Finite speed of light effects in ionization of extended systems

I. A. Ivanov1,∗ Anatoli S. Kheifets2, Kyung Taec Kim1,3

1Center for Relativistic Laser Science, Institute for Basic Science, Gwangju 61005, Korea2Research School of Physics, The Australian National University, Canberra ACT 2601, Australia

3Department of Physics and Photon Science, GIST, Gwangju 61005, Korea

(Dated: June 8, 2021)

We study finite speed of light propagation effects in ionization of extended systems. We presenta general quantitative theory of these effects and show under which conditions such effects shouldappear. As a case study, we consider the molecular hydrogen ion H+

2 interacting with a short laserpulse. The finite speed of light propagation effects are encoded in the non-dipole time-dependentShrodinger equation and display themselves in the photoelectron momentum distribution projectedon the molecular axis. Our modeling shows that the finite light propagation time from one atomiccenter to another can be accurately determined in a table top laser experiment similarly to an earliersynchrotron measurement by Grundmann et al [Science 370, 339 (2020)].

PACS numbers: 32.80.Rm 32.80.Fb 42.50.Hz

Every quantum system evolves on its characteristictime scale which varies widely between molecules (fem-toseconds 10−15 s) [1], atoms (attoseconds 10−18 s) [2]and nuclei (zeptoseconds 10−21 s) [3]. A crossover be-tween these time scales is very rare in nature. It wastherefore quite unexpected to discover an ionization pro-cess in the hydrogen molecule that evolved on a zeptosec-ond time scale [4]. An explanation of this phenomenonappeared to be quite simple. While it takes tens of at-toseconds for an electron to trespass the H2 molecule, theincoming light wave sweeps from one molecular end to an-other orders of magnitude faster. This results in one ofthe constituent hydrogen atoms getting ionized a fractionof the attosecond sooner than its counterpart. Such a tinyionization delay manifests itself quite noticeably in thetwo-slit electron interference that the H2 molecule readilydisplays [5]. To discover a zeptosecond delay in molecularphotoionization, Grundmann et al. [4] needed to deployan extremely bright synchrotron source of highly ener-getic photons [6]. We demonstrate that even a table toplaser experiment is capable of detecting a similar effectmaking it much more readily accessible.

In this Letter we present a general quantitative theoryof the delay due to the finite light propagation speed, andwe show under which conditions such effects may man-ifest themselves. In our demonstration, we consider theH+

2 ion, the simplest molecular system that had beenthe favorite theoretical model since the early days ofquantum mechanics [7]. We subject H+

2 to an attosec-ond laser pulse that can be readily produced in high-order harmonics generation sources [8, 9]. The photo-electron flux encodes the timing information about theionization process. This flux is reconstructed by solv-ing the laser-driven time-dependent Schrodinger equa-tion (TDSE). The quantitative results obtained usingthe TDSE can be interpreted in a transparent qualitative

∗Electronic address: igorivanov@ibs.re.kr

way by considering a very simple heuristic tight-bindingmodel (TBM). This gives us a tool for understanding thetime delay caused by the finite speed of light propagation.We apply this tool to H+

2 and find the time delay of afraction of attosecond that depends sensitively on the ori-entation of the molecular axis relative to the propagationand polarization directions. While we limit the presentinvestigation to H+

2 , similar reasoning and modeling canbe applied to H2 and more complicated molecules.Our approach is based on the numerical solution of the

three-dimensional TDSE

i∂Ψ(r, t)/∂t =[

Hmol + Hint(t)]

Ψ(r, t) , (1)

where Hmol is the field-free Hamiltonian describingH+

2 and Hint(t) describes field-molecule interaction. Weconsider a traveling wave propagating in the positive x-direction and polarized in the z-direction, described bythe vector potential A(t−x/c), where c = 137.036 is thespeed of light in atomic unites (a.u.).

The leading order relativistic corrections to Hint(t)come from the leading terms of the expansion of A(t −x/c) in powers of c−1. We apply the procedure we usedin [10] to study non-dipole effects in atomic photoion-ization. We plug A(t − x/c) into the standard minimal

coupling Hamiltonian [11] Hmin = (p+ezA(x, t))2/2 and

keep the terms linear in c−1:

Hint(r, t) = pzA(t)+pzxE(t)

c+A(t)E(t)x

c+A2(t)

2. (2)

In the above expression, A(t) no longer depends on thecoordinates and E(t) = −∂A(t)/∂t is the electric fieldof the pulse. The last term on the r.h.s. of Eq. (2) is afunction of time only and can, therefore, be removed by aunitary transformation. The interaction Hamiltonian (2)can be related by a gauge transformation to the Kramers-Henneberger Hamiltonian used in [12].

An additional source of relativistic corrections in Hint

is the interaction of the magnetic field of the pulse withthe electron spin. These corrections cannot be obtained

2

by a simple generalization of the minimal coupling Hamil-tonian, to obtain them one has to consider systematicallythe transition of the Dirac equation to the non-relativisticlimit [13]. The inclusion of the electron spin, however, isnot necessary for the present study where we keep onlythe c−1 terms as was done, for instance, in [10, 14]. The

Breit-Pauli relativistic corrections to Hmol, such as theeffects of the relativistic kinematics, spin-orbit interac-tion and the so-called Darwin term are all of the order ofc−2 [11] and can also be safely omitted.The H+

2 ion interacts with a short laser pulse

A(t) = −E0ω−1 sin2(πt/T1) sinωt , (3)

of the total duration T1 = 4T with T = 2π/ω and thebase laser frequency ω = 4.04 a.u. (the photon energyof 110 eV). The peak electric field strength E0 = 0.1 aucorresponds to the intensity of 3.5× 1014 W/cm2

To solve the TDSE (1) we follow the procedure thatwe employed previously in the non-relativistic case [15–17]. A solution of the TDSE is sought in the sphericalcoordinates as an expansion over the spherical harmonicsbasis

Ψ(r, t) =

lmax∑

l,m

flm(r, t)Ylm(r) , (4)

The radial grid is discretized with the step δr = 0.05 a.u.in a box of the size Rmax = 400 au which, together withlmax = 10, ensured a numerical convergence .We will characterize the final photoelectron state by

its asymptotic momentum p. Ionization amplitudes apare obtained by projecting the TDSE solution after theend of the pulse on the ingoing scattering states φ−

p[18]

of H+2 .

On the other hand, for a weak electromagnetic field weemploy, the following perturbation theory (PT) expres-sion is valid for the photoionization amplitude:

ap = −i

∫ +∞

−∞

〈φ−p|Hint(t)|φ0〉ei(Ep−ε0)τ dτ , (5)

where φ0 and φ−pare the initial and final molecular states

with the corresponding energies ε0 and Ep.

We split the ionization amplitude (5) ap = a(0)p + a

(1)p

into the nonrelativistic part a(0)p and the first order rel-

ativistic correction a(1)p . By introducing the Fourier

transforms A(t) = (2π)−1∫

a(Ω)e−itΩ dΩ and A2(t) =(2π)−1

∫

b(Ω)e−itΩ dΩ we obtain for these amplitudes:

a(0)p

= −ia(Ω)〈φ−p|pzφ0〉 , Ω = Ep − ε0 (6)

a(1)p

= Ωc−1(

〈φ−p|a(Ω)pzx|φ0〉 + 〈φ−

p|b(Ω)x|φ0〉/2

)

≈ Ωc−1〈φ−p|a(Ω)pzx|φ0〉 . (7)

The second term on the right hand side of Eq. (7)can be neglected because the ratio of the second termand the first term in this expression is approximatelyb(Ω)/(a(Ω)p), where p is the typical value of the mo-mentum of the ionized electron. For the pulse param-eters that we consider, this value can be estimated asE0/(ωp) ≈ 0.01.

It can be seen from Eq. (7) that the relativistic cor-

rection a(1)p can vanish in certain cases. It is zero, for

instance, for an axially symmetric system with the field-free Hamiltonian invariant under rotations about the z-axis. Indeed, for such systems the scattering state φ−

p(r)

in is a function of the following arguments: r,p,z,pz andρ ·pρ, where z, pz and ρ, pρ are the components of the rand p vectors parallel and perpendicular, respectively, tothe z- axis of the coordinate system we employ. Assum-ing that initial state is a non-degenerate ground state ofthe system, so the wavefunction φ0 in Eq. (7) is invariantunder rotations about the z-axis, we obtain from Eq. (7):

a(1)(px, py, pz) = −a(1)(−px, py, pz). Therefore the am-

plitude a(1)p is zero in the plane px = 0, in particular

in the most important region of the momenta near themaximum of the momentum distribution.

The effect of the propagation correction (7) can bemost easily illustrated within the tight-binding model inwhich the ground state of H+

2 is represented by a Heitler-London wave function

φ0(r) = [φ(r −R/2) + φ(r +R/2)]/√2 .

In the TBM, the overlap of the two terms is small andφ(r) is represented by a spherically symmetric s-state.Under these approximations, and using Born approxima-tion for the scattering state φ−

p, which is justified for the

relatively high electron energies we consider, we obtain:

a(0)p

= ad(p)√2 cos (p ·R/2) , (8)

ad(p) = −ia(Ω)〈p|pzφ〉 (9)

a(1)p

= −Ωc−1(∂/∂px)a(0)p

. (10)

The non-relativistic term a(0)p is represented by the single-

center dipole amplitude (9) modulated by the two-centerinterference factor [5]. Adding the relativistic term leadsto

ap = a(0)p

+ a(1)p

≈ ad(p)√2 cos (p ·R/2 + δ) , (11)

where δ = −ΩRx/(2c) and Rx = R · ex. It is this phasefactor, which has been introduced in [4] using a simplepicture of a double-slit interference, that is responsiblefor the modified interference pattern observed in [4].

3

FIG. 1: (Color online) The photoelectron momentum distributions projected on the (px, py)-plane for different orientationsand inter-nuclear distances of H+

2 illustrated in the top row of panels. The middle row of panels displays the numerical TDSEresults while the bottom row exhibits the corresponding TBM results obtained using Eq. (8).

Appearance of the additional phase factor δ in (11) isdue to an extra propagation time of the light wave fromone end of the molecule to the other. This interpretationbecomes yet more transparent if we use coordinate rep-resentation for the wave-function describing ionized wavepacket:

Ψion(r, t) =

∫

apφ−p(r)e−iEpt dp , (12)

where φ−p(r) are the molecular scattering states. To eval-

uate this integral in the limit t → ∞, we rely on thesaddle-point method (SPM) that is commonly used indescription of scattering [21] or ionization [19, 20] pro-cesses. By writing ad(p) = |ad(p)|eiη(p) in Eq. (11) andemploying the SPM, we obtain from Eq. (12):

limt→∞

Ψion(r, t) = eipm·r

G [r +R/2− pm(t− τ − τ1)]

+ G [r −R/2− pm(t− τ + τ1)]

. (13)

Here pm is the most probable photoelectron momentum,τ = ∂η/∂E is the usual Wigner photoemission time-delay[22], and τ1 = ∂δ/∂E with δ = −ΩRx/(2c) from Eq. (11).The τ1 term represents an additional time that it takesfor the light pulse to cover the distance Rx = R · ex.The wave packet G(r) in Eq. (13) is a Fourier transform

of the dipole amplitude ad(p) in Eq. (11). Importantly,G(r) has a strong peak near the origin. Therefore thetwo terms in Eq. (13) describe two wave packets emittedfrom the two atomic centers r = −R/2 and r = R/2 atthe times τ+τ1 and τ−τ1, respectively. For transparencyof derivation, we omitted in Eq. (13) the Coulomb termswhich would only add slowly varying (logarithmic) cor-rections in the arguments of G(r) [19, 20]. These addi-tional logarithmic terms are the same for the two wavepackets and they would therefore cancel in the time delaydifference between these wave packets.We note that the right-hand side of Eq. (9) can be

represented as a product of two factors, the factor pzresponsible for the angular dependence of the amplitudeand the Gaussian factor exp

−a(p− p0)2

representing

the energy conservation p20/2 = ε0 + ω. Such a Gaussianrepresentation of the ionized wave packets emitted in thesingle-center problems is often used in studying temporaldynamics of atomic ionization [19, 20]. The parameter adetermines the width of the wave packet and depends onthe pulse parameters. One can give analytical expressionfor its value but we will not need it below. By employingthe Gaussian ansatz for ad(p) and Eq. (11) we finallyobtain

ap = A exp

−a(p− p0)2

pz cos (p ·R/2 + δ) , (14)

We use Eq. (14) to evaluate the photoelectron emis-

4

2.0×10-6

6.0×10-6

1.0×10-5

-3 -2 -1 0 1 2 3

P(p x

) (a

.u.)

px (a.u.)

R=2 a.u., parallel orientation

TDSE, c=c0Fit Eq.(12), c=c0

TDSE, c=c0/10Fit Eq.(12), c=c0/10

FIG. 2: (Color online) Fit based on Eq. (15) and TDSE calcu-lations for inter-nuclear distance R = 2 a.u., and the geometryin which molecular axis is parallel to the pulse propagationdirection.

sion pattern and to compare it with the fully ab initio

TDSE calculations for various orientations and differentinter-nuclear distances of H+

2 . This comparison is pre-sented in Figure 1. The top row of panels illustratesthe geometry of the ionization process. It is assumedthat the molecular axis is confined to the xz-plane wherethe propagation and polarization vectors of the pulse lie,making angle θ with the propagation direction. The pho-toelectron momentum distribution is projected on the xzplane and computed as P (px, pz) =

∫

|ap|2 dpy with theamplitude ap obtained by projecting the TDSE solutionon the set of the scattering states of H+

2 . The num-ber and location of the bright spots in Figure 1 reflecta simple two-center interference pattern governed by thecosine term in Eq. (14).Comparison of the TDSE calculations (the middle row

of panels in Figure 1) and results based on Eq. (14) (thebottom row of panels) shows that the TBM reproducesthe spectra very well for the geometries that we consider.We will, therefore, analyze and interpret our TDSE re-sults using this transparent model. We will focus on thephotoelectron momentum distribution projected on the(xz) plane and integrated over the momentum compo-nent p⊥ which is perpendicular to the molecular axis.Such a momentum distribution is a function of the mo-mentum component p|| which is parallel to the molecularaxis. By employing the Gaussian ansatz (14) we obtain:

P (p||) =

∫

P (p) dpydp⊥ (15)

≈ B exp

−Cp2||

cos2(

p||R

2+ δ

)

.

We use analytical expression Eq. (15) with adjustableparameters B, C and δ to fit P (p||) obtained from thenumerical TDSE calculations.The accuracy of the fitting procedure is illustrated

in Figure 2. Here we fix the inter-nuclear distance toR = 2 au and align the molecular axis along the pulse

TABLE I: Propagation delay for different geometries and in-ternuclear distances.

R (a.u.) θ Fit (as) Rx/c (as)2 0 0.46 0.352 π/4 0.17 0.246 0 1.56 1.05

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 2 4 6 8 10 12δ

(a.u

.)c0/c

α=−0.0146 ± 0.0003

α=−0.0398 ± 0.0004

α=−0.133 ± 0.004

R=2 a.u., θ=0R=6 a.u., θ=0

R=2 a.u., θ= π/4Fit, Eq.(7)

FIG. 3: (Color online) Parameter δ in Eq. (15) for differentgeometries and internuclear distances.

propagation direction. In Figure 2 we present the twosets of simulations in which the speed of light is eitherset to its physical value c = c0 = 137.036 a.u. or scaleddown to c = c0/10. In both cases, the analytical fitwith Eq. (15) represents the TDSE results obtained forthe corresponding values of the speed of light quite accu-rately. This allows us to extract the phase shift δ accu-mulated due to the finite speed of light propagation.In Figure 3 we plot thus extracted phase shift δ as

a function of c0/c. According to TBM, the phase shiftshould scale linearly as δ = αc0/c with the slope α =−RxΩ/2/c0. The predicted linear scaling is confirmedvery accurately by the numerical values shown in Fig-ure 3 with only a small error margin. Using TBM wecan now deduce the time delay due to the finite inter-nuclear distance between the ionization events from thetwo H+

2 atomic centers. The time delay values corre-sponding to the physical speed of light c0 are shown inTable I in comparison with the predictions of the TBM∆t = 2Rx/c0. Agreement of the results can be deemedquite satisfactory given the relative simplicity of TBM.More importantly, the linear dependence of the TDSEresults on the parameter c0/c clearly demonstrates ex-istence of the finite speed of light effect in ionization ofH+

2 .Our work was motivated by the synchrotron based

experiment by Grundmann et al. [4] who discovereda zeptosecond time delay in photoionization of the

5

H2 molecule. We aimed to demonstrate that a similardelay, which is caused by the finite speed of light propa-gation from the one constituent atom to another, can bedetected in a much more accessible table top laser set-tings. In doing so, we developed a general theory of thefinite speed of light propagation effects in ionization ofextended systems. As a simple case study, we consideredthe H+

2 molecular ion interacting with a short laser pulse.This target affords a very accurate numerical treatmentwithin the time-dependent Schrodinger equation. At thesame time, an heuristic tight-binding model employingthe Heitler-London molecular ground state produces verysimilar results, giving a simple interpretation of the gen-eral theory. This way we demonstrate that the speedof light delay in ionization of H+

2 can be encoded in theadditional phase factor which can be extracted from thephotoelectron momentum distribution projected on themolecular axis.Our selection of the molecular target and the laser

pulse parameters was motivated primarily by the accu-racy of the numerical TDSE solution which is achievedeasier for a relatively simple target like H+

2 . In the exper-iment, more affordable targets like H2 and other diatomicmolecules can be used and the laser pulse parameters canbe relaxed to lesser intensities and larger pulse durations.This would not affect the physics of the process with thespeed of light delay encoded in the same addition phasefactor modifying the photoelectron momentum distribu-tion projected on the molecular axis.

6

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