An LC resonator has and

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Problem

An LC resonator has

C = 200\,\mathrm{fF}

and

L = 5.0\,\mathrm{nH}

, and is treated as an ideal lossless quantum harmonic oscillator with

\Phi_0 = 2.068\times10^{-15}\,\mathrm{Wb}

. ((a)) Find its resonant frequency

f_r

and characteristic impedance

Z_r = \sqrt{L/C}

. ((b)) Find the zero-point flux fluctuation

\Phi_{\rm zpf} = \sqrt{\hbar Z_r/2}

, quoted also as a fraction of

\Phi_0

. ((c)) Find the mean thermal photon number

\bar n = 1/(e^{hf_r/k_BT}-1)

T = 4.0\,\mathrm{K}

and at

T = 20\,\mathrm{mK}

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