Ivan Gonoskov, Christian Hünecke and Stefanie Gräfe of the Institute of Physical Chemistry at Friedrich Schiller University Jena have developed an analytical framework revealing how entangled light arises from intense laser interactions with matter. The researchers began with a Pauli-Fierz description to model the dynamics of electrons and light, identifying that nonlinearities in the electronic dipole response directly correlate to nonclassicality as measured by Wigner-function negativity.
This work establishes a predictive link between strong-field electron behavior and the quantum properties of emitted light, offering a path toward generating high-photon-number nonclassical radiation in high-order harmonic generation. The results provide a basis for engineering nonclassical states in intense laser-matter interactions.
Pauli-Fierz Hamiltonian Describes Strong Laser-Quantum System Coupling
The analytical framework begins with a Pauli-Fierz Hamiltonian, offering a fully quantum description of charged particles interacting with electromagnetic fields, and departs from previous approaches by directly using the time-dependent Schrödinger equation without conditioning or post-selection techniques. This foundation allows for a specific factorization of the light-matter system into a field-driven electronic state and a quantized light state, a method that implicitly accounts for the influence of a strong driving field on the system.
The resulting model describes the system’s response to the light field as a finite perturbation, defined by a parameter proportional to the amplitude of the j-th light mode. This factorization is not merely mathematical convenience. It provides a physically intuitive framework for understanding and controlling the generation of nonclassical light during intense laser-matter interactions.
The framework’s utility stems from its ability to connect strong-field electron dynamics directly to the quantum-optical properties of the high-order harmonic generation (HHG) radiation produced. Specifically, the researchers demonstrate that the nonlinear dependence of the electronic dipole response on the light-mode coordinate dictates the nature of the emitted radiation; an approximately constant dipole yields coherent radiation, while a linear dependence produces squeezing.
Higher-order nonlinearities in this dipole response are directly linked to Wigner-function negativity, a key indicator of nonclassicality, meaning the emitted light exhibits quantum properties beyond those found in ordinary light. The analytical model was tested using atomic and molecular systems, with parameters including an intense driving laser pulse polarized along the x-axis and a quantization parameter β ranging from 1. 4 x 10-6 to 1. 4 x 10-3 atomic units.
These simulations reveal how the interplay between the laser field and the electronic structure of the emitting material shapes the quantum state of the generated light. Further analysis explored multi-emitter configurations, revealing a pathway toward high-photon-number nonclassical radiation in HHG. The researchers found that scaling the number of emitters can amplify the nonclassical effects, potentially leading to brighter and more robust sources of entangled photons.
The initial conditions for the simulations involved a product state of light and matter, with the system beginning in the second excited state for a hydrogen atom model. Laser parameters included wavelengths of 2227 nm and 3000 nm, pulse durations of 2 cycles and intensities ranging from 1 x 1012 to 1 x 1013 W/cm2.
The Pauli-Fierz Hamiltonian, as presented in equation 1 within the source material, is the starting point for the analysis, allowing for a rigorous treatment of the quantum interactions between light and matter. This approach contrasts with methods that require post-selection, offering a more direct and complete description of the underlying physics. “Our results provide a transparent connection between strong-field electron dynamics and quantum-optical properties of HHG radiation,” the researchers state, emphasizing the clarity and predictive power of their model.
Parametric Factorization Links Electron Dynamics to Light State
A newly developed analytical framework establishes a direct link between the behavior of electrons in strong laser fields and the quantum properties of the light they emit, moving beyond approximations that require post-selection of results. The core innovation lies in identifying how the nonlinear response of the electronic dipole to changes in the light mode dictates the characteristics of the resulting radiation. The approach hinges on recognizing that an unchanging dipole generates coherent light, while a linear dependence between the dipole and the light mode produces squeezing, a reduction in quantum noise.
Higher-order nonlinearities in this relationship directly lead to a definitive signature of nonclassical light, meaning light that cannot be described by classical electromagnetism. This connection provides a predictive pathway for designing systems that generate light with specific, nonclassical properties.
The framework’s mathematical foundation rests on a specific factorization of the combined light-matter wavefunction, allowing for the decoupling of the Schrödinger equation into separate equations for the electronic state and the light mode. This decoupling is achieved through a carefully chosen “gauge” for the parametric factorization, ensuring the equations remain consistent and solvable.
By multiplying both sides of a key equation by the complex conjugate of the electronic wavefunction and integrating over all space, the researchers were able to establish a complete system of equations describing the evolution of the quantum state of the light mode. The resulting equations reveal that the nonlinear dependence on the light mode coordinate arises from the q-dependent dipole moment, a consequence of the parametrically connected electronic state.
Simulations using a one-dimensional model of a calcium atom, subjected to a laser pulse with an intensity of 4 x 1013 W/cm2 and a wavelength of 1945 nm, demonstrate the framework’s ability to accurately predict the emitted light’s quantum state. Further analysis explored how scaling the number of emitters, modeled as 1011 individual CaO molecules, affects the generation of high-photon-number nonclassical radiation. The parametric-connection approximation, they note, is analogous to the Born-Oppenheimer approximation used to simplify electron-nuclear dynamics, highlighting its fundamental nature.
Nonlinear Dipole Response Drives Nonclassical Radiation Emergence
This approach differs from previous methods by directly linking the nonlinear response of electrons to the quantum properties of the emitted light, a connection previously unestablished. The researchers demonstrate that these nonlinear terms, stemming from the oscillating dipole moment of an electron, can generate non-classical states even when starting from a coherent or vacuum initial state.
This is visually represented in their modeling of a calcium atom, where a constant dipole yields a high-photon-number coherent state, a linear dipole produces a squeezed state. A cubic dependence generates a Wigner function with negative regions. The work extends beyond single emitters, analyzing how scaling the number of radiating units impacts the generation of nonclassical radiation in high-order harmonic generation.
Simulations reveal that increasing the number of emitters can amplify these nonclassical effects, suggesting a pathway toward achieving high-photon-number nonclassical radiation. “If the dipole response is approximately independent of the light-mode coordinate, the emitted radiation remains close to a coherent state,” the researchers state, clarifying the conditions necessary for observing these effects. The researchers summarize their work as a rigorous model describing the generation of nonclassical electromagnetic radiation in strongly driven systems.
This model is based on factorizing and separating the time-dependent Schrödinger equation into equations for the field-driven system and the quantized light mode, coupled via the oscillating dipole of the driven system. “With this theory, we provided a mechanism for the emergence of nonclassical (high-harmonic) light generation in terms of a nonlinear dependence of this dipole moment d(β q,t) on the light mode’s coordinate q, which becomes particularly pronounced when the harmonic frequency Ω corresponds to an electronic resonance in the system.” This resonance, they explain, amplifies the nonlinear effects, making the generation of nonclassical light more efficient and observable.
By carefully controlling the nonlinearities in the dipole response, it may be possible to design systems that generate light with specific, desired quantum characteristics. This opens avenues for advancements in quantum technologies and applications requiring entangled photons.
High-Order Harmonic Generation Enables Bright Tunable Radiation
The analytical framework detailed in recent work links the nonlinear dependence of an electronic dipole moment on the light-mode coordinate to the generation of nonclassical states of light, even when starting from coherent or vacuum initial states. This connection bypasses the need to assume initially nonclassical driver or harmonic fields, offering a new route to bright, highly nonclassical radiation through high-order harmonic generation (HHG).
Expanding the time-dependent dipole moment into a series reveals how these effects arise; the team visualized this process using a model calcium atom, showing that higher-order nonlinearities in the dipole’s response directly correlate to Wigner-function negativity, a definitive signature of nonclassicality. The work illustrates that the broad, coherent spectrum inherent to HHG, spanning many octaves, is a pathway to engineer the quantum properties of the emitted radiation.
This precise control over the dipole moment’s dependence on the light mode is essential for tailoring the quantum characteristics of the generated photons. These operators collectively generate the full light-state evolution, demonstrating the potential for designing systems that generate tailored nonclassical radiation.
Wigner-Function Negativity Characterizes Strongly Nonclassical States
A framework now more clearly defines the relationship between a laser’s influence on electrons and the resulting quantum properties of emitted light, identifying a direct link between nonlinearities in the electronic dipole response and the emergence of nonclassical radiation. If that dipole response remains constant, the radiation resembles a coherent state; a linear dependence produces squeezing, and higher-order nonlinearities yield states exhibiting Wigner-function negativity, a hallmark of nonclassicality.
Visualizations of a strongly driven calcium atom model illustrate this mechanism, showing that constant, nearly light-mode independent dipole moments generate high-photon-number harmonic radiation approximating a coherent state. Introducing a slight linear dependence on the light-mode coordinate allows for the generation of squeezed light states, also with a high photon count.
Most strikingly, dipole moments exhibiting a nonlinear dependence on the light-mode coordinate, approximated to cubic terms in one example, produce nonclassical light, evidenced by pronounced negative regions within the corresponding Wigner function. This nonlinearity in the oscillating dipole moment, the researchers emphasize, is a necessary, though not sufficient, condition for generating nonclassical light. Further analysis extended this principle to the 13th harmonic generated from a laser-driven hydrogen atom, demonstrating nonclassical light in the ultraviolet region at a wavelength of 171 nanometers.
The framework’s predictive power extends to multi-emitter configurations, systems with spatially isolated, non-Coulomb interacting emitters. The average number of photons in the output mode scales approximately with the square of the light-mode coordinate, indicating the potential for generating very bright nonclassical states.
The team’s calculations, fitting the dipole moment up to the fifth order, demonstrate that these configurations can indeed produce significant nonclassicality. This work builds upon previous parametric factorization techniques, but importantly identifies the physical implications of this approach for nonclassical radiation. The researchers highlight that they do not need to assume initial coherence, vacuum, or classical fields for the driver or its harmonics. The ability to predict and control these states, based on the nonlinear dependence of the dipole moment, represents a step toward harnessing the unique properties of nonclassical light for advanced quantum technologies.
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