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arXiv:2501.14907

Supersymmetry in nonlinear and linear Quantum Optics: the Kerr-like and multiphoton Jaynes-Cummings models

Ivan A. Bocanegra-Garay, Leonardi Hernández-Sánchez, H. M. Moya-Cessa

Published Jul 17, 2025• 0 citations
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Abstract

Abstract A novel approach is proposed to analyze a rather vast counter-rotating Hamiltonian interaction in the context of quantum optics.The method relies upon the supersymmetric mapping of the corresponding rotating interaction and allows the analysis of thedynamics in the counter-rotating system in a fully general and exact analytical manner by diagonalizing the correspondingHamiltonian. Intriguing features of the counter-rotating system are revealed through the simple supersymmetric transformation.In turn, such interesting attributes have an important range of potential technological applications. In this way, supersymmetryemerges as a useful tool to both connect and construct exactly solvable photonic systems in quantum optics, as well as toanalyze the corresponding physical consequences and their possible technology implementations.

DOI
10.21203/rs.3.rs-6999454/v1

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Abstract A novel approach is proposed to analyze a rather vast counter-rotating Hamiltonian interaction in the context of quantum optics.The method relies upon the supersymmetric mapping of the corresponding rotating interaction and allows the analysis of thedynamics in the counter-rotating system in a fully general and exact analytical manner by diagonalizing the correspondingHamiltonian. Intriguing features of the counter-rotating system are revealed through the simple supersymmetric transformation.In turn, such interesting attributes have an important range of potential technological applications. In this way, supersymmetryemerges as a useful tool to both connect and construct exactly solvable photonic systems in quantum optics, as well as toanalyze the corresponding physical consequences and their possible technology implementations.
The supersymmetric connection that exists between the Jaynes-Cummings (JC) and anti-Jaynes Cummings (AJC) models in quantum optics is unraveled entirely. A new method is proposed to obtain the temporal evolution of observables in the AJC model using supersymmetric techniques, providing an overview of its dynamics and extending the calculation to full photon counting statistics. The approach is general and can be applied to determine the high-order cumulants given an initial state. The analysis reveals that engineering the collapse-revival behavior and the quantum properties of the interacting field is possible by controlling the initial state of the atomic subsystem and the corresponding atomic frequency in the AJC model. The substantial potential for applications of supersymmetric techniques in the context of photonic quantum technologies is thus demonstrated. Published by the American Physical Society 2024
Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system.
Abstract The aim of this work is to show how supersymmetric (SUSY) quantum mechanics can be applied to the Jaynes–Cummings (JC) Hamiltonian of quantum optics. These SUSY transformations connect pairs of Jaynes–Cummings Hamiltonians characterized by different detuning parameters as well as Jaynes–Cummings to anti-Jaynes–Cummings Hamiltonians. Therefore, JC Hamiltonians can be classified in hierarchies or sequences which are connected through SUSY transformations. As a byproduct, the symmetries of JC Hamiltonians are found as well as the special case of a sequence of resonant-like interacting systems having the form of a simple shape invariant JC Hamiltonian hierarchy.
We demonstrate how supersymmetries of Hamiltonians for coupled Bose-Fermi systems can be used to place the Hamiltonians of the Jaynes-Cummings model and Dicke model under the rotating wave approximation in matrix form and provide explicit analytic solutions for their eigenvalues. We then use this supersymmetry to place the Liouvillians of the associated Markovian open systems in matrix form and provide explicit solutions for their eigenvalues. These results are a consequence of the fact that the Hamiltonian of the Jaynes-Cummings model commutes with the linear Casimir invariant of the superalgebra u(1∣1)u(1|1)u(1∣1) and that the Hamiltonian of the Dicke model commutes both with the linear invariant of ∑iui(1∣1)\sum_{i} u_{i}(1|1)∑i​ui​(1∣1) and with the invariant of an additional su(2)su(2)su(2) algebra. Our methods apply to various coupled Bose-Fermi systems with u(1∣1)u(1|1)u(1∣1) and more generally with u(n∣m)u(n|m)u(n∣m) dynamical superalgebras, and may provide efficient tools for studying more complicated examples.
The multiphoton Jaynes-Cummings model is investigated and applications in quantum information science are explored. Considering the strong atom-field coupling regime and an N-photon interaction, a nonlinear driving field can perform an arbitrary rotation in the Fock space of a bosonic mode involving the vacuum and an M-Fock state, with M<N. In addition, driving a bosonic mode with a linear coherent field (superposition of many Fock states), only the cavity states within the Fock subspace {|0⟩,|1⟩,…,|N-1⟩} can be populated; i.e., we show how to implement a Fock state filter, or quantum scissor, that restricts the dynamics of a given bosonic mode to a limited Hilbert space. Such a device can be employed as a generator of finite-dimensional quantum-optical states and also as a quantum-optical intensity limiter, allowing as a special case the generation of single-photon pulses. On the other hand, our system also provides a very rich physics in the weak atom-field coupling regime, in particular, multiphoton electromagnetically induced transparencylike phenomena, inducing a narrow (controllable) reflectivity window for nonlinear probe fields. These results are useful for applications in quantum information processing and also motivate further investigations, e.g., the use of an N-photon Jaynes-Cummings system as a qudit with harmonic spectrum and the exploration of multiphoton quantum interference.