Adds more text about the cutoff and lack of cutoff in Lindhard case

Detail

bibliography.bib

@@ -37,6 +37,20 @@
 	journal = {Applied Physics B}
 }
 
+@article{Henkel2006,
+	doi = {10.1007/s00340-006-2219-9},
+	url = {https://doi.org/10.1007/s00340-006-2219-9},
+	year = {2006},
+	month = apr,
+	publisher = {Springer Science and Business Media {LLC}},
+	volume = {84},
+	number = {1-2},
+	pages = {61--68},
+	author = {C. Henkel and K. Joulain},
+	title = {Electromagnetic field correlations near a surface with a nonlocal optical response},
+	journal = {Applied Physics B}
+}
+
 @article{QubitRelax,
 	doi = {10.1103/physreva.86.010301},
 	url = {https://doi.org/10.1103/physreva.86.010301},
@@ -219,3 +233,13 @@
 	year = {1954},
 	month = {1}
 }
+
+@book{LandauLifshitzElectrodynamics,
+	author        = "Landau, Lev Davidovich and Lifshitz, Evgenii Mikhailovich and Pitaevskii, Lev Petrovich",
+	title         = "{Electrodynamics of continuous media; 2nd ed.}",
+	publisher     = "Butterworth",
+	address       = "Oxford",
+	series        = "Course of theoretical physics",
+	year          = "1984",
+	url           = "https://cds.cern.ch/record/712712",
+}

paper.tex

@@ -13,7 +13,7 @@
 \usepackage{todonotes}
 \usepackage{siunitx}
 
-\title{Title}
+\title{EWJN from a BCS Superconductor}
 
 \addbibresource{./bibliography.bib}
 
@@ -34,25 +34,62 @@
 \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 $\hbar = k_{\mathrm{B}} = 1$.
+Physically, we are interested in the relaxation time of a qubit near the surface of the metal.
+Sufficiently close to the metal, such that the separation between qubit and metal is much smaller than the shortest dimension of the metallic body, we can approximate the metal as a half space.
+This defines a natural coordinate system, and we can allow the metal to take up all points with $z$-coordinate less than zero, extending to infinity in the $x$ and $y$ directions.
 
-For a half space, these spectral densities can be written in terms of the surface's reflection coefficients, as derived by\cite{Henkel1999}.
+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:
+\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 $\hbar = k_{\mathrm{B}} = 1$.
+
+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}
+	\frac{1}{T_1} = \frac{\mu^2}{\epsilon_0} \frac{\omega^3}{c^3} \chi_{i}^{(B)}(z, \omega) \coth\frac{\omega}{2 T}.
+\end{equation}
+
+The spectral densities are in general functions of the geometry and electrodynamic response of the metal object.
+A standard limiting case for the response is that of local, linear electrodynamics, where the relationship between the electric displacement $\vec{D}$ and electric field $\vec{E}$ is a proportionality:
+\begin{equation}
+	\vec{D}(\vec{r}) = \epsilon \vec{E}(\vec{r})
+\end{equation}
+For a metal with conductivity $\sigma$, this dielectric function will typically have a large imaginary value as determined by the Drude expression in Gaussian units:
+\begin{equation}
+	\epsilon = 1 + i\frac{4 \pi \sigma}{\omega}
+\end{equation}
+
+For a half space, these spectral densities can be written in terms of the surface's reflection coefficients, as derived by\cite{Henkel1999} for local electrodynamics.
 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}
+	\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 $r_p$.
+In the local limit, for dielectric constant $\epsilon$, the reflection coefficients will be the Fresnel $r_p$ and $r_s$:
+\begin{align}
+	r_p &= \frac{\epsilon v - \sqrt{\epsilon - u^2}}{\epsilon v + \sqrt{\epsilon - u^2}} \\
+	r_s &= \frac{v - \sqrt{\epsilon - u^2}}{ v + \sqrt{\epsilon - u^2}}
+\end{align}
 
-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}:
+However, as discussed in~\cite{QubitRelax} and~\cite{Henkel2006}, this expression no longer remains accurate for arbitrarily small distances, with \eqref{eq:chi} diverging as $\frac{1}{z^3}$ as $z \rightarrow 0$.
+The divergence here stems from the unphysicality of local electrodynamics at very small scales;
+for $z$ smaller than the electromagnetic coherence length of the metal the full response function defined by
+\begin{equation}
+	\vec{D}(\vec{r, t}) = \int \dd{r'} \dd{t'} \epsilon(r, r', t, t') \vec{E}(\vec{r', t'})
+\end{equation}
+becomes necessary.
+The experiment described by Kolkowitz et al.\cite{Kolkowitz2015} gives an example of a physical situation where the local description is insufficient, with a qubit probe at a distance from a silver film less than the film's mean free path.
+Given that the coherence lengths of typical BCS superconductors can be larger than the distances used in that experiment, we should expect that the purely local description will not work for the superconducting case.
+
+We can keep~\eqref{eq:chi} accurate for sufficiently small $z$ if we make appropriate changes to our reflection coefficients\cite{QubitRelax,Henkel2006}, so 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}} \\
@@ -60,8 +97,8 @@ For the description of the behaviour of~\eqref{eq:chi} to remain accurate for sm
 	\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 quantities $\zeta_p$ and $\zeta_s$ are proportional to the surface impedances of the metal, derived for this case by Ford and Weber\cite{Ford1984}, and with more thorough discussion in~\cite{LandauLifshitzElectrodynamics}.
 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.
@@ -72,6 +109,12 @@ Using the form described by Solyom\cite{SolyomV3},
 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.
 
+The Lindhard dielectric function reflects the important property of having an imaginary part that vanishes for $q$ such that $\abs{\varepsilon_q - \omega} >  q\vf$.
+This is a very generic feature of these types of response functions, and occurs because there are no points on the assumed spherical Fermi surface further than $2 q_{\mathrm{F}}$ apart.
+Thus, there are no available quasiparticle-hole excitations available for energy dissipation (cf discussion in \cite{AGD}, \cite{FetterWalecka} or \cite{SolyomV3}).
+This is a general argument, and it should be expected that a superconducting dielectric function should also have zero imaginary part above some momentum on the order of the Fermi momentum.
+Because the Lindhard dielectric function's imaginary part vanishes above a cutoff $q_c\left(\omega\right)$ and has real part that goes as $\frac{1}{q^2}$, all of the integrals in $\eqref{eq:chi}, \eqref{eq:zp} and \eqref{eq:zs}$ converge.
+
 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.}
 
@@ -136,8 +179,6 @@ which clearly leads to a divergent $\chi_\perp^E$.
 
 This is the same divergence discussed in Langsjoen et al.~\cite{QubitRelax}, where for the normal state this problem does not arise if the Lindhard dielectric function is used.
 Ultimately, this is because for $q > 2 q_{\mathrm{F}}$ the imaginary part of the Lindhard function goes to zero.
-Physically, this happens because there are no points on the assumed spherical Fermi surface further than $2 q_{\mathrm{F}}$ apart, which means there are no available quasiparticle-hole excitations available for energy dissipation (cf discussion in \cite{AGD}, \cite{FetterWalecka} or \cite{SolyomV3}).
-This is a general argument, and it should be expected that a superconducting dielectric function should also have zero imaginary part above some momentum on the order of the Fermi momentum.
 
 This can be handled in two coarse approximations.
 The key is that the integral in \eqref{eq:chi} picks out values around $u = \frac{c}{\omega} \sim \frac{1}{z}$ over most of its range, because of the $u^2 e^{-2 z u}$ factor for $u \gg 1$.