An MQDT Model for the Description of Time Resolved Pump Probe Experiments

Abstract

A method based on multichannel quantum defect theory (MQDT) for the treatment of perturbative and time resolved photoionization in pump probe experiments is presented. Following an introduction to the theory, examples in two regimes of argon are presented. The theory successfully describes the energy distribution and the dependence on probe delay of the photoionization spectrum. We finalize with a discussion about angular distributions and possible further applications of the method.

1 Introduction

In a recent study [1] a new methodology to resolve energy levels in pump probe spectroscopy experiments was proposed and used in argon as a proof of concept. The method increases the possible energy resolution by exploiting the photoelectron signal coming from autoionizing states in combination with the oscillation as a function of probe time delay. This study provides details on the theoretical method utilized to describe the experiment. In view of existing theoretical descriptions based on single active electron methods interacting with the atom with effective potentials [2, 3], we propose a method based on the MQDT wave functions [4] which has a more comprehensive description of the atom that describes inter-channel interactions. Section 2 provides a brief introduction to the main ideas of MQDT and points to references where more complete descriptions of the theory are presented. Section 3 describes the method utilized for the description of argon in different energy regions. This section will explain the theory for both autoionizing state wave packets and purely bound state packets. We conclude with an overview of the future directions of research and other possible systems to be treated with this method.

2 Introduction to MQDT

Multichannel quantum defect theory was developed in its current form by Michael Seaton in the mid 1960s with the intention of leveraging the power of analytic methods for the treatment of complex atomic systems in which the outermost electron interacts with the residual ionic core with a known central potential and there are many interacting channels [5]. Initially used to treat Rydberg atoms, it was then extended by the extensive work of Fano [6], C.H. Greene [7, 8] and their collaborators to treat more general systems.

A complete description of the different uses and details about general multichannel calculations and their application for the description of a plethora of experiments can be found in the work of Aymar et al. [4]. Here we will focus on describing multichannel functions for a Rydberg atom using the eigenchannel description introduced by Fano [6].

The degenerate solutions to the time independent Schrödinger equation at a fixed energy, at electron distances outside the core where the equation can be separated into a fragmentation coordinate (which for our use case will be the distance between the ionic core and the outermost electron) and core coordinates (which are the spin and angular degrees of freedom of the atom, as well as the coordinates of all the core electrons) is a linear combination of the form:

$$\displaystyle \begin{aligned} \psi_\beta = \sum_{i} \varPhi_i(\omega) F_{i\beta}(r)/r \end{aligned} $$

(1)

where the sum is over all the channels. The functions of the fragmentation coordinate will, in general, follow a set of coupled differential equations. Nonetheless, for a large enough value of r, say $r>R_o$, the equations decouple and the solutions are given in terms of a linear combination of functions known analytically. For a Rydberg atom these functions are the well known Coulomb functions.

The two functions, when energy normalized, are commonly referred to as f and g, the former being regular at the origin and the latter diverging as $r\to 0$. For $r>R_o$ the solution has the form:

$$\displaystyle \begin{aligned} \psi_\beta = \sum_{i} \varPhi_i(\omega) (f_i(r)I_{i\beta}-g_i(r)J_{i\beta})/r \quad r>R_o, \end{aligned} $$

(2)

for some coefficient matrices I and J. The functions f and g depend on the channel i and on the independent solution index $\beta$ since each residual ion state might be different, leading to different kinetic energies for the outer electron and different angular momentum. In order to completely determine the solution, we must solve the equation for the region $r<R_o$. Here we would impose continuity of the logarithmic derivative and thus find the value of the matrix of coefficients I and J. To find physically relevant solutions, boundary conditions are imposed at infinity. Therefore it is expected that the coefficient matrices vary smoothly with energy, and that the strong variation of the wave functions on energy will derive from the imposition of $r\to \infty$ boundary conditions.

One particularly useful set of solutions are the so called eigenchannels. This concept, introduced by Fano [6], assumes that there is a set of channels that are approximate eigenstates of the hamiltonian near the core region. The solutions are then characterized by a set of eigen quantum defects $\{\mu _\alpha \}$, representing the elastic quantum scattering phase of each one of the eigenchannels, and a change of basis matrix $U_{i\alpha }$ that transforms from one set of channels into the other. The physical wave functions are then given by:

$$\displaystyle \begin{aligned} {} \psi_\beta = \sum_\alpha \sum_i \varPhi_i(\omega)/r \left[ f_i(r)U_{i\alpha}\cos{}(\pi \mu_\alpha)-g_i(r)U_{i\alpha}\sin{}(\pi \mu_\alpha) \right] A^\beta_{\alpha}. \end{aligned} $$

(3)

In general the parameters $\mu _\alpha$ and $U_{i\alpha }$ depend on the energy E and, in many cases, a linear E-dependence is enough to describe most of the spectra. The fact that near the core the spin orbit effects are diminished and that the total orbital angular momentum and total spin are approximately conserved, implies that for many atoms the eigenchannels are but small deviations from the LS coupled channels. Therefore it is assumed that the eigenchannel transformation factorizes as $U_{i\alpha }=\sum _{\overline {\alpha }} U_{i\overline {\alpha }}V_{\overline {\alpha }\alpha }$, where channels $\overline {\alpha }$ are the LS coupled channels, $U_{i\overline {\alpha }}$ is the analytically known basis that transforms angular momentum from the scheme in the fragmentation channels into LS coupling, and $V_{\overline {\alpha }\alpha }$ is a matrix close to the identity. The matrix V  is generally parameterized as the product of rotation matrices, so that the total number of parameters for n channels is $n(n+1)/2$ or $n(n+1)$ if a linear energy dependence is included.

In order to obtain physically valid wave functions, the coefficients $A_\alpha$ have to be chosen to fulfill the appropriate boundary conditions at infinity. These are: for the closed channels, to vanish at infinity; and for the open channels, to be energy normalized. If angular distribution and/or spin-dependent observables are desired, then it is also necessary to impose the incoming wave boundary condition. In the ionization continuum, there are as many linearly independent physical solutions as there are open channels (channels where the outer electron has a positive energy); those are distinguished by the index $\beta$ and a phase $\tau _\beta$. From the asymptotic forms of the coulomb functions it is possible to summarize the boundary conditions in a set of equations:

$$\displaystyle \begin{aligned} \left.\begin{array}{ll} \sum_\alpha U_{i\alpha} \sin{}(\pi \mu_\alpha) A^\beta_\alpha = \tan{}(\pi \tau_\beta) \sum_\alpha U_{i\alpha} \cos{}(\pi \mu_\alpha) A^\beta_\alpha & \text{if } i\text{ is open} \\ \sum_\alpha \sin{}(\pi(\nu_i+\mu_\alpha))U_{i\alpha}A^\beta_\alpha = 0 & \text{if } i\text{ is closed} \end{array}\right\} \end{aligned} $$

(4)

When all channels are closed, the system has solutions only for certain energies that correspond to the bound state energies. When there are some open channels the solutions come from a generalized eigenvalue problem which has solutions for all energies, but the coefficients depend strongly on energy owing to the resonance structure of the autoionizing states. Finally, when all channels are open the solution of the eigenvalue problem is $A^\beta _\alpha = \delta _{\alpha \beta }$ and $\tau _\beta$ = $\mu _\beta$.

What we have presented is but one of all the possible, and equivalent, ways of expressing the different solutions. Among them are the reaction (K) and scattering (S) matrix approaches [4], perhaps clearer for scattering problems. Nonetheless this form of the theory is particularly popular since the parameters $\mu _\alpha$ and $U_{i\alpha }$ can be optimized to fit experimental data.

3 MQDT of the Argon Atom

Treatments of the noble gas atoms were among the first [9,10,11,12] and most successful applications of the MQDT formalism. Motivated by the experimental data we recognized that in order to describe the multi-photon processes we had to determine wave functions for symmetries $J^\varPi =0^+,1^-,2^+,3^-,4^+$ and $5^-$. We used the eigenchannel formalism described in the previous section, choosing the LS coupled channels as the eigenchannels and $J_{cs}$ coupled channels as the fragmentation channels.

It is not the objective of this study to propose a calculation for the wave function inside the core region. Thus we will assume that the Coulomb solution is valid in all the dissociation coordinate. This is a decent approximation since the excited states are Rydberg states and the outermost electron is mostly outside the core. We regularize the divergence near the origin coming from the irregular Coulomb function by using the Seaton-Burgess cutoff procedure [13].

We must stress that our treatment goes significantly beyond a single active electron model since electron-core collisions and recollisions, that can change the core states and are consistent with angular momentum conservation, are included in the multichannel wavefunction.

3.1 Autoionizing Wave-Packets

In the experiment, a linearly polarized short-pulse laser uses the 9th harmonic to excite the ground state to a linear combination of low-lying bound states. From there the fundamental harmonic excites the autoionizing states of the $2^+$ and $0^+$ symmetry. These encompass five different outgoing waves. Looking at the density of states in the autoionizing continuum shows that the f states are very narrow, whereby the resonances have predominantly bound character and are long lived. In comparison, the p states are much shorter lived. Therefore the final autoionizing wave packet will be effectively described by a bound wave-packet of only f states, except for very short times.

The wave function used here is then of the form:

$$\displaystyle \begin{aligned} \psi_o = \sum_n A_n \psi_n, \end{aligned} $$

(5)

where the $\psi _n$ function represents the bound part of the nth autoionizing f resonances. These have the explicit form

$$\displaystyle \begin{aligned} \psi_n = \frac{u_{n,3}(r)}{r} \xi_{1/2} {\left\vert [(1/2,1/2)1,3]2,0\right\rangle}, \end{aligned} $$

(6)

where the kets represent the channel functions in $J_{cs}$ coupling as ${\left\vert [(j_c,s_e)J_{cs},\ell ] J,M\right\rangle}$ and $\xi _{1/2}$ is the wave function of the core electrons in the $3p_{1/2}$ ion state. In this case only $j_c=1/2$ are present since this is the excited core state that supports bound states (or strictly speaking, Fano-Feshbach resonances) at this energy.

Following the delayed probe, the wave function acquires a component in the continuum so that the final wave-packet has the form:

$$\displaystyle \begin{aligned} \psi = \sum_n A_n(t) \psi_n + \sum_{J\in\{1,2,3\}}\sum_{M=-1}^{1} \int dE B^{J,M}_{E}(t) \psi_{J,M} (E) \end{aligned} $$

(7)

This calculation only includes continuum states as there are no true bound states that lie within the bandwidth of the probe laser. We are interested in the yield of electrons after the laser has passed, so we want to determine the coefficients for $t\to \infty$. Here the continuum states referred to are those that leave the core in the excited $j_c=1/2$ energy level.

Calculations of the dipole matrix element to the continuum shows that these high lying states are only weakly coupled to the $1^-$ and $3^-$ continuum states, allowing for the use of time dependent perturbation theory. Accordingly, the electron potential energy of interaction with the electromagnetic laser field can be treated classically in the length gauge with the operator

$$\displaystyle \begin{aligned} {} V(t) = \mathcal{E}_o \mathrm{exp}\left\{-\left( \frac{t-t_o}{\gamma} \right)^2\right\} \cos{}(\omega t) \epsilon \cdot r. \end{aligned} $$

(8)

In atomic units, the intensity of the laser is given by $I=\mathcal {E}_o^2$ and the duration of the pulse, defined here as the full width at half maximum (FWHM) of the intensity, is given by $\gamma \sqrt {2{\mathrm {ln}}(2)}$.

We use perturbation theory to second order in the electric field strength. To this order the unitary time evolution operator is given by:

$$\displaystyle \begin{aligned} U(t_1,t_2) = 1-i\int_{t_1}^{t_2} d\tau_1 e^{i H_o \tau_1}V(\tau_1)e^{-i H_o \tau_1} \end{aligned} $$

(9)

$$\displaystyle \begin{aligned}-\int_{t_1}^{t_2}\int_{t_1}^{\tau_1} d\tau_1 d\tau_2 e^{i H_o \tau_1}V(\tau_1)e^{-i H_o (\tau_1-\tau_2)} V(\tau_2) e^{-i H_o \tau_2}. \end{aligned} $$

(10)

Then the transition matrix is $T=U(-\infty ,\infty )$, and the photoionization probability can be obtained as the transition matrix element connecting to the initially-excited wave-packet:

$$\displaystyle \begin{aligned} \begin{aligned} B^{J,M}_E(\infty)=&-i\delta_{j_c,1/2} \delta_{J_{cs},1} \sum_n A_n \mathcal{E}_o \sqrt{\pi}\gamma/2 \mathrm{exp}\left\{-\gamma^2 \frac{(\epsilon_{1/2}-\epsilon_n-\omega)^2}{4}\right\}\\ & e^{i(\epsilon_{1/2}-\epsilon_n-\omega)t_o} \left\langle \epsilon_{1/2};\ell \left|\left| r^{(1)} \right|\right| \epsilon_n;3 \right\rangle \times \\ & (-1)^{1+3+J+1} [J][2] \begin{Bmatrix} \ell & J & 1 \\[3mm] 2 & 3 & 1 \end{Bmatrix} \sum_{q}(-1)^q \epsilon_{-q} \begin{pmatrix} 2 & 1 & J \\[3mm] 0 & q & -M \end{pmatrix}. \end{aligned} \end{aligned} $$

(11)

Here $\epsilon _{1/2} = E-E_{1/2}$ is the kinetic energy of the photoelectron relative to the $3p_{1/2}$ argon ionization threshold, $\ell$ is the orbital angular momentum of the photoelectron, $\epsilon _n$ is the energy of the autoionizing state with respect to the same threshold, $\left \langle \cdot \left |\left | r^{(1)} \right |\right | \cdot \right \rangle$ is the reduced matrix element of the electric dipole operator and $[x]\equiv \sqrt {2x+1}$. The possible values for $\ell$ are 2 and 4.

This transition matrix element shows the origin of quantum beats in the photoionization signal as a function of the time delay. Notice also the role of the duration of the laser in limiting the region where these beats take place. The Gaussian factor limits the beats to only those states which can be taken to the same continuum energy with frequencies that are within the bandwidth of the laser.

Conservation of parity implies that the first nonvanishing correction to the initial autoionizing states must be second order in the field strength. For states that are initially populated, this gives a second order correction to the probability deriving from the cross terms involving the initial amplitude:

(12)

This insight is key. It explains why the observed modification of the autoionization signal due to the probe laser is observed at the energies of the f $J=2$ states only. This second order term can be interpreted as a two-photon Raman transition, with the continuum states above the $j_c=1/2$ threshold as the relevant intermediate states.

The expression involves a time ordered integral over the field profile. In addition, to evaluate the exponential of the unperturbed hamiltonian, it is necessary to represent the identity operator in terms of eigenstates. Nonetheless, we can make some approximations that make the significance of this term clearer and have little diminution of the calculation accuracy.

Indeed, the second order term of transition matrix element looks like:

(13)

The sum ${{{\int _\xi }}}{{{\sum }}}\;\;\; dE_\xi$ represents the sum/integral over all energies and values of the total and orbital angular momenta of the atom. A key element is the interplay of the bandwidth of the laser, the third line in Eq. 13, which indicates that the intermediate energy of greatest importance corresponds to the average energy between the two autoionizing states plus one photon at the central frequency of the laser. This approximation yields:

(14)

where the functions $F_2$ and $F_4$ are defined as

$$\displaystyle \begin{aligned} \begin{aligned} F_2(J_f,M_f)=&\sum_{J=1,2,3} (-1)^{1+J_f+2 J+M_f}[J]^2[J_f][2] \begin{Bmatrix} \ell_f & J_f & 1 \\[3mm] J & 2 & 1 \end{Bmatrix} \begin{Bmatrix} 2 & J & 1 \\[3mm] 2 & 3 & 1 \end{Bmatrix} \times \\[2mm] & \sum_{q,q'} (-1)^{q} \epsilon_{-q} \epsilon_{-q'} \begin{pmatrix} J & 1 & J_f \\[3mm] q' & q & -M_f \end{pmatrix} \begin{pmatrix} 2 & 1 & J \\[3mm] 0 & q' & -q' \end{pmatrix} \end{aligned} \end{aligned} $$

(15)

$$\displaystyle \begin{aligned} \begin{aligned} F_4(J_f,M_f)&=(-1)^{J_f+1+M_f}\sqrt{5}[J_f] \begin{Bmatrix} \ell_f & J_f & 1 \\[3mm] 3 & 4 & 1 \end{Bmatrix} \\[2mm] & \sum_{q,q'} (-1)^{q} \epsilon_{-q} \epsilon_{-q'} \begin{pmatrix} 3 & 1 & J_f \\[3mm] q' & q & -M_f \end{pmatrix} \begin{pmatrix} 2 & 1 & 3 \\[3mm] 0 & q' & -q' \end{pmatrix} \end{aligned} \end{aligned} $$

(16)

with the values $J_f=2$ and $M_f=0$.

This expression shows that the second order corrections contain the quantum beats deriving from interfering pathways. Figure 1 compares this theory with the experiment. The spectrograms shown can be viewed as a differential electron spectrogram [1], because they show the difference between the total yield with and without the probe. In the theory this amounts to isolating only the second order correction term.

Fig. 1

Differential electron spectrogram for the autoionizing spectrum of argon. Panel (a) is the experimental data and panel (b) is the theoretical data. Horizontal lines and labels indicate the energy position of the different autoionizing resonances. It is possible to see the how this population transfer sometimes enhances the yield of electrons

3.2 Bound Wave-Packets

After the excitation of the ground state by the $9^{\text{th}}$ harmonic, a wave packet of bound $J^\varPi = 1^-$ remains excited. This packet can be photoionized by the probe laser following a two photon absorption. Unlike the previous case, there is a strong coupling between bound states which requires the time propagation of the state to get accurate values for the photoionization probability. Another difference with the previous case is that the multichannel structure of the bound and continuum states is crucial for understanding the experimentally observed spectrogram.

To incorporate all the dynamics, our calculations included states in all the symmetries referenced at the beginning of this section. But we truncated to a limited Hilbert space that includes only the states that are most relevant for the propagation. The calculation includes 4 bound states in the $1^-$ symmetry and 3 bound states in the $3^-$ symmetry. At one photon energy above and below these states in the opposite parity there are 17 states in the $0^+$, 54 states in the $2^+$ symmetry divided into 24 f states and 30 p states and 26 states in the $4^+$ symmetry. The first 36 autoionizing states are included for the $0^+$ and $2^+$ symmetries. Finally, the calculation included 200 states for each open channel, which are uniformly spaced with a resolution of $3.76$ meV in the continuum, starting from 50 meV above the $3p_{1/2}$ threshold for the $1^-$, $3^-$ and $5^-$ symmetries.

For maximum efficiency, the calculation propagates separately the time evolution associated with the probe and the pump pulses. That enables an efficient computation of the cases without overlap, since we can propagate the probe for each one of the bound states and then make a linear combination using the amplitudes found with the pump propagation. The method utilized for time propagating was the split operator method, using the same shape for the time dependent perturbation as in Eq. 8 for both pump and probe. As suggested by Tarana et al. [14], since the time dependence of the perturbation comes only from the electric field profile it is possible to handle the propagation for a single time step as:

$$\displaystyle \begin{aligned} \psi(t+\delta t) = \mathrm{exp}\left\{-iH_o \delta t/2\right\} O^\dagger \mathrm{exp}\left\{-i \hat{\lambda} \int_{t}^{t+\delta t} \mathcal{E}(\tau)d\tau\right\} O \mathrm{exp}\left\{-iH_o \delta t/2\right\} \psi(t). \end{aligned} $$

(17)

Here the diagonal operator $\hat {\lambda }$ is the diagonalized form of the dipole operator and the matrix O is its eigenvector-matrix.

As discussed in the introduction, any multichannel wave function is composed of an expansion as in Eq. 3. Therefore, under the current approximation, for any two multichannel states $\psi _1$ and $\psi _2$ the dipole element is expanded as

(18)

where the symbol $\mathcal {R}$ stands for the Seaton-Burgess cutoff regularized radial integral

$$\displaystyle \begin{aligned} \mathcal{R}^{i_1\alpha_1i_2\alpha_2} & = \int_0^\infty \left(f_{i_1}\cos{}(\pi \mu_{\alpha_1})-g_{i_1}\sin{}(\pi \mu_{\alpha_1})\right)r\left(f_{i_2}\cos{}(\pi \mu_{\alpha_2}) \right. \\ & \quad \left.-g_{i_2}\sin{}(\pi \mu_{\alpha_2})\right) \end{aligned} $$

(19)

The resulting generated spectrograms, like the ones shown in Fig. 2, capture many of the features observed experimentally. For instance, two features are evidence that indicate ionization with respect to the two different thresholds, even though the pump excites states that are predominantly attached to the $3p_{1/2}$ threshold. This can be observed in this treatment since it utilizes multichannel functions throughout; the small but nonzero mixing of components attached to the $3p_{3/2}$ threshold can accumulate to generate photoelectrons at this energy with considerable probability. Another interesting detail is the fact that the photoelectrons associated with different ionization thresholds have different phases. In this particular atom, the signals are almost $\pi /2$ out of phase. This is also captured by our model as is evident in the figure.

Fig. 2

Panel (a) is the theory and panel (b) the experiment corresponding to the photoionization of a bound wave-packet of argon following two photon absorption. The two main features correspond to ionization with respect to each distinguishable ion states. The beating demonstrates interference of the interfering two photon absorption pathways from the states excited by the pump. Panel (c) shows the position where a monochromatic laser would ionize the aforementioned states and a scaled profile of the mean of the signal. The theory predicts an unobserved excess of electrons leaving the core in the $3p_{3/2}$ state.

4 Conclusions and Discussion

This study develops a model based on multichannel quantum defect theory to treat photoionization in pump-probe experiments in both time resolved and perturbative methods. We have presented a brief introduction of the MQDT developed by M. Seaton [5] and its recasting by Fano and collaborators [6] based on the concept of eigenchannels.

Two regimes of a pump-probe experiment in argon served as two studied applications of this method. The first was an interpretation of the observed quantum beats in the autoionization region of long-lived nf-states. Even though the model included only the bound part of the wave functions, motivated by the narrowness of the autoionizing resonances, it successfully described the observed quantum beats as well as the energy distribution of the electrons.

The second case explored the photoionization, by two photon absorption, of a bound state wave packet. In this case the strong coupling between bound states via the dipole interactions required the use of a nonperturbative time propagation. Through the use of the split operator formula and a limited Hilbert space composed of the most relevant states, the method was able to describe many of the observed features in the experimental spectrograms. Among them, a considerable phase difference in the electron signal from photoelectrons that leave the ionic core in the different spin orbit thresholds.

One of the key elements this method has not yet explained is the structure of the angular distributions in both cases. As is well known [15], in experiments where all the lasers are linearly polarized and aligned the emitted electrons follow a cylindrically symmetric angular distribution. This distribution is a linear combination of Legendre polynomials. The degree of the maximum Legendre polynomial is determined by the number of photons involved.

In the case of the bound wave packet photoionization, the experiment observes beating angular distributions with a strongly dominant term of order 8. That is the reason behind the inclusion of the $3^-$, $4^+$ and $5^-$ symmetries in the Hilbert space, the latter two seemingly unnecessary for a two photon process starting from a $1^-$ bound state. We discovered that the laser might be strong enough to additionally excite a Raman type transition from the intermediate $2^+$ bound states into the $3^-$ states, which are then photoionized to the $5^-$ final state symmetry. This is the only discernible pathway so far identified that could account for this process, but in preliminary explorations the theory predicts it to be much weaker than what has been observed experimentally. We leave this study for a future publication.

Another interesting case suitable for a description as presented here is the neon atom. Recently a study of the properties of neon atom photoionization [2, 3] discovered that there is an interesting dependence of the partial wave composition of the photoelectrons with the intensity and the frequency of an array of two co-propagating lasers with commensurate frequencies. Direct application of the model as presented in this article is not possible, since the energy range of the study reaches states that are close to and even in between the ionization thresholds, which would require a number of states that appears to be intractable owing to the small spin orbit splitting in Ne${ }^+$. Alternative methods that could extend this method to such a situation are currently being explored. One of them is restricting the valid wave functions to those that fit inside a finite box. We leave the report of those findings for a future publication.