quantum dots.The main goal is to identify and characterize the relaxation and thermalization mechanism or mechanisms that bring the system back to equilibrium.A rather common procedure for analyzing the experimental results, has been to focus on the decaying part of the whole process, and to extract the decay times by fitting the experimental results with exponential, biexponential or multi-exponential functions.Although the time evolution of most of the reported experimental data behaves as shown in figure 1, with an initial process of rapid growth of the measured properties and a subsequent decay, it happens that the initial growth step is generally ignored.Here, we deal with the temporal evolution of the stimulated transition probability between two eigenstates of an unperturbed Hamiltonian.Assuming that the initial state has a finite time-life, we derive an explicit time-dependent function that describes both the fast growing and the decaying part of the time evolution of the transition probabilities.We will revisit the time-dependent perturbation theory and derive a one-parameter time distribution function, which, as shown in figure 1, describes the characteristic time evolution of this type of experimental data.The time evolution of the transition probabilities of two-level systems interacting randomly with a heat bath has been addressed and solved previously [32].
The time-dependent perturbation theory of quantum mechanics laid the foundation for the optical transitions formalism and was instrumental in deriving numerous general results.In the calculation of the transition probability per unit time, i.e. in the calculation of the Fermi golden rule, as well as in the calculation of the Einstein absorption and emission coefficients, it has been a common procedure [33] to replace the ratio of the time-dependent sine function and the energy dependent resonant function by a factor 2πht times a delta function δ(E f − E i − hω).The reasoning behind this replacement is appealing and simplifying.Similar arguments were also considered when calculating the transition probabilities from excited states with finite half-life times τ i .[34] The simplifying arguments had practically no consequence on the explanation of the optical responses as functions of the frequency or energy, and on the envision and performance of important applications.An important goal regarding the description of optical response spectra has been to reproduce, as faithfully as possible, all resonant peaks and their frequencies.In recent years, the highly accurate optical response of blue-emitting superlattice diodes, characterized by a large number of peaks, has challenged the theory of optical transitions for describing the peak clustering feature.When this challenge was met, another theoretical challenge became apparent.The theory has not been able to account for the time distribution of the stimulated transition probabilities.This was the original aim of the present study.
It will be shown here that based on the same time-dependent perturbation theory of quantum mechanics, it is possible to further develop the theory and to determine the time dependent transition probability, which explains other temporal features of the optical response spectra that depend also on τ i .

Revisiting the time-dependent perturbation theory
In this section, we outline the most general and well-known formulas and assumptions in the theory of a photon-emitting or absorbing system, in the presence of an electromagnetic field.For the purpose of this paper, we assume that the emitting and absorbing system is described by a Hamiltonian that can be written as where H o describes the non-interacting matter and field parts, and H I represents the charge-field interaction Hamiltonian, which is described by where the Coulomb gauge is assumed, no scalar potential (ϕ = 0) and no spin-field interaction are involved.In terms of the photon annihilation and creation operators, the vector potential A(r, t) is written as (3) with V denotes the volume of the system.For the calculation of the transition probability we consider the time-dependent perturbation theory, and write the wave vector at t = 0 as |Φ j ⟩ 0 = |φ j ⟩ 0 |n ϕj ⟩, with |n ϕj ⟩ the eigenvector of the number operator.We assume that the excited state |Φ i ⟩ 0 has a finite half-life time τ i (also known as life time).In this approximation, the transition probability from an initial eigenstate of H o with energy E i to a final eigenstate with energy E f is given by If we consider only the emission process (E f < E i ) and assume the well-known dipolar approximation, the transition probability becomes If we also write the momentum p as p = pu, the integration on the solid angle can be easily performed [34] and gives 1 4π Taking into account the factor δ λλ ′ and defining 6) can be written as Based on these results, the usual procedures to obtain the transition probability per unit time have been to search for the asymptotic limit as t → ∞, [35,36], or by transition probability expressions in the weak and strong field limits [37], or to replace the last factor in (8) with a function that, as a function of t, has an equivalent area below the main resonance, an area of the order of 2πht, which is then multiplied by a function δ that takes into account the resonant behavior [33,34].None of these procedures will be followed here.

The transition probabilities per unit frequency and unit time
Multiplying the transition probability |a fi (t)| 2 by the density of the final photon states per unit frequency, we obtain the transition probability per unit frequency as It is clear that if we take the time derivative of this function, we have the transition probability per unit frequency and unit time: which is explicitly given by Here we define the energy Γ i = hγ i , better known as the level broadening energy, and the function which comprises the entire dependence on the time and frequency.To obtain the transition probability per unit time, all we need is to integrate upon the frequency ω.Before we perform the integration we will analyze some properties of the function G(ω, t).Even though it seems to be a relatively complex function of time, owing to its resonant behavior, it is possible to obtain an extremely good approximation to G(ω, t), not only to simplify it, but also to unveil the time distribution of the transition probabilities.Due to the resonant factor, the function G(ω, t) is basically concentrated around ω t = (E i − E f )/h.At this frequency and near it, the function G(ω, t) reduces to As shown in figure 2, G(ω, t) and D(ω, t) match perfectly when ω = 1.01ω t .In D(ω, t), the frequency and time functions are decoupled and can be written as where and The function ρ w (ω, τ i ) is the well-known Lorentzian distribution, which reaches its maximum value at ω = ω t = (E i − E f )/h.As can be seen in figure 3, where ρ t (t, τ i ) is plotted for two values of τ i , the function ρ t (t, τ i ) has the form of the Planck distribution and is also its analog in the time domain.The function ρ t (t, τ i ) provides the temporal distribution of the emission probability from level i.This is the main result that we are looking for.It is interesting to see that τ i is to ρ t (t, τ i ) what the temperature is to Plank's frequency distribution of the electromagnetic radiation.In fact, the maximum values of ρ t (t, τ i ) shift when τ i changes.It can easily be shown that the maximum of ρ t (t, τ i ) occurs at For this value of t, the maximum of the distribution is This means that the maximum of Q fi occurs at t = t max = 2τ i ln 2 and, at this instant, the transition probability has its maximum value In terms of the time distribution, the transition probability per unit time and frequency can be written as This is an important result.In the same way as P fi (ω), the probability Q fi (ω, t) can be used to describe the optical response as a function of frequency and time.The next objective is to obtain the transition rate W fi (t) from Q fi (ω, t) by integration upon ω.

The transition probability per unit time
We will now obtain another relevant physical quantity in the analysis of the induced emission, the emission probability per unit of time W fi (t), or the transition rate.This is easily obtained from Q fi (ω, t) by integration upon the frequency ω.The integral of the frequency-dependent part is ˆωdω where π/γ i is the well-known normalization constant of the Lorentzian distribution.After replacing, the transition probability per unit time or transition rate becomes This is, essentially, the Fermi golden rule.It is well-known that if instead of an induced emission process we are dealing with an absorption process, the factor n ph +1 must be replaced by n ph − 1.When n ph is large, both factors can be written simply as n ph .It is well known that for spontaneous emission no electromagnetic radiation is present, and n ph +1 is replaced by 1.To remove the constant γ i , which comes from the integration upon ω, we can redefine the time distribution ρ t (t, τ i ) as ρ t (t, τ i ) = γ i ϱ(t, τ i ), in which case the transition probability is written as with This factor is our main result and defines the time evolution of the transition probability.The procedure followed here led us also to obtain a time dependent transition probability W fi (t), which is the usual transition probability multiplied by the dimensionless time-dependent factor ϱ(t, τ i ).[16].The authors fit the temporal decay with exponential functions (black curves).Magenta and red curves are fittings with ϱ(t, τ i ).In (c) and (d), ultrafast PL carrier relaxation in InN thin films [17], magenta and cyan curves are also fittings with ϱ(t, τ i ).In (e), the PL decay of direct e-h recombination in Ge/Si 0.15 Ge 0.85 multi quantum wells, fitted by the authors with the dashed black line [18].The red curve is the fitting with the function of equation ( 25) defined in the text.

The time distribution and the experimental results
The time distribution of the transition probabilities derived here is an important and general result that will allow a one parameter description of the temporal evolution of the phenomenon under consideration when only one mechanism governs its time evolution.It is remarkable to ascertain that a large number of experimental results published in the literature, which sometimes are fitted with more than one exponential function, can be easily fitted with the one parameter function ϱ(t, τ i ).As will be seen here, with some of many possible examples, the function ϱ(t, τ i ) is able to describe, with only one parameter, the growing and decaying parts of the temporal evolution of the transition probabilities.
Figures 1 and 4 show a small fraction of the numerous ultrafast, femtosecond spectroscopic results.In figure 1, we show the relaxation of transient absorption observed at 1960 nm in the near-infrared spectra of CdSe nanoparticles reported by Burda et al [9].The continuous color curves are the fittings with the function ϱ(t, τ i ).In figure 4, we collect other experimental results.In (a) and (b), Ag emission spectrum measured at different delay times observed by Civìs et al [16].The authors fit (black curves) the temporal decay with exponential functions.The magenta and red curves are fittings with ϱ(t, τ i ).In panels (c) and (d), the time-resolved photoluminescence (PL) measurements in InN thin films by Jang et al [17].These authors report a failure of a single-exponential fitting, which led them to conclude that 'more than one single mechanism of carrier relaxation is involved during the decay process'.The continuous magenta and cyan curves are fittings with the one-parameter function ϱ(t, τ i ).In (d) we show the PL decay of direct e-h recombination in strained Ge/Si 0.15 Ge 0.85 multi quantum wells, reported by Giorgioni et al [18] and fitted with the dashed black straight line, partially hidden by the red curve, which is a fitting with the combination to describe a sequence of two decay mechanisms, with θ(t) being the Heaviside function, A a constant to fix the continuity at t 1 , and t i fractions of 2τ i ln 2, with i = 1, 2, chosen to fit the experimental curve.
In figure 5 we show other experimental results, with the common characteristic that physical processes are explained as two-step processes characterized by two time constants.Also in these cases the distribution function ϱ(t, τ i ) can be successfully applied, with a single lifetime parameter.In panel (a) we reproduce a figure from Olaizabal et al [38] for timeresolved carrier recombination dynamics in InGaN quantum wells; in panel (b) the results reported by Chithambo et al [39] for time-resolved optically stimulated luminescence and spectral emission features of α-Al 2 O 3 :C, and in panel (c) results reported by Klimov et al [40] for ultra-fast time evolution of the PL intensity, at different spectral energies, in CdS  [38] for time-resolved carrier recombination dynamics in InGaN quantum wells; in panel (b), results reported by Chithambo et al [39] for time-resolved optically stimulated luminescence and spectral emission features of α-Al 2 O 3 :C.In this graph, BUS stand for building up signal, LD for luminescence decay and BS for background signal.In panel (c), results reported by Klimov et al [40] for ultrafast time evolution of the PL intensity at different spectral energies.All continuous curves are fits by the authors, in (a) and (c) were done to the two-exponential decay, similar to our one-parameter function ϱ(t, τ i ).All dashed curves are fittings with ϱ(t, τ i ).
nano-crystals dispersed in a glass matrix, studied with femtosecond up-conversion technique.As mentioned before, these results are explained as a two-step process characterized by two time constants, in fact the fitting in (a) was done with the two-parameter function where t r and t d are the rise and decay time constants and A is a constant, and the fitting in (c) was performed with the function where I 0 is the initial intensity change.These functions are, to some extent, close to our function ϱ(t, τ i ).All dashed curves in the graphs of this figure are fittings with ϱ(t, τ i ).We have shown in this section just a subset of the several examples, where the growing and decaying portions of the temporal evolution of the transition probability can simultaneously be described by the function ϱ(t, τ i ).This is an important ability of this function and will certainly 'allow accurate description of the experimental data and lifetimes estimation as well as better understanding of the underlying mechanisms governing the phenomenon under consideration' [41].

Optical response and time distribution in heterostructures
An important class of light emitting and absorbing systems are semiconductor heterostructures.Most of these are layered quasi-two dimensional quantum wells and superlattices.The growth direction in these systems is generally defined as the z-axis and the eigenfunctions |φ i ⟩ 0 are functions of z.For these systems, the transition probability per unit time and unit frequency Q fi (ω, t) in equation ( 20), taking into account that ρ t (t, τ i ) = γ i ϱ(t, τ i ), can be written as where is the well known optical susceptibility, frequently used to calculate the optical response of these devices.When the physical system that absorbs or emits photons has several energy levels from which the transition begins and several energy levels that the electron can reach, the total optical response is obtained by summing over the energy level indices i and f, that is, The semiconductor devices where optical transitions involve transitions from one or more than one energy level in the conduction band to one or more energy levels in the valence band occur, mainly, in the active zone of the optical devices, are quantum wells and superlattices.The blue lasers spectra reported, among others, by Nakamura et al [42], were produced in GaN based superlattices.Since different energy levels imply different half-life times, we should expect for each energy level different time evolution of the transition probability.In figure 6 we plot the time distribution ϱ(t, τ i ) as a function of time for three different values of the life time.It is clear from this figure that at any time t greater than all maxima (t max,i = 2τ i ln 2) (see the blue points), the time distribution ϱ(t, τ i ) for the energy levels with smaller half-life times τ i is also smaller.The same is true for the time distribution ρ t (t, τ i ).Optical response without and with the time distribution factor ϱ(t, τ i ).In (a) the optical response of the blue emitting InxGa 1−x N/InyGa 1−y N superlattice studied in [44] without the time factor.In (b) the optical response with the time factor ϱ(t, τ i ).In Nakamura's experimental spectrum [43] the higher energy peaks (due to transitions between surface state energy levels) are smaller.
In the upper panel of figure 7 we show the optical response reported by Nakamura et al [43].for the blue emitting In x Ga 1−x N/In y Ga 1−y N superlattice, and in panel (a) the theoretical calculation reported in [44], using the optical susceptibility in equation (30).Although the frequency distribution of the optical transitions are faithfully described, the lower wavelength peaks are much larger than the experimental ones.
If we take into account the time distribution ϱ(t, τ i ), we can define the time dependent optical susceptibility Assuming that the life-times of the excited energy levels in the conduction and valence subbands are of the order of the life-times in the harmonic oscillator, and taking into account the time distribution, we have the optical response shown in panel (b).The factor ϱ t (t, τ i ), that represents the temporal distribution of the optical transitions, allows us also to explain the peaks height characteristic in the blue laser spectra.
In the appendix we recover the well-known result in the small time limit.

Conclusion
The time distribution of the stimulated transition probabilities, from an exited state with finite life time to a final state, was determined.It is shown here that based on the well established time-dependent perturbation theory, and by following a slightly different procedure, an analytical function ϱ(t, τ i ), implicit in the theory, can be straightforwardly derived.A function that describes the time dependence of the transition probability.This function resembles Planck's frequency distribution ρ(ω, T), and as shown with several examples, fits well an important class of experimental results, all the way from the stage in which they grow to the stage in which they decay.The distribution function ϱ(t, τ i ) explains also the observed peakheight features in the optical response spectra of light-emitting superlattices.

Figure 1 .
Figure 1.Relaxation of transient absorption observed at 1960 nm in the near infrared spectra for CdSe nanoparticles, reported by Burda et al [9].The continuous curves are fittings with the one parameter-function ϱ(t, τ i ) derived here.

Figure 2 .
Figure 2. The temporal distribution functions G(ω, t) and D(ω, t).The exact functions G(ω, t) and the approximate D(ω, t) describe, practically, the same temporal distribution of the emission probability.The maximum occurs at t = tmax = −2τ i ln 0.5 ≃ 1.4τ i .

Figure 3 .
Figure 3. Temporal distribution of the emission probability for two values of τ i .Each energy level is characterized by a half-life τ i .The distribution ρt shows the behavior of the emission probability from the energy level E i , with time.For t < τ i it grows linearly with t, reaches a maximum when t = tmax = 2τ i ln 2, and for longer times it decays exponentially.

Figure 4 .
Figure 4. Time-resolved PL spectra.In (a) and (b), measured emission spectrum at different delay times observed for Ag lines by Civìs et al[16].The authors fit the temporal decay with exponential functions (black curves).Magenta and red curves are fittings with ϱ(t, τ i ).In (c) and (d), ultrafast PL carrier relaxation in InN thin films[17], magenta and cyan curves are also fittings with ϱ(t, τ i ).In (e), the PL decay of direct e-h recombination in Ge/Si 0.15 Ge 0.85 multi quantum wells, fitted by the authors with the dashed black line[18].The red curve is the fitting with the function of equation (25) defined in the text.

Figure 5 .
Figure 5. Experimental results explained as two step processes or two relaxation mechanisms.In panel (a), results from Olaizabal et al[38] for time-resolved carrier recombination dynamics in InGaN quantum wells; in panel (b), results reported by Chithambo et al[39] for time-resolved optically stimulated luminescence and spectral emission features of α-Al 2 O 3 :C.In this graph, BUS stand for building up signal, LD for luminescence decay and BS for background signal.In panel (c), results reported by Klimov et al[40] for ultrafast time evolution of the PL intensity at different spectral energies.All continuous curves are fits by the authors, in (a) and (c) were done to the two-exponential decay, similar to our one-parameter function ϱ(t, τ i ).All dashed curves are fittings with ϱ(t, τ i ).

Figure 6 .
Figure 6.Temporal distribution ϱ(t, τ i ) for three values of τ i .If t, is greater than the times in which ϱ(t, τ i ) is maximum, the factor ϱ(t, τ i ) is different in each case (blue points) and smaller if τ i is smaller.

Figure 7 .
Figure 7.Optical response without and with the time distribution factor ϱ(t, τ i ).In (a) the optical response of the blue emitting InxGa 1−x N/InyGa 1−y N superlattice studied in[44] without the time factor.In (b) the optical response with the time factor ϱ(t, τ i ).In Nakamura's experimental spectrum[43] the higher energy peaks (due to transitions between surface state energy levels) are smaller.