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Sensing with the harmonic oscillator

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 Publication date 2019
  fields Physics
and research's language is English




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A system obeying the harmonic oscillator equation of motion can be used as a force or proper acceleration sensor. In this short review we derive analytical expressions for the sensitivity of such sensors in a range of different situations, considering noise of thermal and measurement origins and a formalism for dealing with oscillators whose natural frequency $omega_0$ jitters. A special case where the sensitivity can be improved beyond the standard expressions and some applications with examples are also discussed.



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145 - John W. Sanders 2021
It is shown that the classical damped harmonic oscillator belongs to the family of fourth-order Pais-Uhlenbeck oscillators. It follows that the solutions to the damped harmonic oscillator equation make the Pais-Uhlenbeck action stationary. Two systematic approaches are given for deriving the Pais-Uhlenbeck action from the damped harmonic oscillator equation, and it may be possible to use these methods to identify stationary action principles for other dissipative systems which do not conform to Hamiltons principle. It is also shown that for every damped harmonic oscillator $x$, there exists a two-parameter family of dual oscillators $y$ satisfying the Pais-Uhlenbeck equation. The damped harmonic oscillator and any of its duals can be interpreted as a system of two coupled oscillators with atypical spring stiffnesses (not necessarily positive and real-valued). For overdamped systems, the resulting coupled oscillators should be physically achievable and may have engineering applications. Finally, a new physical interpretation is given for the optimal damping ratio $zeta=1/sqrt{2}$ in control theory.
We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-Wojcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low and high temperature regimes, and investigate its time evolution by calculating its first two cumulants.
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Using Schwinger Variational Principle we solve the problem of quantum harmonic oscillator with time dependent frequency. Here, we do not take the usual approach which implicitly assumes an adiabatic behavior for the frequency. Instead, we propose a new solution where the frequency only needs continuity in its first derivative or to have a finite set of removable discontinuities.
147 - Yoni BenTov 2021
I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving $ivarepsilon$ prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
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