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(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]  \\
	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}}

	Nam's expressions are no longer valid for $q \geq q_\mathrm{F}$, and as $q \rightarrow \infty$ exhibit an unphysical $\flatfrac{1}{q}$ dependence, a feature shared by similar expressions in \cite{AGD}.
	In order to use~\eqref{eq:zp} and~\eqref{eq:zs}, this must be corrected to prevent divergences.
	For sufficiently large momenta, the response function should approach the normal state function\todo{add good ref of this}.
	Moreover, $\Im \epsilon$ should go to $0$ when $q \gtrapprox 2 k_{\mathrm{F}}$, as otherwise there will be no available states within the Fermi surface for energy transfer (for more on this point, see the discussion in~\cite{FetterWalecka}).
	In order to account for the first point, for the normal state we can use the Lindhard dielectric function, which has the correct nonlocal behaviour to describe the low $z$ noise\cite{QubitRelax}.

	The effect of the inaccurate large momentum values in $\epsilon$ is an overestimation of the dissipative part $\Im r_p$ of the reflection coefficient.
	To correct for this, we can use~\eqref{eq:lindhardsolyom} and~\eqref{eq:eps} to find $r_{p\mathrm{, Lindhard}}$ and $r_{p\mathrm{, Nam}}$, then choose whichever value has a smaller imaginary part.
	Effectively, this defines an $r_{p\mathrm{, effective}}$.
	For $q$ below some cutoff $q_{uc}$ on the order of $q_{\mathrm{F}}$, this picks the Nam reflection coefficient and reflects the diminished noise in the superconducting state.\todo{Does this need to be justified with a graph of the two functions?}

	The integral in~\eqref{eq:chi} picks out values around $u = \frac{c}{\omega} q \sim \frac{1}{z}$.
	As long as $\frac{1}{z} \ll \frac{\omega}{c} q_{uc}$, the value for the noise will reflect the effects of the superconducting expressions, without picking up the inaccuracies of the transition region.
	This corresponds with physical distances on the order of $\SI{1}{\nm}$ for metals with $\vf \sim \SI{1e6}{\m\per\s}$.
	For $z$ smaller than this, more sophisticated dielectric functions would be necessary to take into account inter-atomic spacing.

\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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