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Register a new userSlow-dissipation limit of the harmonic oscillator with general power-law damping
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Jarrett L. Lancaster
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An approximate solution is presented for simple harmonic motion in the presence of damping by a force which is a general power-law function of the velocity. The approximation is shown to be quite robust, allowing for a simple way to investigate amplitude decay in the presence of general types of weak, nonlinear damping.
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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.
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 analyse the charging process of quantum batteries with general harmonic power. To describe the charge efficiency, we introduce the charge saturation and the charging power, and divide the charging mode into the saturated charging mode and the unsaturated charging mode. The relationships between the time-dependent charge saturation and the parameters of general driving field are discussed both analytically and numerically. And according to the Floquet theorem, we give the expressions of time-dependent charge saturation with the quasiengery and the Floquet states of the system. With both the analytical and numerical results, we find the optimal parameters to reach the best charging efficiency.
Let $H$ denote the harmonic oscillator Hamiltonian on $mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrodinger propagator $U(t)=e^{-itH},$ and find that while $operatorname{singsupp} operatorname{Tr} U(t) subset 2 pi mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d geq 2$ for which the singularities of $operatorname{Tr} U(t)$ at nonzero multiples of $2 pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(lambda^{d-1})$, improving in these cases the $O(lambda^{d-1})$ remainder previously established by Helffer--Robert.
Thermodynamic fluctuations in mechanical resonators cause uncertainty in their frequency measurement, fundamentally limiting performance of frequency-based sensors. Recently, integrating nanophotonic motion readout with micro- and nano-mechanical resonators allowed practical chip-scale sensors to routinely operate near this limit in high-bandwidth measurements. However, the exact and general expressions for either thermodynamic frequency measurement uncertainty or efficient, real-time frequency estimators are not well established, particularly for fast and weakly-driven resonators. Here, we derive, and numerically validate, the Cramer-Rao lower bound (CRLB) and an efficient maximum-likelihood estimator for the frequency of a classical linear harmonic oscillator subject to thermodynamic fluctuations. For a fluctuating oscillator without external drive, the frequency Allan deviation calculated from simulated resonator motion data agrees with the derived CRLB $sigma_f = {1 over 2pi}sqrt{Gamma over 2tau}$ for averaging times $tau$ below, as well as above, the relaxation time $1overGamma$. The CRLB approach is general and can be extended to driven resonators, non-negligible motion detection imprecision, as well as backaction from a continuous linear quantum measurement.
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