The role of reflections in the generation of a time delay in strong field ionization
Daniel Bakucz Canário, Michael Klaiber, Karen Z. Hatsagortsyan, Christoph H. Keitel
TThe role of reflections in the generation of a time delay in strong field ionization
Daniel Bakucz Can´ario, ∗ Michael Klaiber, Karen Z. Hatsagortsyan, † and Christoph H. Keitel Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: January 21, 2021)Against the background of recent attoclock experiments, the problem of time delay in tunneling ionizationis revisited. The origin of time delay at the tunnel exit is analysed, underlining the two faces of the conceptof the tunnelling time delay: the time delay around the tunnel exit and the asymptotic time delay at a detector.We show that the former time delay, in the sense of a delay in the peak of the wavefunction, exists as a matterof principle and arises due to the sub-barrier interference of the reflected and transmitted components of thetunneling electronic wavepacket. We exemplify this by describing the tunnelling ionization of an electron boundby a short-range potential within the strong field approximation in a “deep tunnelling” regime. If sub- barrierreflections are extracted from this wavefunction, then the time delay of the peak is shown to vanish. Thus, weassert that the disturbance of the tunnelling wavepacket by the reflection from the surface of the barrier causes atime delay in the neighbourhood of the tunnel exit.
I. INTRODUCTION
State-of-the-art techniques in attosecond science are able toprovide exceptional time and space resolution, reaching tens ofattoseconds time- and Angstr¨om space-resolution. In particular,the attoclock technique provides time resolution of tunnelingionization [1], and has inspired researchers to experimentallyaddress the challenging problem of tunneling time [2–6], i.e.,how much time, if any, elapses during the quantum tunnellingprocess. This question strikes at the fundamentals of quan-tum mechanics and experiments have often been followed bycontroversial discussion [2–37] in the strong field communityarguing either for, or against, the existence of a tunnellinginduced time delay between ionised electron wavepacket andionising laser field.In this respect one should distinguish two concepts of thetunneling time, namely, the time delay near the tunnel exit(the classically expected coordinate for the tunneled electronto emerge), and the asymptotic time delay. While the latteris relevant to attoclock experiments, the former, known alsoas the Wigner time delay, can be calculated theoretically andmeasured in a Gedanken experiment with a so-called virtualdetector [38, 39].The asymptotic time delay is derived from the asymptoticphotoelectron momentum distribution (PMD). It is defined bythe classical backpropagation of the peak of the photoelectronasymptotic wavefunction up to the tunnel exit [12, 17, 20, 21].A theoretical description of the experimental asymptotic timedelay is challenging because of the entanglement of Coulombfield e ff ects with those of the tunneling delay. This asymptotictime delay was found to be negative and shown to arise dueto interference of direct and the under-the-barrier recollidingtrajectories [22].In the tunnelling region, the tunneling time delay is deducedby following the peak of the electron wavefunction, the so-called Wigner trajectory. A few de-Broglie wavelengths awayfrom the tunnel exit, the electron dynamics are quasiclassical ∗ [email protected] † [email protected] and a classical trajectory accurately describes the eikonal ofthe wave function in the Wentzel-Kramers-Brillouin (WKB)approximation [40, see § a r X i v : . [ phy s i c s . a t o m - ph ] J a n the first-order SFA is not only nonzero, but sizeable. This isin contrast to the asymptotic time delay, where the first-orderSFA result is vanishing and the second-order SFA is requiredto obtain a nonvanishing asymptotic time delay (attributable tosub-barrier recollisions and interference of paths).The structure of the paper is as follows: in Sec. II we in-troduce the basic ionization model, calculate the SFA time-dependent wavefunction, and construct the Wigner trajectoryfrom the latter. To elucidate our method of analysing the con-tribution of reflections to the wavefunction, in Sec. III weconsider a simpler, analytic, model of ionization: an electronin a constant electric field. The analyticity of this model allowsus to relate the reflected components of the wavefunction tothe contribution of the specific saddle points of the integralrepresentation of the wavefunction. We apply this concept tothe SFA wavefunction in Sec. IV, whereby we show that ne-glecting reflections results in a zero Wigner time delay. Lastly,in Sec. V, we summarize our results and discuss the impli-cations of these on the interpretation of attosecond streakingexperiments. The Wigner time delay for scattering by a squarepotential barrier was previously analyzed in [42, 43], wherethe significance of reflections to the time delay was alludedto. In Appendix A this problem is revisited and the role ofreflections is explicitly presented. II. TIME DELAY IN STRONG FIELD IONIZATION
In both classical and quantum mechanics, the passage oftime is always defined with respect to some dynamical vari-able (for instance, the hands of a clock). In the absence of acanonical quantum mechanical time operator, the definitionand measurement of time in quantum mechanics is challeng-ing as dynamical observables are inherently non-deterministicand probabilistic. Despite the conceptual di ffi culties involved,techniques with which to measure time have been developed(see e.g. [44–47] for overviews).One of the main applications of these time measurementprotocols is the study of the time taken for a particle to tunnelthrough a barrier, known as tunnelling time. There exist severaldefinitions of the tunneling time such as the Wigner time [48–50], the Larmor time [51–54], the dwell time, etc. [13]. These,in general, do not coincide and each definition corresponds toa specific aspect of the measurement process.The Wigner time, developed for scattering problems [48–50], considers the time taken for the peak of the wavepacketto travel a given distance. For tunneling through a time-independent barrier, the Wigner time can be derived via theenergy derivative of the phase of the tunnelling wavefunc-tion ψ : τ W ( x ) = − i ∂∂ E ln ψ ( x ) | ψ ( x ) | . (1)The Wigner time corresponds well to the attoclock measure-ment, because in the latter a well-defined peak is formed in theasymptotic PMD, which is then measured and interpreted. Asthe peak of the wavepacket follows the trajectory τ W ( x ), the Wigner group delay velocity of the wave packet can be defined v W ( x ) = (cid:32) ∂τ W ( x ) ∂ x (cid:33) − . (2)When a monochromatic wave is incident on a finite potentialbarrier, the propagation inside the barrier is a superpositionof an exponentially suppressed wave (transmission) with agrowing exponential wave, namely the reflection by the surfaceof the barrier (see e.g. Appendix A). The goal of this paper isto establish the causal e ff ect of wavefunction refections insidethe barrier on the time delay in strong field ionization.In order to do this, we calculate the time-dependent ionizedelectron wavefunction at the position where the electron ap-pears in the continuum, thereby allowing us to deduce the peakof the wavepacket with respect to time. Correspondingly, wederive the Wigner time delay at the tunnel exit explicitly asthe delay of the peak of the wavepacket, rather than from theenergy derivative of the phase of the wavefunction. A. The Model
In the following, we work exclusively in atomic units (a.u.).Consider an electron initially bound in a 1D short-range poten-tial V ( x ) = − κ δ ( x ) (3)with a binding energy I p = κ /
2, and with a correspondingground state wavefunction ψ ( x , t ) = √ κ exp (cid:16) − κ | x | + iI p t (cid:17) .To describe the generation of the peak of the ionizedwavepacket at the tunnel exit, not the usual PMD at a detector,a half-cycle laser field is modelled, avoiding recollisions in thecontinuum. The laser electric field, E ( t ) = − E cos ( ω t ) , (4)is switched on at ω t = − π/
2. The superposition of thelaser field with the atomic potential creates a potential bar-rier, through which an electron can tunnel, with a characteristicclassical tunnel exit x e = I p / E . While this is a rudimentaryestimate, we show later that it is broadly in agreement with atunnel exit derived quantum mechanically.We consider the tunnel ionization regime and keep constantthe Keldysh parameter [55] γ = (cid:115) I p U p (cid:28) , (5)where U p = E / (4 ω ) is the ponderomotive potential. Thus,when varying the field strength E , we vary the frequency ω = γ E /κ accordingly.Moreover, we consider the so-called deep tunneling regime,wherein E (cid:28) E th , and E th is the threshold field of OTBI.This threshold can be estimated as the field strength where thecoordinate-saddle point of the SFA-matrix element, i.e. thestarting point x s ≈ √ κ/ E of the quantum orbit, becomes com-parable the tunnel exit x e , which for the short range potentialcorresponds to the condition E th ≈ κ / - - - - - - - - - - (a) (b) Figure 1. (a)
Ionization amplitude | ψ i ( x , t ) | as defined in Eq. (17). At every cross-section in x , the temporal peak of the wavefunction wasdetermined (plotted in blue) which can be interpreted as the trajectory of the peak. (b) Comparison between the peak (blue), classical (orange)and Wigner (green) trajectories in the region near the tunnel exit, x e . Under the barrier ( x <
10, shaded blue), there is an increasing delay of thepeak w.r.t the laser peak. Outside the barrier, the peak trajectory rapidly converges with the classical and Wigner electron trajectory. - - -
20 0 20 40 602. × - × - × - × - Figure 2. Probability distribution | ψ ( x e , t ) | vs time at the classicaltunnel exit x e = I p / E . This distribution is peaked at a greater time, t m ≈ . ≈
183 as, than the peak of the laser pulse.
B. Time Evolution of the Wavefunction
The ionization dynamics are described by the Schr¨odingerequation i ∂∂ t Ψ ( x , t ) = ( H + H i ) Ψ ( x , t ) , (6)with the atomic Hamiltonian H = − ∂ ∂ x + V ( x ) , (7)and the interaction Hamiltonian with the laser field H i = xE ( t ) (8)The standard SFA is employed for the solution of theSchr¨odinger equation. In the interaction picture with Hamilto-nian H = H + H i , the time evolution operator U ( t , t ) satisfies the Dyson integral equations [57] U ( t , t ) = U ( t , t ) − i (cid:90) tt d t (cid:48) U ( t , t (cid:48) ) H i ( t (cid:48) ) U ( t (cid:48) , t ) , (9)where U is the evolution operator corresponding to the Hamil-tonian H . A perturbation series can be constructed by replac-ing the full evolution operator U ( t , t ) in the Dyson integralwith the evolution operator corresponding to the Hamiltonian H i , namely U f ( t , t (cid:48) ) = (cid:90) d p | Ψ p ( t ) (cid:105) (cid:104) Ψ p ( t (cid:48) ) | , (10)where | Ψ p (cid:105) are the Volkov states [58]. Each order in the per-turbation theory corresponds to a sub-barrier recollision in ourhalf-cycle laser field. As each sub-barrier recollision implies alonger tunnelling path, each term in this series is suppressedby the Keldysh exponent exp( − κ / E ) and so, in contrast to thenear OTBI regime, in the deep tunnelling regime a first orderexpansion in the SFA without sub-barrier recollision su ffi ces.[22].Consequently, the electron state during the interaction takesthe form | ψ ( t ) (cid:105) = | ψ ( t ) (cid:105) + | ψ i ( t ) (cid:105) where | ψ ( t ) (cid:105) is the eigenstateof the atomic Hamiltonian H and | ψ i ( t ) (cid:105) = − i (cid:90) ∞−∞ d p (cid:90) tt dt (cid:48) | Ψ p ( t ) (cid:105) (cid:104) Ψ p ( t (cid:48) ) | H i ( t (cid:48) ) | ψ ( t (cid:48) ) (cid:105) , (11)corresponds to the ionized part of the electron state. Denotingthe electronic kinetic momentum as P ( t ) ≡ p + A ( t ) , (12)with the laser vector potential A ( t ) = − (cid:82) t − t E ( t (cid:48) ) dt (cid:48) , the 1DVolkov state in the length gauge is (cid:12)(cid:12)(cid:12) Ψ p ( t ) (cid:69) = |P ( t ) (cid:105) e − i S ( t ) (13) (a) (b) Figure 3. Log-log plots of the scaling w.r.t. field strength, E , of the (a) Wigner time delay and (b) group velocity at the classical tunnel exit, x e = I p / E , for the time evolved SFA wavefunction (red dots) and the adiabatic constant field wavefunction (blue lines). For the time evolvedwavefunction, the maxima in time, t m , of the probability distribution | ψ ( x e , t ) | and its derivative were calculated; for the adiabatic case, resultsfollow from Eqs. (26) and (27). The agreement in the trends is good, deteriorating as one approaches the OTBI threashold E th ≈ .
25 a.u. with a plane wave component (cid:104) x | p (cid:105) = (2 π ) − exp( i p x ) andthe Volkov-phase S ( t ) = (cid:90) t d τ P ( τ ) . (14)We can expand the overlap in (11) using a resolution of identity in a dummy variable x (cid:48) (cid:104) Ψ p ( t (cid:48) ) | H i ( t (cid:48) ) | ψ ( t (cid:48) ) (cid:105) = e i ( I p t (cid:48) − S ( t (cid:48) ) ) √ π E ( t (cid:48) ) (cid:90) ∞−∞ dx (cid:48) e i P ( t (cid:48) ) x (cid:48) − κ | x (cid:48) | x (cid:48) (15)While this integral is soluble, we do not evaluate it but ratherfirst integrate (11) w.r.t. to p and then x (cid:48) . With the notation ∆ f n ( t , t (cid:48) ) = f n ( t ) − f n ( t (cid:48) ), and defining the integrals f n ( t ) = (cid:82) t dt (cid:48) A ( t (cid:48) ) n , we may perform the initial integral over p whichis a simple Gaussian integral. The resulting expression ψ i ( x , t ) = (cid:90) tt d t (cid:48) (cid:90) ∞−∞ d x (cid:48) x (cid:48) √ κ E ( t (cid:48) ) √ π i ( t − t (cid:48) ) exp (cid:34) i (cid:32) I p t (cid:48) + i κ | x (cid:48) | + x ( A ( t ) − A ( t (cid:48) )) + i (( x − x (cid:48) ) − ∆ f ( t , t (cid:48) )) t − t (cid:48) ) − ∆ f ( t , t (cid:48) )2 ) (cid:33)(cid:35) (16)may then be finally integrated in x (cid:48) to yield ψ i ( x , t ) = (cid:90) tt dt (cid:48) √ κ E ( t (cid:48) ) z − (1 + Erf( z − )) e − z + z − κ + z + (1 − Erf( z + )) e − z − z + κ exp ( i ζ ) ≡ (cid:90) tt dt (cid:48) exp( i Φ ( x , t , t (cid:48) )) (17)where we have defined z ± = (cid:113) i ( t − t (cid:48) )( z ± κ ) for z = − i (cid:32) A ( t (cid:48) ) + x − ∆ f ( t , t (cid:48) ) t − t (cid:48) (cid:33) (18)and where ζ = I p t (cid:48) + x A ( t ) + (19)( t − t (cid:48) ) (cid:32) ( x − ∆ f ( t , t (cid:48) )) t − t (cid:48) ) + κ + z (cid:33) − ∆ f ( t , t (cid:48) )2 (20)Our aim is to calculate the wavefunction and Wigner timedelay in the interaction region around the tunnel exit by study-ing the phase Φ . Thus, in deviation to standard SFA studies, the time integral is calculated up to the finite observation time t , meaning the integration must be performed numerically. Theusual saddle-point integration method is valid only for su ffi -ciently large values of t , placing one well outside the tunnellingregion. C. Numerical Integration
For the numerical integration of the SFA wavefunctionEq. (17) an electric field strength E = .
05 a.u. and Keldyshparameter γ = . I p = κ / = / E th ≈ .
25 a.u. The time integration was carried out with thestandard numerical integration routine of
Mathematica
12 to aprecision of 30 digits.The space-time probability distribution, | ψ i ( x , t ) | , is pre-sented in Fig. 1. From this distribution the Wigner trajectoryis derived as follows: the probability time-distribution at eachspace point x is invoked (see Fig. 2 for the case of the tunnelexit coordinate x = x e ) and the maximum t m ( x ) of this distri-bution is derived. The Wigner trajectory is represented by thefunction t m ( x ), which runs along all maximum points of thespace-time distribution.We note the maximum of the wavefunction displays a timedelay with respect to the peak of the field. The tunnelingtime delay under the barrier ( x <
10) increases when movingtowards the tunnel exit. Outside the barrier, this trajectoryrapidly approaches the classical electron trajectory (beginningat the classical tunnel exit with zero momentum). The slightdeviation is due to quantum mechanical corrections to thequasiclassical wave function not far from the tunnel exit.The main idea advocated in this paper is that the Wignertime delay during tunneling is closely related to reflectionsarising during tunneling dynamics. It is straightforward inthe simple case of tunneling through a box potential to showthat reflections are responsible for the tunneling time delay,as is done in Appendix A. However, unlike in the separablebox potential case, a given wavefunction is not easily decon-structible as a simple superposition of reflected and transmittedcomponents corresponding to simple decaying / growing expo-nentials. Instead, the transmitted and reflected components arecompletely encapsulated in the wavefunction making it rathermore problematic to disentangle. This is ultimately achievedin Sec. V for the SFA wavefunction presented in this paper butin order to highlight the main aspects of the method we discussbeforehand a simpler example, that of tunneling in a constantelectric field. III. TIME DELAY IN A CONSTANT FIELD
The adiabatic model of ionization, namely atomic ionizationin a constant field, will be helpful for our purposes as it isanalytically tractable. Moreover, it will provide us with areference with which to compare our earlier time dependentmodel.
A. Ionisation in an adiabatic field
Consider a bound electron of energy − I p in 1D δ -potentialionized by a constant electric field E . The continuum eigen-state of the electron in this field is given by the solution to thetime-independent Schr¨odinger equation: −
12 d ψ d x + (cid:16) I p − E x (cid:17) ψ ( x ) = , (21) which has as a general analytical solution as a superposition ofthe Airy functions of the first and second kind ψ ( x ) = c Ai( ˜ x ) + c Bi( ˜ x ) , (22)where ˜ x = E (cid:16) I p − E x (cid:17) . (23)The requirement that the wavefunction be a travelling wave as x → ∞ imposes the condition c = − ic so that the wavefunc-tion takes the form ψ ( x ) = T [Bi( ˜ x ) + i Ai( ˜ x )] , (24)where T can be determined by matching this wavefunction tothe bound-state solution of the atom; this prefactor plays a rolein the ionization amplitudes but is irrelevant to the phase ofthe wavefunction. We calculate the Wigner time delay for thiswavefunction via the energy derivative of the phase τ W ( x ) = i ∂∂ I p ln ψ | ψ | = π E x ) + Bi( ˜ x ) . (25)This is interpreted as the trajectory of the peak of a tunnelledelectron wavepacket under a constant electric field, which weuse as a benchmark for our SFA results. The constant fieldmodel allows one to estimate the scaling of the Wigner timedelay at the tunnel exit x e = I p / E : τ W ( x e ) = / Γ (cid:16) (cid:17) / π E / . (26)From the latter we derive the scaling of the Wigner groupvelocity of the electron at the tunnel exit, v W ( x e ) = (cid:16) (cid:17) / √ π Γ (cid:16) (cid:17) Γ (cid:16) (cid:17) E / , (27)and find it consistent with the estimate of the scaling of theelectron momentum p e ∼ E τ W in Ref. [22].We compare the Wigner time delay and the group delayvelocity in the constant field case with the exact time dependentSFA calculations in Fig. 3. We see that the time delay in theSFA case follows the field scaling τ W ( x e ) ∝ / E / similar tothe adiabatic case, however with a constant time shift.This shift stems from the fact that the Wigner trajectoryis derived via the saddle-point approximation of the electronwave packet formation integral, while the time-integrationresponsible for the formation of the ionization wave packetin SFA is calculated exactly numerically. The field scalingof the Wigner group velocity via SFA follows the one of theadiabatic case v W ( x e ) ∝ E / with the error growing withthe field strength, consistent with the simple estimate for thethreshold for OTBI, E th ≈ .
25 a.u.Note that the Wigner trajectory in the constant field caseis determined from the solution of the time-independent - Figure 4. The solution to the electron in a constant field problemis given by a superposition of Airy functions of the first and secondkind, Ai( ˜ x ) and Bi( ˜ x ), plotted in blue and orange respectively. Thesefunctions have very accurate asympotic expansions, shown dashed,which can be derived by considering the saddle points of the Airyintegrals (28)-(29). Under the barrier, x <
10 (shaded blue), theseexpansions show that the wavefunction components Ai( ˜ x ) and Bi( ˜ x )respectively correspond to the reflected and transmitted componentsof the wavefunction. Schr¨odinger equation (energy eigenstate), while in the SFAcase from the time-dependent wavefunction. In the first case,the Wigner trajectory is defined as the derivative of the wave-function phase with respect to the energy, which correspondsto determining the coordinate of the peak of the electronwavepacket at a fixed time moment, i.e. determining the mo-tion of the wavepacket peak. In contrast, in the SFA case weexplicitly determine the maximum of the wavefunction in timefor a fixed coordinate.
B. Reflections in constant field
To investigate the e ff ect of under-the-barrier reflected com-ponents, we need to isolate their contribution to the full wave-function. The analytical wave function of Eq. (24) consistsof a superposition of Airy functions Ai( ˜ x ) and Bi( ˜ x ) whichcan be given interpretation as the under-the-barrier reflectedand transmitted components of the tunneling wavefunction,respectively.This interpretation can be generally understood by consider-ing the Airy functions, as shown in Fig. 4. The Bi-componentof the wavefunction decays exponentially as it approaches thetunnel exit, x = x e , from the atomic core at x =
0, and hencecan be seen to correspond to the transmission component; like-wise, the exponentially growing Ai-component corresponds towavefunction reflection.This interpretation can be formally established by consider- ing the integral representation of the Airy functionsAi( ˜ x ) = π i (cid:90) γ ds exp( ˜ x s − s , (28)Bi( ˜ x ) = π (cid:90) γ − γ ds exp( ˜ x s − s , (29)where the complex plane integration paths γ i are indicated inFig. 5 (a). These integrals converge when their endpoints liein the slices of the complex plane defined by − π < θ < π , π < θ < π , and π < θ < π , where in polar { r , θ } coordinates s = r exp( i θ ). In these regions, shaded blue in Fig. 5, theintegrand vanishes rapidly as r → ∞ .The majority of the contributions to the Airy integrals thuscome from around the saddle points, s ± = ± √ ˜ x , (30)of the argument of integrand. The Airy contours, γ i , can bedeformed into paths that to go through these saddle-pointsand, using the standard technique of saddle-point integrationmethod [59], asymptotic expressions for the Airy integrals canbe determined.The saddle-points, and the respective paths of steepest de-scents, are illustrated in Figs. 5 (b) and (c); since these aredependent on ˜ x the application of the saddle point method isdi ff erent for the two cases of inside ( ˜ x <
0) and outside ( ˜ x > x < γ smoothly into the path of steepestdescents for the saddle point s − and in doing so obtain theasymptotic approximationAi( ˜ x ) = exp (cid:16) − ˜ x (cid:17) √ π ˜ x + O ( ˜ x − ) . (31)Likewise, we may deform the contours γ and − γ to both gothrough the steepest descent path of the saddle point s + andhence deduce the asymptotic form of the Bi( x )-function underthe potential barrierBi( ˜ x ) = exp (cid:16) + ˜ x (cid:17) √ π ˜ x + O ( ˜ x − ) . (32)Thus, from a tunnelling particle’s perspective, one may identifythe Ai( x )-function and its saddle-point s − with the reflectedpart of the wavefunction, and the saddle-point s + , or the Bi( x )function with the transmitted part of the wavefunction: thetransmitted wave decays as it moves to the tunnel exit, while thereflected wave propagates from the exit decaying exponentiallytoward the atomic core, as shown in Fig. 4.Thus, we can relate the integration path and saddle pointsof a wavefunction integral to the real space behaviour of thewavefunction. In the following section we apply this principleto the problem of ionization in a time-dependent field. Werelate the concept of sub-barrier reflection to the contributionof one saddle point of the time-integrand of the amplitude. Thisallows us to investigate the role of sub-barrier reflection in theformation of a tunneling time delay. (a) (b) (c) Figure 5. The Airy integral (cid:82) γ ds exp( ˜ x s − s /
3) is defined on the complex s plane; it converges when the endpoints of its infinite contour, γ , lie in the shaded areas. The canonical contours defining the Airy function of the first ( γ ) and second ( γ − γ ) kind are shown in (a) .Contributions to the Airy integral arise principally from portions of the contour near the saddle-points of the integrand function exp( ˜ x s − s / s ± = ± √ ˜ x (red dots) in the complex plane is shown for positions (b) inside the barrier ( ˜ x < (c) outsidethe barrier ( ˜ x > IV. WIGNER TIME AND REFLECTIONS IN THE SFA
As shown in Sec. III, we may reasonably ascribe the saddle-point of the integrand of the wavefunction to under the barrierreflection or transmission. To reveal the contribution of the re-flection to the tunneling time delay, we investigate the complexcontinuation of the integrand function of the SFA wavefunc-tion, Eq. 17, and use the insight developed previously, namelythe relation of a specific saddle-point of the integrand functionto reflection-like behaviour in the wavefunction, to extract thecontribution of reflections to the wavefunction by modifyingthe path of integration the complex plane.In general, the complex t (cid:48) -plane picture of the phase Φ inEq. (17) has many similarities to that of adiabatic ionisaion.However, it depends continuously not only on the coordinate x but now also on observation time t , making its analysissomewhat more involved. In adiabatic ionisation, there wasonly one degree of freedom, ˜ x , and the saddle points wereeither purely real or imaginary.As shown in Fig. 6, there are also two saddle-points of rel-evance in the SFA integral denoted t ± . Their arrangement inthe complex t (cid:48) -plane is dependent on x and t but, as in the adi-abatic case, the real coordinate x determines whether they arevertically or horizontally aligned (in or outside the barrier); theobservation time t merely shifts the axis of symmetry whichis always observed around the line t (cid:48) = t . A more detailed dis-cussion of the configuration of saddle points in ( x , t ) parameterspace may be found in Appendix B. A. Extracting Reflections
In the adiabatic case, a partial integration over only contourcontaining the saddle point s − would exclude the exponentially growing contributions to the wavefunction, i.e. the reflections,associated with the saddle point s + .We use the same principle in the time dependent case; inFig. 6 we identify the saddle point corresponding to reflections,denoted t + (represented by a circle), and the secondary saddlepoint t − (represented by a triangle). When we consider thewavefunction under the potential barrier, we can split the inte-gration contour in Eq. (17) from negative infinity to the uppersaddle point, t + , and from t + to the real time t .The contributions of these two contours to the wavefunctionare shown in Fig. 7. The former integral has the form of agrowing exponential and so can ostensibly be identified as thereflected part of the wavefunction while the latter is a decayingexponential identifiable as the transmitted part. To neglectcontributions from the reflections to the full wavefunction weidentify the following wavefunction: ψ nr ( x , t ) = (cid:90) t + −∞ dt (cid:48) exp( i Φ ( x , t , t (cid:48) )) x > x e (cid:90) tt + dt (cid:48) exp( i Φ ( x , t , t (cid:48) )) x < x e (33)Our method of distinguishing the reflection contribution isbased on the distinguishability of the saddle-points contribu-tions to the wavefunction integral. The latter is not possiblewhen the two saddle points are so close that the cubic term inthe expansion of the phase Φ ( x , t , t (cid:48) ) becomes non-negligible[60], i.e. when (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂ t (cid:48) Φ ( x , t , t (cid:48) ) (cid:16) ∂ ∂ t (cid:48) Φ ( x , t , t (cid:48) ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:48) = t ± (cid:38) . (34)for either saddle point t ± . This occurs in the region very nearthe barrier boundary and at the peak of the laser pulse, as - ▼ - - - - - - - - Figure 6. Complex t (cid:48) plane of the argument Φ ( x , t , t (cid:48) ) of the wavefunction integral ψ i = (cid:82) tt dt (cid:48) exp( i Φ ), for parameters x = ω t = π/ t + , corresponds to reflections. Thesaddle-point method can be used to solve the wavefunction integral, when the path is taken over both saddle-points. We use a partial path givenby (33) to remove the contribution of reflections by integrating over only one of the saddle-points. Other possible configurations of the saddlepoints are shown in Fig 11. - - - - Figure 7. Log-log plot of the relative contributions to the wave-function amplitude from the portion of the integration path up to thesaddle point (blue) and from the saddle point to the given real time t (orange) for various positions x under the classical barrier (i.e. farfrom the atomic core, but smaller than the tunnel exit). The behavioursare approximately decaying and growing exponentials, respectively,analogous to transmitted and reflected parts of the wavefunction. discussed further in Sec. IV B. In this case, the reflection cannotbe separated in the wavefunction in a meaningful manner.We find that when reflections are extracted from the wave-function using Eq. (33) the Wigner time delay under the barriervanishes, as evident in both panels of Fig. 8. That is, whenreflections are neglected, there is no time delay between thepeak of the electron probability distribution and the peak ofthe laser field. Moreover, outside the barrier, the trajectory ofthe peak rapidly becomes classical.Plotted also in Fig. 8(b) is the trajectory for an electron in aconstant field, given by Eq. (25). We see that, for parametersin the deep tunnelling regime, the SFA trajectory (given bythe peak of the wavefunction), and the classical trajectory, theconstant field trajectory display very similar behaviour as the pulse elapses (the trajectories do not completely coincide faraway due to the time dependence of the field and the need tomatch a tunnel exit, respectively).That is, at a detector infinitely far away such as one in anattoclock experiment, a signature of time delay would not bepresent for the deep tunneling conditions detailed in this work.Thus, it is of importance to distinguish between a time delayof the peak of the wavefunction near the tunnel exit and anasymptotic time delay at a detector. Since in attoclock experi-ments the asymptotic momentum distribution is measured, i.e.,the asymptotic time delay, one expects to find a zero time delayin the deep tunnelling regime. However, near the tunnel exit,quantum mechanical considerations must be taken into accountand a quantum mechanical treatment such as the one exposedhere exhibits an explicit non-zero Wigner delay. B. Tunnel Exit from the SFA
We may also use the saddle point analysis of Eq. (17) toextract relevant physical information about the electron dynam-ics. In particular, and in analogy to the constant field scenario,the topology of the saddle points through the complex t (cid:48) planereveal the functional behaviour of the electronic wavepacket.The paths of these two saddle point, for two representativechoices of time t while varying x , are shown in Fig. 9, closelyresembling those corresponding to Figs. 5 (b) and (c). Muchas in the constant field case where the saddles are purely realor imaginary, see Eq. (30), the pair of saddle points in the SFAis aligned around the line given by Re[ t (cid:48) ] = t ; this alignmentis either vertical or horizontal depending on the coordinate x much as in the constant field case shown in Fig. 5. For a fixedobservation time t , there corresponds a coordinate x of closestapproach between the saddle points. For the peak of the pulse, t = x t where the saddle points mergecompletely, as they do in the case ˜ x = - -
20 0 20 404. × - × - × - × - × - (a) (b) Figure 8. (a)
Temporal probability distributions at the position x = ψ i ,and the pseudo-wavefunction neglecting reflections, ψ nr . The physical wavefunction has accrued a Wigner time delay with respect to the laserfield, whereas the maximum of the pseudo-wavefunction is synchronous with the laser field peak. (b) Trajectories in the ( x , t ) plane via: themaximum of the wavefunction ψ i , the maximum of the wavefunction neglecting reflections ψ nr , and the energy derivative of the constant fieldwavefunction τ W ( x ). The classical electron trajectory, x cl ( t ), starting at the tunnel exit at the peak of the laser field is also shown. Under thebarrier, the probability distribution neglecting reflections | ψ nr | shows zero time delay. In regions where Eq (34) does not apply, mostly near theclassical tunnel exit, we may not identify reflections nor plot a subsequent trajectory. electron.Thus, two lines, t = x = x t , define separatrices forthe behaviour of the saddle points and their contributions tothe integral: the behaviour of the wave function can be seen tochange depending on whether x > x t . This is a phenomenonexactly analogous to the merging of the saddle points for con-stant field case of Sec. III: there, the merging of the saddlepoints occurs at the classical exit and separates the regions ofevanescence and oscillation in the wave function.We may use this parallel to identify the merging point as ameasure of the tunnel exit. In our study, for a field strength of E = .
05 and γ = .
1, we find that this exit takes the value x t ≈ . x e = I p / E =
10 a.u. More preciseagreement is achieved with the probability averaged tunnel exit x K = (cid:82) dt (cid:48) I p | E ( t (cid:48) ) | exp (cid:16) − κ | E ( t (cid:48) ) | (cid:17)(cid:82) dt (cid:48) exp (cid:16) − κ | E ( t (cid:48) ) | (cid:17) ≈ .
35 a.u. , (35)where the integration runs over the whole laser pulse. V. CONCLUSIONS
In this work, we have analyzed the tunneling time delay instrong field ionization for a simple model of an atom with ashort-range potential. The wave function was calculated tofirst-order in the SFA for any intermediate time, enabling aquantum treatment of dynamics in the region where quasiclas-sical descriptions break down, viz. around the classical tunnelexit. For a given coordinate, the peak of the wave packet shows a time delay with respect to the peak of the laser field and thistime delay is positive at the classical tunnel exit.We argue that reflections of the electron wavepacket underthe tunneling barrier are fundamentally responsible for thisnon-zero time delay around the tunnel exit. This is a generalphenomenon, present in any regime of strong field ionization,as well as in any tunneling process. In particular, a simpleshowcase is presented in Appendix A for the case of tunneling - - Figure 9. Positions of the upper and lower saddle-points of theintegral (17) in the complex t (cid:48) plane with varying x and for fixed ω t = ± . π/ x , where the arrows indicate growing values of x . For values of x (cid:46)
7, the lower saddle-point disappears belowthe imaginary axis. Each pair of curves is centred around the lineRe( ω t (cid:48) ) = ω t , shown in dashed. For the case ω t =
0, not shown, thetwo lines meet one point corresponding to x = x t . Appendix A: Tunneling through a box potential barrier
In this appendix we consider the role of quantum reflec-tions in the tunneling time delay for an electron wavepackettunnelling through a one dimensional box potential. This isperhaps the simplest example of tunnelling time delay sincethe wavefunction is readily separable (and of analytic solu-tion) in the regions inside and outside the barrier, allowing thecontributions of reflections in to the time delay to be clearlyidentified.It should be mentioned that this model di ff ers from truestrong field ionization in that it considers the scattering of awavepacket incident on a potential barrier; in actual ionizationthe electron originates from within a potential barrier. Be thatas it may, this study provides a simple, intuitive picture oftunnelling time delay, suitable even for the uninitiated. V x = a x = 0 I II III
Figure 10. Pictorial representation of the square barrier potential foran incident monochromatic wave. The wavefunction is a piecewisesolution of the Schr¨odinger equation for the three regions shown,given by Equations (A2)-(A3).
1. The Box Potential
Consider a wave packet Ψ ( x , t ) = (cid:90) d p f ( p − p ) ψ ( p ) e − i E ( p ) t (A1)with energy E ( p ) = p /
2, incident on a potential barrier V ( x ) = V for 0 ≤ x ≤ a and 0 elsewhere, where f ( p − p ) is somedistribution peaked at p (e.g. a Gaussian), as shown Fig. 10.Each p -component wavefunction obeys the time-independentSchr¨odinger equation with the piecewise solution ψ I ( x ) = e ip x + Re − ip x , (A2) ψ II ( x ) = C e qx + C e − qx , (A3) ψ III ( x ) = T e ipx , (A4)with momenta p = √ E , q = √ V − E ). The amplitude ofthe incoming wave has been set to unity, without loss of gener-ality, and the co-e ffi cients C and C are the typical reflectionand transmission coe ffi cients under the barrier, respectively.Matching the above solutions and their derivatives at theboundaries yields the coe ffi cients C = ( − i χ )(1 + i χ ) e − ξ (1 − i χ ) e ξ − (1 + i χ ) e − ξ , (A5) C = exp(2 ξ ) C , R = C + C −
1, and T = ( − i χ ) e − ipa (1 − i χ ) e ξ − (1 + i χ ) e − ξ . (A6)We have introduced the dimensionless parameters χ = p / q and ξ = q a which, loosely speaking, determine the relative lengthand height of the barrier respectively.
2. Time Delay
The wavepacket after transmission is of the form Ψ III = (cid:90) d p | T ( p ) | exp [ i ( ϕ ( p ) + px − E ( p ) t )] , (A7)1where T = | T | e i ϕ . The maximum of this amplitude occurswhen the phase in Eq. (A7) vanishes, that is when (after somere-arrangement): x = p t − (cid:34) ∂ϕ∂ p (cid:35) p = p (A8)In the absence of a potential barrier, the peak travels at theclassical velocity (in atomic units) p . Equation (A8) showsthat the barrier causes a delay of the peak in reaching a givenposition x , a delay which is given the by the energy derivativeof the transmission phase ϕ . Thus, τ = p (cid:34) ∂ϕ∂ p (cid:35) p = p + ap , (A9)which gives the time delay of the peak after crossing the barrier.For the box potential, one finds τ = ap ξ (cid:16) χ + χ (cid:17) tanh( ξ ) + (1 − χ )2 sech ( ξ )1 + (cid:16) χ − χ (cid:17) tanh ( ξ ) . (A10)Interestingly, as ξ goes to infinity, τ vanishes: the longer thebarrier, the less time it takes the peak to cross it (see also [61]).For longer barriers, ξ (cid:29)
1, the tunnelling time τ becomesindependent of χ ,lim ξ (cid:29) τ ≈ ap ξ = p √ V − E ) , (A11)and dependent only on the barrier height V .
3. Quantum Reflections and Time Delay
In the above problem there are two reflections: the reflectionof the incoming wave from the barrier surface at x =
0, and thereflection of the tunnelled wavepacket from the barrier edgeat x = a . The latter reflection will decrease with increasingbarrier length ( ξ (cid:29) ffi cient Eq. (A5). As shown above, for ξ (cid:29)
1, the timedelay τ vanishes, which hints that the origin of the tunnelingtime is related to the reflection of the tunneling wavepacket.We can show this more explicitly by neglecting the reflectedwave under-the-barrier.Under the barrier, the wavefunction takes the form ofEq. (A3), a superposition of increasing and decaying expo-nentials. Consider the neglecting one of these coe ffi cients,namely C (cid:28) C (feasible when ξ (cid:29)
1) and, as before, calcu-late the transmission co-e ffi cient by matching at the boundaries.Since we have three unknowns, we require only three equa-tions. We forsake continuity of derivatives at x = a . In thisscenario one finds the transmission coe ffi cient: T = e − i ( p + q ) a (cid:32) − i χ − i χ (cid:33) , (A12)leading to a time delay τ = ap ξ =
12 lim ξ (cid:29) τ (A13) which is only half the time delay of Eq. (A11). Thus, thequantum reflection of the tunneling wave packet at the internaledge ( x = a ) is responsible for half the total transmission timedelay. The other half is ostensibly due to the time delay fromentering the the barrier in the first place.The existence of a non-unit reflection co-e ffi cient R is apurely quantum mechanical phenomenon. One expects thisdeviation of R from unity to contribute to a delay of the peak.Indeed, we can see that the longer the barrier ξ , the smaller 1 − R and thus the smaller the time delay due to entering the barrier.We can show this explicitly by calculating the time delay uponentry for the case ξ (cid:29)
1. There, we have R ≈ − (1 + i χ ) / (1 − i χ )and τ ≈ − ip ∂∂ p ln (cid:32) ψ I ( x ) | ψ I ( x ) | (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = = ap ξ =
12 lim ξ (cid:29) τ. (A14)Thus, the total transmission time of an electron wave packetthrough the box barrier consists of the time delay upon enteringthe barrier and of the tunneling time delay arising from thequantum reflection of the tunneling wavepacket inside thebarrier. Appendix B: Configurations of saddle points
The configurations of the saddle points of the argument Φ ( x , t , t (cid:48) ) of the wavefunction integral ψ i = (cid:82) tt dt (cid:48) exp( i Φ ) incomplex t (cid:48) plane, for a large ( x , t ) parameter range are pre-sented as a table in Fig. 11. The laser pulse evolves from left toright in this figure and mostly shifts the reference line aroundwhich the saddle points are centred, namely Re[ t (cid:48) ] = t . Itshould be noted, by the definition in Eq. 17, a singularity isalways to be observed at the point t (cid:48) = t .As one moves down the table, configurations of the complex t (cid:48) plane are shown for spatial coordinates inside, neighbour-ing, and outside the tunnelling barrier ( x = , , and 14 a.u.respectively). There are marked di ff erences between each ob-servation coordinate but the behaviour is reminiscent of thecomplex plane configuration for the Airy integral, displayed inFig. 5. Indeed, the two pictures can easily be reconciled by a π / rad. clockwise rotation.Inside the barrier, the saddle points are vertically alignedalong the line Re[ t (cid:48) ] = t . As one increases the observationcoordinate and approaches the tunnel exit, x ≈
10, the twosaddle points approach each other vertically; as one exits theregion neighbouring the tunnel exit, x (cid:29)
10, the saddles thenseparate from each other on the horizontal.As one approaches the peak of the laser field, t =
0, thedistance of closest approach around the tunnel exit shrinks. At the exact peak, this distance is zero; that is, the saddle pointsmerge. This phenomenon is directly comparable to the mergingof saddle points for the Airy integral at the classical tunnel exit.Thus, we are able identify a new tunnel exit, x t , from thecomplex plane landscape, a topic discussed in Sec. IV B.The identification the exact value for the co-ordinate x t to adesired precision is in principle achievable by a binary searchor global minimization of the distance function for the saddle2 ▼ ▼ ▼ ▼ ▼ ▼ - - - - ▼ - - ▼ - ▼ Figure 11. Configurations of the saddle points of the argument Φ ( x , t , t (cid:48) ) of the wavefunction integral ψ i = (cid:82) tt dt (cid:48) exp( i Φ ) in complex t (cid:48) plane,for parameter ranges x = , ,
14 and ω t = − π/ , , + π,
20. The dashed line corresponds to t (cid:48) = ω t . The scale and colour coding are identical to those of Fig. 6. As x increases the two saddle points approach vertically and,after a closest approach, separate horizontally. For the peak of the pulse, ω t =
0, this closest approach is zero and the two saddle points merge atthe point x t . Otherwise, varying t skews the relative orientation of the saddle points around the line Re[ t (cid:48) ] = t . points. However, such precision was deemed unnecessary for the purposes of this work. 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