⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 043201 Nonadiabatic and Multielectron Effects in the Attoclock Experimental Scheme * Zhi-Lei Xiao (肖智磊)1,2, Wei Quan (全威)1**, Song-Po Xu (许松坡)1,2, Shao-Gang Yu (余少刚)1, Xuan-Yang Lai (赖炫扬)1, Jing Chen (陈京)3,4**, Xiao-Jun Liu (柳晓军)1** Affiliations 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071 2University of Chinese Academy of Sciences, Beijing 100049 3HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100084 4Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088 Received 22 January 2020, online 24 March 2020 *Supported by the National Key Research and Development Program of China (Nos. 2019YFA0307700 and 2016YFA0401100), the National Natural Science Foundation of China (Nos. 11527807, 11774387, 11834015, 11847243, 11804374, 11874392, and 11974383), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB21010400), and the Science and Technology Department of Hubei Province (No. 2019CFA035).
**Corresponding author. Email: charlywing@wipm.ac.cn; chen_jing@iapcm.ac.cn; xjliu@wipm.ac.cn Citation Text: Xiao Z L, Quan W, Xu S P, Yu S G and Lai X Y et al 2020 Chin. Phys. Lett. 37 043201    Abstract The problem of how long it takes for an electron to tunnel from one side of a barrier to the other has been debated for decades and the attoclock is a promising experimental procedure to address this problem. In the attoclock experiment, many physical effects will contribute to the experimental results and it is difficult to extract the tunneling time accurately. We numerically investigate a method of measuring the residual equivalent temporal offset (RETO) induced by the physical effects except for tunneling delay. The Coulomb potential effect, the nonadiabatic effect, the multielectron effect, and the Stark effect are considered in the theoretical model. It is shown that the ratio of the RETO of the target atoms to that of H is insensitive to the wavelength and is linearly proportional to (2$I_{\rm p}$)$^{-3/2}$. This work can help to improve the accuracy of the attoclock technique. DOI:10.1088/0256-307X/37/4/043201 PACS:32.80.Fb, 03.65.Sq, 32.80.Rm © 2020 Chinese Physics Society Article Text Quantum tunneling is one of the fundamental issues of quantum mechanics, which has attracted a great deal of attention recently (see, e.g., Refs. [1–3]). The tunneling time problem,[4–8] i.e., how long it takes for an electron to tunnel from one side of the barrier to the other is a highly debated topic. A lot of experimental techniques, such as the high-harmonic spectroscopy[9] and attoclock,[10,11] have been developed to investigate this problem. The temporal resolutions of these techniques are on the timescale of attoseconds (1 as = $10^{-18}$ s). Such a high temporal resolution is indispensable to resolve the ultrafast dynamics of the photoelectron.[12] The attoclock procedure is promising because it lowers the technical demand by employing a normal elliptically polarized (EP) laser field, instead of the complicated laser fields or attosecond pulses. The principle of this procedure[10] relies on the relationship between the tunneling time and the photoelectron momentum distribution for target atom subject to a strong EP laser field. The combined Coulomb and laser EP electric field creates a rotating barrier, through which the valence electron can be ejected by tunneling. Essentially, the rotating electric field vector serves as the hand of a clock and the emission angle of the ejected electron depends on the instant when the electron escapes. The tunneling instant can be obtained by simply measuring the photoelectron emission angle. Thus, with the comparison of the instant when the strength of electric field gets to the maximum and the instant when the photoelectron orbit with highest weight is launched, the tunneling delay can be achieved. In the last decade, we have witnessed an ongoing debate about whether the tunneling process will cost a definite time delay.[12] The tunneling time problem has been investigated by a lot of works with the attoclock scheme, and inconsistent conclusions have been achieved by two groups of researchers. The first group believes that the tunneling happens instantaneously. The vanishing tunneling delay has been experimentally shown in single ionization[11,13,14] and double ionization[15] of noble gas atoms subject to EP laser fields. In 2015, ab initio numerical simulations have been performed for H[14] and tunneling time of its valence electron is found to be zero. This theoretical prediction has been experimentally verified with the attoclock technique in 2019.[15] The other group argues that a finite tunneling time exists. Experimental evidences of a clear tunneling delay have been recently demonstrated by Landsman et al.[16] and Camus et al.[17] As discussed earlier, the issue of tunneling delay is still an open problem[18–21] and the reason of the inconsistency of the above documented studies may be attributed to the difficulty of accurately extracting the tunneling time from the measurements of attoclock experiments. In fact, with the attoclock experimental procedure, the tunneling time can be obtained accurately only when the physical process can be ideally described with the Simpleman model.[22,23] For a real attoclock experiment, some physical effects, such as the Coulomb potential effect, the nonadiabatic effect, and the multi-electron effect, will contribute significantly to the measurements and make it difficult to accurately extract the tunneling time from the attoclock readings. For example, as documented (see, e.g., Refs. [11,24]), the Coulomb potential will induce a significant angular offset of the photoelectron momentum distributions, which cannot be easily disentangled from the tunneling time. Recently, a procedure has been proposed to measure the Coulomb potential influence and, in this work, the equivalent temporal offset solely induced by the Coulomb potential (TOCP) has been studied systematically with a semiclassical model.[24] However, unfortunately, other physical effects, such as the nonadiabatic effect, and the multi-electron effect, have not been considered in the calculations. Note that the numerical calculation results for the ionization dynamics could be significantly influenced by these physical effects. Indeed, in the semiclassical physical picture, the nonadiabatic effect may give rise to different initial photoelectron spatial and velocity distributions, and the multi-electron effect may alter significantly the evolution of the orbit. In this Letter, we numerically study the wavelength dependence of the photoelectron momentum distributions (PMD) for H and noble gas atoms subject to intense EP laser fields. It is shown that, even with the Coulomb potential influence, the nonadiabatic effect, the multielectron effect and also the Stark effect included, the ratio of the residual equivalent temporal offset (RETO) induced by the physical effects except for tunneling delay of the target atoms to that of H is still insensitive to the wavelength and linearly proportional to (2$I_{\rm p}$)$^{-3/2}$. This result is helpful to extract the RETO of the target atom and significant to further extract accurately the tunneling time for the attoclock scheme. The adiabatic model with the adiabatic approximation (see Refs. [24,25] for details) included is also dubbed classical-trajectory Monte Carlo (CTMC). In this model, the initial spatial coordinates of tunneled photoelectron are derived from Landau's work[26] and each electron orbit is weighted by the ADK ionization rate.[27,28] The laser electric field is given by $$\begin{align} & E(t)=E_{0}f(t)\left [ \cos(\omega t){\hat{\boldsymbol z}}+\varepsilon \sin(\omega t){\hat{\boldsymbol x}} \right ],~~ \tag {1} \end{align} $$ where $\varepsilon$ is the ellipticity, ${\hat{\boldsymbol z}}$ the major axis and ${\hat{\boldsymbol x}}$ the minor axis of the polarization ellipse. In the calculation, the envelope of the laser field is defined by $$ f(t)=\begin{cases}\!\!\! 0,~~~t\leq0\\ \!\!\! 1,~~~ 0 < t\leq10T\\ \!\!\! \cos^{2}\left(\pi\frac{t-10T}{6T}\right),~~~ 10T < t\leq 13T \\ \!\!\! 0,~~~ t> 13T. \end{cases}~~ \tag {2} $$ Based on the strong-field approximation,[29,30] the nonadiabatic effect will give rise to different initial conditions and ionization rate of the tunneled electron.[31,32] Note that the conserved canonical momentum $p$ is related to the initial momentum $p_{0}$ through $p=p_{0}-A(t)$. Here $p_{0}$ can be divided into the longitudinal momentum $p_{\parallel}$ and transverse momentum $p_{\perp}$ with respect to the instantaneous laser polarization direction at the tunnel exit.[31,32] Thus, $$ \cosh(\omega t_{i})=\begin{cases}\!\!\! \frac{1}{a^{4}-\varepsilon ^{2}} \Big\{ \varepsilon (\frac{a\omega }{E_{0}}p_{\perp} -\varepsilon)\pm a^{2} \Big[(\frac{a\omega }{E_{0}}p_{\perp} -\varepsilon)^{2}\\ +(a^{4}-\varepsilon ^{2})(1+\frac{\gamma {_{\rm eff}}^{2}}{a^{2}})\Big]^{1/2} \Big\} ,~~~a^{2}\neq \left | \varepsilon \right |,\\ \!\!\! \frac{1}{2}\big(1\!-\!\frac{a\omega }{\varepsilon E_{0}}p_{\perp }\big)\!+\!\frac{a^{2} (1+\frac{\gamma {_{\rm eff}}^{2}}{a^{2}})}{2\varepsilon ^{2} (1\!-\!\frac{a\omega }{\varepsilon E_{0}}p_{\perp })} ,~ a^{2}= \left | \varepsilon \right |, \end{cases}~~ \tag {3} $$ $$ p_{\parallel }=\frac{(1-\varepsilon ^{2})E_{0}\sin(\omega t_{0})\cos(\omega t_{0})(\cosh(\omega t_{i})-1)}{a\omega },~~ \tag {4} $$ where $a=\sqrt{\cos^{2}(\omega t_{0})+\varepsilon ^{2}\sin^{2}(\omega t_{0})}$ and $\gamma _{\rm eff}=\omega \sqrt{2(I_{\rm p}+p_{y0}^{2}/2)}/E_{0}$.[31,32] The ionization probability can be given by[31] $$\begin{align} {\it\Gamma} =\,&\exp \Big[-2\Big(\frac{p^{2}}{2}+I_{\rm p}+U_{\rm p}\Big)t_{i}+2p_{z}\frac{E_{0}}{\omega ^{2}}\sin(\omega t_{0}) \\ &\cdot\sinh (\omega t_{i}) -2p_{x}\frac{\varepsilon E_{0}}{\omega ^{2}}\cos(\omega t_{0})\sinh (\omega t_{i}) \\ \nonumber & +\frac{E{_{0}}^{2}(1-\varepsilon ^{2})}{4\omega ^{3}}\cos2\omega t_{0}\sinh (2\omega t_{i}) \Big],~~ \tag {5} \end{align} $$ where $U_{\rm p}=(1+\varepsilon ^{2})E{_{0}}^{2}/4\omega ^{2}$ is the ponderomotive energy. As shown above, $t_{i}$, $p_{\parallel}$ and ${\it\Gamma} $ depend on $t_{0}$, $p_{\perp}$, and $p_{y0}$, which are randomly distributed in the following ranges: $0\leq t_{0} \leq 4\,T$, $-1\leq p_{\perp} \leq 1$, and $-1\leq p_{y0} \leq 1$. With the initial conditions and ionization rate given above, the nonadiabatic effect can be well considered. A Hartree–Fock–Slater type potential has been applied to describe the binding potential of a multi-electron atom,[33,34] $$\begin{alignat}{1} &U(r)=-\frac{1}{r}-\frac{{\it\Phi} (r)}{r},~~ \tag {6} \end{alignat} $$ $$\begin{alignat}{1} &{\it\Phi} (r)=b\exp(-cr)+(Z-b-1)\exp(-dr),~~ \tag {7} \end{alignat} $$ where $Z$ is the atomic number, $b$, $c$, and $d$ are chosen carefully to make sure that the bound state energies are consistent with the documented measurements. For argon ($Z = 18$), these coefficients are $b = 5.4$, $c = 1$, $d$ = 3.682,[33] and in the case of krypton ($Z = 36$), the coefficients are $b = 6.42$, $c = 0.905$, $d$ = 4.2.[34] In our model, the static Stark shift has been included, $$ I_{\rm p}=I_{p0}+\frac{1}{2}(\alpha _{\rm N}-\alpha _{\rm I})E{_{0}}^{2},~~ \tag {8} $$ where $I_{p0}$ is the field-free ionization potential, $\alpha _{\rm N}$ and $\alpha _{\rm I}$ are the static polarizability of the neutral atom and the singly charged ion, respectively.[35,36] To further include the multielectron polarization effect of the atomic core, the tunneled electron will evolve in the potential,[37,38] $$ U(r)=-\frac{1}{r}-\frac{{\it\Phi} (r)}{r}-\frac{\alpha _{\rm I}{\boldsymbol E}(t)\cdot {\boldsymbol r}}{r^{3}}.~~ \tag {9} $$ With our semiclassical model, the PMDs for H subject to the laser fields at the intensity of $3.0\times 10^{14}$ W/cm$^{2}$ and the ellipticity of 0.7 at a series of wavelengths are obtained. To investigate the influence of the nonadiabatic effect and the Stark effect, the calculations with and without including the nonadiabatic effect or the Stark effect have also been performed. Note that there is only one electron of the H atom, no multielectron effect is expected.
cpl-37-4-043201-fig1.png
Fig. 1. (a) The calculated PMD for H, which is obtained with the nonadiabatic model. The wavelength is 800 nm, the ellipticity 0.7 and the laser intensity $3.0\times 10^{14}$ W/cm$^{2}$. (b) The photoelectron angular distribution calculated with parameters identical to those of (a). (c) The wavelength dependence of the offset angle. (d) The wavelength dependence of temporal offset, $\Delta t$, which is extracted from the data in (c). The square indicates the calculations by the adiabatic model. The circle indicates the calculations by the model with only the nonadiabatic effect included. The upper triangle indicates the calculations by the adiabatic model with Stark effect included. The lower triangle indicates the calculations by the nonadiabatic model with the Stark effect included.
The typical PMD for H subject to EP laser fields at 800 nm is depicted in Fig. 1(a). According to the Simpleman model, the yields maxima will appear at the minor axis of the polarization ellipse. In contrast, as shown in Fig. 1(a), the PMD has been rotated by an angle of $\Delta \theta $ (see Fig. 1(a)). With closer inspection, the asymmetry of the PMD can be identified and the asymmetry appears even more obvious in the angular distributions of photoelectrons in Fig. 1(b). With the comparison of the adiabatic model applied and the Simpleman model, the deviation of the results from the prediction of the Simpleman model can be attributed to the Coulomb potential effect.[24] In an attoclock experiment, the angular offset obtained from the yield maximum (indicated by A in Fig. 1(b)) is usually employed to extract the angle offset information. Here we employ a more precise procedure.[16] Compared to A, point B can be applied to extract the tunneling time more accurately.[24] The wavelength dependence of the angular offset obtained with the procedure described above is shown in Fig. 1(c), where the angular offset decreases smoothly with respect to wavelength. The laser intensity has been kept to be constant to obtain the wavelength dependence of the RETO. With the equation of $\Delta t=\tan^{-1}(\varepsilon \tan\Delta \theta)/\omega $, the corresponding temporal offset can be obtained, as shown in Fig. 1(d), where the numerical calculations are shown with symbols and the exponential fit results are indicated with solid lines. As shown in this panel, the temporal offset calculated with each model descends gradually with respect to wavelength. Moreover, the slopes of the curves are abrupt at shorter wavelength and becomes moderate smoothly at longer wavelength. With closer inspection, the temporal offset calculated with adiabatic model is the largest and that calculated by the nonadiabatic model with Stark effect included is the lowest. Apparently, the nonadiabatic effect leads to a significant decrease of $\Delta t$ and plays an important role in the attoclock experiment. Meanwhile, the influence of Stark effect is much weaker.
cpl-37-4-043201-fig2.png
Fig. 2. The photoelectron initial longitudinal (a) and transverse (b) momentum distributions at the tunnel exit with respect to the laser electric field phase for H subject to the EP laser field at 800 nm. The local yield maximum for each laser phase [red line in (a) and (b)] and phase dependence of laser field [white line in (a) and (b)] are shown in arbitrary units. The photoelectron initial spatial distributions (c) and the photoelectron angular distribution (d) for the H atom calculated by the semiclassical model with the chosen effects included are also pictured. The arrows in panel (d) indicate the angles at which the photoelectron orbits launched from the field strength maximum will appear.
To shed more light on the physics behind the results presented in Fig. 1, we depict the photoelectron initial longitudinal and transverse momentum distributions with respect to the laser phase in Figs. 2(a) and 2(b), respectively. As shown in Fig. 2(a), the photoelectron initial longitudinal momentum distributions depend sensitively on the laser phase. Nevertheless, the highest yields appear at $p_{\parallel }\approx 0$ and the asymmetry induced by the longitudinal momentum distribution will become invisible for cycle averaged results. For the initial transverse momentum, as shown in Fig. 2(b), the local yield maximum at each laser phase oscillates periodically. This oscillation is out of phase to that of the strength of the laser field and will distort the PMD to some extent. After averaged over the pulse duration, the initial transverse momentum distribution will lead to a distorted irregular donut-shaped PMD, its averaged radius is larger than that of the adiabatic case, and the angle offset will also be reduced to some extent if compared to that calculated by the adiabatic model, as shown in Fig. 2(d). The initial spatial distributions of tunneled electron ionized from the atom of H are presented in Fig. 2(c). The tunneling exit is the smallest if the adiabatic model is applied, where only the Coulomb potential influence has been considered. If these results are compared with those calculated by the adiabatic model with the Stark effect included, then it is found that only a small shift away from the nucleus is further introduced. This reveals the impact of the correction of the ionization potential due to Stark effect. Additionally, compared to the case of the adiabatic approximation, the tunneling exit distribution becomes wider and a significant offset of this distribution can be identified if the nonadiabatic effect has been well considered. The calculated photoelectron angular distributions for H are pictured in Fig. 2(d). Note that the minor axis of the laser ellipse appears at $\theta = 1.57$, the largest angle offset is apparently the result by the adiabatic model. This result can be mainly attributed to the smallest tunnel exit (see Fig. 2(c)). If the Stark effect is further introduced in the model, the offset angle becomes smaller and the influence of the Stark effect is not very significant. For the nonadiabatic effect, several aspects have been considered separately. The results indicated by the lower triangle in Fig. 2(d) are calculated by the adiabatic model where the tunneling exit has been replaced by that calculated by the nonadiabatic model. The influence of the tunneling exit from the nonadiabatic effect can be studied in this way. Compared to the result by the adiabatic model, the influence of the tunneling exit is significant. A similar procedure has also been applied to study the influence of the initial momentum distributions (open star in Fig. 2(d)) and its influence is relatively less significant than that of the tunneling exit. If the nonadiabatic effect has been fully included (the upper triangle in Fig. 2(d)), the calculation result is close to the case of an open star. To study the RETO of complicated noble gas atoms, such as Ne, Ar, and Kr, the multi-electron effects, including the screening and the polarization effect, should be further considered. To simulate the influence of these effects on the electron trajectory, the model potential with careful chosen parameters is applied when calculating the evolution of the electron. The calculated wavelength dependencies of the offset angles and temporal offsets for Ar are presented in Figs. 3(a) and 3(b), respectively. As shown in Fig. 3(a), the offset angle decreases with increasing wavelength and the offset angle is the largest for the results of the adiabatic model. Each of the other physical effects will decrease the offset angle a little bit. The corresponding temporal offsets can be obtained and are presented in Fig. 3(b). With closer inspection, the strongest influence comes from the nonadiabatic effect, which is around 25 as. In contrast, the weakest temporal offset induced by the screening effect is only around 2 as.
cpl-37-4-043201-fig3.png
Fig. 3. (a) Wavelength dependence of the offset angle calculated for Ar with multielectron effect included. The laser parameters are identical to those of Fig. 1(a). (b) The corresponding wavelength dependence of the temporal offsets extracted from the data of (a) and their exponential fits. The laser electric field phase dependences of the forces felt by the electron, which is ejected at $t = T/2$, along the $x$ (c) and $z$ (d) axes are also pictured.
cpl-37-4-043201-fig4.png
Fig. 4. Wavelength dependence of the ratios of RETO of Ne, Ar and Kr to that of H with all the physical effects considered. The ionization potential dependence of the ratio for each atom is depicted in the inset, where the calculation result extracted by the fit procedure is shown with red open circles and the function of (2$I_{\rm p}$)$^{-3/2}$ is given with a black solid line.
To shed more light on the physics behind the results given in Figs. 3(a) and 3(b), the evolution of the electron orbit launched at $t = T/2$ has been analyzed. The phase dependence of the forces felt by the electron along the $x$ and $z$ directions have been pictured in Figs. 3(c) and 3(d). To show clearly the influence of these physical effects, the force induced by the laser electric field has been totally ignored. As shown in these two panels, the impacts of the screening effect and the Stark effect are small, while the ones of the polarization effect and the nonadiabatic effect are relatively strong. In particular, the influences of these physical effects are significant only in the first two cycles after tunneling and decrease with respect to time abruptly. In Fig. 4, the wavelength dependence of the ratios of temporal offsets of Ne, Ar and Kr to that of H are presented. As shown in this figure, all the data show the trend of horizontal lines. We fit the data of each atom with a horizontal line and the obtained ratio is depicted with respect to the ionization potential in the inset. It is found that the ionization potential dependence of the ratio closely follows the function of (2$I_{\rm p}$)$^{-3/2}$. Considering the angular offset and also the temporal offset can be calculated accurately with solving the time-dependent Schrödinger equation (TDSE) for H, these results are useful to extract the RETO of noble gas atoms, which can be further applied to obtain the tunneling time from the attoclock experiments. Further experimental investigations relevant to our theoretical investigations will be very interesting. In conclusion, a semiclassical model with the Coulomb potential effect, the nonadiabatic effect, the Stark effect, and the multielectron effect included has been applied to investigate the RETO for H and noble gas atoms subject to intense EP fields. It is found that the ratio of RETO for the noble gas atom to that of H is a constant with respect to laser wavelength, and the ionization potential dependence of the ratio closely follows the function of (2$I_{\rm p}$)$^{-3/2}$. This work can help to improve the accuracy of attoclock. References Laser-Induced Electron Tunneling and DiffractionPhotoelectron Holographic Interferometry to Probe the Longitudinal Momentum Offset at the Tunnel ExitAdiabaticity in the lateral electron-momentum distribution after strong-field ionizationAttoclock revisited on electron tunnelling timeTunneling times: a critical reviewNote on the Transmission and Reflection of Wave Packets by Potential BarriersValidity of extracting photoionization time delay from the first moment of streaking spectrogramGeneral way to define tunneling timeResolving the time when an electron exits a tunnelling barrierAttosecond angular streakingAttosecond Ionization and Tunneling Delay Time Measurements in HeliumUniversal Attosecond Response to the Removal of an ElectronInterpreting attoclock measurements of tunnelling timesAttosecond angular streaking and tunnelling time in atomic hydrogenAttoclock reveals natural coordinates of the laser-induced tunnelling current flow in atomsUltrafast resolution of tunneling delay timeExperimental Evidence for Quantum Tunneling TimeWigner time delay for tunneling ionization via the electron propagatorKeldysh-Rutherford Model for the AttoclockUnder-the-Barrier Dynamics in Laser-Induced Relativistic TunnelingTunneling Time and Weak Measurement in Strong Field IonizationPlasma perspective on strong field multiphoton ionizationRescattering effects in above-threshold ionization: a classical modelCoulomb potential influence in the attoclock experimental schemeLong-range Coulomb effect in above-threshold ionization of Ne subject to few-cycle and multicycle laser fieldsEnergy and momentum spectra of photoelectrons under conditions of ionization by strong laser radiation (The case of elliptic polarization)Experimental verification of the nonadiabatic effect in strong-field ionization with elliptical polarizationTunneling wave packets of atoms from intense elliptically polarized fields in natural geometryNumerical simulation of high-order above-threshold-ionization enhancement in argonSemiclassical description of high-order-harmonic spectroscopy of the Cooper minimum in kryptonCURRENT METHODS FOR THE STUDY OF THE STARK EFFECT IN ATOMSIonization of oriented carbonyl sulfide molecules by intense circularly polarized laser pulsesStrong-field ionization of polar molecules: Stark-shift-corrected strong-field approximationCoulomb focusing in intense field atomic processes
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