The exact factorization approach, originally developed for electron-nuclear dynamics, is extended to light-matter interactions within the dipole approximation. This allows for a Schrödinger equation for the photonic wavefunction, in which the potential contains exactly the effects on the photon field of its coupling to matter. We illustrate the formalism and potential for a two-level system representing the matter, coupled to an infinite number of photon modes in the Wigner-Weisskopf approximation, as well as to a single mode with various coupling strengths. Significant differences are found with the potential used in conventional approaches, especially for strong couplings. We discuss how our exact factorization approach for light-matter interactions can be used as a guideline to develop semiclassical trajectory methods for efficient simulations of light-matter dynamics.
The exact factorization approach, originally developed for electron-nuclear dynamics, is extended to light-matter interactions within the dipole approximation. This allows for a Schrödinger equation for the photonic wavefunction, in which the potential contains exactly the effects on the photon field of its coupling to matter. We illustrate the formalism and potential for a two-level system representing the matter, coupled to an infinite number of photon modes in the Wigner-Weisskopf approximation, as well as to a single mode with various coupling strengths. Significant differences are found with the potential used in conventional approaches, especially for strong couplings. We discuss how our exact factorization approach for light-matter interactions can be used as a guideline to develop semiclassical trajectory methods for efficient simulations of light-matter dynamics.
This Review is devoted to the presentation of the exact factorization as a framework employed to study a variety of quantum-mechanical many-body problems. Since its original formulation in the 70s, the main applications of the exact factorization were directed towards understanding the properties of multi-component systems, e.g. electron-nuclear systems. Especially in the electron-nuclear case, the exact factorization is viewed as an exactification of the Born-Oppenheimer approximation, thus it was, and still is, largely employed in nonadiabatic dynamics. Nonetheless, as early as the 80s, the formalism was employed to study many-electron interacting systems and, quite recently, i.e. less than a decade ago, it was extended to study the behavior of molecules in the context in cavity quantum electrodynamics. These formulations, perhaps less popular than the electron-nuclear formulation, have attracted a lot of attention over the years. Therefore, we review here the exact electron-only factorization and the exact photon-electron-nuclear factorization.
Simulation of the interaction of light with matter, including at the few-photon level, is important for understanding the optical and optoelectronic properties of materials and for modeling next-generation nonlinear spectroscopies that use entangled light. At the few-photon level the quantum properties of the electromagnetic field must be accounted for with a quantized treatment of the field, and then such simulations quickly become intractable, especially if the matter subsystem must be modeled with a large number of degrees of freedom, as can be required to accurately capture many-body effects and quantum noise sources. Motivated by this we develop a quantum simulation framework for simulating such light-matter interactions on platforms with controllable bosonic degrees of freedom, such as vibrational modes in the trapped ion platform. The key innovation in our work is a scheme for simulating interactions with a continuum field using only a few discrete bosonic modes, which is enabled by a Green's function (response function) formalism. We develop the simulation approach, sketch how the simulation can be performed using trapped ions, and then illustrate the method with numerical examples. Our work expands the reach of quantum simulation to important light-matter interaction models and illustrates the advantages of extracting dynamical quantities such as response functions from quantum simulations.
Quantum light-matter systems at strong coupling are notoriously challenging to analyze due to the need to include states with many excitations in every coupled mode. We propose a nonperturbative approach to analyze light-matter correlations at all interaction strengths. The key element of our approach is a unitary transformation that achieves asymptotic decoupling of light and matter degrees of freedom in the limit where light-matter interaction becomes the dominant energy scale. In the transformed frame, truncation of the matter or photon Hilbert space is increasingly well justified at larger coupling, enabling one to systematically derive low-energy effective models, such as tight-binding Hamiltonians. We demonstrate the versatility of our approach by applying it to concrete models relevant to electrons in crystal potential and electric dipoles interacting with a cavity mode. A generalization to the case of spatially varying electromagnetic modes is also discussed.
This chapter considers electron–photon interaction, i.e., light–matter coupling. It introduces the theoretical treatment of electron–photon interaction for different and mostly strongly simplified cases. In resonator structures, calculation of the appropriate photon modes is required to allow access to the strength of electron–photon coupling. The general guideline is that light may be treated as a wave when propagating in vacuum, yet it has to be quantized in the case of the interaction with matter. In any ensemble of atoms or molecules in interaction with an electromagnetic field, there are three processes occurring: spontaneous emission, absorption, and stimulated emission. In order to get a flavor of a many-body treatment of electron–photon interaction, the derivation of the semiconductor Bloch equations will be discussed briefly. The strength of the light field at the location of the electron determines the interaction strength between light and matter.
The Exact Factorization (EF) theory aims at the separation of the nuclear and electronic degrees of freedom in the many-body (MB) quantum mechanical problem. Being formally equivalent to the solution of the MB Schrödinger equation, EF sets up a strategy for the construction of efficient approximations in the theory of the correlated electronic-nuclear motion. Here we extend the EF formalism to incorporate the case of a system under the action of an electromagnetic field. An important interplay between the physical magnetic and the Berry-curvature fields is revealed and discussed within the fully non-adiabatic theory. In particular, it is a known property of the Born-Oppenheimer approximation that, for a neutral atom in a uniform magnetic field, the latter is compensated by the Berry-curvature field in the nuclear equation of motion (\citet{Yin-92}). From an intuitive argument that the atom must not be deflected by the Lorentz force from a straight line trajectory, it has been conjectured that the same compensation should occur within the EF theory as well. We give a rigorous proof of this property.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.