Looking inside the tunnelling barrier: II. Co- and counter-rotating electrons at the 'tunnelling exit'

In the ARM theory we include the effects of Coulomb potential $U(r)=-Z/r$ in the outer region up to the first order in Z, in the wave function's exponent (action). In the inner region, the Coulomb effects are included fully. It leads to the shift of the saddle points in the time-domain quantum integrals over the ionisation times. In particular the complex ionisation time becomes ${t}^{* }\equiv {t}_{s}^{c}\equiv {t}_{s}+{\rm{\Delta }}{t}_{s}^{c}$ [64, 66, 67, 70]. Since the PPT times, this time is associated with the beginning of tunnelling. Here t_s is the SFA ionisation time for a short-range potential, while ${\rm{\Delta }}{t}_{s}^{c}$ is the Coulomb correction to this time. For the same final electron momentum at the detector, the trajectory starts earlier w.r.t to the SFA (short-range) case. The correction ${\rm{\Delta }}{t}_{s}^{c}$ is given by equation (13) in appendix A of [71], which we rewrite here:

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{s}^{c}=-\displaystyle \frac{{{\bf{v}}}_{{\bf{k}}}({t}_{s})\cdot {\rm{\Delta }}{{\bf{v}}}^{c}}{{{\bf{v}}}_{{\bf{k}}}({t}_{s})\cdot {{\bf{E}}}_{L}({t}_{s})},\end{eqnarray} \tag{ 1 }$$

or, in the equivalent form [66, 67, 70]:

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{s}^{c}=-{\left.\displaystyle \frac{{{d}{W}}_{C}({\bf{k}},{t}_{s})}{{{d}{I}}_{p}}\right|}_{\kappa =\mathrm{const}},\end{eqnarray} \tag{ 2 }$$

where $\kappa =\sqrt{2{I}_{p}}$ and the phase ${W}_{C}({\bf{k}},{t}_{s})$ is the Coulomb correction to the SFA action in the ARM theory:

$$\begin{eqnarray}&&{W}_{C}({\bf{k}},{t}_{s}^{c})=-{\int }_{{t}_{\kappa }}^{t}{d}{t}\,U({{\bf{r}}}_{L}^{s}(t)),{t}_{\kappa }={t}_{s}-{i}{\kappa }^{-2}.\end{eqnarray} \tag{ 3 }$$

This term describes the Coulomb-laser coupling, and amounts to integrating the Coulomb potential along the trajectory ${{\bf{r}}}_{L}^{s}(t)$ of free electron in the laser field:

$$\begin{eqnarray}&&{{\bf{r}}}_{L}^{s}(t)={\int }_{{t}_{s}}^{t}{\bf{k}}+{\bf{A}}(\xi ){d}\xi .\end{eqnarray} \tag{ 4 }$$

Because the Coulomb-laser coupling term originates from the outer region, any further correction to the trajectory would involve second and higher order terms in Z, which are much smaller for direct electrons and field frequency $\omega \ll {I}_{p}\equiv {\kappa }^{2}/2$, where I_p is the ionisation potential.

The initial velocity at complex ionisation time t_s^c enters the expression for the ionisation amplitudes [67] and rates (equations (4) and (5) in [71]) via the term ${e}^{im{\phi }_{v}^{c}({t}_{s}^{c})}$, where ${\phi }_{v}^{c}({t}_{s}^{c})$ is the complex tunnelling angle, $\tan {\phi }_{v}^{c}({t}_{s}^{c})=\tfrac{{v}_{\parallel }({t}_{s}^{c})}{{v}_{\perp }({t}_{s}^{c})}$—the angle at which the electron enters the classically forbidden region. Here ${v}_{\parallel }({t}_{s}^{c})$ and ${v}_{\perp }({t}_{s}^{c})$ are the initial velocities parallel and perpendicular to field direction at time t_s^c. The perpendicular velocity is positive if it has positive projection on the final momentum. Finally, m is the magnetic quantum number of the ionising bound state. For orbitals with m = 0, this Coulomb correction to momentum has no impact on final observables at the detector, i.e. the photoelectron spectrum.

For ${\ell }\ne 0$ and $m\ne 0$ bound states, the Coulomb-laser coupling leads to a momentum shift, ${\rm{\Delta }}{{\bf{v}}}^{c}(T)$ [67]:

$$\begin{eqnarray}{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T) & = & -{\displaystyle \int }_{{t}_{Q}}^{T}{d}{t}\,{\rm{\nabla }}U({{\bf{r}}}_{L}^{s}(t))\\ & & -{{\bf{v}}}_{{\bf{k}}}({t}_{s}),{t}_{Q}={t}_{s}-{i}\displaystyle \frac{Q}{{\kappa }^{3}},\end{eqnarray} \tag{ 5 }$$

${{\bf{v}}}_{{\bf{k}}}^{2}({t}_{s})=-2{I}_{p}$ which modifies the initial velocity at time t_s^c:

$$\begin{eqnarray}&&{\bf{v}}({t}_{s}^{c})={\bf{k}}+{\bf{A}}({t}_{s}^{c})-{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T).\end{eqnarray} \tag{ 6 }$$

Time t_Q is an outcome of the matching procedure. Here Q is the effective charge of the core which describes the behaviour of the bound wave function in the classically forbidden region, ${I}_{p}=Q/2{\nu }^{2}$, where $\nu =\sqrt{Q/2{I}_{p}}$ is the effective quantum number. For short-range potential Q = 0, ν = 0. Note that ${\mathrm{lim}}_{Q\to 0}{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T)\to 0$ (see appendix 2, [67]). The physical meaning of this expression is as follows: the electron momentum ${\bf{k}}$ is fixed at the detector. The first term ${\bf{k}}+{\bf{A}}({t}_{s}^{c})$ describes the SFA velocity at the Coulomb-corrected complex ionisation time t_s^c. The term ${\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T)$ describes the momentum shift accumulated due to the Coulomb-laser coupling on the way from the origin to detector. Because the final momentum is fixed to be ${\bf{k}}$, the initial velocity subtracts ${\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T)$ to compensate this 'future' change in velocity.

The above expressions are valid for arbitrary field geometries (for direct electrons), all electron momenta, and for all values of the Keldysh parameter γ within the applicability of the theory. The latter requires that real transitions to excited states are negligible. In practice, this of course limits the region of available frequencies and laser intensities (usually ω ≪ κ²/2).

In the previous two subsections we defined the complex ionisation time and the electron velocity at this time. Both quantities directly enter the ARM expression for ionisation amplitudes and contribute to the photoelectron spectra.

Now we aim to introduce auxiliary tool—quantum and classical trajectories. They will allow us to interpret observables in terms of the Newtonian mechanics. Both quantum (complex) and classical (real) trajectories are evaluated in the first order with respect to Coulomb potential, to be consistent with the ARM approach.

The quantum trajectory starts at real coordinate ${r}_{Q}\,=Q/{\kappa }^{2}$ (along the axis defined by the laser field at time ${\mathfrak{R}}[{t}_{s}^{C}]$, but in the opposite direction) at the time ${t}_{Q}^{c}={t}_{Q}+{\rm{\Delta }}{t}_{s}^{c}$ with the velocity:

$$\begin{eqnarray}&&{\bf{v}}({t}_{Q}^{c})={{\bf{v}}}_{{\bf{k}}}({t}_{Q}^{c})-{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T).\end{eqnarray} \tag{ 7 }$$

The velocity at any time t > t_Q is:

$$\begin{eqnarray}&&{\bf{v}}(t)={{\bf{v}}}_{{\bf{k}}}(t)-{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T)\\ &&\quad -{\displaystyle \int }_{{t}_{Q}}^{t}{d}{t}\,{\rm{\nabla }}U({{\bf{r}}}_{L}^{s}(t))-{{\bf{v}}}_{{\bf{k}}}({t}_{s})\equiv {{\bf{v}}}^{{FW}}(t).\end{eqnarray} \tag{ 8 }$$

The coordinate r_Q, time t_Q and the respective initial velocity at this time are known due to the matching procedure. These quantities fully specify the quantum trajectory:

$$\begin{eqnarray}&&{{\bf{r}}}^{Q}(t)={{\bf{r}}}_{Q}+{\int }_{{t}_{Q}^{c}}^{t}{{\bf{v}}}_{{\bf{k}}}(\tau )d\tau +{\int }_{{t}_{Q}}^{t}{{\bf{v}}}^{c}(\tau )d\tau ,\end{eqnarray} \tag{ 9 }$$

where the Coulomb velocity is ${{\bf{v}}}^{c}(\tau )\equiv -{\rm{\Delta }}{{\bf{v}}}^{c}({t}_{Q},T)\,-{\int }_{{t}_{Q}}^{t}{dt}\,{\rm{\nabla }}U({{\bf{r}}}_{L}^{s}(t))-{{\bf{v}}}_{{\bf{k}}}({t}_{s})$.

Importantly, the ARM -trajectory ${{\bf{r}}}_{L}^{s}(t)$ which enters the argument of the Coulomb force $-{\rm{\nabla }}U({{\bf{r}}}_{L}^{s}(t))$ in the integral (equation (8)) is always complex, as discussed in the introduction. It remains complex, with constant imaginary part, even after the exit from the barrier during the real time evolution. It has implications for our ability to introduce an equivalent classical trajectory.

Indeed, taking into account the explicit expression for the velocity ${\bf{v}}(t)$ (equation (5)), we can rewrite (equation (8)) in the equivalent form:

$$\begin{eqnarray}&&{\bf{v}}(t)={{\bf{v}}}_{{\bf{k}}}(t)+{\int }_{t}^{T}{dt}\,{\rm{\nabla }}U({{\bf{r}}}_{L}^{s}(t))\equiv {{\bf{v}}}^{{BW}}(t).\end{eqnarray} \tag{ 10 }$$

The last equation simply describes back propagation of the final momentum ${\bf{k}}$ from time T at the end of the pulse to arbitrary time t. The velocities in equations (equations (8) and (10)) propagated forward from time t_Q and backward from time T will coincide at any point t if and only if the back propagation is performed along the complex trajectory ${{\bf{r}}}_{L}^{s}(t)$. This is the case for the quantum trajectory.

If we neglect the imaginary part of the trajectory ${{\bf{r}}}_{L}^{s}(t)$ during back propagation from the detector, as we have to do if we want to introduce a purely classical trajectory, we will have a velocity off-set between the inevitably quantum forward propagation (i.e. using complex ARM trajectories) from the 'under the barrier' region and the desirably classical backward propagation (i.e. using only real parts of ARM trajectories) back from the detector.

Nevertheless, the classical trajectory is also uniquely defined. Its initial coordinate is defined by the real part of the coordinate on quantum trajectory at the exit time: ${\mathfrak{R}}[{{\bf{r}}}^{Q}({t}_{{exit}})]$. But the exit velocity is defined by ${{\bf{v}}}^{{BW}}({t}_{{exit}})$ propagated backwards from the end of the laser pulse along the real trajectory. Such back propagation is necessary to make sure that the classical trajectory propagated from the 'barrier exit' to the detector will have correct final momentum ${\bf{k}}$ at time T. thus, the classical trajectory is uniquely derived from the quantum trajectory as follows:

$$\begin{eqnarray}&&{{\bf{r}}}^{C}(t)={\mathfrak{R}}[{{\bf{r}}}^{Q}({\mathfrak{R}}[{t}_{s}^{c}])]+{\int }_{{\mathfrak{R}}[{t}_{s}^{c}]}^{t}{{\bf{v}}}^{C}(\tau )d\tau ,\end{eqnarray} \tag{ 11 }$$

where ${t}_{{exit}}\equiv {\mathfrak{R}}[{t}_{s}^{c}]$, and the classical velocity is obtained by propagation in real time and space:

$$\begin{eqnarray}&&{{\bf{v}}}^{C}(t)={{\bf{v}}}_{{\bf{k}}}(t)+{\int }_{t}^{T}{dt}\,{\rm{\nabla }}U({\mathfrak{R}}[{{\bf{r}}}_{L}^{s}(t)]).\end{eqnarray} \tag{ 12 }$$

To specify how the Coulomb potential leads to coupling between the real and the imaginary parts of the ARM trajectory and how neglecting the imaginary part of the exit coordinate we remove this coupling, consider the analytical continuation of the Coulomb potential to the complex plane. The procedure has been described in detail in [65]. Here we recall the results to make the discussion more specific. For a real vector ${\bf{r}}$, we have $U({\bf{r}})=-Z/\sqrt{{\bf{r}}\cdot {\bf{r}}}$. For a complex vector, ${\bf{r}}+i{\boldsymbol{\rho }}$, we have

$$\begin{eqnarray}U({\bf{r}}+{\bf{i}}{\boldsymbol{\rho }}) & = & \displaystyle \frac{-Z}{\sqrt{({\bf{r}}+i{\boldsymbol{\rho }})\cdot ({\bf{r}}+i{\boldsymbol{\rho }})}}\\ & = & \displaystyle \frac{-Z}{\sqrt{{r}^{2}-{\rho }^{2}+2i{\bf{r}}\cdot {\boldsymbol{\rho }}}},\end{eqnarray} \tag{ 13 }$$

where $r=| {\bf{r}}{| }^{2}$ and $\rho =| {\boldsymbol{\rho }}{| }^{2}$. We choose the branch cut of the complex square root to be infinitesimally above the negative real axis to ensure that our contour never crosses the branch cut. The real and imaginary parts of the Coulomb potential are:

$$\begin{eqnarray}U & = & -\displaystyle \frac{Z}{\sqrt{2({a}^{2}+{b}^{2})}}(\sqrt{\sqrt{{a}^{2}+{b}^{2}}+a}\\ & & -i\ \mathrm{sgn}(b)\sqrt{\sqrt{{a}^{2}+{b}^{2}}-a}),\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&a={r}^{2}-{\rho }^{2}\quad \mathrm{and}\quad b=2{\bf{r}}\cdot {\boldsymbol{\rho }},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&\mathrm{sgn}(b)=\left\{\begin{array}{l}-1,\quad \mathrm{for}\ b\leqslant 0\\ +1,\quad \mathrm{for}\ b\gt 0.\end{array}\right.\end{eqnarray} \tag{ 16 }$$

The real and imaginary parts of the Coulomb force ${{\bf{F}}}_{C}(t)=-Z{{\bf{r}}}_{L}(t)/{r}_{L}^{3}(t)$:

$$\begin{eqnarray}&&{{\bf{F}}}_{C}(t)=-{\rm{\nabla }}U({\bf{r}}={\bf{r}}+i{\boldsymbol{\rho }})=-Z\displaystyle \frac{{\bf{r}}+i{\boldsymbol{\rho }}}{\parallel {\bf{r}}+i{\boldsymbol{\rho }}{\parallel }^{3}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&=\,-\displaystyle \frac{Z({\bf{r}}+i{\boldsymbol{\rho }})}{{[2({a}^{2}+{b}^{2})]}^{3}}({[\sqrt{{a}^{2}+{b}^{2}}+a]}^{3/2}\\ &&\quad -\,3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}-a}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&-\,i\mathrm{sgn}(b)\{3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}+a}-{[\sqrt{{a}^{2}+{b}^{2}}-a]}^{3/2}\}),\end{eqnarray} \tag{ 19 }$$

describe the evolution of the real and imaginary parts of quantum trajectory ${{\bf{r}}}^{Q}(t)$.

Separating real and imaginary contributions to Coulomb force F_C, we get:

$$\begin{eqnarray}{{\bf{F}}}_{C} & = & -\displaystyle \frac{Z}{{[2({a}^{2}+{b}^{2})]}^{3}}[({[\sqrt{{a}^{2}+{b}^{2}}+a]}^{3/2}\\ & & -3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}-a}){\bf{r}}+\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\boldsymbol{\rho }}\mathrm{sgn}(b)(3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}+a}-{[\sqrt{{a}^{2}+{b}^{2}}-a]}^{3/2})-\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&i\mathrm{sgn}(b)\{(3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}+a}-{[\sqrt{{a}^{2}+{b}^{2}}-a]}^{3/2}){\bf{r}}-\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\boldsymbol{\rho }}({[\sqrt{{a}^{2}+{b}^{2}}+a]}^{3/2}-3b\sqrt{\sqrt{{a}^{2}+{b}^{2}}-a})\}],\end{eqnarray} \tag{ 23 }$$

and we see the contribution of ${\boldsymbol{\rho }}(t)$ and $b(t){\bf{r}}(t)$ to the real part of quantum velocity.

The coupling between the real and the imaginary parts of the ARM trajectory due to the Coulomb force is obvious. Even though the imaginary part of the ARM trajectory is constant after the barrier exit, the real part evolves and therefore both b and a are the functions of time along the ARM trajectory. Thus, the evolution in real time is affected by the presence of the imaginary exit coordinate. Depending on the sign of the parameter $b(t)=2{\bf{r}}(t)\cdot {\boldsymbol{\rho }}(t)$, the Coulomb force either increases or decreases in its attractive strength.

The imaginary exit coordinate is present in quantum trajectory, but absent in the classical trajectory. This mismatch leads to the off-set in their real velocities at the exit point. The presence of the imaginary component of the ARM trajectory is due to the non-adiabaticity of the electron dynamics. Thus, in circularly polarised fields the majority of electrons released from s-orbitals will follow the classical trajectory (classical and quantum trajectories coincide in this case), but the majority of co- and counter-rotating electrons (released from the orbitals with $m\ne 0$) will follow quantum trajectories with non-zero imaginary parts. In this case, the degree of approximation that classical trajectory offers depends on the degree of non-adiabaticity of electron dynamics. We will illustrate this general picture below.

We first consider a few-cycle, circularly polarised laser pulse, defined via its vector potential:

$$\begin{eqnarray}&&{\bf{A}}(t)=-{{ \mathcal A }}_{0}f(\omega t)(\cos \omega t\,\hat{{\bf{x}}}+\sin \omega t\,\hat{{\bf{y}}}),\end{eqnarray} \tag{ 26 }$$

where ${{ \mathcal A }}_{0}$ is the amplitude. The envelope profile used here is $f(\omega t)={\cos }^{4}(\omega t/2{N}_{e})$ with N_e = 2. The peak field strength is defined as ${{ \mathcal E }}_{0}={{ \mathcal A }}_{0}\omega$. The most dominant ionisation event happens near t = 0, which maps onto a peak at ϕ_k = 0 in the SFA theory. Long-range interactions lead to an offset in this mapping between the ionisation time and the momentum peak, which ARM accurately predicts for s-orbitals [66, 70]. The off-set angle is equal to $\omega {\mathfrak{R}}[{t}_{s}^{c}]$ [66, 70].

The velocities at the exit point (time $t={\mathfrak{R}}[{t}_{s}^{c}]={t}_{i}^{c}$) are shown in figure 1, as a function of the final (detected) energy and angle of the photoelectron. We use the example of Yb III, interacting with ${{ \mathcal E }}_{0}=0.065\,\mathrm{au}$, λ = 800 nm, right circular polarised field. Superimposed on these distributions are the contour levels of the final photoelectron momentum distribution at the detector. Red solid curves are for f₋₃ (counter-rotating) orbital, which has the highest ionisation rate in a right-circularly polarised field. The blue dashed contours are for the f₊₃ (co-rotating) orbital, which has weaker photoelectron signal (see the companion paper I). Contours for both ${\rm{A}}{\text{}}\mathrm{RM}$ (upper rows) and SFA (lower rows) are shown.

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We begin with the analysis of the SFA results, see figure 1 (SFA, v_⊥, SFA v_∥). The velocity v_⊥ orthogonal to the instantaneous field at the time of exit is known to define the final energy of the photoelectron at the peak of the distribution (see e.g. [72]). Let us follow the contours, which specify the angle and energy dependence of co- and counter-rotating electrons, superimposed on the colour-coded background representing the 'exit' value of v_⊥ as a function of the final angle and energy of the photoelectron (see figure 1(d)). In agreement with our expectations, we find larger v_⊥ for the co-rotating electron at the peak of its momentum distribution, than for the counter-rotating electron at the peak of its momentum distribution. This non-adiabatic effect is discussed e.g. in [72]. It is responsible for the fact that counter-rotating electron (blue dashed contour) has the peak of its distribution at higher energy than the co-rotating electron (red solid contour). In the quasistatic limit, the perpendicular 'exit' velocity v_⊥ for both co- and counter-rotating electrons tends to zero, removing the difference between their spectra and ionisation rates.

The longitudinal 'exit' SFA velocity v_∥ (see figure 1(c)) varies as a function of the detection angle. It is zero for zero detection angle (where the maxima of both photoelecton distributions are located) and is exactly opposite for positive and negative angles. For positive detection angles, it is directed away from the core, while for negative detection angles, it is directed towards the core. This angular dependence reflects the non-adiababticity of electron dynamics due to the fast change of the magnitude of the laser field in the short pulse. In this respect, the situation becomes similar to linearly polarised fields, where non-adiabaticity of the electron response maps into non-zero longitudinal velocity for all ionisation times within the optical cycle, except for the ionisation time at the peak of the laser field.

This (envelope-related) non-adiabatic feature of the 'exit' value of v_∥ already tells us that when the Coulomb potential is included, the trajectories with the SFA velocities directed towards the core can be more efficiently decelerated by the Coulomb field (and may even be trapped), than the trajectories with the SFA velocities directed away from the core. As a result, it is clear that when the Coulomb field is included, the final distributions will be more distorted (with respect to their SFA counterparts) for positive angles and less distorted for the negative angles. This is exactly what we see in figure 1(a), where the final momentum distributions are shown. They are superimposed on top of the colour-coded 'exit' velocities.

Overall, the main effect of the Coulomb potential is to 'skew' the final and the initial momentum distributions while mostly preserving the range of the 'exit' velocities arising in the short-range potential. This observation is in agreement with the standard assumptions of the Coulomb-corrected three-step models.

The range of variation of the transverse velocity component is reduced in ARM, compared to SFA, but is shifted to larger positive velocities. This is the influence of ${\rm{\Delta }}{{\bf{v}}}^{c}({t}_{i})$. The SFA transverse velocity sharply depends on energy, whereas the ARM velocity spreads out gradually. Long-range interactions therefore lead to bunching of the trajectories just after emerging into the continuum.

We next consider initial coordinate distribution at the barrier exit, figure 2, where the coordinates parallel and perpendicular to the electric field direction are shown. These values of the exit coordinates correspond to quantum trajectories. In both SFA and ARM case, counter-rotating (f₋₃) electrons have smaller parallel component of the coordinate than the co-rotating electron (f₊₃) (figures 2(a) and (c)). Compare this to the perpendicular coordinate distribution (figures 2(b) and (d)): counter-rotating electrons are liberated farther from the core in terms of their perpendicular displacement, than the co-rotating electrons.

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The extent of the parallel displacement is much stronger than perpendicular displacement, and thus, overall, the co-rotating electrons are liberated at a larger distance from the core than the counter-rotating electrons.

With this information on initial velocity and coordinates, we can develop a schematic of electron trajectories after tunnelling through the barrier.

The corresponding SFA quantum orbits and long-range quantum and classical trajectories for majority of electrons are shown in figure 3, for the time duration corresponding to two cycles of the laser field after ionisation (the laser field switches off after one cycle). The curve with red solid triangles represents the SFA trajectory for f₋₃ orbital and the curve with the blue dashed triangles is the SFA trajectory for the f₊₃ orbital. The two trajectories are very close to each other, with f₊₃ ending up slightly farther from f₋₃, due to its higher momentum, as expected.

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When the long-range Coulomb interaction is switched on, however, we see a significant difference between the trajectories for the f₋₃ and f₊₃ electrons, travelling in very different directions (as one would expect from the full picture [71]). While the f₊₃ electrons still maintain greater distance from the core than their f₋₃ counterparts, the f₋₃ electrons show larger steering by the laser field compared to f₊₃ electrons.

This leads to larger offset angle in the photoelectron spectra at the detector [71]. The SFA trajectories fail to capture this essential feature.

As discussed in detail in section 2.3, we can either define the real initial conditions at the barrier exit (real part of coordinate on quantum trajectory at the exit time, real velocity obtained by back propagation from the final momentum ${\bf{k}}$ to real exit coordinate, see equations (11) and (12)) and then propagate them to the detector in real time, leading to the curve with with squares in figure 3(a) (classical), or we can perform propagation directly from the 'entrance to the barrier' (in complex space and time), guided by the matching scheme and leading to the quantum trajectories (equations (9) and (10)). The difference between the two after the exit point is due to imaginary exit coordinate missing in classical trajectory and present in the quantum trajectory and is a result of the coupling between the real and imaginary components of the quantum trajectories.

The coupling can be understood by considering the Coulomb force (see section 2.3) on the imaginary trajectory. Its real part depends on the parameter b(t), proportional to the imaginary part of the exit coordinate:

$$\begin{eqnarray}&&b(t)=2\left(-k{\tau }_{s}+\displaystyle \frac{{A}_{0}}{\omega }\sinh \omega {\tau }_{s}\right)\left({kt}-\displaystyle \frac{{A}_{0}}{\omega }\sin \omega t\right),\end{eqnarray} \tag{ 27 }$$

for any real t ≥ t_exit, τ_s is the imaginary part of complex SFA ionisation time t_s.

For circularly polarised fields $\left(-k{\tau }_{s}+\tfrac{{A}_{0}}{\omega }\sinh \omega {\tau }_{s}\right)=0$ (hence b(t) = 0) at optimal momentum, corresponding to the majority of s-electrons. However, the spectral peaks corresponding to co- and counter-rotating electrons are on the opposite sides from the peak of photoelectron spectrum for s-electrons. Thus, b > 0 for counter- and b < 0 for co-rotating electron, reflecting the fact that these electrons are not born at the peak of the laser field, i.e. their dynamics is manifestly non-adiabatic. The deviation between the classical and the real part of the quantum trajectories in the long-range potential is due to this effect.

In elliptical fields $b\ne 0$ for s-orbitals and therefore co- and counter-rotating electrons are 'asymmetrically' off-set from b = 0. In this case the degree of mismatch between the classical and quantum trajectories in long-range potential depends on the magnetic number m of the orbital (i.e., whether it is co-rotating or counter-rotating w.r.t. the field).

Non-adiabaticity of ionisation is the key reason for the deviations between the classical and quantum trajectories. It is enhanced due to the application of short pulses, Coulomb-laser coupling and is, probably, best detected by looking at orbitals with large l, m. However, the ellipticity of the field offers an additional control knob: non-adiabaticity of ionisation is also enhanced by decreasing the ellipticity, because the sub-cycle variation of the field comes into play. It is particularly relevant for long pulses, where the ionisation dynamics in circularly polarised fields becomes largely adiabatic. Analysing the initial conditions for various orbitals, the differences in the evolution of quantum and classical trajectories we can clearly identify the 'degree' of non-adiabaticity of strong field ionisation. We will now focus on orbitals with lower m, l, such as s and p orbitals and longer laser pulses. As discussed earlier, $b(t)\ne 0$ already for the majority of s electrons in elliptical fields, leading to m-dependent deviations between classical and quantum trajectories for $\$l\ge 1\$$, which we illustrate here.

We extend the method we recently employed in [73], to include initial coordinate with Coulomb corrections due to the under the barrier dynamics.

We consider elliptically polarised short pulse with the vector potential defined as:

$$\begin{eqnarray}&&{\bf{A}}(t)={A}_{0}f(t)[\cos (\omega t+{\phi }_{\mathrm{CEP}})\hat{{\bf{x}}}+\varepsilon \sin (\omega t+{\phi }_{\mathrm{CEP}})\hat{{\bf{y}}}],\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&\mathrm{with}\ \mathrm{envelope},\,f(t)={\cos }^{2}\left(\displaystyle \frac{\omega t}{2{N}_{e}}\right).\end{eqnarray} \tag{ 29 }$$

Here we choose ${{ \mathcal E }}_{0}=0.115\,\mathrm{au}$, λ = 780 nm, $\varepsilon =0.61$, ${\phi }_{\mathrm{CEP}}=3\pi /2$, and N_e = 6 cycles. For Ar II, the ionisation potential is I_p = 27.630 eV.

The electric field defined from the vector potential of equation (29) has its major axis along the x-axis, which means that the peak of the photoelectron distribution is close to the y-axis, as is also evident from the contour plots in figures 4 and 5. The symmetry breaking around ϕ_k = 90° is the well-known effect of the Coulomb interaction [74, 75].

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Figure 6 shows the perpendicular and parallel components of the exit coordinate and velocity (real parts of quantum trajectories and their respective velocities at the exit time), respectively, as a function of ellipticity for the majority of p⁻ and p⁺ electrons liberated from Ar II.

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Brief analysis shows that the parallel displacement of p⁻ electron is always higher than that of p⁺ electrons. Counter-rotating electrons also have higher perpendicular and parallel exit velocity (figure 6(b)). This is in contrast to circular fields, where parallel displacement for p⁺ electrons is higher than for the p⁻ electrons.

The result for exit parallel velocities shown in figure 6 is particularly instructive. The parallel velocity is zero in short-range potential in the adiabatic regime, it always applies to the majority of electrons in SFA. Figure 6 shows that in long-range potentials, the parallel velocity for the majority of the p⁺ and p⁻ electrons is small, but not equal to zero. The non-zero velocity is due to the non-adiabaticity of electron dynamics and subsequent sub-barrier Coulomb-laser coupling. Note that the respective result for the majority of s electrons (with the same I_p) would be squeezed in between the green and black curves for p⁺ and p⁻ electrons. First, we see that for elliptically polarised fields and ellipticity in the range $\varepsilon \simeq 0.4\mbox{--}0.8$, co-rotating p⁺ electrons have the largest non-zero initial velocities, i.e. they are the most non-adiabatic in elliptical fields. The s-electrons follow next and then, the most 'adiabatic' in elliptical fields are the p⁻ electrons. In contrast, recall that in circularly polarised fields, the most adiabatic are the s-electrons. Indeed, for ellipticity $\varepsilon =1$, the exit parallel velocity is very close to zero for s and p⁻ electrons, because the pulse we are using (for given frequency and intensity) is already sufficiently long to significantly suppress the non-adiabaticity of ionisation along the major field axis for the majority of s and p⁻ electrons in circularly polarised fields, but not for p⁺ electrons. Just like in short-range potential, the non-zero parallel exit velocity reflects non-adiabaticity of ionisation dynamics in long-range potentials and the parallel velocity is equal to zero in the adiabatic limit. In adiabatic (long-wavelength) limit the ARM predictions for the photoelectron angular and energy distributions in attoclock set-up (strong field ionisation in circularly polarised fields) are remarkably accurate [70], further supporting our conclusions.

In figure 8, we show the (a) real and (b) imaginary parts of the trajectories corresponding to the peak of the photoelectron spectra for the p⁻ and p⁺ electrons, for $\varepsilon =0.61$, for the SFA (triangles), classical (squares) and quantum (diamond) trajectories in the long-range potential. We take into account the impact of the Coulomb momentum shift term ${\rm{\Delta }}{{\bf{v}}}^{c}(t)$ for the ARM quantum orbits. The difference between the SFA and the ARM trajectories is the deflection of the trajectories in the long-range potential. The effect depends on the magnetic number m of the initial bound state: counter-rotating electrons (p⁻) experience greater deflection by the Coulomb potential compared to the co-rotating (p⁺) electrons. Both p⁻ and p⁺ electrons emerge from under the barrier at almost the same point in space, but p⁻ electrons are released after the peak of the pulse (see Figure 7) and are more likely to pass closer to the core (see Figure 8). We can also see some difference between the ARM quantum trajectories and ARM classical trajectories, especially for the p⁺ orbital (blue dashed curves in figure 8) discussed above.

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