Observation of Rabi dynamics with a short-wavelength free-electron laser

Rabi oscillations are periodic modulations of populations in two-level systems interacting with a time-varying field1. They are ubiquitous in physics with applications in different areas such as photonics2, nano-electronics3, electron microscopy4 and quantum information5. While the theory developed by Rabi was intended for fermions in gyrating magnetic fields, Autler and Townes realized that it could also be used to describe coherent light–matter interactions within the rotating-wave approximation6. Although intense nanometre-wavelength light sources have been available for more than a decade7–9, Rabi dynamics at such short wavelengths has not been directly observed. Here we show that femtosecond extreme-ultraviolet pulses from a seeded free-electron laser10 can drive Rabi dynamics between the ground state and an excited state in helium atoms. The measured photoelectron signal reveals an Autler–Townes doublet and an avoided crossing, phenomena that are both fundamental to coherent atom–field interactions11. Using an analytical model derived from perturbation theory on top of the Rabi model, we find that the ultrafast build-up of the doublet structure carries the signature of a quantum interference effect between resonant and non-resonant photoionization pathways. Given the recent availability of intense attosecond12 and few-femtosecond13 extreme-ultraviolet pulses, our results unfold opportunities to carry out ultrafast manipulation of coherent processes at short wavelengths using free-electron lasers.

Rabi oscillations are periodic modulations of populations in two-level systems interacting with a time-varying field 1 .They are ubiquitous in physics with applications in different areas such as photonics 2 , nano-electronics 3 , electron microscopy 4 , and quantum information 5 .
While the theory developed by Rabi was intended for fermions in gyrating magnetic fields, Autler and Townes realized that it could also be used to describe coherent light-matter interaction within the rotating wave approximation 6 .Although intense nanometer-wavelength light-sources have been available for more than a decade [7][8][9] , Rabi dynamics at such short wavelengths have not been observed directly.Here we show that femtosecond extremeultraviolet pulses from a seeded free-electron laser 10 can drive Rabi oscillations between the ground state and an excited state in helium atoms.The measured photoemission signal revealed an Autler-Townes doublet as well as an avoided crossing, phenomena that are both trademarks of quantum optics 11 .Using theoretical analyses that go beyond the strong-field approximation 12 , we found that the ultrafast build-up of the doublet structure follows from a quantum interference effect between resonant and non-resonant photoionization pathways.Given the recent availability of intense attosecond 13 and few-femtosecond 14 extremeultraviolet pulses, our results offer opportunities to carry out ultrafast manipulation of coherent processes at short wavelengths using free-electron lasers.
The advent of free-electron laser (FEL) facilities, providing femtosecond light pulses in the gigawatt regime at extreme-ultraviolet (XUV) or X-ray wavelengths, has opened up new prospects for experiments in isolated atoms and molecules in the gas-phase 15,16 .Over the last decade, pioneering results concerning multi-photon ionization of atoms 17 , and small molecules 18 were ob-tained using pulses from self-amplified spontaneous emission (SASE) FEL sources 8 .However, these pulses are prone to low degree of coherence and poor shot-to-shot reproducibility due to the inherent instability pertinent to the SASE process.As a result, despite theoretical predictions to observe Rabi oscillations at short wavelengths [19][20][21][22] , effects from these oscillations on the measured spectra were only indirect 23,24 .Instead, XUV pulses from a SASE FEL have been used as pumps that allowed subsequent ultrafast Rabi dynamics to be driven by laser pulses at nearinfrared wavelengths 25 .In this regard, XUV pulses from a seeded FEL, such as FERMI 10 , with its high temporal and spatial coherence, and large peak intensity can permit studying coherent light-matter interactions 26 as well as phase-dependent interference effects of the wavefunction 2 .
According to the Rabi model 1 , if a two-level atom initially in its ground state |a , is subjected to an interaction with a field of frequency ω that couples it to the excited state |b , the probability for excitation varies sinusoidally in time as: The oscillating population leads to a symmetric doublet structure in the frequency domain, known as an Autler-Townes (AT) doublet.The splitting is given by the generalized Rabi frequency: 2 , where ∆ω = ω − ω ba , is the detuning of the field with respect to the atomic transition frequency, ω ba .
The Rabi frequency for light-matter interaction within the dipole approximation is: Ω = eE 0 z ba /h, with E 0 being the electric field amplitude, z ba the transition matrix element, h the reduced Planck's constant and e the elementary charge.In addition to the periodic population transfer P b (t), the coherent dynamics is further associated with sign changes of the oscillating amplitudes for the two states.For fermions, such sign changes of the wavefunction can be connected to rotations in real space 1 that have been measured for neutron beams in magnetic fields 27 .Analogous sign changes in quantum optics were studied using Rydberg atoms to determine the number of photons in a cavity 28 .Recently, the sign changes in Rabi amplitudes have been predicted to strongly alter AT doublet structures in photo-excited atoms, when probed by attosecond XUV pulses 29 .
Here, we investigate the Rabi dynamics at XUV wavelengths in helium atoms induced by an intense pulse from the FERMI seeded FEL that couples the two levels: |a = 1s provided that the electric field can be approximated as a flat-top shape in time.The sign changes associated with these Rabi amplitudes are essential to understand the ultrafast build-up of AT doublets from |a or |b , by absorption of two or one resonant XUV-FEL photons.The AT doublet emerges due to a destructive interference effect between photoelectrons ejected before and after the first sign change, which is found to occur at 1/2 and 1 Rabi period for the amplitudes: a(t) and b(t) with ∆ω = 0, respectively.This is in agreement with the results from the analytical model presented in Fig. 1b.
Measured photoelectron spectra, displayed in Fig. 2a, exhibit an AT splitting of hΩ = 80 ± 2 meV (see Methods for details about the blind deconvolution procedure used here).The corresponding Rabi period: 2π/Ω ≈ 52 fs, given its proximity to the FWHM of the XUV-FEL pulse, suggests that the experiment was performed in a regime of ultrafast AT doublet formation close to a single Rabi cycle.A slight blue detuning of the XUV light by almost 11 meV, relative to the atomic transition, is required to record a symmetric AT doublet (black squares in Fig. 2a).A strong asymmetry is observed when the FEL frequency is detuned to the red (red circles in Fig. 2a) or blue (blue diamonds in Fig. 2a) side of the symmetric doublet.The asymmetry of the AT doublet is quantitatively well-reproduced by ab initio numerical simulations for helium within the time-dependent configuration-interaction singles (TDCIS) approximation 33 , as shown in Fig. 2b.
Gaussian pulses were used with parameters chosen to match the experimental conditions with effective intensity of 2 × 10 13 W/cm 2 (as obtained from Ω) and pulse duration (FWHM) = 56 fs (see Methods for details).It is worth noting that the Rabi dynamics is sensitive to the exact shape of the driving pulse.For instance, a Gaussian pulse can induce more Rabi oscillations than a flat-top pulse with same FWHM by a factor of π/(2 ln 2) ≈ 1.5.Thus, the calculated photoelectron spectra from the analytical model using flat-top pulses in Fig. 1b for 3/2 Rabi periods agree well with those from the TDCIS calculations using Gaussian pulses having FWHM close to a single Rabi period in Fig. 2b.Clearly, the AT doublet manifests itself between 1 and 3/2 Rabi periods.
The difference in kinetic energy (∼ 0.4 eV) of the symmetric AT doublet between experiment and theory (see Fig. 2a and 2b, respectively) is attributed to electron correlation effects not included in the TDCIS calculations that increase the binding energy beyond the Hartree-Fock level.The observed asymmetry in the AT doublet cannot be explained by a breakdown of the RWA because the experiment is performed at a resonant weak-coupling condition 6 : ω ≈ ω ba and Ω/ω ba = 0.34%.
Instead, we express the Rabi amplitudes from Eq. 1 in terms of their frequency components and find that a(t) has two asymmetric components that are proportional to (1 ± ∆ω/W ), while b(t) has two symmetric components that are proportional to ±Ω/W .Using the analytical model with 3/2 Rabi periods, we confirm that the AT doublet from |b is symmetric, while that from |a is asymmetric, as shown in Fig. 2c and 2d, respectively.Quite remarkably, the observed asymmetry in the experiment suggests that the photoelectron signal contains significant contributions from the two-photon ionization process from |a .How is this possible given that the electric field amplitude: E 0 = 0.02388 a.u.implies an ionization-probability ratio of 10 4 : 1 in favor of the one-photon process from |b ?
We propose that the two-photon signal from |a can compete with the one-photon signal from |b due to constructive addition of non-resonant intermediate p-states (see grey bound and continuum states in Fig. 3a).This leads to a giant localized wave: |ρ =b , in comparison with the normalized wavefunction for |b , as shown in Fig. 3b.The scaling factor due to atomic effects is estimated to be ∼ 1 : 10 4 in favor of the two-photon process (see Extended Data Table 1 for the matrix elements).Thus, we can explain why the XUV-FEL pulse is intense enough for the nonresonant two-photon process from the ground state to be comparable with the one-photon process from the resonant excited state.In general, addition of two pathways leads to quantum interference that depends on the relative phase.From Fig. 3b we can notice that the giant wave oscillates out of phase with the excited state close to the atomic core, which affects the signs of the matrix elements (see Extended Data Table 1).The ultrafast build-up of the AT doublet can be used to study the resulting interference phenomenon in time.In order to understand this phenomenon, we have used the analytical model to perform calculations where the one-and two-photon contributions are added coherently to simulate the angle-integrated measurements.In Fig. 3c we show how the resonant case (∆ω = 0) leads to a strongly asymmetric AT doublet after one Rabi period.Fig. 3d indicates that a blue detuning (∆ω = 62 meV) leads to the symmetric AT doublet at an earlier time between 0.5 and 1 Rabi period.The advancement of the AT doublet in time follows from the faster Rabi cycling at the rate of generalized Rabi frequency.The required blue shift for the symmetric AT doublet is the signature of an interplay between the one-photon and two-photon processes that depends on the exact pulse form.A general loss of contrast in the AT doublet structures is found by considering the effect of an extended gas target in our model.However, the two-photon doublet was found to be less sensitive to the volume averaging effect when compared to the one-photon doublet (see Methods for details), allowing us to clearly observe the AT doublet in the measured photoelectron signal.
To provide further evidence in support of the coherent interaction between the helium atoms and the XUV-FEL pulses, we show that the ultrafast emergence of the AT doublet can be interpreted in terms of the dressed-atom picture with coupled atom-field energies: ± = ( a + b +hω±hW )/2.
One photon energy above these coupled energies implies final photoelectron kinetic energies: 5873 eV is the ionization potential of helium.In Fig. 4a, kinetic energies are labeled with the uncoupled atom-field states 34 , |a, 1 and |b, 0 .The experimental results in Fig. 4b can be understood as one photon above |a, 1 at large detuning of the XUV-FEL pulse.This is because the interaction there is weak and the atom remains mostly in its ground state, |a , such that two photons are required for ionization.In contrast, both coupled energies appear close to the resonance to form an avoided crossing in kinetic energy.This is a clear signature of coherent Rabi dynamics taking place in the two-level atom.Note that the avoided crossing appears at a blue detuning from the resonant transition, ∆ω = 0 (denoted by the dashed vertical line), revealing the quantum interference between the two pathways from the ground state |a and the excited state |b .Similar results were obtained from the TDCIS simulations (Fig. 4c) and the analytical model with contributions from both |a and |b for 3/2 Rabi periods (Fig. 4d).
The observed blue detuning for the experimental avoided crossing (∼ 11 meV) is very well reproduced by TDCIS calculations (∼ 14 meV).The enhanced shift of the AT doublet to blue detuning in the analytical model is an effect of the pulse-envelope that can be reproduced with TDCIS using smoothed flat-top pulses.
Our results show that it is now possible to simultaneously drive and interrogate ultrafast coherent processes using XUV-FEL pulses.Prior attempts to understand Rabi dynamics at short wavelengths have relied on the strong-field approximation, where the influence of the atomic potential is neglected, leading to an inconsistent AT doublet when compared with numerical simulations 19,20 .
In contrast, our analytical model includes the full effect of the atomic potential and Rabi dynamics in the two-level subspace, while the remaining transitions to and within the complement of the Hilbert space are treated by time-dependent perturbation theory.Consequently, we could establish a unique mechanism in the form of a giant Coulomb-induced wave from the ground state to explain why the non-resonant two-photon process can compete with the resonant one-photon process and generate quantum interference effects at the high intensities provided by the XUV-FEL beam.With this model, we now understand how Rabi oscillations can prevail at short wavelengths despite photoionization losses from the neutral atom.Together with our experimental approach of using two-photon ionization as an in-situ probe of the coherent population transfer, which does not rely on any additional laser probe field, the scheme proposed here becomes applicable to other quantum systems as well.With the ongoing developments of seeded FEL facilities around the world 35,36 capable of providing coherent light pulses down to few-ångstrom wavelength, our findings can inspire future studies involving core-level electrons in multi-electronic targets, such as molecules, and nano-objects at ultrafast timescales.

Experiment
The experiment was carried out at the Low Density Matter (LDM) beamline of FERMI 37 .A pulsed Even-Lavie valve, synchronized with the arrival of the FEL pulse served as the target source.The target gas jet was estimated to be a cone with 2 mm diameter at the interaction region.We measured the photoelectron spectra at and around the 1s 2 → 1s4p transition in helium, using a 2-m long magnetic bottle electron spectrometer (MBES).The gas jet, FEL beam, and magnetic bottle axes are mutually perpendicular, with the first two being on the horizontal plane of the laboratory, and the last one in the vertical direction.Before entering into the flight tube of the MBES, the photoelectrons were strongly retarded to below 1 eV kinetic energy to achieve high spectral resolution (E/∆E ≈ 50).To suppress any short-term fluctuation arising from the instability of the FEL, we performed a 'round-trip' scan across the wavelength range, 52.50 ↔ 51.80 nm.Empirically, the FWHM of the XUV-FEL pulse duration (τ xuv ) can be approximated 32 to be in between τ seed /n 1/2 and 7τ seed /6n 1/3 .Here, τ seed ≈ 100 fs is the duration (FWHM) of the seed pulse (wavelength: 261.08 nm) and n = 5 is the harmonic order for the undulator.It leads to τ xuv = 56 ± 13 fs that matches very well the FWHM of ∼ 66 fs, obtained from the simulation of the FEL dynamics using PERSEO 38 .The spectral bandwidth (FWHM) of the pulse was estimated using PERSEO to be around 0.13 nm at the central wavelength of λ = 52.216nm.Extended Data Fig. 1a and 1b displays the simulated temporal and spectral profiles of the FEL pulse, respectively.At best focus, the spot-size (FWHM) was estimated to be 12 µm.We measured the energy per pulse at the output of the FEL undulator to be around 87 µJ, which refers to the full beam including all photons contained in the transverse Gaussian distribution.In order to consider those, we used 4σ as the FEL beam diameter at best focus, where σ = 12/2.355≈ 5.1 µm.Hence, the beam waist (w 0 ) is given by w 0 = 2σ = 10.2 µm, along with a Rayleigh length of: πw 2 0 /λ ≈ 6.3 mm.

Data analysis
To filter the measured photoelectron spectra on a shot-to-shot basis, we used the photon spectrum recorded by the Photon Analysis Delivery and REduction System (PADRES) at FERMI to determine the bandwidth (FWHM) of the XUV pulse.Any shot without the photon spectrum was rejected: out of 355000 shots, 354328 shots were retained.All the shots having more than 65 meV FWHM width were discarded (see, Extended Data Fig. 2a).Note that the simulated value of the photon bandwidth (59 meV) lies within the filtering window of 20 -65 meV.Additionally, we chose only the shots with integrated spectral intensities ranging from 0.8 × 10 5 to 1.6 × 10 5 in arbitrary units (see, Extended Data Fig. 2b).The filtered shots were sorted into 30 photon-energybins, uniformly separated from each other by ∼ 13 meV and covering the entire photon energy window of the wavelength scan (see, Extended Data Fig. 2c).Overall, only 304192 shots (filtering ratio of 0.857) out of the raw data were retained.The measured photoelectron spectra, following shot-to-shot filtering, are shown in Extended Data Fig. 3.The avoided crossing is only faintly visible here.To obtain the clear avoided crossing from Fig. 4b, we deconvoluted the photoelectron spectra for three photon energies near the 1s 2 → 1s4p transition using the Richardson-Lucy blind iterative algorithm 39 .To reduce the noise introduced during the deconvolution we incorporated the Tikhonov-Miller regularization procedure into the algorithm 40 .The outcomes are shown in Ex-tended Data Fig. 4. Following deconvolution, the values of FWHM for the Gaussian instrumentresponse-functions were found to be: 70.9 ± 1.2 meV, 69.6 ± 2.4 meV and 69.4 ± 1.4 meV, for the three photon energies.These values match well the combined resolution of ∼ 65 meV, obtained from the photon bandwidth and the kinetic energy resolution of the MBES.No filter, either metallic or gaseous, was used along the path of the FEL beam.Hence, a minor contribution (< 5%) from the second-order light can be noticed as an asymmetric tail close to 22.8 eV kinetic energy (see, Extended Data Fig. 4a and 4b).To rule out any artifact from the fluctuations of the FEL pulse properties, we used another filtering criterion for the photon bandwidth (0 -45 meV) and the integrated spectral intensity (1 × 10 5 -3 × 10 5 arb.u.).The corresponding deconvoluted photoelectron spectra at 23.753 eV is shown in Extended Data Fig. 5, along with that from Fig. 2a of the main text.No significant change in the AT doublet structure due to change in filtering criteria could be seen.Finally, for a transform-limited Gaussian pulse, τ xuv can vary between 30 -90 fs from shot to shot that encompasses its empirical value: 56 ± 13 fs.Since τ 2 xuv is significantly higher than the absolute value of the simulated group-delay-dispersion of the FEL pulse: −690 fs 2 , no effect due to the linear chirp was considered in the theoretical calculations.

Intensity averaging over macroscopic interaction volume
To better understand the experimental results, we perform an intensity averaging procedure of the analytical model results (see Supplementary Information for details about the model).We assume a Gaussian beam profile with intensity varying as with ρ 2 = x 2 + y 2 , w 0 the beam waist at focus, and w(z) = w 0 1 + (z/z R ) 2 , where z R is the Rayleigh length of the beam.We estimate that w 0 ≈ 10 µm and z R ≈ 6.3 mm (see, Experiment).
Since the target gas is spread out over a finite volume, the experimental signal will contain the response of atoms subject to a range of different intensities according to Eq. 2. The total signal strength is given by where c( , I) is the spectral amplitude for energy calculated with the model at intensity I, and V gas represents the extent of the gas target.Since the beam waist is much smaller than the extent of the target in the transverse directions, we treat the target as a box shape with an extent of L = 2 mm along the z-axis.In the transverse direction we consider contributions that are closer than 5w 0 from the axis of the beam.The total signal is then The integrals are evaluated using numerical routines from the SciPy library 41 .
Extended Data Fig. 6a contains a comparison of the shape of the intensity averaged spectrum and the spectrum for a single atom subject to the peak intensity, at the detuning where the peaks appear symmetric (∆ω = 62 meV).Extended Data Fig. 6b and 6c contains the same type of curves for the separate contributions from the two-and one-photon process, but at zero detuning (where they are symmetric), illustrating that the two processes are affected differently by the intensity averaging procedure.The spectra in Extended Data Fig. 6b and 6c are normalized to the maximum of each simulation type (single-atom or macroscopic average).The shape of the two-photon signal is less affected by intensity averaging than the one-photon signal, since most of the signal comes from areas with high intensity due to the quadratic dependence on the electric field in the amplitude.On the other hand, the one-photon signal should have significant contributions from a larger volume of the target, since the one-photon amplitude scales linearly with the strength of the electric field.This means that the shape of the one-photon signal might become more distorted, as seen in Extended Data Fig. 6c, but that its relative contribution compared to the two-photon process should increase, when compared to the single-atom case.

Numerical simulations using TDCIS
The ab initio numerical simulations are performed using the Time-Dependent (TD) Configuration-Interaction Singles (CIS) method 33,[42][43][44] in the velocity gauge.The CIS basis for helium is constructed using Hartree-Fock (HF) orbitals that are computed using B-splines.Exterior complex scaling is used to dampen spurious reflections during time propagation of TDCIS 45 .The vector potential of the XUV-FEL pulse is defined as, The Extended Data Table 1.Values of the dipole transition elements z i computed using CIS functions.To compute the dipole transition elements to the continuum, CIS continuum functions were used, and the estimated values for a photoelectron kinetic energy of 23.3076 eV can be found in Extended Data Table 1.
Combining the factor from one extra interaction with the electric field, E 0 /2, with the ratio of the dipole elements it is possible to estimate the ratio of the amplitudes for the one-and two-photon processes where z i is the transition matrix element from state i to a final continuum state with orbital angular momentum .At an intensity of 2 × 10 13 W/cm 2 , which corresponds to an electric field amplitude: E 0 = ωA 0 = 0.023880 a.u., we estimate R s = 0.13546 and R d = 1.1958.
While the one-photon transition is clearly dominant to the s-wave, the two contributions to the d-wave are more comparable in magnitude.The total signal mainly comes from the two-photon transition to the d-wave, but interference with the one-photon transition to the d-wave can not be neglected since it will lead to a blue shift of the symmetric AT doublet.In the main article, the photoelectron spectra shown for the model contain the combined signal for both s and d final states.All atomic parameters required for the model are taken from the ab initio TDCIS method described in Methods.

Figure 1 :
Figure 1: Rabi oscillations induced by an XUV-FEL pulse.a, The sinusoidal population transfer between the XUV-FEL coupled states: |a and |b (black horizontal lines) is associated with sign changes between adjacent Rabi cycles.Photoelectrons can be ejected from excited state |b , by one photon, or by two photons from |a via intermediate states, |c (grey horizontal lines).This results in the build up of an ultrafast Autler-Townes doublet structure.b, The build up of an ultrafast Autler-Townes doublet for 1/2, 1 and 3/2 completed Rabi periods is shown for onephoton ionization from |b (dashed, magenta line) and two-photon ionization from |a (solid, black line) using the analytical model described in the Supplementary Information.

Figure 2 :
Figure 2: Asymmetry of the ultrafast Autler-Townes doublet.a, Deconvoluted experimental photoelectron spectra with symmetric AT doublet (black squares) at 23.753 eV photon energy, and the asymmetric ones at ±13 meV detuning (blue diamonds and red circles, respectively).b, Ab initio photoelectron spectra using TDCIS at three photon energies with symmetric AT doublet at 24.157 eV (dashed, black line) and the asymmetric ones at ±13 meV detuning.Red (blue) curve corresponds to red (blue) detuned light.c-d, Same as b, but using the analytical model for 3/2 Rabi periods in case of one-photon ionization from |b and two-photon ionization from |a , respectively.

Figure 3 :
Figure 3: Quantum interference with a giant wave.a, Energy level diagram for the photon transitions that lead to quantum interference.The summation of contributions from the non-resonant (grey) states leads to the formation of a giant wave, |ρ =b , shown in b.The excited state, |b is shown for comparison with a magnification factor of 10 (dotted, black line in b).Both wavefunctions are computed for helium using CIS.c, Photoelectron spectra from the total analytical model containing contributions from both ground |a and excited |b states with resonant atomic excitation: ∆ω = 0. d, same as c, but with ∆ω = 62 meV.The dashed, white lines denote the expected kinetic energy (23.3076 eV) of a photoelectron that has absorbed two resonant photons.

Figure 4 :
Figure 4: Avoided crossing phenomena in the energy domain.a, Photoelectron kinetic energies, for one photon above the energy of the dressed-atom states, as a function of detuning.Photoelectron spectra as a function of the photon energy retrieved b, experimentally, c, using TDCIS, and d, using the total analytical model for 3/2 Rabi periods.In each case, the dashed white line corresponds to the photon energy for the 1s 2 → 1s4p transition in helium.The shifts in energy scales between a-b, and c-d, are due to the difference between experimental and Hartree-Fock ionization potential.

5 - 1 . 6 ×
Photoelectron spectra generated with the analytic model for a single atom (solid line), and for a macroscopic sample (dashed line).A pulse length of 3/2 Rabi periods, and detuning of ∆ω = 62 meV was used.b-c, Same as a, but for the individual contributions of the two-and one-photon processes, respectively.The results in both b, and c are calculated for ∆ω = 0 meV, where the spectra are symmetric.The separate contributions are normalized to the maximum of the onephoton spectra for both the single-atom and intensity averaged signals.
leads to constructive interference and to a large dipole transition element from the intermediate states to the final state as compared to the transition from the |b = 1s4p state to the final state.Only two intermediate states (1s2p and 1s3p) add destructively to the perturbed wavefunction since they have less energy than the 1s4p state (as shown in Fig. 3(a) of the main text).
30 1 S 0 ) and |b = 1s4p ( 1 P 1 ), with hω ba = b − a = 23.742eV30.The dynamics is probed in-situ by recording 31otoelectrons ejected from the state |b or |a during the ultrashort interaction, with one or two XUV-FEL photons, as illustrated in Fig.1a.In order to interpret this non-linear dynamics, we have developed an analytical model based on a Dyson series for the two-level system undergoing Rabi oscillations (see Supplementary Information for details).The resulting AT doublet structure depends on whether the photoelectron is originating from the ground state, |a , or the excited state, |b , as shown in Fig.1b.The narrow spectral bandwidth of the XUV-FEL pulse (20 -65 meV; see Methods) enables efficient coupling of |a and |b with the dipole element: z ba = 0.1318a 0 , a 0 being the Bohr radius31.It is possible to drive the transition coherently, because 1. Rabi, I. I. Space Quantization in a Gyrating Magnetic Field.Phys.Rev. 51, 652 (1937).
3. Vijay, R. et al.Stabilizing Rabi oscillations in a superconducting qubit using quantum feed-