## Abstract

We propose a straightforward approach to directly probe the tunneling time by observing the transition of photoelectron wave packets in strong-field ionization processes, where Coulomb potentials do not affect the results. A circularly polarized laser pulse is used to avoid the impact of scattering electrons on the direct ionization electrons, and a pure transmission photoelectron wave packet can be obtained. Then, a positive tunneling time is extracted. The results demonstrate that the tunneling time is dominated mainly by the laser frequency in some laser intensity range. At the same time, we also investigate the tunneling time by analyzing the instantaneous ionization rate, and the consentaneous results are obtained.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The tunneling ionization is one of the most fundamental and ubiquitous quantum phenomena that depart from classical processes. The schematic can be described as: an electron tunnels out of the atom horizontally through the barrier formed from the Coulomb potential (CP) bended by an additional electric field as shown in Fig. 1(a) [1]. In the tunneling regime, the Keldysh parameter $\gamma =\omega \sqrt{2{I}_{\mathit{ion}}({\epsilon}^{2}+1)}/{F}_{0}$ is less than one, where *F*_{0} is the peak amplitude of the laser electric field, *ω* is the angular frequency, *I _{p}* is the ionization potential, and

*ε*is the ellipticity [2]. However, the question of whether tunneling through the barrier takes a finite time or is instantaneous has been debated over the past 80 years [3,4]. Up to now, many efforts have been made to define [5,6] and measure [7, 8] the tunneling time. But the concept of tunneling time is not well-defined since time is not a quantum operator, and many conflicting definitions based on more or less classical concept have been reported in independent physical regimes. So far, a cathedratic conclusion is still lacking for the quantum tunneling time.

The tunneling time problem raised as early as quantum mechanics is still a highly debated subject. And there have been repeated attempts to address this question since MacColl [4]. In this community, there are mainly four definitions of the tunneling time named as Larmor time [9,10], Büttiker-Landauer time [11], Eisenbud-Wigner time [12], and Pollack-Miller time [13]. The first two definitions depend on the height of the potential and the tunneling times have been called as the resident time; while the other two depend on the incident energy of the particle and the tunneling time have been called as the passage time [3]. On the other hand, tunneling time can also be viewed as average values, rather than deterministic quantities [14, 15]. Landsman et. al. have predicted the time by the probability distribution of tunneling times constructed by using a Feynman Path Integral formulation [16]. Then, the quantum mechanical theory of weak measurement attaches increasingly importance to investigate the tunneling time [17,18].

In recent decades, experiments for measuring tunneling time have been developing. Uiberacker et.al. gave the first real-time observation of light-induced electron tunneling experimentally [8]. The attoclock technique had been used for the prior measurement of tunneling time in strong field ionization of a helium atom [7,19], and got an upper limit on tunneling time of 34 attoseconds which is far less than their theoretical prediction. Afterwards, Keller group also investigated this question using the same technique in helium and argon extended towards higher intensities [20] and lower intensities [16]. Recently, Torlina et. al. reported that no tunneling delays arise in the ionization of single-electron atom, but for the double-electron or multi-electron systems, the interaction of different electrons leads to additional delays [21]. And Camus et. al. had verified the nonzero tunneling time of Argon and Krypton atom theoretically and experimentally [22]. In order to avoid the multi-electron effect, Sainadh et. al. investigated the tunnelling time in atomic hydrogen in their latest work [23].

The attoclock definition of tunneling time relies on the electron momentum vector which is compared to the “watch hand” in the polarization plane of circularly polarized light. In polar coordinates, the angular coordinate directly gives the time information. Thus, tunneling time can be extracted from the angular shift of the “watch hand” from the direction of the electric field vector at the moment of ionization [7,19], as shown in Fig. 1(b). However, the momentum distribution obtained at the end of the pulse is influenced by all the effects in the interaction process, such as the interplay between the external field and the CP, and the tunneling time.

In this work, we investigate the tunneling time by supervising the transition of photoelectron wave packet which we regard as the photoelectron group in the interacting process. It is based on the well-known theoretical assumption that the highest probability for the electron to tunnel occurs at the peak of electric field if the ionization yield is far less than the total electron quantity [19,24]. If the tunneling process needs time, the peak of photoelectron wave packet will appear later. Thus we can obtain the tunneling time from the time delay of the peak [25]. In tunneling processes, the electrons rest in the inner wall of barrier *x _{in}* before tunneling, and are driven into the forbidden zone at one moment by the external electric field. Afterwards they pass the tunnel exit

*x*and fly away as illustrated in Fig. 1(a). There will be a maximum probability that electrons enter into the barrier at one moment because of the electric field. We probe the character of the electron group integrally, thus all the effects are included in the process, such as the electron recollisions under the barrier [26]. Here, we defined the group delay time as resident time that the photoelectron spends under the potential barrier, i.e., the tunneling time. And we assume the electron is ionized as soon it leaves the tunnel exit.

_{exit}We choose hydrogen as the target atom exposed in circularly polarized field which is a new powerful tool for investigating electron dynamics [27]. Here, the full quantum method based on solving the time-dependent Schrödinger equation (TDSE) in the length gauge is used [28–30]. The theoretical method is reviewed in section II. Then we give the results and discussions in Section III. Unless otherwise stated, atomic units are utilized.

## 2. Theoretical method

Considering the circular polarization of the laser pulse and the symmetry of hydrogen atom in strong-field ionization, we choose the three-dimensional spherical coordinate system to investigate the dynamics [31]. The time-dependent wave function is expanded in spherical harmonics as [32],

*χ*

_{l,m}(

*r*,

*t*) is represented in the sine-type discrete variable representation [33]. And the TDSE is solved in the length gauge,

**H**

_{0}represents the field-free Hamiltonian, and

*V*

^{(F)}=

**r**·

**F**is the laser-atom interaction under dipole approximation. The second-order split-operator scheme is employed to propagate the wave function fast and efficiently [34], and an exponential absorption potential is introduced at the boundaries.

In order to solve the three dimensional TDSE efficiently, we take advantage of the Wigner rotation technique [35,36]. As the circularly polarized light can be deemed to the rotation of the linearly polarized light, we make a linearly polarized field to interact with the time-dependent wave function and then rotate the wave function by Wigner rotation matrix in each propagation time step [35,37]. Thus, the linearly polarized laser is equivalent to a circularly polarized laser to the revolving atomic system. The total Hamiltonian used in circularly polarized laser field [38,39] is combined by linear Hamiltonian and Wigner rotation matrix.

In the time propagation, the amount of unionized electron is defined by projecting the time-dependent wave function at evolution time *ψ*(**r**, *t*) onto the bound eigenstates *ψ _{j}*(

**r**) which are obtained by solving the time-independent Schrödinger equation by diagonalizing the field-free Hamiltonian. Therefore, the ionization yield can be written as,

*P*=

_{rat}*∂P*(

_{ion}*t*)/

*∂t*, which can be used to evaluate the dynamics of electron with the laser pulse in real time.

At the end of the propagation, the two-dimensional photoelectron momentum distribution is obtained by projecting the final wave function onto the scattering states,

*F*

_{k,l}(

*r*) is related to the energy function

*χ*

_{E,l}(

*r*) via ${F}_{k,l}(r)={F}_{k,l}(r)=\sqrt{k}{\chi}_{E,l}(r)$, where

*χ*

_{E,l}(

*r*) is the reduced radial function corresponding to the energy

*E*.

*σ*(

_{l}*k*) = arg[Γ(

*l*+ 1 −

*i*/

*k*)] and

*δ*(

_{l}*k*) = arctan[

*K*(

_{l}*k*)] are Coulomb phase shift and non-Coulomb phase shift, respectively, and

*K*(

_{l}*k*) is called the K-matrix element. Inserting the scattering wave function into Eq. (4) gives the momentum distribution,

*r*range of 0.0–300 a.u., the expansion in spherical harmonics is truncated at

*l*≤

*l*= 55, and the results are checked for convergence. The initial sate is the ground eigenstate of hydrogen atom. A propagation time step of 0.02 a.u. is used. The circularly polarized laser pulse is assumed to oscillate in the xz plane in Cartesian coordinate system, that is,

_{max}*ϕ*= 0 or

*π*in spherical coordinate system, and the circularly polarized laser electric field includes two vectors:

**F**

*(*

_{x}*t*) =

*F*

_{0}

*f*(

*t*) cos(

*ωt*+

*φ*)

*e**and*

_{x}**F**

*(*

_{z}*t*) =

*F*

_{0}

*f*(

*t*) sin(

*ωt*+

*φ*)

*e**, with*

_{z}*φ*the carrier-envelope phase (CEP), $f(t)={\text{sin}}^{2}(\omega t/2N)/\sqrt{2}$ the envelope and

*N*the number of the optical cycles.

## 3. Results and discussion

As is known, the highest probability for the electron to tunnel occurs at the peak of laser field and the direction of photoelectron momentum is consistent with that of the electric field vector at the moment when electron instantaneously tunnels out [19], based on the quasistatic approximation. However, there is an angular shift of momentum distribution (Fig. 1(b)) which is influenced by the interplay between the external field and the CP and the tunneling time. In Fig. 2, we calculate the momentum angular distributions clearly, which are obtained by integrating the momentum distributions over the radial direction in circularly polarized laser pulse with different intensities. The intensities are 1.0 × 10^{14} W/cm^{2}, 1.5 × 10^{14} W/cm^{2} and 2.0 × 10^{14} W/cm^{2}, respectively. The wavelength is 800 nm and duration is 3 cycles. We find that the corresponding angular shift indicated by Δ*θ*_{1}, Δ*θ*_{2} and Δ*θ*_{3} decreases with the intensity increasing (Δ*θ*_{1} > Δ*θ*_{2} > Δ*θ*_{3}) which agrees with the results in Ref. [21]. As the momentum is vector which has direction as well as magnitude, and the direction is related to the CP [21] closely, the angular shift of photoelectron momentum distribution cannot be used to measure the tunneling time accurately, which is affected by both tunneling time and CP.

In fact, the photoelectron momentum distribution includes another observable characteristic, i.e. the ionization probability, which can be used to investigate the tunneling process [40]. It is a scalar, so the CP do not affects the results for extracting the time information. Similarly, we investigate the tunneling time by recording the transition of photoelectron wave packet during the interacting process. The density of wave function (|*ψ*(**r**, *t*)|^{2}) in the radial and time coordinate is shown in Fig. 3(a). For a certain radial coordinate when *r* ≥ *x _{in}*, there is a peak (the blue curve in the figure) which implies the highest probability of electron entering the barrier and ionized in the time process. This is related to the only peak intensity of circularly polarized pulse. What is more, the peak appears late at a larger radial place, so electrons need a real time to transfer from the entry of potential barrier to the exit or the farther ionization zone. In order to get a pure motion information of the ionization electrons, we extract the transmission density of wave function (|

*ψ*(

**r**,

*t*)|

^{2}− |

*ψ*(

**r**,

*t*= 0)|

^{2}) in the radial and time coordinates in Fig. 3(b). There are also peaks at different radial places in the ionization process, and the peaks appear late at large radial coordinate. What is more, there is an another obvious feature in Fig. 3(b) that there are peaks all the time at a certain radial place (as shown the black dash line), which is defined as the entry of the barrier.

To investigate the character of entry of the potential barrier clearly, we obtain the photoelectron density distribution, extracted from the transmission density of wave function in Fig. 3(b), as a function of radial coordinate at different time as shown in Fig. 4(a). The results are extracted at the moment of 105 a.u., 115 a.u., 125 a.u., 135 a.u., 145 a.u., 155 a.u. and 165 a.u. in the time propagation. And all the density peaks appear at a fixed location, i.e. the entry (*x _{in}*) of potential barrier. In addition, we calculate the photoelectron density distribution in the same laser field but with different intensities in Figs. 4(b) and 4(c). The laser intensities are 1.0 × 10

^{14}W/cm

^{2}(a), 1.5 × 10

^{14}W/cm

^{2}(b) and 2.0 × 10

^{14}W/cm

^{2}(c), respectively. We obtain similar results in Figs. 4(b) and 4(c) with those in Fig. 4(a), i.e. the peaks appear at the entry of the barrier. What is more, the sites of the barrier entrance are the same for the three cases. Thus, the location of the entry of potential barrier has nothing to do with the external field intensity, and is only dominated by CP.

The peak of the transmission density of wave function represents the highest ionization probability. We extract the moments when the peaks appear at different radial locations, and constitute a curve which implies the most likely track of photoelectrons. In Fig. 5, we obtain the ionization peak curves of hydrogen exposed in 3 cycles duration, 800 nm wavelength circularly polarized laser fields with intensity of 1.0 ×10^{14} W/cm^{2}, 1.5×10^{14} W/cm^{2} and 2.0×10^{14} W/cm^{2}, respectively. The slash shadow zones indicate the potential barriers, where the entry of the barrier is obtained from the above analysis in Fig. 4, and the width of the barrier is given approximately as $\sqrt{2}{I}_{p}/{F}_{0}$ in the strong-field approximation (SFA) [2]. Thus, the time difference Δ*t* between the moments when electrons appear at the entry and exit of the barrier means the tunneling time taken by electrons passing through the potential barrier, marked as Δ*t*_{1}, Δ*t*_{2} and Δ*t*_{3} in Figs. 5(a)–5(c) for the three cases. Furthermore, the three time gaps are almost equivalent (13 a.u.), which demonstrates that the tunneling time is almost not influenced by the laser intensity. Obviously, the value is larger than the results obtained by attoclock experiments, due to the CP do not influence the calculated tunneling time in our straightforward approach. But the experimental results cannot avoid this influence, where the tunneling time is defined by extracting the angular shift of photoelectron momentum distribution which is affected by the CP.

In order to check the influence of CP which affects the angular offset of momentum and brings ambiguity in deducing tunneling time [21], we attach an absorption function of CP starting at 3 a.u. ∼ 50 a.u. in our calculations to attenuate the CP [41]. The curve of the peak of transmission density obtained using this approach are almost invariable when adding the absorption function at 3, 8, 14, 20, 30 a.u., respectively, as shown in Fig. 6. Thus, the CP do not affect the results when we extract the tunneling time. However, the photoelectron momentum angular shift is affected by the CP [21]. And the CP will affect the motion as soon the electrons enter into the barrier, which can not be avoided in experimentation. That is why there are small difference between the experimental results and our calculated tunneling time [7,23].

What is more, we calculate the tunneling time by analyzing the instantaneous ionization rate in Fig. 7 [40,41] with the same laser fields as those in Fig. 5, to verify the relationship between tunneling time and the external field intensity. Here, we obtain the instantaneous ionization rate by deriving from the ionization yield which is defined by projecting time-dependent wave function onto the bound eigenstates in the whole interaction space in Eq. (3). So the quantum model excludes any classical concept, such as the width, height of the barrier. We defined the tunneling time by supervising the time delay between the emergence instant of the largest ionization rate and the moment when the electric field is the strongest. It is obvious that the time delay do not change in the external fields with different intensity in our calculation. This confirms the conclusion obtained in Fig. 5.

In addition to this, the peaks of the transmission electron density are also extracted with different laser wavelengths and intensities in tunnel regime in Fig. 8. The wavelength are 1200 nm (a–c) and 1600 nm (d–f) and the intensities are 1.0 ×10^{14} W/cm^{2} (a,d), 1.5×10^{14} W/cm^{2} (b,e) and 2.0 ×10^{14} W/cm^{2} (c,f), respectively. When we change the laser intensity with a certain wavelength, the tunneling time is almost invariable agreeing with the results in Fig. 5 and Fig. 7. However, the tunneling time decreases with the wavelength increasing (Δ*t*_{1200} = 10.5 a.u. > Δ*t*_{1600} = 9.6 a.u.). This feature implies that the photoelectron dynamics process under the barrier is closely related with the laser frequency which primarily dominates the tunneling time.

## 4. Conclusion

In summary, we investigate the tunneling ionization theoretically. First, the photoelectron momentum angular distribution of hydrogen exposed in a circularly polarized laser field is calculated. And an angular shift is obtained, but it is influenced by CP at the same time, thus, it cannot be used to measure the tunneling time accurately. Then, we extract the tunneling time by supervising the transition of photoelectron wave packet in the interacting process, where the CP do not affect the results. In our calculation, we get a positive tunneling time, and the results demonstrate that the tunneling time is dominated mainly by the laser frequency and almost uninfluenced by the laser intensity in some small range. In order to verify the results, we calculate the tunneling time by analyzing the instantaneous ionization rate again, and the same conclusion is obtained.

## Funding

National Natural Science Foundation of China (NSFC) (11504412, 11674363).

## Acknowledgments

We acknowledge valuable discussions with Xue-Bin Bian, Mu-Zi Li, Tao-Yuan Du and the Supercomputing Environment furnished by Ke-Li Han Group, DICP, CAS.

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