text:
\section{Introduction}
Optical lattices arising from the interference of laser
beams have proved to be a versatile tool for probing the physics of periodic
systems \cite{jessen1996}. When loaded with cold atoms or molecules they can
be used to investigate paradigms ranging from single particle Bloch physics
\cite{jessen1996} to coherent \cite{morsch2006} and strongly correlated
many-body systems \cite{lewenstein2006}. Compared to their counterparts in
the solid-state, optical lattices can be modulated easily in time as well
as space, possess no phonons and are nearly free from defects.

The most easily available laser modes are Gaussian (G), whose phase structure
is essentially that of plane waves. Their interference therefore gives
rise to optical lattices with \textit{discrete} symmetries, which have been
explored quite intensely \cite{jessen1996,morsch2006,lewenstein2006}. Cold
atoms loaded into these lattices can be used to simulate other physical
systems which share the same discrete symmetries, such as the Hubbard model
on a honeycomb lattice \cite{parmekanti2006}, or quasi-crystals
\cite{grynberg1997}. However there are also interesting physical systems
which possess \textit{continuous} symmetries, such as a Bose-Einstein
condensate (BEC) in a toroidal potential expected to display a persistent
current \cite{kurn2005,juha1998} or a particle in a helical waveguide
expected to be bound geometrically \cite{exner2007}. In order to simulate
these scenarios with cold atoms optical lattices with continuous symmetries
are required. Such lattices can be made by interfering laser modes with
azimuthal phase structure such as Laguerre-Gaussian (LG) beams which
have recently become available (\cite{nOAMbook} and references therein).

Some work has been done regarding continuous optical lattices (see below);
however they are yet to be explored in as much detail as discrete lattices.
In this Communication we examine the structure of the optical lattice arising
from the interference of two LG beams. While it is generally known that this
produces a helical potential for ultracold matter, the waveguiding properties
of this potential have hitherto been unexplored. Here we provide an elementary
demonstration of the fact that the lattice can indeed work as a waveguide.
Further we show that the confinement in both transverse directions in the
continuous helical lattice can be related to that in the more familiar
one-dimensional discrete optical lattice via simple geometrical factors.
Lastly, we present a compact expression for the aspect ratio of the waveguide.
This enables us to prove that the waveguide is always anisotropic. It also
provides a convenient but realistic limit to spectral geometers analysing the
potential for its ability to support geometrically bound states \cite{exner2007}.

To place our work in perspective we point out that theoretically the
superposition of LG beams has been considered in the context of discrete
circular lattices \cite{nienhuis2004}, ring lattices without \cite{arnold2006}
and with \cite{amico2005} tunable boundary phase twists and twisted optical molasses
\cite{carter2006}. Toroidal traps \cite{ndholakia2001} and atom waveguides
\cite{dholakia2000} have been proposed using LG beams and also seem achievable
with the use of microfabricated optical elements \cite{ertmer2001}.
Experimentally LG modes have been used for waveguiding a BEC \cite{sengstock2001}
and their interference has been used to trap microparticles \cite{macdonald2002}
as well as to create vortex states in a BEC \cite{Phillips2007,staliunas2002}.

\section{Interference of Laguerre-Gaussian beams}

We consider the intensity pattern arising from the interference of two identical
linearly polarized LG beams with charge $l$ and a single radial node at the origin,
counter-propagating along the cylindrical $z-$axis
\begin{equation}
\label{eq:inet} I=4I_{l}(\rho)\cos^{2}\left(kz-l\phi
\right)\cos^{2}(\omega t),
\end{equation}
where $I_{l}(\rho)$ is the intensity due to either beam, $k$ the wavevector
and $\omega$ the optical frequency. In writing Eq.(\ref{eq:inet}) we have
considered displacements much smaller than the Rayleigh range, i.e.
$z/z_{R} \ll 1$, where $z_{R}=\pi \omega_{0}^{2}/\lambda$, $\omega_{0}$ being
the beam waist and $\lambda$ the wavelength of light.

\section{The helical dipole potential}
We now examine what happens when a cold atom is brought into a region
illuminated by light with the intensity distribution (\ref{eq:inet}). Without
loss of generality we consider an isolated ground-excited transition for the
atom. We let the frequency $\omega$ of the LG beams be red-detuned
($\Delta < 0$) by $\Delta\gg \Gamma$ from the atomic transition, where
$\Gamma$ is the linewidth of the excited state. In this case the cold atom
experiences a nondissipative attractive potential \cite{meystrebook}
\begin{equation}
\label{eq:potential}U=U_{l}(\rho)\cos^{2}\left(kz-l\phi\right),
\end{equation}
where $U_{l}(\rho)=\frac{\hbar
\Gamma^{2}}{2\Delta}\left[\frac{I_{l}(\rho)}{I_{Sat}}\right] < 0$,
and $I_{Sat}$ is the saturation intensity for the transition. The
non-resonant potential $U$ attracts atoms to the most intense part of its
intensity distribution. $U$ can trap sufficiently cold molecules as well.

\subsection{The geometry of the potential}
The minima of the potential (\ref{eq:potential}) lie on $2l$ intertwined
helices of radius \cite{macdonald2002}
\begin{equation}
\label{eq:radius}
R=\omega_{0}\sqrt{l/2},
\end{equation}
and pitch
\begin{equation}
\label{eq:pitchwithr}
 P=l\lambda\left[1-\left(\frac{3l+2}{4}\right)\theta^{2}\right]^{-1}.
\end{equation}
The pitch of each helix is practically $\sim l\lambda$ ; the correction to it has been
calculated by including terms linear in $z/z_{R}$ and has been defined
in terms of the asymptotic half-angle of divergence of either LG beam
\cite{siegmanbook}
\begin{equation}
\label{eq:halfangle} \theta = \frac{\lambda}{\pi w_{0}},
\end{equation}
as shown in Fig.\ref{fig:Helix} which displays the potential for the case
$l=1$. For regular laser beams $\theta$ is quite small, about $10^{-3}$.
We note that $\theta = 1$ corresponds to the diffraction limit
\cite{siegmanbook}.

The helices are displaced $\lambda/2$ parallel to each other along the
$z$-axis, and rotated $\pi/l$ radians away from each other on any plane
where $z$ is constant. For high enough laser power because of the red
detuning $(\Delta < 0)$ a cold atom moves along a particular minimum of
the potential Eq.(\ref{eq:potential}) and effectively sees a potential
that is helical in symmetry.
\begin{figure}
\includegraphics*[width=0.49 \textwidth]{Helixpix1.eps}
\caption{\label{fig:Helix}The double-helical lattice, arising from the
optical interference of two Laguerre-Gaussian $l=1$ beams.
Two right-handed helical waveguides drawn using solid red and dotted
blue curves are formed, each with radius proportional to the beam
radius $\omega_{0}$ and with pitch equal to the optical wavelength
$\lambda$. They are translated by $\lambda/2$ and rotated by $\pi$
with respect to each other. The half-angle of divergence of either
beam $\theta$, and the pitch angle or \textit{chirality} of each
helical strand $\alpha_{R}$ are depicted in the figure. Also shown
are the orthogonal unit vectors of the right-handed Frenet frame
local to the helix, consisting of the tangent $(\hat{\tau})$,
normal $(\hat{\nu})$ and binormal $(\hat{\beta})$ at a point.}
\end{figure}

\subsection{Atomic motion along the waveguide}
Here we establish the fact that $U$ can act as a waveguide. Without loss of
generality we restrict ourselves to a single strand of the ensemble. For
an atom in such a helix an orthogonal co-moving coordinate system is
supplied by the Frenet frame consisting of the unit vectors
$(\hat{\tau},\hat{\nu},\hat{\beta})$, which are the tangent, normal and
binormal to the helix respectively at a point
$(\rho,\phi,z)$ (Fig.\ref{fig:Helix}) \cite{aminovbook}
\begin{eqnarray}
\label{eq:Frenetframe}
\begin{array}{lll}
\hat{\tau}&=&\sin \alpha
\hspace{0.02in} \hat{z}+\cos \alpha \hspace{0.02in} \hat{\phi}\\
\hat{\nu}&=&\cos \alpha
\hspace{0.02in} \hat{z}-\sin \alpha \hspace{0.02in} \hat{\phi}\\
\hat{\beta}&=& \hat{\rho}, \\
\end{array}
\end{eqnarray}
where the transformation from the cylindrical unit vectors involves the
pitch angle $\alpha$ of a helix of radius $\rho$ defined by
\begin{equation}
\label{eq:pitchangle} \tan \alpha = \frac{P}{2\pi \rho},
\hspace{.1in} 0 \leq \alpha < \pi/2.
\end{equation}
If the laser detuning ($\Delta$) from the atomic transition is large
enough, photon scattering due to spontaneous emission is negligible
and the potential Eq.(\ref{eq:potential}) is essentially conservative. This
implies that we can calculate the force due to the potential simply as
\begin{equation}
\label{eq:force}
F=- \nabla U.
\end{equation}
Expressing Eq.(\ref{eq:force}) in the Frenet frame Eq.(\ref{eq:Frenetframe})
we find that the component of force along the $\hat{\tau}$ direction
vanishes identically. However there are constraining forces in the
directions $\hat{\nu}$ and $\hat{\beta}$. The potential $U$
therefore acts as a waveguide that confines the atom in the locally
transverse directions while allowing free motion along the local
longitudinal direction. It is worth noting that this is true
everywhere in the potential, and not only at its minimum.

\subsection{Atomic motion transverse to the waveguide}
We now turn to a detailed consideration of the atomic motion in the transverse
directions $\hat{\nu}$ and $\hat{\beta}$. From Eq.(\ref{eq:pitchangle})
we find that at the minimum $(\rho=R)$ of the potential $U$ the pitch angle,
or the \emph{chirality} of the helix (Fig.\ref{fig:Helix}),
is $\alpha_{R}$ where
\begin{equation}
\label{eq:alphaR}
 \tan \alpha_{R} = \frac{P}{2\pi R} = \theta \sqrt{\frac{l}{2}}.
\end{equation}
To investigate the nature of the potential seen in the transverse
direction by the atom, we use Eq.(\ref{eq:alphaR}) and the curvilinear
coordinates local to the Frenet frame
\begin{eqnarray}
\label{eq:ccords}
\begin{array}{ll}
\nu=&(kz-l\phi)/k, \\
\beta=&\rho-R. \\
\end{array}
\end{eqnarray}
Using these quantities we Taylor expand Eq.(\ref{eq:potential}) near its
minimum (where $\nu \sim 0, \beta \sim 0$, and $\alpha \sim \alpha_{R}$) and obtain
\begin{equation}
\label{eq:appu}
U \sim U_{l0}(R)+\frac{1}{2}m\omega_{\nu}^{2}\nu^{2}+\frac{1}{2}m\omega_{\beta}^{2}\beta^{2}.
\end{equation}
Thus to lowest order in $\nu$ and $\beta$ the transverse confinement is simple
harmonic. Let us consider the trapping frequencies $\omega_{\nu}$ and
$\omega_{\beta}$. The trapping frequency in the $\hat{\nu}$ direction is
given by
\begin{equation}
\label{eq:onu}
\omega_{\nu}=\frac{1}{\hbar}\left[ 4 E_{R} U_{l0}(R)\sec \alpha_{R}\right]^{1/2},
\end{equation}
where we have used the single photon recoil energy
\begin{equation}
\label{eq:Erec}
 E_{R}=\frac{\hbar^{2}k^{2}}{2m}
\end{equation}
to write $\omega_{\nu}$ in a transparent form. In Eq.(\ref{eq:Erec}) $m$ is
the atomic mass. Importantly, Eq.(\ref{eq:onu}) allows us to relate the helical
lattice to the more familiar one-dimensional optical lattice if we recall that $l=0$
corresponds to the interference of G beams. If we put $l=0$ in Eq.(\ref{eq:alphaR}),
the tangent vanishes. This implies $\sec \alpha_{R}=1$, and Eq.(\ref{eq:onu})
reduces to
\begin{equation}
\label{eq:onu1d}
\omega^{1d}=\frac{1}{\hbar}\left[ 4 E_{R} U_{00}(R)\right]^{1/2},
\end{equation}
which is the trapping frequency for an atom in the well of a 1-d lattice
constructed from G beams. In this case the $\hat{\nu}$ direction points
along the $z-$axis. Now for LG beams, $l \neq 0$. Nevertheless since
$\theta \sim 10^{-3}$ is usually a small quantity, the tangent in
Eq.(\ref{eq:alphaR}) is still close to zero and hence $\sec \alpha_{R}$ is
not much greater than 1. Therefore we may not expect the trapping
frequency in the normal direction for a helical lattice to be very
different than that for the one-dimensional G lattice for
comparable laser parameters ($U_{00} \sim U_{l0}$).

The trapping frequency in the $\hat{\beta}$ direction can be written as
\begin{equation}
\label{eq:obeta}
\omega_{\beta}=\theta \frac{\left[ 4 E_{R} U_{l0}(R)\right]^{1/2}}{\hbar} \sim \theta \omega^{1d}.
\end{equation}
Eq.(\ref{eq:obeta}) shows that for $l \neq 0$ and comparable parameters
($U_{00} \sim U_{l0}$) the confinement along the binormal in the helical
waveguide is typically weaker than in the corresponding one-dimensional G
lattice by a factor of $\theta \sim 10^{-3}$.

Relating the continuous helical LG lattice to the discrete G lattice
allows us to transfer some of the well-established physical intuition from
the latter to the former. For example the sub-wavelength confinement of
atoms, characteristic of deep discrete optical lattices \cite{meystrebook}
can typically be obtained only along the $\hat{\nu}$ direction in a helical
lattice.

The ratio of the two transverse frequencies Eqs.(\ref{eq:onu}) and (\ref{eq:obeta}), i.e.
the aspect ratio $A$ of the waveguide, is given by
\begin{equation}
\label{eq:anisotropy} A = \frac{\omega_{\nu}}{\omega_{\beta}} =
\frac{1}{\theta} \left(1+\frac{l\theta^{2}}{2}\right)^{1/4}.
\end{equation}
Eq.(\ref{eq:anisotropy}) enables us to examine the cross section of the helical
waveguide quite generally. For example perfect isotropy ($A=1$) requires
$\theta = 1$ and $l=0$. The first condition corresponds to the diffraction
limit and the second to the absence of any helical structure. This implies
that perfect isotropy for the waveguide is not possible even in principle.
Staying in the diffraction limit and allowing for just a single pair
of helices ($l=1$) introduces a $10\%$ anisotropy: $A=1.1$. For typical
experimental parameters however $\omega_{\nu} \sim 10$ MHz,
$\omega_{\beta} \sim 10$ KHz and $A \sim 10^{3}$, i.e. the anisotropy is quite
large. An important implication of this anisotropy is the ability of
the waveguide to serve as a three, two or one-dimensional structure for
microscopic objects. For sufficiently cold atoms or molecules
($T < \hbar \omega_{\nu}/k_{B} \sim 10\mu$K) the motion along the $\hat{\nu}$
direction can be frozen out and the waveguide is rendered essentially
two dimensional. Similarly, for even lower temperatures
($T < \hbar \omega_{\beta}/k_{B} \sim 10 $nK) the atomic motion along the
$\hat{\beta}$ direction can be frozen out and the waveguide made
one-dimensional. Here $k_{B}$ is Boltzmann's constant.

\section{Applications}

The helical potential (\ref{eq:potential}) allows access to a less-explored
symmetry, and may reveal interesting physics such as negative group
velocities \cite{calvo1978} and Berry's phase \cite{nbitter1987} for particles
moving under its influence. Below we outline a possible application to quantum
transport.

The availability of a waveguide that bends and twists naturally suggests a
problem that has been considered (as with much of optical lattice physics)
initially in the condensed matter literature. This is the phenomenon of
\textit{geometrically bound states} relevant to electron transport through
nanoscopic wires in quantum heterostructures (\cite{londerganbook} and
references therein). The basic idea can be understood by considering the
transmission of a quantum particle through a straight waveguide of uniform
cross-section. A low energy cut-off for the transmission exists because the
transverse wavelength of the particle cannot be larger than the width of the
waveguide. Now imagine a bulge in the waveguide, which locally allows for a
longer wavelength, i.e. an energy lower than the cut-off. In an infinitely long
waveguide this would correspond to the localization of the particle at the bulge,
or a bound state. In a waveguide of finite extent, this shows up as an exponentially
decaying resonance : the particle eventually leaks out of the waveguide. The
same effect can be achieved even in a waveguide of constant width, if the
waveguide is bent \cite{goldstone1992}. In fact it has been shown quite
generally that curvature in a waveguide is equivalent to the presence of
an attractive potential \cite{exner1989}. Classically analogous
`trapped' electromagnetic modes have experimentally been observed in bent
microwave waveguides \cite{murdock1992}.

Motivated by our work, Exner \textit{et al.} have examined a curved helical
potential for the presence of bound states and suggested an answer in the
affirmative \cite{exner2007}. Their analysis implies that for
large aspect ratios $(A\gg 1)$ and small pitch angles ($\alpha_{R} \sim 0)$
appropriate to the optical potential $U$ described here, particles can be
bound by the waveguide if the radius goes through a local maximum. It
would be interesting to test this prediction as in practice the pitch angle of $U$
can be adjusted using $\theta$ and $l$ while local extrema in the radius can
easily be obtained using (de)focussing lenses. One experimental signature of the
bound state would be a minimum in the transmission through the waveguide at
the binding energy \cite{londerganbook}.

\section{Conclusion}
We have investigated the waveguiding properties of the optical lattice
arising from the interference of two identical counter-propagating
Laguerre-Gaussian beams. We have provided a quantitative connection to the more
familiar Gaussian lattices. We have also provided a basic description of atomic
motion in the waveguide.
\section{Acknowledgements}
It is a pleasure to thank P. Meystre for both intellectual and material
support and E. Wright for stimulating discussions. This work is supported
in part by the US Army Research Office, NASA, the National Science
Foundation and the US Office of Naval Research.


text:
\section{Introduction\label{Chap:Intro}}
Much SRF research and development is presently concentrating on finding means to reduce the losses in the cavity for next-generation accelerator applications. It was found that the surface resistance \rs can be impacted by a number of different treatments. \rs consists of a temperature dependent BCS term \rbcs and a residual resistance $R\sub{res}$, both of which may be impacted by the treatment. For example, it was discovered that the annealing of high RRR niobium with titanium or nitrogen can significantly change \rs and especially the dependence of \rs on the RF magnetic field \cite{Grassellino2013,Dhakal2013}.

We expand here on studies \cite{Vogt2013a,Kugeler2009,Kugeler2013,Eichhorn2013, Gonnella2015} to minimize the additional residual term in the cavity systems introduced by thermal currents based on thermal cycling. During a thermal cycle the superconducting cavity is warmed up to temperatures around $20\,\kelvin$ which exceeds the transition temperature $T\sub{c}$. Afterwards the cavity is cooled down again to the operating temperature. Depending on the temperature \textit{difference} between the two ends of the cavity tank system during the superconducting (sc) phase transition, significant variations of the surface resistance are observed. In the best case the residual resistance of the $1/9\,\pi$ mode is decreased from $10.6\,\nano\ohm$ to $1.4\,\nano\ohm$ which is more than a factor of seven.

In contrast to annealing techniques, the cycling procedure should not change \rbcs because the material properties such as mean free path and coherence length are not influenced. Hence we attribute the observed modification of \rs for different cycles to a change in residual resistance. This assumption is borne out by extracting the residual resistance from quality factor versus temperature measurements (see Section \ref{sec:rres}).

\subsection{Thermoelectric effect in the cavity tank system}

The TESLA-type cavity reported on here received a heavy BCP (about $150\,\micro\meter$) prior to a $2\,\hour$ bakeout at $800^{\circ}$C.  A light BCP etch followed the heat treatment. Before the helium tank was welded onto the cavity a quality factor of about $2\cdot10^{10}$ in the $\pi$ mode at $2\,\kelvin$ was measured in a vertical test which corresponds to a residual resistance of $\approx\,1.2\,\nano\ohm$ if one assumes that \rbcs did not change between vertical and horizontal test (fitting parameters for \rbcs in horizontal test are listed in Section \ref{sec:rres}). In all tests presented in this paper, the cavity was installed horizontally in the HoBiCaT facility \cite{Kugeler2010}, similar to an installation in a cryomodule.

In a previous study \cite{Vogt2013a}, we suggested that thermoelectrically induced currents and their associated magnetic flux is responsible for the change of \rs upon thermal cycling. It is dependent on the temperature difference along the cavity helium tank system during cooldown. In the horizontal setup, the system is fabricated of two materials: Niobium (cavity) and titanium (helium tank). If a temperature gradient is applied along the system, a voltage arises which drives a current along the cavity and back through the tank:
\begin{eqnarray}
V &=& (S\sub{Nb} - S\sub{Ti}) \cdot (T\sub{1} - T\sub{2}) \label{eq:seb} \\
&=& I \cdot R\sub{system, DC}
\end{eqnarray}
where $S$ is the thermopower or Seebeck coefficient of the respective material, $T\sub{1,2}$ are the temperatures of the two joints, $I$ is the thermocurrent and $R\sub{system, DC}$ is the DC resistance of the system which will be dominated by the resistivity of the helium tank, especially once part of the cavity goes superconducting.

The magnetic flux associated with the current can then be trapped in the niobium when it becomes superconducting and hence produces additional RF losses \cite{Vallet1992,Aull2012,Gurevich2013}.

A number of questions were left open in our previous paper. For one, the thermopower for niobium and titanium were poorly known so reliable estimates of the thermocurrents near \tc were not possible. Furthermore, it was not clear how magnetic flux at the RF surface can be generated by a cavity tank system whose geometry is cylindrically symmetric. Finally, a systematic study of the surface resistance versus (controlled) temperature difference during the superconducting transition was still lacking.

Here we present a far more systematic and in-depth study that addresses these questions. In the first part (Section \ref{Chap:Cav}) we discuss more extensive measurements of the quality factor of a TESLA cavity for many temperature differences. Heaters were installed to generate temperature differences between the cavity ends that are typically encountered during cooldowns in HoBiCaT. These measurements indeed confirmed a direct correlation between the residual resistance and the temperature difference (see Section \ref{sec:correl}). We also demonstrated that a reduced temperature difference ($10\--20\,\kelvin$) between the cavity ends during the very first cooldown of the cavity to the sc state can produce high quality factors without an additional thermal cycle being required. These results were confirmed for several modes of the passband and suggest that Q factors up to $10^{11}$ are possible in our test cryostat, especially for the center cells where the external magnetic shield is most effective.

In the second part (Section \ref{sec:sims}) we discuss a quantitative analysis of the system to determine if indeed the thermoelectric effect can explain the observed cavity behavior. Here, we turned to thermoelectric simulations of the cavity with helium tank. For these, accurate values are required for the temperature dependent thermoelectric properties of both niobium and titanium. Measurements presented in Section \ref{sec:thermopower} now provide more reliable values of the thermopower that were used in simulations.

The simulations allow one to calculate the thermoelectric currents in the cavity tank system. Mechanically, the system exhibits toroidal symmetry (ignoring misalignment issues). Assuming perfect rotational symmetry one would then expect a magnetic field between cavity and tank in azimuthal direction. Its magnitude is zero at the inner cavity RF surface, rising inside the wall, having a maximum at the outer surface of the cavity and falling back to zero on the outer surface of the tank. Hence it should not influence the RF properties. However, what counts is the symmetry of the current flowing in the system which in fact can be broken due to the temperature dependence of the material properties, as discussed in Section \ref{sec:sym}. Due to the broken symmetry, magnetic flux actually exists at the RF surface and within the cavity volume that potentially can be trapped. Our simulations, whose results are presented in Section \ref{sec:simres}, confirm that the expected thermocurrent distribution is consistent with the observed variations in cavity quality factor.

\subsection{Trapping of ambient magnetic field} \label{sec:trapp}

While we focus on the impact of thermocurrents and explain why it is a plausible explanation for the observed variation of \rs in our experiments, there is an additional effect that was intensively investigated during the last months: The change of the efficacy of the Meissner effect depending on the cooling rate and temperature distribution for slow, homogeneous cooling \cite{Aull2012,Romanenko2014c,Romanenko2014b,Gonnella2014,Gonnella2015}. This effect can therefore also play a role in determining the residual resistance in cavities.

Since our instrumentation is limited to sensors outside the helium tank we cannot make any statement on the exact temperature \textit{distribution} of the cavity. However, we believe that a change in flux trapping efficiency of the residual field has only a minor impact on our particular results. Our cavity was shielded from the earth's magnetic field by a double (warm/cold) shield combination, and the ambient field at the cavity position, as measured at room temperature, is low (typically a fair amount less than $0.5\,\micro\tesla$ at the center cells, about $1\,\micro\tesla$ maximum in the end cells \cite{Kugeler2010}). The inner cold shield was manufactured from Cryoperm, so that the shielding efficacy should theoretically be even better during cold operation.  We measured the permeability of the material \cite{Kugeler2011} and actually found little temperature dependence, so that the warm results should in fact be a reasonable estimate of the residual field during cavity operation.  This is borne out by the best residual surface resistances of only $1.4\,\nano\ohm$ observed in the center cells of the cavity when the temperature difference during cooldown is small and the thermoelectric effect should play no role.

Furthermore we stress that the cavity we tested was not nitrogen doped. While doped as well as undoped cavities trap a higher percentage of ambient field when the cooling rate is decreased below a certain value, the same amount of trapped flux causes considerably less RF losses in an undoped cavity \cite{Gonnella2014}. Hence the difference between trapping all of the ambient flux in the center cells (at most $0.5\,\micro\tesla$) and not trapping any of the flux would lead to a decrease of at most $1.75\,\nano\ohm$ in \rs for the $1/9\,\pi$ mode using the approximation of $3.5\,\nano\ohm / \,\micro\tesla$ \cite{Aune2000}. This is considerably less than the observed change of $9.2\,\nano\ohm$ between thermal cycles. For the $\pi$ mode the contribution may be larger and this is one of the reasons we measured several modes of the passband.  We show later (e.g., Figure \ref{fig:ModeCompare}) that indeed we observe no significant impact of a changing Meissner efficacy on our measurement results.

\begin{figure}
\includegraphics[width=\linewidth]{setup3.eps}
\caption{Setup of TESLA cavity inside HoBiCaT cryostat with positions of Cernox temperature sensors 1 -- 6 and heaters.\label{fig:setup}}
\end{figure}

\section{Thermal cycling experiments\label{Chap:Cav}}

\subsection{Experimental setup and cooling procedures}

A fully equipped TESLA-type cavity welded into a titanium tank and with a TTF-3 input coupler installed was mounted horizontally inside the HoBiCaT cryostat. The cavity was equipped with Cernox sensors on the helium vessel head and beam pipes near the Nb-Ti joints, as well as helium inlet and outlet. Furthermore two heaters were attached, one on each beam pipe. The setup including the helium supply is sketched in Figure \ref{fig:setup}.

HoBiCaT can cool the cavity with different schemes. The cryoplant fills the helium tank via two valves: The filling valve (FV) at the bottom left and the Joule-Thomson valve (JTV) which fills the tank via the 2-phase-pipe from the top right. We used three different cooling schemes, the initial cooldown, the thermal cycle and the parked cooldown, which will be explained in the next sections.

During the \textbf{initial cooldown} (whose temperature profile is shown in Figure \ref{fig:Ts}a), mainly the FV at the bottom of the cavity was used for filling the helium tank with $4.2\,\kelvin$ helium. The coldest spot was always the one closest to the FV. This lead to high spatial temperature gradients along the cavity \textit{and} from bottom to top. The JTV was $25\,\%$ open as well to cool down the heat exchanger but no liquid helium reached the cavity.

\begin{figure}
\includegraphics[width=\linewidth]{Temperatures.eps}
\caption{Temperatures measured by Cx 1 -- Cx 5 during initial cooldown (a), a thermal cycle (b) and parked cooldown (c). The dotted line indicates the transition temperature of $9.2\,\kelvin$. The dashed line indicates the time when the first sensor dropped below transition temperature and a $5\,\minute$ interval before that transition. \label{fig:Ts}}
\end{figure}

For a subsequent \textbf{thermal cycle} (as in Figure \ref{fig:Ts}b) the FV was closed and remained so during the whole process. As the first step the JT valve was closed and the heaters on both ends were used to evaporate the helium in the tank and create a temperature difference between the cavity ends if desired. The targeted difference could be adjusted by varying the heater power. Values chosen were typical of those encountered during normal cooldowns. Cernox sensor 5 (purple) measured a jump in temperature once the tank was empty and the cavity was normal conducting. In addition, we monitored the temperature of the evaporated gas with Cernox sensor 6 (black) which exhibited a characteristic rise in temperature (sensor is not displayed in Figure \ref{fig:Ts} for clearness). Now, the JT valve was opened again to start gas flow through the system and to restart cooling while the FV remained closed.

To pinpoint the time of the phase transition we measured the bandwidth of the $\pi$ mode during one complete warm-up/cooldown cycle.  This allowed us to correlate the sc transition with the temperatures shown by Cernox sensors. The first sensor at the ends read close to $9.2\,\kelvin$ within less than one minute of the time when the first cells of the cavity went superconducting.

Note that cycling scheme may lead to a reversed spatial temperature difference compared to the initial cooldown if not balanced by the heaters because the cold gas and liquid started to pour from the top right. As soon as the first liquid entered the tank a drop in temperature was registered by Cernox sensor 5 as can be seen in Figure \ref{fig:Ts}.

During one test series we performed a \textbf{``parked cooldown''} from room temperature which combines properties of both the initial cooldown and the thermal cycle (Figure \ref{fig:Ts}c). The cooling procedure of the initial cooldown was adapted to stop well before the sc phase transition. The cryoplant was balanced to maintain a constant temperature at the FV for $48\,\hour$. The set point was first set to $30\,\kelvin$ and then continuously lowered to $14\,\kelvin$ during this period. Both valves, the FV and the JTV, were used. After all temperature sensors were clearly in equilibrium (no temperature change with time) the set point was further lowered towards $1.8\,\kelvin$ and the cavity tank system transitioned with a small $\Delta\,T\,<\,10\,\kelvin$ into the sc state.

Altogether, we performed one initial cooldown from room temperature, followed by 11 cycles, one parked cooldown from room temperature and another two cycles. After each cycle we performed a measurement of quality factor $Q$ versus temperature at a ($\pi$ mode) gradient of $4\,\mega\volt/\meter$ to extract the residual resistance for all three modes.

The coupling during the RF cold tests was in most cases close to critical and always between $0.3$ and $3.5$ which results in a low error margin below $10\,\%$ \cite{Powers2005}. A three stub tuner was installed in front of the input antenna and used to ensure optimal coupling for each mode.

\begin{figure}
\includegraphics[width=1.0\linewidth]{beadpull_vs_measTESLA4_color.eps}
\caption{Field distributions of three passband modes of a TESLA cavity: $\pi$ mode, $1/9\,\pi$ mode and $8/9\,\pi$ mode computed with CST and measured with a bead pull setup using an Al sphere.\label{fig:Fields}}
\end{figure}

\subsection{Cavity modes}
For the evaluation of the residual resistance in the various cells of the cavity we measured the quality factor in three different passband modes: The $\pi$ mode ($1299\,\mega\hertz$), the $8/9\,\pi$ mode (1298\,\mega\hertz) and the $1/9\,\pi$ mode (1274\,\mega\hertz). Each measurement is therefore an average over the cells which are exposed to the RF field in the respective mode (${R\sub{s}}\,=\,G/Q\sub{0}$). The geometry factor is taken from CST simulations:
\begin{eqnarray*}
 G_{\pi}&=&271.2\,\ohm\\
 G_{8\pi/9}&=&271.5\,\ohm\\
G_{\pi/9}&=&268.3\,\ohm.
\end{eqnarray*}
The $\pi$ mode exhibits a homogeneous field distribution along the cavity. The $1/9\,\pi$ mode has the maximum field in the center cells and low field in both end cells whereas the $8/9\,\pi$ mode has minimum field in the mid cell and maximum field in the end cells.

Figure \ref{fig:Fields} shows a calculation of the field distribution in the cavity computed with CST microwave studio. The plot is normalized to the same stored energy. We validated the computed field distribution with a bead pull measurement after all RF measurements were completed (Figure \ref{fig:Fields}).

The bead pull measurement confirmed the overall field distribution of the three passband modes. Hence, they were used to evaluate contributions to the surface resistance depending on the location. We were able to distinguish between end cell region (dominant in $8/9\,\pi$) and mid cell region (dominant in $1/9\,\pi$) but we could not determine the cell number because all modes are mirror symmetric with respect to the cavity center plane in the axial direction.

The RF measurements confirmed that \rbcs did not vary between the three modes (see Section \ref{sec:rres}). In contrast, the measured \rres was influenced by localized contributions like the spatial variation of HoBiCaTs ambient magnetic flux due to inhomogeneous shielding at the end cells and maybe material defects. A higher surface resistance in the end cells lead to a lower $Q$ in the $8/9\,\pi$ and $\pi$ mode than in the $1/9\,\pi$ mode.

\begin{figure}
\includegraphics[width=\linewidth]{BCS_example.eps}
\caption{Arrhenius plots of cycle 8 (highest $\Delta T$) and the parked cooldown (lowest $\Delta T$). The thick lines indicate the BCS fits with $A\,=\,31.8\,\micro\ohm$ and $B\,=\,15.7\,\kelvin$ for the stated $R\sub{res}$. The thin lines indicate the range of fits that are covered by the uncertainty of $A$ and $B$. \rs was calculated using $R\sub{s}\,=\,G/Q\sub{0}$ and $G_{\pi}\,=\,271.2\,\ohm$, $G_{8\pi/9}\,=\,271.5\,\ohm$, $G_{\pi/9}\,=\,268.3\ohm$.
\label{fig:BCS_example}}
\end{figure}

\subsection{Extraction of residual resistance} \label{sec:rres}
The temperature dependent BCS contribution to the surface resistance \rbcs was separated from the temperature independent residual contribution \rres using \cite{Ciovati2014}:
\begin{eqnarray}
R\sub{s}(T) &=& A \cdot \exp\left(\frac{-B}{T}\right) + R\sub{res}\\
\mbox{with}\quad B &=& \frac{U}{k\sub{B}}
\end{eqnarray}
For all measured modes, the fit parameters $A$ and $B$ overlapped within the error margin confirming that \rbcs remained constant for different cycles. Hence we determined overall BCS-parameters of $A\,=\,(31.8\,\pm\,2.2)\,\micro\ohm$ and $B\,=\,(15.7\,\pm\,0.2)\,\kelvin$ corresponding to $U\,=\,(1.70\pm0.02)\,k\sub{B} \tc\,=\,(1.35\pm0.02)\,\milli\electronvolt$. With these parameters we extracted the residual resistance from all measurements. Figure \ref{fig:BCS_example} shows two examples for BCS fits and \rres extraction: Cycle 8 (highest temperature difference during phase transition) and the parked cooldown (lowest temperature difference during phase transition). The error bars of the RF measurement are within the symbol size.

\subsection{Correlation of \rres with cooling parameters} \label{sec:correl}

Table \ref{tab:Qs} provides examples of the measured residual resistances. We observed that cooldowns with a low temperature difference along the cavity result in lowest residual resistances. For the $1/9\,\pi$ mode (field dominant in center cells) this amounted to a residual resistance of $1.4\,\nano\ohm$. Presumably the quality factor then is dominated by the ambient field in HoBiCaT which is about $0.2 - 0.5\,\micro\tesla$ along the center cells causing a residual resistance of $0.7 - 1.75\,\nano\ohm$ \cite{Aune2000}. Other, non-flux contributions may also add to the residual resistance.

\begin{figure}
\includegraphics[width=\linewidth]{RsVT.eps}
\caption{Residual resistance measured with the three passband modes as a function of the temperature difference along the cavity at the onset of sc phase transition. The lowest residual resistance was achieved in the center cells following the cycle 1 and the parked cooldown ($1.4\,\nano\ohm\,\widehat{=}\,Q\sub{0}> 10^{11}$ at $1.5\,\kelvin$).\label{fig:RvsDT}}
\end{figure}

\begin{table}
\setlength{\tabcolsep}{2.8mm}
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabularx}{\columnwidth}{m{2.5cm}ccc}
\hline
$R\sub{BCS}\,\approx\,0.9\,\nano\ohm$ & \multicolumn{3}{c}{$R\sub{res}$}\\
at $1.5\,\kelvin$ & $\pi$ mode & $8/9\,\pi$ mode & $1/9\,\pi$ mode\\
\hline
Initial cooldown & 13.6 & 14.2 & 10.6\\
\hline
Thermal cycle 8 (high $\Delta T$) &  13.3 & 18.6 & 8.0 \\
\hline
Parked cooldown & 6.4 & 9.9 & 1.4 \\
\hline
Thermal cycle 7 (low $\Delta T$)& 6.5 & 9.9 & 1.9\\
\hline
\end{tabularx}
\end{center}
\caption{Examples of measured residual resistances of the three passband modes.}
\label{tab:Qs}
\end{table}

Figure \ref{fig:RvsDT} displays the extracted residual resistances for all cooldowns and cycles as a function of temperature difference $\Delta T$. This is defined as:
\begin{eqnarray}
\Delta T\,=\,\left|\frac{T\sub{Cx1}+T\sub{Cx2}}{2}\,-\,\frac{T\sub{Cx3}+T\sub{Cx4}}{2}\right|
\end{eqnarray}
The temperature difference between the temperature sensors on the two ends of the tank is relevant for the thermocontact voltages. This parameter is a measure for the induced thermoelectric voltage and calculated at the instance when the first of the four sensors drops below the transition temperature.

Figure \ref{fig:RvsDT} shows that the residual resistance of the cavity decreases with $\Delta T$. The curves for the three passband modes run parallel indicating that all nine cells are similarly affected. Thus the change in \rres is a global effect consistent with the thermocurrent model.

Furthermore, we see that the $1/9\,\pi$ mode has a significantly reduced surface resistance compared to the two other modes while the residual resistance of the $8/9\,\pi$ mode is elevated.

We believe this is due to an increased ambient field in HoBiCaT near the end cells. Measurements of the magnetic shielding in HoBiCaT yielded that the magnetic field inside the end parts of the shield is increased due to cut outs for coupler and geometry effects (up to about $1\,\micro\tesla$) \cite{Kugeler2010}.

Note that we were limited to only four temperature sensors outside the tank which only provides an incomplete picture of the temperature distribution. Two sensors were near heaters which results in an additional offset in $\Delta T$. For more reliable values more sensors along the setup are needed.

\begin{figure}
\includegraphics[width=\linewidth]{ModeCompare.eps}
\caption{\rres obtained for the $8/9\,\pi$ and $1/9\,\pi$ mode as a function of \rres obtained for the $\pi$ mode. Linear regression leaving out the initial cooldown results in: \\$R_{8/9\,\pi}\,=\,1.2\cdot R_{\pi} +1.8\,\nano\ohm$ and\\$R_{1/9\,\pi}\,=\,1.0\cdot R_{\pi} -4.7\,\nano\ohm$.\label{fig:ModeCompare}}
\end{figure}

Figure \ref{fig:ModeCompare} compares the average surface resistance of the modes to that of the $\pi$ mode. A linear dependency is visible for the cycles and the parked cooldown. The initial cooldown for the $8/9\,\pi$ does not fit into the linear curve which is already visible in Figure \ref{fig:RvsDT}. This might be explained by the fact that the cooling dynamics of initial cooldown are different from the other coolings. The effect would be largest for the cells in the higher ambient magnetic field region, consistent with the observation that the $8/9\,\pi$ mode differs significantly.

Leaving out the data point of the initial cooldown, the two graphs in Figure \ref{fig:ModeCompare} exhibit a linear behavior. The slopes are $1.2$ ($8/9\,\pi$ mode) and $1.0$ ($1/9\,\pi$ mode) which is close to $1$ meaning that all cells are affected by the thermocurrent in a similar way as would be expected since the current is constant along the length of the cavity. The current and resulting trapped magnetic flux adds a constant residual resistance to each cell.

The results in Figure \ref{fig:RvsDT} and Figure \ref{fig:ModeCompare} also demonstrate that the efficacy of the Meissner effect in expelling the constant ambient field plays no major role in our observed changes in $R\sub{s}$.   Otherwise the $8/9\,\pi$ mode should change its residual resistance more than the $1/9\,\pi$ mode.  Then the three curves in Figure \ref{fig:RvsDT} would not run parallel and the slopes of the curves in Figure \ref{fig:ModeCompare} would not be linear with a slope near $1$. The exception is the $8/9\,\pi$ mode result after the initial cooldown ($\Delta R\sub{s}\,\approx\,5\,\nano\ohm$) which may be due to the fact that the cooling conditions during initial cooldown significantly differ from the conditions during the thermal cycles and the parked cooldown resulting in different Meissner efficacies. This would most affect the $Q$ value of the $8/9\,\pi$ mode since the measurement averages over the high-ambient-field end cells.

The intercepts in Figure \ref{fig:ModeCompare} are $+1.8\,\nano\ohm$ ($8/9\,\pi$ mode) and $-4.7\,\nano\ohm$ ($1/9\,\pi$ mode). They are a measure of the different residual resistance that is not caused by thermoelectrically induced magnetic flux but is always present. The values indicate that the end cells of the cavity have higher average surface resistance. Sources can be either the increased ambient magnetic field near the end cells in HoBiCaT as discussed above or particulate contamination (e.g. from the nearby input coupler).

The $1/9\,\pi$ mode which has the least field in the end cells exhibits a residual resistance down to $1.4\,\nano\ohm$ in the best case. The $\pi$ mode with equal field distribution in each cell shows an intermediate \rres of $6.4\,\nano\ohm$ while the $8/9\,\pi$ mode with maximum field in the end cells has the highest \rres of $9.9\,\nano\ohm$. If the low level of magnetic field in the center of the shielding would extend to the end cells, the low surface resistance, corresponding to a quality factor of more than $10^{11}$ at $1.5\,\kelvin$, might be achieved in the $\pi$ mode as well.

In conclusion we can say that we observed a significant contribution to the residual resistance that increases with the temperature difference between the ends of the cavity tank system that is installed in a horizontal test stand. Given a high-Q cavity with an overall surface resistance of order $10\,\nano\ohm$ at $2\,\kelvin$ this effect could dominate the losses.

\section{Simulation of thermoelectric induced magnetic flux\label{sec:sims}}

\subsection{Setup for simulations: Symmetry and symmetry breaking \label{sec:sym}}
\begin{figure}
\includegraphics[width=\linewidth]{COMSOLsetup2.eps}
\caption{Simplified geometry for COMSOL simulations. Dimensions are in millimeters.\label{fig:COMSOLsetup}}
\end{figure}
To investigate the experimental results further and to make a quantitative analysis of the thermocurrents we turned to numerical simulations of the system. In particular it is important to understand the distribution of the currents and the magnetic flux. In our previous paper, we made a rough estimate of the induced current and the corresponding magnetic flux. Now, we want to use temperature dependent material properties to yield a more reliable number on the amount of flux which is to be expected on the RF surface. For that purpose we conducted COMSOL simulations which will we discuss in the next section.

We modeled a simplified geometry consisting of two coaxial cylinders (Figure \ref{fig:COMSOLsetup}). The inner cylinder which resembled the cavity had the typical cavity wall thickness of $2.8\,\milli\meter$ and was defined as niobium. The outer cylinder and the two end plates resembled the helium vessel and helium vessel heads and were defined as titanium with a thickness of $5\,\milli\meter$ (vessel) or $10\,\milli\meter$ (vessel heads) respectively. The $1\,\meter$ length of both cylinders was the approximate length of a TESLA cavity in a tank. The model exhibited perfect cylindrical symmetry.

The simulations were performed in several steps. For each step different temperature boundary conditions were applied. We defined ten areas on the outer surface of the tank (Figure \ref{fig:boundaries}). Each area could either be assigned a fixed temperature for the simulation or left undefined as the rest of the setup. Thereby we created temperature distribution similar to our experimental situation.

In the first step, we defined the temperatures only on the end plates, areas 1 ($10\,\kelvin$) and 6 ($100\,\kelvin$). In this case the isothermals were perpendicular to the cylinder axis. The resulting current distribution was cylindrically symmetric and the azimuthal magnetic flux on the RF surface was zero though it increased rapidly as one moves into the wall. This situation is as expected from Gauss' Law.

In the next steps, the areas 2 -- 5 and 7 -- 10 were assigned fixed temperatures as well, leading to asymmetric temperature distributions. The isothermals were not orthogonal to the cylinder axis anymore but distorted. Thereby the electrical symmetry was broken due to the temperature dependence of the electrical resistance. This resembles the cavity cooldown when liquid helium started to collect first at the bottom of the tank.

Simulations where the inner cylinder is replaced by the actual cavity geometry are not presented in this paper but are in preparation.

\begin{figure}
\includegraphics[width=.8\linewidth]{Skizze_Sims.eps}
\caption{Different areas in which thermal boundaries could be applied in COMSOL simulations. No boundary condition was applied to the rest of the setup.\label{fig:boundaries}}
\end{figure}

\subsection{Input data} \label{sec:thermopower}
For these simulations the temperature dependent material properties are needed. We used literature data as given in references \cite{Padamsee2009,Jensen1980,Merio2011} for the temperature dependent electrical conductivity. The thermal conductivity was set constant to improve the convergence of the COMSOL simulations. We choose $\kappa\sub{Nb}\,=\,50\frac{\watt}{\meter \cdot \kelvin}$ and $\kappa\sub{Ti}\,=\,22\frac{\watt}{\meter \cdot \kelvin}$ as approximations \cite{Kes1974,Jensen1980,Merio2011}.

On the thermopower only poor information is available. Hence we performed sample measurements of the thermopower on niobium ($RRR\,=\,300$) and titanium (grade 2). The dimension of the samples were approximately $2\,\milli\meter \times 5\,\milli\meter \times 30\,\milli\meter$ and $2\,\milli\meter \times 2.3\,\milli\meter \times 30\,\milli\meter$.

The experiments were performed with a physical property measurement system by Quantum Design \footnote{\url{www.qdusa.com/sitedocs/productBrochures/1070-002.pdf}} using the thermal transport option \footnote{\url{http://www.qdusa.com/sitedocs/productBrochures/tto_rev7-06.pdf}}.

The system allowed high precision measurements of the thermopower over a wide temperature range. We measured all four samples (two niobium, two titanium) in the range from $4.5\,\kelvin$ to $300\,\kelvin$ several times.

\begin{figure}
\includegraphics[width=.95\linewidth]{Ti_Thermopower.eps}
\includegraphics[width=.95\linewidth]{Nb_therm.eps}
\caption{PPMS thermopower data of niobium and titanium for the small ($2\,\milli\meter \times 2.3\,\milli\meter \times 30\,\milli\meter$) and the large sample ($2\,\milli\meter \times 5\,\milli\meter \times 30\,\milli\meter$). \label{fig:S}}
\end{figure}

Figure \ref{fig:S} shows the measured thermopower $S\,[\micro\volt/\kelvin]$ as a function of temperature. The data is in agreement with literature data \cite{Blatt1976,Nystrom1959,Weinberg1966}. For the titanium data, the small and the large sample were the same. The curve was fitted as shown in Figure \ref{fig:S}. A table was extracted and entered into COMSOL.

\begin{figure*}
\includegraphics[width=\linewidth]{Ergebnisse_BundT_2.eps}
\caption{Results of the COMSOL simulations. The left cylinders display the magnetic field distribution inside the cylinders at $r\,=\,102\,\milli\meter$. The legend is given on the left reaching from $\approx 50\,\nano\tesla$ (green) to $\approx 13\,\micro\tesla$ (red). The right figures display a cut through the temperature distribution inside the cylinders. The legend for the temperature reaches from $10\,\kelvin$ to $100\,\kelvin$.  \label{fig:SimResB}}
\end{figure*}

For niobium, the two samples did not give the same result. In general, both curves are similar and have the same zero-crossing but the amplitude was higher for the large sample compared to the small sample. The COMSOL simulations were first performed with the fitted data for the small sample as shown in Figure \ref{fig:S} to obtain a conservative lower limit. Note that the disagreement in the measured thermopower in the temperature range up to $75\,\kelvin$ is only in the $30\,\%$ range. The simulations were also repeated with the less conservative Seebeck values for a sensitivity check, but we found that the impact was not dramatic (see later Figure \ref{fig:Bsim}). Nevertheless we intend to repeat the measurements in the future to provide more accurate results.

\subsection{Magnetic flux at the RF surface} \label{sec:simres}

We performed five simulations:

\begin{enumerate}
	\item A symmetric test distribution as a consistency check for the simulations. Surface 1 was set to $10\,\kelvin$ and surface 6 to $100\,\kelvin$.
	\item Asymmetric studies: Surfaces 1 and 10 at $10\,\kelvin$ and surfaces 5 and 6 at $100\,\kelvin$
	\item Surfaces 1, 10 and 9 at $10\,\kelvin$ and surfaces 4, 5 and 6 at $100\,\kelvin$
	\item Surfaces 1, 10, 9 and 8 st $10\,\kelvin$ and surfaces 3, 4, 5 and 6 at $100\,\kelvin$
	\item Surfaces 1, 10, 9, 8 and 7 at $10\,\kelvin$ and surfaces 2, 3, 4, 5 and 6 at $100\,\kelvin$
\end{enumerate}

The simulations were performed as two steps. First, the temperature distribution and the resulting electric current distribution was computed. In the second study, this input data was used to calculate the magnetic field distribution.

The RF surface is at a radius of $r\,=\,103.3\,\milli\meter$. Figure \ref{fig:SimResB} shows the solution for the magnetic field \textit{inside} the cylinders close the RF surface at $r\,=\,102\,\milli\meter$ to ensure that it is inside the cylinder and not in the cylinder wall due to meshing. The arrows display the direction of the magnetic field at the same radius. The arrow size is proportional to the field strength. Generally the field is near perpendicular to the surface in high field regions.

In addition to the magnetic field distribution, Figure \ref{fig:SimResB} displays cuts with the temperature distributions inside the cylinders.

Figure \ref{fig:SimBCut} displays the magnetic field at a cut through the walls of the two cylinders at $z\,=\,0.5\,\meter$. For the symmetric case, no field is inside the inner cylinder, the field increases in the inner cylinder wall and decreases towards the outer cylinder. Outside of the two cylinders there is no magnetic field anymore. This is just as expected from Gauss' Law. For studies 2 to 5, the increasing asymmetry of the current density leads to an increased magnetic field inside the cylinders. This field can get trapped during superconducting phase transition and produce RF losses as we have observed in the experiment.

\begin{figure}
\includegraphics[width=\linewidth]{Bcut.eps}
\caption{Absolute magnetic field along a cut trough the cylinders at $z\,=\,0.5\,\meter$. \label{fig:SimBCut}}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{Balongz.eps}
\includegraphics[width=\linewidth]{Talongz.eps}
\caption{Top Figure: Absolute magnetic field along $z$ at $x\,=\,102\milli\meter$ and $y\,=\,0\,\milli\meter$. The peaks at $z\,=\,0\,\milli\meter$ and $z\,=\,1\,\meter$ are due to the sharp corners when the cylinder passes into the end plates. Bottom Figure: Temperature difference between top and bottom of the inner cylinder.  \label{fig:SimZ}}
\end{figure}

Figure \ref{fig:SimZ} displays a cut of the RF surface along the z-axis. The position of the cut was chosen to be at $x\,=\,102\,\milli\meter$, $y\,=\,0\,\milli\meter$. Furthermore the Figure displays the temperature difference between top and bottom of the inner cylinder. Where the temperature difference is maximum, the inhomogeneity in the electric current distribution is largest and hence the most magnetic field penetrates the RF surface.

The results clearly illustrate that a non symmetric temperature distribution leads to a nonzero magnetic field on the RF surface. The highest fields are obtained where the temperature difference between top and bottom part are largest. In study 5, this effect leads to a more or less constant magnetic flux along the cylinder while in study 3 where the temperature gradient from bottom to top is less distinct the magnetic field increases towards the center of the cylinders ($z\,=\,0.5\,\meter$). In the experiment, the highest degree of broken symmetry is obtained when the first superconducting path through the cavity establishes below $9.2\,\kelvin$. This corresponds to an extreme example of case 5 where the DC resistance of the bottom part of the inner cylinder drops to zero while the rest of the cavity is still normal conducting and hence the thermocurrent is forced into the sc path. In this case all cells would be affected by the additional induced flux in the same way which is in agreement with the linear increase in Figure \ref{fig:ModeCompare} with slope $\approx\,1$. Temperature mapping data of a dressed multicell cavity in a horizontal test would give more insight in the real temperature distribution.

Finally, we varied the temperature difference in the simulation and set the warmer boundaries of study 5 to the temperatures $80\,\kelvin$, $60\,\kelvin$, $40\,\kelvin$ and $20\,\kelvin$ while the other surfaces remained fixed at $10\,\kelvin$. The result is given in Figure \ref{fig:Bsim} together with the experimental findings. Since the measurements of the thermopower of niobium allowed for $30\,\%$ higher values, we added bars indicating the magnetic field at the RF surface for a $30\,\%$ higher thermopower of niobium. The results show that the magnetic field increases with the temperature difference. While we do not claim that the quantitative values should be directly compared, the trend is very similar.

In conclusion, we assess that the COMSOL simulations support the assumption that thermoelectrically induced magnetic field penetrates the RF surface due to an inhomogeneous temperature distribution. We see that even for a moderate asymmetry the flux is locally of order several $\micro\tesla$. The RF losses resulting from the increased magnetic field will depend on the fraction of the nine cell cavity that is affected by the thermocurrent due to more pronounced asymmetry, the exact location of the induced magnetic field relative to the RF field and the amount of flux that eventually gets trapped in the material. However the variation of about $10\,\nano\ohm$ seen in the cavity tests are consistent with the thermocurrent hypothesis.

\begin{figure}
\includegraphics[width=\linewidth]{Bsim.eps}
\caption{Magnetic field at the RF surface simulated for different temperature differences in combination with the data of Figure \ref{fig:RvsDT}. The red bars indicate the magnetic field that is obtained assuming a $30\%$ higher thermopower, corresponding roughly to the uncertainty resulting from the discrepancy depicted in Figure \ref{fig:S}. \label{fig:Bsim}}
\end{figure}

\section{Summary and Outlook\label{Chap:Sum}}

We demonstrated that poorly controlled cooling conditions can significantly deteriorate the quality factor of SRF cavities due to the generation of thermocurrents. While already a simple misalignment of the cavity to the helium tank could break the symmetry of the cavity tank system, although the effect on \rs should be small \cite{Crawford2013} even a mechanically cylindrically symmetric system will generate magnetic flux on the RF surface. The presented simulations explain how the electrical symmetry is broken due to the temperature dependance of the electrical conductivity.

We also demonstrated that an initial cooldown through \tc with small $\Delta\,T$ can in principle yield residual resistances at $1\,\nano\ohm$ (parked cooldown), \textit{provided the external magnetic shielding is very effective in eliminating external residual flux}.

What we did not investigate in this paper is the influence of changed cooling conditions on the efficacy of the Meissner effect as we explained in section \ref{sec:trapp} since our magnetic shield eliminated most of the residual field.

Temperature mapping during dressed cavity horizontal test could improve the understanding on the real temperature distribution during cooldown and cycles. Furthermore it would help with characterization of cooling parameters like cooling speed, temperature gradients in all spatial directions and homogeneity.

\begin{acknowledgments}
We would like to thank Dr.~Axel Neumann and Dr.~Adolfo Velez for providing the simulations of the field distribution of the TESLA cavity and the accompanying bead pull measurement. We thank Michael Schuster, Andre Frahm, Sascha Klauke, Dirk Pfl{\"u}ckhahn, Stefan Rotterdam and Axel Hellwig for experimental support. Furthermore we thank the group of Dr. Klaus Kiefer for the support with the thermal transport measurements, Bernd Spaniol (Hereaus) for providing the niobium and our machine shop for support especially with last minute modification of the samples.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv