From 4a90dfcd44136bb2b523605a27a2b6f9d98420c9 Mon Sep 17 00:00:00 2001 From: dmallubhotla Date: Tue, 2 Feb 2021 17:31:15 -0600 Subject: [PATCH] Discussion from Meeting. --- paper.tex | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/paper.tex b/paper.tex index 439b3d4..d73207e 100644 --- a/paper.tex +++ b/paper.tex @@ -41,14 +41,12 @@ This defines a natural coordinate system, and we can allow the metal to take up For a charge qubit with level separation $\omega$ and dipole moment $\vec{d}$, the relaxation rate $\frac{1}{T_1}$ depends on the qubit's distance from the surface $z$, as well as its orientation $i$. The vacuum wavelength $\lambda = \frac{c}{\omega}$ is a natural unit for this distance $z$, so we wil measure $z$ in units of $\lambda$. The electromagnetic field fluctuations that contribute to qubit relaxation have been described in~\cite{QubitRelax} and~\cite{Henkel1999}. -Both use Fermi's golden rule and the fluctuation-dissipation theorem to relate the relaxation rate to the spectral density of the field fluctations, and obtain the following expression: +Both use Fermi's golden rule and the fluctuation-dissipation theorem to relate the relaxation rate to the spectral density of the field fluctuations, and obtain the following expression: \begin{equation} \frac{1}{T_1} = \frac{d^2}{\epsilon_0} \frac{\omega^3}{c^3} \chi_{i}^{(E)}(z, \omega) \coth\frac{\omega}{2 T}. \end{equation} -\todo{All the Nam stuff is in Gaussian units, so should pick one unit system and stick with it. -Doesn't affect results so far, as \chi is unitless and only depends on quantities that are the same in SI / Gaussian. -Still bad though.} -Here and elsewhere we take $\hbar = k_{\mathrm{B}} = 1$. +%\todo{All the Nam stuff is in Gaussian units, so should pick one unit system and stick with it. Doesn't %affect results so far, as $\chi$ is unitless and only depends on quantities that are the same in SI / %Gaussian. Still bad though.} +%Here and elsewhere we take $\hbar = k_{\mathrm{B}} = 1$. Similarly, for spin qubits with dipole moment $\vec{\mu}$, both~\cite{QubitRelax} and~\cite{Henkel1999} have a similar expression with a different spectral density expression: \begin{equation}