Virtual-detector approach to tunnel and tunneling times

Nicolas Teeny,∗ Christoph H. Keitel, and Heiko Bauke† Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

Tunneling times in atomic ionization are studied theoretically by a virtual detector approach. A virtual detector is a hypothetical device that allows one to monitor the wave function’s density with spatial and temporal resolution during the ionization process. With this theoretical approach, it becomes possible to define unique moments when the enters and leaves with highest probability the classically forbidden region from first principles and a tunneling time can be specified unambiguously. It is shown that neither the moment when the electron enters the tunneling barrier nor when it leaves the tunneling barrier coincides with the moment when the external electric field reaches its maximum. Under the tunneling barrier as well as at the exit the electron has a nonzero velocity in the electric field direction. This nonzero exit velocity has to be incorporated when the free motion of the electron is modeled by classical equations of motion.

PACS numbers: 03.65.Xp, 32.80.Fb

1. Introduction 0.0 0.2 − (a.u.) 0.4

4 − In a seminal work, MacColl [1] studied the time that may be / 0.6 E

, − ) 0.8

associated with the process of a particle approaching from far ξ ( − ξ ξ 1 1.0 in exit away a potential barrier of a height larger than the particle’s V − 1.2 energy and eventually tunneling through the barrier. A series of − subsequent works led to various definitions of tunneling times 0.0 0.2 and different physical interpretations [2–4]. Also many efforts − (a.u.) 0.4

4 − have been directed towards measuring [5–10] tunneling times. / 0.6 E

, − A related still open and actively studied problem of atomic ) 0.8 η ( − η physics is the question how long it takes to ionize an via an 2 1.0 0 V − electron tunneling through the potential barrier [11–28] which 1.2 − is formed by the electron’s binding energy and the Coulomb po- 0 1 2 3 4 5 ξ, η (a.u.) tential bent by the time-dependent electric field’s potential; see Fig. 1. Employing the angular streaking technique the tunneling FIG. 1: Tunneling potentials V1(ξ) and V2(η) (solid lines) and the dynamics can be studied on subcycle time scales [14, 15] in the ground state energy level E (dashed lines) for the two-dimensional so-called attoclock experiments, aiming to determine when the Coulomb problem with (black) and without (light blue) accounting for electron ionizes from a bound state. The attoclock experiments the Stark effect in parabolic coordinates ξ and η for = 1 a.u. Inter- E stimulated renewed efforts towards defining a well-founded sections between the potential lines and the energy level determine the tunnel ionization time and determining the tunnel ionization borders between classically allowed classically forbidden regions, i. e., time. A consensus on a suitable definition of tunneling time the tunneling entry ξin, tunneling exit ξexit, and η0. See the Appendix and the interpretation of experimental results is, however, still for details. lacking [29]. In the case of tunnel ionization the experimental determi- possible to place a detector close to the atomic tunneling bar- nation of a tunneling time is complicated by the fact that it rier. Thus, information about the tunneling dynamics has to is notoriously difficult to determine the starting (and ending) be inferred from measurable asymptotic quantities, e. g., the moment of the tunnel dynamics. There is no apparent reason to arXiv:1605.06411v2 [quant-ph] 4 Aug 2016 momentum distribution of the photo ionized . In at- assume that the electron enters the barrier at the instant of the toclock experiments, an electron is ionized by an elliptically electric field maximum. In fact, we are going to demonstrate polarized few-cycle pulse. This quasi-free electron is acceler- that one has to carefully distinguish between the tunneling time, ated in the rotating electric field, and in this way the instant of i. e., the time to cross the tunneling barrier, and the tunneling ionization texit is mapped to the final angle of the momentum delay, i. e., the time delay until the electron becomes quasi-free vector in the polarization plane. Ignoring Coulomb effects as with respect to some external reference, e. g., the moment of well as non-dipole effects, the electron’s final momentum p(t ) maximal electric field strength. f and the instant of ionization texit are related according to the Experimentally the quantum dynamics in the vicinity of two-step model [30–33] via the tunneling barrier cannot be studied directly, i. e., it is not Z tf p(tf ) = p(texit) + q E(t) dt , (1) texit

∗ [email protected] where E(t) denotes the electric field, which vanishes for t tf , [email protected] → † q denotes the electron’s charge, and p(texit) is its initial momen- 2 tum, which is usually assumed to be zero. Assuming that the where Ψ(r, t)∗ indicates the complex conjugate of Ψ(r, t). The ionized electron becomes free at the instant of the electric field dynamics of the density %(r, t) is associated with the probability maximum t0, the formula (1) predicts a final momentum with current a specific direction [27]. Deviations of an experimentally ob- served final momentum direction from this prediction may be 1  j(r, t) = Re Ψ(r, t)∗ ( i~ qA(r, t)) Ψ(r, t)) . (4) due to a delay between the instant of maximal field strength and m − ∇ − the instant of ionization, i. e., t0 , texit. But also Coulomb ef- r fects [34], a nonzero initial momentum p(texit) [28], nondipole Expressing the wave function Ψ( , t) as effects, or quantum effects, which may play a role for the dy- p namics at the vicinity of the tunneling exit, cause deviations Ψ(r, t) = %(r, t) exp(iϕ(r, t)) , (5) from the simple two-step model (1). Thus, the interpretation of attoclock experiments requires a precise model of the electron’s the probability current j(r, t) may be written as a product of motion from the barrier exit to the detector. The value of a the probability density %(r, t) and some local velocity, viz., possible tunneling delay depends crucially on the theoretical model [25], which is employed to calibrate the attoclock. %(r, t) j(r, t) = (~ ϕ(r, t) qA(r, t)) , (6) In view of the difficulties of determining tunneling times m ∇ − from asymptotic momentum measurements, we consider a com- plementary theoretical approach based on ab initio solutions which also occurs in the framework of Bohmian mechanics of the time-dependent Schrödinger equation and virtual de- [37]. The probability density and the probability current fulfill tectors [35, 36], which allows us to link tunneling times to the continuity equation observables related to the quantum dynamics in the vicinity of the tunneling barrier. This paper is organized as follows: To ∂%(r, t) = j(r, t) . (7) keep the presentation self-contained, the concept of a virtual − ∂t ∇ · detector is summarized in Sec. 2. In Sec. 3 numerical results for tunneling times as determined by the virtual detector ap- For any compact subspace V of the physical space with a piece- proach are presented and discussed. Mainly, we answer the wise smooth boundary S the continuity equation (7) yields by questions: When does the electron enter the barrier and exit integration over the subspace the tunneling barrier; and, accordingly, how much time does Z Z the electron spend under the barrier? Additionally, we compare d %(r, t) dV = j(r, t) n dS , (8) the quantum trajectory of the electron to the one predicted by − dt · the two-step model [30–33]. Finally, we conclude in Sec. 4. V S where n denotes the outward pointing unit normal field of the boundary S. Equation (8) holds also for unbounded subspaces if %(r, t) decreases fast enough when r . Thus, the rate 2. Virtual detectors of the change of the probability to find| the| → electron ∞ within the space region V is given by the probability current through the 2.1. Fundamentals volume’s surface Z DS(t) = j(r, t) n dS . (9) Ionization from a binding potential means that an electron S · moves away from the vicinity of the binding potential’s mini- mum. A virtual detector allows us to quantify this dynamics. A virtual detector is a hypothetical device that measures More specifically, a virtual detector is a hypothetical device DS(t). In other words, a virtual detector placed at the surface that determines how the probability to find a particle within S determines how much probability passes from one side of some specific space region changes over time. In the following, the surface to the other per unit time. Placing S at sufficient we present its mathematical foundations. distance around a bound wave packet, which is ionized by a The quantum mechanical evolution of an electron’s wave time-dependent electric field E(t) with limt E(t) = 0, then r, R →±∞ function Ψ( t) with mass m and charge q is governed by the ∞ DS(t) dt equals the total ionization probability. Further- Schrödinger equation more,−∞ if D (t) is strictly nonnegative or nonpositive D (t) ∆t S | S | ! can be interpreted as the probability that the electron moves ∂Ψ(r, t) 1 i~ = ( i~ qA(r, t))2 + qφ(r, t) Ψ(r, t) , (2) from one side of the surface S to the other within the short time ∂t 2m − ∇ − interval [t, t + ∆t]. For one-dimensional systems the surface S degenerates to a single point. For this case, it can be shown where φ(r, t) and A(r, t) denote the electromagnetic potentials. within the framework of Bohmian mechanics that DS(t) is The probability density to find the electron at position r at time up to normalization the probability distribution of the| time| of t is given by arrival at the point S [38, 39]. In general, DS(t) is proportional to the probability that the electron arrives| at the| surface S at %(r, t) = Ψ(r, t)∗Ψ(r, t) , (3) time t. 3

classically forbidden region

classically allowed region

tunneling region

FIG. 2: Schematic of tunneling time determination via virtual detectors at the tunneling entry and the tunneling exit. The lines ξ = ξin,out and η = η0 which determine tunneling region, the classically forbidden, and the classical allowed regions are indicated by the solid lines; see Ref. [40] and the Appendix for details. The black solid lines indicate the positions of the virtual detectors. The wave-function’s probability density Ψ(x, y, t)∗Ψ(x, y, t) at the instant of electric field maximum, i. e., t = t , is represented by colors for = 1.1 a.u. and for the Keldysh 0 E0 parameter γ = 0.25. Arrows indicate the probability current j(x, y, t), where the arrows’ length is proportional to the absolute value of the current and the arrows’ opacity scales with the wave function’s probability density.

2.2. Tunnel ionization from a where t denotes the instant of maximal electric field (t ) = 0 E 0 E0 two-dimensional Coulomb potential and τ = √2/ω is the time scale of the raise and decay of the electricE field. Cartesian coordinates are not the most suitable choice to In the following, we will study tunnel ionization from a two- deal with the Coulomb problem with an external homogeneous dimensional Coulomb potential. The restriction to two dimen- electric field. Parabolic coordinates are the natural coordinate sions is mainly because this system resembles tunnel ionization system for this problem. In particular, it allows us to define a from hydrogen-like ions while keeping the computational de- tunneling barrier [45]. As one can show, the Hamiltonian Hˆ mands small. In the long-wavelength limit, i. e., when the dipole in Eq. (10) separates in parabolic coordinates 0 ξ < and approximation is applicable, the three-dimensional Coulomb 0 η < , which are related to the Cartesian coordinates≤ ∞ x potential with an external linearly polarized electric field has and≤ y via∞ rotational symmetry, which makes this system effectively two- ξ η dimensional. Furthermore, the two-dimensional Coulomb prob- x = − , (12a) lem is of interest in its own and has been used to model some 2 p semiconductor systems [41–43]. It can also be derived as a y = ξη , (12b) limit of the Hamiltonian of a three-dimensional in a planar slab as the width of the slab tends to zero [44]. into two independent one-dimensional Schrödinger-type Hamil- ξ η η Applying the dipole approximation and choosing the coor- tonians with specific potentials V1( ) and V2( ). While V2( ) ξ dinate system such that the linearly polarized external electric is purely attractive, V1( ) represents the tunneling barrier, and, ξ field (t) points into the (negative) x direction, the Schrödinger therefore, denotes the tunneling direction. Intersections of / equationE reads in atomic units, which are applied for the re- the potentials with the energy value E 4 of the one-dimensional mainder of this article, Hamiltonians define the borders between the classically al- lowed, the tunneling, and the classically forbidden regions. ∂Ψ(x, y, t) Here, E denotes the Stark-effect corrected ground state energy i = Hˆ Ψ(x, y, t) = ∂t of the two-dimensional system. The classically allowed region, ! 1 ∂2 1 ∂2 1 the tunneling region, and the classically forbidden region are x (t) Ψ(x, y, t) . (10) characterized by −2 ∂x2 − 2 ∂y2 − √x2 + y2 − E / > ξ , / > η , In order to study the ionization dynamics without undesirable E 4 V1( ) E 4 V2( ) (13a) artifacts, i. e., to avoid multiple ionization and rescattering, we E/4 > V1(ξ) , E/4 < V2(η) , (13b) choose an electric field pulse with a unique maximum, E/4 < V1(ξ) , E/4 < V2(η) , (13c) 2 2 ! ω (t t0) respectively. The tunneling barrier is confined by the three (t) = 0 exp − , (11) E E − 2 parabolas at ξ = ξin, ξ = ξexit, and η = η0, which are functions 4 of the applied electric field strength (t). See the Appendix for 1.2 6 E (t) Dξin(t) details and Figs. 1 and 2. 1.0 E 5 D (t) The virtual detectors are placed at ξ = ξin and ξ = ξexit ξexit 0.8 4 as given for the maximal electric field strength 0 at t = t0. E (a.u.) )

In two dimensions, the surface integral in Eq. (8) becomes a 0.6 3 t ( exit (a.u.) line integral of a vector field j(x, y, t). Thus, the probability ξ ) 0.4 2 t D ( ,

ξ = ξ ) current over the entry line in, denoted by Dξin (t), and the E t 0.2 1 ( in current over the exit line ξ = ξexit, denoted by Dξexit (t), are given ξ explicitly by 0.0 0 D 0.2 1  ∂x(ξ,η)  − − Z η0    ∂η  25 20 15 10 5 0 5 10 15 Dξ (t) = j(x(ξ, η), y(ξ, η), t)   dη − − − − −  ∂y(ξ,η)  t t0 (a.u.) 0 ·   − ∂η  ∂x(ξ,η)  Z η 0  ∂η  FIG. 3: The probability current Dξ (t) over the entry line ξ = ξin + j(x(ξ, η), y(ξ, η), t)   dη , (14) in  ∂y(ξ,η)  and the current over Dξexit (t) the exit line ξ = ξexit and the amplitude 0 − ·   ∂η of the external electric field (t) as functions of time t. The three − E vertical lines indicate from left to right the moments of the maximal where we have taken into account that Eq. (12) covers only the probability current over the line ξ = ξin, of the maximal electric upper half of the coordinate system. Monitoring the probability field, and of maximal probability current over the line ξ = ξexit. The current at these fixed entry and exit lines can be justified as parameters of the electric field correspond to a maximal field strengths of = 1.1 a.u. and a Keldysh parameter γ = 0.25. follows. For static fields the tunneling probability is maximal E0 for maximal electric fields, and it is exponentially suppressed for lower fields. Furthermore, the applied electric field (11) is quasistatic for t t0 < τ : confirms that parabolic coordinates are the natural choice to | − | E define a tunneling barrier for the Coulomb problem.  2 2  4 4  (t) = 0 1 (t t0) /τ + O (t t0) /τ . (15) The action of the external electric field induces a probability E E − − E − E current, which flows over the tunneling barrier’s entry line Therefore, the tunneling barrier is also quasistatic if times close ξ = ξin and its exit line ξ = ξexit. This means that probability is to t0 are considered. As it will be shown later, the extracted carried over from the center region of the Coulomb potential tunneling times are short compared to τ indeed. E into the tunneling barrier, from where it can escape into the

classically allowed region. The quantities Dξin (t) and Dξexit (t) grow and decay over time with the applied external electric 3. Numerical results and discussion field, as shown in Fig. 3. As a central result of our numerical solution of the Schrödinger equation, however, the position

of the maxima of Dξin (t), Dξexit (t), and the electric field (t) 3.1. Entry time, exit time, and tunneling time do not coincide. The probability current over the tunnelingE

entry Dξin (t) reaches its maximum before the maximum of the The Schrödinger equation (10) is solved numerically by em- electric field is attained. Furthermore, the probability current ploying a Lanczos propagator [46, 47] and fourth-order finite over the tunneling exit Dξexit (t) may reach its maximum before differences for the discretization of the Schrödinger Hamil- or after the maximum of the electric field is attained depending tonian. The ground state of the two-dimensional Coulomb on the electric field strength 0. For the rather strong electric E potential was chosen as an initial condition at t t0 = 5τ , field of the parameters of Fig. 3, Dξexit (t) reaches its maximum i. e., when the external electric field is negligible small.− The− so-E slightly before the electric field. Turning off the external electric called Keldysh parameter γ = ω √ 2E0/ 0 [11] characterizes field, the flow at the tunneling entry becomes negative and then the ionization process as dominated− byE tunneling for γ 1 oscillates rapidly around zero after switching off the electric and by multiphoton ionization for γ 1. Here, E denotes the field. These oscillations are a result of the excitation of the the  0 ground state binding energy, which equals E0 = 2 a.u. for the bound portion of the wave function during the action of the two-dimensional Coulomb problem; see also the− Appendix. In electric field. The final bound state is a superposition of several the following, the electric field amplitude and the frequency eigenstates causing a nonsteady probability current. In contrast, E0 ω are adjusted such that γ = 0.25 < 1. the quantity Dξexit (t) remains positive as reflections are absent Figure 2 illustrates the ionization dynamics in the vicinity at the tunneling exit and behind the tunneling barrier [28]. of the tunneling barrier by presenting the electron’s probability Monitoring the probability current at ξ = ξin and ξ = ξexit density Ψ(x, y, t)∗Ψ(x, y, t) and the probability current j(x, y, t) allows us to determine the moments when the electrons enters at the instant of maximal field strength t = t0. The external or leaves the tunneling barrier with maximal probability. The electric field induces a probability current opposite to the di- probability to cross the tunneling barrier entry at ξ = ξin or rection of the electric field. This flow is nonnegligible only in the exit at ξ = ξexit during the small time interval [t, t + ∆t] the region η < η0, i. e., in the classically allowed region and is ∆t Dξin (t) and ∆t Dξexit (t), respectively. This motivates us to the tunneling region. Thus, the shape of the probability current introduce the times of maximal flow over the tunneling entry 5

5 To explain the reason for this behavior, we define in analogy τexit 4 to Eq. (14) the two integrals over the probability density τtsub Z η0 3 %

(a.u.) Dξ (t) = %(x(ξ, η), y(ξ, η), t) dη 2 0 tsub

τ Z η0 , 1 + %(x(ξ, η), y(ξ, η), t) dη (20) exit

τ − 0 0 1 and over the velocity field − 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4  ∂x(ξ,η)  Z η0   0 (a.u.) ϕ  ∂η  E D∇ (t) = ϕ(x(ξ, η), y(ξ, η), t)   dη ξ  ∂y(ξ,η)  0 ∇ ·   ∂η FIG. 4: τ τ The tunneling delay tsub and the tunneling time exit as func-  ∂x(ξ,η)  Z η tions of the electric field amplitude 0. The frequency ω is adjusted 0  ∂η  E + ϕ(x(ξ, η), y(ξ, η), t)   dη , (21) such that the Keldysh parameter is fixed to γ = 0.25.  ∂y(ξ,η)  0 ∇ − ·   − ∂η which are motivated by fact that the probability current can be and exit, i. e., the positions of the maxima of the arrival-time written as a product of the probability density and the local distributions at the tunneling entry and exit: velocity; see Eq. (6). D% (t) gives the probability density inte- ξin grated along the entry line ξ = ξin as a function of time, while t = arg max D (t) (16) ϕ in ξin D∇ (t) denotes the integrated velocity. As shown in Fig. 5, the ξin ϕ velocity along the entry line D∇ (t) increases with the elec- and ξin tric field, reaches a maximum at the instant of electric field maximum t = t0, then decreases with the electric field, and texit = arg max Dξexit (t) . (17) it becomes even negative due to reflections from the tunnel- ϕ ing process. Finally, the integrated velocity D∇ (t) oscillates ξin Then the delay between the instant of the maximum of the around zero indicating the excitation of the bound state. The % driving electric field t0 and the exit time texit, the tunneling probability density along the entry line Dξ (t) is nonzero even delay, is in at t t0 because the wave function penetrates the tunneling barrier even for = 0. Increasing the electric field, the proba- τ = t t , (18) E exit exit − 0 bility density flow from the core into the barrier increases the probability density along the entry line. Although the local and the tunneling time velocity at the tunneling entry is maximal at t = t0 in sync with the electric field, the probability flow along the entry line % % Dξ (t) reaches its maximum earlier. The quantity Dξ (t) goes τtsub = texit tin (19) in in − asymptotically to a constant value less than the initial value because some of the total probability has tunneled. Because the can be introduced with the definitions above. A measurement probability flow is the product of the probability density and the of the delay τ is implemented in the attoclock experiments. exit local velocity, the maximum of the integrated probability flow The time τtsub denotes the delay between the instants when the Dξin (t) is reached between the instants of the maxima of the probability flow is maximal at the tunneling barrier’s entry and % ϕ integrated density D (t) and the integrated velocity D∇ (t); at the exit. Thus, τtsub may be interpreted as the typical time ξin ξin which an electron needs to pass the tunneling barrier. this means tin < t0. The time texit is usually larger than t0, i. e., τ > 0, especially for weak external fields. The times τexit and τtsub are presented in Fig. 4 for the con- exit After the electron has passed the entry line of the tunneling sidered setup and varying electric field amplitudes 0. An increasing electric field amplitude reduces the widthE of the barrier, it needs some time to cross the barrier and to reach the tunneling exit. This means, t > t , but not necessarily tunneling barrier. Consequently, the tunneling time τtsub de- exit in t > t . The latter may be understood by realizing that the creases with increasing electric field amplitude 0, as shown exit 0 E above explanation for t < t is not specific to the coordinate in Fig. 4. The time τtsub remains, however, always positive. in 0 This means that the probability current reaches its maximum ξ = ξin. In particular, it also applies to ξ = ξexit. Consequently, also the integrated flow at the exit coordinate D% (t) may reach at the entry of the tunneling barrier before the current becomes ξexit maximal at the exit, i. e., the electron enters the barrier before its maximum before the maximum of the electric field, as, for it exits the barrier. Furthermore, it follows from the data in example, for the parameters in Fig. 3. In general, however, we ϕ τ > τ < find that D∇ (t) is maximal at an instant very close to t . Fig. 4 that tsub exit and consequently tentry t0, because ξexit 0 τtsub τexit = t0 tentry. This means that the electron always Taking the results above into consideration, the time tin enters− the tunneling− barrier before the electric field reaches its should be regarded as a reference for the tunneling time, but maximum. not t0. The latter leads to the delay τexit. Choosing t0 as a 6

20 50 1.3 1.2 15 40 1.1 30 10 1.0

(a.u.) 20 (a.u.) ) ) t t 0.9

( 5 ( (a.u.)

ϕ 10 in in v % ξ ∇ ξ 0.8 D D 0 0 0.7 5 10 − − 0.6 20 10 − 0.5 − 25 20 15 10 5 0 5 10 15 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 − − − − t t−(a.u.) (a.u.) − 0 E0

FIG. 5: The probability density D% (t) (red dashed line, right scale) FIG. 6: Average velocity v of the motion of the probability current’s ξin ϕ maximum from the tunneling barrier entry to the exit as a function of and the local velocity Dξ∇ (t) (blue dot-dashed line, left scale) along in the applied maximal electric field strength . the entry line ξ = ξin are plotted as a function of time for the electric E0 field strength = 1.3 a.u. The solid black line denotes the instant E0 of maximal Dξin (t), vertical dashed lines indicate the position of the 4 % ϕ maxima of D (t) and D∇ (t). ξin ξin 3 2 ξq(t) 1 reference can lead to negative delays τexit for large electric field (a.u.) ξc(t) 0 strengths, e. g., for 1.0 a.u. for the setup that was applied t 0 ξc 0 − c (t)

E ≥ t for Fig. 4. This, however, should not be misinterpreted as a 1 τ − negative tunneling time because exit is not related to the time 2 0 = 0.9 a.u. 0 = 1.0 a.u. which an electron needs to pass the tunneling barrier. This time − E E 3 is given by τ . As τ is always larger than τ , the delay − tsub tsub exit 4 τexit may be used as a lower bound of the tunneling time. It has been speculated that tunnel ionization may be instanta- 3 neous [14, 25], which would be associated with a superluminal 2 traversal of the tunneling barrier. Our results, however, clearly 1 (a.u.) 0

τ t indicate that there is a nonvanishing tunneling time tsub. This 0 − leads to a finite average velocity of the motion of from ξ = ξin t 1 to ξ = ξexit: − 2 0 = 1.1 a.u. 0 = 1.2 a.u. − E E ξexit ξin v = − . (22) 3 2τtsub − 0 5 10 15 20 25 30 0 5 10 15 20 25 30 ξ (a.u.) ξ (a.u.) Here we have taken into account that the lines of constant ξ are parabolas in the x-y plane and the probability current flows FIG. 7: ξ mainly at y 0. As shown in Fig. 6, there is a monotonous The quantum trajectory of the electron in the coordinate ≈ ξq(t), compared to the classical trajectory according based on the two- relation between the applied maximal electric field strength step model ξc(t). Both are also compared to the classical trajectory 0 and the velocity v. The velocity presented in Fig. 6, how- ξc E corrected according to quantum initial conditions c (t). For more ever, is more than two orders of magnitude below the speed of information see the main text. The shadowed area marks the positions light. Identifying the tunneling motion of the electron over the ξ corresponding to under the barrier. tunneling barrier with the motion of the probability-current’s maximum, it is justified to say that tunneling ionization neither is instantaneous nor moves the electron at superluminal speed. step model ξc(t). This classical trajectory is determined by the Newton equation d2ξc(t) 8 3.2. Quantum and classical trajectories = + 2 (t) , (24) dt2 −ξc(t)2 E The probability currents over different lines of constant ξ reach where the reduction to one dimension is justified as the most their maxima at different times, which allows us to define a probable trajectory is along the x direction at y 0. The initial quantum trajectory ξq(t) by inverting conditions for the two-step model are ≈

c tξ = arg max Dξ (t) . (23) ξ (t0) = ξexit , (25a) c q dξ (t) The quantum trajectory ξ (t) defined by Eq. (23) can be com- = 0 , (25b) pared to the trajectory predicted by a Coulomb-corrected two- dt t=t0 7

3.5 actual asymptotic momentum and the prediction of the classical 3.0 Coulomb-corrected two-step model is just the electron’s initial momentum at the barrier exit. For the two-dimensional model 2.5 q as considered in Fig. 7 the condition ξ (t0) = ξexit is fulfilled 2.0 for an electric field strength 0 = 1.0 a.u. (a.u.) E 1.5 exit v 1.0 0.5 4. Conclusions 0.0 0.6 0.8 1.0 1.2 1.4 (a.u.) We studied the dynamics of tunnel ionization by analyzing the E0 quantum mechanical wave-function’s probability current via FIG. 8: The quantum trajectory’s initial velocity v plotted for dif- virtual detectors. Virtual detectors are a capable concept for exit determining tunneling times because they allow us to identify ferent electric field strengths 0 for Keldysh parameter γ = 0.25. E well-defined moments when the ionized electron enters and leaves the tunneling barrier. meaning that the electron exits at the instant of electric field The numerical solutions of the time-dependent Schrödinger maximum t = t0 with zero initial momentum at the turning equation and the dynamics of the probability current at the tun- point x = ξexit/2. As shown in Fig. 7 for different electric neling barrier indicate that the electron spends a nonvanishing field strengths, the classical trajectory ξc(t) deviates from the time under the potential barrier. This corresponds to a crossing quantum trajectory ξq(t) not only near the barrier exit but also velocity that is much lower than the speed of light. Further- at a far away detector, i. e., for ξ ξ . more, the electron may enter the classical forbidden region  exit From the above studies we have seen that the electron ex- on average before the instant of the maximum of the driving electric field. Therefore, the instant of electric field maximum its the tunneling barrier at texit, which differs from t0. It has been shown by quantum mechanical calculations [28] that the should not be considered as an initial reference time of the electron exits also with an initial momentum in the electric tunneling process. Nevertheless, for electric field strengths field direction. This initial velocity can be inferred from the well below the threshold regime the electron exits the barrier quantum trajectory, via after the instant of electric field maximum. For such strengths a nonvanishing delay τexit between the electric field maximum dξq(t) and the emergence of the electron behind the tunneling barrier v = , (26) exit exists as a signature of the time spent under the barrier. The dt t=texit actual time span that the electron has spent in the classically which yields results that are consistent with the results of forbidden region, however, will be larger than τexit. Vanish- c Ref. [28]. The classical trajectory ξ (t) can be corrected by the ing or negative time delays τexit should not be interpreted as quantum initial conditions: instantaneous tunneling. Under the barrier as well at the tunneling exit, the electron c ξc (texit) = ξexit , (27a) has a nonzero velocity in electric field direction. After the c tunneling exit, the quantum trajectory, which is induced by dξc (t) = vexit . (27b) the wave-function’s probability current, agrees well with a dt t=t exit trajectory as given by classical equations of motion when the c As shown in Fig. 7 the corrected classical trajectory ξc (t) agrees electron’s initial velocity at the tunneling exit is taken into well with the quantum trajectory ξq(t). Nevertheless, there is account properly. still a slight discrepancy between both trajectories at a far away detector, which can be attributed to that ξq(t) originates from the motion of a broad wave packet which experiences a spatially Acknowledgments varying Coulomb force [48]. The initial velocity vexit is plotted in Fig. 8 for different electric field strengths. In agreement with Ref. [28], the initial momentum is slightly dependent on the We gratefully thank Enderalp Yakaboylu for valuable discus- electric field strength also in the considered two-dimensional sions. system. The initial momentum is of the order of the width of the ground state distribution in momentum space. The initial velocity vexit is difficult to measure directly in an experiment. But it also affects the asymptotic velocity of A. The Coulomb problem and the the ionized electron. Note that one can choose an electric field q Strak effect in two dimensions strength such that the quantum trajectory is at ξ = ξexit at t = t0, i. e., the electron leaves the tunneling barrier at the moment of electric-field maximum, which is one assumption of the two- The eigen-equation of the two-dimensional Coulomb problem step model. In this case, the difference between the electron’s [42, 49] with some additional homogeneous electric field of 8 strength is given in Cartesian coordinates x and y and em- For a vanishing external electric field, i. e., = 0, ploying atomicE units by Eqs. (A.4a) and (A.4b) are identical. Introducing theE variable ! transformation ξ = η = ζ/ √ 2E gives in this case 1 ∂2 1 ∂2 1 − x Ψ(x, y) = EΨ(x, y) , 2 ! −2 ∂x2 − 2 ∂y2 − √x2 + y2 − E ∂ 1 ∂ β1,2 ζ ζ + + f1,2(ζ) = 0 . (A.8) (A.1) ∂ζ2 2 ∂ζ √ 2E − 4 where Ψ(x, y) denotes an eigen-function with energy E. This − eigen-equation separates in parabolic coordinates ξ and η [49– Making the ansatz f1,2(ζ) = exp( ζ/2)g1,2(ζ) this differential − 51], which are related to the Cartesian coordinates x and y equation yields the equation via Eq. (12). This coordinate system is particularly useful 2 ! !! ∂ 1 ∂ 1 β1,2 because here the two-dimensional Schrödinger equation can ζ + ζ g1,2(ζ) = 0 (A.9) be separated into two one-dimensional Schrödinger equations, ∂ζ2 2 − ∂ζ − 4 − √ 2E which allows us to define a one-dimensional tunneling barrier. − for g (ζ), which can be identified as the confluent hypergeo- The calculations in this section follow Ref. [52], where the 1,2 metric equation [53] with three-dimensional Coulomb problem is considered in a similar way. 1 β1,2 1 The Laplacian becomes in the new coordinate system a = , b = . (A.10) 4 − √ 2E 2 ! − ∂2 ∂2 1 ∂2 ∂ ∂2 ∂ + = 4ξ + 2 + 4η + 2 . (A.2) The nonsingular solution of the confluent hypergeometric equa- ∂x2 ∂y2 ξ + η ∂ξ2 ∂ξ ∂η2 ∂η tion is usually called confluent hypergeometric function and denoted by F (a; b; ζ) or M(a; b; ζ). Thus we have found the Expressing (A.1) in parabolic coordinates yields after some 1 1 solution algebraic transformations !  ξ  1 β 1 ! f (ξ) exp √ 2E M 1 ; ; ξ √ 2E ∂2 1 ∂ E 1 ∼ −2 − 4 − √ 2 − ξ + + ξ + Eξ2 Ψ(ξ, η)+ 2E ∂ξ2 2 ∂ξ 2 4 − (A.11) ! f η f ξ f η ∂2 1 ∂ E and similarly for 2( ). These functions 1( ) and 2( ) can be η + + η Eη2 Ψ(ξ, η) = Ψ(ξ, η) . (A.3) normalized only if the first argument of the confluent hyper- 2 ∂η ∂η 2 2 − 4 − geometric function is a negative integer or zero. In this case the confluent hypergeometric function coincides (up to normal- Substituting the ansatz Ψ(ξ, η) = f1(ξ)f2(η) into the last equa- tion end separating the variables ξ and η we obtain ization) with the associated Laguerre polynomials. Thus, the quantization conditions 2 ! ∂ 1 ∂ E 2 ξ + + ξ + ξ + β ξ = , 1 β , 2 E 1 f1( ) 0 (A.4a) 1 2 ∂ξ 2 ∂ξ 2 4 = n1,2 (A.12) 4 − √ − 2 ! 2E ∂ 1 ∂ E 2 − η + + η Eη + β2 f2(η) = 0 , (A.4b) ∂η2 2 ∂η 2 − 4 with n1,2 = 0, 1, 2,... will have to be fulfilled. Together with the relation (A.5) the quantization conditions yield where the separation constants β1 and β2 are related by 1 = , E 2 (A.13) β1 + β2 = 1 . (A.5) −2(n1 + n2 + 1/2) ! 1 The tunneling barriers are obtained by substituting f1(ξ) = β1,2 = n1,2 + √ 2E . (A.14) 1/4 1/4 4 − g1(ξ)/ξ and f2(η) = g2(η)/η , which gives the equations for the new functions as Normalizing f1(ξ) and f2(η) to unity finally gives the 2 ! 1 ∂ g1(ξ) 3 β1 ξ E bound eigen-states of the two-dimensional Coulomb problem + g1(ξ) = g1(ξ) , (A.6a) Ψ (ξ, η) = f (ξ) f (η) with −2 ∂ξ2 −32ξ2 − 2ξ − 8E 4 n1,n2 1;E,n1 2;E,n2 2 ! s 1 ∂ g2(η) 3 β2 η E ! + + g (η) = g (η) . (A.6b) √ 2E  ξ  1 2 2 2 2 √ √ −2 ∂η −32η − 2η 8E 4 f1;E,n1 (ξ) = − exp 2E M n1; ; ξ 2E , 1 + 4n1 −2 − − 2 − These two equations represent Schrödinger-type eigen- (A.15a) s equations with the potentials ! √ 2E  η  1 f , (η) = − exp √ 2E M n ; ; η √ 2E , 3 β1 ξ 2;E n2 + n − − − 2 − V1(ξ) = , (A.7a) 1 4 2 2 2 −32ξ2 − 2ξ − 8E (A.15b) 3 β η V (η) = 2 + , (A.7b) 2 −32η2 − 2η 8E and the energy E given by (A.13). Taking into account the Stark effect, i. ,e., > 0, leads to and the energy E/4. a modification of the eigen-states, of the eigen-energiesE E, as 9

2.0 for β , β , and E can be calculated via perturbation theory [52]. − 1 2 2.1 In case of the ground state with n1 = n2 = 0, we get in second − 2.2 order − (a.u.) 2.3 E − β = β(0) + β(1) + β(2) 2.4 1,2 1,2 1,2 1,2 (A.16) − 2.5 − with 0.65 p 0.60 β(0) = E/8 , (A.17a) 0.55 1,2 − β1 Z   (a.u.) 0.50 (1) ξ

2 β , 2 β = f , E, (ξ) f , E, (ξ) dξ = E , (A.17b) 1 0.45 1,2 1 2; 0 1 2; 0 β ∓4E ∓4E 0.40 1 0.35 (2) β , = 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 2 √ 2E × (a.u.) − E X Z   2 ∞ 1 ξ   f1,2;E,n(ξ) f1,2;E,0(ξ) dξ FIG. 9: The ground state energy E of the two-dimensional Coulomb 1 1 ∓4E n=1 4 n + 4 problem with an external electric field and the separation parameters − 2 β and β as functions of the electric field strength . 1 2 E 0.2004642410 √ 2E E . (A.17c) ≈ − E3

Equations (A.16) and (A.5) determine the parameters β1, β2, well as of the separation constants β1 and β2. In this way, the and E uniquely, and the results of a numerical solution of these tunneling potential (A.7a) also changes. The resulting values equations are shown in Fig. 9.

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