1. Introduction

The tunneling of an electron through a potential barrier is one of the most important quantum processes in strong-field ionization. The question of whether tunneling through the barrier takes a finite time or is instantaneous has been debated over the past 80 years [1, 2]. The concept of tunneling time is not well-defined in quantum mechanics since time is not a quantum operator. Many definitions based on classical physics model have been reported in independent physical regimes [3, 4]. In tunneling ionization, an electron tunnels out of the atom horizontally through the barrier formed from the Coulomb potential bended by additional electric field [5, 6]; the picture is described in Fig. 1(a). The barrier width is given approximately as I_p/F in the strong field approximation (SFA) [7], where I_p and F are the ionization potential and the electric field strength, respectively. So far, cathedratic conclusions are still lacking for the quantum tunneling time.

 

Fig. 1 (a) The effective potential V(r) is formed from the Coulomb potential (CP) bended by additional electric field (EF). An electron, which is represented by wave function ψ₀, can escape the atom through the potential barrier in tunneling regime. x_in and x_exit are the entry and exit of the barrier. The ionization rate (blue dash line) should get the largest at the peak of the EF (red solid line) (b) or when the EF (red solid arrows) points to the maximum of bound electron density (color scale map) (c). If the tunneling takes some time, however, the maximum ionization rate (blue solid line) appears late (Δt) after the moment when the electric field gets the strongest or points in the maximum bound electron density direction.

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Experiments for measuring tunneling time have been developing. A prior one is called attoclock experiment which takes the angular streaking technique in close-to-circularly polarized laser ionization [8] and a tunneling time with an upper limit of 34 as is obtained in the given range of laser intensity (2.3 × 10¹⁴ to 3.5 × 10¹⁴ W/cm²). However, the results are far less than their theoretical prediction, and they drew a conclusion that “there is no real tunneling time”. Recently, Keller et al. employed the same experimental method including Stark shifts and multi-electron effects in stronger field, and did not resolve any tunneling time [9], but they got a real tunneling time over a large intensity regime in their following work [10, 11]. Meanwhile, another experiment investigated the tunneling time in high harmonic generation and no tunneling time could be inferred as well [12, 13]. Nonetheless, these experiments did not give a detailed suitable theoretical explanation or explicit concept from the point of view of quantum mechanics, although more recent work reported the vanishing tunneling delays based on a quasiclassical model [14], or using a quantum mechanical solution combined with classical back-propagation [15].

At present, most theoretical definitions of tunneling time are related to the barrier. For instance, the widely used Larmor time [16, 17] and Büttiker-Landauer time [18] which have been called as the resident time depend on the height of the potential, Eisenbud-Wigner time [19] and Pollack-Miller time [20] depend on the incident energy of the particle, which have been called as the passage time. What is more, the tunneling ionization can also be described by the wavepacket approach, but this method is beset by the fact that we have to resolve the wavepacket technically. For instance, the arriving wavepacket peaks to the barrier do not turn into the transmitted wavepacket completely [21]; the packet may include components with a high energy above the barrier, and this part transmits more effectively, so the delay time is actually unconnected to tunneling [22]; the time of electron wavepackets arriving ahead or behind the barrier is not measured easily [23]. Summarily, this method relies on the tricks in viewing the wavepacket in the coordinate frame of the barrier [24]. Therefore, a precise quantum definition avoiding classical concept is requisite.

In this work, we introduce a quantum method to investigate the tunneling time by supervising the instantaneous ionization rate. It is based on the two well-known theoretical assumptions which are serviceable to interpret experimental results of tunneling ionization. One, 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 as shown by the dash line in Fig. 1(b) [25, 26]; The other one, the ionization rate can also get largest when the electric field points in direction where the bound electron density is maximum [27], such as the 3p electron state of argon atom exposed in a circular polarized laser pulse as shown in Fig. 1(c). If the tunneling process takes some time, the maximum ionization rate [blue solid line in Figs. 1(b) and 1(c)] appears late, i.e., there is a delay time Δt between the instant of the largest ionization and the moment when the electric field gets strongest or points in the maximum bound electron density direction. Thus we can obtain the delay time by observing the time shift of the largest ionization rate. In tunneling process, the electrons rest on 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_exit and fly away [Fig. 1(a)]. There will be a maximum electron probability at the time entering into the barrier because of the electric field or the bound electron density peak, and a corresponding subsequent ionization peak after tunneling. Then, the delay time can be deemed as transit time the photoelectron spends through the potential barrier, that is, the tunneling time [28].

2. Theoretical methods

As well known, the electronic dynamics of atom in strong laser field is described by the three dimensional time-dependent Schrödinger equation (TDSE) [29] under single-active-electron model (atomic units are utilized)

In the time propagation, the unionized electron amount is defined by projecting time-dependent wave function ψ(r, t) at evolution time t onto the bound eigenstates ψ_j(r). Therefore, the ionization yield can be written as follows:

The eigenstates can be obtained by solving the time-independent Schrödinger equation by diagonalizing the field-free Hamiltonian in spherical coordinate frame with equidistant grids of 4096 points in the spatial radial, a grid step of 0.26 a.u. The expansion in spherical harmonics is truncated at l_max = 55. We perform the wavepacket-propagation with a time step of 0.02 a.u., starting from the ground state wavefunction of atom in free field. Then, the instantaneous ionization rate is derived from the ionization yield, P_rate = ∂P_ioni(t)/∂t.

3. Results and discussion

We investigate the hydrogen atom tunneling dynamics exposed to a half cycle linearly polarized laser pulse. The half cycle is used to avoid the photoelectron recollision. Here the F_z is only included in the electric field. In the tunneling regime, the calculation is performed at γ = 0.25 and γ = 0.35, as shown in Fig. 2(a). The Keldysh parameter $\gamma =\omega \sqrt{2I_{ion}}/F_0$ is used to characterizes the tunneling γ ≪ 1 and multiphoton ionization γ ≫ 1 [7]. The standard potential V^(a)r = 1/r is utilized in this three-dimensional model. In the two cases, both of the peaks in ionization rate curves appear later than the instant of electric field maximum t₀. The delay time Δt is much less than the pulse duration time $T_d=2n\pi \sqrt{2I_{ion}}/(\gamma F_0)$ (n is the cycle number). Thus, the potential barrier is deemed to be constant in the tunneling process. We fix the laser intensity but increase γ, the delay time Δt increases (Δt₂ > Δt₁). As the intensity is constant, the wavelength will decrease with the γ increasing. This implies the delay time Δt is related to the interaction of electric field and tunneling electron. The wave function of the tunneling electron driven through the potential barrier is shown in Fig. 1(a). For shorter wavelengths, the electron absorbs less energy from the laser field in the tunnel process and hence takes longer time to cross the potential barrier.

 

Fig. 2 (a) Ionization rate as a function of time. The delay times are marked by Δt₁ and Δt₂ for the conditions with Keldysh parameter of γ = 0.25 (T_d = 262 a.u.) and γ = 0.35 (T_d = 186 a.u.), respectively. (b) Our determined quantum delay time τ_Q versus the probability current delay time τ_J by [28], and the Wigner time τ_W by [34, 35] for different electric field intensity. The Keldysh parameter takes an invariable value γ = 0.35.

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Our determined delay times have values in the same order as previous reports [28] shown in Fig. 2(b) at Keldysh parameter γ = 0.35. The quantum delay time τ_Q is comparable with the probability current delay time τ_J which is defined by the photoelectron probability current through a virtual detector at the barrier exit [28]. Both of the results show the same trend. Our calculation also agrees with the Wigner time τ_W which is extracted from the quantum mechanical Wigner trajectory t_W(x) [34, 35], where τ_W = t_W(x_exit) − t_W(x_in)stands for the lagging of the trajectory traveling from entry x_in to exit x_exit of the barrier. Here, our defined quantum delay time does not rely on the classical concept.

To avoid rescattering effects led by multiple returns of the electron towards the core in a multi-cycle linear polarization pulse [36], we employ an atom with anisotropic electronic distribution on the ground state exposed to a circularly polarized laser pulse. The circularly polarized pulse has many advantages for investigating electron dynamics [37] and has been used wildly in recent work [38–40]. In this case, the duration is related to the envelope f(x), and the total electric field intensity varies slowly, i.e., there is only one maximum in the whole duration. Therefore, we have to apply this kind of pulse to an atom with anisotropic electronic distribution, e.g., an argon atom with the initial ground state (3p_z) which looks like a pea and have two maximums along z axis, as exhibited in Fig. 1(c). The three dimensional atomic potential is taken from Ref. [41]. There are two peaks in ionization rate curve corresponding to the two bound electron density maximums in a single cycle period as shown in Fig. 3(a). When the electric field points to the ±z axis (F_x = 0 and $F_z=F_{\text{total}}=F_0/\sqrt{2}$) where the bound electron density is maximal, the ionization peaks come to appear.

 

Fig. 3 (a) The time-dependent ionization rate and laser electric field components of 9 optical cycles circularly polarized laser pulse. (b) The shifts of peaks in time-dependent ionization rate curve from the time when the electric field points to the maximum of bound electronic density. the triangle blue line denotes quantum delay time and the circle red line denotes the delay time extracting from the photoelectron probability current at the barrier exit. (c) Potential barriers formed by CP and EF at one moment. The cutting positions of CP are taken to be 4.1, 6.1, 8.1, 10.1, 12.1 a.u., respectively. (d) And the corresponding delay times of peaks in time-dependent ionization rate curve in a 9 cycles flat-top circularly polarized laser. The laser intensity is 1.8 × 10¹⁴ W/cm² and wavelength is 800 nm.

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Obviously, there is a time offset for every peak in the curve, and the corresponding delay time can serve as an indication for the tunneling time as shown in Fig. 3(b). What cannot be neglected, however, there are greater offset values at front (positive) and end (negative) of the pulse, which is dominated by the Coulomb potential at both ends of the pulse where the laser intensity is weak. In a weak field, the shift occurs just like the angular offset of momentum distribution from the interaction between the photoelectron and the nucleus [14]. In order to avoid the influence of the ends of pulse, we can focus on the middle peaks [the moments a, b and c in Fig. 3(a)] to investigate the delay time in the propagation [around peak 6 in Fig. 3(b)]. what is more, we also calculate the the instantaneous photoelectron probability current at the barrier exit (The barrier width is estimated by SFA), and get the same characteristic. In Fig. 3(b), we add the delay times of the middle peaks, and it is comparable with quantum delay time τ_Q. In a word, our method is different from the attoclock experiment which obtains the delay time by analyzing the angular offset of the final momentum at the end of the pulse [8]. In their method, the final results include the influence of Coulomb potential at both end of the pulse.

The delay time determination by ionization rate is immune to the long-range Coulomb potential which affects the angular offset of momentum and brings ambiguity in deducing tunneling time [14]. We attach an absorption function starting at 4.1 a.u. ~ 12.1 a.u. in our calculations to attenuate the long-range Coulomb potential. The effective potentials are shown in Fig. 3(c), and the corresponding delay times of the ionization peaks are estimated in Fig. 3(d). The results are invariable when we change the absorption position, which indicates that delay time is not influenced by the long-range Coulomb potential.

Therefore, the quantum delay time τ_Q which characterizes the time delay of the peak of ionization rate after the maximum electric field in tunneling regime can be interpreted as the tunneling time when the linearly polarized pulse duration is long enough. However, this method does not work any more for ultrashort pulses where the form of electric field is closely related to the wavelength. For example, a half optical cycles laser has various electric field forms at wavelengths from 800 nm to 2400 nm as shown in Fig. 4(a). Therefore, the delay time decreases towards zero quickly with the wavelength getting shorter, as shown in the top layer of Fig. 4(c). The delay time can not give an acceptable value for tunneling time until at longer enough wavelength, for example, 2700 nm and 3800 nm as used in Fig. 2(a). It might be caused by the fact that the electric field intensity oscillates violently during the electrons entering and spreading in the barrier. It is a non-adiabatic effect occurring at short wavelengths [42]. Anyway, the delay time cannot represent the tunneling time in short pulses.

 

Fig. 4 (a) The linearly polarized electric fields with various wavelengths (top layer) and the electric field envelop of circularly polarized laser with constant duration (low layer). (b) The electric fields of a circularly polarized laser pulse with various wavelengths impacting the initial state (3p_z) of argon atom. (c) The delay time in linearly polarized lasers can not be used to estimate the tunneling time in short wavelength range while the circularly polarized one can give a reasonable tunneling time estimation in a wide wavelength range, i.e., 800–2400 nm, agreeing with Ref. [28]. (d) The transmission norm-squared of wave function (|ψ_t|² − |ψ₀|²) at the end of tunneling at wavelength 800 nm for circular polarization pulses in (c). The black dash line corresponds to at moment of quantum delay time τ_Q = 7.1 a.u. and blue dot dash line at the probability current delay time τ_J = 5.9 a.u.. The initial wave function norm (red gradual map) is displayed in the radial coordinate and the gray filled area stands for the potential barrier.

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In order to detour around the influence of pulse duration, we choose the circularly polarized pulse acting on argon atom instead. As shown in the low layer of Fig. 4(a), we use an identical electric field envelop even the wavelength varies from 800 nm to 2400 nm, corresponding to 3 to 1 cycle(s), respectively, where the maximum intensity of the electric field points in direction where the bound electron density is maximum (Fig. 4(b)). For the pulse duration of T_d = 331 a.u., far longer than the delay time, the determined delay time from the tunneling ionization process is really the tunneling time. In these circularly polarized laser fields, our determined quantum delay times τ_Q are well consistent with the delay time τ_J by the probability current method [28], as shown in the low layer of Fig. 4(c). Both of them show an uniform tendency, similar to the results of hydrogen exposed in linearly polarized light with long pulse duration [Fig. 2(b)]. In circular polarization case, the laser pulse with shorter wavelength takes less time to rotationally point to the area of high bound electron density [Fig. 4(b)], namely, a shorter time of interaction with the electron, and hence more time to tunnel through the barrier completely.

We have a look at the wave function at the end of tunneling dynamics at wavelength 800 nm for circularly polarized pulse in Fig. 4(c), where the determined tunneling times are slightly different. Our method gives a value of τ_Q = 7.1 a.u. while the probability current method gives a value τ_J = 5.9 a.u.. At both moments, the distributions of transmission norm-squared of wave function (|ψ_t|² − |ψ₀|²) are very close. They are shown in Fig. 4(d). The black dash line corresponds to at moment of quantum delay time and the blue dot dash line at the probability current delay time. The agreement indicates that both methods can give a consistent physical picture for the tunneling process.

4. Conclusions

To summarize, we propose a quantum method to estimate the tunneling time by supervising the instantaneous ionization rate. As an example, we calculate the quantum tunneling time τ_Q for a hydrogen exposed to linearly polarized pulses and for an argon atom exposed to circularly polarized laser pulses. The determined tunneling times are in the same order with the estimated tunneling time τ_W by Wigner trajectory and τ_J by probability current method. The consistence verifies that the electron takes real time to tunnel through the barrier in the photoionization. Our study also shows a circular polarized pulse acted on an atom with anisotropic electronic distribution on the ground state can supply a clean environment to measure the quantum tunneling time unambiguously.