nam_paper/paper.tex

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\begin{document}

\maketitle

\section{Introduction \label{sec:intro}}
\begin{itemize}
	\item Motivate with~\cite{Tenberg2019} and~\cite{Kolkowitz2015}
	\item Existing work with Lindhard expression~\cite{QubitRelax}
	\item Variety of existing superconducting dielectric functions like in~\cite{AGD, llv9, Zimmermann1991, Mattis, Tinkham}.
	Advantage of Nam expression\cite{Nam1967} is that it's got the worked out dependence on momentum and impurities, both of which we need.
\end{itemize}
\section{Methods \label{sec:methods}}

The electromagnetic field fluctuations that contribute to qubit relaxation haven 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:
\begin{equation}
	\frac{1}{T_1} = \frac{d_{(E,B)}^2}{\epsilon_0} \frac{\omega^3}{c^3} \chi_{i}^{(E,B)}(r, \omega) \coth\frac{\omega}{2 T}.
\end{equation}
Here, $\vec{d}_{(E,B)}$ is the dipole moment of a point qubit at position $\vec{r}$, with $i$ the direction of the qubit's dipole moment, with $(E,B)$ represents an electric or magnetic qubit (and correspondingly, an electric or magnetic spectral field density).
The frequency $\omega$ corresponds to the separation between energy levels of the qubit, and here and elsewhere we take $\hbar = k_{\mathrm{B}} = 1$.

For a half space, these spectral densities can be written in terms of the surface's reflection coefficients, as derived by\cite{Henkel1999}.
Considering for now a qubit pointing in the direction perpendicular to the half-space, we can write
\begin{align}
	\chi_\perp^E(z, \omega) &= \Re \int_0^{+\infty} \dd{u} \frac{u^3}{v} e^{2 i z v} r_p(u), \label{eq:chi}
\end{align}
with $z$ the distance to the qubit from the half-space measured in terms of the vacuum wavelength $\lambda = \flatfrac{c}{\omega}$.
The integration variable $u$ effectively represents a momentum in units of $\flatfrac{1}{\lambda}$, with $v = \sqrt{1 - u^2}$.
If $v \geq 1$, we take the positive square root $v = i \sqrt{u^2 - 1}$.
The magnetic spectral density is the same, except with an additional factor of $\flatfrac{1}{c^2}$ and with $r_s$ instead of $r_p$.

For the description of the behaviour of~\eqref{eq:chi} to remain accurate for small $z$, we use the reflection coefficients in Ford and Weber~\cite{Ford1984}:
\begin{align}
	r_p(u) &= \frac{\pi v - \zeta_p(u)}{\pi v + \zeta_p(u)} \\
	r_s(u) &= \frac{\zeta_s(u) - \frac{\pi}{v}}{\zeta_s(u) + \frac{\pi}{v}} \\
	\zeta_p(u) &= 2i \int_0^\infty \dd{y} \frac{1}{\kappa^2} \left( \frac{y^2}{\epsilon(\frac{\omega}{c}\kappa, \omega) - \kappa^2} + \frac{u^2}{\epsilon_(\frac{\omega}{c}\kappa, \omega)} \right) \label{eq:zp} \\
	\zeta_s(u) &= 2i \int_0^\infty \dd{y} \frac{1}{\epsilon(\frac{\omega}{c}\kappa, \omega) - \kappa^2} \label{eq:zs} \\
	\kappa^2 &= u^2 + y^2
\end{align}
The treatment in~\cite{QubitRelax} compares the difference between these expressions and the simpler Fresnel reflection coefficients.
In effect, using the Fresnel reflection coefficients for a metal for some constant conductivity corresponds to a local limit $u \rightarrow 0$.

The dielectric function $\epsilon(q, \omega)$, then, contains the information needed to describe the electromagnetic properties of the surface near the qubit.
For the normal state, the dielectric function derived by Lindhard~\cite{Lindhard} used in~\cite{QubitRelax} describes the non-local electromagnetic response of a metal.
Using the form described by Solyom\cite{SolyomV3},
\begin{equation}
	\epsilon_{\mathrm{Lindhard}}(\vec{q}, \omega) = 1 + \frac{q_{TF}^2}{q^2}\frac{\displaystyle 1 + \frac{\omega + \flatfrac{i}{\tau}}{2 \vf q} \ln(\frac{\omega - \vf q + \flatfrac{i}{\tau}}{\omega + \vf q + \flatfrac{i}{\tau}})}{\displaystyle 1 + \frac{\flatfrac{i}{\tau}}{2 \vf q} \ln(\frac{\omega - \vf q + \flatfrac{i}{\tau}}{\omega + \vf q + \flatfrac{i}{\tau}})}. \label{eq:lindhardsolyom}
\end{equation}
Here, $q_{TF}$ is the Thomas-Fermi wavevector $q_{TF}^2 = 3 \flatfrac{\omega_p^2}{\vf^2}$, $\omega_p$ is the plasma frequency $\sqrt{\flatfrac{4 \pi n e^2}{m}}$, $\tau$ is the collision time and $\vf$ is the Fermi velocity.
This can be shown to reduce to the Drude dielectric function in the $q \rightarrow 0$ limit.

We use the expressions from Nam in~\cite{Nam1967} to represent the superconducting response function.
This extends the previous models by Mattis and Bardeen~\cite{Mattis} and Abrikosov, Dzyaloshinskii and Gorkov\cite{AGD} to give expressions that allow for broader ranges of impurity values.\todo{Including the full expressions from Nam here is a bit space-prohibitive, but it may be important to show exactly what our assumptions encode to in his notation? ex: by assuming no magnetic impurities, our renormalisation factor becomes simpler.}

Here, we look at Nam's expressions in the weak coupling limit, for no magnetic impurities and an isotropic material.
\begin{equation}
	\epsilon_{\mathrm{Nam}}(\vec{q}, \omega) = 1 + \frac{3 \pi}{\omega^2} \frac{n e^2}{m} \left[\int_{\Delta - \omega}^{\Delta}\dd{\omega'} \tanh(\frac{\omega + \omega'}{2 T}) I_1 + \int_{\Delta}^{\infty} \dd{\omega'} \left( \tanh(\frac{\omega + \omega'}{2 T}) I_1  - \tanh(\frac{\omega'}{2 T})I_2 \right) \right], \label{eq:eps}
\end{equation}
with
\begin{align}
	I_1 &= F(q, \Re[\sqrt{(\omega + \omega')^2 - \Delta^2} - \sqrt{\omega'^2 - \Delta^2}]) (g + 1) \nonumber\\
	&\quad + F(q, \Re[-\sqrt{(\omega + \omega')^2 - \Delta^2} - \sqrt{\omega'^2 - \Delta^2}]) (g - 1) \\
	I_2 &= F(q, \Re[\sqrt{(\omega + \omega')^2 - \Delta^2} - \sqrt{\omega'^2 - \Delta^2}]) (g + 1) \nonumber\\
	&\quad + F(q, \Re[\sqrt{(\omega +  \omega')^2 - \Delta^2} + \sqrt{\omega'^2 - \Delta^2}]) (g - 1) \\
	F(q, E) &= \frac{1}{q \vf} \left[2 S(E) + (1 - S(E)^2)\ln(\frac{S(E) + 1}{S(E) - 1})\right] \label{eq:NamF} \\
	S(q, E) &= \frac{1}{q \vf} \left( E - i \left(\Im[\sqrt{(\omega + \omega')^2 - \Delta^2} + \sqrt{\omega'^2 - \Delta^2}] + \frac{2}{\tau} \right) \right) \\
	g &= \frac{\omega' \left(\omega + \omega'\right) + \Delta^2}{\sqrt{\omega'^2 - \Delta^2}\sqrt{(\omega + \omega')^2 - \Delta^2}}.
\end{align}
The assumption of isotropy suppresses the $q$ dependence for $\Delta$, which then is just a function of temperature, and can be described using the well-known BCS expression $\Delta \approx 3.06 \sqrt{T_c(T_c - T)}$ (see for example \cite{Tinkham}).

\section{Numerical Techniques \label{sec:technical}}

The noise integral \eqref{eq:chi} can be calculated numerically, with proper care taken to handle the integrand's behaviour across the entire range.
For small momenta, where $\vf q \ll \omega$, both \eqref{eq:lindhardsolyom} and \eqref{eq:eps} can be series expanded to give explicit expressions.
The Lindhard dielectric function, up to $\mathcal{O}(q^2)$, becomes
\begin{gather}
	\epsilon_{\mathrm{Lindhard}}(\vec{q}, \omega) = 1 - \frac{\omega_p^2}{\omega^2} \left(\frac{\omega}{(\omega + \frac{i}{\tau})} + (\vf q)^2  \frac{9 \omega + 5 \frac{i}{\tau}}{15 (\omega + \frac{i}{\tau})^3} \right). \label{eq:lindhardsmallkseries}
\end{gather}
As expected for a description of the normal state, for $q \rightarrow 0$ this reduces to the Drude expression.

All of the momentum dependence in the Nam expression is contained within the $F(q, E)$ function in \eqref{eq:NamF}.
Expanding this out to second order in the momentum gives
\begin{align}
	F = \frac43 \frac{1}{\eta} + (\vf q)^2\frac{4}{15} \frac{1}{\eta^3},
\end{align}
where
\begin{equation}
	\eta = E - i \left(\Im[\sqrt{(\omega + \omega')^2 - \Delta^2} + \sqrt{\omega'^2 - \Delta^2}] + 2 \frac{1}{\tau} \right).
\end{equation}
This, and other limiting forms, were stated by Nam as well~\cite{Nam1967}.
Inserting this in $\eqref{eq:NamF}$ suffices to obtain the small $q$ values in a more numerically stable way.

By comparison to the $q \rightarrow 0$ case, the large momentum dependence requires more careful thought.
For sufficiently large $q$, the logarithm in $\eqref{eq:NamF}$ goes to $i \pi$, and $F \rightarrow \frac{i \pi}{q \vf}$.
This behaviour leads to an unphysical divergence in \eqref{eq:chi}, as 

\section{Experiments \label{sec:experiments}}
\begin{itemize}
	\item Show that expression is of right order of magnitude to describe~\cite{Tenberg2019}.
	\item Look at temperature at which~\cite{Tenberg2019} should show SC effects (not far off!)
	\item Compare to~\cite{Kolkowitz2015}?
	Much further from relevant temperature range.
	\item Metal choices?
	Obviously large $T_c$ is most important to get noise reduction benefits.
	What else is most relevant for usable metals?
\end{itemize}
\section{Conclusions \label{sec:conclusions}}

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