Maximizing hole coherence in ultrafast photoionization of argon with an optimization by sequential parametrization update

Photoionization with attosecond pulses populates hole states in the photoion. Superpositions of hole states represent ideal candidates for time-dependent spectroscopy, for example via pump-probe studies. The challenge consists in identifying pulses that create coherent superpositions of hole states while satisfying practical constraints. Here, we employ quantum optimal control to maximize the degree of coherence between these hole states. To this end, we introduce a derivative-free optimization method with sequential parametrization update (SPA optimization). We demonstrate the versatility and computational efficiency of SPA optimization for photoionization in argon by maximizing the coherence between the $3s$ and $3p_0$ hole states using shaped attosecond pulses. We show that it is possible to maximize the hole coherence while simultaneously prescribing the ratio of the final hole state populations.

Figure 1

(a) Comparison of a fixed parametrization (black line, $n_p=13$) and a sequential parametrization update (SPA optimization, colored lines) in optimization using the principal axis method of Brent. SPA optimization converges significantly faster and yields a better hole coherence. (b),(c) Comparison of SPA optimization using the principal axis method of Brent (b) and Nelder-Mead simplex search (c). The same initial guess field was utilized in both cases. SPA optimization with the principal axis method converges significantly faster and yields a better hole coherence than with the Nelder-Mead simplex search.

Figure 2

Optimized fields obtained with SPA optimization using the principal axis method (same color code as in Fig. 1). The field with 26 parameters, shown in panel (d), yields a degree of coherence of $g_{3s,3p_0}\left(T\right)=0.989$.

Figure 3

Parameter scanning prior to optimization: frequency scan with fixed peak amplitude for several pulse durations (FWHM of the intensity) (a) and joint frequency and peak amplitude scans for fixed pulse durations $\tau =6\mathrm{a}.\mathrm{u}.$ (b) and $\tau =23\mathrm{a}.\mathrm{u}.$ (c). Favorable parameters for the initial guess field can clearly be identified.

Figure 4

Channel-resolved PES obtained from the transform-limited Gaussian pulses studied in Fig. 3 for a maximal field amplitude of $E_0=0.02\mathrm{a}.\mathrm{u}.$

Figure 5

Channel-resolved PAD corresponding to the PES shown in Fig. 4: panels (a) and (b) display the contribution of $3s$ photoelectrons to the energy-integrated PAD; panels (c) and (d) that of $3p_0$ photoelectrons.

Figure 6

SPA optimization with the principal axis method, using favorable initial parameters in the guess pulse: the convergence is significantly accelerated (a). Guess and optimized fields are shown in (b) and (c).

Figure 7

Degree of coherence as a function of time obtained with guess (left) and optimized (right) fields: (a) randomly chosen initial parameters ($n_p=26$); (b) corresponding optimized field with $g_{3s,3p_0}\left(T\right)=0.989$ ($n_p=26)$; (c) initial guess field consisting of two time-delayed Gaussians ($n_p=8$); (d) corresponding optimized field for which the degree of coherence oscillates between $g_{3s,3p_0}\left(t\right)=0.97$ and 0.75 with a final value of $g_{3s,3p_0}\left(T\right)=0.80$ ($n_p=16)$; (e) initial monochromatic guess field with favorable parameters identified by parameter scan; (f) corresponding optimized field $[n_p=7$, the same as shown in Fig. 6] for which the degree of coherence oscillates between $g_{3s,3p_0}\left(t\right)=0.98$ and $g_{3s,3p_0}\left(T\right)=0.90$.

Figure 8

Photoionization probability, obtained in terms of the absorbed part of the ion density matrix, cf. Eq. (8), as a function of time for the three optimized fields depicted in Figs. 7, 7, and 7. The color code is the same as in Fig. 7.

Figure 9

Degree of coherence (a) and hole populations (b),(c) as a function of time, obtained with the optimized field shown in Fig. 7.

Figure 10

Maximizing the degree of coherence between the $3p_0$ and $3s$ hole states while simultaneously optimizing for a hole population ratio of one: degree of coherence (a), hole populations (b), and optimized electric field (c) as a function of time.

Figure 11

Maximizing the degree of coherence between the $3p_0$ and $3s$ orbitals while simultaneously optimizing for a hole population ratio of ${\rho }_{3s,3s}/{\rho }_{3p_0,3p_0}=0.7$: degree of coherence (a), hole populations (b), and optimized electric field (c) as a function of time.

Figure 12

Maximizing the degree of coherence between the $3p_0$ and $3s$ hole states while simultaneously optimizing for a hole population ratio of ${\rho }_{3p_0,3p_0}/{\rho }_{3s,3s}=0.7$: degree of coherence (a), hole populations (b), and optimized electric field (c) as a function of time.

Figure 13

Maximizing the degree of coherence between the $3p_0$ and $3s$ hole states while simultaneously optimizing for a given hole population ratio: spectra of the optimized fields for the three different hole population ratios, $\mathcal{R}={\rho }_{3s,3s}/{\rho }_{3p_0,3p_0}$.