In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.
In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.
High harmonic generation (HHG) is an extremely nonlinear effect generating coherent broadband radiation and pulse durations reaching attosecond timescales. Conventional models of HHG that treat the driving and emitted fields classically are usually very successful but inherently cannot capture the quantum-optical nature of the process. Although prior work considered quantum HHG, it remains unknown in what conditions the spectral and statistical properties of the radiation depart considerably from the known phenomenology of HHG. The discovery of such conditions could lead to novel sources of attosecond light having squeezing and entanglement. Here, we present a fully-quantum theory of extreme nonlinear optics, predicting quantum effects that alter both the spectrum and photon statistics of HHG, thus departing from all previous approaches. We predict the emission of shifted frequency combs and identify spectral features arising from the breakdown of the dipole approximation for the emission. Our results show that each frequency component of HHG can be bunched and squeezed and that each emitted photon is a superposition of all frequencies in the spectrum, i.e., each photon is a comb. Our general approach is applicable to a wide range of nonlinear optical processes, paving the way towards novel quantum phenomena in extreme nonlinear optics.
Simulations of harmonic generation in the single-atom limit usually describe the harmonic radiation as proportional to the squared magnitude of the Fourier transform (FT) of the expectation value of either the dipole moment or the dipole acceleration. However, no derivation of an explicit single-atom representation of the typical experimental measurement (i.e., the number of harmonic photons produced versus the frequency of the harmonic) has been given. By means of the first principles of quantum electrodynamics a formula for this ``harmonic photon-number spectrum'' is derived, which is proportional to the double FT of the time-autocorrelation function of the dipole velocity of the whole system. Within the long-wavelength approximation the time-correlation function can be recast as the FT of the squared magnitude of the expectation value of the dipole velocity of a single representative atom.
We develop a theoretical framework for high-harmonic generation (HHG) driven by quantum states of light based on a temporal-mode expansion of the electromagnetic field. This approach extends previous single plane-wave mode treatments to realistic pulse configurations and arbitrary multi-mode states of light, resolving conceptual inconsistencies arising from non-normalizable infinite plane waves and establishing consistency between analytical and numerical methods. We derive a correction factor that quantifies deviations from the diagonal approximation (in which the yield becomes a statistical average over classical-field simulations) both for the response of a single atom and in the many-atom regime. Our results confirms that the HHG spectrum for atoms driven by any quantum state of light in free space is accurately described by averaging semi-classical calculations over the Husimi distribution, with no observable genuine quantum effects in the spectrum. We also demonstrate that in the many-atom regime, the mean-field coherent-state approximation underlying this treatment does not preserve probabilities, although unitarity is restored by in the diagonal approximation. The absence of genuine quantum effects in the HHG yield is attributed to the large photon numbers (∼1011) required to reach HHG intensities in free space, which render quantum fluctuations negligible. We discuss nanophotonic environments with ultrasmall mode volumes as potential platforms where few-photon strong-field processes could exhibit genuine quantum signatures.
We present an analytical framework for the emergence of nonclassical radiation in strongly laser-driven quantum systems, with a focus on high-order harmonic generation (HHG). Starting from a Pauli-Fierz description, we employ a parametric factorization of the coupled light-matter wavefunction that reduces the dynamics to coupled equations for a field-driven electronic state and a quantized light mode. Within this framework, we identify a simple and predictive mechanism for nonclassicality: it originates from the nonlinear dependence of the electronic dipole response on the light-mode coordinate. An approximately constant dipole yields coherent radiation, a linear dependence produces squeezing, and higher-order nonlinearities give rise to Wigner-function negativity. We illustrate this mechanism for atomic and molecular model systems and analyze its scaling in multi-emitter configurations, indicating routes toward high-photon-number nonclassical radiation in HHG. Our results provide a transparent connection between strong-field dynamics and quantum-optical properties of emitted light, offering a basis for engineering nonclassical states in intense laser-matter interactions.
The interaction of electrons with strong laser fields is usually treated with semiclassical theory, where the laser is represented by an external field. There are analytic solutions for the free electron wave functions, which incorporate the interaction with the laser field exactly, but the joint effect of the atomic binding potential presents an obstacle for the analysis. Moreover, the radiation is a dynamical system, the number of photons changes during the interactions. Thus, it is legitimate to ask how can one treat the high order processes nonperturbatively, in such a way that the electron-atom interaction and the quantized nature of radiation be simultaneously taken into account? An analytic method is proposed to answer this question in the framework of nonrelativistic quantum electrodynamics. As an application, a quantum optical generalization of the strong-field Kramers-Heisenberg formula is derived for describing high-harmonic generation. Our formalism is suitable to analyse, among various quantal effects, the possible role of arbitrary photon statistics of the incoming field. The present paper is dedicated to the memory of Prof. Dr. Fritz Ehlotzky, who had significantly contributed to the theory of strong-field phenomena over many decades.
The generation of higher-order harmonic radiation originating from the interaction of intense laser pulses with matter is typically described semiclassically: While the electronic structure and dynamics of matter is described quantum mechanically, the intense light field is described classically---and accordingly the generated harmonic radiation. However, pioneering experiments on quantum optical properties of high-order harmonic generation (HHG) in atomic gases from the group of P. Tzallas [I. Gonoskov et al., Sci. Rep. 6, 32821 (2016) and N. Tsatrafyllis et al., Nat. Commun. 8, 15170 (2017)], and theoretical investigations from the group of M. Lewenstein [M. Lewenstein et al., Nat. Phys. 17, 1104 (2021) and P. Stammer et al., Phys. Rev. Lett. 128, 123603 (2022)] have impressively demonstrated that light's quantum properties are observable in the strong-field realm. In this paper, we develop a quantum optical description of HHG in a bulk semiconductor originating from the nonlinear Bloch current. This mechanism of HHG, known as the intraband current, constitutes the major contribution to the emission spectrum for harmonic with energies of quanta roughly below the bandgap. Under certain approximations, employing a quantum description of both light and matter, we obtain analytical solutions, which allow us to analyze the classical and quantum optical characteristics of the fundamental mode of light and the harmonic modes. We find intricate but sufficiently mild modifications of the photon statistics of the fundamental mode and coherent displacements depending on the parameters of the driving laser field. Similar to high-harmonic generation in atoms, the fundamental and emitted harmonic field modes are entangled. Moreover, our analytical model predicts parameter ranges where these quantum optical properties will be most pronounced, allowing protocols for quantum information processing with high photon numbers over a large range of frequencies.