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arXiv:1907.01469

Multi-Frequency Atom-Photon Interactions

Ben Yuen, Christopher J. Foot

Published Jul 2, 2019• 0 citations
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Abstract

We present a formalism that enables the analytic calculation of the interaction of a spin-half particle with a polychromatic electromagnetic field. This powerful new approach provides a clear physical picture even for cases with highly degenerate energy levels, which are complicated to interpret in the standard dressed-atom picture. Typically semi-classical methods are used for such problems (leading to equations that are solved by Floquet theory). Our formalism is derived from quantum electrodynamics and thus is more widely applicable. In particular it makes accessible the intermediate regime between quantum and semi-classical dynamics. We give examples of the application to multi-frequency multi-photon processes in strong fields by deriving the Hamiltonians of such systems, and also to the dynamics of weak fields at long times for which semi-classical methods are insufficient.

DOI
10.48550/arxiv.1907.01469

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We present a formalism that enables the analytic calculation of the interaction of a spin-half particle with a polychromatic electromagnetic field. This powerful new approach provides a clear physical picture even for cases with highly degenerate energy levels, which are complicated to interpret in the standard dressed-atom picture. Typically semi-classical methods are used for such problems (leading to equations that are solved by Floquet theory). Our formalism is derived from quantum electrodynamics and thus is more widely applicable. In particular it makes accessible the intermediate regime between quantum and semi-classical dynamics. We give examples of the application to multi-frequency multi-photon processes in strong fields by deriving the Hamiltonians of such systems, and also to the dynamics of weak fields at long times for which semi-classical methods are insufficient.
A vast collection of light-matter interactions are described by the single-frequency Rabi model. However, the physical world is polychromatic, and until now there is no general method to find analytic solutions to the multi-frequency Rabi model. We present the first general method to solve the Rabi model with N frequency modes. Analytic expressions are derived for a spin-half particle in a multi-frequency coherent field, and accurately describe the time evolution even when the interactions are strong. The dynamics are solved in an extended dressed atom picture whereby the spin is progressively dressed by each frequency mode. For weak fields, the closed form expressions for the time evolution of atoms or their analogues take a particularly simple form. These results analytically solve problems in a wide range of fields including quantum information processing, coherent control and resonant nonlinear optics.
Analytic expressions are derived for the time evolution of a spin half in a multi-frequency coherent field, described by a polychromatic Rabi model. We transform to a picture where the spin is dressed progressively by each frequency component of the field. In this picture the time evolution operator takes a simple approximate form, and can be transformed back to the undressed picture where it contains resonant and off-resonant terms. Furthermore, we simplify these transformations to leading order in the field amplitudes, and describe the some of the dominant effects in such multi-frequency processes.
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.
The nonlinear dynamics of an open quantum system containing an arbitrary number of two-level atoms coupled with a classical polychromatic electromagnetic field and a quantised mode of the electromagnetic field is studied. Two particular cases of the elastic and inelastic interactions are considered. In the first case, the quantised mode is resonant with the transition between the quasi-energy levels corresponding to the same quasi-energy state, while in the second case, the levels involved in the interaction correspond to different quasi-energy states. For the elastic interaction, which can appear only in open quantum systems, an analytic solution of the Heisenberg equations is obtained. The time dependences of the population of quasi-energy states, the number of photons in the quantised mode, and photon statistics are numerically analysed for the inelastic interaction and at the crossing point of quasi-levels, when both types of the interaction are simultaneously present.
For pt.I see Chem. Phys. Lett. vol.96, p.464 (1983). The non-linear multiphoton excitation dynamics of two-level systems under the influence of two strong linearly polarised monochromatic fields is studied exactly for the first time along the semiclassical many-mode Floquet theory. The present approach relaxes the restrictions of the conventional generalised rotating-wave approximation (GRWA) and allows correct treatment of various non-linear features such as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending the nearly degenerate perturbative method of Salwen (1955), approximate analytical expressions for transition probabilities are obtained for multiphoton resonance processes for which (n+1) photons of one field are absorbed, whereas n photons of the other field are emitted, where n is an arbitrary integer. Detailed comparison of the analytical, the GRWA and the numerically exact results is given. Also presented is a case study of the time evolution of spin 1/2 systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.
Elements of a new quantum electrodynamic theory which might enable one to obtain a better understanding of the linear and nonlinear interaction between atomic systems in near-field contact is presented. To follow the space-time dynamics in the atomic near-field zone an electromagnetic propagator formalism with polychromatic photons is constructed. A first-quantized description based on the photon energy wave function for free polychromatic photons is extended in such a manner that the birth process of the photon, which takes place in the near-field zone of the source, can be followed. In order to be able to describe harmonic generation processes in near-field optics polychromatic photons of the simple wave-train type are used. The wave mechanical (first-quantized) description based on wave-train photons is upgraded to a field-theoretic (second-quantized) formalism using a new so-called propagator gauge which is closely related to the Poincaré gauge underlying the multipole description of quantum electrodynamics.