Controlling Floquet states on ultrashort time scales

The advent of ultrafast laser science offers the unique opportunity to combine Floquet engineering with extreme time resolution, further pushing the optical control of matter into the petahertz domain. However, what is the shortest driving pulse for which Floquet states can be realised remains an unsolved matter, thus limiting the application of Floquet theory to pulses composed by many optical cycles. Here we ionized Ne atoms with few-femtosecond pulses of selected time duration and show that a Floquet state can be observed already with a driving field that lasts for only 10 cycles. For shorter pulses, down to 2 cycles, the finite lifetime of the driven state can still be explained using an analytical model based on Floquet theory. By demonstrating that the amplitude and number of Floquet-like sidebands in the photoelectron spectrum can be controlled not only with the driving laser pulse intensity and frequency, but also by its duration, our results add a new lever to the toolbox of Floquet engineering.

: (a) Raw SB normalized intensities as extracted from the photoelectron spectra obtained with a quasi-monochromatic IR pulse. (b) Absolute electron counts calibration factor for the two different sets of TOF voltages used in the experiment. energy region around the corresponding SB peak. It is important to notice that since the photoelectron spectra are acquired with a TOF spectrometer, the absolute value for the photoelectron counts is not given. To circumvent this problem we normalized the SB signal by the total area of the XUV-only photoelectron spectrum, I 0 . The results reported in Fig. S1(a) show the expected behaviour with I IR , even if the absolute value of the SB normalized intensity is lower than what predicted by the theory. This discrepancy originates from a non-flat energy transfer function of the TOF spectrometer which usually depends on the actual voltages applied to its electrostatic lenses and on the geometry of the interaction. To estimate the non-flat response of the TOF we proceeded as follows. For a chosen SB order n, we considered separately the raw data in Fig. S1(a) which have been taken under the same experimental conditions. Then, using a fitting procedure, we calculated the parameter α n which minimizes the rms distance between the raw data and the theoretical prediction A 2 n =J n −p n E 0 ω 2 0 , − Up 2ω 0 . The values of α n as a function of the SB order for the two sets of lenses used in the experiment are reported in Fig. S1(b). The vertical error bars indicate the confidence interval associated with the estimation of α n . We find that the calibration factor α n grows almost linearly with n between 0.5 and 4. Once the calibration factor has been evaluated, the experimental data can be normalized before being directly compared to the theoretical curves (Fig. 2c of the main manuscript). For a given SB, the total relative error associated to the estimation of α n , represented by the shaded areas in Fig. 2c of the main manuscript, is calculated by summing the squared confidence intervals of the two sets reported in Fig. S1(b).

S1.2 Short IR pulses
To generate short IR pulses of controlled duration, the IR beam is sent to a hollow-core fiber compression setup before passing through the broad-band λ/2-waveplate and the polarizer.
Also in this case, the time duration of the IR pules is measured with the FROG technique ( Fig.1e of the main manuscript) while the intensity is set to ∼ 5 × 10 11 W/cm 2 by adjusting the transmitted power with the λ/2-waveplate. We performed several pump-probe experiments by collecting the photoelectron spectra while the XUV-IR delay was changed with a step of 3-4 fs (smaller step were used with shorter IR pulses). The resulting spectrograms were mediated to obtain the final spectrogram and reduce the experimental noise. A photoelectron spectrum without the IR was measured each 5 delay steps by blocking the beam with a mechanical shutter during the scan. This XUV-only spectrum is used both to normalize the SB signal as discussed in the main text and to estimate: (i) the variation of the XUV photon flux during the measurement, (ii) the electronic noise which produces a non-zero background below the SB peaks. The raw delay-dependent SB signal was obtained by integrating the spectrogram in energy after the effects (i) and (ii) had been corrected for. The SB intensity is then given by the maximum of the delay-dependent signal which can be estimated, for example, with a Gaussian fit of the energyintegrated delay-dependent SB profile. The raw results are reported in Fig. S2. While each SB signal qualitatively follows the theoretical prediction of Eq. (4) of the main manuscript, we notice some discrepancies which are related to the not perfect Gaussian shape of the pulses and to the residual variations of the IR central wavelength and intensity from the nominal values.
This effects can be accounted for as described in the next session. Figure S2: (a) Raw SB normalized intensities as extracted from the photoelectron spectrograms obtained with short IR pulses. (b) Theoretical prediction for perfect IR Gaussian pulses centered around 800 nm and with I IR = 5 × 10 11 W/cm 2 .

S1.2.1 Pulse temporal reconstruction
From the shape of Eq. (4) of the main manuscript it appears to be clear that small variations of the IR central wavelength and intensity can influence the value of Λ n . Therefore, it is crucial to estimate the exact experimental parameters to be able to recast the results over a common nominal IR intensity and wavelength. One possibility is to use ptychographic reconstruction algorithms to extract both the IR and XUV characteristics directly from the experimental spectrogram (2,5). Before applying the reconstruction algorithms one has to characterize the finite spectral resolution of the TOF spectrometer and acquisition system, which may artificially broaden the photoelectron spectrum. To estimate the TOF response, we changed the XUV spectral bandwidth between 200 and 400 meV (full width half maximum, FWHM) by varying the aperture of the central slit of the TDCM, and measured the associated XUV-only photoelec- Figure S3: Photoelectron spectral width measured with the TOF spectrometer as a function of the radiation spectral width measured with the XUV spectrometer (black full markers). The solid line represent the expected behavior obtained by convoluting the radiation bandwidth with the TOF energy resolution (reported in the legend).
tron spectra (black markers in Fig. S3). For an ideal spectrometer the two quantities should correspond (black dashed line). In our case we observe that the photoelectron FWHM is generally bigger than the FWHM measured by the XUV spectrometer at the end of the beamline, suggesting that the TOF response can be modeled by a Gaussian bell with a FWHM width of 250 meV (yellow curve in Fig. S3). As the photoelectron spectrum significantly deviates from the XUV spectrum for such harmonic pulses, the application of a standard reconstruction algorithm, which assumes the two quantities to be identical, will strongly underestimate the time duration of the XUV pulse and overestimate the duration of the IR pulse. To avoid this issue, we developed a reconstruction procedure based on the SFA formulation of the photoelectron signal (Eq. (S1)) and capable to account for the actual XUV spectrum and the TOF response function.
In brief, following a common approach used in the reconstruction of FROG-like measurements, the photoelectron spectrum is described by the Fourier transform of the product between a pulse (the XUV field) and a pure phase gate (the exponential which depends on the IR). Starting from an educated guess of pulse and gate, the algorithm corrects these quantities while imposing the amplitude of the simulated spectrogram to be identical to the experimental one. To improve the accuracy, in the present case we forced the code to assume the IR temporal envelope to be the one independently measured with a second-harmonic FROG, the TOF response function to be Gaussian with a width of 250 meV, and the XUV spectral intensity to be identical to the one measured by the XUV spectrometer (assumed to be more reliable as this spectrometer has a resolution ≤ 50 meV in this range). Moreover, to minimize the effect of the experimental noise, the algorithm directly runs over the differential spectrogram (pump-probe trace minus the XUV-only spectrum). An example of reconstruction is reported in Fig. S4 for the same data of Fig  As expected the IR wavelength, λ IR ( Fig. S5(a)), observes a non-negligible blue shift while moving towards shorter pulses. The IR intensity on target ( Fig. S5(b)) ranges between 4.6 and 10.6 × 10 12 W/cm 2 , while the reconstructed XUV FWHM time duration, σ x , is constant across the different measurements (σ x = 12.3 ± 0.4 fs). In order to set the correct pulse energy, the IR intensity is estimated in real time before performing the pump-probe scan by measuring the IR temporal and spatial profiles. While the temporal measurement is sufficiently accurate, the FROG setup has been calibrate and tested against the results obtained on target, the measurement of the spatial profile is affected by a considerable background due to the limitations of the detection system. As a results, the IR intensity that is estimated in real time in the lab suffers by a poor accuracy. Furthermore, the effective IR intensity felt by the Ne atoms depends on the degree of spatial overlap at the target position. For these reasons the reconstruction procedure allows us to obtain a more accurate estimation of the effective I IR and correctly calibrate the results. It is important to stress that while both the reconstruction procedure and our model start from the SFA description of the photoelectron spectrogram, they are not based on the same approximations. For example, the reconstruction algorithm applies a global CMA (p → p c ) instead of considering the SB momenta p n . The approximations introduced in the reconstruction algorithm have been tested and proved to hold under the experimental conditions used in this work, thus justifying its employment.

S2 Theoretical methods
In this section we describe in details the numerical models used for the calculations and simulations. Atomic units are used throughout the document.

S2.1 SFA model
Within the strong-field approximation (SFA), the collection of photoelectron spectra obtained by ionizing an atom of ionization energy I p with an XUV field E x (t) and an IR field where we assumed the IR and XUV fields to be linearly polarized with parallel polarization and we considered only the photoelectrons emitted along the polarization direction. We note that in atomic units ω = p 2 /2.
If we assume the dipole to be constant in the momentum range under consideration, by exploit- ) 2 dt ′ results in a constant phase term which does not affect that final spectrogram of Eq. (S1), we can write: All the simulations performed in this work are obtained by numerical evaluation of Eq. (S1) and assuming the IR and XUV pulses to be Gaussians of the form: where the complex quantities γ x = 2 c 2 2 − i βx 2 and γ r = 2 a 2 2 − i βr 2 depend on the group-delay dispersion (GDD) coefficients β x and β r , and the XUV and IR intensity FWHM time durations, σ x = c 2 log(2) and σ r = a 2 log(2).

S2.2 Floquet (Volkov) states
Let's consider a free electron described by the following plane wave: In the presence of an additional monochromatic IR field, the wave function of the free electron is described by the Volkov wave (7): where p is the electron momentum and r is the position vector. Following what reported by Madsen (8), and assume that we collect electrons whose momentum p is parallel to the polarization of the IR pulse and XUV pulses, the Volkov wave can be rewritten as: whereJ n indicates the generalized Bessel function defined as the following sum of products of ordinary Bessel functions:J If we neglect the spatial dependence, the time-dependent part of the Volkov state can be rewritten in the form: (S11) Using the dipole approximation, the photoelectron spectrum resulting from the ionization of an atomic state |0⟩ = e iIpt ϕ 0 (r) with a field E x (t) into a Volkov state ψ V (r, t) is then given by (8): (S12) If the Fourier transform of the spatial part of the initial atomic state ϕ 0 (r) does not vary significantly in the energy region of interest, i.e. if the atomic dipole d n = ⟨ e ip·r (2π) 3/2 |µ|ϕ 0 (r)⟩ ≃ 1, we get: which yields the same result of the SFA formula, as we will discuss in the next Section.

S2.3 Link between the two models
Starting from Eq. (S2) and assuming a monochromatic IR field of the form of Eq. (S6), we can developed integral in the phase term into two contributions of the form: and where we introduced the ponderomotive energy U p = E 2 0 /(4ω 2 0 ). We can thus rewrite Eq. (S2) as: (S16) Using the Jacobi-Anger expansion, e iz sin θ = ∞ n=−∞ J n (z)e inθ , for the two sinusoidal phase terms in the above integral we get (8): whereJ n (x, y) indicates the generalized Bessel function of order n defined by Eq. (S9). It is now possible to rewrite Eq. (S16) as: (S18) Since − sin(θ) = sin(−θ), starting from the definition of Eq. (S17), it is easy to show that Therefore, if we substitute k = −n, Eq. (S18) becomes which proves that the SFA approach and Eq. (S13) yield the same result: with A n (p, E 0 , ω 0 ) defined by Eq. (S11).

S2.4 Photoelectron SB signal
If we concentrate on the positive part of the spectrum, use the XUV field definition of Eq. (S4) and consider that in atomic units p 2 /2 = ω, we can rewrite Eq. (S21) as: where we have introduced the SB energy ω n = ω x +nω 0 −U p −I p . If the XUV pulse bandwidth is narrow enough, the square modulus of the sum in the above equation corresponds to the sum of the square moduli. Furthermore, within each SB the momentum dependence of A n can be neglected by substituting p → p n = √ 2ω n . Therefore, the photoelectron spectrum is given by S(ω) = ∞ n=−∞ SB n (ω) where the quantity SB n (ω) is the modulus square of the Fourier transform of the XUV field envelope, multiplied by the Floquet ladder amplitude evaluated at the SB central momentum: This quantity represents the n-th SB signal which depends, for monochromatic IR fields, only on the final photoelectron energy ω.
If the XUV bandwidth is narrow enough, from Eq. (S22) we get: is the Fourier transform of the XUV envelope shifted in ω n . In this case the time evolution of the photoelectron wavepacket originating solely from the IR dressing, s(t), can be estimated as: and therefore which represents the time evolution of the scattering amplitude without considering the XUV temporal properties.

S2.5 Driving pulses of finite duration
In case of IR pulses of finite duration the vector potential is given by: and the associated electric field is: If the slowly varying envelope approximation (SVEA) can be applied, i.e. if the pulse duration is bigger than its oscillation period σ IR ≫ 2π/ω 0 , and dA 0 (t)/dt ∼ 0, then where E 0 (t) = A 0 (t) ω 0 .
Referring to Eq. (S2), using the fact that the pulse is finite (A 0 (−∞) = 0) and under the SVEA (dA 0 (t)/dt ∼ 0), the integral in the IR phase term yields the following two contributions: where we have introduced an "instantaneous" ponderomotive energy of the form: U p (t) = E 2 0 (t)/(4ω 2 0 ). The integral in Eq. (S2) can now be rewritten as: If the pulse envelope E 0 (t) evolves on a slower time scale than the carrier period T IR = 2π/ω 0 , following the two-time approach used in Ref. (9) (see Sec. S2.9), it is possible to show that: If we extend this approach to the second complex exponential and use the generalized Bessel functions, the spectrogram becomes: which is identical to Eq. (S20) besides the time dependence in the argument of the generalized Bessel functions. Focusing only on the positive spectrum and expressing the XUV field as: where we omitted the pure phase term e iωxτ as it disappears with the | · | 2 . Also in this case, if the XUV bandwidth is small when compared to the IR photon energy, the total spectrum can be seen as the sum of spectrally separated contributions: After applying the central momentum approximation, the SB signal is hence given by: where the symbol F denotes the Fourier transform.

S2.6 Low-intensity limit
Let's analyze the behavior of the generalized Bessel functionsJ n (x, y), for both arguments that tend towards zero. For a standard Bessel function of the first kind it holds: where sg(x) donetes the sign of x. We can use the approximation of Eq. (S41) to obtain the following limit forJ n (x, y): If the field amplitude and the electron final momentum are not too big, than Up(t) 2ω 0 ≪ pE 0 (t) , and The weight of the terms in the above equation decreases rapidly with |k|, and the terms which give the major contributions are the ones for which 0 < k < n/2 for n > 0 and with n/2 < k < 0 for n < 0. If n ̸ = 0, the time evolution of those major terms follows E 0 (t) |n−2k|+2|k| = E 0 (t) |n| = E |n| 0 g(t) |n| , where we have expressed the pulse envelope E 0 (t) as the product of a normalized function g(t) and an amplitude E 0 . In view of this limit, we can rewriteJ n as follows: where U p = E 2 0 /(4ω 2 0 ). In this limit the temporal evolution of the n-th generalized Bessel follows the n-th power of the pulse normalized envelope times the amplitude of the n-th state of the Floquet ladder (Eq. (S11)) for a monochromatic pulse of amplitude E 0 = E 0 (t = 0) (corresponding to the pulse maximum amplitude). For n = 0 the generalized Bessel can instead be approximated with: While Eq. (S45) has been derived for pE 0 (t)/ω 2 0 → 0, we found Eq. (S46) to be a good approximation on a broader range, i.e. until the Bessel functions in Eq. (S42) are monotonically increasing in their arguments. Figure S7 shows the square of the first order generalized Therefore the approximation gets worse (Figs. S7(g), (h)).
It is important to underline that if the approximation holds forJ 1 , all the more reasons it will hold for the other orders. Indeed, with increasing order the maximum of the generalized Bessel function is reached later in intensity ( Fig. S8(a)). For n = −1, instead, the central SB momentum p−1 is lower (corresponding to the 22 nd harmonic instead of the 24 th ). Figures S8(b), (c) show the square ofJ 1 as a function of time I IR , calculated for the same pulse as in Fig S7 with the exact formula (Fig. S8b) and with the approximated expression of Eq. (S46) (Fig. S8c).  In the range where the approximation is valid, Eq. (S38) thus becomes: If the XUV bandwidth is small compared to the IR photon energy then the SB signal (n ̸ = 0) of Eq. (S40) is now given by: which is the same as Eq. (3) of the main manuscript.
In this limit, the time evolution of the scattering amplitude without considering the XUV temporal properties goes as: where A ′ 0 (t) is given by Eq. (S47). Figure S9

S2.7 Gaussian pulses
If both pulses have a Gaussian envelope of the form of Eqs. (S3) and (S4), the SB signal of Eq. (S49) becomes: where γ r,n = γ r / |n|. The sideband signal in the experimental spectrogram is given by where γ xr = γ 2 r,n γ 2 x γ 2 r,n + γ 2 If we now evaluate its maximum (i.e. compute the value at τ = 0), and normalize the result by the area of the XUV-only photoelectron spectrum, we find the expression of Eq. (4) of the main manuscript: This proves that it is possible to extract the Floquet ladder amplitude A n (Eq. (S11)) directly from the measurements if we correct the normalized SB intensity for the XUV and IR time durations. The higher the SB order, i.e. the higher n in Eq (S58), the stronger the amplitude reduction. Therefore high-order ladder states are more affected by the finite time duration of the IR pulse. As expected, at the long-pulse limit we get: and the ladder amplitudes coincide with the ones of Eq. (S11) for the monochromatic case.
Please notice that the same limit is reached with σ x → 0, but in this case XUV spectral bandwidth goes to infinity and Eq. (S39) is no longer valid.

S2.8 Comparison with SFA calculations
To test the result of Eq. (S58), we performed SFA calculations as described in Sec. S2.1 and compare the value of Λ 2 n predicted by the model with the one directly extracted from the simulated spectrogram. To mimic the experimental conditions we simulated the single harmonic spectrogram (SHS) generated by harmonic 23 in Neon with an XUV time duration σ x = 11 fs and an IR time duration, σ r , that varies between T IR and 22T IR with T IR = 2.6685 fs (i.e. corresponding to a wavelength of 800 nm). We repeated the calculations for different IR intensities in the range between 10 8 and 10 12 W/cm 2 . An example of different SHSs at the experimental IR intensity of 5 × 10 11 W/cm 2 and for transform limited (TL) IR and XUV pulses is reported in Fig. S10. As expected, the numerical simulations confirm that the strength of the SBs depends on the IR time duration. The quantity Λ 2 n is extracted from the simulated spectrograms by inte- Figure S10: SHSs calculated for H23 in Neon with σ x = 11 fs (TL). The IR average intensity is set to I IR = 5 × 10 11 W/cm 2 while σ IR = 1T IR (a), 2T IR (b), 4T IR (c) and 22T IR (d).
grating the n-th SB signal around its nominal energy position in a 1.1-eV energy window and evaluating the maximum of the resulting delay-dependent curve. At low intensity, the maximum is located at τ = 0 fs. As done for the experimental data, the extracted value is normalized by the area of the XUV spectrum, following what defined in Eq. (S58). Figure S11 reports a comparison between the amplitudes Λ 2 n extracted from the SFA calculations and calculated with the Figure S11: Comparison between the normalized SB amplitude Λ 2 n /A 2 n calculated with SFA or with Eq. (S58) for σ r /σ x between 0.5 and 5.33 and 10 8 < I IR < 10 12 W/cm 2 . In all the calculations σ x = 11 fs while the IR duration is scanned between 2T IR and 22T IR . Each couple of figures corresponds to a SB order. analytical expression of Eq. (S58). In order to allow a direct comparison between the SB amplitudes obtained at different intensities, we decided to plot the quantity Λ 2 n /A 2 n (Figs. S11(a), (c), (e), (g)). As it is possible to notice the SFA calculations (colored surfaces and curves) nicely agree with the theoretical model (black surface and curve) at the low intensity and long pulse limit. For short IR pulse durations or for IR intensities higher than 10 11 W/cm 2 the SFA results are generally higher than what predicted by the analytical model. The larger error between SFA and theory is observed for the first SB (n = 1, Figs. S11(e) and (f)). Nevertheless, at the average IR intensity (5 × 10 11 W/cm 2 ) and for the σ r /σ x range used in the experiment, the relative error, (Λ 2 n − A 2 n )/A 2 n , stays below 8% (Fig. S11(f)). For the second SB (n = 2, Fig. S11(g)) the error stays below 5% (Fig. S11(h)). While all the above calculations have been performed with transform-limited pulses, the validity of Eq. (S58) has been tested against a finite GDD of both pulses. The degree of agreement found is comparable to what reported in Fig. S11 for TL pulses. If the actual time duration of the pulse is considered, the model predicts the correct SB strength behavior also for higher chirp orders, unless the spectral chirp is so severe to cause a clear deviation of the pulse temporal envelope from a Gaussian shape.

S2.9 Two-time approach
We here show how the approach used in Ref. (9) can be used to justify Eq. (S36). Let's start by .
(S61) are calculated by considering the periodicity ofV (t, t ′ ) in t ′ . We get: where indeed Using Eq. (S61) and Eq. (S63) we can now writẽ from which, using the Jacoby-Anger expansion we get