Sub-barrier pathways to Freeman resonances
Michael Klaiber, Karen Z. Hatsagortsyan, Christoph H. Keitel
SSub-barrier pathways to Freeman resonances
Michael Klaiber, ∗ Karen Z. Hatsagortsyan, † and Christoph H. Keitel Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: September 16, 2020)The problem of Freeman resonances [R. R. Freeman et al. , Phys. Rev. Lett. , 1092 (1987)] when strongfield ionization is enhanced due to transient population of excited states during the ionization, is revisited.An intuitive model is put forward which explains the mechanism of intermediate population of excited statesduring nonadiabatic tunneling ionization via the under-the-barrier recollision and recombination. The theoreticalmodel is based on perturbative strong-field approximation (SFA), where the sub-barrier bound-continuum-boundpathway is described in the second order SFA, while the further ionization from the excited state by an additionalperturbative step. The enhancement of ionization is shown to arise due to constructive interference of contributionsinto the excitation amplitudes originating from di ff erent laser cycles. The applied model provides an intuitiveunderstanding of the electron dynamics during a Freeman resonance in strong field ionization, as well as meansof enhancing the process and possible applications to related processes. I. INTRODUCTION
The enhancement of strong field ionization due to transientexcitation of Stark-shifted bound states is well-known fromexperiments in multiphoton regime of ionization and is termedas Freeman resonances [1–13]. It is assumed that the exci-tation at Freeman resonances happens due to a bound-boundmultiphoton transition, when the electron wave function dur-ing transition is localized within the binding potential. Withincreased laser intensity, the tunneling through the laser sup-pressed Coulomb barrier becomes dominant and the boundelectron moves from the ground state immediately to the con-tinuum. Strong field approximation (SFA) [14–16] describessuccessfully direct strong field ionization in tunneling and mul-tiphoton regimes as well as in the intermediate nonadiabaticregime [17, 18], when the electron gains energy during the tun-neling [19]. The quantum orbit picture [20, 21] which stemsfrom the SFA description, applying saddle-point approxima-tion (SPA) in calculation of integrals in S-matrix amplitude,provides intuitive understanding of strong field ionization pro-cesses. Can the quantum orbit picture be extended to interpretthe electron dynamics at Freeman resonances?In the tunneling regime the atom excitation due to bound-bound transitions is not probable, because it is overwhelmedby electron tunneling into the continuum. In the nonadiabatictunneling excitations can happen only when the electron revis-its the atomic core, i.e. at recollisions. However, the commonrecollisions via excursion in the real continuum [22], are ac-companied by a large spreading of the electron wave packet,which reduces significantly the recollision probability. Re-cently, it has been recognized that recollision can happen alsowithin the sub-barrier dynamics during tunneling [23]. Thelatter may contribute to the electron transition to the excitedstate as long as the electron gains su ffi cient energy during thenonadiabatic tunneling.In this paper we develop a theory for Freeman resonancesin the nonadiabatic tunneling regime which is based on the ∗ [email protected] † [email protected] concept of the under-the-barrier recollision. We employ SFA,treating the recollision with the atomic core within a perturba-tive approach. The resonant channel of ionization is describedwithin the next-order perturbation of SFA. The given descrip-tion allows for an interpretation of the process as taking placevia sub-barrier recollision with increasing energy in the nona-diabatic regime and transition to the excited state, with furtherionization after some time delay, see the interaction scheme inFig. 1. The proposed model provides a physical explanationvia quantum orbit picture for the resonantly enhanced strongfield ionization involving excited states at a Freeman resonance.The paper is organized as follows. The theoretical modelis described in Sec. II. The half-cycle and multi-cyle contribu-tions to the Freeman resonances are discussed in Secs. III A - I p - I p * ˜ coordinate ene r g y FIG. 1. The scheme of the resonantly enhanced nonadiabatic tun-neling (Freeman resonance): the sub-barrier recollision (describedby a quantum trajectory propagating from the bound state up to thebarrier surface, reflected, and tunneled back to the core) may yield torecombination to the Stark-shifted excited state of the atom, whichis followed by further ionization. The electron energy in the excitedstate is up-lifted due to the laser dressing during dwelling in the ex-cited state before further ionization. Dashed line shows the path ofthe direct nonadiabatic ionization. a r X i v : . [ phy s i c s . a t o m - ph ] S e p and III B, respectively. The photoelectron spectra within thepresent model are presented in Sec. III C, and our conclusionis given in Sec. IV. II. THEORETICAL MODEL
Our main aim is to provide an intuitive picture within thenew scheme for Freeman resonances. For this purpose it isimportant to have analytical theory and, therefore, we illustratethe new scheme in a simple and transparent one dimensional(1D) model. We expect the picture to hold also in 3D, becausethe under-the-barrier recollision is virtually one dimensionalalong the parabolic coordinate even in the full 3D consideration.The ionization dynamics of an atom in a strong laser field isdescribed by the Hamiltonian H = ˆ p / + V ( x ) + H I ( t ) , (1)where ˆ p is the momentum operator, V ( x ) is the potential of theatomic core, and H I ( t ) = xE ( t ) is the laser-electron interactionHamiltonian, with the laser electric field E ( t ).The theoretical treatment is based on SFA. We begin withthe exact ionization amplitude m ( p ) for the photoelectron witha final momentum p : m ( p ) = − i (cid:90) dt (cid:104) ψ Vp ( t f ) | U ( t f , t ) H I ( t ) | φ ( t ) (cid:105) , (2)where U ( t f , t ) is the exact time-evolution operator (TEO), withan asymptotic time t f , ψ Vp ( x , t ) = √ π exp[ i ( p + A ( t )) x + iS ( t )]is the Volkov wave function [24], with the contracted classicalaction S ( t ) = (cid:82) ∞ t ds [ p + A ( s )] /
2, and φ ( x , t ) = φ ( x ) exp( iI p t ) isthe wave function of the atomic bound state, with the ionizationpotential I p , κ = (cid:112) I p is the atomic momentum. The linearlypolarized laser pulse is described by the vector potential A ( t ) = ( E /ω ) f ( t ) sin( ω t ), with the field amplitude E , the frequency ω , E ( t ) = − ∂ A ∂ t , and the slowly varying pulse envelope f ( t ).We describe the strong field ionization via resonant excita-tion during nonadiabatic tunneling. This pathway includes anunder-the-barrier recollision due to which a transition to theexcited state happens, from where the electron is readily ion-ized via tunneling or an over-the-barrier passage. To model thedescribed pathway, in Eq. (2) we need to approximate the exactTEO U ( t (cid:48) , t (cid:48)(cid:48) ), which is designed to describe the laser drivensub-barrier dynamics of the electron, including an intermediaterevisiting the atomic core. For this reason we represent theexact TEO symbolically as follows U ( t (cid:48) , t (cid:48)(cid:48) ) = (cid:88) n | ˜ φ n ( t (cid:48) ) (cid:105)(cid:104) ˜ φ n ( t (cid:48)(cid:48) ) | , (3)with the sum running over the exact basis set | ˜ φ n ( t ) (cid:105) , repre-senting the exact solutions of the Schr¨odinger equation in thelaser and the atomic potential fields. As U ( t (cid:48) , t (cid:48)(cid:48) ) recounts thedynamics via direct ionization and through the laser-dressedexcited state, we extend the sum in Eq. (3) over continuumstates and bound states. Taking into account that for the directionization the influence of the potential is negligible and duringthe resonance only one excited state | ˜ φ ∗ ( t ) (cid:105) is important in the sum of Eq. (3), the one which has an energy that fits to theenergy of the recolliding electron, we approximate: U ( t (cid:48) , t (cid:48)(cid:48) ) ≈ | ˜ φ ∗ ( t (cid:48) ) (cid:105)(cid:104) ˜ φ ∗ ( t (cid:48)(cid:48) ) | + U V ( t (cid:48) , t (cid:48)(cid:48) ) , (4)where U V ( t (cid:48) , t (cid:48)(cid:48) ) = (cid:82) dw | ψ Vw ( t (cid:48) ) (cid:105)(cid:104) ψ Vw ( t (cid:48)(cid:48) ) | is the Volkov-TEO.Using Eqs. (2) and (4), we derive the SFA amplitude: m ( p ) = − i (cid:90) dt (cid:104) ψ Vp ( t ) | H I ( t ) | φ ( t ) (cid:105)− i (cid:90) dt (cid:104) ψ Vp ( t f ) | ˜ φ ∗ ( t f ) (cid:105)(cid:104) ˜ φ ∗ ( t ) | H I ( t ) | φ ( t ) (cid:105) . (5)Further, we neglect direct bound-bound transitions during thelaser dressing of the bound state, assuming that the laser-dressed bound state emerges from the corresponding barebound state due to the action of the Volkov-Dyson expansion: | ˜ φ ∗ ( t ) (cid:105) ≈ − i (cid:90) t dt (cid:48) U V ( t , t (cid:48) ) VU V ( t (cid:48) , t i ) | φ ∗ ( t i ) (cid:105) , (cid:104) ˜ φ ∗ ( t ) | ≈ − i (cid:90) t dt (cid:48) (cid:104) φ ∗ ( t f ) | U V ( t f , t (cid:48) ) VU V ( t (cid:48) , t ) , (6)where t i is the initial time when the laser field is turned on, and | φ ∗ ( t ) (cid:105) is the corresponding bare atomic eigenstate. The poten-tial V is accounted for perturbatively to describe recombinationto the excited state during the sub-barrier rescattering. Hereit is assumed that the zeroth order term yields an unphysicalboundary term and is neglected. Hence the amplitude reads m ( p ) = m ( p ) + m ( p ) (7) m ( p ) = − i (cid:90) dt (cid:104) ψ Vp ( t ) | H I ( t ) | φ ( t ) (cid:105) (8) m ( p ) = i (cid:90) dt (cid:90) t dt (cid:48) (cid:90) dt (cid:48)(cid:48) (cid:90) dq (cid:90) dw (cid:90) dv ×(cid:104) ψ Vp ( t (cid:48)(cid:48) ) | V | ψ Vv ( t (cid:48)(cid:48) ) (cid:105)(cid:104) ψ Vv ( t i ) | φ ∗ ( t i ) (cid:105)(cid:104) φ ∗ ( t f ) | ψ Vw ( t f ) (cid:105)×(cid:104) ψ Vw ( t (cid:48) ) | V | ψ Vq ( t (cid:48) ) (cid:105)(cid:104) ψ Vq ( t ) | H I ( t ) | φ ( t ) (cid:105) , (9)with the direct ionization amplitude m ( p ), and the ionizationamplitude with a Freeman resonance m ( p ). To simplify thecalculation of the high-order amplitude m ( p ), we model theatom by a short-range potential.Dressing of the bound states emerges in Eq. (7) due to tran-sitions to intermediate Volkov states given by the integrationsover the momenta v and w . The mathematical structure of thedressing of the excited state consists of two integrals of theform (cid:90) dx (cid:90) d p exp (cid:34) − i (cid:90) tt i ds ( p + A ( s )) / − ipx − κ ∗ | x | (cid:35) = π exp (cid:34) − i (cid:90) tt i ds ( i κ ∗ + ( − k A ( s )) / (cid:35) , (10)which were solved with the two-dimensional SPA. Here, thefactor ( − k corresponds to the k th half-cycle and arises fromthe derivative of | α ( t ) | , with the excursion coordinate α ( t ) = (cid:82) t A ( s ) ds . The integration over the momenta v and w resultsin dressing of the excited state with the vector potential ofthe laser field via i κ ∗ → i κ ∗ + A ( s ), e ff ectively shifting theexcited state energy from − I ∗ p to − I ∗ p + U p , with the laserponderomotive potential U p = E / ω , κ ∗ = (cid:113) I ∗ p , and theionization potential of the excited state I ∗ p . The remainingintegrals in m ( p ) over the times t , t (cid:48) , t (cid:48)(cid:48) and q are calculated bySPA.We gain a physical insight of the excitation process fromthe saddle-point conditions. The saddle points for these fourvariables t , t (cid:48) , t (cid:48)(cid:48) and q are determined from the followingequations[ q ( t (cid:48) , t ) + A ( t )] / = − κ / q ( t (cid:48) , t ) = − α ( t (cid:48) ) − α ( t ) t (cid:48) − t (12)( i κ ∗ + ( − j A ( t (cid:48) )) / = ( q ( t (cid:48) , t ) + ( − j A ( t (cid:48) )) / p + ( − i A ( t (cid:48)(cid:48) )) / = ( i κ ∗ + ( − i A ( t (cid:48)(cid:48) )) / , (14)with the factors ( − i and ( − j corresponding to the processesin the i th and j th half-cycles. The Eqs. (11)-(14) describe theelectron dynamics during the Freeman resonance. The ion-ization path begins at t from the ground state, see Eq. (11).The electron revisits the atomic core at t (cid:48) , when the interme-diate momentum q ( t (cid:48) , t ) fulfills Eq. (12), which may lead tothe electron recombination into the dressed excited state, seeEq. (13). We choose the recolliding trajectory with sub-barrierexcursion dynamics. In this case all saddle points are com-plex with similar real parts, but di ff erent imaginary parts, andthe interpretation of the sub-barrier motion is in order. In theconsidered nonadiabatic tunneling regime the electron gainsenergy during tunneling which allows for the transition to thedressed excited state, according to Eq. (13). Finally, the elec-tron is ionized from the dressed excited state at time t (cid:48)(cid:48) , givenby Eq. (14). III. RESULTSA. The half-cycle contribution to the yield
Firstly, we examine the excitation of a Rydberg state viasub-barrier recollision. Let us analyze the contribution to theexcitation yield Y ex during a half-cycle of the laser field. Wedefine Y ex ≡ (cid:82) | m ex ( p ) | d p , with the excitation amplitude dueto sub-barrier rescattering m ex = i (cid:90) dt (cid:90) t dt (cid:48) (cid:90) dq (cid:90) dw (cid:104) φ ∗ ( t f ) | ψ Vw ( t f ) (cid:105)×(cid:104) ψ Vw ( t (cid:48) ) | V | ψ Vq ( t (cid:48) ) (cid:105)(cid:104) ψ Vq ( t ) | H I ( t ) | φ ( t ) (cid:105) , (15)which is derived from Eq. (9), dropping the amplitudes ofionization from the excited state. The ratio of the resonantexcitation to the direct ionization yield, Y ex / Y , is shown inFigs. 2(a) and (b), where Y = (cid:82) | m ( p ) | d p . We see that theexcitation during a half-cycle is quite small. The excitationprobability is significantly damped at large I ∗ p , and at largefields, which can be explained as follows. The process takesplace at the laser field maximum, when the spatial distribu-tion of the dressed excited state is concentrated at the distance ( a ) - - - I p * / ω Y e x / Y ( b ) - - - - E [ a.u. ] Y e x / Y ( c ) - - - I p * / ω Y / Y FIG. 2. Ratio of the excitation yield to that of the direct ionization Y ex / Y (blue) from a single half-cycle: (a) vs I ∗ p /ω , for E = . E for κ ∗ = .
23 a.u.; ω = . κ = γ =
2, (orange) the scaling ∼ exp( − κ ∗ α ). (c) Ratio Y / Y of the ionization yields due to the sub-barrier recollision with( Y ) and without ( Y ) excited state, respectively, for E = .
035 a.u.(blue), E = .
025 a.u. (orange) and E = .
015 a.u. (green). α ∼ E /ω away from the core, with the width ∼ /κ ∗ , mean-while, for recombination the recolliding electron arrives at thecore, because momentum transfer from the core is needed forrecombination. As a results the recombination into the excitedstate is suppressed by a factor exp( − κ ∗ α ), see Figs. 2(a) and(b).How the availability of the intermediate excited statechanges the probability of the sub-barrier path is demonstratedin Fig. 2(c), where the ratio Y / Y of the ionization yield dur-ing an half-cycle period due to the sub-barrier recollision with - - - - I p * / ω Y e x / Y FIG. 3. Ratio of the excitation yield to that of the direct ionization Y ex / Y from 10 cycles, for E = .
025 a.u, ω = .
05 a.u., ( γ = κ = U p + I ∗ p − I p ) = (cid:96)ω ,with an integer (cid:96) , for the given excited state ( P e =
1) are indicated byvertical lines. ( Y = (cid:82) | m | d p ) and without ( Y = (cid:82) | m | d p ) excited stateis shown. Here, the sub-barrier recollision is described in thesecond-order SFA by the matrix element m = − (cid:90) dt (cid:90) t dt (cid:48) (cid:90) dq (cid:104) ψ Vp ( t (cid:48) ) | V | ψ Vq ( t (cid:48) ) (cid:105)(cid:104) ψ Vq ( t ) | H I ( t ) | φ ( t ) (cid:105) . During the half-cycle there is no resonance enhancement inthe excitation, which is created only due to multi-cycle in-terference. Nevertheless, we see that even in that case theintermediate bound state increases the ionization probabilityseveral times at small energies of the excited state I ∗ p and weakfields. There are two reasons for this enhancement. Firstly, theelectron gains energy during the dwelling in the excited state,which increases further ionization probability. For instance,the energy of the electron at the recombination t (cid:48) is approxi-mately − . ω = .
05 a.u. for all values of the Keldyshparameter γ = κω/ E , whereas the laser-dressed energy atthe tunneling from the excited state at t (cid:48)(cid:48) is significantly larger,approximately − I ∗ p for small final momenta. Secondly, the elec-tron wave packet spreading is suppressed during the dwellingtime in the excited state. The spreading factor is dominating,and it is larger for larger Keldysh parameters, therefore the en-hancement due to absence of spreading is also larger for larger γ . At large fields and large ionization energies of the excitedstate the recombination into the excited state is suppressed bythe factor ∼ exp( − κ ∗ α ), which suppress severely the excita-tion probability and the ionization via Freeman resonances.We note that the enhancement of the ionization yield due to thetransient excitation is not large in the half-cycle contribution,but significantly boosted due to multi-cycle interference, asdiscussed below. B. Multi-cycle interference
A conspicuous resonance e ff ect emerges in a long laser pulsewhen interference of contributions in the ionization amplitudesfrom di ff erent laser cycles is included. The structure of the m amplitude is a product of an amplitude of direct ionizationand a recombination amplitude into the dressed excited state.The three-dimensional saddle points are, therefore, decoupledand recombination can happen in a di ff erent half-cycle thanthe direct ionization. Consequently we add: m = (cid:88) i m ∗ , i (cid:88) j ≤ i m ex , j (16)where the ionization from the excited state takes place in the i th half-cycle, and the recombination in the j th one, where thetime ordering ionization after recombination is insured by j ≤ i .The phase of the excitation amplitude has the form Φ j = ( − j κ ∗ (cid:2) α ( t (cid:48) ) − α ( t i ) (cid:3) − (cid:34) κ ∗ t (cid:48)(cid:48) − β ( t (cid:48)(cid:48) ) + β ( t i ) (cid:35) (17) − (cid:90) t (cid:48) t (cid:48)(cid:48) ds (cid:2) q ( t (cid:48) , t (cid:48)(cid:48) ) + A ( s ) (cid:3) + κ t (cid:48)(cid:48) + − P e i log[ α ( t )] , where j is the half-cycle number, ˙ β ( t ) = A ( t ) /
2, and P e is theparity of the excited state. Two consecutive half-cycles with t (cid:48) → t (cid:48) + π/ω and t (cid:48)(cid:48) → t (cid:48)(cid:48) + π/ω have the phase di ff erence ∆Φ = π ( U p + I ∗ p − I p ) /ω + π (1 − P e ) /
2. The interference of excitationamplitudes in the second sum of Eq. (16) is constructive, andthe yield is enhanced, if the resonance condition is fulfilled: U p + I ∗ p − I p = (cid:96)ω, (18)with an integer (cid:96) ; even (cid:96) corresponds to the case of the sameparity of the ground and the excited states, and odd (cid:96) to theopposite parities. For the given excited state, the resonancepeaks with respect to the state energy (or the laser intensity)have 2 ω separation, see Fig. 3. It is due to constructive in-terference of the excitation amplitudes originating from eachhalf-cycle. Comparison of the multi-cycle yield in Fig. 3 withthat of a single-half cycle one of Fig. 2(a) shows that the yield Y ex / Y scales roughly quadratically with the number of lasercycles. This is because the resonance comes from the coher-ent contributions of half-cycle terms in the sum (cid:80) j m ex , j inEq. (16). Our model includes only one (lowest) excited state,more resonances are possible when higher excited states areconsidered. C. Photoelectron spectra
We analyze the signature of the resonant transient excita-tion during strong field ionization in the photoelectron spectra.Photoelectron energy distribution in a 10-cycle laser pulse isshown in Fig. 4 in the case of the resonant excitation. We seethat our model of excitations via sub-barrier recollision is ableto describe the typical photoelectron spectra at Freeman reso-nances. Double peak structures arises in spectrum, see Fig. 4.One peak in series corresponds to the direct ionization from the - - - - - - ϵ / ω Y ( ϵ ) ω FIG. 4. Photoelectron energy spectrum in a 10-cycle laser pulsein a logarithmic scale: (blue) direct ionization via m and (orange)ionization at a Freeman resonance via m + m ; E = .
025 a.u., ω = .
05 a.u., κ ∗ = .
23 a.u., κ = γ =
2. Grid lines indicatethe ionization from the ground state n − ( I p + U p ) /ω , and from theexcited state n − ( I ∗ p + U p ) /ω ; U p + I ∗ p − I p = . ground state with the energy conservation n ω = p / + U p + I p .The second peak in series is due to the Freeman resonance. Itcorresponds to the multiphoton transition from the excited statewith the energy − I ∗ p + U p to the continuum with the energy p / + U p , with the energy conservation n ω = p / + I ∗ p , seegrid lines in Fig. 4. IV. CONCLUSION
We have proposed an intuitive model for Freeman reso-nances in the nonadiabatic tunneling ionization. Using specificquantum orbits provided by strong field approximation theory,we show a concrete pathway leading to the transient populationof intermediate excited states during strong field ionization.What is interesting, the pathway along which the electron gains energy necessary for the transfer to the excited state, mostlytravels under-the barrier, reflects from the outer surface of thebarrier, propagates back to the core, and recombines to theexcited bound state. All this sub-barrier recollision takes placeduring imaginary time within a single half-cycle, visualizingthe electron vertical transition in the strongly driven atom [18].We found that the available excited bound state can increasethe probability of the sub-barrier recolliding pathway, evenduring the singe half-cycle contribution. The latter is mostlydue to the suppressed spreading of the electron wave packet,during the dwelling time in the excited state. Although the tran-sition probability during an half-cycle is small, it is resonantlyenhanced due to constructive interference of contributions tothe excitation amplitude emerging from di ff erent half-laser cy-cles, proportional to the square of the number of half cycles.As each half-cycle gives an interfering contribution, the reso-nance condition of di ff erent orders for a ceratin excited statein this model are separated by twice of a photon energy. Thedescribed sub-barrier pathway of the Freeman resonance is rel-evant in the nonadiabatic tunneling regime, when the electrongains energy during tunneling, enabling transition to the laserdressed excited state.As an outlook beyond the scope of this paper, let us noteon a possible application of the presented model of the Free-man resonances via sub-barrier recollision on the strong fieldelectron-positron pair production problem in ultrastrong laserfields. When pairs are produced during the impinging of thelaser beam on a nucleus (ion, or other atomic system), then theproduced electron from vacuum due to multiphoton processcan be captured into the bound state in the Coulomb poten-tial. This bound-free channel of pair production has beenthoroughly investigated in Refs. [25–29]. However, rather thanreal bound-free pair production, the state of a bound-free paircan emerge virtually as a transition state, which finally mayend up with a free electron and positron state. 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