Optical parametric free-electron--photon quantum interaction

TL;DR

The paper develops a Bogoliubov-based theoretical framework for parametric free-electron–photon interactions driven by degenerate parametric down-conversion, revealing two detuned phase-matching channels and squeezing-enhanced coupling. It shows that squeezed vacuum seeding yields heralded squeezed Fock states via electron-sideband postselection, while bare vacuum seeding enables high-fidelity Schrödinger cat states through parity-based photonic projections. A key application is quantum parametric dielectric laser accelerators, which can achieve near-unity acceleration probability and relax temporal synchronization requirements. The work outlines feasible on-chip nonlinear photonics implementations and discusses extensions to multimode and multi-electron settings, highlighting a versatile platform for quantum shaping of electrons and photons.

Abstract

Optical parametric processes underpin quantum photonics, while free-electron--photon interactions offer agile pathways to generate nontrivial quantum photonic states. These threads have so far largely progressed independently, whereas placing free electrons in a driven nonlinear system can potentially activate coherent parametric interaction channels for joint state engineering of both types of particles. Here we unify these paradigms by developing a general theoretical framework for parametric free-electron--photon interactions in a nonlinear optical system driven by degenerate parametric down-conversion. Unlike free electrons in a linear bath, here they can couple to Bogoliubov quasiparticles through two detuned phase-matching channels, where the parametric process and free-electron interactions can quantum amplify each other. Seeding the interaction with squeezed vacuum yields gain-only or loss-only electron energy spectra, and enables electron-heralded squeezed Fock states; with bare vacuum, postselecting electron energy sidebands generates high-fidelity Schrödinger cat states. Our results show how optical parametric interactions can quantum shape free electrons and photons, potentially enabling a quantum parametric dielectric laser accelerator that mitigates the need for temporal phase synchronization, thereby allowing acceleration probabilities to approach unity even for phase-random electrons.

Figures (4)

• Figure 1: Theoretical framework.a. Schematic of free-electron--photon interaction driven by DPDC. A nonlinear cavity or waveguide with mode frequency $\omega_0$ is parametrically driven by an undepleted pump at frequency $\omega_{\rm{p}}$ through nonlinear $\chi^{(2)}$-mediated down-conversion. A quantum source seeds the cavity with an initial photonic state of either a squeezed vacuum (source on) or the bare vacuum (source off). A free electron interacts with the nonlinear system and is then measured with an energy spectrometer. The output photon state is characterized via homodyne detection in coincidence with the postselected electron energy sideband. DM--Dichroic mirror. BS--Beam splitter. LO--Local oscillator. b. Parametric phase-matching condition. Existing QPINEM requires the phase velocity of the mode at $\omega_0$ to match the free-electron velocity $v_z$, i.e., the phase-matching momentum is $k_0 = \omega_0/ v_z$ (gray arrow). With DPDC driving, the phase-matching momentum is shifted to $k^{(+)}$ (blue arrows) with detuned momentum $\abs{\Omega - \Delta}/v_z$, or to $k^{(-)}$ (red arrows) with detuned momentum $\abs{\Omega + \Delta}/v_z$, where $\Omega$ is the squeezed mode frequency. c.$g^{(\pm)}$ as a function of $g_\mathrm{Q}^{(\pm)}$ and squeezing parameter $r$. The interaction strength between the free electron and the squeezed mode is given by $g^{(+)} = g_\mathrm{Q}^{(+)} \, \cosh{r}$ or $g^{(-)} =g_\mathrm{Q}^{(-)} \, \sinh{r}$ for the phase-matching condition $k^{(+)}$ or $k^{(-)}$, respectively, both of which exhibit an exponential enhancement for $r>1$. d. Squeezed Fock state preparation. When the interaction is seeded by squeezed vacuum, postselection on free-electron sidebands [either gain-only sidebands for $k^{(+)}$ (top left) or loss-only sidebands for $k^{(-)}$ (bottom left)] heralds the generation of squeezed Fock states (right). e. Cat state preparation. When the interaction is seeded by bare vacuum, postselection on free-electron sidebands [gain or loss sidebands for either $k^{(+)}$ (top left) or $k^{(-)}$ (bottom left) phase matching] heralds the generation of Schr√∂dinger cat states (right; an odd cat state with magnitude $\alpha = 5.431$). Even and odd electron energy sidebands correspond to even and odd cat states, respectively. In (d) and (e), ${g_\mathrm{Q}^{(\pm)}}=1$, the squeezing parameter ${r}=1.15$ ($\approx10dB$), detuning $\Delta=0.2\omega_0$ is chosen for better visualizing the electron energy shifts, and energy spectra are normalized to $\hbar\omega^{(\pm)}$ with a Lorentzian broadening of $0.3\omega^{(\pm)}$.
• Figure 2: Free-electron parametric interaction with squeezed vacuum and its generation of squeezed Fock states.a-b. Electron energy spectra as a function of squeezing parameter ${r}$ for phase-matching conditions $k^{(+)}$ (a) and $k^{(-)}$ (b). For $k^{(+)}$, only electron energy loss occurs, whereas for $k^{(-)}$, only electron energy gain is observed. c-d. Joint probability distribution between the squeezed Fock state and electron energy loss $-n\hbar\omega^{(+)}$ (c) or energy gain $n\hbar\omega^{(-)}$ (d), with nonzero probability only along the anti-diagonal (c) and diagonal (d). The right insets show the corresponding Poisson probability distributions of the squeezed Fock states. The squeezing parameter is $r=1.15$; plots (a) and (c) use $g_\mathrm{Q}^{(+)}=1$ [$k^{(+)}$], and plots (b) and (d) use $g_\mathrm{Q}^{(-)}=1$ [$k^{(-)}$]; the detuning is fixed at $\abs{\Delta} = 0.2\omega_0$ and the Lorentzian peak broadening is $0.1\omega^{(\pm)}$.
• Figure 3: Free-electron parametric interaction with bare vacuum and its generation of Schr√∂dinger cat states.a-b. Electron energy loss and gain spectra versus squeezing parameter $r$ for the two phase-matching conditions $k^{(+)}$ (a) and $k^{(-)}$ (b). The detuning is $\abs{\Delta}=0.2\omega_0$ and the Lorentzian broadening is $0.1\omega^{(\pm)}$, and electron energy change is normalized to $\omega_0$. c. The joint probability distribution between the Fock state and the electron energy change $n \hbar \omega^{(+)}$ for $k^{(+)}$, exhibiting a checkerboard-like pattern. The right inset shows the corresponding traced near-Poissonian photon statistics. d. Postselection of even or odd loss/gain band enables the selection of even and odd photon Fock space (e.g., postselecting $n=-1$ in Fig. (c) yields an odd-parity photon-number distribution), enabling the preparation of even or odd cat states. The right inset plots the Wigner function for the photonic state generated by postselecting $n=-1$. e. Cat amplitude $\alpha$ as a function of interaction strength $g_\mathrm{Q}^{(+)}$ and squeezing parameter $r$ for the $n=-1$ electron energy loss sideband. Inset: High fidelity of cat state preparation for various electron sideband heralding under different levels of squeezing parameter $r$. In all the plots, $g_\mathrm{Q}^{(+)}=1$. In (a)-(d), $r=1.15$, and in (a)-(b), the detuning is $\abs{\Delta}=0.2\omega_0$, the Lorentzian broadening is $0.1\omega^{(\pm)}$, and electron energy change is normalized to $\omega_0$ .
• Figure 4: Probability of electron acceleration in quantum parametric DLAs. Insets: Electron energy-level diagrams for existing DLAs (upper-left) and the quantum parametric DLAs (lower-left), respectively. Right: Electron energy-gain probability $\mathrm{P}_{\mathrm{acc}}$ versus interaction strength $g$ for DLAs and $g^{(-)}$ for the quantum parametric DLAs. The probability of electron arrival is assumed to be uniformly distributed within an optical cycle. $\mathrm{P}_{\mathrm{acc}}$ saturates at 0.5 for DLAs, whereas it can reach unity for quantum parametric DLAs.