# Thomas–Reiche–Kuhn Correction for Truncated Configuration Interaction
Spaces:
Case of Laser-Assisted Dynamical Interference

Mattias Bertolino1 Stefanos Carlström1 Jasper Peschel1 Felipe Zapata1 Eva
Lindroth2 Jan Marcus Dahlström1 1Department of Physics, Lund University, Box
118, SE-221 00 Lund, Sweden
2Department of Physics, Stockholm University, AlbaNova University Center,
SE-106 91

###### Abstract

The Thomas–Reiche–Kuhn sum rule is used to form an effective potential that is
added to the time-dependent configuration interaction singles (TDCIS)
equations of motion in velocity gauge. The purpose of the effective potential
is to include virtual coupling from singles to doubles, which is required for
size-consistent velocity gauge TDCIS results. The proposed method is compared
to length gauge TDCIS results for laser-assisted photoionization. Finally, a
dynamical interference effect controlled by two-color fields is predicted for
atomic targets.

##  I Introduction

There are several frontiers of research in attosecond physics [1], including
studies of charge migration in biorelevant molecules [2], generation of
attosecond pulses in solid-state targets [3] and atomic delay measurements in
laser-assisted photoionization [4]. It has been shown that attosecond
precision measurements can be performed by photoelectron interferometry, using
various forms of extreme ultraviolet (XUV) and infrared (IR) pulses, in atoms
[5, 6, 7, 8, 4], molecules [9, 10] and solids [11, 12]. While this type of
non-linear interferometry can be qualitatively understood using the Strong-
Field Approximation (SFA), as we have recently reviewed in Ref. [13] for
numerous experiments with light from High-order Harmonic Generation (HHG) and
Free-Electron Laser (FEL) sources, a more detailed interpretation requires use
of advanced many-body theory. In cases where the interaction with the fields
is “weak” it is possible to employ perturbation theory to interpret attosecond
delays from atoms [14, 15], molecules [16, 17, 18] and laser-stimulated
transitions via autoionizing states [19, 20, 21, 22, 23, 24]. However, when
the interaction with fields increase in strength, time-dependent simulations
are essential to understand the correlation effects that accompany the laser-
driven dynamics. Brute force approaches for photoionization studies are
possible only in two-electron systems, such as He and H-, where the
Schrödinger equation can be directly propagated [25, 26]. In other systems,
time-dependent simulations of multi-electron dynamics are more costly and one
must rely on approximations that balance precision against the numerical cost.
A few examples include the Time-Dependent Configuration Interaction Singles
(TDCIS) [27, 28], $R$-matrix [29, 30], X-CHEM [31, 24] and self-consistent
field theories [32, 33, 34]. The latter theories rely on time-dependent Slater
determinants, which form eigenstates of approximate time-dependent
Hamiltonians. The simplest such example is the Time-Dependent Hartree Fock
(TDHF) theory, where a single Slater determinant is constructed from time-
dependent occupied orbitals.

In addition to the level of electron correlation included in the simulations,
care must also be given to the question of gauges that describe the
electromagnetic interaction with the electrons. If no approximations were
made, the Schrödinger equation could have equivalently been expressed in
either gauge. However, the truncation of the basis makes the physical
observables gauge-dependent [35, 27]. This raises the question of which gauge
happens to give (i) the best correspondence to experiment, (ii) the most
convenient numerical properties. While the Hartree–Fock equations (HF) have
been shown to be gauge invariant [36], such a formulation requires the
occupied orbitals to be gauge-transformed dynamically in the presence of time-
dependent fields. This implies that self-consistent field theories [32, 33,
34] are highly desirable in this regard, but, unfortunately, such theories are
costly to propagate numerically. According to Kobe’s gauge theory [37], the
length gauge is unique with expansion coefficients that are genuine
probability amplitudes. This implies that dynamic gauge transformations are
less essential, and that more efficient basis truncations can be performed in
the length gauge, as compared to any other gauge. However, the length gauge is
notorious for being difficult to converge for strong-field processes with low-
frequency laser pulses [38, 39, 40, 41]. Indeed, a relatively low number of
angular momenta is required for convergence of TDCIS [27, 28] in velocity
gauge to describe laser-assisted photoionization [42, 13]. On the other hand,
length gauge TDCIS calculations have empirically proven better than velocity
gauge calculations [43], meaning that the converged TDCIS results in length
gauge are more accurate than the converged velocity gauge results, in
agreement with the gauge theory of Kobe [37, 36]. To overcome this gauge-
problem in TDCIS, Sato et al. have developed a modification to TDCIS in
velocity gauge by successively rotating the orbitals in each time step,
effectively defining time-dependent orbitals, which are necessary for gauge
invariance with the length gauge TDCIS results [44, 36, 43].

In this article, we investigate the gauge-dependence problem in TDCIS, from a
different perspective than Sato et al. [43], by bridging the space of single
(S) excitations to that of virtual double (D) excitations. This is done by
systematically adjusting for the lack of “core” polarization of excited states
in the TDCIS. In Section II, a review of polarization effects in $N$-electron
atoms is presented by usage of the Thomas–Reiche–Kuhn (TRK) sum rule, (see
e.g. Ref. [45]). We then introduce an effective potential of TRK-type to the
equations of motion for velocity gauge TDCIS in Section III. In Section IV,
results for laser-assisted photoionization are presented for velocity gauge
TDCIS and the proposed TDCIS-TRK theory. It is shown that TDCIS-TRK provides
accurate results where velocity gauge TDCIS fails. We also revisit dynamical
interference in photoionization [46, 47, 48], and predict that the phenomenon
is experimentally feasible in a new setting with two-color fields. Finally, in
Section V we conclude our findings.

##  II Theory

An atom subject to an external laser field will have its dynamics ruled by the
minimal coupling Hamiltonian

$\hat{H}(\mathbf{r},t)=\hat{H}_{0}+\hat{V}_{1}(\mathbf{r},t)+\hat{V}_{2}(\mathbf{r},t),$
(1)

where the zeroth term $\hat{H}_{0}$ is the atomic Hamiltonian, which accounts
for the kinetic energy and the Coulomb interaction with the nucleus and among
the electrons. For a wavelength substantially larger than the atom,
$\lambda\gg a_{0}$, the spatial dependence of the field can be neglected. This
is referred to the dipole approximation, and when applied to the minimal
coupling Hamiltonian, we refer to it as the velocity gauge. The interactions
with the electromagnetic field are hence given by a term linear in the time-
dependent vector potential:

$\hat{V}_{1}(t)=-\frac{q}{m}\mathbf{A}(t)\cdot\sum_{ij}\mathbf{p}_{ij}\hat{c}_{i}^{\dagger}\hat{c}_{j},$
(2)

and a term quadratic in the vector potential:

$\hat{V}_{2}(t)=\frac{q^{2}A^{2}(t)}{2m}\hat{n}.$ (3)

Here, $q=-e$ is the electronic charge and $\mathbf{p}_{ij}=\langle
i|\hat{\mathbf{p}}|j\rangle$ denotes the matrix element for momentum between
the canonical orbitals $i$ and $j$, which are eigenstates of the mean-field
(Fock) operator $\hat{f}|i\rangle=\varepsilon_{i}|i\rangle$. The operators
$\hat{c}_{i}^{\dagger}$ and $\hat{c}_{i}$ creates respectively annihilates the
canonical orbital $i$. The $\hat{V}_{2}$ operator acts as a scalar, due to any
$N$-body state being an eigenstates to the number operator,
$\hat{n}=\sum_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$, such that
$\hat{n}|\Psi\rangle=N|\Psi\rangle$.

###  II.1 Energy shifts in an oscillating potential

Consider a monochromatic vector potential $A(t)=A_{0}\cos(\omega t)$,
polarized linearly along the $z$-axis. The potential term linear in the vector
potential is given by

$\hat{V}_{1}(t)=\hat{V}_{1}^{(+)}\exp(-i\omega
t)+\hat{V}_{1}^{(-)}\exp(i\omega t)$ (4)

with

$\hat{V}_{1}^{(+)}=\hat{V}_{1}^{(-)}=-\frac{q}{2m}A_{0}\sum_{ij}(p_{z})_{ij}\hat{c}_{i}^{\dagger}\hat{c}_{j},$
(5)

where the superscripts $(+)$ and $(-)$ correspond to interactions that induce
absorption and emission of a photon, respectively. The quadratic potential
amplitude is given by

$\hat{V}_{2}(t)=\frac{q^{2}A_{0}^{2}\cos^{2}(\omega t)}{2m}\hat{n}.$ (6)

The second-order energy correction to any $N$-electron state,
$|\Psi_{0}\rangle$, due to $\hat{V}_{1}(t)$ is given by

$\langle\Delta E_{1}^{(2)}(\omega)\rangle=\lim_{\eta\to
0^{+}}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{n\neq
0}\left(\frac{\langle\Psi_{0}|\hat{V}_{1}^{(+)}|\Psi_{n}\rangle\langle\Psi_{n}|\hat{V}_{1}^{(-)}|\Psi_{0}\rangle}{E_{0}-\hslash\omega-
E_{n}+i\eta}+\frac{\langle\Psi_{0}|\hat{V}_{1}^{(-)}|\Psi_{n}\rangle\langle\Psi_{n}|\hat{V}_{1}^{(+)}|\Psi_{0}\rangle}{E_{0}+\hslash\omega-
E_{n}+i\eta}\right),$ (7)

which can be interpreted as a level shift of the state averaged over time
[49]. The TRK sum rule states that the sum of all oscillator strengths,
$f_{nn^{\prime}}$, from a particular atomic eigenstate, $|\Psi_{n}\rangle$, is
equal to the number of electrons:

$\sum_{n^{\prime}}f_{nn^{\prime}}=\sum_{n^{\prime}}\frac{2m}{\hslash^{2}}(E_{n}-E_{n^{\prime}})|\langle\Psi_{n^{\prime}}|\hat{z}|\Psi_{n}\rangle|^{2}=N.$
(8)

The application of TRK theory to (7) requires that the commutation between the
atomic Hamiltonian and the dipole operator equals the momentum operator,
$[\hat{H_{0}},\hat{z}]=-i\hslash\hat{p}_{z}/m$, (see e.g. Eq. (61.1) in Ref.
[50]). Since this condition is satisfied for any local potential, the TRK sum
rule can be used to perform exact calculations in $N$-electron atoms, where
all interactions are inherently local. The time-averaged energy-shift is
derived by Taylor expanding the denominators in Eq. (7) with
$\hslash\omega\ll|E_{n}-E_{0}|$ to yield an expression that is valid up to
second order in $V_{1}(t)$:

$\langle\Delta
E_{1}^{(2)}(\omega)\rangle=-\frac{A_{0}^{2}}{4}\left(\frac{q^{2}}{m}N+\omega^{2}\alpha_{0}\right)+\mathcal{O}\left(\omega^{4}\right),$
(9)

where

$\alpha_{0}=\frac{q^{2}\hslash^{2}}{m}\sum_{n\neq
0}\frac{f_{n0}}{(E_{0}-E_{n})^{2}}$ (10)

is the atomic polarizability, as defined in Ref. [51]. The time-averaged
energy shift of the first-order energy correction due to the quadratic term,
$\hat{V}_{2}(t)$, is simply given by

$\langle\Delta E_{2}^{(1)}(\omega)\rangle=\frac{q^{2}\langle
A^{2}(t)\rangle}{2m}N=\frac{q^{2}A_{0}^{2}}{4m}N.$ (11)

This shows that, in the case of a monochromatic field, the energy corrections
due to $\hat{V}_{1}(t)+\hat{V}_{2}(t)$ to second order in the vector potential
cancel such that the explicit $N$-dependence in the energy shift energy
disappears. The remaining energy shift in velocity gauge is

$\langle\Delta
E^{(2)}(\omega)\rangle=-\frac{\omega^{2}A_{0}^{2}\alpha_{0}}{4}+\mathcal{O}\left(\omega^{4}\right)=-\frac{E_{0}^{2}\alpha_{0}}{4}+\mathcal{O}\left(\omega^{4}\right),$
(12)

which can be found more easily in the length gauge with electric field
$E(t)=-\frac{dA}{dt}=E_{0}\sin(\omega t)$ by second-order perturbation theory
with the dipole operator $\hat{V}_{1}^{(\mathrm{len.})}=-qE(t)\hat{z}$. While
length gauge is simpler in this regard, because it does not rely on a detailed
cancellation effect of two different perturbation terms, the energy shifts are
found to be equivalent in both gauges. We stress that this exact result is
general, as it applies to any state in atomic or molecular targets, but it
fails in cases where the considered state resonates with another state at the
applied frequency: $\hslash\omega\approx|E_{n}-E_{0}|$. Thus, as pointed out
by Bucksbaum et al. [52], it is typically an excellent approximation for the
ground state of an atom, or molecule, subject to a low-frequency laser field,
$\hslash\omega\ll|E_{n}-E_{0}|$, but it is a bad approximation for excited
states, where other Rydberg states are likely to be close in energy. We note
that the TRK rule has been used in solids to reduce the necessary number of
energy bands in velocity gauge while ensuring convergence [53]. In the present
work however, we will make use of the TRK sum rule in the numerical
propagation to correct for size inconsistency in virtual excitations that are
missing due to truncations of the many-body configuration-interaction
expansion.

###  II.2 Configuration Interaction (CI)

Many-body effects can be described by Time-Dependent Configuration Interaction
(TDCI) methods, where the total electron wavefunction is expanded as

$|\Psi(t)\rangle=\alpha_{0}(t)|\Phi_{0}\rangle+\sum_{ap}\alpha_{a}^{p}(t)|\Phi_{a}^{p}\rangle+\sum_{abpq}\alpha_{ab}^{pq}(t)|\Phi_{ab}^{pq}\rangle+...,$
(13)

where $|\Phi_{0}\rangle$ denotes the Hartree–Fock (HF) reference Slater
determinant, while $|\Phi_{a}^{p}\rangle$ and $|\Phi_{ab}^{pq}\rangle$ denote
single (S) and double (D) excitations, respectively. The labels $a,b,...$
index the occupied orbitals, and the labels $p,q,...$ index the unoccupied
orbitals in the initial reference state. While the orbitals are found
variationally, without the laser-action part of the Hamiltonian, the expansion
coefficients $\alpha(t)$ are time-dependent quantities in TDCI methods. The
expansion in Eq. (13) can be truncated in several ways. If the expansion is
truncated so that the CI space is limited to the level of single excitations
it is known as TDCIS [27, 28]. The singly excited states
$|\Phi_{a}^{p}\rangle$ are in the non-relativistic case restricted to the
spin-singlet state character constructed using second quantization as

$|\Phi_{a}^{p}\rangle=\frac{1}{\sqrt{2}}\\{\hat{c}_{p+}^{\dagger}\hat{c}_{a+}+\hat{c}_{p-}^{\dagger}\hat{c}_{a-}\\}|\Phi_{0}\rangle.$
(14)

The operator $\hat{c}_{p\sigma}^{\dagger}$ creates an electron in the virtual
orbital $p$ with spin $\sigma$ and the operator $\hat{c}_{a\sigma}$ creates a
hole in the core with orbital $a$ and spin $\sigma$. While TDCI methods imply
that occupied orbitals are static eigenstates of a laser-free mean-field
Hamiltonian, electronic excitations at TDCIS-level can be interpreted as time-
dependent wave packets in the form of superpositions of initially unoccupied
orbitals [27]. Recently, implementations of TDCIS that are not restricted to
spin-singlet configurations have been realized, e.g. one in the form of two-
component orbitals from relativistic pseudopotentials [54, 55], and another
based on the four-component orbitals from the Dirac–Fock equation [56]. In the
following work we will consider only the spin-singlet configurations and
linearly polarized fields, such that the magnetic quantum number is conserved:
$m_{p}=m_{a}$, where in addition the gerade symmetry of TDCIS can be utilized
to further reduce the number of correlated channels [57]. Further truncation
in CI can be performed to limit the allowed excitations, e.g. “freezing”
occupied orbitals corresponding to tightly bound electrons cf. Refs. [58, 59,
32, 33, 34].

###  II.3 Application of TRK to TDCIS

Truncation of the Hilbert space implies that the TRK sum rules will not hold.
Nonetheless, we consider the CI space to be truncated at the level of single
excitations (CIS), where the states in Eq. (7) are given by the space spanned
by the HF ground state and the CIS singly excited states,
$\\{|\Psi_{0}\rangle,|\Psi_{n}\rangle\\}\to\\{|\Phi_{0}\rangle,|\Phi_{a}^{p}\rangle\\}$,
respectively. We denote the space of no excitations by $\mathcal{P}_{0}$,
single excitations by $\mathcal{P}_{1}$ and double excitations by
$\mathcal{P}_{2}$.

Figure 1: Goldstone diagrams of the energy contributions within (a–d) CIS and
(e–g) CID. The diagram in (a) describes the energy shift of the core, while
the diagrams in (b–g) describe the energy shift of an excited CIS state. The
background shading color indicates the CI space where the atom resides. Time
flows upwards in these Goldstone diagrams.

In Figure 1, we illustrate some processes from perturbation theory,
represented as Goldstone diagrams, that give rise to energy shifts due to
second-order action of $\hat{V}_{1}(t)$. The first diagram: Fig. 1 (a), can be
interpreted as an uncorrelated (bare) energy correction of the HF ground
state, $|\Phi_{0}\rangle$ in $\mathcal{P}_{0}$, due to a virtual transition
via the singly excited state, $|\Phi_{a}^{p}\rangle$ in $\mathcal{P}_{1}$.
Since in TDCIS theory both $\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ are
contained in the truncated Hilbert space, the polarization of the HF ground
state will be reasonably well described. The singly excited states can also
shift their energies due to second-order action with $V_{1}(t)$ within
$\mathcal{P}_{1}$ and via $\mathcal{P}_{0}$. These effects are described in
TDCIS theory, as shown in Fig. 1 (b–d). However, some perturbative processes
that also change the energy of the singly excited states are missing, since
they require a virtual transition via the doubly excited states in
$\mathcal{P}_{2}$, as shown in Fig. 1 (e–g). The former diagram: Fig. 1 (e),
shows that the laser field can polarize the core of the atom, similar to how
the ground state was excited: Fig. 1 (a), which leads to an energy shift $\sim
N$, in accordance with Eq. (9). However, the atomic core is missing one-
electron exchange processes: Fig. 1 (f,g) that impose the Pauli exclusion
principle, which will reduce the core energy shift to $\sim N-1$.
Additionally, the reversed time order of Fig. 1 (d) is also missing in CIS and
further many-body polarization effects, induced by Coulomb interactions, must
be carefully considered by Many-Body Perturbation Theory (MBPT) for accurate
energy shifts.

Since the diagrams in Fig. 1(e–g) are non-existent in TDCIS, the time
evolution will be subject to a spurious effects. While the polarization
effects are reasonably accounted for in the ground state, the excited states
in CIS have an unbalanced polarization that lacks size consistency. Since
transitions between the ground state and the excited states are essential in
TDCIS theory this inconsistency will result in unphysical relative energy
shifts for excitations. These type of relative energy shifts are usually
avoided in MBPT, by the linked diagram theorem, (see e.g. Ref. [60]), but in
time-dependent simulations, such as TDCIS, the problem must be treated in a
different way. We will introduce an effective potential that corrects the
relative energy shifts based on TRK theory. In order to understand this better
we first consider the energy shifts computed for the HF ground state of neon
with various approximations using MBPT.

Due to the non-locality of the Hartree–Fock exchange potential, when static or
frozen occupied orbitals are used, the second-order energy correction is not
exactly given by Eq. (9). Instead, we define the ground state energy shift

$\langle\Delta\tilde{E}_{1}^{(2)}(\omega)\rangle=-\frac{A_{0}^{2}}{4}\left(\frac{q^{2}}{m}\tilde{N}+\omega^{2}\tilde{\alpha}_{0}\right)+\mathcal{O}\left(\omega^{4}\right),$
(15)

where $\tilde{N}$ and $\tilde{\alpha}_{0}$ denote the corresponding value of
$N$ and $\alpha_{0}$ for a given approximation: Lowest-Order Perturbation
(LOP), Configuration Interaction Singles (CIS) and Random Phase Approximation
with Exchange (RPAE). The effective number of active electrons, which is a
frequency independent quantity, is computed as

$\tilde{N}\equiv-\left(\frac{8m}{A_{0}^{2}q^{2}}\right)\sum_{a}^{\mathrm{occ.}}\langle
a|\hat{V}_{1}|\rho_{0,a}^{(+)}\rangle,$ (16)

where $a$ label all occupied orbitals in from the HF ground state and
$|\rho_{0,a}^{(+)}\rangle$ is an associated perturbed wavefunction that
depends on the level of approximation. In the case of RPAE, the perturbed
wavefunctions are (in atomic units) constructed as [61]

$|\rho_{\omega,a}^{(\pm)}\rangle=\sum_{p}^{\mathrm{exc.}}\frac{|p\rangle}{\varepsilon_{a}-\varepsilon_{p}\pm\omega}\bigg{[}\langle
p|\hat{V}_{1}|a\rangle-\sum_{b}^{\mathrm{occ.}}\bigg{(}\langle
b,p|r_{12}^{-1}|a,\rho_{\omega,b}^{(\pm)}\rangle-2\langle
b,p|r_{12}^{-1}|\rho_{\omega,b}^{(\pm)},a\rangle+\langle\rho_{\omega,b}^{(\mp)},p|r_{12}^{-1}|a,b\rangle-2\langle\rho_{\omega,b}^{(\mp)},p|r_{12}^{-1}|b,a\rangle\bigg{)}\bigg{]},$
(17)

where the superscripts $(+)$ and $(-)$ denote forward- and backward-
propagating perturbed wavefunctions. Note that it is the $(+)$-function at
zero-frequency, $\omega=0$, that is inserted into Eq. (16). The first
$(+)$-term on the right-hand side of Eq. (17) corresponds to uncorrelated
(bare) excitation, which we label as LOP. Including also the second and third
$(+)$-terms correspond to forward-propagating electron–hole correlations,
which we label CIS, because it is the level of correlation obtained by solving
TDCIS (also known as Tamm–Dancoff, (see e.g.[61]). The fourth and fifth terms
are exclusive to RPAE and they correspond to ground state correlation effects
(direct and exchange) by self-consistent solution of both $(+)$ and
$(-)$-terms. More details about the use of perturbed wavefunctions in
attosecond physics, and how they correspond to dressing of the lower vertex in
Fig. 1 (a), are found in Refs. [14, 15]. In Table 1 the numerically obtained
values of $\tilde{N}$ for LOP, CIS, and RPAE are presented together with the
number of active electrons in the core of neon atoms. RPAE is exceptional
since it corresponds to the linear response of the TDHF approximation, which
is a self-consistent field theory, and therefore does obtain the correct
number of electrons. However, when the active core is truncated to
$N_{\mathrm{A}}<N$, RPAE does not provide the correct number of active
electrons. It is interesting to note that when only $2p$ orbitals are active,
the CIS approximation outperforms the RPAE approximation in this aspect.

Table 1: Effective number of active electrons $\tilde{N}$ within LOP, CIS, and RPAE, for the neon atom $(1s^{2}2s^{2}2p^{6})$ given the number of electrons in the active core: $N_{\mathrm{A}}$. Active | $\tilde{N}_{\mathrm{LOP}}$ | $\tilde{N}_{\mathrm{CIS}}$ | $\tilde{N}_{\mathrm{RPAE}}$ | $N_{\mathrm{A}}$
---|---|---|---|---
2p | 5.4091 | 6.1758 | 7.2461 | 6
2s,2p | 6.2712 | 7.2558 | 8.3022 | 8
1s,2s,2p | 7.8528 | 8.8858 | 10.0000 | 10

###  II.4 Energy shifts in a static potential

Alternatively, static energy shifts of the HF ground state,
$|\Phi_{0}\rangle$, in response to an external potential, $V_{1}$, can be
obtained by diagonalizing the Hamiltonian, $\hat{H}_{0}+\hat{V}_{1}$ expressed
within CIS. In this case a constant static vector potential,
$A^{(\mathrm{stat.})}=A_{0}$, in the linear interaction term in Eq. (2) is
used. Since the action of the quadratic term is trivial in this case it is
omitted in the following discussion. This approach has the advantage that it
is not restricted to the second-order interaction, but includes all higher
order corrections with the field directly. Diagonalization of the CIS
Hamiltonian can also be used to study non-linear DC Stark shifts of excited
(Rydberg) states. This feature is however of limited use since laser fields
with finite frequencies will induce AC Stark shifts that differ significantly
from the DC results due to resonances in the Rydberg series.

Figure 2: Calculated polarization of the HF ground state of neon with an
applied static vector potential in velocity gauge with truncated space and
full CIS space. Perturbation theory results are included as guiding lines.

In Figure 2, we plot half static energy shifts, $E_{1}^{\mathrm{(stat.)}}/2$,
as a function the ponderomotive energy of an electron,
$U_{p}=\frac{q^{2}A_{0}^{2}}{4m}$, in an oscillating vector potential with the
corresponding amplitude, $A^{(\mathrm{osci.})}(t)=A_{0}\cos(\omega t)$. The
half static energy shifts of the HF ground state follow the time-averaged
energy shifts, predicted by Eq. (9) with $N$ substituted by
$\tilde{N}_{\mathrm{CIS}}$ from Table. 1, as shown by solid lines,
$-\tilde{N}_{\mathrm{CIS}}U_{p}$. There is a factor of two difference between
the static energy shifts and those predicted by Eq. (7) for monochromatic
pulses because energy conservation is respected only by acting once with
$V^{(-)}$ and once with $V^{(+)}$, in any time order, but not with twice the
action of the same $V^{(\pm)}$. In the limit $\omega\to 0$, however,
conservation of energy is fulfilled for all four cases, yielding a factor of
two compared to the time-averaged energy shift,

$\Delta E_{1}^{(\mathrm{stat.})}\approx
2\left<\Delta\tilde{E}_{1}^{(2)}(\omega\rightarrow 0)\right>.$ (18)

At high ponderomotive energies, higher-order interactions with the static
field lead to a slight disagreement with the perturbative results,
$-\tilde{N}_{\mathrm{CIS}}U_{p}$, as can be observed in Fig. 2. Energy shifts
equal to $-N_{\mathrm{A}}U_{p}$ are shown as dashed lines for reference. The
significant discrepancy between the diagonalized ground state energy shift in
CIS and the exact TRK theory, $-N_{\mathrm{A}}U_{p}$, originates from the
truncations performed in the CI expansion, which leads to a non-local
potential in TDCIS theory.

##  III Method

###  III.1 Effective potential

Effective potentials have found several important applications in atomic,
molecular, and optical physics. The key point is to reduce the complexity of a
problem by replacing some degrees of freedom with a potential that depends on
energy. Effective potentials can be employed to perform self-energy correction
to holes, due to virtual Auger or shake-up processes, or, more generally, to
include virtual many-body interactions into a subspace $\mathcal{P}$ from a
complement space $\mathcal{Q}$ [60, 61]. Similarly, optical potentials are
used in scattering theory to include virtual processes in the target that are
induced by the impinging particle [62]. Effective potentials can also be used
to include laser dressing effects, such as AC Stark shifts into a two-level
system ($\mathcal{P}$), due to virtual interactions with the complement of the
truncated Hilbert space ($\mathcal{Q}$) [63, 64]. In this work, we introduce
an effective potential that corrects for the lack of core polarization effects
in the velocity gauge formulation of TDCIS theory due to virtual coupling to
double excitations in the time domain.

We make the following assumptions for the core of $N-1$ electrons that remain
after a single excitation from the ground state:

1\. Markovian process: The excitations from singles (S) to doubles (D) are
virtual processes that only depend on the instantaneous value of the vector
potential: $A(t)$, with no lasting memory effects in the state of the core
over time.

2\. Non-resonant dynamics: The energy shift of the core is described by non-
resonant dynamics that can be evaluated by TRK theory as:
$-(N-1)q^{2}A^{2}(t)/2m$, at each given time, $t$, by virtual coupling from
$\mathcal{P}=\mathcal{P}_{1}$ to $\mathcal{Q}=\mathcal{P}_{2}$.

3\. Non-local correspondence: The effect of exchange interactions in TDCIS are
adopted into the core energy shift using the substitution:
$N-1\rightarrow\tilde{N}-1$, where $\tilde{N}$ is the effective number of
active electrons in the HF ground state.

4\. Uniform action: The core polarization energy affects all singly exited
states in the same way independent of their individual energy and symmetry.

Using this set of assumptions, we present an effective potential for singles
(S):

$\displaystyle\hat{V}_{\mathrm{TRK}}^{\mathrm{(S)}}(t)={\mathcal{V}}_{\mathrm{TRK}}(t)\hat{P}_{1}=-\frac{q^{2}}{2m}(\tilde{N}-1)A^{2}(t)\hat{P}_{1},$
(19)

where

$\hat{P}_{1}=\sum_{p}^{\mathrm{virt.}}\sum_{a}^{\mathrm{occ.}}|\Phi_{a}^{p}\rangle\langle\Phi_{a}^{p}|,$
(20)

is a $\mathcal{P}_{1}$-projector that acts on the singly excited states. The
effective potential in Eq. (19) could be further improved, by adding the
polarization energy of the core, $-\alpha_{0}^{(N-1)}E^{2}(t)/2$, from Eq.
(12) for the $N-1$ subsystem. We have not done this because that type of
correction requires knowledge of the core system at a level that is beyond the
CIS framework. We stress that such core polarization effects are also missing
in the length gauge formulation of TDCIS [see Fig. 1 (e)]. We mention that it
is straightforward to widen the present concept for generation of effective
potentials that correct for virtual triples from doubles:
$\hat{V}_{\mathrm{TRK}}^{\mathrm{(D)}}(t)$ etc., but since such potentials are
beyond TDCIS they are not considered in the present work.

###  III.2 Equations of motion

We define the CIS-TRK Hamiltonian as

$\displaystyle\hat{H}_{\mathrm{TRK}}$
$\displaystyle\equiv\hat{H}_{\mathrm{CIS}}+\hat{V}^{\mathrm{(S)}}_{\mathrm{TRK}},$
(21)

where $\hat{H}_{\mathrm{CIS}}$ is the original CIS Hamiltonian for velocity
gauge [28, 65]. Due to the closure of the CIS-space,
$\hat{I}_{\mathrm{CIS}}=\hat{P}_{0}+\hat{P}_{1}$, the effective potential can
be moved to instead act exclusively on ground state. This is done by
subtracting a time-dependent term from the CIS-TRK Hamiltonian:

$\displaystyle\hat{H}^{\prime}_{\mathrm{TRK}}\equiv\hat{H}_{\mathrm{TRK}}-{\mathcal{V}}_{\mathrm{TRK}}(t)\hat{I}_{\mathrm{CIS}}$
$\displaystyle=\hat{H}_{\mathrm{CIS}}-{\mathcal{V}}_{\mathrm{TRK}}(t)\hat{P}_{0},$

where $P_{0}=|\Phi_{0}\rangle\langle\Phi_{0}|$. This substitution,
$\hat{H}_{\mathrm{TRK}}\rightarrow\hat{H}_{\mathrm{TRK}}^{\prime}$, can
formally be interpreted as a time-dependent gauge transformation that will
modify the phase of the wavefunctions, but not alter any physical observables
generated from the CIS-TRK theory.

The TDCIS-TRK′ equations of motions are (in atomic units) written as

$\begin{split}i\dot{\alpha}_{0}(t)&=\sqrt{2}A(t)\sum_{ap}\langle
a|\hat{p}_{z}|p\rangle\alpha_{a}^{p}(t)+\frac{\tilde{N}-1}{2}A^{2}(t)\alpha_{0}(t)\\\
i\dot{\alpha}_{a}^{p}(t)&=(\varepsilon_{p}-\varepsilon_{a})\alpha_{a}^{p}(t)+\sum_{bq}[2\langle
bp|r_{12}^{-1}|qa\rangle-\langle bp|r_{12}^{-1}|aq\rangle]\alpha_{b}^{q}(t)\\\
&+A(t)\bigg{(}\sqrt{2}\langle
p|\hat{p}_{z}|a\rangle\alpha_{0}(t)+\sum_{q}\langle
p|\hat{p}_{z}|q\rangle\alpha_{a}^{q}(t)-\sum_{b}\langle
b|\hat{p}|a\rangle\alpha_{b}^{p}(t)\bigg{)},\end{split}$ (22)

which differ only by the addition of the effective potential in the ground
state amplitude equation, when compared with the original TDCIS formulations
[28, 65]. In writing Eq. (22) we have omitted the $\hat{V}_{2}(t)$-operator,
because it will affect all CIS states in the same way within the dipole
approximation. As a result, it is possible to remove it without affecting any
physical observable generated from the theory. We stress that the cost of
implementing the TDCIS-TRK′ theory in Eq. (22) is only one scalar
multiplication per time step in the numerical propagation. In Section IV, we
will show that this seemingly minor correction to the equations of motion have
important implications for the TDCIS theory.

##  IV Results

In order to validate the action of the effective potential in Eq. (19), we
perform truncated TDCIS simulations for laser-assisted photoionization for
neon atoms with realistic pulse parameters corresponding to an XUV pulse and
an IR laser pulse, as defined in Eq. 25. The pulse duration is set to
$\tau=35{}\,\mathrm{fs}$, which is a typical value for XUV pulses generated by
HHG or FEL. The properties of the IR field are chosen to correspond to a
Ti:Sapph. laser system. In general, the photoelectron peak position depends on
the detailed pulse forms of both the IR and XUV fields in laser-assisted
photoionization. This is because the ponderomotive energy,

$U_{p}(t)\equiv U_{p}^{0}f^{2}(t)\approx
q^{2}{\mathcal{A}}_{\omega,0}^{2}(t)/4m,$ (23)

with ${\mathcal{A}}_{\omega,0}(t)=A_{\omega,0}f(t)$, is dominated by the IR
field, while the probability of one-photon ionization by the central XUV
photon, $\Omega$, depends on time through the squared envelope of the XUV
field:

$\dot{P}\sim|{\mathcal{A}}_{\Omega,0}(t)|^{2}=|{A}_{\Omega,0}f(t)|^{2}.$ (24)

In this way, the two fields have clearly defined roles: the IR field controls
the dressing of the continuum, while the XUV field controls the flux of
ejected electrons.

###  IV.1 Energy shift of the photoelectron

Figure 3: Normalized photoionization spectra using velocity gauge TDCIS in
neon with (a–c) the 2p-orbital active and (d–f) the 2s- and 2p-orbitals
active. The solid lines show measured peak position and the dashed lines show
the expected peak position given by $\Delta E_{\mathrm{kin}}=-U_{p}$. The
middle column (b,e) displays velocity gauge TDCIS-TRK with the effective
potential $\hat{V}_{\mathrm{TRK}}^{(\mathrm{S})}$ and the rightmost column
(c,f) displays velocity gauge TDCIS-TRK with the substitution
$\tilde{N}-1\to\tilde{N}$.

In this subsection we consider the case where both the XUV and IR pulses have
flat-top envelopes as defined in Eq. (26). In Figure 3 (a), we show
photoelectron probability distributions for absorption of one XUV photon from
the neon ground state. Truncated TDCIS theory is used with only the $2p$
orbital active. The intensity of the IR field is kept in the range
$I=10^{11}\,\mathrm{W}/\mathrm{cm}^{2}{}$ to
$I=10^{12}\,\mathrm{W}/\mathrm{cm}^{2}{}$, such that its main consequence is
to assist the XUV photoionization process with the formation of sidebands (not
shown), but also to shift the photoelectron structure to lower kinetic
energies due to an increased ponderomotive potential [52]. A classical
estimate for the kinetic energy of the photoelectron is given by:
$\left<E_{\mathrm{\mathrm{kin}}}\right>\approx\hslash\Omega+\varepsilon_{2p}-U_{p}^{0}$
(dashed sky blue line), where $\varepsilon_{2p}<0$ is the energy of the $2p$
orbital. As will be shown, this estimate is valid provided that both IR and
XUV fields are flat-top pulses of the same duration, but with truncated TDCIS
we instead observe a shift of the peak position of roughly
$E_{\mathrm{kin}}\approx\hslash\omega+\varepsilon_{2p}-N_{\mathrm{A}}U_{p}^{0}$
(solid orange line), where $N_{\mathrm{A}}\approx 6$ because only the $2p$
orbital is open. Obviously, this TDCIS shift is much too large when compared
to the classical estimate. In order to understand the convergence properties
of TDCIS, we expand the active space to include both $2s$ and $2p$ orbitals.
This leads to an even stronger effect, as shown in Fig. 3 (d), and thus worse
agreement with the classical estimate. Finally, in the case where all orbitals
are active, we find that the agreement with the classical prediction is
further worsened (not shown). This proves that extending the active core of
TDCIS to full TDCIS does not resolve the issue, but rather confirms that the
major issue with velocity gauge TDCIS is a synthetic AC Stark shift due to
size inconsistency.

In Fig. 3 (b) and (e), we show the results of TDCIS-TRK theory, with the
equation of motion defined in Eq. (22), for the truncated active cores:
$\\{2p\\}$ and $\\{2s,2p\\}$, respectively. It is observed that the
photoelectron peak for single XUV absorption follows the expected classical
prediction closely (dashed line). Furthermore, the two sets of simulations
yield graphically equivalent results, which implies that the addition of the
active $2s$ orbital was not essential to describe the physical process.

In order to study the role of the effective potential further, we make the
substitution: $\tilde{N}-1\rightarrow\tilde{N}$ in Eq. 22, and present the
corresponding photoelectron peaks in Fig. 3 (c) and (f), with active cores,
$\\{2p\\}$ and $\\{2s,2p\\}$, respectively. Quite remarkably, it is found that
the photoelectron peak now remains fixed at the same kinetic energy
independent of the laser intensity. This (unphysical) substitution corresponds
to neglecting that one electron is removed from the atom in the process of
photoionization and it shows that the exact value of effective number of
active electrons, $\tilde{N}$ in Table 1, is crucial to understand the
ponderomotive shift of photoelectrons. In this way we have verified that the
assumption of non-local correspondence between the $(N-1)$-body ionic core and
the $N$-body atom is correct.

###  IV.2 Comparison with length gauge

In this subsection we consider the case where the pulses have more realistic
time-dependent envelopes, specifically given by the truncated Gaussian
envelopes defined in Eq. (27). Photoelectron probability distributions for
one-photon ionization by an XUV field with an assisting laser field are shown
using TDCIS-TRK theory in Figure 4. A classical estimate of the mean kinetic
energy shift of the photoelectron gives: $\left<\Delta
E_{\mathrm{kin}}\right>\approx-\Delta U_{p}^{0}/\sqrt{2}$, provided that both
IR and XUV fields are Gaussian pulses of the same duration. The classical
estimate for the photoelectron shift is marked with an arrow and shows good
agreement with the numerical simulations. As expected from the classical
estimates, the photoelectron peak shifts less when Gaussian pulses, instead of
flat-top pulses, are employed.

Figure 4: One-photon peak of laser-assisted photoionization with a truncated
Gaussian envelope using both TDCIS-TRK velocity gauge and length gauge.
Expected classical energy shift for a Gaussian envelope is shown with an arrow
labeled $\Delta U_{p}/\sqrt{2}$.

The quantum mechanical result shows a slightly smaller shift than the
classical Gaussian estimate. Finally, we have verified that the photoelectron
spectra from length gauge TDCIS simulations (dashed lines) are in excellent
agreement with the corresponding TRK results for the normalized yield. On the
blue side of the high-intensity peak
($I_{\omega}=10^{12}\,\mathrm{W}/\mathrm{cm}^{2}$), we observe an interference
ripple in both gauges. We propose that this phenomenon is the onset of
dynamical interference due to ionization at the rising and the falling sides
of the field, which is discussed in more detail in the following section.

###  IV.3 Laser-assisted dynamical interference

In Ref. [47] two criteria to observe dynamical interference are put forward:
(i) the relative AC Stark shift between the initial and final states need to
be larger than the bandwidth of the pulse and (ii) the ionization rate should
not be so large that the initial state is depleted on the rising side of the
field. In practice this makes the experimental observation of dynamical
interference very challenging, as it must rely on atomic stabilization
mechanisms, due to the inherently small ponderomotive shifts of short-
wavelength radiation [46, 48]. Here, we propose that the usage of two-color
fields, composed of XUV and IR parts, make the study of dynamical interference
experimentally feasible with XUV fields from HHG or FEL sources. The reason
for this is that the flux and ponderomotive shift of photoelectrons are not
constrained by a single field, as previously considered in Refs. [46, 47, 48,
66], but that they can be separately controlled by tuning the IR and XUV
fields independently. In Figure 5 (a) we show that dynamical interference is
observed when the IR intensity is increased from $1\times 10^{11}$ to $5\times
10^{12}$ W/cm2.

Figure 5: Demonstration of laser-assisted dynamical interference numerically
using a single XUV pulse by (a) varying intensity of the IR field, and (b)
mixing flat-top and truncated Gaussian envelopes of the XUV and IR fields.

The laser-assisted dynamical interference pattern can be studied further by
choosing two different envelopes for the XUV and IR pulses, as shown in Fig. 5
(b), with a peak IR intensity of $5\times 10^{12}$ W/cm2. The dramatic change
of the interference pattern is the result of the different ponderomotive
potentials of the IR field and the time-dependent photoionization triggered by
the XUV field. The case with a flat-top XUV and Gaussian IR field, shown in
green in Fig. 5 (b), gives an interference pattern that is rather similar to
the usual dynamical interference phenomenon, shown in Fig. 5 (a), with peaks
on the blue side that decrease monotonically in magnitude and increase in
oscillation rate with kinetic energy. The case with a Gaussian XUV and flat-
top IR field, shown in yellow in Fig. 5 (b), displays a more significant shift
of the main peak, as expected for flat-top IR fields using classical
arguments. The interference fringes on the blue side, however, exhibit a
qualitatively different behaviour with non-monotonic peaks. We interpret this
effect as Ramsey-like fringes from unshifted photoemission by the XUV pulse
before and after the flat-top IR field is present. The observed Ramsey fringes
are $\hbar\Delta\omega_{\mathrm{R}}\approx 60$ meV, given
$2\pi=\Delta\omega_{\mathrm{R}}T$ this implies a time separation of $T\approx
64$ fs. We note that this separation is larger than the pulse duration of the
flat-top IR field ($35$ fs), by almost a factor of two, which we interpret as
a result of the temporal spread of the XUV Gaussian flank distributions (from
$\pm 17.5$ fs to $\pm\infty$) with mean values at $\pm 29$ fs.

##  V Conclusions

In this article, we discussed the velocity-gauge problem of TDCIS theory. It
was shown that the velocity gauge is size inconsistent, with respect to the
number of active core orbitals, due to the lack of virtual transitions to
doubles (D). An effective potential was introduced to account for non-resonant
core polarization by usage of TRK theory. The simulated photoelectrons from
TDCIS-TRK theory were found to agree well with those from length gauge TDCIS,
with a low number of angular momenta being required for numerical convergence
of laser-assisted photoionization. Formally, there is no justification to
claim that gauge invariance is achieved, but the TDCIS-TRK theory seems to
perform at a level comparable to the length gauge TDCIS, with reasonable
energy shifts obtained for photoelectrons. Contrary to previous attempts to
achieve full gauge invariance in TDCIS between length and velocity gauge of
TDCIS [43], the present correction can be implemented with negligible
numerical cost in propagation with a spectral representation of the
wavefunction, which may prove important for efficient implementations of TDCIS
for more complex targets, such as molecules or heavy atoms.

Interestingly, the TDCIS-TRK simulations were essentially converged with the
active core space limited to $\\{2p\\}$ in neon atoms at the considered field
parameters. While this may seem reasonable at first glance, because laser-
assisted photoionization processes from $2p$ are known to be weakly correlated
with $2s$ in the length gauge [14], the present theory was derived from the
velocity gauge, where the convergence of many-body effects in laser fields are
much more subtle [15]. As an example, we showed that convergence of the
effective number of active electrons: $\tilde{N}\rightarrow N$, was only
reached for the full active core within RPAE (Table 1).

In summary, the TDCIS-TRK theory allows for a reduced number of angular
momenta and an efficient frozen space in time-dependent simulations of laser-
assisted photoionization. The theory was used to predict a new type of
dynamical interference phenomenon for atoms, under experimentally viable
conditions, by separately controlling the flux and the ponderomotive shift by
two-color fields. In future works, the TDCIS-TRK theory may prove useful for
studies of other observables, such as attosecond time-delays in
photoionization with strongly-driven resonant transitions, HHG with multiple
active channels, interferometric above-threshold ionization processes and
photoionization dynamics by intense attosecond pulse trains generated by
seeded FELs.

##  Appendix A Numerical specifications

Our calculations of laser-assisted photoionization are performed using the
vector potential

$A(t)=\left[A_{0,\Omega}\sin(\Omega t)+A_{0,\omega}\sin(\omega t)\right]f(t),$
(25)

with $\hslash\omega=1.53067{}\,\mathrm{eV}$ and
$\hslash\Omega=27.211386{}\,\mathrm{eV}$. To maintain the notion of a constant
ponderomotive energy, while alleviating the spectral profile of the electric
field, we use a flat-top pulse with smooth but sufficiently rapid truncation,
given by

$f(t)=\begin{cases}1,\quad&|t|\leq\frac{\tau}{2}\\\
\exp\left[-\tan\left(\pi\frac{|t|-\frac{\tau}{2}}{t_{\mathrm{max}}-\tau}\right)^{2}\right],\quad&\frac{\tau}{2}<|t|\leq\frac{t_{\mathrm{max}}}{2}\\\
0,&\textrm{otherwise.}\end{cases}$ (26)

This envelope is equal to unity for a width of $\tau$ and is smoothly
suppressed between $|\tau|/2$ and $|t_{\mathrm{max}}|/2$. The width of the
pulses is set to $\tau=35{}\,\mathrm{fs}$ and the smooth suppression starts at
$\tau_{\mathrm{off}}=33.88{}\,\mathrm{fs}$.

In our last example, we however make use of more realistic truncated Gaussian
pulses, adapted from Ref. [67]. In this case, the vector potential envelope is
given by

$f(t)=\begin{cases}\exp[-\alpha t^{2}],\quad&|t|\leq t_{\mathrm{off}}\\\
\exp\left[-\alpha\left(t_{\mathrm{f}}+\frac{2}{\pi}(t_{\mathrm{max}}-t_{\mathrm{f}})\tan(\frac{\pi}{2}\frac{|t|-t_{\mathrm{f}}}{t_{\mathrm{max}}-t_{\mathrm{f}}})\right)^{2}\right],\quad&\frac{\tau}{2}<|t|\leq
t_{\mathrm{max}}\\\ 0,&\textrm{otherwise}\end{cases},$ (27)

where $\alpha=2\log(2)/\tau^{2}$ is chosen such that the pulse length,
expressed as full width at half maximum, is specified for the intensity
profile of the pulse. This is valid in the long-wavelength limit. The vector
potential follows a Gaussian profile within the width, specified in terms of
standard deviations of the intensity profile, $\sigma=T/(2\sqrt{2\log{2}})$,
of $|t|\leq t_{\mathrm{off}}=4\sigma$, and is truncated at
$t_{\mathrm{max}}=6\sigma$.

To resolve the photoelectron spectra we use the time-dependent surface flux
[68, 42] (t-SURFF) and infinite-time surface flux [69, 54] (iSURF) methods. A
limiting assumption of t-SURFF is that the total wavefunction be separable
into a bound ionic part and a free electronic part. This assumption breaks due
to the long-range Coulomb potential, which scales with the inverse of the
radial distance in space. However, the photoelectrons are recorded at
$R_{0}=100\,\mathrm{Bohr}$, where the effect of the Coulomb potential is
sufficiently small to not affect our results.

###### Acknowledgements.

JMD acknowledges support from the Swedish Research Council: 2018-03845, the
Olle Engkvist Foundation: 194-0734 and the Knut and Alice Wallenberg
Foundation: 2017.0104 and 2019.0154. We thank Andrea Idini for fruitful
discussions about many-body treatment, and Per Eng-Johnsson and Anne
L’Huillier for fruitful discussions about laser-assisted dynamical
interference.

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