Relativistic features and time delay of laser-induced tunnel-ionization

Relativistic features and time delay of laser-induced tunnel-ionization

Abstract

The electron dynamics in the classically forbidden region during relativistic tunnel-ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge invariant total energy operator. Introducing position dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel-ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser’s propagation direction. This is compatible with results from a strong field approximation calculation where the binding potential is assumed to be zero-ranged. Further, the tunneling time delay, stemming from Wigner’s definition, is investigated for model configurations of tunneling and compared with results obtained from the exact propagator. By adapting Wigner’s time delay definition to the ionization process, the tunneling time is investigated in the deep-tunneling and in the near-threshold-tunneling regimes. It is shown that while in the deep-tunneling regime signatures of the tunneling time delay are not measurable at remote distance, it is detectable, however, in the latter regime.

pacs:
32.80.Rm, 31.30.J-, 03.65.Xp

I Introduction

The investigation of relativistic regimes of laser-atom interactions, in particular the strong field ionization of highly charged ions Moore et al. (1999); Chowdhury et al. (2001); Dammasch et al. (2001); Yamakawa et al. (2003); Gubbini et al. (2005); DiChiara et al. (2008); Palaniyappan et al. (2008); DiChiara et al. (2010), is feasible with current laser technology Yanovsky et al. (2008); Piazza et al. (2012). Strong field multiphoton atomic processes in the relativistic domain are governed by three parameters Becker et al. (2002) which can be chosen to be the Keldysh parameter Keldysh (1964), the barrier suppression parameter , and the relativistic laser field parameter , with the ionization potential , the atomic field , the laser’s electric field amplitude , the angular frequency , and the speed of light (atomic units are used throughout). At small Keldysh parameters () the laser field can be treated as quasistatic and the ionization is in the so-called tunneling regime up to intensities with , while for higher intensities over-the-barrier ionization dominates Augst et al. (1989). In the nonrelativistic case, the quasistatic tunnel-ionization is a well-established mechanism which is incorporated as a first step in the well-known simple-man three-step model of strong field multiphoton ionization Corkum (1993). In the first step of this intuitive picture the bound electron tunnels out through the effective potential barrier governed by the atomic potential and the scalar potential of the quasistatic laser field as , where indicates the moment of quasistatic tunneling. In nonrelativistic settings, the effect of the magnetic field component of the laser field can be neglected and the quasistatic laser field is described solely in terms of the scalar potential . In the second step the ionized electron propagates in the continuum quasiclassically and the third step is a potential recollision of the laser driven electron with the ionic core that will not be considered here.

When entering the relativistic regime at , however, the laser’s magnetic field modifies the second step via an induced drift motion of the continuum electron into the laser propagation direction. For even stronger laser fields, that is when , the laser’s magnetic field can also not be neglected anymore during the first step. Here, the description by a sole scalar potential is not valid anymore and the intuitive picture of tunneling fails. The presence of a vector potential which generates the associated magnetic field led to a controversy over the effective potential barrier ?. Hence we ask, can the tunneling picture be remedied for the application in the relativistic regime and can it be formulated in a gauge-independent form? These questions are addressed in this paper and it is shown that for a quasistatic electromagnetic wave, it is possible to define a total energy operator where the tunneling barrier can be identified without ambiguity in any gauge.

A further interesting and controversial aspect of tunneling and hence tunnel-ionization is the issue of whether the motion of the particle under the barrier is instantaneous or not. The question of whether the tunneling phenomenon confronts special theory of relativity has been raised Nimtz (2011). The main difficulty in the definition of the tunneling time delay is due to the lack of a well-defined time operator in quantum mechanics. For the generic problem of the tunneling time MacColl (1932) different definitions have been proposed and the discussion of their relevance still continues Eisenbud (1948); Wigner (1955); Smith (1960); Landauer and Martin (1994); Sokolovski (2007); Steinberg (2007); Ban et al. (2010); Galapon (2012); ?. Recent interest to this problem has been renewed by a unique opportunity offered by attosecond angular streaking techniques for measuring the tunneling time during laser-induced tunnel-ionization Eckle et al. (2008a, b); Pfeiffer et al. (2012); Maurer et al. (2013). Here we investigate the tunneling time problem for ionization in nonrelativistic as well as in relativistic settings. For tunnel-ionization by electromagnetic fields there is a direct relationship of the tunneling time to the shift in coordinate space of the ionized electron wave packet in the laser’s propagation direction at the appearance in the continuum Klaiber et al. (2013a). Within the quasiclassical description, using either the Wentzel-Kramers-Brillouin (WKB) approximation or a path integration in the Euclidean space-time along the imaginary time axis Perelomov et al. (1966); Popov (2004, 2005); Popruzhenko et al. (2008); Popruzhenko and Bauer (2008), the under-the-barrier motion is instantaneous. Thus, we address the time delay problem by going beyond the quasiclassical description. In the present paper, we adopt Wigner’s approach to the tunneling time Eisenbud (1948); Wigner (1955); Smith (1960) which, in simple terms, allows to follow the peak of the tunneled wave packet. We find conditions when a non-vanishing Wigner time delay under the barrier for the tunnel-ionization is expected to be measurable by attosecond angular streaking techniques.

One of the theoretical tools applied in this paper is the relativistic strong field approximation (SFA) Reiss (1990a, b). Neglecting the atomic potential for the continuum electron and approximating its dynamics with a Volkov state is the main approximation of the SFA Faisal (1973); Keldysh (1964); Reiss (1980). Consequently, the prediction of the SFA is much more accurate for a zero-range potential than for a more realistic long-range potential as the Coulomb potential. SFA calculations for the tunnel-ionization modeled with a zero-range potential show that there is a momentum shift along the laser’s propagation direction due to the tunneling step. We find that this shift can also be estimated via a WKB analysis when a Coulomb potential is used and that it is measurable in a detector after the laser field has been turned off.

The structure of the paper is the following. In Sec. II the parameter domain of the relativistic tunneling dynamics is estimated. In Sec. III gauge invariance in quantum mechanics is discussed and the gauge independence of the tunneling barrier is established in nonrelativistic as well as in relativistic settings. The intuitive picture for the tunnel-ionization is discussed in Sec. IV reducing the full problem to a one-dimensional one. In Sec. V the SFA formalism is presented and the momentum distribution at the tunnel exit is calculated. In Sec. VI the tunneling time delay and its corresponding quasiclassical counterpart are investigated and in Sec. VII it is applied to tunnel-ionization. Our conclusions and further remarks are given in Sec. VIII.

Ii Relativistic parameters

Let us estimate the role of relativistic effects in the tunnel-ionization regime which is valid for and for the intensities up to . The typical velocity of the electron during the under-the-barrier dynamics can be estimated from the bound state energy as (for a hydrogenlike ion with charge and in the ground state it follows ). The nonrelativistic regime of tunneling is defined via . This relation is valid for hydrogenlike ions with nuclear charge up to where . For ions with charge the relativistic regime is entered because the velocity during tunneling is not negligible anymore with respect to the speed of light. However, even for an extreme case of with it is , i. e., the dynamics is still weakly-relativistic and a Foldy-Wouthuysen expansion of the relativistic Hamiltonian up to order is justified. The expansion yields

(1)

where is the binding potential, the four-vector potential is given by and the phase of the electromagnetic wave is , with and and the laser’s propagation direction .

In the tunneling regime, the typical displacement along the laser’s propagation direction can be estimated as with the Lorentz force and the typical ionization time (Keldysh time) . Identifying the Lorentz force along the laser’s propagation direction as (), the typical distance reads . Hence, electric as well as magnetic non-dipole terms are negligible since , i. e., the typical width of the electron’s wave packet is small compared to the laser’s wavelength.

Furthermore, the leading spin term in Eq. (1) is the spin-magnetic field coupling Hamiltonian . Its order of magnitude can be estimated as . Therefore, in the tunneling regime the spin related terms and the Darwin term in Eq. (1) can be neglected because . In summary, the electron’s under-the-barrier dynamics is governed in the Göppert-Mayer gauge, see Sec. III, by the Hamiltonian

(2)

with the free atomic Hamiltonian , the electric-dipole , the magnetic-dipole , the relativistic kinetic energy correction , and finally . For the electron’s under-the-barrier dynamics the relative strengths of the various terms of the Hamiltonian (2) are , , , and for typical displacements along the polarization direction.

In the reminder of the article, we consider tunnel-ionization from a Coulomb potential or zero-range potential by a monochromatic plane wave in the infrared ( a.u.) and we use two extreme but feasible sets of parameters which ensure that we are in the tunneling regime, viz.  and for the deep-tunneling regime and for the near-threshold-tunneling regime. Formal definitions of these two regimes will be given in Sec. VII.4.

Iii Electrodynamics, gauge freedom, and tunneling

In this section we demonstrate how the tunneling barrier in the presence of electromagnetic fields can be defined in a gauge invariant manner. For this purpose, we briefly summarize the gauge theory in the light of Mills (1989); Weinberg (2013).

iii.1 Gauge theory

In classical electrodynamics the Maxwell equations allow to express the physical quantities, the electric and magnetic fields, in terms of a scalar potential and a vector potential

(3)
(4)

Gauge invariance is the feature of electrodynamics that any other pair of a scalar and a vector potential that is related by a so-called gauge transformation describes the same electromagnetic fields. More precisely, the transformation

(5a)
(5b)

induced via the gauge function leaves the electric field and the magnetic field invariant. Consequently, all physically measurable electrodynamic quantities, the Maxwell equations, and the Lorentz force law are gauge invariant. This means they do not depend on the choice for the electromagnetic potentials. Furthermore, the Schrödinger equation for a particle in electromagnetic fields is invariant under the transformation (5) provided that the state vector transforms with the gauge transformation as

(6)

iii.2 Gauge invariant energy operator

Besides the elegance of the gauge theory, all the physical quantities, i. e., the experimental observables cannot depend on the choice of the gauge function. Let us discuss if a physical tunneling potential barrier can be defined in a gauge independent manner.

Each physical operator that corresponds to a measurable quantity must be gauge independent. For example, the canonical momentum operator , transforms under the gauge transformation as

(7)

The kinetic momentum , however, obeys

(8)

Here, the canonical momentum which generates the space translation and satisfies the canonical commutation relation is not a physical measurable quantity, it is the kinetic momentum that is measured in the experiment. In general, any operator that satisfies the transformation

(9)

is called as a physical operator. For instance, the Hamiltonian for a charge particle interacting with an arbitrary electromagnetic field in the nonrelativistic regime

(10)

transforms under the gauge transformation as

(11)

Hence, it cannot be a physical operator (because is not equally transformed to ), while is the physical operator which guarantees the invariance of the Schrödinger equation under a gauge transformation.

In contrast to the Hamiltonian , the total energy of a system has to be a gauge invariant physical quantity. Therefore, we have to distinguish two concepts: the Hamiltonian and the total energy. The Hamiltonian is the generator of the time translation, while the total energy is defined as a conserved quantity of the dynamical system under a time translation symmetry of the Lagrangian. As a consequence, if the Hamiltonian is explicitly time independent, then the Hamiltonian coincides with the total energy operator.

For a time independent electromagnetic field there exists a certain gauge where the Hamiltonian is explicitly time independent. The identification of the Hamiltonian as a total energy operator implies, then, that both the vector potential and the scalar potential associated to the constant electromagnetic field have to be time independent. This leads to the fact that

(12)

where we have used Eq. (3). In this gauge, the Hamiltonian which coincides with the total energy operator in the presence of any external potential reads

(13)

where the time independent vector potential generates the associated magnetic field via Eq. (4).

Accordingly, if we identify Eq. (13) as a definition of the gauge independent total energy operator, it reads in an arbitrary gauge

(14)

where we have used the transformation (9). The first term on the right hand side of (14) is the kinetic energy for an arbitrary vector potential that appears in the corresponding Hamiltonian. The second term should not be regarded as a scalar potential, but defines the potential energy. The energy operator (14) fulfills the conservation law

(15)

which can be prooven in a straightforward calculation. We find

(16)
(17)

and hence

(18)

is obtained. The definition (14), then, suggests to introduce the gauge independent effective potential energy as

(19)

In conclusion, the electron dynamics during ionization can be described as tunneling through a potential barrier if the total energy of the electron is conserved. Then, the potential energy and the tunneling barrier can be identified unambiguously. Thus, for ionization in a laser field we have to identify the quasistatic limit such that the tunneling picture becomes applicable. The tunnel-ionization regime in a laser field is determined by the Keldysh parameter . It defines the so-called tunneling formation time , see Sec. V.3. The tunneling regime corresponds to situations when the formation time of the ionization process is much smaller than the laser period. Consequently, the electromagnetic field can be treated as quasi-static during the tunneling ionization process and the electron energy is approximately conserved. Therefore, the gauge-independent operator for the total energy in a quasistatic electromagnetic field can be defined and from the latter the gauge-independent potential energy can be deduced, which in the case of tunnel-ionization constitutes the gauge-independent tunneling barrier. Therefore, Eq. (19) defines the gauge independent tunneling barrier in the tunnel-ionization regime. In the long wavelength approximation it yields

(20)

where is the moment of ionization and is the binding potential for the tunnel-ionization.

As an illustration of the gauge independence of the tunneling barrier, let us compare two fundamental gauges used in strong field physics to describe nonrelativistic ionization. In the length gauge where , , the nonrelativistic Hamiltonian for a constant uniform electric field is given by

(21)

Here, the Hamiltonian coincides with the physical energy operator. In the velocity gauge, however, where , , the same dynamics is governed by the Hamiltonian

(22)

In Eq. (22) it seems as if there is no potential barrier. However, the conserved energy operator

(23)

reveals the tunneling barrier . Thus, for arbitrary time independent (quasistatic) electromagnetic fields, the gauge-independent tunneling barrier can be defined without any ambiguity.

The physical energy operator and the tunneling barrier can be generalized to the relativistic regime straightforwardly by using the relativistic Dirac Hamiltonian

(24)

From Eq. (14) we deduce the physical energy operator in the relativistic case as

(25)

One possible generalization of the length gauge into the relativistic regime is the Göppert-Mayer gauge

(26)

Taking into account that the dipole approximation for the laser field can be applied inside the tunneling barrier, the Hamiltonian which coincides with the total relativistic energy operator in the Göppert-Mayer gauge reads

(27)

where is the laser phase at the moment of ionization.

Figure 1: The electron density (solid line) along the laser polarization direction at the instant of maximal field strength at the atomic core. The shaded area represents the classical forbidden area, gray dashed lines are exponential fits on the wave function density. The electron’s wave function is obtained by solving the two-dimensional Dirac equation for an electron in a soft-core potential interacting with an external laser pulse, laser parameters are the same as in Klaiber et al. (2013a).

The tunneling barrier results from an interpretation of the individual mathematical terms of the quasistatic energy operator (27). It has, however, also a physical significance as it can be demonstrated by an ab initio numerical simulation of the tunneling process in a highly charged ion in a laser field of relativistic intensities based on the Dirac equation Bauke and Keitel (2011); Klaiber et al. (2013a). Figure 1 shows the gauge-independent electron density along the laser polarization direction at the instant of maximal field strength at the atomic core. The electron density can be divided in two parts that are characterized by two different decay rates. The switchover region includes the tunneling exit that is defined by the tunneling barrier. The decay of the density under the barrier is related to damping due to tunneling, i. e., approximately , whereas outside the barrier it is dominated by transversal spreading. As the change of slopes occurs close to the tunneling exit the tunneling barrier is real and physical and not just a result of an interpretation in a particular gauge.

Iv Intuitive picture for the relativistic tunnel-ionization process

Having identified the gauge invariant tunneling barrier, we elaborate in this section on the intuitive picture for the tunnel-ionization process in the relativistic regime. For the reminder of the article we choose our coordinate system such that the laser’s electric field component is along the direction, the laser’s magnetic component is along the direction and the laser propagates along the direction. Since we assume to work in the quasistatic regime, the the Göppert-Mayer gauge is applied for the Hamiltonian such that it coincides with the energy operator. In the nonrelativistic limit the latter agrees with the Schrödinger Hamiltonian in the length gauge.

iv.1 Nonrelativistic case

In the nonrelativistic limit the intuitive picture for the tunnel-ionization is well-known. In this picture the magnetic field and nondipole effects can be neglected, and the Hamiltonian reads

(28)

with . Introducing the potential

(29)

one can define the classical forbidden region. The tunneling probability increases with decreasing width of the barrier, thus, the most probable tunneling path is concentrated along the electric field direction as indicated by the dashed line in Fig. 2. Therefore, it is justified to restrict the analysis of the tunneling dynamics along the laser’s polarization direction 1.

Figure 2: (Color online) The potential barrier for quasistatic tunnel-ionization. The laser field is along the direction. The most probable tunneling path is indicated by the black dashed line.
Figure 3: The nonrelativistic tunneling probability versus the momentum along direction. The maximum tunneling probability occurs at . The applied parameters are and . Without loss of generality, was chosen.
Figure 4: (Color online) Schematic picture of nonrelativistic (sub-figures (a) and (b)) and relativistic (subfigure (c)) tunneling from a bound state into the continuum: (a) the atomic potential is aproximated by a zero-range potential; (b) and (c) the Coulomb potential case. The potential barrier (solid, blue) and the energy levels (dashed, red) for (in the nonrelativistic cases) and for the most probable transversal momentum (relativistic case) are plotted against the longitudinal tunneling coordinate . The shaded area can be interpreted as a measure for the tunneling probability. The electron wave packet is indicated in black.

In this one-dimensional picture the barrier for the tunnel-ionization follows as

(30)

The momentum components and along the and the direction are conserved and tunneling along the direction is governed by the energy

(31)

The wave function of the electron and the corresponding transition probability can be derived within the Wentzel-Kramers-Brillouin (WKB) approximation. The zeroth order WKB wave function is given by

(32)

with the classical action

(33)

and the momentum’s component

(34)

The WKB tunneling probability follows as

(35)

where and are the entry point and exit point of the barrier such that . The dependence of the tunneling probability on the momentum is shown Fig. 3. The tunneling probability is maximal for , because the energy level (31) decreases with increasing . From this it follows that the exit coordinate increases with increasing .

In summary, nonrelativistic tunneling from an atomic potential can be visualized by an one-dimensional picture given in Fig. 4(a) and (b). The area between the barrier and the energy level represents a messure for the probability of the process. The larger the area the less likely the ionization.

iv.2 Relativistic case

In the relativistic regime, the largest correction to the nonrelativistic Hamiltonian comes from the magnetic dipole term. Let us consider the role of the magnetic dipole interaction in the laser field for the tunneling picture. The corresponding time-independent Schrödinger equation reads

(36)

Similar to the nonrelativistic case, an approximate one-dimensional description is valid for the most probable tunneling path along the electric field direction. Restricting the dynamics along the electric field direction and neglecting the dependence of the ionic core’s potential on the transverse coordinate, we have , and the momentum along the polarization direction is given by Eq. (34) with the barrier (30), which is the same as in the nonrelativistic case. The energy, however, is modified by the magnetic dipole term

(37)

The energy level (37) depends on the coordinate. This is because the electron’s kinetic momentum along the laser’s propagation direction changes during tunneling due to the presence of the vector potential (magnetic field). As a consequence, the tunneling probability in the relativistic regime is maximal at some non-zero canonical momentum in the laser’s propagation direction. For instance, the kinetic momentum with maximal tunneling probability at the tunneling entry is , whereas at the exit it is for the Coulomb potential, see Fig. 5. During the under-the-barrier motion the electron acquires a momentum kick into the laser’s propagation direction due to the Lorentz force, which can be estimated as

(38)

with the barrier length . Thus, relativistic tunneling can be visualized by an one-dimensional picture given in Fig. 4(c). The area between the barrier and the position dependent energy level is larger than in the nonrelativistic case due to the non-vanishing transversal kinetic energy, indicating the reduced tunneling probability in the relativistic description.

The WKB analysis can be carried out also in a fully relativistic way. Taking into account the relativistic energy-momentum dispersion relation, one obtains for the momentum and the ionization energy along the polarization direction

(39)
(40)

which determines the fully relativistic tunneling probability via Eq. (35). The latter is shown in Fig. 5. For comparison it shows also the results for the nonrelativistic case, the calculation using the magnetic dipole correction, and the calculation with the leading relativistic kinetic energy correction 2.

As demonstrated in Fig. 5 the shift of the kinetic momentum along the laser’s propagation direction that maximizes the WKB tunneling probability is determined mainly by the magnetic dipole correction to the Hamiltonian. This correction also decreases the tunneling probability, since the Lorentz force due to the laser’s transversal magnetic field transfers energy from the tunneling direction into the perpendicular direction hindering tunneling. Taking into account further relativistic effects does not change the behavior qualitatively but increases the tunneling probability. This can be understood intuitively by noticing that in the reference frame of the relativistic electron the length of the barrier is contracted and in this way enhancing the tunneling probability. The mass correction term is more important in the zero-range-potential case than in the Coulomb-potential one, since the typical longitudinal velocities are smaller in the latter case. Furthermore, Fig. 5 indicates that the calculation including only the leading relativistic kinetic energy correction reproduces the fully-relativistic approach satisfactorily. Thus, the magnetic dipole and the leading order mass shift are the only relevant relativistic corrections.

Figure 5: (Color online) Tunneling probability vs. the kinetic momentum along the laser’s propagation direction at the tunnel entry (lines with peak on the left) and tunnel exit (lines with peak on the right). Results for calculations including magnetic dipole correction are indicated in blue while for the red lines also the leading relativistic correction to the kinetic energy are taken into account. The dashed-black lines correspond to a fully relativistic calculation, which are very close to the ones including leading relativistic corrections to the kinetic energy. The densities are normalized to the maximum density in the nonrelativistic case. A similar comparison was made in Klaiber et al. (2013a) for the case of a zero-range potential via SFA.

The value of the kinetic momentum shift at tunnel exit varies significantly with respect to the barrier suppression parameter in the case of a Coulomb potential of the ionic core, as shown in Fig. 6, while it does not depend on the laser field in the case of zero-range atomic potential. The main reason for the decreased momentum shift in the Coulomb potential case is that the length of the Coulomb-potential barrier is reduced approximately by a factor compared to the barrier length of the zero-range potential. According to Eq. (38), this barrier length reduction leads to smaller momentum kick due to the magnetic field.

Figure 6: The kinetic momentum shift at the tunnel exit versus the barrier suppression parameter for an electron bound by a Coulomb potential (dashed) and a zero-range atomic potential (solid).

Furthermore, we compare the prediction of the WKB approximation with the results obtained by an ab initio numerical calculation solving the time-dependent Dirac equation Bauke and Keitel (2011). For this purpose the tunnel-ionization from a two-dimensional soft-core potential was simulated yielding the time-dependent real space wave function . A transformation into a mixed representation of position and kinetic momentum via

(41)

allows us to determine the kinetic momentum in direction as a function of the coordinate and in this way at the tunnel exit , see Fig. 7. Both, the solution of the fully relativistic Dirac equation and the WKB approximation predict a momentum distribution with a maximum shifted away from zero. The momentum shifts are in a good agreement.

Figure 7: (Color online) Electronic density in the mixed space of position and kinetic momentum at the moment of maximal field strength at the atomic core (left panel). The electron’s wave function has been obtained by simulating the tunnel-ionization from a two-dimensional soft-core potential by solving the time-dependent Dirac equation with all numerical parameters as in Fig. 1. The right panel shows the normalized kinetic momentum distribution of the tunneled electron at the tunnel exit (indicated by the white line in the left panel) as obtained by solving the Dirac equation and by the WKB approximation.

V Tunnel-ionization with a zero-range potential model

The intuitive considerations of the previous section on the relativistic under-the-barrier motion during tunnel-ionization led us to the conclusion that relativistic tunneling induces a momentum kick along the laser’s propagation direction. The aim of this section to prove this conclusion by a rigorous calculation based on SFA, to show how this momentum shift arises during the under-the-barrier motion, and to find out how this relativistic signature is reflected in the electron momentum distribution in far distance at the detector.

The SFA is based on an S-matrix formalism. The ionization is described by the Hamiltonian

(42)

where is the field-free atomic Hamiltonian including the atomic potential and denotes the Hamiltonian of the laser-atom interaction. Initially at time , the electron is in the bound state . In SFA the influence of the atomic core potential on the free electron and the influence of the laser field on the bound state are neglected. This allows us to express the time evolution of the state vector in the form Becker et al. (2002)

(43)

where is the Volkov propagator which satisfies

(44)

with the Volkov Hamiltonian . The SFA wave function in momentum space of the final state reads

(45)

where denotes a Volkov state Volkov (1935). The ionized part of the wave function in momentum space in Eq. (45) can be expressed also in the form Becker et al. (2002)

(46)

As the SFA neglects the effect of the atomic potential on the final state, the SFA gives an accurate prediction when the atomic potential is short ranged. For this reason, we will model the tunnel-ionization with a zero-range potential in the following. The nonrelativistic tunneling scheme for this case is visulized in Fig. 4(a). The barrier has triangular shape which simplifies the analytical treatment of the tunneling dynamics.

v.1 Nonrelativistic case

Let us start our analysis with the nonrelativistic consideration when the Hamiltonian for an atom in a laser field is given by Eq. (42) with

(47)
(48)

where is the zero-range atomic potential. In SFA the ionized part of the wave function far away after the laser field has been turned off reads Klaiber et al. (2013b)

(49)

where

(50)

is the contracted action, is the kinetic momentum, , , and is the bound state of the zero-range potential. The time integral in Eq. (49) can be calculated via the saddle point approximation (SPA). The saddle point equation

(51)

yields the kinetic momentum at the saddle point time . Then, the wave function in momentum space reads in the quasi-static limit

(52)

for the vector potential and with and . From expression (52) it follows that the density of the ionized wave function is maximal at for any value of . The coordinate space wave function

(53)

is obtained by a Fourier transform of Eq. (49). From the SPA it follows that the main contribution to the integral over in Eq. (53) originates from momenta near the momentum which fulfills the saddle point condition . The latter defines the trajectories which contribute to the transition probability with amplitudes depending on . For the most probable final momentum , the trajectory which start at the tunneling entry at time is given by

(54)

The line integral in Eq. (54) is along a path connecting the complex time with the real time . The complex saddle point time can be determined by solving for . The corresponding kinetic momentum is

(55)

In Fig. 8 the complex trajectory (54) and the complex kinetic momentum (55) along the tunneling direction are shown. The spatial coordinate is real under the barrier as well as behind the barrier, whereas the kinetic momentum is imaginary during tunneling and becomes real when leaving the barrier, which corresponds to the time . The tunneling exit coordinate is which is consistent with the intuitive tunneling picture. The momentum in the tunneling direction is when tunneling starts, whereas it is at the tunnel exit.

Figure 8: Coordinate (sub-figure (a)) and momentum (sub-figure (b)) along the electric field direction of the complex SFA trajectory for tunnel-ionization from a zero-range potential. During the under-the-barrier motion the momentum is imaginary whereas the coordinate is real. Tunneling finishes when the kinetic momentum becomes real.

v.2 Relativistic case

Our fully relativistic consideration is based on the Dirac Hamiltonian with a zero-range atomic potential

(56)

where and are standard Dirac matrices Lifshitz et al. (1996) and the Göppert-Mayer gauge (26) is employed. The ionized part of the momentum wave function in SFA yields

(57)

Here is the ground state of the zero-range potential with the ground state spinor and ; is the relativistic Volkov wave function in the Göppert-Mayer gauge for a free electron in a laser field, which is obtained from the Volkov wave function in the velocity gauge Volkov (1935) with further gauge transformation via the gauge function , . Furthermore, is the normalization constant,

(58)

is the quasiclassical action and

(59)

with , , and the free particle spinor . After averaging over the spin of the initial electron as well as over the spin of the ionized electron the wave function of the ionized electron reads

(60)

Here, a coordinate transformation is employed and

(61)

is the contracted action with the field-dressed electron momentum in the laser field

(62)

For a zero-range potential the inner product is only dependent. Further, the -integral in (58) can be calculated using SPA and the saddle point equation yields as in the nonrelativistic regime Klaiber et al. (2013c). Then, the wave function of the ionized electron in momentum space yields

(63)

for the vector potential with the laser’s propagation vector , where

(64)
(65)
(66)
Figure 9: (Color online) The momentum distribution of the ionized electron: (a) in infinity at the detector; (b) at the tunnel exit, depending on the tunnel exit time . The maximum tunneling probability occurs at . The applied parameters are , and .

The relativistic momentum distribution of the ionized electron of Eq. (63) differs qualitatively from the nonrelativistic one. In the nonrelativistic case the maximum of the distribution is at at any . In the relativistic case, however, the momentum distribution has a local maxima along the parabola which can be approximated as

(67)

see Fig. 9(a). The global maximum of the tunneling probability is located at , while in the nonrelativistic case it is at . This shift of the maximum is connected with the first step of the ionization, the tunneling, whereas the parabolic wings are shaped in the second step, the continuum dynamics. These wings are located around with the ponderomotive potential .

The momentum distribution at the tunnel exit can be calculated via back propagation of the final momentum space wave function (63). Thus, the wave function at the tunnel exit is

(68)

with the tunnel exit time and final time where the interaction is turned off, hence the Volkov wave function reduces to the free particle wave function. Because holds at the tunnel exit holds, the exact Volkov propagator

(69)

can be simplified expanding the phase dependent functions around , which yields

(70)

with the exit momentum and the phase

(71)
(72)

respectively. As a result, the momentum space wave function at the tunnel exit reads in terms of the final wave function

(73)

with

(74)

The transversal momentum distribution at the tunnel exit can be calculated via replacing the momentum in the wave function Eq. (63) with Eq. (74), which can be seen in Fig. 9(b). The comparison of Figs. 9(a) and (b) indicates that the relativistic shift of the peak of the transverse momentum distribution at the tunnel exit is maintained in the final momentum distribution. The parabola can for example be calculated from classical trajectories. The kinetic momentum at the exit is connected with the final momenta via

(75)

and the relation

(76)

follows.

Figure 10: (Color online) Comparison of the nonrelativistic (blue) and relativistic (black) complex SFA trajectories: (a) coordinates and (b) momentum components. The dashed and the solid lines correspond to the under-the-barrier motion and the motion after the tunneling, respectively. In the relativistic regime, the trajectory enters the tunneling barrier at , it is complex under the barrier and becomes real again when it leaves the tunneling barrier. The trajectory enters the barrier with the transversal momentum and leaves the barrier with . The applied parameters are , .

We investigate the trajectory of the electron and its momentum during the tunneling in the relativistic regime. The coordinate wave function can be obtained via a Fourier transform of momentum space wave function (60). Then, the stationary phase condition gives the quasiclassical trajectories at the most probable momentum given by Eq. (67). The results are plotted in Fig. 10. It shows that in the relativistic regime most probable trajectory is the trajectory where the electron enters the barrier with the transversal momentum and reaches the exit with . This is in accordance with our intuitive discussion in Sec. IV.

For the most probable momentum, the trajectory starts at the real axis, obtains complex values during tunneling and has to return to the real axis after tunneling Fig. 10(a). Here an analogy to relativistic high-harmonic generation can be drawn, where also a return condition has to be fulfilled: the recollision of the ionized electron to the atomic core in the presence of the drift motion induced by the laser’s magnetic field Klaiber et al. (2007). Similarly, the electron has to start with a momentum of the order of against the laser propagation direction (cf.  at the entering the ionization barrier) which compensates the drift motion and facilitates the recollision to the atomic core with a momentum along the laser’s propagation direction (cf.  at the exit of the ionization barrier).

The shift of the electron’s momentum distribution along the laser’s propagation direction in the relativistic regime is possible to detect by measuring the final momentum distribution of the ion Smeenk et al. (2011). The ionized electron acquires momentum along the laser’s propagation direction in the laser field because of the absorbed momentum of the laser photons. However, part of the momentum of laser photons is transferred to the ion. The energy conservation law provides a relationship between the number of absorbed photons and the electron momentum

(77)

where is the laser frequency, the ionization potential, and the energy of the electron. The kinetic energy of the ionic core can be neglected due to the large mass of the ion. Additionally, the momentum conservation law gives information on the sharing of the absorbed photon momentum between the ion and the photoelectron