Emergence of nonclassical radiation in strongly laser-driven quantum systems

Abstract

Nonclassical light sources are central to emerging quantum technologies, yet current platforms offer limited tunability and typically operate at low photon numbers. In parallel, strong-field physics provides widely tunable, bright coherent radiation through high-order harmonic generation (HHG), but its quantum optical character has remained largely unexplained. While recent experiments have revealed signatures of entanglement, squeezing, and quantum-state modification in both the driving and generated fields, a unified theoretical framework capable of identifying the origin and controllability of these effects has been missing. Here we introduce a fully quantum, analytically tractable theory of intense light-matter interaction that rigorously captures the emergence of nonclassicality in HHG. Our approach employs a parametric factorization of the coupled electron-field system into a driven electronic state and a dynamically perturbed quantum optical field, derived directly from the time-dependent Schrödinger equation without requiring conditioning, homodyne detection, or mode-selection techniques. We show how quantum correlations, squeezing, and Wigner-function negativity arise intrinsically from the interaction dynamics, and we identify the precise conditions under which specific nonclassical features are amplified or suppressed. The theory enables predictive design of bright, high-photon-number quantum states at tunable frequencies, and we demonstrate its utility by outlining realistic conditions for generating bright nonclassical ultraviolet light. Our results establish a comprehensive foundation for strong-field quantum optics and open new avenues toward tabletop quantum light sources for sensing, communication, and photonic quantum information processing.

Figures (6)

• Figure 1: Visualization of the emergence of nonclassicality in high-order harmonic generation to result from the nonlinear $q$- dependence of the oscillating dipole moment. The upper row displays the dipole moments expanded into a series (truncated after the terms as labeled) (eq. \ref{['eq:dipole-series']}) while the lower row shows the resulting output state of light in phase space, represented by its Wigner function. (a) Constant, almost $q$-independent dipole moment leads to (below): emitted light represented by a Wigner function of an almost coherent state (here $\Omega=13$th harmonic). (b) Linear $q$-dependence of dipole moment can lead to Wigner function of a squeezed state ($\Omega=13$th harmonic). (c) Dipole moment containing terms nonlinear in $q$, here for an approximately quadratic dependence on $q$, can lead to the generation of nonclassical light, as represented by a Wigner function with negative values/regions ($\Omega=13$th harmonic). Initial conditions: product state of light and matter, with the initial system being in the superposition of the ground and first excited state for the H-atom; Laser parameters: $F_{class}(t)\;$ ($\lambda$ = 2227 nm, T$_{p}$ = 6 cycles, $I = 1\cdot 10^{13}$ W/cm$^{2}$). The quantization parameter is $\beta = 0.41 \text{au.}$.
• Figure 2: Gallery of Wigner functions after the interaction with a laser pulse for different systems and initial conditions. (a, d) atomic system mimicking the Ca atom; (b, e) molecular system with fixed internuclear distance; (c, f) hydrogen atom-like potential. The upper row shows the results when starting with the electronic ground state of the driven system while the lower row the corresponding results when starting with a superposition state: (a) Atom initially in the electronic ground state; (d) initial atomic state corresponding to a superposition state of the ground state and excited state. The following parameters are employed: $\Omega= 7; F_{class}(t):\;$$\lambda$ = 3750 nm, T$_{p}$ = 2 cycles, $I = 2\cdot 10^{12}$ W/cm$^{2}$ and $\beta = 0.55 \text{au.}$; (b, e) Same for the corresponding molecular model system, with $\Omega= 3; F_{class}(t):\;$$\lambda$ = 3750 nm, T$_{p}$ = 2 cycles, $I = 1\cdot 10^{12}$ W/cm$^{2}$ and $\beta = 0.82$ a.u.; (c, f) for the H-atom model with $\Omega= 13; F_{class}(t):\;$$\lambda$ = 2227 nm, T$_{p}$ = 2 cycles, $I = 1\cdot 10^{13}$ W/cm$^{2}$ and $\beta = 0.41 \text{au.}$
• Figure 3: Nonclassical output states for $\Omega=5$th harmonic generated from the multi-emitter case with $N_e$ spatially isolated (non-Coulomb interacting) emitters. Each emitter is a 1D model CaO model molecule). The parameters of the driving laser pulse are $F_{class}=3.4\cdot{}10^{-2}\,\text{(a.u.)}$, $I = 4\cdot 10^{13}$ W/cm$^{2}$, $\lambda=1945\,\text{nm}$. The solid line corresponds to the solution based on the parametric factorization. The dashed line shows the corresponding coherent state for comparison (no $\beta{q}$-dependence in the dipole). (a) The initial state is a vacuum (Gaussian) state and $\beta=10^{-8}$ considering $N_e=10^{11}$ emitters. (b) Initial state consisting of a weakly nonclassical state, represented by a vacuum (Gaussian) with several percent of 1-photon (7%), 2-photon (3%), and 3-photon (1.7%) states, with $\beta=10^{-7}$ and $N_e=10^{9}$ emitters. The calculations are performed by fitting the dipole up to the 5-th order (eq. \ref{['eq:dipole-series']})). As the average number of photons in the output mode is $N_{\Omega}\approx{\bar{q}^{2}/2}$, it can be seen that very bright nonclassical states can be obtained.
• Figure 4: Nonclassical harmonic generation from various model systems discussed in the main text: 1D electronic system coupled to one mode of quantized light with harmonic frequency $\Omega$ as labeled. (A) Left: 1D soft-core model potential (blue) mimicking the Ca-atom and corresponding electron wavefunctions of the ground state (orange) and a superposition of the ground and excited electronic state (Ca atom: I$_{p}$ = 6.11 eV). Right: Solution of the electronic Schrödinger equation (eq. \ref{['eq:TDSE-x']}), displaying the time-dependent electron dipole moment (eq. \ref{['eq:dipole']}). The results of two calculations are shown, where the field-driven electronic Schrödinger equation is coupled to a light mode of $\Omega=3$ and $\Omega=5$. For comparison, the dashed line displays the (classical) driving laser, with parameters $F_{class}(t)\;$ ($\lambda$ = 3750nnm, T$_{p}$ = 2 cycles), $I = 2\cdot 10^{12}$ W/cm$^{2}$$\beta = 0.55 \text{au.}$. (B) same for an asymmetric diatomic molecule model (1D CaO model with I$_{p}$ = 6.6 eV), however with $\beta$ = 0.82 au., $I = 10^{12}$ W/cm$^{2}$. The gray area shows the modes' behavior after the interaction with the driving laser.
• Figure 5: As in Figure \ref{['fig:overview']}, visualization of the emergence of nonclassicality in high-order harmonic generation to result from the nonlinear $q$- dependence of the oscillating dipole moment, however for the multi-emitter case, demonstrating that high photon numbers can be obtained. The upper row displays the dipole moments expanded into a series (truncated after the terms as labeled) (eq. \ref{['eq:dipole-series']}) while the lower row shows the resulting output state of light in phase space, represented by its Wigner function. (a) Constant, almost $q$-independent dipole moment leads to (below): emitted light represented by a Wigner function of an almost coherent state with high photon number (here $\Omega=5$th harmonic). (b) Linear $q$-dependence of dipole moment can lead to Wigner function of a squeezed state with high photon number ($\Omega=5$th harmonic). (c) Dipole moment containing terms nonlinear in $q$, here for an approximately quadratic dependence on $q$, can lead to the generation of nonclassical light, as represented by a Wigner function with negative values/regions ($\Omega=5$th harmonic). Initial conditions: product state of light and matter, with the initial system being in the a)3rd excited state for the H-atom, b) ground state for the H-atom and c) superposition of the ground and third excited state for CaO; Laser parameters: (a) and (b): $F_{class}(t)\;$ ($\lambda$ = 2227 nm, T$_{p}$ = 2 cycles, $I = 10^{13}$ W/cm$^{2}$). The quantization parameter is $\beta = 1.4\cdot 10^{-6} \text{au.}$ in (a) and $\beta = 1.4\cdot 10^{-3} \text{au.}$ in (B). The corresponding parameters for (c) are $F_{class}(t)\;$ ($\lambda$ = 3000 nm, T$_{p}$ = 2 cycles, $I = 1\cdot 10^{12}$ W/cm$^{2}$) and $\beta = 0.22 \text{au.}$ The average number of photons in the output state is approximately $N_{\Omega}\approx{\bar{q}^{2}/2}$