# How is "linewidth" defined for a damped driven oscillator?

Suppose we have a damped driven harmonic oscillator governed by the following equation of motion: $$\ddot \phi(t) + 2 \beta \dot \phi(t) + \omega_0^2 \phi(t) = J(t) \, .$$ In the case that $$J(t) = A \cos(\Omega t)$$, we can show that $$\phi(t) = \text{Re} \left[e^{i \Omega t} \underbrace{\frac{-A}{\Omega^2 - \omega_0^2 - i 2 \beta \Omega}}_\text{response function} \right] \, .$$ As explained in this other question, we define the resonance frequency as the frequency where power flows only into the oscillator, which happens when the response function is purely imaginary. That's because when the response function is imaginary, the drive and the oscillator position are 90 degrees out of phase so the drive and oscillator speed are in phase and so the work done is always positive. The response function is imaginary when $$\Omega = \omega_0$$.

How do we define the oscillator's linewidth?

• If the Wikipedia article on Oscillator linewidth is anything to go by, we would need a source of "phase noise" in the signal. Your oscillator doesn't have any noise? Mar 16 '19 at 9:00
• @GingerBreadMan we don't need phase noise here because we can vary $\Omega$. Mar 16 '19 at 9:14
• In the high-Q limit and near resonance the amplitude transfer function can be approximated by a (complex) Lorentzian, $T\sim \frac{1}{\Delta + i\Gamma}$ with squared magnitude (energy transfer function, regular Lorentzian) of $|T|^2 \sim \frac{1}{\Delta^2 + \Gamma^2}$ . I've always seen the linewidth defined as the FWHM of that Lorentzian: $2\Gamma$. Sometimes it's the HWHM and sometimes there's other factors of 2 kicking around here and there. Mar 16 '19 at 9:31
• I'll point out that the FWHM of the energy transfer function is the energy decay rate and the HWHM of the energy transfer function is the amplitude decay rate. Are you looking for a definition that works outside of the high-Q, near resonance limit? More clarity on the factors of two and possible convention choices? Mar 16 '19 at 9:32
• I agree with jberger - I would associate 'linewidth' with the FWHM of the Lorentzian, and I would definitely frown on any work that used the term differently without defining it explicitly. But I suspect there's several other choices that make sense - I don't think there'll be a single universal definition. Mar 16 '19 at 11:44