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Physica Scripta

Parametric resonances: from the Mathieu equation to QASER*

Goong Chen, Jing Tian, Bandar Bin‐Mohsin, Reed Nessler, Anatoly A. Svidzinsky, Marlan O. Scully

Published Jun 6, 2016• 8 citations
Publisher

Abstract

Abstract Combination resonances produced by periodic modulation of parameters can generate substantial gains in new light amplifiers, acronymed QASER (quantum amplification by superradiant emission of radiation). Here we study such gains and investigate their parametric resonance spectral patterns. We use the Floquet theory and develop a projection method that can properly capture gains near the primary resonance and subharmonic frequencies for the Mathieu equation and QASER.

DOI
10.1088/0031-8949/91/7/073004

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Abstract Combination resonances produced by periodic modulation of parameters can generate substantial gains in new light amplifiers, acronymed QASER (quantum amplification by superradiant emission of radiation). Here we study such gains and investigate their parametric resonance spectral patterns. We use the Floquet theory and develop a projection method that can properly capture gains near the primary resonance and subharmonic frequencies for the Mathieu equation and QASER.
The new concept of Quantum Amplification by Superradiant Emission of Radiation for generating high-frequency coherent light has produced several open questions, and one of the most important is how to determine the gain in two coupled oscillators with a parametric amplifier. In this article, we analytically derive the equation for the gain expression and state sufficient conditions to obtain gain. We also draw connections to single oscillator topics, including the Mathieu equation and the Kronig–Penney model.
Lasers and masers typically require population inversion. But with phase coherent atoms (phasers), we get lasing without inversion (e.g. 10% of the atoms excited). However, in recent work we found that it is possible to get coherent light emitted with no atoms excited, via Quantum Amplification of Superradiant Emission of Radiation (QASER). In particular, we found that by utilizing collective superradiant emission, we can generate coherent light at high frequency in the UV or x-ray bands by driving the atomic system with lower frequency source. Here, we present a simple analysis based on near-resonant QASER operation and on a multi-photon Hamiltonian obtained by, e.g. a canonical transformation.
With Maxwell-Bloch equations how the superradiance can lead to amplification and gain at a frequency much larger than the pumping frequency has been shown. This remarkable effect has been examined in terms of a simpler model involving two coupled oscillators, with one of them parametrically driven. We show that this coupled-oscillator model has a hidden parity-time (PT\mathcal{PT}PT) symmetry for quantum amplification by superradiant emission of radiation (QASER); we thus bring PT\mathcal{PT}PT symmetry to the realm of parametrically coupled resonators. Moreover, we find that the QASER gain arises from the broken-PT\mathcal{PT}PT-symmetry phase. We then quantize the simplified version of QASER using quantum Langevin equations. The quantum description enables us to understand how the system starts from quantum fluctuations.
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically unstable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically instable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.