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High-order harmonic generation from the dressed autoionizing states

In high-order harmonic generation, resonant harmonics (RH) are sources of intense, coherent extreme-ultraviolet radiation. However, intensity enhancement of RH only occurs for a single harmonic order, making it challenging to generate short attosecond pulses. Moreover, the mechanism involved behind such RH was circumstantial, because of the lack of direct experimental proofs. Here, we demonstrate the exact quantum paths that electron follows for RH generation using tin, showing that it involves not only the autoionizing state, but also a harmonic generation from dressed-AIS that appears as two coherent satellite harmonics at frequencies ±2Ω from the RH (Ω represents laser frequency). Our observations of harmonic emission from dressed states open the possibilities of generating intense and broadband attosecond pulses, thus contributing to future applications in attosecond science, as well as the perspective of studying the femtosecond and attosecond dynamics of autoionizing states.


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In our calculations, the spectrum is found via numerical solution of the 3D time-dependent Schrödinger equation

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(TDSE) for the model system in the laser field. The numerical approach is presented in Ref. 1 and the model Sn + 19 potential is described in Ref. 2. Only short quantum path contribution to HHG is presented, and the long quantum 20 path contribution is suppressed with the method suggested in Ref. 3. The peak laser intensity is 2x10 14 Wcm -2 and 21 the driving laser wavelength is 1.81 μm. We are using specific softly-truncated Gaussian temporal envelope of the 22 laser pulse, namely, the field is: In our calculations, the FWHM of the Gaussian (2τ ) is 18 fs, front τ is 3 optical cycles (18.1 fs) and top τ is 8 optical 2 2 / top ≈ so the pulse intensity is softly truncated when it is less than 1% from its maximum.

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Thus, the used field has long temporal wings, in contrast to 2 sin envelope widely used in numerical calculations. In 32 this sense our field at least qualitatively reproduces the experimentally used one. We will show below that it is very 33 important to reproduce the presence of the laser pulse temporal wings in the calculations of the dressed AIS states.

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The truncation of the Gaussian is necessary to have the field exactly equal to zero at the start of the numerical TDSE 35 solution. Moreover, the used field satisfies the

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The calculated spectrum is shown in the main article in Fig. 3 (b).
To study the temporal dynamics of the XUV emission, we use Gabor analysis. Namely, using the TDSE solution we 38 find the microscopic dipole d(t) and calculate its spectrum: Then, we find the inverse Fourier transform of this spectrum within the spectral range from ω 1 to ω 2 (the Gabor 41 transform): In Supplementary Fig. 3 (a & b) we present the intensity of this signal

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As we have shown in the main article in Fig. 3 (b), additional peaks appear in the spectrum at the resonant 46 frequency and at this frequency ±2Ω. To understand their origin, we make the Gabor analyze, namely we select 47 several peaks from the spectrum and study when they are emitted. In Supplementary Fig. 3 (a), we see that

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Taking into account this behavior, one can explain the emission of the satellites at the resonant frequency ±2Ω by 55 the resonance with the dressed AIS. Indeed, the AIS is mainly populated at the falling edge and after the pulse (see 56 usual resonant emission, black line in Supplementary Fig. 3 b) but the dressed states exist only inside the laser 57 pulse, not after it. Thus the emission due to the resonance with the dressed AIS should be temporally confined 58 within the falling edge of the pulse but not after it.

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The correct reproduction of the edges of the laser pulse is thus very important for the calculation of the emission 60 from the dressed AIS. Using the 2 sin temporal envelope in the calculations, we found far less pronounced emission 61 from the dressed AIS, because the temporal interval where the AIS is populated and the laser field dresses it is much 62 shorter. Our softly-truncated Gaussian envelope is closer to the real laser pulse. However, the wings of the laser 63 pulse can be more pronounced (of even some pedestal of the laser pulse can be important), so this explains the 64 difference between the experimental and calculated spectra.
Note that alternatively the peaks at the resonant frequency ±2Ω can be attributed to the resonance between the 66 (usual) AIS and the dressed ground state. However, the population of the ground and dressed ground state can be 67 easily tracked in the TDSE numerical solution. These populations differ by more than 3 orders of magnitude, while 68 the intensity of the usual resonant emission (black line in Fig. 3 (b) of the main text and "resonant emission 69 involving dressed states" (red and violet lines)) are closer, namely they differ by approximately 2 orders. This allows 70 us to conclude that the dressed ground state cannot explain the observed emission. Moreover, in general the AIS 71 should be more affected by the laser than the ground state, so the dressed AIS is more populated than the dressed 72 ground state.  envelope. Every curve is renormalized to its maximum. The spectral ranges used for the Gabor transforms are shown 79 in Fig. 3(b), in the main text.