Information-theoretic perspective on energy conservation in high harmonic generation

Abstract

The use of energy conservation arguments is ubiquitous in understanding the process of high harmonic generation, yet a complete quantum optical description of exact photon number exchange remained elusive. Here, we solve this gap in description by introducing the energy conserving subspace in high harmonic generation in which many photons of the driving field are absorbed to generate a single photon of higher energy. The presented solution to energy conservation in quantum optical high harmonic generation naturally results in highly entangled states of light with non-classical properties in their marginals and photon statistics. This new technique can be seen as an information-theoretic approach to the problem of photon exchange between field modes, providing a new kind of selection rule imposed on the quantum optical state by the structure of the Hilbert space. In addition to providing the quantum state satisfying exact energy conservation, it allows to explain recent experimental results for quantum state engineering of optical cat states.

Figures (3)

• Figure 1: Photon number distribution of the energy conserved state.$P_{IR}(n_1)$ of the fundamental field when constrained to energy conservation on the subspace $\Pi^{(N\omega)}$, for the different energy subspaces $N \in [14, 15]$. The solid line corresponds to the Poisson distribution of the unconstrained state $\ket{\psi}$ (the coherent state amplitudes are given by $\alpha + \delta \alpha = 3.0$, and $\chi_q = 1.5$).
• Figure 2: Non-classical states in HHG. Left: Marginal of the Wigner functions $W(\beta)$ for different energy conserving subspaces $\Pi^{(N \omega)}$ of Eq. \ref{['eq:fundamental_EspaceN']}, with total energy $N \omega = 3 \omega$ (solid line), $8 \omega$ (dotted) and $15 \omega$ (dashed line). For the respective subspaces the coherent state amplitudes of the state in Eq. \ref{['eq:state_wavepicture']} are $\{\alpha = 1.2, \delta \alpha = -0.3, \chi_q = 0.1 \}_3$, $\{4.2, -1.3,0.3 \}_8$, and $\{ 5.2, -2.3, 0.8 \}_{15}$. Right: Mandel $Q$-parameter of the photon number distribution from Fig. \ref{['fig:photon_number']} as a function of the energy subspace $N\omega$, indicating clear sub-Poissonian statistics.
• Figure 3: Wigner function $W(\beta)$johansson2012qutip of the normalized photon number superposition in Eq. \ref{['eq:fock_superposition_small']}, showing clear negativities and strong agreement with the previously reported experimental Wigner functions in Refs. lewenstein2021generationrivera2022strong. The coherent state amplitudes are given by $\alpha = 2.5$, $\delta \alpha = -0.1$ and $\chi_q = 0.1$. The state shown here has a fidelity of $F = 0.998$ with the cat state of Eq. \ref{['eq:cat']}.