Tunneling ionization of atoms and molecules induced by intense laser pulses contains the contributions of numerous quantum orbits. Identifying the contributions of these orbits is crucial for exploring the application of tunneling and for understanding various tunneling-triggered strong-field phenomena. We perform a combined experimental and theoretical study to identify the relative contributions of the quantum orbits corresponding to the electrons tunneling ionized during the adjacent rising and falling quarter cycles of the electric field of the laser pulse. In our scheme, a perturbative second-harmonic field is added to the fundamental driving field. By analyzing the relative phase dependence of the signal in the photoelectron momentum distribution, the relative contributions of these two orbits are unambiguously determined. Our results show that their relative contributions sensitively depend on the longitudinal momentum and modulate with the transverse momentum of the photoelectron, which is attributed to the interference of the electron wave packets of the long orbit. The relative contributions of these orbits resolved here are important for the application of strong-field tunneling ionization as a photoelectron spectroscopy for attosecond time-resolved measurements.

## 1.

## Introduction

Laser-induced tunneling ionization of atoms and molecules is a fundamental process in strong-field physics. It is the first step for various intriguing phenomena in attosecond science, such as high-order above-threshold ionization,^{1} high-order harmonic generation,^{2} and enhanced double/multiple ionization.^{3}^{–}^{5} Therefore, detailed understanding of the tunneling step is of fundamental importance for attosecond science. In the past decades, great efforts have been made to reveal the dynamics of strong-field tunneling. For instance, the questions of how long it takes the electron to tunnel through the potential barrier^{6}^{–}^{8} and when the tunneling electron appears at the outside of the barrier have been surveyed in depth.^{9}^{–}^{11} The features of the tunneling electron wave packet have also been widely studied.^{12}^{–}^{16} Theoretically, tunneling ionization of atoms and molecules in a strong laser field could be understood with the quantum-orbit (QO) model.^{17} In the QO model, the relationship between the ionization time and the final momentum of photoelectrons for each orbit was established. For plenty of the time-resolved measurements, the attosecond dynamics were extracted based on this concept.^{18}^{–}^{22} The concept of the QO model also provides a transparent picture for dynamics of various tunneling-triggered processes.

One intrinsic issue in the QO model of tunneling ionization is that for the same final momentum multiple quantum orbits are involved.^{17}^{,}^{23}^{,}^{24} The interferences of these orbits obscure the insights into the attosecond time-resolved information recorded by the photoelectrons. Thus, identifying the quantum orbits is a prerequisite for understanding the dynamics of the strong field processes and for the time-resolved measurements. Specifically, there are two types of orbits within one optical cycle for strong-field tunneling ionization, referred to as the long and short orbits, which correspond to the electrons tunneling being ionized during the adjacent ascending and descending quarter cycles of electric field, respectively.^{25}^{,}^{26} For the electron ionized during the descending quarter cycle (the long orbit), the electron could return back to the parent ion, resulting in high-order harmonic generation or high-energy photoelectrons, while the electron ionized during the ascending quarter cycle (the short orbit) usually does not contribute to these signals. Thus, for the measurements based on these signals, only the long orbit is involved. However, for the photoelectrons with low energy (lower than $2{U}_{p}$, where ${U}_{p}$ is the ponderomotive energy), which has also been widely used to extract ultrafast electron dynamic information,^{18}^{–}^{22} both the long and short orbits contribute. Due to the Coulomb focusing effect after tunneling, it is accepted that the long orbit dominates over the short orbit for the signals in the photoelectron momentum distribution (PEMD), and thus temporal information is usually obtained based on the assumption that the long orbit contributes exclusively. Obviously, this approximation is questionable. In a recent experiment, it has been shown that the extracted ionization time with this approximation deviates significantly from the theoretical prediction.^{27} Recently, with the scheme of an orthogonal two-color field, the relative contributions of the long and short orbits were theoretically estimated.^{28}^{,}^{29} In that scheme, the relative contributions were obtained with two time-to-momentum maps of the short and long orbits. However, the effect of the Coulomb interaction on the ionization time in the orthogonal two-color field cannot be measured,^{30} and thus the maps used there are calculated based on a strong-field approximation where the ionization time is inaccurate. Instead, the parallel two-color field could record the Coulomb effect on the ionization time, and it can be employed to identify the contributions of the long and short orbits.^{30} So, in this study, we perform a combined experimental and theoretical study to determine the relative contributions of the long and short orbits in tunneling ionization with the parallel two-color field. We monitor the photoelectron yield as a function of the relative phase at each momentum of the PEMD. This relative phase dependence of the signal is determined by the ionization time, and thus it is different for the long and short orbits. With the Coulomb-corrected strong-field approximation (CCSFA), we calculate the relative phase dependences of the photoelectron yields for the long and short orbits separately. By comparing the measured relative phase dependence of the photoelectron yield with these theoretical data, the individual contributions of the long and short orbits in tunneling ionization are qualitatively determined. The obtained results show that their contributions depend on both the longitudinal and transverse momenta, and in certain regions of the PEMD, the contributions from the short orbit could be even higher than that of the long orbit. This information is crucial for accurately determining the ionization time of photoelectron in the PEMD, which is the basis of the time-resolved measurements with the photoelectron from tunneling.

## 2.

## Methods

In our scheme, the parallel two-color field consists of a strong 800 nm field and a much weaker second harmonic (SH) field, and it is polarized along the $x$ direction. In the experiment (see the Supplemental Materials for details about our experiment and the numerical calculation with CCSFA, which includes Ref. 31), the SH pulse was produced by frequency doubling a femtosecond laser pulse (25 fs, 800 nm, 1 kHz) from a Ti:sapphire laser system using a $300\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$-thick $\beta $-barium-borate crystal. The parallel two-color pulse was generated in a phase-locked Mach–Zehnder interferometer scheme. The relative phase $\phi $ was finely controlled by the phase-locking system.^{32}^{,}^{33} The parallel two-color pulse was tightly focused onto a supersonic gas jet of Ar by a concave silver mirror ($f=15\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{cm}$) in an ultrahigh vacuum chamber of a velocity map imaging setup.^{34} The PEMDs were then recorded as a function of the phase delay between the FM and SH fields, using a step size of $0.1\pi $. The peak intensities of FM and SH fields were $\sim 1.2\times {10}^{14}$ and $\sim 0.3\times {10}^{11}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, respectively. The relative phase of the two-color field is calibrated by comparing the photoelectron yield obtained from the experiment with that obtained by solving the time-dependent Schrödinger equation (see the Supplemental Materials for details).

## 3.

## Results and Discussion

Figures 1(a)–1(c) show the measured two-dimensional PEMDs for the parallel two-color field with relative phases $\phi =0$, $0.5\pi $, and $\pi $, respectively. As mentioned above, numerous quantum orbits contribute to the PEMDs, giving rise to the interference structures such as the above-threshold ionization rings^{35} and the spider-like holographic pattern.^{18}^{,}^{36} The interference of the long and short orbits that we will focus on is not visible due to the laser focal volume effect. This does not affect the determination of their relative contributions in this study. In our experiment, the laser intensity of the SH field is three orders of magnitude weaker than that of the FM field, and thus its perturbation on the PEMDs is tiny. To reveal the effect of the weak SH more clearly, we calculate the normalized difference (ND) of the PEMDs. The ND is defined as

## Eq. (1)

$$\mathrm{ND}(\mathbf{p};\phi )=\frac{Y(\mathbf{p};\phi )-{Y}_{\mathrm{avg}}(\mathbf{p})}{Y(\mathbf{p};\phi )+{Y}_{\mathrm{avg}}(\mathbf{p})},$$To quantitatively characterize the relative phase dependence of the signal, we further analyze the NDs employing the phase-of-the-phase spectroscopy,^{37}^{,}^{38} which has been proposed to reveal the different orbits in strong-field ionization.^{39}^{–}^{41} In this method, the higher-order Fourier component of NDs is very small, and thus it is canceled out safely here. Figure 2(a) shows an example of the signal at the momentum $\mathbf{p}=(-0.6,0.1)$ a.u. as a function of the relative phase. It suggests that the photoelectron yield on the relative phase $\phi $ at momentum $\mathbf{p}$ can be described as

## Eq. (2)

$$\mathrm{ND}(\mathbf{p};\phi )=P(\mathbf{p})\text{\hspace{0.17em}}\mathrm{cos}[\phi -{\phi}_{m}(\mathbf{p})],$$In the two-color field, it is intuitive that the photoelectron yield at momentum $\mathbf{p}$ maximizes at the relative phase where the instantaneous electric field at ionization maximizes.^{30} Thus, the optimal phase is determined by the ionization time. However, the situation is more complex when multiple orbits contribute to the signal in the PEMDs. In Fig. 3(a), we illustrate the two orbits (the long and short orbits) within one optical cycle that contribute to the PEMDs. Due to the different ionization times of the long and short orbits, the photoelectron yields corresponding to these two orbits should maximize at different relative phases. The experimentally measured optimal phase provides us with a way to assess the contributions of these two orbits.

To this end, the key information is the optimal phase for each orbit. In the previous studies with a scheme of an orthogonal two-color field, this information is obtained with the strong-field approximation, where the Coulomb effect on the tunneling ionized electron is neglected.^{28}^{,}^{29} For more accurate optimal phases, it is necessary to include the effect of Coulomb interaction on the relationship of the ionization time and final momentum. This is crucial for our parallel two-color field. Thus, we calculate the modulation of the photoelectron yield as a function of the relative phase $\phi $ at each momentum with the CCSFA (see Supplemental Materials for details), the accuracy of which has been well confirmed.^{42}^{–}^{45} Figures 3(b) and 3(c) show the calculated NDs as functions of the relative phases for the long and short orbits, respectively. Here, we display the signals at zero transverse momentum. It is shown that for the long orbit the phase where the ND maximizes varies from $1.85\pi $ to $2.05\pi $ as ${p}_{x}$ increases from $-1.0$ to $-0.2$ a.u., and for the short orbit, the phase changes from $1.45\pi $ to $1.25\pi $. For each momentum, we repeat our calculation in the PEMD and extract the optimal phase ${\phi}_{m}$. The results of the long and short orbits are shown in Figs. 3(d) and 3(e), respectively.

In the measured PEMDs, both the long and short orbits contribute, and the relative phase ${\phi}_{m}$ should locate within the phases window determined by the long and short orbits. Two examples are shown in Figs. 4(a) and 4(b). For $\mathbf{p}=(-0.5,0)$ a.u. in Fig. 4(a), the optimal phases ${\phi}_{m}$ determined by the long and short orbits are $2.05\pi $ and $1.3\pi $. As expected, the optimal phase of the measured data is located between these values, and it is closer to the value of the long orbit, implying that the long orbit has the dominant contributions. For $\mathbf{p}=(-0.5,0.2)$ a.u., the phase of the experimentally measured data shifts toward the short orbit, meaning that the contributions of the short orbit increase.

The measured phase dependence of the ND at each momentum can be formulated as (see Supplemental Materials for details)

## Eq. (3)

$$\mathrm{ND}(\mathbf{p};\phi )=\alpha \text{\hspace{0.17em}}\mathrm{cos}[\phi -{\phi}_{m}^{\mathrm{L}}(\mathbf{p})]+\beta \text{\hspace{0.17em}}\mathrm{cos}[\phi -{\phi}_{m}^{\mathrm{S}}(\mathbf{p})],$$^{18}

^{,}

^{36}At the interference maxima or minima, the contributions of the long orbit are enhanced or suppressed, leading to the modulation of the ratio $\alpha /\beta $ with the transverse momentum.

In the time-resolved measurements using photoelectrons from strong-field tunneling ionization, a map of time-to-momentum is necessary for extracting attosecond dynamics. In previous studies, this map was established using the long orbit, because it was assumed that the contributions of the long orbit dominate the signal in the PEMDs.^{27} Due to the constructive holographic interference, it is true that for the signal in the PEMD along the laser polarization direction, the contribution of the long orbit is larger than that of the short orbit. However, the ratio is only about four times higher, as shown in Fig. 4(c). Neglecting the contributions of the short orbit leads to the inaccuracy of the time-to-momentum map. Specifically, at the minimum of the holographic interference, the contributions of the short orbit could be even larger than that of the long orbit, as shown in Fig. 4(d). So, the time-resolved studies of revealing attosecond dynamics with the time-to-momentum map solely based on the long orbit are invalid. The inaccuracy of this map induced by the multiple orbits has also been realized in a very recent study.^{41} Furthermore, we should mention that in our scheme the multiple cycle pulses are used. For the multiple cycle pulses, the electric field during the adjacent rising and falling quarter cycles is the same, and the relative contributions of these two orbits are the same from cycle to cycle. However, in the few cycle pulses, the electric field of the adjacent quarter cycles is different, and the relative contributions of these orbits are different from cycle to cycle. Thus, for the few cycle pulses the scheme will break down.

## 4.

## Conclusions

We have experimentally determined the contributions of the long and short orbits in strong-field tunneling ionization. Due to the holographic interference of the electrons of the long orbits, the relative contributions of the long and short orbits depend on both the transverse and longitudinal momenta. At the holographic interference minimum, the contributions of the long orbit are even less than that of the short orbit. In previous studies, the ionization time of the photoelectron is usually determined based on the assumption that the long orbit dominates the photoelectron yields. Our results indicate this assumption is invalid, and it will induce inaccuracy in determining the time-to-momentum map. Our study provides a complete map to quantify the relative contributions of different quantum orbits for the signal at each momentum in the PEMDs. This information is necessary for establishing accurate correspondence between the ionization time and the final photoelectron momentum in the PEMDs, and thus it is important for the application of strong-field tunneling ionization as a photoelectron spectroscopy to probe atom and molecule dynamics in attosecond scale.

## Acknowledgments

This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFA0308300) and the National Natural Science Foundation of China (Grant Nos. 11874163, 12021004, 11627809, 11934006).

## References

## Biography

**Jia Tan** is currently a PhD graduate student with Professor Yueming Zhou at Huazhong University of Science and Technology (HUST), China. Her main research interest is ultrafast dynamics of atoms and molecules in strong laser fields.

**Shengliang Xu** is a PhD student with Professor Qingbin Zhang at HUST. Before that he received his bachelor’s degree in physics from Nanchang University. His main research interest is ultrafast dynamics in strong laser fields.

**Xu Han** is a PhD student with Professor Qingbin Zhang at HUST, where he received his bachelor’s degree in physics. His main research interest is related to molecules and nanostructures exposed to strong laser fields.

**Yueming Zhou** is a professor at the School of Physics, HUST, Wuhan, China. He received his PhD from HUST and did his postdoc as a JSPS research fellow at the University of Electro-Communications. He was selected as an outstanding young scholar of NSFC. His research focuses on the ultrafast electron dynamics of atoms and molecules in strong laser fields.

**Min Li** is a professor at the School of Physics, HUST, Wuhan, China. He received his PhD from Peking University in 2015. He was selected as an outstanding young scholar of NSFC. His research focuses on the tunneling ionization of atoms and molecules induced by strong laser fields, quantum control, and molecular dissociation.

**Wei Cao** is a professor at the School of Physics, HUST, Wuhan, China. He received his PhD in physics from Kansas State University in 2014. From 2014 to 2017, he worked as a postdoctoral fellow at Lawrence Berkeley National Lab in the United States. His research focuses on the generation of the attosecond pulse and its application for ultrafast process diagnosis.

**Qingbin Zhang** is a professor at the School of Physics, HUST, Wuhan, China. He received his PhD from HUST in 2011. From 2016 to 2017, he was a visiting scholar at Wayne State University in the United States. His current research interest is related to high-resolution intense field photoelectron full-dimensional detection technology as well as ultrafast optical field manipulation based on the above-mentioned innovative technology.

**Peixiang Lu** is a professor and the vice director at Wuhan National Laboratory for Optoelectronics, HUST, Wuhan, China. He received his PhD from Shanghai Institute of Optics and Fine Mechanics. He was selected as a Cheung Kong Scholar Chair Professor, the distinguished young scholar of NSFC, and he was selected as OSA fellow in 2016. His research interest is strong ultrafast optics.