Intermodal entanglement in a quantum optical model of HHG due to the back-action on the driving field

Abstract

Preparation of nonclassical light with special quantum properties is essential for quantum technologies. High-harmonic generation (HHG) is a process which not only enables the creation of attosecond pulses but also has the potential to generate light with intricate quantum properties. In a recent experiment [1], nonclassical inter-harmonic correlations have been measured from a HHG source. In this work, we theoretically investigate entanglement between different harmonics within an effective quantum optical model. This model implements a signifcant degree of simplifcation regarding the processes within the target material, treating the material through susceptibilities, as it is usual in quantum optics. Such an approach yields a general description of HHG, permitting the implications that can be derived within it to hold broadly. We find that entanglement is produced as a result of the often neglected back-action. We can qualitatively reproduce experimentally measured nonclassicalities, which suggests that intermodal entanglement can, to an extent, be considered a universal phenomenon associated with HHG, rather than a result of using specific material targets.

Figures (8)

• Figure 1: Deviation in the Wigner function of the 10'th harmonic, $\tfrac{\rho'^{(2)}_{10}(t)}{\text{Tr}[\rho'^{(2)}_{10}(t)]}-|0\rangle_{10}\langle0|_{10}$, based on [Eq. (\ref{['densitm']})], for driving pulse of 2 optical cycles. Parameters are chosen as $|\alpha_0|=10^2$, $t=2\tfrac{2\pi}{\omega}$ and $\chi_n=\tfrac{1}{\sqrt{n}|\alpha_0|^n}$, with harmonic orders being in $n\in{3,...10}$.
• Figure 2: Single-mode physical quantities characterizing the excitation and harmonics. a) Time-evolution of $\langle N_1(t)\rangle$ shown as a function of $t/T$, where $T=2\pi/\omega$ is the optical cycle. b) Time-evolution of the photon correlations: $\gamma_{1,1}(t)$ (black), $\gamma_{3,3}$ (red); $\gamma_{5,5}(t)$ (orange); $\gamma_{7,7}(t)$ (yellow); $\gamma_{9,9}(t)$ (green); $\gamma_{11,11}(t)$ (blue). The parameters are chosen such that $|\alpha_0|=1$, $\chi_3=0.02$, $p=0.3$, with a cutoff order of $N=11$. Dashed lines show the chosen interaction time $t=1.5T$, applied in Fig. (\ref{['fig:vazlat3b']})
• Figure 3: Two-mode physical quantities. Intermodal correlation function values $\gamma_{nm}(1.5T)$ and $\gamma_{1n}(1.5T)$ a) between harmonic modes and b) between the excitation mode and harmonic modes, respectively. c) Upscaled logarithmic negativity between harmonic modes $10^5\times E_{nm}(1.5T)$.
• Figure 4: Time-evolution of the photon correlation functions, shown as a function of $t/T$, where $T=2\pi/\omega$ is the optical cycle. $\gamma_{1,1}(t)$ is shown as black, $\gamma_{3,3}$ red; $\gamma_{5,5}(t)$ orange; $\gamma_{7,7}(t)$ yellow; $\gamma_{9,9}(t)$ green; $\gamma_{11,11}(t)$ blue curves. Dashed lines show the corresponding interaction time $t=2T$. Parameters are $|\alpha_0|=\sqrt{20}$, $\chi_3=0.0002$, $N=11$.
• Figure 5: Two-mode physical quantities. Intermodal correlation function values $\gamma_{nm}(2T)$ and $\gamma_{1n}(2T)$ a) between harmonic modes and b) between the excitation mode and harmonic modes, respectively. c) Upscaled Logarithmic negativity between harmonic modes $10^5\times E_{nm}(2T)$.