Attosecond quantum optical interferometry

Abstract

In this work, we explore the scheme of attosecond quantum interferometry (AQI), the quantum optical version of classical attosecond interferometry, which allows to measure quantum optical properties on the attosecond time-scale. We develop how the scheme of AQI can be used for quantum state engineering of the emitted harmonics, by varying the relative phase of a two-color driving field, and further enables one to manipulate the field correlations as well as their entanglement characteristics. In addition, this scheme allows us to learn properties of the phase-space distribution of the harmonic quantum state, by means of measuring an attosecond quantum tomography trace. This serves as a new type of protocol for in situ attosecond measurements of quantum optical observables. With this, we achieve to further connect all-optical attosecond measurement schemes with quantum optics, allowing for a rich manifold of observations.

Figures (9)

• Figure 1: Attosecond quantum interferometry. A strong classical pump field at frequency $\omega$ is combined with a perturbative $2\omega$ field exhibiting well-defined squeezing signatures. By varying the relative phase between the two fields, one can control the photon statistics, phase-space distribution, and intermode correlations of the generated harmonics.
• Figure 2: Quantum state engineering. Wigner functions of the harmonic modes $q=12$ [(a)-(d)] and $q=16$ [(e)-(h)] for varying two-color phase $\phi$. Minimum [(i)] and maximum [(j)] values of the quadrature variances for three even harmonic orders. Calculations were performed with $E_{\omega} = 0.053$ a.u., $E_{2\omega} = 10^{-2}E_{\omega}$, $I_{\text{squ}} = 10^{-6}$ a.u., $\omega = 0.057$ a.u. and $I_p =0.5$ a.u., with a field duration of 5 optical cycles. Amplitude squeezing for the $2\omega$ field is considered here although results are not affected much by the specific type of squeezing.
• Figure 3: Field correlations. (a) Second order correlation function between harmonics $q_1$ and $q_2$, with the black line quantifying the autocorrelations ($q_1 = q_2$). (b) Value of the CSI difference. (c) Linear entropy as a function of $\phi$. The same parameters as those in Fig. \ref{['fig:state_engineering']} have been considered here.
• Figure 4: Learning the quantum state through attosecond quantum tomography (AQT). (a), (b) Reconstructed AQT-distribution for the 12th and 16th harmonic orders with $\theta = 0$. (c), (d) AQT-traces for the 12th harmonic mode for two different quadrature operators $\hat{X}_{\theta} = \hat{a} e^{-i\theta} + \hat{a}^\dagger e^{i\theta}$. (e) Variance of $\hat{X}_{\theta}$ for the 12th (green) and 16th (purple) harmonics, computed from the AQT-traces for $\theta = 0$ (lighter curves) and $\theta = \pi/2$ (darker curves). The same parameters as in Fig. \ref{['fig:state_engineering']} are used here.
• Figure 5: Vector potential dependence with time in the original frame of reference. Here, we change the values of $E_{2\omega,x}$ and $E_{2\omega,y}$, defined in through $\gamma_x$ and $\gamma_y$, while keeping the mean field strength constant.