Compensation of high order harmonic long quantum-path attosecond chirp

Attoscience is now recognized as a discipline on its own right since the first experimental observation of attosecond (as) pulse train [1] in 2001 and isolated attosecond pulses (IAPs) [2, 3]. Pulse durations routinely reach the 100 as regime which enables to resolve numerous ultra-fast phenomena in quantum systems ranging from atoms [4–7] and molecules [8–11] to periodic solid structures [12, 13]. Attosecond pulses are produced via high harmonic generation (HHG) with a typical spectral distribution exhibiting a plateau region and a sharp cut-off in the extreme ultra violet (XUV) region, now even pushed to the x-ray region [14]. In the intuitive picture of HHG [15–17] at the single atom level, the generating system's potential is distorted by an intense and short infrared (IR) laser field creating an oscillating time-dependent barrier. An electronic wavepacket initially bound to the ground state of the system can then reach the continuum by tunnel effect. Once in the continuum, this electronic wavepacket is driven back by the laser field where it finally recombines to the core, emitting high order harmonics of the IR driving frequency. The radiating time-dependent dipole can be expanded over a Fourier sum of electronic quantum-path contributions: the cut-off quantum-path leads to the highest harmonic photon energy in the spectrum profile. To each emitted frequency corresponds two quantum paths followed by the electron in the continuum. They are characterized by different electronic wavepacket excursion durations, commonly dubbed as long and short trajectories. While the long and short trajectories merge at the cut-off region in the harmonics spectrum, both short and long trajectories contribute to the same photon energy in the plateau region. Measurements of their interferences now constitute a powerful probe of the dynamical processes occurring, for instance, in molecular cations [18, 19]. However, most attosecond experiments uses the short trajectory contribution for which phase-matching conditions can be found to maximize the far-field emission [20] and the inherent attosecond positive chirp can be compensated for. The compensation is achieved via chirped mirrors [21], metallic filters [22, 23] and phase-matching conditions [24]. For the long quantum-path, our past investigations demonstrated experimental methods to enhance its yield contribution to the HHG signal by controlling recollision events employing synthesized laser fields [25]. Producing pulses from the long quantum-path is therefore only limited by the capability to compensate for the associated attosecond negative chirp as the spectral bandwidth, similarly to the short path, affords attosecond duration. If compensated, the long path will provide the important advantage of resulting in attosecond Fourier transform limited pulses while being emitted at times greater than the cut-off ones. Whereas short trajectories encode information over a typical 100 as to 1.7 femtoseconds (fs) time window, the long trajectory however covers a time window from 1.7 fs to several fs. In other words, both trajectories encode dynamical information in two distinctive time windows. Most electronic dynamics are expected to take place in the attosecond time scale, whereas the nuclear ones are expected to take place in the femtosecond time scale. More importantly the correlation between these dynamics can only be captured if one can observe both windows and if one can manipulate these. Our compensation approach to the long trajectory therefore opens the use of this extra window to bridge the gap of information. Ultimately, producing XUV attosecond emission from both the short and long quantum-paths while covering a time scale from 100 as to few femtosecond will greatly enhance XUV attosecond sources capabilities. Hence, here we propose the first scheme to compensate for the attosecond negative chirp associated to the long quantum-path. Such a method is based on the combination of an IAP generated from the short quantum-path of a primary HHG source, together with the IR few-cycle laser pulse to generate harmonics according to the long-path scheme in a secondary HHG source. Note however that the IR is responsible for the harmonic emission in both the primary and secondary source. The IAP, issued from the short trajectory of the primary source, will be used to manipulate coherently the wave packet issued from the long trajectory of the secondary source. Previous work have shown conceptually the ability to use attosecond pulse train to assist the production of high harmonics via a controlled quantum path selection [26]. Here, we propose a novel route which use an IAP assisting the compensation of the phase associated with the long quantum path. This provides a new route to attosecond pulse production from long trajectory as an attosecond source on its own right. This compensation of the attosecond chirp relies principally on the fact that short and long quantum paths issued from the same source have near identical but opposite signed chirps. Note however that the IR is responsible for the harmonic emission in both the primary and secondary source. The IAP, issued from the short trajectory of the primary source, will be used to manipulate coherently the wave packet issued from the long trajectory of the secondary source. Moreover, the process highlighted is independent on the system and laser field frequencies. We investigate this scheme theoretically by resolving the time-dependent Schrödinger equation (TDSE) in a helium model atom. We show how such combined fields can be used to compensate for the long path attosecond chirp which in turn leads to the production of long path attosecond XUV pulses.

We use atomic units (${\rm{a}}.{\rm{u}}.$) unless stated otherwise.

We consider the case where the IAP is generated in a primary source by a fundamental few-cycle IR pulse in Helium. We used a standard method to numerically solve the 1D TDSE in the single active electron approximation on a uniform grid with a regularized Coulombic potential (soft-core) adjusted to the ground state energy of He −0.904 ${\rm{a}}.{\rm{u}}.$ with a regularization parameter a set to 0.6945 [27]. The box size is set to 300 ${\rm{a}}.{\rm{u}}.$ with a discretization spacing equals to 1/30 ${\rm{a}}.{\rm{u}}.$ and we used a Crank–Nicolson algorithm with 32768 time-steps to propagate $| {\rm{\Psi }}(t)\rangle$ under the influence of the electric field. The resulting time dependent dipole is computed in the acceleration gauge ${D}_{{\rm{A}}}(t)=\langle {\rm{\Psi }}(t)| \hat{{\bf{a}}}| {\rm{\Psi }}(t)\rangle$ with $\hat{{\bf{a}}}=-\tfrac{{\rm{d}}}{{\rm{d}}x}V(x)$ at each time step [28]. The HHG spectra is deduced by Fourier transforming the dipole $S(\omega )={ \mathcal F }\{{D}_{{\rm{A}}}(t)\}$. The total electric field is written under the form of a sum over the electric fields of the IR pulse and the IAP.

$$\begin{eqnarray}E(t) & = & {E}^{{\rm{IR}}}(t){\rm{\sin }}({\omega }^{{\rm{IR}}}t+{\phi }^{{\rm{IR}}})\\ & & +\displaystyle \sum _{i}{E}^{{\rm{IAP}}}(t+{\rm{\Delta }}\tau +{\rm{\Delta }}{{ \mathcal T }}_{i}){\rm{\sin }}({\omega }_{i}t+{\phi }^{{\rm{IAP}}}),\end{eqnarray} \tag{ 1 }$$

where ${E}^{\mathrm{IR}}(t)$ and ${E}^{\mathrm{IAP}}(t)$ are ${\sin }^{2}$ temporal envelopes of both the IR pulse and the IAP. The sum over $i\,=\{{\rm{H}}23,\ {\rm{H}}25,\ {\rm{H}}27,\ {\rm{H}}29,\ {\rm{H}}31\}$ (H stands for harmonic) corresponds to the total XUV field composed of five frequencies ${\omega }_{i}$ coinciding with its spectral continuum [29] from H23 to H31. We denote ${\rm{\Delta }}\tau$ the global delay between the two temporal envelopes of the IR and IAP pulses and ${\rm{\Delta }}{{ \mathcal T }}_{i}$ the negative/positive relative delay in the emission time of frequencies within the IAP spectral range (${\omega }_{{\rm{H}}23}$ to ${\omega }_{{\rm{H}}31}$) with respect to the central frequency ${\omega }_{{\rm{H}}27}$, e.g. ${\rm{\Delta }}{{ \mathcal T }}_{i}={t}_{{\rm{H}}27}^{e}-{t}_{i}^{e}$ (owing that we arbitrarily set ${t}_{{\rm{H}}27}^{e}=0$). The respective carrier envelope phases (CEP) of each pulses are ${\phi }^{\mathrm{IR}}$ and ${\phi }^{\mathrm{IAP}}$. Computations are performed with IR pulse duration with four cycles, i.e. a full duration ${T}^{\mathrm{IR}}\simeq 10$ fs and typical IAP duration full width half maximum (FWHM) is 600 as. The intensities of both the IR and the IAP are set to fit possible experimental conditions, ${I}^{\mathrm{IR}}={2\times 10}^{14}$ W cm⁻² and ${I}^{\mathrm{IAP}}\,={5\times 10}^{8}$ W cm⁻² and we present in figure 1 an example of such fields.

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Starting from this combined fields input, and in order to maintain coherence throughout the full process, a replica of the very same few-cycle IR pulse is used in a second stage target (HHG secondary source). The IAP attosecond chirp is set from typical short quantum-path chirp of about ${\rm{\Delta }}{{ \mathcal T }}_{i}=\pm \ 4$ ${\rm{a}}.{\rm{u}}.$ (100 as between each harmonic emission time) around ${t}_{{\rm{H}}27}^{e}$ [29]. IAP spectral range above ${\omega }_{{\rm{q}}}/{\omega }_{\mathrm{IR}}=27$, carry a positive chirp whereas spectral range below ${\omega }_{{\rm{q}}}/{\omega }_{\mathrm{IR}}=27$ carry a negative chirp. Moreover, the IAP is delayed with ${\rm{\Delta }}\tau =67.37$ ${\rm{a}}.{\rm{u}}.$ (≃1.67 fs) such that its ${t}_{{\rm{H}}27}^{e}$ IAP delay coincides with the ${t}_{{\rm{H}}27}^{e}$ emission time in the long-path contribution to HHG in the secondary source (see arrow in figure 1). Finally, the relevant free parameter left to control the effect of the combined field onto the long-path HHG is the relative CEP between the two pulses ${\rm{\Delta }}{\rm{\Phi }}={\phi }^{{\rm{IAP}}}-{\phi }^{\mathrm{IR}}={\phi }^{{\rm{IAP}}}$ since we set ${\phi }^{\mathrm{IR}}=0$ as the origin of phases.

In figure 2, we present a time–frequency Gabor analysis of the resulting dipole for four values of ${\rm{\Delta }}{\rm{\Phi }}=n\tfrac{2\pi }{10}=n{\phi }_{1}$ with $n=0,3,6,9$. In past investigations, such analysis were successfully performed in the context of HHG [31, 32] and a typical spectrogram generally depicts two branches per half optical IR cycle, which correspond to short and long quantum-paths contribution. These branches converge at high harmonic orders towards the single cut-off trajectory. In the plateau, the slope associated with each of the branches is the attosecond XUV chirp. If the slope is minimized (vertical), all frequencies in this spectral range are emitted at the same time and results in a Fourier transform limited temporal profile. In our case, we use Gabor analysis to exhibit the effect of the combined fields onto the slope and therefore onto the chirp associated with the long trajectory. In figure 2, one can clearly see that in the spectral region covered by both the IR laser pulse and the IAP (delimited by the dash lines), the phase effect due to the combined fields is clearly evidenced. In cases where ${\rm{\Delta }}{\rm{\Phi }}=3{\phi }_{1}$ and ${\rm{\Delta }}{\rm{\Phi }}=6{\phi }_{1}$, figures 2(b) and (c), the Gabor analysis signal corresponds to destructive and constructive interferences respectively. Interestingly, when ${\rm{\Delta }}{\rm{\Phi }}=0$ and ${\rm{\Delta }}{\rm{\Phi }}=9{\phi }_{1}$ meaning intermediate conditions, the chirp of the long quantum-path is compensated for by the presence of the IAP, giving rise to a vertical structure in the Gabor signal, visible in figures 2(a) and (d). This indicates that the compensation of the chirp associated with the long quantum-path is achieved hence throughout this spectral range, frequencies are synchronously emitted. We need now to answer two questions: would this compensation survive near field to far field propagation? Could the long-path harmonics emitted with phase compensation build up macroscopically to be detectable? The first question relates to the coherence of the emission and is the necessary condition for detection. The second question concerns phase matching conditions for which the macrocopic field builds up through spatial propagation to become detectable.

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First, we calculated the substraction of the dipole contributions of the IR and the IAP to the total dipole induced by the combination of the two,

$$\begin{eqnarray}&&{D}_{{\rm{A}}}(t)={D}_{{\rm{A}}}^{{\rm{IR}}+{\rm{IAP}}}(t)-{D}_{{\rm{A}}}^{{\rm{IR}}}(t)-{D}_{{\rm{A}}}^{{\rm{IAP}}}(t).\end{eqnarray} \tag{ 2 }$$

The resulting dipole is non-zero which indicates that the IAP interacts coherently with the long-trajectory electronic wavepacket. The compensation scheme is therefore not a pure 'photonic interference' as described in [30] but a coherent transfer of phase between the IAP and the electronic wavepacket of the long quantum-path in the final HHG emission.

To answer the second question, we have computed the so-called α parameter, which represents the conjugated variable to the intensity in a time–frequency type analysis [31–33]. In the following, we compare α values when the IR intensity varies from 0 to 10¹⁵ W cm⁻² for ${\omega }_{{\rm{H}}27}$ within the spectral compensation window and ${\omega }_{{\rm{H}}45}$ which lies far out the spectral window. In each situation, the IAP is either absent or present with ${\rm{\Delta }}{\rm{\Phi }}=\pi /2$ and ${\rm{\Delta }}{\rm{\Phi }}=\pi$ respectively. These two values of ${\rm{\Delta }}{\rm{\Phi }}$ correspond to reconstructed long path attosecond pulses with FWHM either close to the minimum (165 as) or to the maximum (235 as), as shown in figure 3. In figure 4(a), when the IAP is absent, one can see the typical 'fork-tuning' shape already thoroughly described and analyzed in [32]. The branches around $\alpha =0$ and $\alpha =20$ are signatures of the short and long quantum paths respectively. Interestingly, a non negligible wiggling component around $\alpha =10$ is present as the result of beatings between the two trajectories. When the IAP is present (see figures 4(b) and (c)), one can see that the α signal below $1.5\times {10}^{14}$ W cm⁻² completely disappears for the long trajectories and the cut-off. Instead, a spot is created around $\alpha =0$ (circled in figures 4(b) and (c)) with an intensity depending on ${\rm{\Delta }}{\rm{\Phi }}$. These observations are consistent whilst analyzing at frequencies ${\omega }_{{\rm{H}}23}$ and ${\omega }_{{\rm{H}}31}$ (not shown here) where the fork-tuning shape is still visible and the displacement of the long and cut-off trajectories to the short one is also clear. This reveals that ${\rm{\Delta }}{\rm{\Phi }}$ is effectively the main control knob over shaping the α pattern in the compensated region (see figures 4(b) and (c)).

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When the same analysis is performed onto ${\omega }_{{\rm{H}}45}$ (see figure 5), i.e. far out the compensated spectral region, no such displacement effect is present. Instead, a unique spot is observable around $\alpha =0$ with changing intensity when ${\rm{\Delta }}{\rm{\Phi }}$ varies. In figure 5(a), we attribute this spot to a low intensity background signal, giving rise to a $\alpha =0$ contribution that disappears when the harmonic signal becomes dominant, around 10¹⁴ W cm⁻². This effect is not shown for the other ${\omega }_{{\rm{q}}}$ as their signal is already strong enough to mask it. In figures 5(b) and (c), the spot is more pronounced (circled) due to the Rayleigh diffusion of the IAP.

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Finally, in all situations where the displacement effect is visible, e.g. ${\omega }_{{\rm{H}}23}$, ${\omega }_{{\rm{H}}27}$, ${\omega }_{{\rm{H}}31}$ but more generally all frequencies within the compensated spectral region, the generated spot at low α contains two main contributions. First, from the compensated long trajectories having now the same characteristics as short trajectories due to phase compensation and second from the IAP diffusion itself. Indeed, this means that this signal could be experimentally detectable using similar on-axis short trajectory based setups.

Next, we address the question of the discrimination of these two contributions using the quantum path interference (QPI) technique [18].

To explore the possibility to experimentally control, detect and discriminate such attophase control, we investigate QPI [16, 18] which is one of the few techniques allowing to disentangle contributions to HHG from different trajectories. QPI signals are presented in figure 6 for different harmonics and IAP phases. In reference cases where the IAP is absent (see red lines in figure 6), the QPI signal is modulated with an expected $2\pi /{\rm{\Delta }}\alpha$ periodicity with ${\rm{\Delta }}\alpha ={\alpha }_{\mathrm{long}}-{\alpha }_{\mathrm{short}}$. In cases depicted in figures 6(a)–(c), the ${\alpha }_{\mathrm{long}}$ values are known to dominate QPI modulations [18], the ${\alpha }_{\mathrm{lshort}}$ values approaching 0 leading to very slow periodicities. In figure 6(d) the considered intensity range above 10¹⁴ W cm⁻² does not allow to observe any modulation, the QPI is still in the cut-off region where the distinction between short and long trajectories becomes irrelevant. When the IAP is present with a ${\rm{\Delta }}{\rm{\Phi }}=\pi /2$ (see blue lines in figure 6), the QPI signal at low intensity is orders of magnitude higher than without the IAP and therefore undoubtedly contributes to the spot located around $\alpha =0$ visible in figures 4, 5(b) and (c) and commented before. However, the different values of ${\rm{\Delta }}{\rm{\Phi }}$ (see blue and black lines) give rise to detectable shifted modulations which is unsurprisingly not present at ${\omega }_{{\rm{H}}31}$. Indeed, the latter is situated far away from the compensation range and its characteristics are independent of ${\rm{\Delta }}{\rm{\Phi }}$. Once again, ${\rm{\Delta }}{\rm{\Phi }}$ appears to be the control knob to the compensation scheme. Finally, in the case of a photonic interference, while changing the laser intensity, the QPI signal should be shifted and enhanced. This amplitude enhancement is directly related to the IAP intensity (first order interference). This is not what we observe in figure 6, where only a shift is present.

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In this work, we proposed a full optical compensation scheme for the long trajectory. This scheme is based on the production of long-quantum path HHG from a combination of two fields: the fundamental driving IR fields and generated IAP from the short quantum path. We have shown that we can reach a compensation of the long-quantum path chirp to produce attosecond XUV light. We show that this compensated long path HHG emission has macroscopic characteristics similar to the short path by analyzing the dipole phase with Gabor extraction. Hence, the compensated long path HHG is expected to follow the same type of macroscopic build-up as the short path one, which makes it compatible with standard experimental setup. We proposed to observe such a compensation by measuring the quantum-path interference signal which depicts a clear signature of the coherent phase transfer characteristic of this compensation scheme.

This work was supported by UK Royal Society project international exchange scheme IE120539 and UK-EPSRC project EP/ J002348/1 'CADAM'. We acknowledge financial support from the LABEXs MiCheM and Plas@Par (ANR-11-IDEX-0004-02) and the program ANR-15-CE30-0001-01-CIMBAAD