Dynamic interference as signature of atomic stabilization

Wei-Chao Jiang; Joachim Burgdörfer

doi:10.1364/OE.26.019921

1. Introduction

The photoelectric effect, i.e. the emission of a bound electron following the absorption of single quantum of light, the photon, is one of the most important and fundamental process of light-matter interaction. As explained by Einstein in 1905 [1], if the energy of photon ω is larger than the ionization potential E_I, the photoelectron departs with kinetic energy E = ω − E_I (atomic units are used unless explicitly stated), and the emission rate is proportional to the intensity I of the electromagnetic field. These two key features are nowadays recognized as signatures of the first-order perturbative quantum effects in the interaction of matter with the electromagnetic field.

With the advent of novel intense light sources in the XUV and X ray frequency region [2–11], the door is open to study novel phenomena beyond the perturbative limit. With the increase of the peak intensity, highly nonlinear interactions between X-ray laser pulse and matter are expected [11–14]. Depending on the frequency, intensities up to 10²⁰ W/cm² are within reach. One of the more remarkable and counter-intuitive consequences of strong-field interaction at high frequencies exceeding the ionization energy E_I is the effect of atomic stabilization, theoretically intensely investigated for almost 30 years [15–22]. In the stabilization region, the ionization rate no longer linearly increases with intensity as predicted by first-order perturbation theory, but is strongly suppressed resulting in the survival rather than ionization of atomic bound states. Observation of this stabilization turns out to be a considerable challenge requiring suitable pulse shaping with carefully tailored envelopes so that ionization remains suppressed during the rising (falling) edge of the pulse before (after) the pulse intensity has reached the stabilization region. Direct experimental evidence of the atomic stabilization of ground state atoms appears to be still missing. However, atomic stabilization of Rydberg states has been reported [23–26].

For moderately intense XUV pulses, the kinetic energy relation E = ω − E_I of the photon effects will be modified by the AC Stark effect [27–30]. As a result, the single peak in the spectrum of the photon electron with peak width 1/T (T : pulse duration) has been theoretically predicted to be shifted or replaced by a multi-peak structure [31–37]. This modulation of the spectrum has been termed dynamic interference [34–37] and explained by a simple temporal two-path interference scenario (Fig. 1): the electron wave packets ejected on the rising edge of the pulse can interfere with the electron wave packets ejected on the falling edge of the pulse, reaching the same final energy when the instantaneous ac Stark shift of the initial state E₀(t) coincide at two different times. A similar temporal interference pattern has been earlier predicted [38, 39] and observed [40–42] in the low-frequency multi-photon region. Distinct effects originating from electron emission on the rising and falling flank of the pulse have been also observed in molecular high-order harmonic generation [43] and molecular holography [44]. Within a perturbative treatment [37], a dynamic interference pattern in the one-photon ionization spectrum of ground-state atom at moderately high frequency (2E_I ≤ ω ≤ 20E_I) has recently been found to be absent. Earlier numerical data showing modulations appears to be an artifact of non-convergent calculations in the length gauge. In the present contribution we explore dynamic interference in a different, i.e. strongly non-perturbative region. Surprisingly, going beyond linear response we find dynamic interference for ground state atoms to be prominently present and closely intertwined with the onset of atomic stabilization. In fact, our results show that dynamic interference can only be observed when stabilization becomes operational. Peak modulations of the one-photon ionization spectrum may thus become a hallmark for the onset of atomic stabilization.

Fig. 1 Two-path interference in one-photon ionization of an atomic ground state. In the nonlinear regime the ground state, strongly Stark shifted by a significant fraction of the binding energy E_I, opens up a temporal double slit for one-photon ionization by pulse with center frequency ω to the same final energy E (Interferings paths marked by dash-dotted lines).

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2. Theoretical methods

Our theoretical description of dynamic interference and atomic stabilization is gauged by comparing with fully convergent numerical solutions of the time-dependent Schrödinger equation (TDSE) in the dipole approximation for ground states of hydrogen and helium, the latter in single-active electron (SAE) approximation. Details can be found in our previous publication [45]. In brief, we expand the wave function of the electron in terms of spherical harmonics. The radial wave function is discretized by the finite-element discrete variable representation (FEDVR) method [46–48]. The recently developed split-Lanczos algorithm is used to propagate the wave function in time. Both the reduced-velocity gauge [H_I(t) = −iA(t) · ∇, where A(t) is the vector potential] and the length gauge [H_I(t) = r · F(t), where F(t) is the electric field] have been used to describe the interaction potential of the electron in the laser field to check for gauge independence. In the velocity gauge, the wave-splitting method [49] has been employed to avoid the use of very large configuration space boxes. Typically, a box with linear dimension of 4800 a.u. is taken in the length gauge while a box of 320 a.u. is taken in the velocity gauge in the present calculations.

Fig. 2 Photoelectron spectra for ionization of ground state of hydrogen in the perturbative regime. The pulse intensity is 5 × 10¹⁶ W/cm², the pulse length T = 2.164 fs, and the photon energy is 53.6 eV. The prediction from the direct amplitude a^D [see Eq. (4) and the text] is given as green dashed line. The TDSE results for both velocity gauge and length gauge with different number of partial waves l = 1 and l = 200 included are shown. The peak position of photoelectron energy predicted by the Einstein formula E = ω − E_I is marked as blue vertical dashed line.

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In our calculations, we use a Gaussian envelope for the vector potential

A (t) = A_{0} g (t) sin (ω t),

with envelope

g (t) = exp (- t^{2} / T^{2}) .

The full width at half maximum (FWHM) T_FWHM of the pulse is related to the pulse length T by

T_{FWHM} = \sqrt{2 ln 2} T

. In the moderately high-frequency region 2E_I ≤ ω ≤ 10E_I, ultrashort pulses of length T ∼ 2fs subtend of the order of ∼100 optical cycles.

Our analysis of the numerical result starts from the exact expression for the probability amplitude a_E (t) for populating the continuum state with energy E,

a_{E} (t) = - i e^{- i E t} \sum_{j} \int p_{E j} \cdot \int_{- \infty}^{t} A (τ) a_{j} (τ) e^{i E τ} d τ,

where a_j(t) is the time-dependent amplitude of the eigenstate |Ψ_j〉 of the field-free Hamiltonian, and p_Ej represents the transition matrix element, i.e., p_Ej = 〈Ψ_E| − i∇|Ψ_j〉. Equation (3) can be derived from the TDSE without any approximation and is equivalent to the post-form of the time-dependent S-matrix element [50]. For frequencies ω ≤ 300 eV non-dipole and relativistic corrections are still negligible [15, 51]. From Eq. (3), a hierarchy of different approximations can be derived. Assuming that the direct (D) transition from the initial state (j = 0) to the final continuum state dominates while neglecting the transitions from all other channels, then Eq. (3) is reduced to

a^{(D)} (t) = - i e^{- i E t} p_{E 0} \cdot \int_{- \infty}^{t} A (τ) a_{0} (τ) e^{i E τ} d τ,

with a₀(τ) the time-dependent projection amplitude of the exact TDSE solution, 〈Ψ₀|Ψ(t)〉. The accuracy of the direct single-channel approximation [Eq. (4)] can be verified by the comparison with the full TDSE solution for the spectrum (Fig. 2). In the perturbative regime |a^(D)|² agrees with both the TDSE result in length and velocity gauge to within the graphical accuracy. Convergence in length gauge requires, however, the inclusion of a much larger number of partial waves than in velocity gauge. An insufficient number of partial waves gives rise to modulation of the spectrum resembling dynamic interference (Fig. 2), which is, however, an artifact of the angular momentum cut-off.

The exact amplitude a₀(τ) can be parameterized in terms of its modulus f(t) and phase ϕ(t)

a_{0} (t) = f (t) e^{- i ϕ (t)},

where both f(t) and ϕ(t) are real functions of t. Numerical results for f(t) and ϕ(t) can be compared with frequently employed models entering various approximations. In first-order perturbation theory, a₀(τ) = exp(−iE₀τ), i.e. f(t) = 1 and ϕ(t) = −E₀t. Inclusion of ac Stark shift and depletion within second-order perturbation theory yields the linear-response (lr) expressions

f^{lr} (t) = exp (- \frac{1}{2} γ U_{p} \int_{\infty}^{t} g^{2} (τ) d τ)

ϕ^{lr} (t) = δ U_{p} \int_{- \infty}^{t} g^{2} (τ) d τ - E_{I} t,

where

U_{p} = A_{0}^{2} / 4

is the ponderomotive energy, and both δ and γ are time-independent but frequency-dependent parameters characterising the strength of the Stark shift and decay. Previous investigations of dynamic interference [34,37] employed perturbation theory [Eqs. (6) and (7)]. Dynamic interference was found to be absent for ground-state atoms. For excited states, spectral modulations appear near frequencies where the dynamical susceptibility undergoes a change of sign due to resonant dipole coupling to lower-lying states [37]. These interference effects appearing for excited states and in the perturbative regime (I > 5 × 10¹⁴ W/cm²) are to be distinguished from the dynamic interference for ground-state atoms in the atomic stabilization regime (I > 10¹⁸ W/cm²) discussed in the following.

The temporal two-path interference between emission from the rising and falling edge of the pulse follows from the decomposition of the amplitude as

a^{D} (\infty) = a_{rising} + a_{falling},

with

a_{rising} = - i e^{- i E t} p_{E 0} \cdot \int_{- \infty}^{0} A (τ) a_{0} (τ) e^{i E τ} d τ

and

a_{falling} = - i e^{- i E t} p_{E 0} \cdot \int_{0}^{\infty} A (τ) a_{0} (τ) e^{i E τ} d τ .

Evaluation in stationary-phase approximation (SPA),

\frac{d Φ (t_{i})}{d t} = 0

, with

Φ (t) = (E - ω) t - ϕ (t)

yields partial amplitudes for the wavepackets emission during the rising and falling edge with moduli

{| a_{E} (t_{i}) |}^{SPA} = | \frac{A_{0}}{2} p_{E 0} | f (t_{i}) g (t_{i}) \sqrt{\frac{2 π}{| {\dot{E}}_{0} (t_{i}) |}}

and relative phase

Δ Φ^{SPA} = \int_{t_{1} (E)}^{t_{2} (E)} [E_{0} (t) - E_{0} (t_{1})] d t + \frac{π}{2},

where

E_{0} (t) = \frac{ϕ (t)}{d t}

is Stark-shifted energy of the initial state, and t_i are the points of stationary phase given as the solutions of E =ω + E₀(t). We note that the phase contribution π/2 due to the caustics [

\frac{d}{d t} E_{0} (t) < 0

] has been omitted in earlier treatments [31–33] but is important to account for the correct peak position.

3. Dynamic interference in the high-intensity regime

The strong-field regime for single-photon ionization (ω > E_I) is reached for field strengths so large that the quiver amplitude exceeds the atomic dimension,

α_{0} = F_{0} / ω^{2} > 1 a . u . .

For the ground state of atomic hydrogen or helium this corresponds to intensities of

I \geq I_{c} = \frac{c ω^{4}}{8 π}

of the order of 10¹⁷ W/cm² (e.g., for one-photon ionization with ω ≈ 50 eV). In this domain, new phenomena appear (Fig. 3) that are absent in the perturbative linear-response regime (Fig. 2). The exact photoionization spectrum P(E) develops interference oscillations [Fig. 3(a)] which can be quite accurately accounted for by the direct single-channel model [Eq. (4)]. They can thereby be unambiguously identified as temporal interference between the emission on the rising and the falling edges of the pulse [Eqs. (8–10)]. The temporal evolution of the modulus of the exact initial state amplitude |a₀(t)| [Eq. (5)] displays a rapid decrease on the rising flank of the pulse [Fig. 3(b)]. However, during the center part of the pulse, the amplitude stabilizes before further decaying on the falling flank. This is a clear signature for the onset of atomic stabilization, which can be quantitatively characterized by its logarithmic derivative, the effective time-dependent ionization rate

Γ (t) = - 2 \frac{d}{d t} ln f (t)

. Γ(t) features two maxima at the rising and falling edges of the pulse. While the pulse is near its maxima strength Γ(t) even becomes slightly negative, indicative of population transfer back to the initial state. It is the temporal confinement of ionization to the rising and falling edges and the stabilization in between which is at the core of the appearance of the dynamic interference oscillations in the spectrum for the atomic ground state as identified here for the first time. By contrast, linear response [Eqs. (6) and (7)] predicts a rapid, irreversible and complete depletion as the pulse is ramped up (Fig. 3) thereby preventing temporal interference.

Fig. 3 Ionization of ground state 1s of hydrogen at high intensity 1 × 10¹⁹ W/cm². (a), ( oe-26-16-19921-i001 ) photoelectron spectra from TDSE; ( oe-26-16-19921-i002 ) direct single channel approximation [Eq. (4)]; ( oe-26-16-19921-i003 ) stationary phase approximation, Eqs. (12) and (13). (b), the time-dependence of the modulus f(t) = |a₀(t)|. the ionization rate Γ(t) and the pulse envelope g(t). f(t) is shown at three levels of approximations: ( oe-26-16-19921-i004 ) exact TDSE with sub-cycle oscillations; ( oe-26-16-19921-i005 ) smoothed TDSE applying a mean filter; and ( oe-26-16-19921-i006 ) linear response f^lr [Eq. (6)]. Vertical bar indicates the t_c at which γ(t) reaches its maxima and f^lr(t_c) approximates the value of f(t) = f₀ in the stabilization plateau reasonably well.

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The time double slit originating from ionization bursts during the emission peaks at the rising and falling flank can analytically be treated by the stationary phase approximation [Eq. (12)]. The interference predicted by the SPA agrees qualitatively quite well with the direct single-channel approximation [Eq. (4)] and the exact TDSE calculation. The interference phase ΔΦ [Eq. (13)] governed by the non-perturbative Stark shift of the initial state, $E_{0} (t) = \frac{d}{d t} ϕ (t)$ , displays fast oscillations on the attosecond scale with twice the frequency of the XUV field, a slower modulation with period of about 35 a.u. (∼ 0.7fs) when the Stark shift is of the order of the ionization potential, and finally a much slower variation that follows the envelope of the pulse (Fig. 4). Only the slowest modulation is relevant in the present context, as the ionization dynamics adiabatically follows the cycle-averaged Stark shift. Accordingly, the time integral over time dependent energy shift [Eq. (4)] can be replaced by that of the cycle-averaged shift determined by a mean filter without any significant error. The phase is accumulated between the two points in time t₁ and t₂ for which the same final energy E can be reached by a one-photon ionization and accurately predicts the periodic modulation of the spectrum P(E).

Fig. 4 The time-dependent energy E₀(t) of the ground state in the high-frequency strong field pulse (ω = 53.6 eV, I = 10¹⁹ W/cm², T = 2.164 fs). The data from TDSE are determined through the phase variation dϕ/dt of the ground state amplitude a₀(t) [Eq. (5)]. Full TDSE ( oe-26-16-19921-i007 ) displays attosecond sub-cycle oscillations; smoothed TDSE after application of a mean filter ( oe-26-16-19921-i008 ) traces the envelope of the pulse. Only for weak intensities in the flanks of the pulse, the second-order perturbation theory for the AC Stark shift linear response[Eq. (7)] agrees with the smoothed TDSE results for E₀(t).

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4. Criteria for dynamic interference in the stabilization regime

Approximate criteria for the appearance of dynamic interference associated with atomic stabilization can be derived from the comparison between the exact numerical TDSE solution and analytic perturbative results [Eqs. (6) and (7)]. The onset of stabilization at t = t_c coincides with the peak in Γ(t_c), beyond which the ionization rate rapidly decreases. Up to this point in time, t_c, the modulus of a₀ is still reasonably well approximated by the perturbative linear-response expression, a₀ ≈ f^lr [Eq. (6)] (see Fig. 3). Expressing the ionization rate γ in terms of the first-order ionization cross section σ, $γ = \frac{σ I}{ω U_{p}}$ where I is the maximum intensity and $U_{p} = F_{0}^{2} / 4 ω^{2}$ is the ponderomotive energy, f(t_c) can be rewritten for the Gaussian pulse [Eq. (2)] as

f^{lr} (t_{c}) = exp {- \sqrt{\frac{π}{32}} \frac{I T σ}{ω} [1 + erf (\frac{\sqrt{2} t_{c}}{T})]}

Expressing t_c in terms of the instantaneous intensity I_c at t_c,

I_{c} = I exp (- \frac{2 t_{c}^{2}}{T^{2}}),

the modulus f(t_c) reads

f^{lr} (t_{c}) = exp {- \sqrt{\frac{π}{32}} \frac{I T σ}{ω} [1 + erf (- \sqrt{ln \frac{I}{I_{c}}})]},

Expansion of the error function [52] gives to linear order in I_c/I,

f^{lr} (t_{c}) \approx exp [- \frac{I_{c} T σ}{2 ω} (0.2037 + 0.1632 \frac{I_{c}}{I})] .

In order to observe dynamic interference the modulus at the stabilization plateau must be sufficiently large such that the second ionization burst on the falling edge is still strong enough to effectively create a second arm of the interferometer. Requiring a₀ ≈ f^lr(t_c) > f₀ (e.g. f₀ ≈ 0.2), the condition for dynamic interference reads

\frac{I_{c} T σ}{2 ω} (0.2037 + 0.1632 \frac{I_{c}}{T}) < ln \frac{1}{f_{0}} .

For a given atomic ground state with its cross section σ and given XUV frequency, Eq. (20) provides an upper bound on the pulse length. Its maximum value T_max is reached for very high intensity (I_c/I ≪ 1) as

T_{max} = \frac{10 ω ln \frac{1}{f_{0}}}{I_{c} σ}

For sufficiently short pulses T ≤ T_max, the Fourier width Δω ≈ 2π/T ≫ 2π/T_max becomes sizable. In addition to Eq. (20), the appearance of interference modulation requires that the phase difference between the two phases, ΔΦ^SPA(E) [Eq. (13)], exceeds 2π for energies E within the spectral width [E_min, E_max] of the photo-ionization spectrum. The latter is governed by both the Fourier width of the ultrashort pulse Δω and the non-linear strong-field Stark shift of the ground state ΔE_Stark = E₀(0) − E₀(−∞) which can be of the order of the ionization potential (see Fig. 4). Accordingly, we find

\begin{array}{l} Δ Φ (E_{max}) - Δ Φ (E_{min}) \\ = (E_{0} (0) - E_{0} (- \infty) + Δ ω) (t_{2} - t_{1}) \\ = (Δ E_{Stark} + \frac{2 π}{T}) (t_{2} - t_{1}) ≫ 2 π, \end{array}

where t₁ ≃ t_c and t₂ ≃ −t_c are the peak positions of the ionization rate. For example, for the photoelectron spectrum shown in Fig. 3(a), E_max ≈ 47 eV and E_max ≈ 39 eV or E_max − E_min ≈ 0.3 a.u., and t₂ − t₁ ≈ 200 a.u.. Consequently, (ΔΦ(E_max) − ΔΦ(E_min)) /2π ≈ 9 allows for up to 9 oscillations consistent with the direct single channel result of Fig. 3(a).

The intensity threshold above which stabilization becomes operational follows from the level of the stabilization plateau f(t_c) ≈ f₀. The inequality (20) also provides an lower bound for the peak intensity I_min (I_min > I_c),

I_{min} = \frac{0.8 I_{c} T}{T_{max} - T}

with T < T_max. For ω = 50 eV and T = 2 fs, I_min is estimated to be 1 × 10¹⁸ W/cm². This value is consistent with the TDSE results shown in Fig. 5, where dynamic interference is observed in both hydrogen and helium for intensities above 1 × 10¹⁸ W/cm² and pulse duration below ≈ 2 fs. Eqs. (21)–(23) provide simple and intuitive criteria for the appearance of dynamic interference in the stabilization regime.

Fig. 5 Photoelectron spectra for the ground state of hydrogen (a, b) and helium (c, d) and pulses with photon energies (a, b) ω = 53.6 eV and (c, d) 54.6 eV. In (a, c) the photoelectron distributions in the (E, I) plane is displayed; (b, d) show lineouts for different I and T. The corresponding peak intensities and pulse lengths are labeled in each panel. The position of the photoelectric peak E = ω − E_I is marked in each frame by a purple line.

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5. Conclusions

We have presented numerically accurate solution of the TDSE for one-photon ionization from atomic ground states in the regime of high-intensity pulses (I ≈ 10¹⁶ ∼ 10²⁰) W/cm² and moderately high frequency above the ionization potential. Our results confirm recent observations [37] that in the perturbation regime where the atomic ground state of the dynamic Stark shift and XUV pulse induced depletion in the XUV pulse is still well approximated by second-order perturbation theory dynamic interferences are absent. Previously reported indications in terms of spectral modulation of the one-photon emission spectrum are most likely an artefact of non-convergent calculations in the length gauge. At higher intensities, however, we have identified a novel scenario beyond linear response: dynamic interference in the one-photon emission spectrum appears prominently for intensities in the atomic stabilization regime. In fact, our analytical estimations, confirmed by simulations, indicate that dynamic interference and atomic stabilization are closely intertwined. It is the enhanced survival by stabilization during the high-intensity plateau of the temporal envelope of the pulse that enables the presence of a second ionization burst of sufficient intensity on the falling edge of the pulse thereby opening up a second path for interference in the one-photon spectrum. This effect appears when standard linear response or second-order perturbation theory fails. Our results therefore suggest as new route to detection of the atomic stabilization so far elusive for ground-state atoms: spectral modulations of the one-photon emission spectrum whose frequencies depend on both the pulse duration and the intensity may serve as an unambiguous indication for the onset of stabilization. For the photon energies considered in the present work, XUV intensities of 10¹⁶ W/cm² have been achieved already a decade ago [11]. With upcoming X-ray FEL facilities [7–9] and high-harmonic based facilities, in particular within the extreme light infrastructure (ELI) [53,54], intensities considered in the present work should come within reach.

Funding

National Natural Science Foundation of China (No. 11747013); FWF-SFB049(Nextlite) and FWF-SFB041(VICOM) funded by the Austrian Science Fund (FWF); Natural Science Foundation of SZU (grant no.2017066)

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