Dynamic Stark induced vortex momentum of hydrogen in circular fields

Meng Li; Guizhong Zhang; Xuelian Kong; Tianqing Wang; Xin Ding; Jianquan Yao

doi:10.1364/OE.26.000878

1. Introduction

When an atom or a molecule is exposed to an intense laser field, ionization or dissociation will occur, the direct outcome of the ionization process is that the displaced electron will manifest a spectrum of either momentum or energy. Therefore, momentum distributions of the ionized electron are extensively investigated in the realm of atomic, molecular and optical physics [1–3]. Pertaining to the ionizing electric field of the strong laser pulse, a variety of pulse combination have been deployed: single color or two colors [4–7]; linearly or circularly polarized [8–12], few cycle or many cycles [13–15], and infrared or ultraviolet [16–19]. A few years ago, Ramsey-type interference of laser-induced electron wavepackets was found to show a totally new feature: vortex-shaped momentum distribution appears when two time-delayed circularly polarized weak-to-intermediate-intensity laser pulses were used in atomic or molecular ionization, i.e., the induced momentum distributions of the ionized electron in the plane of the laser polarization [20] showed spiral structure or vortex shape [21–24]. Counter-rotating circularly polarized pulses would lead to spiral momentum pattern while co-rotating ones produced no vortex pattern at all. More importantly, this kind of vortex momentum distribution has been confirmed experimentally using pottasium atoms [25]. In addition, Yu investigated the semiclassical dynamics of electron wavepackets carrying orbital angular momentum [26]. In 2016, Djiokap numerically studied the single ionization of helium atom subject to circularly polarized laser pulses. Vortex momentum patterns of zero-start, one-start, three-start and four-start were discovered for UV carrier frequency of 15eV and intensity of 1 × 10¹²W/cm² [23].

On the other hand, dynamic Stark effect [27–30], also termed a.c. or optical Stark effect, discovered a long time ago and serving as a proof for quantum mechanics, turns out to be a prominent process in almost all laser-atom and laser-molecule interactions. This phenomenon has been studied both experimentally and theoretically for many decades [31], and it still remains an interesting topic today. With the advent of ultrashort XUV laser pulses [32] the dynamic Stark effect shows novel features. The pioneering work by Demekhin et al. [33] found that strong interference fringes emerge as new spectral feature in photoelectron spectra of hydrogen atoms excited with a XUV laser pulse of carrier frequency 53.6 eV which is far beyond the hydrogenic ionization potential of 13.6 eV. The dynamic interference effect was revealed to come from the dynamic Stark effect of a hydrogen atom mapped to the continuum via excitation of a single photon of XUV wavelength. Under appropriate conditions, the energy level shift induced by the dynamic Stark effect just follows in time the square of the envelope of the electric field of a laser pulse regardless of the carrier frequency. Usually, the value of the second-order Stark shift is negative, meaning that the energy level is lowered in energy. For a hydrogen atom in its ground state, the dynamic Stark shift is, however, positive.

Inspired by the pioneering work on vortex-shaped momentum distribution of inert atoms [20–25] and on interference photoelectron energy spectra [33] of hydrogen atom, we investigate the distortion and mechanism of the vortex-shaped momentum distribution of hydrogen atom exposed to two time-delayed oppositely circularly polarized attosecond laser pulses. We deploy the strong field approximation (SFA) theory in our numerical simulation of the dynamic-Stark-induced vortex-shaped momentum pattern. Our extensive numerical computation is carried out for a variety of laser parameters and three different Stark coefficients. We found that the vortex-shaped momentum distribution is sensitively distorted compared to the non-Stark situation, and the distorted vortex momenta are manifested in either intensity or orientation of the vortex arms. This observation may serve as an alternative sensitive tool for studying the dynamic Stark effect in atoms and molecules.

2. Numerical methods

We use the strong field approximation theory [34–40] to simulate the momentum distribution of hydrogen atom ionized by two time-delayed circularly polarized attosecond pulses. The electric field associated with the laser pulses is supposed to be polarized in the x-y plane of the Cartesian coordinates. Its two components are written as (the first pulse has left helicity and the second right helicity):

{\begin{matrix} E_{x} = \frac{\sqrt{2}}{2} F (t) \cdot E_{0} \cdot \cos (ω t) + \frac{\sqrt{2}}{2} F (t + τ) \cdot E_{0} \cdot \cos (ω (t + τ)) \\ E_{y} = \frac{\sqrt{2}}{2} F (t) \cdot E_{0} \cdot \sin (ω t) - \frac{\sqrt{2}}{2} F (t + τ) \cdot E_{0} \cdot \sin (ω (t + τ)) \end{matrix}, (- \frac{T}{2} < t < \frac{T}{2})

And the vector potential of the field,

\vec{A}

, is also in the x-y plane.

τ

is the time delay,T indicates the pulse duration.

F = e^{- (\frac{t}{T})}^{2}

is the envelope function. The amplitude of the electron being ionized to a final momentum P is expressed by the SFA formula [1]:

f (\vec{p}) = - i \int_{- \infty}^{\infty} d t^{'} \exp (i φ) [\frac{2^{3.5} \times {(2 I_{p})}^{\frac{5}{4}}}{π} [\vec{p} / {({\vec{p}}^{2} + 2 I_{p})}^{3}]] \cdot \vec{E}

where the so-called classical action phase is given by:

φ = \int_{t^{'}}^{\infty} d t^{″} [I_{p} + Δ S + \frac{{[\vec{p} + \vec{A}]}^{2}}{2}]

with I_p being the ionization potential of the hydrogen atom.

In order to investigate the dynamic Stark effect, we introduce a term in the expression of the classical action phase, $Δ S$ , to quantitatively simulate the temporal variation of the ionization potential due to the Stark effect. The dynamic Stark effect up-shifts the ground-state energy level according to the envelope square, which is equivalent to a time-dependent down-shift of the ionization potential.

Δ S = - I_{p} \cdot Δ \cdot (F^{2} (t^{'}) + F^{2} (t^{'} + τ))

where a coefficient,

Δ

, is introduced to measure the strength of the dynamic Stark effect. It is tentatively referred to as the Stark coefficient.

We use the atomic units (a.u.) in the concrete numerical simulation while we also use the absolute values for some physical quantities for clarity in the text. The intensity I₀ (corresponding to peak field amplitude E₀), carrier frequency ω, and time delay τ have been varied within a relatively large range to perform an extensive simulation.

3. Simulation results and discussion

We present, in Fig. 1, a schematic of the dynamic Stark effect which will induce the up-shift of the ground-state energy level of hydrogen atom, and the ionization process which will induce interference of the ionized electron wavepackets. The former is equivalent to a lowering of the ionization threshold, while the latter alone will generate axially symmetric vortex-shaped momentum spectra with the ionization proceeding via the states of $| s, 0 〉 \to | p, 1 〉 \to | d, 2 〉$ and $| s, 0 〉 \to | p, - 1 〉 \to | d, - 2 〉$ . Where the first and the second symbols in the Dirac notations are the orbital and magnetic quantum numbers. The simultaneous action of these two processes will, however, result in a distortion of the vortex-shaped momenta. In Fig. 2, we plot a few typical vortex-shaped momentum distributions for different helicities and carrier frequencies of two time-delayed attosecond pulses, taking into account of the dynamic Stark effect. We have performed extensive numerical simulation for a range of laser parameters such as intensity, carrier frequency, pulsewidth and pulse-pulse delay. These simulations reveal that the vortex-shaped momentum distribution can be achieved over a relatively wide range of these parameters. In Fig. 2, the carrier frequencies are 4eV, 8eV and 16eV, the pulsewidths are two optical cycles (o.c.), the intensity is 3.5 × 10¹⁴W/cm², and the time delay (defined between the two envelope peaks) is τ = 3 o.c. The general observation is that the helicity of the vortex momenta follows that of the second time-delayed pulse: the condition that the first pulse is left polarized and the second pulse is right polarized gives rise to a right polarized vortex momentum. As discussed in [21,23,25], the number of the spiral arms in these vortex patterns doubles the number of the photons required to ionize the electron in hydrogen: for instance, if the carrier frequency is 8eV, then 2 photons are required to overcome the ionization threshold of 13.6eV for hydrogen, and therefore, the number of the spiral arms is 4, as shown by the middle plots in both the top and bottom panels in Fig. 2. This observation is independent of the polarization of the two time-delayed pulses. Figure 2 clearly shows that the vortex momentum patterns will be created provided that the two laser pulses are oppositely polarized.

Fig. 1 Schematic illustration of the dynamic Stark effect and ionization process leading to distorted vortex-shaped momentum distribution in hydrogen exposed to two time-delayed oppositely circularly polarized attosecond pulses of ω = 8eV. The upshift of the ground-state energy level is mapped to a lowering of the ionization potential.

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Fig. 2 Vortex-shaped momentum distribution of hydrogen atom exposed to two time-delayed attosecond pulses: carrier frequencies ω = 4, 8, 16eV, pulsewidth T = 2 o.c., intensity I = 3.5 × 10¹⁴W/cm², the time delay is τ = 3 o.c.. The top row is for ‘left/right’ polarization: the first pulse is left circularly polarized and the second one is right circularly polarized. The bottom row is for ‘right/left’ polarization.

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The vortex-shaped momentum distribution will become distorted, or lost in the axial symmetry, if the dynamic Stark effect is taken into account. This phenomenon can be seen in Fig. 3 for either the left- or right-helicity momentum vortices. For clarity, we only present the momentum distribution for the following condition: carrier frequencies ω = 8eV, pulsewidth τ = 2 optical cycles, intensity I = 3.5 × 10¹⁴W/cm², and time delay τ = 3 o.c. The dynamic Stark effect was simulated for three Stark coefficients: Δ = 0.01, 0.02, and 0.04, as expressed in Eq. (4). As expected, the vortex-shaped momentum distribution becomes more and more distorted with intensifying dynamic Stark effect which is elucidated by our Stark coefficient. A faint feature is that the intensity of some of the vortex arms in the momentum pattern starts to decrease even for Δ = 0.01, as shown by the first plots in both the top and bottom panels in Fig. 3, considering thatΔ = 0.01 means that the maximal Stark shift of the ground-state energy level of hydrogen is only 0.01 × 13.6eV = 0.136eV.

Fig. 3 Distorted vortex-shaped momentum distribution of hydrogen atom ionized by two time-delayed attosecond pulses: carrier frequencies ω = 8eV, pulsewidth T = 2 o.c., intensity I = 3.54 × 10¹⁴W/cm², the time delay is τ = 3 o.c., when the dynamic Stark effect is taken into account: Stark coefficient Δ = 0.01, 0.02, and 0.04. The top row is for ‘right/left’ polarization. The bottom row is for ‘left/right’ polarization.

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We focus on the left-helicity situation of ω = 8eV in the following. In order to discuss the mechanism behind the distortion of the vortex-shaped momentum distribution, we define in Fig. 4 the numbering of the vortex arms: the first vortex arm is the one mostly centered on the positive P_x-axis, and so on.

Fig. 4 Top row shows the numbering of the vortex arms and the transition of the vortex-shape from Δ = 0 (a) to Δ = 0.04 (b) considering the dynamic Stark effect. The black arrow in the second plot (top row) illustrates the corresponding shift of the peak position of the fourth vortex arm. The bottom row illustrates the variations of the intensity (c), absolute momentum value (d) and the polar angle (between −90 and 270 degrees) (e) of the peak of each of the four vortex arms. Red, green, blue and black colors correspond to vortex arms of No.1 to No.4.

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The first observation from the second plot in the top panel in Fig. 4 is that the axially symmetric spiral pattern becomes asymmetric and the intensities of the four vortex arms become profoundly uneven if the dynamic Stark effect is counted. A careful examination of the seemingly irregular vortex pattern can still unveil some rules to follow. It is clear from the top panel in Fig. 4 that the vortex arm #1 shrinks to the origin while its intensity is the strongest; the vortex arm #2 is greatly weakened while its position is approximately maintained; the vortex arm #3 behaves similarly as vortex arm #2; and the vortex arm #4 as #1. We utilize three quantities to describe the vortex arm variation in detail: (1) peak intensity of vortex arm, (2) absolute momentum value corresponding to the vortex-arm peak, and (3) the polar angle subtended by the vortex arm. The second quantity is simply the distance between the peak position of each vortex arm and the origin. The variation of these three quantities with Stark coefficient is plotted in the bottom panel in Fig. 4. It can be seen from the intensity curve (the first plot in the bottom panel in Fig. 4) that the intensities of the second and third vortex arms show strikingly monotonic decrease with intensifying dynamic Stark effect; the momentum curve (the second plot) shows that the first and fourth vortex arms shrink towards the origin while the second and third ones swirl away from the origin when the dynamic Stark effect is considered; the shrinking of the vortex arms #1 and #4 is corroborated by the decreasing of their polar angles, as shown by the third plot in the bottom panel in Fig. 4.

The additional phase caused by the dynamic Stark effect plays a unique role in forming the distorted vortex-shaped momentum distributions. Equation (3) expresses the total phase (standard classical action phase plus the Stark-induced phase) which is dependent upon time and the final momentum value of ionized electron. In Fig. 5, we plot the time variation of the total phase for a momentum value corresponding to the peak position of the vortex arm #4, compared to that without the dynamic Stark effect. The second plot in Fig. 5 shows the additional Stark phase which is purely dependent on time, as given by Eq. (4). One can see that the additional phase takes maximal values at time instants corresponding to the crests of the two time-delayed pulse envelopes. The third plot is the total phase vs time curve for all the four vortex arms with momentum values corresponding to their crests. The curves in the third plot in Fig. 5 show that the total phase variation is different for different vortex arms, thus giving rise to different spiral variation. However, the proximity of the total phases for vortex arms #1 and #4 is manifested by the similar trend of spiral variation, as discussed above. The time segment over which the total phase considering the dynamic Stark effect deviates from that neglecting this effect is approximately between the second and eighth optical cycles, as shown by the first plot in Fig. 5. Therefore, we choose this segment between 2 and 8 o.c. for the integration in Eq. (2) to reproduce the momentum distribution, and the resultant momentum pattern is given in Fig. 6. Also, the field is weak outside the 2-8 o.c. so that the corresponding contribution to the integration is neglected. The excellent similarity between the thereby reproduced momentum distribution and that in Fig. 4 (the right plot in the top panel, integration over full time) proves that the additional dynamic Stark phase plays the key role in the formation of the distorted vortex-shaped momentum distribution.

Fig. 5 Time variation of various kinds of phase. Left panel: the total phase (classical action plus the Stark ones) for Δ = 0 (red curve) and Δ = 0.04 (blue curve) for the peak of vortex arm #4; middle panel: the Stark phase only; right panel: the total phase (shown only for 4.4 to 5.0 optical cycle for clarity) for vortex arms #1 (red), #2 (green), #3 (blue), and #4 (black). As described in the text, the phase values are shifted by multiples of 2π into the range of 0 to 2π.

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Fig. 6 Distorted vortex-shaped momentum distribution reproduced by integrating over only a segment of the pulse duration, other parameters are the same as in Fig. 4 (left plot). The vortex-shaped momentum distribution shows no distortion if the additional phase is instead a linear function of time (right plot). See the text for details.

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In order to further interrogate the role of the dynamic Stark phase, we change the time-dependent envelope function in Eq. (4) into a constant of unity and re-simulate the momentum distribution. The thus obtained momentum pattern is also shown in Fig. 6. It is clear that this vortex-shaped momentum distribution is not distorted and is exactly reproduced as without considering the dynamic Stark effect. As a matter of fact, we found that any additional linear phase (linear function of time) will result in an un-distorted vortex-shaped momentum distribution (same momentum pattern as without the Stark effect). Therefore, it is the nonlinearly time-dependent Stark phase that induces the prominent distortion in the vortex-shaped momentum distribution.

4. Conclusion

By deploying the strong field approximation theory, we have performed extensive numerical simulation on the distortion of the vortex-shaped momentum distribution of ionized hydrogen atom. We found that two time-delayed oppositely circularly polarized attosecond pulses would induce distorted vortex-shaped momentum distribution when the dynamic Stark effect is taken into account. We deciphered this phenomenon with the additional Stark phase factor. We anticipate that the sensitive evolvement of the vortex pattern of the momentum distribution with the dynamic Stark effect could be useful in atomic and molecular studies.

Funding

National Natural Science Foundation of China (Nos. 11674243 and 11674242); Fundamental Research Funds for the Central Universities (No. 3122016D014).

References and links

1. T. Brabec, Strong Field Laser Physics, (Springer, 2008), pp. 166–168.

2. A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express 18(4), 3660–3671 (2010). [CrossRef] [PubMed]

3. S. Hu, J. Chen, X. Hao, W. Li, L. Guo, and S. Han, “Effect of two-center interference on molecular ionization in multiphoton ionization regime,” Opt. Express 25(19), 23082–23092 (2017). [CrossRef] [PubMed]

4. M. H. Xu, L. Y. Peng, Z. Zhang, Q. Gong, X. M. Tong, E. A. Pronin, and A. F. Starace, “Attosecond Streaking in the Low-Energy Region as a Probe of Rescattering,” Phys. Rev. Lett. 107(18), 183001 (2011). [CrossRef] [PubMed]

5. J. W. Geng, W. H. Xiong, X. R. Xiao, L. Y. Peng, and Q. Gong, “Nonadiabatic Electron Dynamics in Orthogonal Two-Color Laser Fields with Comparable Intensities,” Phys. Rev. Lett. 115(19), 193001 (2015). [CrossRef] [PubMed]

6. M. Li, J. Geng, M. Liu, X. Zheng, L. Peng, Q. Gong, and Y. Liu, “Spatial-temporal control of interferences of multiple tunneling photoelectron wave packets,” Phys. Rev. A 92(1), 013416 (2015). [CrossRef]

7. Y. Zhou, C. Huang, A. Tong, Q. Liao, and P. Lu, “Correlated electron dynamics in nonsequential double ionization by orthogonal two-color laser pulses,” Opt. Express 19(3), 2301–2308 (2011). [CrossRef] [PubMed]

8. L. Peng, E. A. Pronin, and A. F. Starace, “Attosecond pulse carrier-envelope phase effectson ionized electron momentum and energy distributions: roles of frequency, intensity and an additional IR pulse,” New J. Phys. 10(2), 025030 (2008). [CrossRef]

9. P. L. He, C. Ruiz, and F. He, “Carrier-Envelope-Phase Characterization for an Isolated Attosecond Pulse by Angular Streaking,” Phys. Rev. Lett. 116(20), 203601 (2016). [CrossRef] [PubMed]

10. M. Klaiber, E. Yakaboylu, and K. Z. Hatsagortsyan, “Above-threshold ionization with highly charged ions in superstrong laser fields. I. Coulomb-corrected strong-field approximation,” Phys. Rev. A 87(2), 023417 (2013). [CrossRef]

11. C. Figueira de Morisson Faria, H. Schomerus, X. Liu, and W. Becker, “Electron-electron dynamics in laser-induced nonsequential double ionization,” Phys. Rev. A 69(4), 043405 (2004). [CrossRef]

12. M. Abu-samha and L. B. Madsen, “Interrogation of orbital structure by elliptically polarized intense femtosecond laser pulses,” Phys. Rev. A 84(2), 023411 (2011). [CrossRef]

13. L. Peng and A. F. Starace, “Attosecond pulse carrier-envelope phase effects on ionized electron momentum and energy distributions,” Phys. Rev. A 76(4), 043401 (2007). [CrossRef]

14. N. Douguet, A. Grum-Grzhimailo, E. Gryzlova, and E. Staroselskaya, “Photoelectron angular distributions in bichromatic atomic ioniza polarized VUV femtosecond pulses,” Phys. Rev. A 93, 033402 (2016). [CrossRef]

15. D. Milošević, G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Express 11(12), 1418–1429 (2003). [CrossRef] [PubMed]

16. S. X. Hu, “Boosting Photoabsorption by Attosecond Control of Electron Correlation,” Phys. Rev. Lett. 111(12), 123003 (2013). [CrossRef] [PubMed]

17. L. Y. Peng and A. F. Starace, “Application of Coulomb wave function discrete variable representation to atomic systems in strong laser fields,” J. Chem. Phys. 125(15), 154311 (2006). [CrossRef] [PubMed]

18. A. N. Grum-Grzhimailo, A. D. Kondorskiy, and K. Bartschat, “Controlling the angular distribution of atomic photoelectrons in the region of laser-induced continuum structure in the femtosecond time domain,” J. Phys. At. Mol. Opt. Phys. 39(22), 4659–4671 (2006). [CrossRef]

19. X. Yu, Q. Bian, Z. Chang, P. B. Corkum, and S. Lei, “Femtosecond laser nanomachining initiated by ultraviolet multiphoton ionization,” Opt. Express 21(20), 24185–24190 (2013). [CrossRef] [PubMed]

20. E. Hasović, W. Becker, and D. B. Milošević, “Electron rescattering in a bicircular laser field,” Opt. Express 24(6), 6413–6424 (2016). [CrossRef] [PubMed]

21. J. M. Ngoko Djiokap, S. X. Hu, L. B. Madsen, N. L. Manakov, A. V. Meremianin, and A. F. Starace, “Electron Vortices in Photoionization by Circularly Polarized Attosecond Pulses,” Phys. Rev. Lett. 115(11), 113004 (2015). [CrossRef] [PubMed]

22. K. Yuan, S. Chelkowski, and A. D. Bandrauk, “Photoelectron momentum distributions of molecules in bichromatic circularly polarized attosecond UV laser fields,” Phys. Rev. A 93(5), 053425 (2016). [CrossRef]

23. J. M. Ngoko Djiokap, A. V. Meremianin, N. L. Manakov, S. X. Hu, L. B. Madsen, and A. F. Starace, “Multistart spiral electron vortices in ionization by circularly polarized UV pulses,” Phys. Rev. A 94(1), 013408 (2016). [CrossRef]

24. M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106(4-6), 161–166 (1994). [CrossRef]

25. D. Pengel, S. Kerbstadt, D. Johannmeyer, L. Englert, T. Bayer, and M. Wollenhaupt, “Electron Vortices in Femtosecond Multiphoton Ionization,” Phys. Rev. Lett. 118(5), 053003 (2017). [CrossRef] [PubMed]

26. K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori, “Semiclassical Dynamics of Electron Wave Packet States with Phase vortices,” Phys. Rev. Lett. 99(19), 190404 (2007). [CrossRef] [PubMed]

27. S. H. Autler and C. H. Townes, “Stark Effect in Rapidly Varying Fields,” Phys. Rev. 100(2), 703–722 (1955). [CrossRef]

28. A. M. Bonch-bruevich, N. N. Kostin, V. A. Khodovoi, and V. V. Khromov, “Changes in the atomic absorption spectrum in the field of a light wave. I.,” Sov. Phys. JETP 29, 82 (1969).

29. N. B. Delone and V. P. Krainov, “AC Stark shift of atomic energy levels,” Phys. Uspekhi 42(7), 669–687 (1999). [CrossRef]

30. W. L. Zhang, X. M. Wu, F. Wang, R. Ma, X. F. Li, and Y. J. Rao, “Stark effect induced microcavity polariton Solitons,” Opt. Express 23(12), 15762–15767 (2015). [CrossRef] [PubMed]

31. C. Yu and L. B. Madsen, “Sequential and nonsequential double ionization of helium by intense XUV laser pulses: Revealing ac Stark shifts from joint energy spectra,” Phys. Rev. A 94(5), 053424 (2016). [CrossRef]

32. C. Yu and L. B. Madsen, “Ionization of helium by intense XUV laser pulses: Numerical simulations onchannel-resolved probabilities,” Phys. Rev. A 93(4), 043412 (2016). [CrossRef]

33. P. V. Demekhin and L. S. Cederbaum, “Dynamic Interference of Photoelectrons Produced by High-Frequency Laser Pulses,” Phys. Rev. Lett. 108(25), 253001 (2012). [CrossRef] [PubMed]

34. M. Klaiber, E. Yakaboylu, and K. Z. Hatsagortsyan, “Above-threshold ionization with highly charged ions in superstrong laser fields. I. Coulomb-corrected strong-field approximation,” Phys. Rev. A 87(2), 023417 (2013). [CrossRef]

35. C. A. Mancuso, D. D. Hickstein, K. M. Dorney, J. L. Ellis, E. Hasović, R. Knut, P. Grychtol, C. Gentry, M. Gopalakrishnan, D. Zusin, F. J. Dollar, X.-M. Tong, D. B. Milošević, W. Becker, H. C. Kapteyn, and M. M. Murnane, “Controlling electron-ion rescattering in two-color circularly polarized femtosecond laser fields,” Phys. Rev. A 93(5), 053406 (2016). [CrossRef]

36. M. Klaiber, J. Danek, E. Yakaboylu, K. Z. Hatsagortsyan, and C. H. Keitel, “Strong-field ionization via a high-order Coulomb-corrected strong-field approximation,” Phys. Rev. A 95(2), 023403 (2017). [CrossRef]

37. C. Yu and L. B. Madsen, “Above-threshold ionization of helium in the long-wavelength regime: Examining the single-active-electron approximation and the two-electron strong-field approximation,” Phys. Rev. A 95(6), 063407 (2017). [CrossRef]

38. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

39. F. H. M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B 6(4), L89–L92 (1973). [CrossRef]

40. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22(5), 1786–1813 (1980). [CrossRef]

Dynamic Stark induced vortex momentum of hydrogen in circular fields

Abstract

1. Introduction

2. Numerical methods

3. Simulation results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (4)

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