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Bulletin of the Russian Academy of Sciences Physics

Atom–Photon Cluster in Nonlinear and Quantum Optics

Ashat M. Basharov

Published Jun 1, 2024• 9 citations
Publisher

Abstract

Abstract Models of radiative quantum systems were constructed as analogs of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of cavity mode quanta. A distinctive feature of the models is the description of an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan–Schwinger representation of the su(2) algebra, so that one can talk about an atom–photon and/or photon clusters. The arising algebras are mathematically unsolvable, in contrast to the Heisenberg–Weyl algebra. This makes it possible to implement and observe fundamental effects of coherent nonlinear optics, such as photon echo, optical nutation, and superradiance, in ensembles of the described clusters.

DOI
10.1134/s1062873824706706

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Abstract Models of radiative quantum systems were constructed as analogs of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of cavity mode quanta. A distinctive feature of the models is the description of an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan–Schwinger representation of the su(2) algebra, so that one can talk about an atom–photon and/or photon clusters. The arising algebras are mathematically unsolvable, in contrast to the Heisenberg–Weyl algebra. This makes it possible to implement and observe fundamental effects of coherent nonlinear optics, such as photon echo, optical nutation, and superradiance, in ensembles of the described clusters.
We described the constructed models of radiative quantum systems analogous of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of resonator mode quanta. A distinctive feature of the models is the possibility to describe an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan-Schwinger representation of the su(2) algebra, that could point to atomic-photon and/or photon clusters. The algebras are arising that are mathematically insoluble, in contrast to the Heisenberg — Weyl algebra, which makes it possible to implement and observe fundamental effects of coherent nonlinear optics, such as photon echo, optical nutation, and superradiance in ensembles of the described clusters.
The interaction of an atomic ensemble localized in a microcavity with external electromagnetic fields under Raman resonance conditions with an optically forbidden atomic transition involving photons of the microcavity mode has been described in terms of third-order polynomial algebra. It has been shown that atoms and photons localized in the microcavity under these conditions form a united object, an atom-photon cluster, on the states of which the irreducible representations of polynomial algebra are implemented. Classical coherent and quantum broadband electromagnetic fields are considered as external fields. The effective Hamiltonian, effective dipole moment operator, and relaxation operator of the atom-photon cluster are expressed in terms of the generators of polynomial algebra, which is the algebra of the dynamical symmetry of the problem. The developed mathematical technique has been applied to describe the main radiative processes—spontaneous emission and nutation effect—on atom-photon clusters. All of these effects are peculiar and differ from similar phenomena on two-level atoms, but only simple cases of the mentioned radiative processes have been considered.
It is shown that our description of the dynamics of interaction between atoms with microcavity mode quanta and an external electromagnetic field under conditions of two-photon resonance follows from a simple theory of one-photon resonance interaction between two-level atoms and the same external electromagnetic field, if we substitute su (2) algebra elements of the one-photon resonance theory for third-order elements of polynomial algebra. Commutation relations of polynomial algebra determine the features of processes with the participation of a single object, i.e. an atom-photon cluster localized in a microcavity. Its states are the basis for irreducible representations of this polynomial algebra.
We show how analogs of a large number of well-known nonlinear-optics phenomena can be realized with one or more two-level atoms coupled to one or more resonator modes. Through higher-order processes, where virtual photons are created and annihilated, an effective deterministic coupling between two states of such a system can be created. In this way, analogs of three-wave mixing, four-wave mixing, higher-harmonic and -subharmonic generation (i.e., up- and down-conversion), multiphoton absorption, parametric amplification, Raman and hyper-Raman scattering, the Kerr effect, and other nonlinear processes can be realized. In contrast to most conventional implementations of nonlinear optics, these analogs can reach unit efficiency, only use a minimal number of photons (they do not require any strong external drive), and do not require more than two atomic levels. The strength of the effective coupling in our proposed setups becomes weaker the more intermediate transition steps are needed. However, given the recent experimental progress in ultrastrong light-matter coupling and improvement of coherence times for engineered quantum systems, especially in the field of circuit quantum electrodynamics, we estimate that many of these nonlinear-optics analogs can be realized with currently available technology.
Within the framework of a two-dimensional microscopic, purely quantum mechanical model, we analyse the dynamics of single-photon wave packets interacting with optical elements (beam splitters, mirrors), modelled as systems of two-level atoms. That is, we utilize a two-dimensional cavity to simulate the quantum behaviour of simple optical components and networks made thereof. The field is quantized using the canonical procedure, and only the basis states with one unit of excitation are included. This, however, covers linear optical phenomena. The field is taken to interact with localized atoms through a dipole interaction. Using different configurations of atoms, and choosing their frequencies to be resonant or off-resonance, we can model mirrors, beam splitters, focusing devices and multicomponent systems. Thus we can model arbitrary linear networks of optical components. We show the time evolution of a photon wave packet in an interferometer as an example. As the state of the field is known at each instant, spectral properties and spatial coherence can immediately be obtained from the simulations. We also know the states of the two-level atoms constituting the components, which allows us to consider their quantum behaviour. Here the decay of an excited atom into the vacuum state of the electromagnetic field in the two-dimensional cavity is studied.