Introduction

The tunneling ionization of an atom in a strong laser field is the first simple step in many strong-field phenomena1, and a key element in attoscience2,3. Notwithstanding its simplicity, the under-the-barrier dynamics during strong-field tunneling from an atom or molecule still holds surprises and offers new perspectives. From this key perspective, the full characterization of the tunneled electron wavepacket is very desirable.

Various methods of attosecond photoelectron interferometry have been developed which allow for coherent measurement of electronic wavepackets4. In particular, using an attosecond pulse synchronized with an infrared laser field, the coherent reconstruction of the bound-state wavepacket is realized5,6. Within a similar pump-probe scheme, the angular components of the photoelectron wavepacket are fully characterized7. Photoelectron interferometry also allowed the extraction of the amplitude and phase of a photoelectron wavepacket created through a Fano autoionizing resonance8.

Recently, attosecond photoelectron interferometry has been extended to characterize the wavefunction of the electron in the most fundamental step of strong-field phenomena, field-induced tunneling. In a two-color setup of circularly polarized laser fields, the phase and amplitude of the tunneling wavepacket are measured using a special two-color attoclock interferometry9,10, which later has been generalized employing a combination of circularly and linearly polarized laser fields in a two-color setup11. The use of phase-of-phase spectroscopy for this purpose is also demonstrated12. In a further development, the method of attosecond-gated interferometry has been put forward13, where the evolution of an electron wavefunction during the under-the-barrier motion is probed using an attosecond pump and attosecond probe scheme. In this way, spatiotemporal shaping of the electron wavefunction during tunneling and its full characterization were demonstrated. Note that the controlled shaping of free electron wavepackets finds important applications in photon-induced near-field electron microscopy, see e.g.14,15.

The phase of the tunneled electron wavefunction is closely related to the tunneling time delay. Attoclock techniques aimed here at measuring this subtle time delay16,17,18,19,20,21, with difficulties in interpretation of the attoclock signal because of the entanglement of the time delay and Coulomb field effects22,23,24,25,26,27,28,29. From the principal point of view, the nonnegligible phase shift of the wavefunction during sub-barrier dynamics arises due to the interference of the direct ionization path with the under-the-barrier recolliding one29,30, which in the time domain is exhibited as a shift of the peak of the tunneled electron wavepacket and can be interpreted as a tunneling time delay. The one dimensional calculation of the tunneling wavefunction in the case of the atomic short-range potential provides a quite large phase29,30,31 equivalent to a tunneling time of about 1 a.u. However, in a three-dimensional case with the atomic short-range potential, the transverse spreading of the electron wavepacket during the sub-barrier recolliding motion suppresses the amplitude of the recolliding path, and in this way damps the tunneling time. The question arises if the damping role of the transverse spreading for the tunneling time delay will persist also in the more realistic case for a Coulomb potential of the atomic core.

In this paper, we derive the analytical relationship between the tunneling wavepacket phase and the tunneling rate in the adiabatic regime of ionization. With this information at hand, we investigate the role of the Coulomb field of the atomic core for the enhancement of the phase shift of the photoelectron wavepacket and the tunneling time delay, that alleviates the negative effect of the electron wavepacket spreading. We begin with the one-dimensional (1D) problem and using the exact wavefunction for the adiabatic tunneling problem from a short-range potential derive the analytical relationship between the tunneling wavepacket phase and the tunneling rate, as well as provide the interpretation for the tunneling wavepacket phase within the under-the-barrier recollision picture, employing a modified strong field approximation (SFA) in high-orders. Further, we prove the relationship between the photoelectron phase and the tunneling rate in the 3D case of a short-range atomic potential, and finally, for the realistic tunneling ionization problem in the adiabatic regime accounting for the Coulombic atomic potential. In the latter cases, we employ the quasistatic dressed bound state described in the eikonal approximation in parabolic coordinates. We prove that the Coulomb correction enhances by orders of magnitude the recollision probability and renders the tunneling time delay observable in strong field ionization measurements. While the attosecond photoelectron interferometry9,10,11,12,13 provides a way for the phase characterization of the sub-barrier electron wavepacket, our findings illuminate the physical origin of this phase, relating it to the under-the-barrier recollision effect.

Results

Theoretical approach

Here we discuss tunneling ionization in the adiabatic regime (in the limit of small Keldysh parameters32, γ ≪ 1) with focusing on the phase of the photoelectron wavepacket and the related tunneling time delay. The ionization of an atom from the ground state is described by the Schrödinger equation with the Hamiltonian

$$H=-\frac{1}{2}{{{\boldsymbol{\nabla }}}}^{2}+{{\bf{r}}}\cdot {{\bf{E}}}(t)+V({{\bf{r}}}),$$
(1)

where V(r) is the atomic potential, and E(t) the laser field. Atomic units are used throughout. The laser field is described as a constant field \({{\bf{E}}}(t)=-\hat{{{\bf{x}}}}{E}_{0}\), referring to the adiabatic regime. According to the Dyson equation, the solution of the Schrödinger equation can be expressed via the exact time-evolution operator (TEO)33:

$$U(t,{t}_{i})={U}_{f}(t,{t}_{i})-i\int_{{t}_{i}}^{t}d{t}^{{\prime} }{U}_{f}(t,{t}^{{\prime} })VU({t}^{{\prime} },{t}_{i}),$$
(2)

where Uf(t, ti) is TEO only in the laser field (Volkov propagator), and ti is the initial time when the electron is in the bound state, before the interaction with the laser field is switched on. Using the Dyson equation above, the exact ionization amplitude m in a constant electric field is expressed via the exact wavefunction ψa(t):

$$m(\varepsilon )=-i\int\,dt\left\langle {\psi }_{0}^{f}(t)| V| {\psi }^{a}(t)\right\rangle ,$$
(3)

where \({\psi }_{0}^{f}\) is the Volkov-state, and \({\psi }^{a}(x,t)={\psi }^{a}(x)\exp (-i\varepsilon t)\) the exact energy eigenstate of the time-independent Hamiltonian of Eq. (1) in a constant electric field, corresponding to the laser field dressed quasistationary bound state with the complex quasienergy ε = ε0 − iΓ(ε0) and the ionization rate Γ(ε0).

The tunneling time delay can be defined by the peak of the asymptotic photoelectron wavepacket and backpropagating to the tunnel exit23. This definition is in accordance with the time delay derived from the phase of the tunneled wavepacket via the Wigner derivative29,34:

$$\tau =-\frac{\partial \phi }{\partial \varepsilon }.$$
(4)

Our results are relevant for the adiabatic case, which assumes a small Keldysh parameter γ ≪ 1. As the main physical quantity discussed in this paper is the tunneling time delay, we restrict the ionization to the tunneling regime and apply intensities below the over-the-barrier ionization regime \({E}_{0} \, < \, {E}_{{{\rm{OTBI}}}}={I}_{p}^{2}/(2Z)\), with the ionization energy Ip, and Z ionic charge. EOTBI is calculated in parabolic coordinates and is twice larger than the 1D value of ref. 35. We apply also the quasiclassical condition E0/Ea ≪ 136,37, with the atomic field Ea = κ3, \(\kappa =\sqrt{2{I}_{p}}\), using expansion over the parameter E0/Ea.

Exact solution in 1D case

Let us illustrate the relation of the phase of the tunneled electron wavepacket to the tunneling rate in the simple 1D case of a zero-range potential (ZRP) V(x) = − κδ(x), which allows for the exact wavefunction in a constant electric field38. In this case the exact ionization amplitude via Eq. (3) is obtained, see Methods section “Exact solution in 1D case":

$$m(\varepsilon )=\frac{i{\kappa }^{3/2}}{{2}^{1/6}\sqrt{\pi }{E}_{0}^{2/3}}\,{{\rm{Ai}}}\,\left(-\frac{{2}^{1/3}\varepsilon }{{E}_{0}^{2/3}}\right)\left(1-\frac{5}{4}{f}^{2}\right).$$
(5)

The quasienergy ε = ε0 − iΓ(ε0) consists of the Stark-shifted bound state energy:

$${\varepsilon }_{0}=-({\kappa }^{2}/2)\left(1+5\,{f}^{2}/4\right),$$
(6)

and the ionization rate:

$$\Gamma ({\varepsilon }_{0})=\frac{{\kappa }^{2}}{2}\left(1-\frac{5\,f}{3}\right)\exp \left(-\frac{2}{3\,f}\right),$$
(7)

is given with the f ≡ E0/Ea correction. Inserting the quasienergy into Eq. (5), and expanding over f, the analytical expression for phase of the wavefunction is derived

$$\phi =\arg [m(\varepsilon )]=-\frac{1}{2 \,f}\left(1-\frac{7 \,f}{6}\right)\exp \left(-\frac{2}{3 \,f}\right).$$
(8)

The comparison of Eqs. (7) and (8) shows that the phase expression is in accordance with the relation in leading order

$$\phi \approx i\Gamma {t}_{s},$$
(9)

with the Keldysh time

$${t}_{s}=\frac{i\kappa }{{E}_{0}}=\frac{i\gamma }{\omega },$$
(10)

where ω is the laser frequency, and ts is given by the saddle point of the time-integration in the ionization amplitude of Eq. (3)39.

The relation (9), one of the main results of this paper, which establishes a link between the tunneling decay rate Γ and the phase of the tunneling wavepacket ϕ (or the tunneling time delay τ). We have derived it by explicitly calculating the phase and the decay rate. However, a more general derivation of this relation can be provided, taking into account that in the tunneling regime the decay rate is exponentially suppressed Γ ≪ ε0. In this case, expanding the amplitude expression:

$$m({\varepsilon }_{0}-i\Gamma ({\varepsilon }_{0}))\approx m({\varepsilon }_{0})-i\Gamma ({\varepsilon }_{0}){\partial }_{\varepsilon }m({\varepsilon }_{0}),$$
(11)

and taking into account that the amplitude m(ε0) for the ionization problem has a real value, we derive

$$\phi =\arg [m(\varepsilon )]=-\arctan \left[\Gamma {\partial }_{\varepsilon }\ln (m({\varepsilon }_{0}))\right]\approx -\Gamma {\partial }_{\varepsilon }\ln (m({\varepsilon }_{0})).$$
(12)

From the comparison of Eqs. (9) and (12), we find a mapping relationship \({t}_{s}=i{\partial }_{\varepsilon }\ln (m({\varepsilon }_{0}))\).

The tunneling time (4) we derive from the expression

$$\tau =-\frac{\partial \phi }{\partial \kappa }\frac{\partial \kappa }{\partial {\varepsilon }_{0}}\approx 2\phi \frac{\kappa }{{E}_{0}},$$
(13)

using the Stark-shifted energy (6). The second approximate expression is correct in the leading order with respect to f, bearing in mind that the leading dependence of the phase on \(\kappa =\sqrt{-2\varepsilon }\) is via the tunneling exponent: \(\phi \propto \exp [-2/(3\,f)]\).

In Fig. 1(a) we show that the analytical f-scaling of the phase given by Eq. (8) accurately describes the phase, coinciding with the numerical result, derived from Eq. (5). The importance of the f-correction term in the scaling is also highlighted.

Fig. 1: The phase and the tunneling time delay in a 1D ZRP.
figure 1

a The phase ϕ of the tunneling wavepacket in a 1D ZRP, normalized by the leading term \(-\frac{1}{2f}{e}^{-\frac{2}{3f}}\) of Eq. (8): (blue cycles) the exact numerical result via the phase of Eq. (5), (green cycles) the analytical scaling of Eq. (8), (orange boxes) leading order of Eq. (8). b The momentum shift due to the tunneling time delay δp = − E0τ = − 2κϕ: (blue) the exact numerical result via the phase of Eq. (5), (orange) the approximate result of ref. 31 using the SFA with the low-frequency T-matrix approximation for the under-the-barrier recollision, (red cycles) the numerical solution of the time-dependent Schrödinger equation (TDSE) of ref. 29.

Full size image

In Fig. 1b the field dependence of the momentum shift of the peak of the photoelectron distribution due to the tunneling time delay δp = − E0τ is shown. Our exact result via Eq. (5) is in qualitative accordance with the numerical solution of the time-dependent Schrödinger equation of ref. 29. Figure 1b also demonstrates that the result of the approximate calculation of ref. 31 using the SFA with the low-frequency T-matrix approximation for the under-the-barrier recollision is qualitatively correct, with a slight deviation in the field scaling.

Recollision picture

While the relation between the tunneling rate and the tunneled electron phase of Eq. (9) shows a straightforward way to calculate the phase of the tunneled electron wavepacket, it does not explain how this phase emerges. In the following we will confirm with high accuracy that the phase and the tunneling time delay emerge due to the under-the-barrier recollision as suggested in refs. 29,30. Let us first show that Eq. (8) with E0/Ea corrections are in accordance with the recollision picture.

The recollision picture in SFA is described representing the exact ionization amplitude as the sum of the direct ionization amplitude m1 and the recolliding one mR:

$$m(p)= \, {m}_{1}(p)+{m}_{R}(p),\\ {m}_{1}(p)= -i\int\,dt\left\langle {\psi }_{0p}^{f}(t)| {H}_{i}| {\psi }_{0}^{a}(t)\right\rangle ,\\ {m}_{R}(p)= -\int\,ds{\int}^{s}dt \left\langle {\psi }_{0p}^{f}(s)| V| U(s,t)| {H}_{i}| {\psi }_{0}^{a}(t) \right\rangle ,$$
(14)

with the exact TEO U. In the conventional second-order SFA the exact TEO is approximated by the Volkov-TEO U ≈ U f, treating the recollision in Born approximation. For the 1D ZRP the amplitudes can be calculated with the help of the saddle point approximation, see Methods section “Recollisions under the barrier":

$${m}_{1}(p)=i\frac{\kappa }{\sqrt{{E}_{0}}}\exp \left(-\frac{1}{3\,f}\right),$$
(15)
$${m}_{2}(p)=-\frac{\kappa }{2\sqrt{{E}_{0}}}\exp \left(-\frac{1}{f}\right).$$
(16)

However, for the under-the-barrier recollision, the description of the rescattering, with the momentum  ~ κ, needs improvement. Rather than to apply the 3rd order SFA leading to involved calculations, we resort to the adiabatic transition approach40. We approximate the exact TEO by the combined atomic and field TEO, considering it as an improvement to the case of the mere Volkov-TEO (yielding m2), see Methods section “Recollisions under the barrier":

$$U(s,t)\approx {U}^{a}(s,{t}_{m}){U}^{f}({t}_{m},t).$$
(17)

In the latter, we switch the laser driven propagation to the atomic one at a specific matching time before recollision, assuming that near the core at the recollision the atomic potential again dominates the dynamics. Approximating further the atomic TEO as \({U}^{a}(s,t)\approx \vert {\psi }_{0}^{a}(s)\rangle \langle {\psi }_{0}^{a}(t)\vert\) the modified recollision matrix element via the adiabatic transition is obtained

$$\begin{array}{l}{m}_{2}^{{{\rm{m}}}}=-\int\,dq\int\,ds{\int}^{s}dt\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0}^{a}(s)\right\rangle \left\langle {\psi }_{0}^{a}({t}_{m})| {\psi }_{0q}^{f}({t}_{m})\right\rangle \left\langle {\psi }_{0q}^{f}(t)| {H}_{i}| {\psi }_{0}^{a}(t)\right\rangle .\end{array}$$
(18)

The switching time tm, when the adiabatic transition from the laser driven state to the atomic one takes place, is defined by the condition of the matching quasienergies: \({\varepsilon }_{0q}^{f}=-{\kappa }^{2}/2\). The intermediate momentum and s-time integrations are carried out via the saddle point method which yields, see Methods section “Recollisions under the barrier":

$${m}_{2}^{{{\rm{m}}}}=\frac{{\kappa }^{4}}{2{E}_{0}^{3/2}}\exp \left(-\frac{1}{f}\right).$$
(19)

Thus, we obtain the full recollision matrix element

$${m}_{R}={m}_{2}+{m}_{2}^{{{\rm{m}}}},$$
(20)

which determines the change of the phase ϕ of the tunneling wavepacket due to recollision: \({m}_{1}={\rho }_{1}{e}^{i{\varphi }_{1}}\), m = m1 + mR = ρeiφ, ϕ = φ − φ1. As mR ≪ m1, in the considered case of E0 ≪ Ea, we have ρ ≈ ρ1, ϕ ≪ 1, and m/m1 ≈ eiϕ ≈ 1 + iϕ, thus

$$\phi \approx \,{{\rm{Im}}}\,\left(\frac{{m}_{R}}{{m}_{1}}\right)=\left(-\frac{1}{2\,f}+\frac{1}{2}\right)\exp \left(-\frac{2}{3\,f}\right).$$
(21)

The phase expression above from the modified SFA, based on the under-the-barrier recollision picture, coincides with the exact analytical expression of Eq. (8) (with the slight deviation of the correction factor  ~ f 0). Thus, we conclude that the tunneling wavepacket phase emerges due to the under-the-barrier recollision and can be estimated via ϕ ≈ iΓts.

We note that the leading contribution to the matrix element of the under-the-barrier recollision of Eq. (20) comes from \({m}_{2}^{{{\rm{m}}}}\), which corrects the rescattering amplitude over the result of the Born approximation (described by m2). It is specific for the under-the-barrier rescattering that the leading term of the Born approximation fails to correctly describe it.

3D zero-range-potential

In the 3D problem, there is the spreading effect of the wavepacket in the lateral direction which decreases the tunneling rate, as well as the recollision probability, and in this way reduces the phase of the tunneled electron wavepacket. We calculate the 3D amplitude of ionization via Eq. (3). In the case of the 3D ZRP, we use the wavefunction of the bound state in a constant electric field in the quasiclassical approximation inside the barrier41,42:

$${\psi }^{a}({{\bf{r}}})=\sqrt{\frac{\kappa }{2\pi }}\frac{\exp \left(-\kappa r\right)}{r}\left(1-\frac{{f}^{2}}{4}\right),$$
(22)

with the complex quasienergy

$$\varepsilon = -{\kappa }^{2}/2(1+{f}^{2}/4)-i\Gamma ,\\ \Gamma = \frac{{\kappa }^{2}f}{4}\exp \left(-\frac{2}{3\,f}\right)\left(1-\frac{17}{12}f\right).$$
(23)

The ionization amplitude is then calculated analytically, yielding

$$m(\varepsilon )=-i\frac{{2}^{1/3}\sqrt{\kappa }}{{E}_{0}^{2/3}}\,{{\rm{Ai}}}\,\left(-\frac{{2}^{1/3}\varepsilon }{{E}_{0}^{2/3}}\right)\left(1-\frac{{f}^{2}}{4}\right).$$
(24)

Using the quasienergy (23) in Eq. (24), the phase of the wavefunction is obtained after the expansion over f:

$$\phi =\arg [m(\varepsilon )]=-\frac{1}{4}\left(1-\frac{11}{12}f\right)\exp \left(-\frac{2}{3\,f}\right),$$
(25)

which in the leading order coincides with the iΓts-estimation. Comparing the phase of the tunneling wavepacket in the 3D case to the 1D one with the ZRP, cf. Eqs. (8) and (25), we see that in the 3D case the phase is smaller by the factor f, which is due to the spreading during the under-the-barrier recollision. In the SFA recollision picture, mR in the 3D case is decreased by the factor f with respect to 1D due to the spreading, which yields the decrease of the phase via Eq. (21).

In Fig. 2 related to the case of 3D ZRP, we show the analytical f-scaling of the phase given by Eq. (25), comparing it with the exact numerical result. In the 3D case, the analytical field scaling deviates slightly from the numerical result at high fields. We show in Fig. 2b the field dependence of the momentum shift of the peak of the photoelectron distribution due to the tunneling time delay, which indicates that in the 3D case, the low-frequency approximation is less accurate at high fields.

Fig. 2: The phase and the tunneling time delay in a 3D ZRP.
figure 2

a The phase ϕ of the tunneling wavepacket in a 3D ZRP, normalized by the leading term of Eq. (25), \(-\frac{1}{4}\exp [-2/(3f)]\): (blue cycles) the exact numerical result, (green cycles) the analytical scaling of Eq. (25), (orange boxes) leading order of Eq. (25). b The momentum shift due to the tunneling time delay δp = − E0τ, (blue) the exact numerical result, (orange) the approximate result of ref. 31 using SFA with the low-frequency T-matrix approximation for the under-the-barrier recollision.

Full size image

Tunneling time for a hydrogen atom

We apply the method developed for a ZRP to calculate the phase of a tunneling wavepacket in the case of a hydrogenlike atom, with the charge Z. We incorporate the solution of the Schrödinger equation in the Coulomb and constant electric fields in parabolic coordinates43,44 into the ionization amplitude of Eq. (3), and investigate the role of the Coulomb effects for the wavepacket phase and tunneling time delay.

In Eq. (3), we use the laser field dressed bound state \({\psi }^{a}(t)={\tilde{\psi }}^{a}\exp (-i\varepsilon t)\), with the complex quasienergy ε, and the energy eigenstate \({\tilde{\psi }}^{a}\) of the time-independent Hamiltonian in the constant field, fulfilling the stationary Schrödinger equation:

$$\varepsilon {\tilde{\psi }}^{a}=-\Delta {\tilde{\psi }}^{a}/2-x{E}_{0}{\tilde{\psi }}^{a}+V{\tilde{\psi }}^{a},$$
(26)

with V(r) = − Z/r. Solving the latter in parabolic coordinates with the ansatz \({\tilde{\psi }}^{a}={\psi }_{\eta }(\eta ){\psi }_{\xi }(\xi )\), the variables are separated, see Methods section “Role of Coulomb potential". We apply the WKB approximation with respect to the electric field effect, using the unperturbed atomic bound state as the wavefunction limit at the vanishing electric field. Accordingly, the following ansatz is employed:

$${\psi }_{\eta }= \, \sqrt{\frac{{\kappa }^{3}}{\pi }}\exp [-\kappa \eta /2+i{S}_{0}^{\eta }(\eta )+i{S}_{1}^{\eta }(\eta )],\\ {\psi }_{\xi }= \, \exp [-\kappa \xi /2+i{S}_{0}^{\xi }(\xi )+i{S}_{1}^{\xi }(\xi )],$$
(27)

which yields the WKB solutions S0 and S1, corresponding to the leading- and the first-order expansion terms over the Plank constant ℏ:

$${S}_{0}^{\eta }(\eta )= \frac{{\left(2\varepsilon +{E}_{0}\eta \right)}^{3/2}}{3{E}_{0}}-\frac{i}{2}\kappa \eta ,\\ {S}_{0}^{\xi }(\xi )= -\frac{{\left(2\varepsilon -{E}_{0}\xi \right)}^{3/2}}{3{E}_{0}}-\frac{i}{2}\kappa \xi ,\\ {S}_{1}^{\eta }(\eta )= -\frac{Z}{\sqrt{2\varepsilon }}\,{{\rm{arctanh}}}\,\left[\sqrt{\frac{2\varepsilon +{E}_{0}\eta }{2\varepsilon }}\right]+\frac{i}{4}\ln \left[(2\varepsilon +{E}_{0}\eta ){({E}_{0}\eta )}^{2}\right],\\ {S}_{1}^{\xi }(\xi )= -\frac{Z}{\sqrt{2\varepsilon }}\,{{\rm{arctanh}}}\,\left[\sqrt{\frac{2\varepsilon -{E}_{0}\xi }{2\varepsilon }}\right]+\frac{i}{4}\ln \left[(2\varepsilon -{E}_{0}\xi ){({E}_{0}\xi )}^{2}\right].$$
(28)

Using the derived wavefunction for the laser dressed bound state in Eq. (3), along with the complex quasienergy ε = ε0 − iΓ44:

$${\varepsilon }_{0}= -\frac{{\kappa }^{2}}{2}\left(1+\frac{9}{2}{f}^{2}\right),\\ \Gamma = \frac{2{\kappa }^{2}}{f}\exp \left[-\frac{2{(-2{\varepsilon }_{0}/{\kappa }^{2})}^{3/2}}{3\,f}\right]\left(1-\frac{53}{12}f\right).$$
(29)

The phase of the tunneling wavepacket is derived via

$$\phi =-\,{{\rm{Im}}}\,\left(i\Gamma \frac{{m}^{{\prime} }({\varepsilon }_{0})}{m({\varepsilon }_{0})}\right).$$
(30)

The results for the phase and tunneling time, when using the quasienergy (29) and calculating the matrix element in (30) numerically, are shown in Figs. 3 and 4.

Fig. 3: The Coulomb effects for the phase of the tunneling wavepacket.
figure 3

a Phase of the tunneling wavefunction vs electric field strength via the phase of Eq. (30) (blue) compared with the 3D-ZRP-case (orange). b The phase ϕ of the tunneling wavepacket in the Coulomb potential, normalized by the leading term \(-2/{f}^{2}{e}^{-\frac{2}{3\,f}}\) of Eq. (31): (blue) the exact numerical result, (green) the analytical scaling of Eq. (31), (orange) leading order of Eq. (31).

Full size image

We can give an analytical estimate for the phase, using the relation ϕ = − Γκ/E0, where Γ in the leading order of f coincides with the PPT-rate, and the next to leading order correction to Γ is given by Eq. (29), additionally we take into account that the difference in the next to leading order corrections in ϕ and Γ is f/2, in analogy to the short-range case, which finally yields:

$$\phi =-\frac{1}{2}{\left(\frac{4}{{f}^{2}}\right)}^{Z/\kappa }\exp \left(-\frac{2}{3\,f}\right)\exp \left(-\frac{101}{12}f\right).$$
(31)

With the relation of Eq. (4) the tunneling time delay in the leading order is estimated for an hydrogenlike atom:

$$\tau \approx -\frac{1}{f{\kappa }^{2}}{\left(\frac{4}{{f}^{2}}\right)}^{Z/\kappa }\exp \left(-\frac{2}{3\,f}\right).$$
(32)

Discussion

We treat the sub-barrier Coulomb enhancement effect for the direct path to the ionization amplitude, which is identical to the adiabatic Coulomb correction of the Perelomov-Popov-Terent’ev (PPT) theory36,37, as well as the similar corrections for the under-the-barrier recolliding path (the PPT Coulomb correction in the nonadiabatic regime of ref. 45 is not relevant to our quasistatic theory). This effect arises due to the modification of the tunneling barrier by the Coulomb field, which enhances the tunneling amplitude and the phase of the tunneling wavepacket at the tunnel exit. It should not be confused with the Coulomb focusing effect46,47,48,49,50, well-known in strong field physics, which emerges due to multiple rescattering of the ionized electron during excursion in the continuum and is not related to the electron phase at the tunnel exit.

Note that the Coulomb effects increase the tunneling time delay by the factor 8/f 2 with respect to the case of the 3D ZRP, see Fig. 3a. Intuitively this enhancement can be understood by Coulomb correction factors \({{\mathcal{C}}} \sim 1/f\) for tunneling ionization36,37. The Coulomb corrections enhance the tunneling amplitude because the height of the tunneling barrier is decreased by the Coulomb field, and the bound state wavefunction is larger in the Coulomb case at the tunneling emergence point xs inside the barrier. The absolute value of the phase is given by the ratio mR/m1, where m1 describes the direct trajectory which tunnels once [(1) from the core to the tunnel exit], whereas mR describes the recolliding trajectory that tunnels three times: [(1) from the core to the tunnel exit, (2) from the exit back to the core, and (3) again from the core to the exit]. Thus, the Coulomb enhancements for the amplitudes are \({m}_{1} \sim {{\mathcal{C}}}\) and \({m}_{R} \sim {{{\mathcal{C}}}}^{3}\), i.e. \(\phi ={m}_{R}/{m}_{i} \sim {{{\mathcal{C}}}}^{3}/{{\mathcal{C}}} \sim 1/{f}^{2}\). Note that the \({{\mathcal{C}}} \sim 1/f\) scaling follows from the fact that at the point of ionization \({x}_{s} \sim 1/(\sqrt{f}\kappa )\), both the Coulomb continuum, as well as the Coulomb bound states are larger than in the short-range one by the factor \(1/\sqrt{f}\). The Coulomb enhancement counteracts the spreading effect, which suppresses the phase in the 3D case by the factor f. Then, the tunneling time in the Coulomb potential case exceeds that in the 1D ZRP case by the factor 1/f.

In Fig. 3b the analytical scaling of Eq. (31) is tested with respect to the numerical calculation via Eq. (30). The leading term of the analytical scaling in Eq. (31) is already quite close to the numerical result, however, the full analytical scaling reproduces more closely the decreasing feature of the phase at high fields, but it underestimates the exact phase.

The comparison of the tunneling time delays in 1D ZRP and 3D ZRP with real atomic cases of hydrogen and krypton are shown in Fig. 4. The tunneling time delay is negative. While in 3D ZRP it is negligible ( ~ 0.1 a.u.), in hydrogen and krypton it can be quite sizable ( ~ 3.5 a.u.). Note that the 1D ZRP result for the tunneling time ( ~ 1 a.u.) is closer to the atomic case than the 3D ZRP one. This is because of the compensating contributions of the spreading and Coulomb effects.

Fig. 4: Tunneling time delay in the case of ZRP and for different atomic species.
figure 4

Tunneling time delay as a function of the applied electric field E0: (a) for 1D ZRP, (blue, solid) via the analytical expression (8), (orange, dot-dashed) via ϕ = iΓts with (7); (b) for 3D ZRP, via the analytical expression (25), (orange, dot-dashed) via ϕ = iΓts with (23); (c) for hydrogen atom, (d) for krypton atom. In panels (c, d): (blue, solid) exact numerically, (orange, dashed) via the analytical estimate (31). In each panel the field maximum is restricted by E0≤0.75EOTBI.

Full size image

For ZRP in Fig. 4, we compare the tunneling time delay derived from the direct analytical calculation of the phase with that via the estimate ϕ = iΓts, demonstrating the qualitative correctness of this estimate. The ionization time delay for hydrogen and krypton in Fig. 4 are derived directly from the numerical calculation of the amplitude, which is compared with the analytical estimate via (32), indicating the qualitative character of the latter estimation.

Conclusions

We calculated the phase of the tunneling wavepacket and the corresponding tunneling time delay in the adiabatic regime using the wavefunction of the dressed bound state in the laser field. The main result of the paper is the direct relationship between the phase of the tunneling wavepacket and the tunneling rate. This relationship is important for several reasons: First of all, the tunneling rate is well investigated and documented theoretically and experimentally for a wide range of atomic species. In particular, the Coulomb enhancement effect for the ionization amplitude and the tunneling rate is well established via the PPT theory. Therefore, employing our relationship one can deduce the best parameters for the observation of the tunneled wavepacket phase or the related tunneling time delay.

Moreover, this relationship provides an easy way to quantitatively estimate the tunneling time delay and the phase of the photoelectron wavepacket in different ionization processes. By shaping the tunneling barrier in strong-field ionization employing a combination of laser fields of different polarizations and frequencies, one can control the dynamics under the barrier, in particular, the amplitude of the under-the-barrier recollision, and achieve the creation of tailored photoelectron wavepackets with phases gauged by the corresponding rates.

We demonstrate also with the accuracy of high-order field corrections, the validity of the interpretation of the tunneling wavepacket phase within the under-the-barrier recollision picture, when the phase of the tunneling wavepacket is induced due to the interference of the direct and recolliding paths and can be observed as a tunneling time29, or by other photoelectron interferometric methods11. When the Coulomb field is neglected (or in the case of a negative ion detachment) the lateral spreading damps fully the recolliding path and the wavepacket phase disappears. In contrast, the Coulomb field of the atomic core helps to counteract the spreading and increases the phase and the tunneling time up to an observable level.

Up to now the attoclock technique has been applied for the measurement of the tunneling time delay16,17,18,19,20,21, struggling to isolate pure tunneling time signals because of its entanglement with the classical Coulomb effect in the continuum. Rather than the tunneling time, one could suggest to obtain equivalent information from the measurement of the tunneled electron phase in more refined interferometric measurements, like those in refs. 9,10,11,12,13, and in this way to obtain insight on how the under-the-barrier recollisions behave when altering the tunneling barrier with an additional probe laser field.

Our results are applicable in the adiabatic regime at small Keldysh parameters γ ≪ 1, with the quasiclassical description E0 ≪ Ea, and in the tunneling regime below the over-the-barrier ionization E0 < EOTBI. The generalization of our results to the nonadiabatic regime of tunneling is feasible and desirable to describe the process at large Keldysh parameters, but it will require the derivation of an approximate quasienergy wavefunction of the electron in the laser and Coulomb fields.

Methods

Exact solution in 1D case

We consider the 1D case of a zero-range potential (ZRP) V(x) = − κδ(x), which allows for the exact wavefunction in the laser field. In this case Eq. (3) reads:

$$m(\varepsilon )=i\kappa \int\,dt{\psi }_{0}^{f}(0,t){\psi }^{a}(0)\exp (-i\varepsilon t).$$
(33)

The exact solution of the Schrödinger equation ψa in the 1D case for the electron bound state in a δ-potential and a constant electric field, with the effective potential \(\tilde{V}(x)=-\kappa \delta (x)-x{E}_{0}\) is known38:

$${\psi }_{+}^{a}(x)= \, {c}_{+}\left(\,{{\mbox{Bi}}}\,\left[-\frac{{2}^{1/3}(\varepsilon +{E}_{0}x)}{{E}_{0}^{2/3}}\right]+i\,{{\rm{Ai}}}\,\left[-\frac{{2}^{1/3}(\varepsilon +{E}_{0}x)}{{E}_{0}^{2/3}}\right]\right),x \, > \, 0, {\psi }_{-}^{a}(x) \\ = \, {c}_{-}\,{{\rm{Ai}}}\,\left[-\frac{{2}^{1/3}(\varepsilon +{E}_{0}x)}{{E}_{0}^{2/3}}\right]\,\,\,x \, < \, 0,$$
(34)

where the boundary conditions are imposed corresponding to the ionization problem, namely, the current density is positive at x → ∞, describing an outgoing wave, and the probability is vanishing at x → − ∞. The latter along with the continuity conditions for the wavefunction at the origin

$${\psi }_{+}^{a}(0)={\psi }_{-}^{a}(0)$$
(35)
$${\partial }_{x}{\psi }_{+}^{a}(0)-{\partial }_{x}{\psi }_{-}^{a}(0)=-2\kappa {\psi }_{+}^{a}(0),$$
(36)

provide the complex eigenenergy

$$\varepsilon ={\varepsilon }_{0}-i\Gamma ({\varepsilon }_{0}),$$
(37)

as well as the ratio of the coefficients c±:

$$\frac{{c}_{+}}{{c}_{-}}=\frac{\,{{\rm{Ai}}}\,\left[-{2}^{1/3}\varepsilon /{E}_{0}^{2/3}\right]}{\,{{\rm{Bi}}}\,\left[-{2}^{1/3}\varepsilon /{E}_{0}^{2/3}\right]+i\,{{\rm{Ai}}}\,\left[-{2}^{1/3}\varepsilon /{E}_{0}^{2/3}\right]}.$$
(38)

From Eqs. (33–35) we obtain the ionization amplitude of Eq. (5). Rather than to obtain the complex quasienergy exactly from Eq. (36) via numerical solution, we calculate it perturbatively, obtaining the Stark-shifted bound state energy and the ionization rate Γ: Γ(ε0) = E0∣m(ε0)∣2/2, which yields Eq. (7).

Recollisions under the barrier

In the conventional second-order SFA the exact TEO is approximated by the Volkov-TEO U ≈ U f, treating the recollisions in Born approximation. It yields for the recollision amplitude

$${m}_{2}(p) =-\int\,dq\int\,ds{\int}^{s}dt\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0q}^{f}(s)\right\rangle \left\langle {\psi }_{0q}^{f}(t)| {H}_{i}| {\psi }_{0}^{a}(t)\right\rangle .$$
(39)

After a partial integration in m1 and m2, the interaction term Hi = xE(t) can be replaced by the atomic potential V = − κδ(x) and the amplitudes can be calculated with the help of the saddle point approximation in the case of a constant field E(t) = − E0:

$${m}_{1}(p)=\sqrt{-\frac{2\pi }{{\zeta }_{1}^{{\prime\prime} }({t}_{s})}}\exp [{\zeta }_{1}({t}_{s})]=i\frac{\kappa }{\sqrt{{E}_{0}}}\exp \left(-\frac{1}{3\,f}\right),$$
(40)
$${m}_{2}(p) = \frac{1}{2}\sqrt{\frac{{(-2\pi )}^{3}}{\det {\partial }_{q,s,t}^{{2}}{\zeta }_{R}^{{{\rm{SFA}}}}(q,s,t)}}{\left. \, \exp [{\zeta }_{R}^{{{\rm{SFA}}}}(q,s,t)]\right\vert }_{({q}_{s},{s}_{s},{t}_{s})}, \\ = -\frac{\kappa }{2\sqrt{{E}_{0}}}\exp \left(-\frac{1}{f}\right),$$
(41)

with \({\zeta }_{1}(p,t)=\ln [-i\langle {\psi }_{0p}^{f}(t)| V| {\psi }_{0}^{a}(t)\rangle ]\), \({\zeta }_{R}^{{{\rm{SFA}}}}(p,q,s,t)=\ln [-\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0q}^{f}(s)\rangle \langle {\psi }_{0q}^{f}(t) | V| {\psi }_{0}^{a}(t)\rangle ]\) and (qs, ss, ts) = ( − 2iκ, iκ/E0, 3iκ/E0).

In the matching method, the modified recollision matrix element via the adiabatic transition is obtained

$${m}_{2}^{{{\rm{m}}}} =-\int\,dq\int\,ds{\int}^{s}dt\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0}^{a}(s)\right\rangle \left\langle {\psi }_{0}^{a}({t}_{m})| {\psi }_{0q}^{f}({t}_{m})\right\rangle \left\langle {\psi }_{0q}^{f}(t)| {H}_{i}| {\psi }_{0}^{a}(t)\right\rangle .$$
(42)

Applying the partial integration in the latter, the amplitude reads

$${m}_{2}^{{{\rm{m}}}}=-\int\,dq\int\,ds{\int}^{s}dt\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0}^{a}(s)\right\rangle \left\langle {\psi }_{0}^{a}({t}_{m})| {\psi }_{0q}^{f}({t}_{m})\right\rangle \left\langle {\psi }_{0q}^{f}(t)| V| {\psi }_{0}^{a}(t)\right\rangle .$$
(43)

The switching time tm, when the adiabatic transition from the laser driven state to the atomic one takes place, is defined by the condition of the matching quasienergies: \({\varepsilon }_{0q}^{f}=-{\kappa }^{2}/2\), yielding

$${t}_{m}(q,x)=-\left(q+i\sqrt{{\kappa }^{2}-2{E}_{0}x}\right)/{E}_{0}.$$
(44)

Inserting it in Eq. (42), renders the x-dependence Gaussian and analytically integrable:

$${m}_{2}^{{{\rm{m}}}} = -i\int\,dq\int\,ds{\int}^{s}dt{\int}_{0}dx\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0}^{a}(s)\right\rangle \\ {\psi }_{0}^{a* }(0,{t}_{m}(q,0)){\psi }_{0q}^{f}(0,{t}_{m}(q,0))\exp [-{E}_{0}{x}^{2}/(2\kappa )] \left\langle {\psi }_{0q}^{f}(t)| V| {\psi }_{0}^{a}(t)\right\rangle \\ = -i\int\,dq\int\,ds{\int}^{s}dt\left\langle {\psi }_{0p}^{f}(s)| V| {\psi }_{0}^{a}(s)\right\rangle \\ \exp \left[\frac{i{q}^{3}+3iq{\kappa }^{2}-2{\kappa }^{3}}{6{E}_{0}}\right]\frac{\kappa }{2\sqrt{{E}_{0}}}\left\langle {\psi }_{0q}^{f}(t)| V| {\psi }_{0}^{a}(t)\right\rangle .$$
(45)

Note that the matching method has to be corrected by an i-factor, as a comparison with the first-order SFA shows40. The intermediate momentum integration in Eq. (45) is carried out via the saddle point method (with the saddle point qs = iκ − E0t) which yields

$${m}_{2}^{{{\rm{m}}}}=-\frac{i{\kappa }^{7/2}}{2\sqrt{2\pi }}\int\,ds{\int}^{s}dt\exp \left(-\frac{2{\kappa }^{3}}{3{E}_{0}}+i\frac{{\kappa }^{2}}{2}s+i\frac{{E}_{0}^{2}{s}^{3}}{6}\right).$$
(46)

Further, dropping the boundary terms at infinity for the t-integration, and with the s-integration by the saddle-point method (the saddle point ss = iκ/E0), results in Eq. (19).

Role of Coulomb potential

The energy eigenstate in the constant electric field \({\tilde{\psi }}^{a}\) fulfills the stationary Schrödinger equation:

$$\varepsilon {\tilde{\psi }}^{a}=-\Delta {\tilde{\psi }}^{a}/2-x{E}_{0}{\tilde{\psi }}^{a}+V{\tilde{\psi }}^{a},$$
(47)

with V(r) = − Z/r. The latter in parabolic coordinates

$$x=(\eta -\xi )/2,\,\,\,\,\,y=\sqrt{\eta \xi }\cos (\phi ),\,\,\,\,\,z=\sqrt{\eta \xi }\sin (\phi ),$$
(48)

reads:

$$-\frac{2}{\eta +\xi }\left[{\partial }_{\eta }(\eta {\partial }_{\eta }\tilde{{\psi }^{a}})+{\partial }_{\xi }(\xi {\partial }_{\xi }{\tilde{\psi }}^{a})\right] +\left[-\frac{2Z}{\eta +\xi }-\frac{(\eta -\xi ){E}_{0}}{2}-\varepsilon \right]{\tilde{\psi }}^{a}=0.$$
(49)

With the ansatz \({\tilde{\psi }}^{a}={\psi }_{\eta }(\eta ){\psi }_{\xi }(\xi )\), the variables are separated

$$-\frac{1}{2\eta }-\varepsilon \eta -\frac{{E}_{0}{\eta }^{2}}{2}-\kappa -\frac{2\eta {\psi }_{\eta }^{{\prime} }(\eta )}{{\psi }_{\eta }(\eta )}-\frac{2\eta \,{\psi }_{\eta }^{{\prime\prime} }(\eta )}{\left.{\psi }_{\eta }(\eta )\right)}=0,$$
(50)
$$-\frac{1}{2\xi }-\varepsilon \xi +\frac{{E}_{0}{\xi }^{2}}{2}-\kappa -\frac{2\xi {\psi }_{\xi }^{{\prime} }(\xi )}{{\psi }_{\xi }(\xi )}-\frac{2\xi \,{\psi }_{\xi }^{{\prime\prime} }(\xi )}{\left.{\psi }_{\xi }(\xi )\right)}=0,$$
(51)

where Z = κ. We apply the WKB approximation with respect to the electric field effect, using the unperturbed atomic bound state as the wavefunction limit at the vanishing electric field. Accordingly, the following ansatz is employed:

$$\begin{array}{rcl}{\psi }_{\eta }&=&\sqrt{\frac{{\kappa }^{3}}{\pi }}\exp [-\kappa \eta /2+i{S}_{0}^{\eta }(\eta )+i{S}_{1}^{\eta }(\eta )],\\ {\psi }_{\xi }&=&\exp [-\kappa \xi /2+i{S}_{0}^{\xi }(\xi )+i{S}_{1}^{\xi }(\xi )],\end{array}$$
(52)

which yields the following equations in the leading order

$$2\varepsilon +{E}_{0}\eta +{\left(\kappa -2i{{S}_{0}^{\eta }}^{{\prime} }(\eta )\right)}^{2}=0,$$
(53)
$$2\varepsilon -{E}_{0}\xi +{\left(\kappa -2i{{S}_{0}^{\xi }}^{{\prime} }(\xi )\right)}^{2}=0,$$
(54)

with solutions:

$${S}_{0}^{\eta }(\eta )=\frac{{\left(2\varepsilon +{E}_{0}\eta \right)}^{3/2}}{3{E}_{0}}-\frac{i}{2}\kappa \eta ,$$
(55)
$${S}_{0}^{\xi }(\xi )=-\frac{{\left(2\varepsilon -{E}_{0}\xi \right)}^{3/2}}{3{E}_{0}}-\frac{i}{2}\kappa \xi .$$
(56)

The next corrections fulfill the equations

$$-4i\varepsilon -3i{E}_{0}\eta -2Z\sqrt{2\varepsilon +{E}_{0}\eta }+4\eta (2\varepsilon +{E}_{0}\eta ){{S}_{1}^{\eta }}^{{\prime} }(\eta )=0,$$
(57)
$$-4i\varepsilon +3i{E}_{0}\xi -2Z\sqrt{2\varepsilon -{E}_{0}\xi }+4\xi (2\varepsilon -{E}_{0}\xi ){{S}_{1}^{\xi }}^{{\prime} }(\xi )=0,$$
(58)

with solutions

$${S}_{1}^{\eta }(\eta )=-\frac{Z}{\sqrt{2\varepsilon }}\,{{\rm{arctanh}}}\,\left[\sqrt{\frac{2\varepsilon +{E}_{0}\eta }{2\varepsilon }}\right]+\frac{i}{4}\ln \left[(2\varepsilon +{E}_{0}\eta ){({E}_{0}\eta )}^{2}\right],$$
(59)
$${S}_{1}^{\xi }(\xi )=-\frac{Z}{\sqrt{2\varepsilon }}\,{{\rm{arctanh}}}\,\left[\sqrt{\frac{2\varepsilon -{E}_{0}\xi }{2\varepsilon }}\right]+\frac{i}{4}\ln \left[(2\varepsilon -{E}_{0}\xi ){({E}_{0}\xi )}^{2}\right],$$
(60)

with the complex quasienergy44:

$$\begin{array}{rcl}\varepsilon &=&{\varepsilon }_{0}-i\Gamma \hfill\\ {\varepsilon }_{0}&=&-\frac{{\kappa }^{2}}{2}-\frac{9}{4}\frac{{E}_{0}^{2}}{{\kappa }^{4}} \hfill\\ \Gamma &=&\frac{2{\kappa }^{5}}{{E}_{0}}\exp \left[-\frac{2{(-2{\varepsilon }_{0})}^{3/2}}{3{E}_{0}}\right]\left(1-\frac{53}{12}\frac{{E}_{0}}{{\kappa }^{3}}\right).\end{array}$$
(61)

Note that the solutions are not normalized, since this is not necessary for the phase calculation.

The time integral in the SFA-amplitude in Eq. (3) is calculated analytically via

$$\int\,dt{\psi }_{0}^{f* }(x,t)\exp (-i\varepsilon t)=\frac{1}{\root 6 \of {2}\sqrt{\pi }{E}_{0}^{2/3}}\,{{\rm{Ai}}}\,\left(-\frac{\root 3 \of {2}\left(\varepsilon +x{E}_{0}\right)}{{E}_{0}^{2/3}}\right),$$
(62)

and the final coordinate integration in Eq. (3), with V = − 2Z/(η + ξ) and d3r = (η + ξ)/4dηdξdϕ, is carried out numerically. Using the Coulomb eigenstate of Eq. (52) we arrive at the phase of Eq. (31).