Quantum estimation in strong fields: In situ ponderomotive sensing

We develop a theoretical framework to optimize and understand uncertainty from in situ strong-field measurements of laser field parameters. We present a derivation of quantum and classical Fisher information in attoscience for an electron undergoing strong-field ionization. This is used for parameter estimation and to characterize the uncertainty of the ponderomotive energy, directly proportional to laser intensity. In particular, the quantum and classical Fisher information for the momentum basis displays quadratic scaling over time. This can be linked to above-threshold ionization interference rings for measurements in the momentum basis and to a “ponderomotive phase” for the optimal quantum measurements. Preferential scaling in uncertainty is found for increasing laser pulse length and intensity. We use this to demonstrate for in situ measurements of laser intensity that high-resolution momentum spectroscopy has the capacity to reduce the uncertainty by more than 25 times compared to measurements employing the ionization rate, while using the optimal quantum measurement would reduce it by a further factor of 2.6. A minimum uncertainty of the order $2.8\times {10}^{-3}\%$ is theorized for this framework. Finally, we examine previous in situ measurements, formulating a measurement that matches the experimental procedure, and suggest alterations to the measurement scheme that could reduce the laser intensity uncertainty.

Figure 1

Solutions of the saddle point Eq. (10) for three Gaussian pulses of two, four and eight cycles (top [(a)–(c)], middle [(d)–(f)], and bottom [(g)–(h)] rows). Left column: the real part of the time of ionization on the electric field for a near-zero final momentum. Middle column: real parts of ionization times for varying $p_{\left|\right|}$ momentum. Right column: The imaginary part of ionization times for varying $p_{\left|\right|}$ momentum. Dotted lines are drawn vertically to denote the full-width at half-maximum of each field. The peak intensity and wavelength of all fields shown is $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2 [U_p=0.44$ a.u.] and $\lambda =800$ nm $[\omega =0.057$ a.u.], respectively. An ionization potential $I_p=0.5$ a.u. is used for the zero-range potential. The CEPs are all $\varphi =\pi /2$, to make the pulses symmetric. The colored shapes mark the real part of solutions of the ionization Eq. (10) on the laser field for a final momentum of $(p_{\parallel },p_{\perp })=(0.002,0.1)$ a.u. The times of ionization in the middle and right columns are plotted using $p_{\perp }=0.1$ a.u., where $p_{\perp }\mathrm{is}$ the momentum coordinate perpendicular to the laser field polarization. Note the color and shape marked on the laser field corresponds to the color and shape used for the real and imaginary plots of the solutions in the middle and right columns.

Figure 2

Momentum-dependent transition amplitude computed for the same target and field parameters as Fig. 1, which corresponds to two-, four-, and eight-cycle Gaussian pulses from top to bottom. Further examples of such distributions can be found in Ref. [80].

Figure 3

The quantum (a), classical full momentum basis (b), yield basis (c), coarse basis (d), full spectral (e), and coarse spectral (f) Fisher information for a single ionization channel. The computation has been done for three Gaussian pulses each with full-width at half-maximum of 5, 10, and 20 laser cycles and a monochromatic laser for reference. The peak intensity and wavelength of all fields shown is $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ ($U_p=0.44$ a.u.) and 800 nm ($\omega =0.057$ a.u.), respectively. An ionization potential $I_p=0.5$ a.u. is used for the zero-range potential. The CEPs are all $\varphi =\pi /2$, to make the pulses symmetric. The coarse momentum and spectral measurements use resolutions of 0.1 and 0.05 a.u. respectively. In the case of the coarse momentum measurement, this means the regions and/or bins ${\mathcal{R}}_i$ from Eq. (27) are squares with sides of 0.1 a.u.

Figure 4

The quantum and classical Fisher informations as in Fig. 3 except computed with five channels of ionization included to assess the effect of interference. The same field and target parameters are used as in Fig. 3.

Figure 5

The uncertainty in laser intensity for coarse measurements with different resolutions. On the left is a three-cycle pulse; on the right is a five-cycle pulse. The laser intensity is $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

The quantum and classical Fisher information with an increasing number of laser cycles. In the monochromatic case, the number of cycles corresponds to the number of ionization events included. The laser intensity is $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The remaining field and target parameters are the same as in Fig. 3.

Figure 7

Uncertainty with increasing laser intensity derived from the quantum and classical Fisher informations for a five-cycle and monochromatic laser field. The remaining field and target parameters are the same as in Fig. 3.

Figure 8

Total ionization yield versus intensity for different length Gaussian pulses, in figure legend. This is computed by integrating the momentum probability distribution. A log scale has been used for the total yield. The red line marks 10% total ionization. For any laser parameters that lead to a yield above the Fisher information, computations will start to break down. The remaining field and target parameters are the same as in Fig. 3.