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arXiv:2403.04383 · Journal of the Optical Society of America B

Jaynes-Cummings interaction with a traveling light pulse

Victor Rueskov Christiansen, Mads Middelhede Lund, Fan Yang, Klaus Mølmer

Published May 28, 2024• 4 citations
arXivHTMLPDF

Abstract

The Jaynes-Cummings model provides a simple and accurate description of the interaction between a two-level quantum emitter and a single mode of quantum radiation. Due to the multimode continuum of eigenmodes in free space and in waveguides, the Jaynes-Cummings model should not be expected to properly describe the interaction between an emitter and a traveling pulse of quantum radiation. In this article, we review a cascaded quantum system approach that accurately describes the interaction of a quantum system with an incident quantum pulse of radiation. This approach leads to different formulations of the theory, each of a similar structure as the Jaynes-Cummings model but with important modifications.

DOI
10.1364/josab.523292

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The Jaynes-Cummings model provides a simple and accurate description of the interaction between a two-level quantum emitter and a single mode of quantum radiation. Due to the multimode continuum of eigenmodes in free space and in waveguides, the Jaynes-Cummings model should not be expected to properly describe the interaction between an emitter and a traveling pulse of quantum radiation. In this article, we review a cascaded quantum system approach that accurately describes the interaction of a quantum system with an incident quantum pulse of radiation. This approach leads to different formulations of the theory, each of a similar structure as the Jaynes-Cummings model but with important modifications.
This paper presents a general master equation formalism for the interaction between traveling pulses of quantum radiation and localized quantum systems. Traveling fields populate a continuum of free-space radiation modes and the Jaynes-Cummings model, valid for a discrete eigenmode of a cavity, does not apply. We develop a complete input-output theory to describe the driving of quantum systems by arbitrary incident pulses of radiation and the quantum state of the field emitted into any desired outgoing temporal mode. Our theory is applicable to the transformation and interaction of pulses of radiation by their coupling to a wide class of material quantum systems. We discuss the most essential differences between quantum interactions with pulses and with discrete radiative eigenmodes and present examples relevant to quantum information protocols with optical, microwave, and acoustic waves.
Introduction In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T , they found that the field mode reached a thermal equilibrium with temperature T f = Ω T /ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission. An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.
We studied a very simple model of the interaction between radiation and matter, named Jaynes-Cummings model. Matter is represented by an atom, treated as a two level system, which interacts with one single mode of the electromagnetic field. We looked at the problem in two different ways: first, treating both systems as quantum and later, considering the field as classical and the atom as quantum. In particular,we concentrate in obtaining the population inversion in both cases. In the context of semiclassical theory, we still used two distinct approachs: one, taking into account the field reaction of the atom and, the other, ignoring any effect of the atom onto the field. We verified that the first gives a more realistic description than the second because it accounts for spontaneous emission by the atom. We still included the environment on the description of both quantum and classical dynamics.
The interaction of a traveling pulse of quantum light with a localized quantum system shows non-Markovian features when scattering takes place in a structured environment. Here, we devise a non-Markovian input-output theory by coherently coupling the scatterer to a pseudomode, which decays into a Markovian reservoir. This incorporates memory effects of a structured environment without altering the cascaded nature of the Lindblad master equation, whose solution provides the quantum state of the output field of any desired mode. We apply our theory to the stimulated emission by a two-level atom, and to transmission of a Gaussian pulse through a cavity. We observe that the non-Markovian revival of coherence in the scatterer distorts the single-mode nature of the incoming pulse, thus resulting in a multimode output field.
Strong coupling between a single resonator mode and a single quantum emitter is key to a plethora of experiments and applications in quantum science and technology and is commonly described by means of the Jaynes-Cummings model. Here, we show that the Jaynes-Cummings model only applies when the cavity does not significantly change the emitter's emission rate into free-space. Most notably, the predictions made by the Jaynes-Cummings model become increasingly wrong when approaching the ideal emitter-resonator systems with no free-space decay channels. We present a Hamiltonian that provides, within the validity range of the rotating wave approximation, a correct theoretical description that applies to all regimes. As minimizing the coupling to free-space modes is paramount for many cavity-based applications, a correct description of strong light-matter interaction is therefore crucial for developing and optimizing quantum protocols.