#### 2.1. Time-Dependent Perturbation Theory

In this subsection, let us do a simple analysis on how the relative importance of the resonant and nonresonant paths depends on pulse width, based on the second-order time-dependent perturbation theory within the common rotating wave approximation. The dynamic Stark effect is negligible for pulse parameters used in the present study. We consider the process where a laser pulse with a central frequency

ω and a pulse envelope

f(

t), linearly polarized in the

z direction, promotes an atomic electron from an initial state |

i〉 to a final continuum state |

f〉 through two-photon absorption. The complex amplitude

c_{f} of the final state after the pulse in the interaction picture can be written as,

where

μ_{mn} denotes the dipole transition matrix element between states

m and

n, ∆

_{m} =

ω_{m} − (

ω_{i} +

ω), ∆

_{f} =

ω_{f} − (

ω_{m} +

ω) with

ω_{m} being the energy eigen-value of state

m, and the sum runs over all the intermediate bound and continuum states

m. One can equivalently express

Equation (1) by

Equation (3) of [

45] using the Fourier transform of

f(

t). Although a rectangular pulse is often assumed in previous work [

2] and textbooks, we take, as a more realistic choice, a Gaussian profile

$f(t)={E}_{0}{e}^{-{t}^{2}/2{T}^{2}}$, with

E_{0} and

T being the field amplitude and the pulse width, respectively. More precisely,

T is related to the full-width-at-half-maximum (FWHM) pulse width

T_{1/2} as

${T}_{1/2}=2\sqrt{\mathrm{ln}2}T$.

For the case of

ω_{f} =

ω_{i} + 2

ω, in particular, one can perform the integrals in

Equation (1) analytically to obtain a physically transparent expression:

where

$F(x)={e}^{-{x}^{2}}{\displaystyle {\int}_{0}^{x}{e}^{{t}^{2}}}dt$ denotes Dawson’s integral [

60], which tends to

x near the origin and 1/2

x for

x →

∞. Only resonant states within the spectral width of the pulse contribute to the first term, corresponding to the resonant path. In particular, if a single intermediate level

r is resonant with the pulse spectrum,

Equation (2) can be rewritten as,

On the other hand, the asymptotic behavior of

F (

x) suggests that all the intermediate states except for the exact resonance (∆

_{m} = 0) participate in the second term, as expected for nonresonant paths. While either term dominates for a relatively long pulse (ps and ns), assuming that resonant excitation is not saturated, we can expect that the two terms are comparative for sufficiently short pulses and that their relative importance, which may be expressed as arg

c_{f}, varies with

T. In such a situation, the amplitude ratio

c_{S}/

c_{D} between the final

S and

D continuum states is complex, since the branching ratio

${\mu}_{Sm}/{\mu}_{Dm}$ of the transitions from the intermediate

P states

m to each state depends on

m. While the actual outgoing wave packets involve the contribution from the final states with

ω_{f} ≠

ω_{i} + 2

ω, it is instructive to write arg

c_{S}/

c_{D} using

Equation (3) as follows:

with

${a}_{f}(f=S,D)={({\mu}_{fr}{\mu}_{ri})}^{-1}{\displaystyle {\sum}_{m(\ne r)}{\mu}_{fm}{\mu}_{mi}/{\mathrm{\Delta}}_{m}}$. Here we have assumed that

F (∆

_{m}T) ≈ (2∆

_{m}T)

^{−1} for all the other intermediate states. Hence, the competition between the resonant and nonresonant paths affects the interference between the outgoing

S and

D wave packets and manifests itself in the photoelectron angular distribution that depends on the pulse width.

#### 2.2. Photoelectron Angular Distribution

The photoelectron angular distribution from 2PI of H and He is given by [

12],

where

σ is the total cross section,

θ is the angle between the laser polarization and the electron velocity vector, and

β_{2} and

β_{4} are the anisotropy parameters associated with the second- and fourth-order Legendre polynomials, respectively. The interference of the

S and

D wave packets produces a photoelectron angular distribution,

where

${\tilde{c}}_{S}$ and

${\tilde{c}}_{D}$ are the real amplitudes that have absolute values of |

c_{S}| and |

c_{D}|, respectively, and can be either positive or negative in principle, and

δ_{l} is the phase of the partial wave, or the

apparent phase shift. The apparent phase shift difference,

consists of a part

δ_{sc} intrinsic to the continuum eigen wave-functions (scattering phase shift difference), which has previously been studied both theoretically [

61–

63] and experimentally [

29], and the extra contribution

δ_{ex} = arg

c_{S}/

c_{D} (if

${\tilde{c}}_{S}/{\tilde{c}}_{D}>0$),

π − arg

c_{S}/

c_{D} (if

${\tilde{c}}_{S}/{\tilde{c}}_{D}<0$) from the competition of the two paths. This situation presents a contrast to the case of the photo-ionization from photo-excited states [

29], where the nonresonant path is absent and only

δ_{sc} is present (

δ =

δ_{sc}).

The amplitude ratio

$W\equiv {\tilde{c}}_{S}/{\tilde{c}}_{D}$ and

δ are related with the anisotropy parameters as [

64],

It should be noticed that, when one extracts

W and

δ from

β_{2} and

β_{4}, one cannot distinguish between (

W,

δ) and (−

W,

π −

δ). One cannot distinguish between

δ and −

δ, either. Hence, in what follows, let us take

W as positive,

i.e.,

W = |

c_{S}/

c_{D}| and define the value of

δ within the range [0

, π]. Then,

δ_{sc} in

Equation (7) should be replaced by |

δ_{sc}| or

π − |

δ_{sc}| for appropriate interpretation in some cases (see the discussion on

Figure 1 below).

#### 2.3. Hydrogen Atom

Let us now verify the above qualitative idea for the case of a hydrogen atom, for which an analytical expression of the Coulomb phase shift

δ_{sc,l} for an azimuthal quantum number

l is at hand as,

where Γ(

z) denotes the gamma function, and

k the wave number.

In principle, one could calculate

c_{S} and

c_{D} using the analytical expression for the dipole transition matrix elements in

Equation (2), but it is not trivial to perform an integration over the continuum intermediate states, since the Dawson’s integral has a long tail (∼ 1/

x). Instead, we take an approach in the time domain, namely, direct numerical solution of the time-dependent Schrödinger equation (TDSE) in the length gauge,

where we have assumed that the field is linearly polarized in the

z direction.

Equation (10) is numerically integrated using the alternating direction implicit (Peaceman–Rachford) method [

9,

65–

73]. Sufficiently long (typically a few times the pulse width) after the pulse has ended, the ionized wave packet moving outward in time is spatially well separated and clearly distinguishable from the non-ionized part remaining around the origin. We calculate

β_{2} and

β_{4} by integrating the ionized part of |Φ(r)|

^{2} over

r and

ϕ, from which one obtains

W and

δ by solving

Equation (8). We have monitored the convergence of the results by calculating the PAD at different times. The calculation has been done for a Gaussian pulse envelope with a peak intensity of 10

^{10} W/cm

^{2}, at which we have confirmed that the interaction is still in the perturbative regime.

The pulse-width dependence of

δ and

W for

ħω = 10.2 eV corresponding to the 2

p resonance is shown in

Figure 2 (a). The calculations have been done at different values of full-width-at-half-maximum (FWHM) pulse width

${T}_{1/2}=2\sqrt{\mathrm{ln}2}T$ between 1 and 21 fs. As expected, both

δ and

W substantially change with pulse width, especially when the pulse is shorter than 10 fs.

δ approaches

$\left|{\delta}_{sc}\right|\equiv |{\delta}_{sc,0}-{\delta}_{sc,2}|=\frac{\pi}{2}$ for a larger pulse width. Accordingly, the PAD also varies as shown in

Figure 3 (a). One finds that the distribution to the direction perpendicular to the laser polarization,

i.e.,

θ ≈ 90°, 270° decreases as the pulse is shortened, which is more prominent for He as we will discuss later [

Figure 3 (b)]. This can be understood as follows: roughly speaking,

δ changes from

$\frac{\pi}{2}$ to ∼

π as

T_{1/2} varies from 21 fs to 1 fs. Thus,

c_{s}/

c_{d} is approximately real and negative in the short-pulse limit, which leads to the cancellation between

Y_{00}(

θ,

φ) and

Y_{20}(

θ,

φ) around

$\theta =\frac{\pi}{2}$.

In

Figure 1, we plot the photon energy dependence of the phase difference

δ and the amplitude ratio

W for 7 fs and 14 fs pulse widths. By noting that the spectral width is 0.26 eV and 0.13 eV for each pulse width, respectively, we can identify several photon energy ranges.

First, if the pulse is not resonant with any level and the first term in

Equation (3) is negligible,

Equation (4) can be approximated as,

which is real. In our plots, we define the value of

δ within the range [0

, π]. Hence, the value of

δ is either |

δ_{sc}| [thin black line in

Figure 1(a)] or

π − |

δ_{sc}| (thin dashed black line), depending on the sign of

c_{S}/

c_{D}. This situation corresponds to

ħω ≲ 9.8 eV and 10.6 eV ≲

ħω ≲ 11.6 eV. The relative phase

δ between the

S and

D partial waves does not depend on the pulse width [

Figure 4(a)], except for the case of very short pulses where the broadened spectrum begins to be resonant with a 2

p or 3

p level, which belong to the second category. We can also see from

Figure 1 that the sign of

c_{S}/

c_{D} changes, e.g., around

ħω = 10.9 eV and

ħω = 11.7 eV, where

W sharply peaks or approaches to zero [

Figure 1(b)]. The jump in

δ is not completely step-function-like, since the contribution from the resonant path can be neglected no longer there.

Second, if the pulse is resonant with a single excited level (near

ħω = 10.2 eV and 12.1 eV for the 2

p and 3

p resonance, respectively), the competition between the resonant and nonresonant ionization paths leads to the value of

δ that deviates from its intrinsic value (|

δ_{sc}| or

π −|

δ_{sc}|) and varies with pulse width, as we have already seen above [

Figure 2(a)] and can also see in

Figure 4(b). It tends to either |

δ_{sc}| or

π − |

δ_{sc}| for the longer pulse width. Interestingly, at

ħω = 11.7 eV [blue line in

Figure 4(b)],

δ first decreases with increasing pulse width, approaching |

δ_{sc}| = 1.40 (

i.e., the resonant path is dominant), but then, for

T_{1/2} ≳ 8 fs, increases again, tending to

π − |

δ_{sc}| = 1.74 (

i.e., the nonresonant paths are dominant). Similar features can be observed for 9.0 and 11.0 eV in

Figure 4(a) as well.

Then, above the 3

p resonance, the pulse is always resonant with one or, more often, multiple levels. In this case, again, the value of

δ deviates from its intrinsic value and varies with pulse width [see the lines for 12.75 and 13.0 eV in

Figures 4(a) and 4(c)], while its photon-energy dependence is somewhat complicated. A new feature is that

δ is roughly constant at

T_{1/2} ≲ 4 fs and 7 fs for 12.75 and 13.0 eV, respectively, which we will discuss below.

Finally, when the pulse excites a Rydberg manifold (

ħω ≳ 13.0 and 13.3 eV for 7 and 14 fs, respectively),

δ varies smoothly with photon energy even across the ionization threshold (13.6 eV), indicating that the Rydberg manifold behaves similarly to the continuum for the case of an ultrashort pulse (note that a similar smooth transition can also be seen in the 2PI photoelectron spectrum [

73]). Moreover, the relative phase is different from its intrinsic value, but surprisingly constant as a function of

T_{1/2} within the pulse width range investigated here [

Figure 4(c),(d)]. These features, seen also for He [

Figure 4(g)] and presumably general for all the atoms, can be understood as follows. Let us assume that the spectral width of the pulse contains a sufficient number of levels that the sum in

Equation (1) can be approximately replaced by an integral,

where

s(∆

_{m}) denotes the density of states multiplied with

${\mu}_{fm}{\mu}_{mi}$. By noting that

F (∆

T) ≈ (2∆

T)

^{−1}, thus,

F (∆

T) ≈

F (∆)/

T is a good approximation in most region of ∆

T,

Equation (12) can be rewritten as,

Since

s(∆/

T) in the first term in the integrand can usually be approximated as

s(0) for the central frequency, the ratio

c_{S}/

c_{D}, and thus

δ and

W, becomes roughly independent of

T. Also, due to the continuity of the oscillator strength distribution across the ionization threshold [

74], one can see that

c_{S}/

c_{D} also changes smoothly as a function of

ħω. It should be noticed that, for the case of two-photon above-threshold ionization [

37] and in the long-pulse limit, the first (resonant path) and second (nonresonant path) terms of

Equation (12) tend to, e.g., the second and first terms of Equation (3.2) of [

37], respectively.

How dense should the levels be within the spectral width for the above discussion to be valid? One can estimate it by investigating how well the integral

$\int}_{-\infty}^{\infty}{e}^{-{x}^{2}$ can be approximated by the sum

${\sum}_{n=-\infty}^{\infty}{e}^{-{x}_{n}^{2}}\mathrm{\Delta}x$ of the values at discrete points, where

x_{n} = (

n +

δn)∆

x, with ∆

x(> 0) and

δn being the increment of

x and the offset of

n, respectively. In

Figure 5 we plot the ratio,

as a function of ∆

x for

$\delta n=0,\frac{1}{4}$, and

$\frac{1}{2}$, where

θ_{a}(

u, q) denotes the theta function [

60]. The approximation is surprisingly good even with ∆

x = 1, though the FWHM of

${e}^{-{x}^{2}}$ is 2 ln 2 (= 1.66511 ⋯). This explains why ∆ varies smoothly even in the range of

ħω where the level spacing is not negligibly small compared with the spectral width in

Figure 1.

#### 2.4. Helium Atom

Let us now turn to a helium atom. As we have emphasized in Section 1, He is much more relevant to experiments than H, since it is much easier to handle experimentally, and its excitation energies lie well within the typical wavelength range of HHG and EUV FEL sources.

We use direct numerical solution of the full-dimensional two-electron TDSE in the length gauge [

54]:

with the atomic and interaction Hamiltonian,

We solve

Equation (15) using the time-dependent close-coupling method [

50–

54]. The numerically obtained excitation energies for the 1

s2

p ^{1}P and 1

s3

p ^{1}P states are 21.220 and 23.086 eV, respectively, in fair agreement with the experimental values (21.218 and 23.087 eV [

48], respectively). Similarly to the case of a hydrogen atom, sufficiently long after the pulse has ended, we calculate

β_{2} and

β_{4} by integrating the ionized part of |Φ(r

_{1}, r

_{2})|

^{2} over

r_{1}, r_{2}, θ_{2},

ϕ_{1}, ϕ_{2}, from which one obtains

W and

δ by solving

Equation (8). We use the values of

δ_{sc} from [

63] to calculate

δ_{ex} =

δ −

δ_{sc}. The calculation has been done for a Gaussian pulse envelope with a peak intensity of 10

^{11} W/cm

^{2}, at which we have confirmed that the interaction is in the perturbative regime. Our preliminary investigation indicates that the interaction begins to deviate slightly from this regime around 10

^{12} W/cm

^{2} if we increase the intensity, but the correction is still small below 10

^{13} W/cm

^{2}, which is the typical focal intensity of SCSS [

40].

The pulse-width dependence of

δ and

W for

ħω = 21.2 eV close to the 1

s2

p resonance (21.218 eV) is shown in

Figure 2(b). The calculations have been done at different values of pulse width

T_{1/2} between 500 as and 21 fs. As expected from the discussions in Sections 2.1 and 2.2, and similarly to the case of a hydrogen atom (Subsection 2.3), both

δ and

W substantially change with pulse width, especially when the pulse is shorter than 10 fs. Accordingly, the PAD also varies as shown in

Figure 3(b). One finds that the distribution to the direction perpendicular to the laser polarization again decreases as the pulse is shortened. As stated earlier, strictly speaking,

Equation (4) is applicable only to

ω_{f} =

ω_{i} + 2

ω, and the actual PAD involves integration over

ω_{f}. Nevertheless, the results in

Figure 2(b) can well be described by

Equation (4) [

44], except for

δ in the ultrashort pulse regime

T_{1/2} ≲ 1 fs, where the spectrum becomes broader than the level spacing. If we compare

Figure 2(a) for H and (b) for He, both for the 2

p resonance, the phase-shift difference

δ has a similar pulse-width dependence. As for the amplitude ratio

W = |

c_{s}/

c_{d}|, it is smaller than unity,

i.e., |

c_{d}| > |

c_{s}| regardless of pulse width for H, while its variation is larger for He; |

c_{d}| > |

c_{s}| in the short-pulse limit, and |

c_{d}| < |

c_{s}| in the long-pulse limit. Accordingly, the variation in PAD is more prominent for He than for H (

Figure 3).

It is interesting at this stage to examine some limiting cases. With increasing pulse duration,

δ approaches the scattering phase shift difference

δ_{sc}, and the PAD changes only slowly with

T_{1/2} [

Figure 3(b)]. When the pulse is resonant (∆

_{r}T ≪ 1) and sufficiently long (

T ≫

a_{S}, a_{D}) at the same time, assuming that the resonant excitation is not saturated, one can approximate the extra phase shift as

hence, it is proportional to the spectral width, which can be confirmed in

Figure 3 of [

44].

On the other hand, if we plot

δ as a function of spectral width,

δ tends to an asymptotic value in the wide-spectrum,

i.e., short-pulse limit. Correspondingly, the PAD does not change much with the pulse width for

T_{1/2} ≲ 1 fs [

Figures 3(b) and

6(b),(c)]. This is because the pulse becomes resonant with multiple levels; the spacing between the 1

s2

p and 1

s3

p is 1.9 eV. As we have shown in the previous subsection, when many neighboring states are resonantly excited by the pulse, the extra phase shift difference

δ_{ex} does not much depend on the pulse duration. Similarly to the case of the hydrogen atom, this especially applies when the photon energy lies in the Rydberg manifold, and exceeds the ionization potential (24.59 eV),

i.e., in the case of above-threshold two-photon ionization.

In

Figures 4(e)–(g) and

6 we compare the pulse-width dependence of

δ_{ex} and the PAD, respectively, for different values of

ħω, where we find trends similar to the case of H. While

δ ≈

δ_{sc} (

δ_{ex} ≈ 0) and the PAD are nearly independent of

T_{1/2} for nonresonant pulses (

ħω = 20.3 eV,

T_{1/2} ≳ 3.5 fs), when the pulse is close to resonance with an excited level (

ħω = 21.2, 21.3, and 23.0 eV),

δ_{ex} and PAD rapidly change with

T_{1/2}. On the contrary,

δ_{ex} is finite and nearly constant for

ħω = 24.3, 24.6, and 25.0 eV [

Figure 4(g)]; accordingly, the PAD hardly varies with pulse width. At

T_{1/2} ≲ 1 fs, the spectrum is so broad that

δ_{ex} restarts to change slightly. One also sees that the transition across the ionization potential is smooth [

Figure 4(g)], as has been seen for a hydrogen atom in

Figures 1 and

4(c),(d). It should be pointed out that the extra phase shift difference due to free-free transitions plays a significant role in the recently observed time delay in photoemission by attosecond EUV pulses [

75–

79].

Figure 7 shows the photon energy dependence of the phase difference

δ and the amplitude ratio

W for a pulse width of 7 fs. Although the number of data points is limited, we can see features basically similar to those in

Figure 1 for H.