Polarisation response of delay dependent absorption modulation in strong field dressed helium atoms probed near threshold

We present the first measurement of the vectorial response of strongly dressed helium atoms probed by an attosecond pulse train (APT) polarised either parallel or perpendicular to the dressing field polarisation. The transient absorption is probed as a function of delay between the APT and the linearly polarised 800 nm field of peak intensity 1.3 × 10 14 W cm − 2 . The APT spans the photon energy range 16–42 eV, covering the first ionisation energy of helium (24.59 eV). With parallel polarised dressing and probing fields, we observe modulations with periods of one half and one quarter of the dressing field period. When the polarisation of the dressing field is altered from parallel to perpendicular with respect to the APT polarisation we observe a large suppression in the modulation depth of the above ionisation threshold absorption. In addition to this we present the intensity dependence of the harmonic modulation depth as a function of delay between the dressing and probe fields, with dressing field peak intensities ranging from 2 × 1012 to 2 × 1014 W cm − 2 . We compare our experimental results with a full-dimensional solution of the single-atom time-dependent (TD) Schrödinger equation obtained using the recently developed abinitio TD B-spline ADC method and find good qualitative agreement for the above threshold harmonics.


Introduction
Dressing an atom with a laser field changes its response to light in diverse ways. Weak resonant fields drive transitions between the field-free atomic states, enabling resonant atomic coherent phenomena such as electromagnetically induced transparency [1,2], Autler-Townes splitting [3,4] and related phenomena such as slow light [5] to be observed. Stronger non-resonant dressing fields can modify the optical response without inducing population transfer. By recording the absorption spectrum of an extreme ultraviolet (XUV) attosecond pulse or pulse train in the dressed medium, the response can be studied on timescales well within the period of the dressing field. This technique of attosecond transient absorption spectroscopy (ATAS) [6][7][8][9] has recently been used to study phenomena ranging from electronic coherence in atoms, to ultrafast switching in dielectrics and the lifetime of autoionising states. In general this method holds great promise for monitoring the dynamics of the response of a laser dressed system on a sub-or few-femtosecond timescale in both gas and condensed phase media.
Despite the simplicity of its structure helium exhibits rich dynamics and the laser dressed response has received much attention. The double excitation around 60 eV serves as a model system for observing and manipulating a two-electron wavepacket [10]. Even single excitations around the first ionisation threshold present a complex array of effects due to the various bound state resonances. The XUV pulses used in ATAS, produced by high-order harmonic generation (HHG), are typically too weak to induce multi-photon effects by themselves. Therefore, the diverse phenomena reported thus far can be broadly categorised by the order of the interaction induced by the dressing field. At the lowest intensity, new two photon XUV+IR transitions (to single-photon dipole forbidden states) have been observed at intensities as low as´-0. 5

W cm
At slightly higher dressing field intensities three photon XUV+2IR transitions occur, enhanced by bound-state resonances, and can undergo 'which-way' quantum interference with single photon XUV transitions [13][14][15] or optical interference with incident light [16,17]. Additionally the sub-cycle AC Stark shift modulates the frequency of excited states [18] and reduces their lifetime. Both of these effects have been detected at »´-3 10 W cm 12 2 . More recently, processes involving four IR photons have been observed [13,19,22]. Earlier work [20,21] has addressed the polarisation dependence of the field dressing in the perturbative regime, but here we investigate the polarisation dependence of the dressing field on the atom in the strong field regime.
There have been few experimental studies of intensities above 5×10 12 -W cm 2 . Due to the potential for higher order processes, one might expect the physics to be even richer. One possibility is for XUV initiated HHG (XiHHG), in which a continuum electron is ionised by the XUV pulse, and undergoes acceleration in the laser field before recombining with the ion [23,24]. It has also been shown that HHG from an excited atomic state may lead to efficient control over the rescattering efficiency and harmonic polarisation [25]. XiHHG has been reported at 5×10 13 -W cm 2 [26], and has also been proposed as a probe of core dynamics such as Auger decay [27]. At these intensities, resonant enhancements [22] are expected to be weakened by appreciable broadening of the bound states [28]. At sufficiently high intensities, a trajectory-based view is likely to become applicable, in which the excited state dynamics are dominated by acceleration in the laser field in a similar fashion to the widely used strong-field approximation [29][30][31] for conventional HHG.
In this article we explore the response of helium atoms probed by transient absorption around the ionisation threshold in this high intensity regime (up to intensities of2 10 14 -W cm 2 ). We investigate the polarisation sensitivity of the response by measuring the delay dependent absorption modulation for dressing fields parallel and perpendicular to that of the probe field. By using an attosecond pulse train (APT), our measurements are sensitive to the components of the response with the same half-cycle periodicity as the driving laser [22]. Alongside the vectorial dependence we also present the intensity response of the delay dependent modulation depth amplitude over a broad intensity range extending from 2×10 12 -W cm 2 , in the perturbative regime, to 2×10 14 -W cm 2 , just below the onset of strong-field ionisation. Our experimental results are compared with a full-dimensional (3D) time-dependent (TD) Schrödinger equation calculation performed using the recently developed ab initio TD B-spline ADC method [32]. Figure 1 shows a schematic of the experimental setup used to measure the transient absorption. The input pulses, of central wavelength 800 nm, full-width at half maximum duration (FWHM) 30 fs and energy ∼3 mJ, were supplied by a 1 kHz, Ti:Sapphire CPA laser (KMLabs Red Dragon). An annular mirror (AM1) split the incoming beam. The transmitted portion was used to produce the XUV APT using HHG in an effusive gas jet (GJ), with a 100 μm diameter nozzle backed by 1.5 bar krypton. The residual IR was blocked by an aluminium filter (AF). The beam reflected from (AM1) was used to dress the helium atoms. The intensity was controlled using a half-wave plate (HWP1) and polariser (P), and the polarisation was switched between vertical and horizontal by a further half-wave plate (HWP2). An insertable beam block (BB) was used to block the dressing field beam for reference (field-free absorption) spectra. The time delay between the XUV and dressing arms was controlled with either a delay translation stage (TS) or piezo-driven mirror (PM). An auxiliary interferometer (AI) provided high-resolution tagging for the delay between the two arms The arms were recombined using a further annular mirror (AM2) and refocused with a toroidal mirror (TM) into a 2.6 mm diameter helium-filled tube target (T). The transmitted XUV spectrum was dispersed by a 1200 lines mm −1 flat-field grating (FFG) and detected on a micro-channel plate (MCP) with phosphor coating, monitored by a CCD camera. The measured spectral range was 16-42 eV, corresponding to harmonic orders 11-27. In this range, the individual harmonics were well resolved. The signal contribution for each individual harmonic was obtained by integrating over the brightest region around each harmonic peak. The regions of integration were kept fixed when analysing the data for each figure.

Experimental methods
The peak intensity of the dressing field in the target plane was inferred from power and beam profile measurements, onto which the beam was directed by an insertable pick-off (PO). The FWHM of the dressing beam focus was ≈100 μm from focal spot imaging. The spot size of the XUV beam was ≈70 μm FWHM, measured using a knife edge method. The uncertainty in the absolute intensity was±50%, however, the uncertainty in the relative intensity between measurements was an order of magnitude less. The experimental pulse duration of the dressing field was estimated to be ∼50 fs, where increasing positive delay corresponds to a later arrival of the XUV pulse. The absolute pump-probe delay was not measured. Typical backing pressures used for the helium tube target were around 50 mbar.
The temporal resolution was taken as the root-mean-square fluctuation of the delay between the XUV and dressing arms as measured in the AI. Over the duration of one exposure of the MCP camera this was ∼150 as placing an upper limit on the observable modulation frequency of To convert our delay-dependent spectra into absolute absorption cross sections, we first determined the density-length product η of the target from its field-free absorption and its known absorption cross section s FF [33], using the Beer-Lambert law. The determined density-length products for harmonics 17 and 19 were averaged. With the dressing field applied, the measured delay-dependent absorption was converted into a delaydependent cross section s t ( ) via the Beer-Lambert law. This was decomposed into the field-free absorption and the additional absorption resulting from the dressing field s t ( ) where I FF and t ( ) I are the measured field-free and dressed-atom harmonic intensities respectively.

Theoretical methods
We calculated the single-atom delay-dependent absorption using B-spline time-dependent (TD) algebraicdiagrammatic construction (ADC) method [32]. The basis set consists of spherical harmonics for the angular part and B-splines B i (r) for the radial coordinate. The single particle basis functions y ilm used in this calculation are therefore expressed as The TD problem is solved within TD-ADC making the following ansatz for the TD many-electron wavefunction where the coefficients ( ) C t 0 and C n (t) refer to the ground-state Y ñ | 0 and to the correlated excited states (CES) Y ñ | n of the ADC theory respectively. These configuration basis states include single, double, etc. excitations with respect to the ground state of the system; the maximum number of electrons which are allowed to be excited at (b) Example comparison of the source harmonics (blue) generated in the effusive gas jet (GJ) with no gas in the target (T) compared with the field-free absorption (red) of the source harmonics in the undressed helium target (T). The harmonic transmission of the laser dressed helium is also shown (black) for a given delay and peak intensity´-1.3 10 W cm 14 2 . The above ionisation energy harmonics (17)(18)(19)(20)(21)(22)(23)(24)(25) are shown in the zoomed insert. the same time, i.e. the point at which the expansion of equation (3) is truncated, defines the order n of the ADC (n) hierarchy. In the following calculation we have used the first-order method of the ADC-hierarchy, namely ADC(1), in which only single excitations are included in the expansion of the wavefunction. This is a good approximation to the current application as in He the threshold for double excitation is above 60 eV and the photon energies of the harmonics investigated here are much lower. The presented results have been calculated making explicit use of the atomic spherical symmetry and they are in principle exact as long as double excitations do not play an important role in the dynamical process of interest. The TD Schrödinger equation (TDSE) for the unknown coefficients C C , n 0 is solved via the Arnoldi-Lanczos algorithm. A complex absorbing potential (CAP) m W(cap), has been employed in order to eliminate wave-packet reflection effects from the grid boundaries. The form of the CAP used is this work reads as where the absorbing radius r CAP defines the size of the inner region and the strength coefficient η regulates the smoothness and steepness of the CAP profile. The CAP used in the calculations starts at a radius = r 120 CAP a.u. and has a strength h = 0.0005. With the addition of the CAP term the form of the total TD Hamiltonian of the system reads . Convergence of the results with respect to the basis set parameters has been checked. With this choice the typical number of singly excited configurations included in the simulation is of the order of a few tens of thousands depending on the polarisation of the dressing field relative to the APT. Since the IR induced couplings are sensitive to the APT spectral amplitudes and phases of the contributing harmonics, the calculated results have a dependency on the XUV pulse shape used. For this reason the pulse shapes of the XUV and IR dressing field used in the ADC(1) calculations were generated to approximate those in the experiment. The calculation of the XUV spectrum was required to access the harmonic phases that could not be measured directly in the experiment. The XUV field was calculated numerically by simulating the dominant processes in the experimental XUV generation process. This included a 2D axisymmetric simulation of the macroscopic HHG process within the krypton GJ, and then vacuum propagation, filtering and focussing of the generated XUV field. The XUV field at the point of maximum on-axis fluence was used in the single atom calculations. A bandwidth limited cos 2 envelope was used for the spectrum of the IR dressing field, with a FWHM pulse duration set to be 50 fs.
The frequency dependent absorption w t ( ) S ; d is calculated from the expectation value of the electric dipole moment of the atom z(t) and the incident XUV field ( ) where tilde denotes the Fourier transform from the time domain to XUV frequency ω,  denotes complex conjugation, and t d is the dressing IR-XUV delay. This quantity is then Fourier transformed with respect to t d to even multiples of the dressing field frequency w 1 . The generalised cross-section, s w t ( ) ; d was calculated using the following equation [34] s w t where α is the fine structure constant. W cm 2 (∼3.8 mJ input pulses) with parallel polarised XUV and dressing fields. Clear modulation is observed for harmonics 13-21, (20-32.5 eV), spanning the ionisation potential of helium. The Fourier transform along the delay axis, shown in figure 2(b), reveals that while the modulation is predominately at twice the dressing field frequency, harmonics 17 and 19 also have significant modulation at w 4 1 . Since the modulation frequency is determined by the difference in the frequencies of the harmonics coupled by the dressing field [22], we infer that there is strong coupling between all adjacent harmonics. Given H11 was blocked by the AF, the entire w 2 1 component of H13 must be due to coupling with H15, whereas the w 2 1 components of the other harmonics are coherent combinations of the couplings between both adjacent harmonics. The w 4 1 component implies weaker but significant coupling between H17 and H13 or H21 (or both) and also between H19 and H15 or H23 (or both).

Results and discussion
Presented in figure 3 is the experimentally measured absorption cross section for the above ionisation threshold harmonics (17 and 19) as a function of delay for dressing field and APT (a) parallel and (b) perpendicularly polarised. Our experimental dressing-field intensity was estimated to be1.3 10 14 -W cm 2 . Changing the relative polarisation of the dressing and XUV fields from parallel (Θ=0°) to perpendicular (Θ=90°) causes a significant reduction in the modulation amplitude. In addition to the total dynamic cross section, we also plot the field-free cross sections (FF) [33] as dashed lines for each harmonic. The decreased modulation amplitude of the above threshold harmonics as the relative polarisation is changed from parallel to  perpendicular can be understood by considering the direction of the dipole induced by the XUV (figure 3(c)). If the two fields are aligned parallel, the XUV-induced dipole will experience maximum modulation from the IRinduced distortion to the bound potential. When the two fields are aligned perpendicularly, the XUV-induced dipole will experience minimum modulation of the bound potential resulting in a reduction in the absorption modulation amplitude. Figure 4 again shows the comparison of the delay dependent absorption cross section modulation for different polarisation alignments however this figure also includes the below ionisation threshold harmonics (panels (e) and (f)) as well as a comparison to theory (panels (c), (d), (g) and (h)). The theoretical and the experimental data assume a similar peak intensity of the dressing field (1.2 10 14 -W cm 2 and 1.3 10 14 -W cm 2 respectively). The significant reduction in modulation amplitude for the above ionisation threshold harmonics H17 and H19 (a) and (b) when the polarisations of the dressing and XUV fields are varied from parallel (left column) to perpendicular (right column) is reproduced qualitatively by the theoretical model. In addition, the approximate phase difference between the modulation peaks for each pair of harmonics in the experiment and the theory is similar. (The absolute phase cannot be compared between experiment and theory.) By contrast, the experimentally measured modulation amplitude of the below threshold harmonics is not significantly altered by changing the relative polarisation alignment (panels (e) and (f)). This behaviour is partially reproduced by the model (panels (g) and (h)). The modulation amplitude for harmonic 13 is qualitatively reproduced. However, the behaviour of harmonic 15 is less well captured. The partial disagreement for the below threshold numerical results may be attributed to the use of the single electron excitation approximation. Truncating the ADC hierarchy to n=1 causes a small (∼0.4-0.5 eV) discrepancy between the calculated and actual energy levels. Harmonic 15 lies just below the ionisation energy of helium, in the vicinity of many bound states. Discrepancy between these energy values may therefore significantly change the calculated absorption cross section. Other disagreements between theory and experiment are consistent with this hypothesis. The offset of the modulation relative to the field-free cross section is reproduced best by the theory for harmonics 13 and 19. These harmonic energies are situated respectively well below and well above the ionisation threshold of helium. For the 13th and 19th harmonics we would expect minimal effects from any inaccuracies in the calculation of the ionisation potential and problems capturing the diffuse Rydberg states for these harmonics. As for the difference in modulation depth between the experiment and theory, the theoretical calculation is a single atom treatment so does not capture any of the field propagation modifications other than Representative error bars are shown. The dashed lines indicate the experimental field-free (FF) absorption [33], and the theoretical field-free absorption from the calculation. Dashed lines are absent from the below threshold harmonics since the field-free absorption for harmonics 13 and 15 should be zero. the attenuation of the APT. As such it is to be expected that the visibility of the experimental data is by comparison reduced.
At certain time delays and for a small range of intensities, the absorption of harmonic 13 was negative, meaning that transmitted light was stronger than if the target were absent. This is examined in greater detail in figure 5. Note the relative phases of modulations between different intensity data sets cannot be reliably established in these measurements, but the relative phases of harmonics at each intensity is well defined. We focus on harmonics 13 and its neighbour 15 (harmonic 11 was blocked by the AF). At´-2.3 10 W cm 12 2 (dark grey), the absorption is negligible for harmonic 13 as it does not overlap with any weakly dressed bound states. At this dressing-field intensity harmonic 15 has a small absorption offset as it lies resonant with the 1s3p electronic state [13,28]. Increasing the intensity to´-6.8 10 W cm 12 2 introduces a two-IR-photon coupling between these harmonics, causing both to modulate with a half-cycle period of the driving laser. Harmonic 13 continues to oscillate around zero indicating a continued absence of available absorption channels via either single-photon or multi-photon pathways. Harmonic 15 however acquires an increased cycle-averaged offset totaling approximately 3.7 Mb. This is attributed to an increase in the absorbance of the 1s3p state with increasing IR peak intensity up to roughly 1×10 13 -W cm 2 [13]. Increasing the intensity further (2.0 10 13 -W cm 2 ) strengthens the coupling between the two harmonics and hence the modulation amplitude, but also opens up an absorption channel for harmonic 13 which offsets the modulation so that the absorption cross section becomes positive for all delays. This absorption channel could be attributed to the XUV+1IR multi-photon transition to the previously dipole-forbidden 1s3s state [13].
Further insight is obtained from the dependence of the modulation amplitudes on the dressing field intensity. Figures 6(a) and (c) show the measured amplitude of the w 2 1 and w 4 1 components respectively for harmonics [13][14][15][16][17][18][19]. Across the full intensity range of2.3 10 12 -2.3 10 14 -W cm 2 we detected w 2 1 modulations. We also detected w 4 1 modulations above 10 13 -W cm 2 in harmonics 13 and 17, with a spike of w 4 1 modulation in harmonic 15 and possibly harmonic 19 around´-1.5 10 W cm 13 2 . The results of the corresponding theoretical calculations are shown in figures 6(b) and (d). The theory, which covers up tó -1.2 10 W cm 14 2 , reproduces qualitatively several aspects of the experiment: the double peak structure of the harmonic 15 w 2 1 modulation, the minima of the harmonic 13 w 2 1 modulation depth towards the highest intensity dressing fields and the general flat response of harmonic 19 w 2 1 modulation depth. Considering the w 4 1 response we again see many of the general shapes from the experimental data reproduced by the theory. The modulation depth of harmonic 17 rises and falls throughout this intensity range, and the sharp peak in the w 4 1 response for harmonic 15 and possibly harmonic 19 is reproduced around a similar dressing field intensity by the theory. More generally, the theory also predicts the higher intensity onset of the w 4 1 modulation contributions compared with the w 2 1 response. In general, the theoretical amplitudes are much larger than those observed. Possible reasons for the discrepancy include the spatial variation of the dressing field intensity over the XUV beam, and the propagation in the target medium which can reduce or eliminate modulations [22].
Another perspective on these measurements comes from considering electron trajectories and XiHHG, which can be recast in a multi-photon picture. In our experiment, the cutoff of the incident APT extended above the highest observed harmonic. The harmonics emitted by recombining electrons were therefore already present in the incident field. In this situation, the emitted and incident light interfere producing delay-dependent absorption with modulation frequency determined by the frequency difference between the initiating and emitting harmonics. The highest order modulation observed is therefore equal to the maximum kinetic energy gained by an electron between ionisation and recombination. In our case, this was w 4 1 . Our wavelength and typical laser intensity were such that the available kinetic energy gain using the three-step model [30] was w » U 3.2 12 p 1 , and our best estimates of the temporal resolution yielded a maximum resolvable modulation frequency of w 8 1 . Likely reasons why we did not observe higher order modulation emerge through signal to noise considerations. Modulation components are detected as peaks after Fourier transforming along the delay axis (2(b)). In our case, for the harmonics under study (13)(14)(15)(16)(17)(18)(19) the noise in the Fourier domain is caused by both detector noise and fluctuations in the XUV intensity (driven by laser fluctuations) between steps in our delay scans. Together, these obscure weak modulation components. By contrast, Gademann et al [26] observed XiHHG at photon energies not present in the incident APT, so their measurements were observed on a zero background. For these harmonic orders, interference between incident and generated harmonics therefore did not occur, and XUV fluctuations would not have affected the detection threshold. At similar intensities to our experiment, the IR-induced coupling they observed was equivalent to a photon energy gain of at least 6 IR photons.
The agreement between experiment and theory will be influenced by the reshaping of the pulse in the medium. At maximum IR intensity, the absorption of the harmonics was ∼80% or more. Such a degree of absorption is accompanied by significant phase slip between the dressing field IR and the harmonics, and order dependent dispersion of the harmonics. This has been shown to be capable of modifying or even removing certain modulation components [22] and changing absorption line shapes of bound state resonances [35]. Physically, different dynamics are initiated and probed throughout the target length. Another source of discrepancy was the spatial intensity variation of the dressing field in the transverse plane. This is caused by the XUV focal spot size being a significant fraction of the dressing field focal spot size, non-uniformity in the input beam profile as well as uncertainty in the spatial and temporal overlap between the XUV and IR beams.
Future work could address some of the difficulties in the experimental measurement. It may be possible to increase the incident XUV flux through target modification or the use of a softer generation gas, or alternatively by further expanding the generation beam prior to focussing. Achieving higher flux could improve the signal to noise and enable lower helium pressures to be used in the interaction target, reducing the pulse reshaping through the target. Tighter focussing of the XUV generation could reduce the spatial variation of the dressing field intensity across the APT focus by offering a more favourable ratio of focal spot sizes. Another possibility is to record scans over the whole range of temporal overlap interaction, reducing the uncertainty in the peak intensity. Scans of this duration would take significantly longer to run, and as such we would need to ensure that the laser and other experimental parameters were sufficiently stable across this time-frame.

Conclusion
We have presented the first measurements of the vectorial response of the transient absorption of singly excited helium dressed by a laser field with polarisation oriented either parallel or perpendicular to the polarisation of the probing APT, and intensity approaching the strong-field ionisation threshold of the ground state. We observed delay-dependent absorption with half and quarter-cycle periodicity with a strong dependence on the relative polarisations of the dressing and probing fields. Several aspects of our results were reproduced by singleatom TDSE calculations. We have studied the dependence of field induced absorption modulation on the relative polarisation of dressing and probe fields. This shows a strong modulation for the case of parallel polarisation, but the modulation is greatly suppressed in the case of perpendicular polarisations for the above threshold harmonics. The behaviour can be deduced qualitatively from a heuristic strong field picture of the laser dressed atom and the geometric dependence of the distortion as a function of angle with respect to the strong field polarisation axis. The full-dimensional calculation reproduces the main features of these results. Little difference in the modulation amplitude was observed between the parallel and perpendicular polarisations for the below threshold harmonics. This behaviour is reproduced by the theory for harmonic 13, but is not so well captured for the 15th harmonic. We understand this to be due to small offsets in the calculated field-free bound energy states in helium arising from the use of a truncated ADC hierarchy, i.e. n=1.
The observations we have made extend our knowledge of strongly laser-dressed helium. We have observed the dependence of the XUV absorption modulations on the dressing field polarisation state and intensity, bridging the gap between the perturbative and strong-field intensity regimes. Our results are a step towards understanding the strongly driven dynamics in more complex systems.