Coulomb time delays in high harmonic generation

Measuring the time it takes to remove an electron from an atom or molecule during photoionization using newly developed attosecond spectroscopies has been a focus of many recent experiments. However, the outcome of such measurement depends on measurement protocols and specific observables available in each particular experiment. One of such protocols relies on high harmonic generation. First, we derive rigorous and general expressions for ionization and recombination times in high harmonic generation experiments. We show that these times are different from, but related to ionization times measured in photo-electron spectroscopy, i.e. using attosecond streak camera, RABBITT and atto-clock methods. Second, we use the Analytical R-Matrix theory (ARM) to calculate these times and compare them with experimental values.


I. INTRODUCTION
The problem of time-resolving the removal of an electron during photoionization and studying time-delays associated with this process is an intriguing one [1]. Besides the fundamental implications for our understanding of atom-light interaction, such measurements have the capacity to serve as a sensitive probe of multielectron dynamics [2,3] and have an important role to play in calibrating attosecond recollision-based pump-probe experiments [4].
The reconstructed times are inherently very sensitive to the way in which this is done.
In the one-photon case, the theoretical basis for doing so is now well-established (e.g. see [1,22]). Ionization is thought of as a half-scattering process and time delays are defined in terms of the Eisenbud-Wigner-Smith (EWS) time [23,24]: the derivative of the scattering phase-shift of the photoelectron with respect to its energy In the multiphoton regime, however, the situation is notably less straightforward. If we think of ionization as a tunnelling process, we are presented with a number of different possible definitions for the tunnelling time, and it is not clear from the outset which is the 'correct' time to use (e.g. see [25][26][27]). What's more, the use of any of these definitions in the context of strong field ionization necessarily restricts us to the tunnelling limit. It is not clear 'a-priori' how to take non-adiabatic effects into account or investigate how time delays are affected as we move towards the few-photon limit [28].
Alternatively, the concept of ionization time arises in another way within certain analytical approaches to strong field ionization. These include the widely used and broadly successful strong field approximation (SFA) -which we use here as an umbrella term for the related work of Keldysh [29], Perelomov, Popov and Terent'ev [30,31], Faisal [32] and Reiss [33] -as well as the more recently developed analytical R-matrix (ARM) method [34][35][36][37][38][39]. The latter is a fully quantum theory that is able to accurately describe the long-range interaction between the outgoing electron and the core -the absence of which is the main limitation of the SFA. In each of these approaches, it is necessary to integrate over a time variable that describes the instant at which the bound electron first interacts with the field.
In analytical approaches, the integral is typically evaluated using the saddle point method, and the real part of the complex saddle point solution t i = Re[t ι ] can then be interpreted as the most probable time of ionization -that is, the time at which the electron appeared in the continuum. The concept of saddle point time as ionization time also features in the Coulomb-corrected SFA approach (CCSFA) [40,41] (see also recent review [42] and recent update of the theory [43]), which takes the SFA ionization amplitude as its starting point Recently, the above idea was applied to analyse [3] the results of the attoclock experiments [6,7,14]. The basic premise here relies on using short pulses of circularly or nearly-circularly polarized light to induce ionization and deflect electrons in different directions, depending on their time of ionization [6,7]. In order to reconstruct the ionization time t i , it is necessary to establish its relationship to the observed angle and momentum at the detector (θ, p). This was possible in ARM using the saddle point analysis, yielding the following expression [3]: where φ C is the phase accumulated by the outgoing electron due to its interaction with the ionic core, I p is the ionization potential of the bound state from which the electron escaped, and ∆t env (θ, p) is a small correction due to the shape of the pulse envelope. Expression 3 has also been derived within the ARM method in Ref. [37]. Applying this relationship to the results of ab initio numerical experiments, it was possible to show that there are no time delays associated with the tunnelling process itself, at least for the simplest case of the hydrogen atom. However, delays ∆t C due to the electron-core interaction could be important and generally cannot be neglected, especially in the case when two-electron excitations lie close to the first ionization threshold [3].
In fact Eq.(3), which we can think of as the delay accumulated in a long-range potential compared to a short-range potential, coincides with an expression for ionization delay derived in an entirely different way. In [28], the idea of the Larmor clock -originally proposed in the context of tunnelling times -was applied to ionization. It was shown to reduce to the EWS time in the single-photon limit and reproduce Eq.(3) in the strong field regime.
Thus, the different definitions for times are not unrelated: there is a link between the saddle point-based ionization time, the Larmor tunnelling time and the EWS single-photon delay.
The attoclock, however, is not the only means by which ionization times can be measured in the strong field regime. Two-colour high harmonic spectroscopy experiments offer another elegant and powerful approach to this problem, using a weak probe pulse to perturb the ionization dynamics in a controlled way [4]. In light of this, a number of questions naturally arise. Are the times measured in HHG the same as those measured by the attoclock? If not, how are they related? How does the electron-core interaction imprint itself on the ionization and recombination times in this case?
Here, we address these questions by extending the saddle point analysis discussed above to HHG. After briefly reviewing the basic approaches for describing time in HHG in Section II, we present a general analysis that incorporates electron-core interaction in Section III. In Section IV, we discuss how the calculation can be implemented in practice using ARM. We present and discuss the results of such a calculation in Sections V,VI. Section VII concludes the work.

II. IONIZATION AND RECOMBINATION TIMES IN HHG: A BRIEF OVERVIEW
A. The classical model The simplest theoretical description of ionization time in the context of HHG comes from the classical model, which describes the process in terms of three steps: tunnel ionization, classical propagation in the continuum and recombination. In the standard classical, socalled three-step model [44], (see also pertinent refs. [45][46][47][48]) electron-core interaction is neglected during the classical propagation step, and it is assumed that the electron starts its continuum motion with a velocity of zero. If we also assume that recombination occurs when the electron returns to its starting point and relate its kinetic energy at this instant with the energy of the emitted photon, we obtain the following three equations: where v(t) = p + A(t) is the electron velocity, A is the vector potential that describes the laser field E = − ∂A ∂t , and E ω = N ω is the energy of the emitted photon. For any given photon energy E ω , the above equations can be solved for the ionization time t i , the recombination time t r and the canonical momentum p. As such, we can associate an ionization and recombination time to each harmonic number: this defines the mapping between experimental observable and time in this case. However, although excellent as a first approximation, this model is clearly rather crude. It artificially matches a quantum mechanical description of the ionization step with classical propagation, assumes very simple initial conditions for the electron's continuum motion and ignores the influence of the positively charged ionic core. It is not surprising, therefore, that a comparison of the above predictions with times obtained in high harmonic spectroscopy experiments revealed a notable discrepancy [4].
B. The strong field approximation (SFA) As mentioned in the introduction, the SFA offers a more sophisticated quantum approach to the problem of time in strong field ionization, which is based on saddle point analysis.
In essence, the key approximation of the SFA is to neglect the interaction between the electron and its parent ion after the instant ionization or, equivalently, to assume that the core potential is short range. Doing so, the induced dipole can be expressed as [49]: where S V is the Volkov phase and P is a prefactor that varies relatively slowly. The times t and t over which we integrate can be associated with ionization and recombination respectively.
The presence of a large phase, which leads to rapid oscillations of the integrand, makes it possible to evaluate the above integral using the saddle point method. It tells us that the integral will be accumulated predominantly in the vicinity of points where the derivative of the phase vanishes, which are defined by the saddle point equations [49,50]: We shall denote the solutions to these equations by t 0 ι , p 0 s , t 0 ρ . In fact, it is easy to check that Eq.(10) and Eq.(11) coincide exactly with Eq.(5) and Eq. (6)  Using this model, it was shown that agreement with times reconstructed from high harmonic spectroscopy experiments is notably improved [4]. However, what both the classical model and the SFA have in common is the neglect of the electron-core interaction throughout the electron's motion in the continuum. In the context of the attoclock, we have seen that this is not sufficient: ignoring the influence of the core potential on saddle point solutions leads to qualitatively and quantitatively incorrect results [3]. Motivated by this, let us now consider how the above solutions are modified if we allow for an electron-core interaction term.

III. INFLUENCE OF ELECTRON-CORE INTERACTION ON TIMES IN HHG:
A GENERAL ANALYSIS Within ARM, we know that the SFA expression for the ionization amplitude -the analogous quantity to the induced dipole given by Eq. (7) -is modified by the addition of a Coulomb phase e −iW C [34]. Let us now consider what a term of this nature would imply for times in HHG.
In particular, suppose that the integral in Eq.(7) now includes some additional factor e −iF (t ,t,p) , which accounts for the interaction between the active electron and the core.
Although ARM provides us with an explicit expression for F , which we shall return to in Section IV, we shall keep the analysis general for the time being. We assume only that F is small compared to S V and allow it to be complex in general. The real part of F then specifies the phase, which we shall denote by φ.
The saddle point equations (9)-(11) are modified as follows and since F is small compared to S V , we can search for solutions of the form (t 0 ι + ∆t ι , p 0 s + ∆p s , t 0 ρ +∆t ρ ), where (t 0 ι , p 0 s , t 0 ρ ) satisfy Eq. (9)- (11). Expanding about the SFA saddle points, keeping only first order terms in F , and using the chain rule to rewrite the derivatives, we arrive at the following result (see Appendix): In general, since F is complex, these corrections will also be complex. However, as before, we can assign an interpretation to their real parts: ∆t i = Re[∆t ι ] and ∆t r = Re[∆t ρ ] encode the delays due to the electron-core interaction imprinted upon ionization and recombination times respectively: There are a few things worth noting about the above results. First, notice that the expression for the correction to the recombination time, ∆t r = −∂φ/∂E ω , is reminiscent of the EWS time given by Eq.(1). Indeed, we can understand this if we recall that recombination is simply single-photon ionization run in reverse. Varying with respect to the photon energy is directly equivalent to varying with respect to the kinetic energy of the recombining electron.
Second, note that the correction to ionization time in HHG, has an extra term compared to its counterpart in strong field ionization given by Eq. (3).
Subtracting the above expressions for ∆t r and ∆t i yields In other words, the derivative of the phase with respect to the ionization potential in HHG tells us how much more (or less) time the electron will spend in the continuum before it recombines, as a consequence of electron-core interaction.

IV. COULOMB CORRECTIONS IN HHG USING ARM
Keeping the function F unspecified, this is as far as we can go -if we want to determine the corresponding time delays in practice, we must evaluate F explicitly. Luckily, this is precisely what ARM allows us to do.
In particular, ARM tells us that in strong field ionization e −iF is replaced by p is the electron momentum measured at the detector, T is the time of observation, U (r) is the core potential and r s is the Coulomb-free electron trajectory, W SFI C is the electron action assosiated with the Coulomb-laser coupling [53,54]. The lower limit of the integral in Eq. (20), t κ = t ι − i/κ 2 , is determined by the boundary-matching procedure for the outgoing electron [34,37].
The purpose of the matching procedure is to avoid using W SFI C outside the limits of its applicability range, i.e. close to the core. This procedure enables a smooth merger of the asymptotic tail of bound electron wave-function with the quasicalssical wave-function of the escaping electron, driven by the laser field. The dominant fraction of electrons liberated by strong field ionization arrive to the detector without revisiting the core and therefore the matching is done only once, when electron departs from the core. However, in our case we have to consider these, not-very-likely return events, since only they give raise to HHG. It means that we need to perform the matching procedure once again, when electron returns back to the core, linking the phase accumulated due to Coulomb-laser coupling between ionization and recombination to the field-free continuum solution for the returning electron.
Fortunately, an equivalent boundary-matching problem has already been solved. In [17], single photon ionization in the presence of a probing infrared field was analysed in the context of the attosecond streak camera. There, a matching argument was used to show that the effective starting point for an electron trajectory with initial velocity v 0 is given by where γ E is Euler's constant and Noting that recombination in HHG is simply the reverse of this process (that is, the emission (rather than absorption) of a photon in the presence of an infrared field), we can apply this result directly to determine t end . In particular, we now think of r 0 as the end point of our electron trajectory and set r 0 = r s (t end ). The corresponding velocity v 0 is the velocity at recombination, For any given photon energy E ω , we then have which can be used to solve for t end , using Eq.(22)- (25) to evaluate r 0 (v r ).
In doing so, it should be noted that r s (t) is complex in general for real times, whereas r 0 is always real. Consequently, in order to satisfy the above equation, we must allow t end to be complex as well. This tells us that, in contrast to ionization, our integral for W C no longer ends on the real axis: both start and end points are now complex. The generalization, however, is straightforward. When describing Coulomb effects in strong field ionization, the integration contour was chosen in two parts: first, down from t κ to Re[t ι ] = t i on the real axis, and then along the real axis up to time T [34,38]. These two legs were interpreted in terms of tunnel ionization and the electron's motion in the continuum respectively, following PPT [31]. For HHG, we simply add a third leg: down from Re[t end ] on the real axis to t end (see Fig.1).
We note that, in general, the interpretation of the real part of the complex saddle point as In the long wave-length limit, the attoclock observable -the so-called off-set angle -is equal to the real part of the saddle point [28]. What's more, in Ref [28] we have introduced an alternative method for deriving the ionization time. This method uses neither the concept of tunneling, nor the concept of trajectories, nor does it rely on the saddle method. The result for the ionization time remains the same.
To extend Eq. 20 to describe HHG, there are only two major changes we need to make: 1. the momentum at the detector p should be replaced by the saddle point solution p s (E ω ), and 2. the observation time T should be replaced by the time t end . That is, where We can readily determine the value of p s (E ω ) from Eq.(9)-(11).

V. RESULTS: COULOMB TIME DELAYS IN HHG
Having determined t end and chosen a contour, we have all the ingredients we need in order to evaluate the correction W C as given by Eq. (27). In itself, this tells us the first order effects on HHG spectra due to the electron-core interaction. However, as we saw in Section III, we need one further step to learn about times: we must differentiate the Coulomb phase W C with respect to I p and E ω to find the corrections to the saddle point solutions. In practice, this can be done numerically by evaluating W C for two or more closely spaced values of I p , We shall now compare these results with the times reconstructed from high harmonic spectroscopy measurements [4]. The reconstruction procedure is described in Ref. [4] (see also SI of [4]) and analysied in detail in Ref. [19]. For the benefit of the reader here we briefly outline the main idea of the reconstruction.
Suppose HHG is driven by a strong laser field at the fundamental frequency ω and is described by the vector potential: A ω (t) = e x A 0 sin(ωt). The idea of detection of the ionization time is very simple. It relies on the application of an additional perturbative control field at the frequency 2ω, A 2ω (t) = e y A 0 sin(2ωt+φ) phase locked to the fundamental field, and polarized in orthogonal direction to the fundamental field. This control field modulates HHG yield as a function of the relative phase φ between the two fields, i.e. as a function of the two-color delay φ. This field kicks the electron in lateral direction, once it leaves the bound atomic state and exits from the tunneling barrier at the ionization time t i . The magnitude and the sign of this kick is controlled by the two-color delay. The HHG signal maximizes for a specific two-color delay, φ max , when the lateral kick is equal to zero, i.e. when the electron displacement between ionization and recombination is minimized. To a good approximation, this requires the vector potential of the control field at the moment of ionization to be close to zero, A 2ω (t i ) = e y A 0 sin(2ωt i +φ max ) ≈ 0, thus the ionization time is Corrections accounting for the kick during tunneling are accounted for in the full reconstruction. Note that the second harmonic field also breaks the symmetry in electron dynamics between the two consecutive laser half cycles, leading to the generation of even harmonics. The asymmetry maximises for those values of φ, for which the lateral velocity of the electron upon recombination is maximal. Recombination times are reconstructed following the maximal HHG signal for even harmonics, as suggested and performed in Ref.
[4] and augumented in Ref. [55] by including complex, rather than real SFA recollision times.  attoseconds, decreasing slightly with N . The recombination times are notably less affected overall, though they display a stronger dependence on the harmonic number: the shift in this case is between ∼ 5 and ∼ 19 attoseconds, again decreasing with N . Putting these two facts together, we see that the total amount of time the electron spends in the continuum (given by ∂φ/∂I p ) increases by ∼ 18-28 attoseconds.
Comparing our results with times reconstructed from high harmonic spectroscopy measurements, we find that ARM offers a notable improvement over the SFA, where electron-core interaction was neglected. Although the corrections to ionization and recombination times are only of the order of tens of attoseconds, they are nevertheless clearly within the resolution of current state-of-the-art HHG experiments.
We note that our method also allows us to analyse the Coulomb corrections to the imaginary times. This has been done in Ref. [13], where imaginary ionization times where reconstructed from the experimental measurements.

VI. WHY COULOMB TIME DELAYS ARE SO SMALL?
We have derived the times in HHG using the saddle point method and iterative procedure for finding saddle point solutions, thus the derived times correspond to solutions of saddle point equations. Iterations treat the electron action associated with the motion in the Coulomb field (so called Coulomb-laser coupling term) as a perturbation to the action due to the laser field. The zero-order iteration yields the well-known SFA (Strong Field Approximation) results for the times, when the Coulomb-laser coupling is neglected. To obtain a small parameter of the iterative procedure, we will compare the results of subsequent iterations: |∆t ι |/ |∆t 0 ι |, where |∆t ι | is the complex ionization time in the first order and |∆t 0 ι | is the complex ionization time in the SFA (zero-order iteration). The Coulomb correction to ionization time in HHG has two components ∆t ι = t SF ι +t SF W S : First, we discuss the small parameter for the first term. To obtain simple analytical results for the small parameter we consider the low-frequency (tunnelling) limit. In this case |∆t 0 is the well-known tunnelling time (the imaginary component of the saddle point solution), E 0 is the strength of the laser field. The real part of the saddle point can be chosen to be zero in the tunnelling limit. Now we need to discuss the absolute value of complex first order correction to saddle point time. One can obtain a very simple analytical expression for the real part of this correction: t SF ι ≈ Z/I 3/2 p , where Z is the charge of the core, I p is ionization potential (see derivation in Refs [36,37]). Note that the absolute value of the saddle point solution is determined by its real component, because the correction to the imaginary component is zero in the first order iteration, in the tunnelling limit. We obtain p is the effective principal quantum number of a given quantum state and ImS SF A is the imaginary part of the electron action associated with its dynamics in the laser field only. Note that 2ImS SF A = 2Ec 3E 0 is the famous exponent that appears in the equation for the tunnelling rate: Γ = exp(−2ImS SF A ). It is well known that this formula is only applicable if ImS SF A >> 1. Thus, we have our small parameter ζ 1 = n * ImS SF A 2 3/2 3 . The quantum number n * characterizes the action of the electron in the bound state, while S SF A characterizes the action of the electron, driven by the laser field. If the action due to the laser-driven dynamics exceeds the action in the ground state (which is usually the case in strong field ionization), the ionization time |t SF ι | will remain small. This condition is essentially the same as the condition of applicability of most of the Coulomb-corrected SFA theories.
The strong field ionization time |t SF ι | is only a first part of the expression for ionization time in HHG: ∆t ι = t SF ι + t SF W S . We now discuss the second contribution, which is also equivalent to the first order correction to recombination time. t SF W S = − ∂F ∂Eω is the analogue of the well-known Wigner-Smith ionization time, albeit the phase F entering this expression is not the field-free scattering phase, but is the phase that includes the effects of the laser field and describes the Coulomb-laser coupling. As discussed in the previous section, |t SF W S | is smaller than |t SF ι |. In the case |t SF W S | we face an interesting effect of partial cancellation of the purely Coulombic (truly Wigner-Smith field free delays) with the delays induced by the laser field. This effect is described in detail in Ref. [17]. The small parameter in this case is the shift of electron momentum ∆q due to Coulomb-laser coupling (see [54]) with respect to velocity of the returning electron: ζ 2 = ∆q vr , where v r = 2 1/2 (E ω − I p ) 1/2 , E ω is the harmonic energy. Using the expression for the momentum shift ∆q due to the Coulomb-laser coupling from [54], we find ζ 2 = Z|E(tr)| |vr||2 3/2 (Eω−Ip) 3/2 |) . Note that the expressions for small parameters ζ 1 and ζ 2 have similar structure: ζ 1,2 = ZE/v 4 , where for ζ 1 we should use the electron velocity in the bound state |v| = (2I p ) 1/2 and the laser field at the moment of ionization, but for ζ 2 we need to use the electron velocity v r = 2 1/2 (E ω − I p ) 1/2 and the laser field at the moment of recombination. Thus, in general, Coulomb corrections to recombination times are smaller than Coulomb correction to ionization times due to larger velocity of the returning electron than departing electron and smaller value of instantaneous laser field at ionization than at recombination for majority of electron trajectories, except the ones corresponding to low harmonic numbers.
We note that small values of the Coulomb shifts in HHG times make their detection challenging. In particular, they have not been seen in numerical simulations of Ref. [55].

VII. CONCLUSIONS AND OUTLOOK
We have shown how ionization and recombination times in HHG are modified when the long-range interaction between the active electron and the ionic core is taken into account.
The resulting corrections are closely related to the delays in strong field and single photon ionization respectively, though they are not identical. In particular, the expression for ionization delay in HHG, ∆t HHG i = −∂φ/∂I p − ∂φ/∂E ω , contains an additional term (with respect to strong field ionization time ∆t SF I i = −∂φ/∂I p , derived earlier in [3,28,37]) that factors in the recombination step. The origin of the "additional" ionization delay is related to different measurement protocols used in HHG and strong-field ionization experiments.
While the latter detects photoelectrons, the former detects photons, where both ionization and electron recombination precede the measurement. Therefore, in case of HHG the "delayline" on the way to the detector is associated not only with ionization (which includes propagation in the continuum), but also with recombination, leading to production of XUV light that is further detected to extract ionization delays.
Comparing the predictions of the ARM theory -in which the Coulomb interaction is accounted for -with times measured in high harmonic spectroscopy experiments, we find that the agreement is excellent. The fit is visibly better than for the SFA, where such effects are omitted. Thus, although relatively small, we can conclude that the electroncore interaction leaves a measurable and distinct signature on times in HHG. As such, it should be taken into consideration when calibrating attosecond recollision-based pump-probe experiments and interpreting experimental data.

VIII. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of Deutsche Forschungsgemeinschaft, project SM 292/2-3. We thank H. Sofier and N. Dudovich for providing the experimental data from [4], shown in Figure 2. We thank V. Serbinenko, F. Morales and M. Ivanov for discussions.

Appendix A: Appendix
In this section we present the derivation of Eq.(15)- (16). To do so, it will be convenient to introduce the following vectors and the vector-valued function The gradient ∇ will be defined as a row vector, and ∇ s 0 F will occassionally be used as shorthand for ∇ s F (s 0 ).
Using this notation, the saddle point equations (12)- (14) can be expressed succinctly as where we have added a constant X → 0 to the LHS of the second saddle point equation for convenience in both cases.
If we now expand the solution to Eq.(A6) about s 0 , we obtain (to first order in ∆s and where J s is the Jacobian with derivatives taken with respect to s. Using Eq.(A7) and the fact that the Jacobian of a gradient is the Hessian, we have Since the Hessian is symmetric and invertible (in this case), we can rewrite this as In principle, if we could evaluate ∇ s 0 F , we would be done. However, we do not have direct control over the value of the complex saddle point s when we do the calculation numerically, which makes this a difficult quantity to work with. Instead, what we can do is vary the parameters E and see how F changes as a result -this makes it possible to evaluate ∇ E F numerically. With this in mind, we would like to rewrite Eq.(A10) in terms of ∇ E F instead of ∇ s F .
To do so, we note that Eq.(A7) establishes a functional relationship between s 0 and E. In principle, we could solve this equation to find s 0 (E). Taking the gradient of F with respect E and applying the chain rule gives so and so (again making use of the fact that the Jacobian of a gradient is the Hessian), Taking the inverse and substituting this into Eq.(A12) gives Finally, this allows us to rewrite Eq.(A10) as which simplifies to In our case, J E f is very simple: and [ This allows us to write down a solution for ∆s in terms of ∇ E F :