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\section{Directions to explore}  \subsection{ Stochastic resonance}   Here is an article on quantum stochastic resonance \cite{L_fstedt_1994}  \subsection{ Measurements of energy barriers}.  \subsection{ Vortex manipulations}   %\cite{Roditchev_2015}  %\cite{Doderer_1995}  %\cite{Fazio_2001}  %\cite{Naor_1982}  %\cite{Zimanyi_1995}  %\cite{Sachdev}  %\cite{Sachdev_2009}  %\cite{Andreev_1996}  %\cite{Sknepnek_2004}  %\cite{Kabanov_2004}\section{Introduction}  \section{Introduction}  \textit{General introduction.}  Single-valuedness of superconducting order parameter has profound consequences.   One of them is that multiply-connected SC structures have discreet fluxoid states.   This allows superconducting rings to support metastable dissipationless currents, which is one of the defining phenomenological properties of superconducting state of matter \cite{Leggett2006}.   Transitions between fluxoid states are realized by phase slips, when the phase winding changes by multiple of $2 \pi$.  This topological protection gives fluxoid states robustness to small perturbations, and makes them attractive as principal degrees of freedom for superconducting devices, memory etc.  For this reasons superconducting structures with multiply-connected geometry became progenitors of many superconducting devices of great practical importance such as SQUIDs, superconducting qubits etc.  Fluxoid states in more complicated multiply-connected devices might find future applications in electronics (flux-tronics).   In superconducting films, all said above translates from fluxoid states to vortices, which are topological defects of superconducting order parameter in continuous case, and, thus, are of fundamental interest.   Vortex core, being a singularity, hosts quasiparticles, and if combined with topological insulator might host Majorana mode.   Finally, in some cases, in superconductors, $h/2e$ periodicity is predicted to be replaced by $h/4e$ or $h/e$ (half-quantum vortices in SRO, s-wave rings with $R<\xi$, maybe some other examples??).  Testing those hypotheses experimentally requires careful observation of vortices/fluxoid states.   To sum it up, the ability to set, read and control fluxoid states in superconducting structures is one of great interest for both fundamental physics and applications.  \cite{Roditchev_2015}  \cite{Doderer_1995}  \cite{Fazio_2001}  \cite{Naor_1982}  \cite{Zimanyi_1995}  \cite{Sachdev}  \cite{Sachdev_2009}  \cite{Andreev_1996}  \cite{Sknepnek_2004}  \cite{Kabanov_2004}\section{Introduction} \textit{Overview of the progress in area and summary of the results.}   MFM technique is one of the oldest scanning probe techniques and the first one for magnetic imaging.   Concieved as a technique to characterize magnetic memory, it became popular for imaging of magnetic domain walls and vortices in superconductors.   Its main advantage is simplicity and high spatial resolution.   Shortcomings include difficult quantitative interpretation, coupling to magnetic field gradient rather than field itself, and exposing sample to magnetic field of the tip, which could possibly perturb the state of sensitive samples.  It is worth mentioning a few of active measurements, in which the magnetic field of the tip was crucial.   These include measurements of penetration depth and pinning strength for vortices by dragging them (reference is needed).  In this paper we demonstrate how MFM can be used to map and control fluxoid or vortex states of superconducting structures.  This is entirely based on magnetomechanical interaction between MFM tip and dynamics of single vortex (or phase slip in 1D superconductor).  Technique exploits inhomogeneous magnetic field created by MFM tip.   It enables to address the vortex states which are inaccessible by application of global homogenous magnetic field.  This mode of MFM has strong similarities with single electron electric force microscopy e-EFM  \cite{Woodside2002, Zhu2005, Stomp2005, Dana2005, Azuma2006, Zhu2008, Bennett2010, Cockins2010}, in which cantilever couples to the motion of single electron on/off quantum dot.  By analogy, we propose to call it $\Phi_0$-MFM.   Apart from mapping and controlling fluxoid states we show that it enables to measure phase slip rate.  It is done by taking advantage of stochastic resonance interaction between dynamics of cantilever and thermally activated phase slips.  First, we will demonstrate the principles of technique on example of superconducting ring.   We discuss model of stochastic resonance interaction of thermally activated phase slips and the motion of cantilever.  Measured phase slip rates are compared to Langer-Ambegaokar theory.   After that, we apply it to a ring with crossbar, which has richer fluxoid states and their dynamics.   It is demonstrated how $\Phi_0$-MFM allows to map and characterize distinct phase-slip processes.  \section{Method}   \paragraph{Description of the setup.}   The schematics of the setup is shown on Fig.\ref{SetupDiagram}.  MFM cantilever is made by gluing a SmCo magnetic particle onto a silicon cantilever and giving it desired shape with FIB, fig.\ref{SetupDiagram}.  We use ulta-soft silicon cantilever with spring constant about $1.8 \times 10^{-4}$ N/m ($2.5\times 10^{-4}$ N/m when measured in tester, must be checked) and resonant frequency $\sim 7675$~Hz.  We use home-build MFM operated in a frequency detection mode\cite{Albrecht1991}.   Laser interferometer is used to detect the motion of the cantilever.  Cantilever is self-oscillated at its resonant frequency.   Feedback loop with automatic gain control(AGC) is used to mountain the desired oscillation amplitude and monitor changes in damping.   The shift of the resonant frequency is measured by phase-lock-loop(PLL).  The resonant frequency shift and increase of damping allows to measure the amplitude of external forces that are in-phase and out-of-phase with the motion of cantilever, accordingly.   \[ f_{in} \approx \frac{2 \Delta f}{f_0} \cdot k x_0,\qquad  f_{out} = - \frac{2(\Delta \gamma/2 \pi)}{f_0}\cdot k x_0,\quad\text{where}\quad\gamma_0=\omega_0/2 Q\]  %f_{out} = - \frac 1 Q \frac{G_{signal}-G_{idle}}{G_{idle}}\cdot {k x_0}\]  %where $\gamma=\omega_0/2 Q$  \textcolor[rgb]{0.65,0.16,0}{(Comment: Here, there are several possibilities for units to measure signals 1) use $f/kx_0$ - dimensionless force units; 2) use $\delta f$ (Hz) and $\delta \gamma/2 \pi$ (Hz) , where $\gamma=\omega_0/2Q$. Also compare to $\gamma=\omega_0/Q$ which is also sometimes used in literature.)}  Cantilever in our setup is oriented vertically, and the magnetic moment of the SmCo paticle is normal to the surface of the sample.   The horizontal oscillations of the cantilever produce ac magnetic field on the surface of the sample with spatially varying amplitude.  The amplitude is zero right under the tip and have the opposite signs in front and behind the tip.  (Comment: Maybe, it's worth including plots of ac modulation amplitude).  \paragraph{Concentric ring pattern explanation.}  To demonstrate the idea of the technique let us consider a superconducting ring.   For each position of the ring, MFM tip will apply flux trough the ring $\Phi (r_{tip})$ which is a function of the position of the tip.  As the tip moves, the fluxoid state that minimizes the energy of the ring in field can change.   In points where two fluxoid states with the lowest energies become degenerate, small oscillations of the tip can drive the ring so that the current in the ring repeatedly switches.  This is possible when the energy barrier for phase slip is low enough to allow thermally activated PS.  The switching of supercurrent which is synchronized (at least statistically) with the motion of cantilever generates force acting on MFM tip, which is stronger than other forces that come from interaction of SC ring and MFM tip.  The averaged over oscillations of the cantilever force in general has an amplitude and phase offset, that depends on the properties of the dynamics of driven phase slips.  By measuring resonant frequency shift and change of damping we can find both in-phase and out-of-phase component of this force.  The points of scan that correspond to switching of fluxoid states form concentric circles around the center of the ring.   For a thin SC ring, for which magnetic screening can be neglected, the fluxoid transition lines correspond to points where half integer number of flux quanta is applied through the ring $\Phi_{tip}=(n+1/2)\Phi_0$, where n is integer.  This fact can be used to recover the distribution of the magnetic field on the surface created by MFM tip.   The simplicity of the procedure and the fact that it requires only a thin superconducting ring makes it potentially useful for calibrating MFM tips.   After calibration, the tips can be used as a source of local magnetic field in combination with transport or other measurements.   \paragraph{Stochastic resonance model.}  %$E_n=\epsilon_0 (h-n)^2,\quad \epsilon_0=\frac{\Phi_0^2}{16\pi^2 \lambda^2} \frac {S} {R}$  Two fluxoid states with winding numbers N=n and N=n+1 cross around $h=n+1/2$, where $h=\Phi/\Phi_0$ is flux through the ring in units of flux quanta.  For applied flux such that $|E_{n+1}(h)-E_n(h)|  The dynamics of the probability $P_n$ of the system to stay in state $n$ is governed by the relaxation rate $\nu_r=\tau_n^{-1}+\tau_{n+1}^{-1}$, where $ \tau_{n, n+1}$ -average time in state $n, n+1$ between phase slip events:   \[dP_n/dt=- \nu_r \cdot P_n +\tau_{n+1}^{-1}\]  At the point of energy level crossing $\tau_n=\tau_{n+1}$ and phase slip rate $\nu_{PS}=2 \cdot \nu_r$.  Small flux modulation due to MFM tip oscillations $h(t) =h_0+\delta h \sin(\omega t)$ produce small periodic changes in $E_n(h)$ and $E_n(h)$.   When frequency of the cantilever $\omega$ becomes comparable to the relaxation rate of fluxoid dynamics $\nu_r$ the average state of the system becomes statistically synchronized with the oscillations of the cantilever.  This can be described by a time varying part of the probability to find ring in state $n$:  \begin{align}  &P_n(t)=P_n(h_0) +\frac{ \nu_r}{\sqrt{\nu_r^2+\omega^2} } \cdot \frac{dP^{eq}}{dh}\cdot \delta h \cdot \sin(\omega t+\theta) \\  &\theta=\arctan\left(-\frac{\nu_r}{\omega}\right)\\  &P^{eq}(h)=\frac{1}{1+\exp(-(E_{n+1}(h)-E_n(h))/k_B T)}  \end{align}  where $dP^{eq}(h)$ - equilibrium probability of being in state $n$.  Depending on fluxoid state the current in the ring exerts a force on cantilever $F=\alpha I_n$, where $\alpha $ is magnetic coupling between current in the ring and MFM tip.   When $\xi< 2R$ the cubic term in flux dependance of current is weak (of the order of $\sim 0.25 (\xi/R)^2$) and we can consider the current difference between two fluxoid states to be independent on flux $I_{n+1}(h)-I_{n}(h)\approx \Delta I$.   As a result of statistically synchronized switching of supercurrent stochastic force acting on cantilever have the following time-average amplitudes of in-phase and out-of phase components:  \[f_{in} =- \frac{ \nu_r^2}{\nu_r^2+\omega^2} \cdot \frac{dP^{eq}}{dh}\cdot \alpha \cdot\Delta I\cdot \delta h\]  \[f_{out} = \frac{\omega \cdot \nu_r}{\nu_r^2+\omega^2} \cdot \frac{dP^{eq}}{dh} \cdot \alpha \cdot\Delta I\cdot \delta h \]  Let us notice that the ratio of in-phase and out-of-phase component of the force gives the relaxation rate in units of cantilever resonant angular frequency:  \[\frac{ \nu_r}{\omega}=\frac{f_{in}}{f_{out}}=\frac{2 \pi \Delta f}{\Delta \gamma}\]  this allows to measure $\nu_r$ even without knowing absolute calibration for coupling of currents to cantilever.  \section{Results}  \subsection{Ring}  \textcolor[rgb]{0.65,0.16,0}{\textcolor[rgb]{1,0.41,0.13}{Is it worth to introduce names for samples like 'Sample 1' for 3um ring and so on, to make the notations shorter afterward ??}}  All samples were made from aluminium by lift-off process.  45~nm Al layer was deposited on top of $5\,\text{nm}$ Ti layer.  Sample 1 has a shape of the ring with radius $R=1.40\,\mu m$, wall width $w=212 \pm 15\, \text{nm}$ (Fig.~\ref{3um_SEM}).   Superconducting transition temperature is $T_c=1.163$~K (Fig.~\ref{3um_Tc}).   MFM images of the frequency shift for Sample 1 at several different tip-sample separations taken at $T=1.1425$~K shown on figure \ref{3um_heights}.  Dark circles correspond to transitions between fluxoid states.   The frequency shift fades in the region of horizontal diameter of the ring, because the modulation of flux becomes very small for corresponding positions of the tip.  By accepting the furthest from the center visible transition line as one between states N=0 and N=1, we can enumerate all of them.  In order to study temperature dependance of the signal that corresponds to transition between fluxoid states with N=3 and N=4, we took a series of short line-scans ( marked by red circle on figure \ref{3um_transition}).  Small scans across the transition at several different temperatures are shown on Fig.~\ref{3um_peaks}.  The temperature dependance of the amplitudes of in-phase and out-of-phase forces was measured for the transition between fluxoid states with N=3 and N=4 (Fig.~\ref{3um_SR}).   The out-of-phase force reaches its maximum at T=1.1387~K (t=0.98), 24~mK below the transition.   The ratio of in-phase and out-of-phase signals $\Delta f/\Delta \gamma$ can be calculated with reasonable error between 1.1372~K and 1.1445~K in 7.3~mK range.   Corresponding increase of $\nu_r$ is from $0.2\omega_0$ to $255 \omega_0$ or from 8.7~kHz to 12.3~MHz, which gives dynamic range of 32~dB.  In order to model the temperature dependance of the phase slip rate we use Langer-Ambegaokar-McCumber-Halperin theory \cite{Langer1967, McCumber1970}.  It was proposed by \citet{Langer1967} that the rate of thermally activated phase slips in 1D superconducting wires is   $\Gamma (T)= \Omega (T) \exp (-\Delta F (T)/k_B T)$, where $\Delta F$ is energy barrier for phase slips.   In case of 1D wire:  \[ \Delta F_{wire}= \frac{8\sqrt 2}{3} \xi wd \cdot \frac{H_c^2 }{8\pi}\]  In case $2 \pi R\gg \xi$ this result for $\Delta F$ can be used for a 1D ring and it becomes:   \[ \Delta F= \frac{8\sqrt 2}{3} \xi wd \cdot \frac{H_c^2 }{8\pi}-2 \pi R wd \frac{H_c^2}{8 \pi} \cdot 2\frac{\xi^2}{R^2} (h-n)^2 \]  \[ \Delta F= \xi wd \cdot \frac{H_c^2 }{8\pi} \left( \frac{8\sqrt 2}{3} - \frac{\pi \xi}{R}\right),\quad\text{ for $h=1/2$}\]  For Sample~1 at $T=1.14$~K we can estimate $\xi=768$~nm, so $\xi/2\pi R=0.088$.   For the case $2 \pi R \approx \xi$, the expression for energy barrier was refined for the case of 1D ring by \citet{Zhang1997}.  It was later suggested by \citet{McCumber1970} that $\Omega = (L/\xi)(\Delta F/k_BT)^{0.5}/\tau$, where $\tau=\pi \hbar / 8 k_B (T_c-T)$.  From the suppression of superconducting transition by external magnetic field, we estimate coherence length to be $\xi(0) =108 \pm 8 \, \text{nm}$ (Fig.~\ref{3um_xi}) (here uncertainty comes mostly from the uncertainty of width, because of the rough edges).  By using known critical field of aluminium of $H_c(0)=100$~Oe we can find $\lambda(0)=216$~nm (by analogy with \cite{Zhang1997}).  Let us use $\xi=\xi(0)/\sqrt{1-t}$ and $\lambda=\lambda(0)/\sqrt{1-t}$ (Comment: here the exact functional form of $\lambda(T)$ does not matter, since the resonance occure very close to $T_c$, where higher order terms are very small).  The theory fit of both peaks shape at several temperatures and temperature dependance of peak height at $h=0.5$ ($h=3.5$??) is shown on figure~\ref{3um_SR}.  The only fitting parameters are $\xi(0)=108$~nm (measured), $\lambda(0)=214$~nm (slightly adjusted from 216~nm), geometrical dimensions of the ring and coupling of currents to cantilever.  The measured signal has amplitude about 2.5 times higher then the estimated one ( in the next paragraph).  Also damping is about 12\% lower from what we would expect in comparison to the frequency shift.  One possible explanation is that the measurement of damping relies on measurement of quality factor.   Here we used the estimate for quality factor from ringdown time 36.2~k.   At the same time, Q measured from thermal oscillations is 31.6~k.  Such discrepancy is, probably, due to interaction of cantilever with laser interferometer, which is amplitude dependent.  If we take into account this systematic error of measuring Q, it explains 12\% discrepancy.   From estimated value of $\lambda (0)=214$~nm, we can find that $\Delta I=39\,\mu A\,(1-t)$.   At resonance(t=0.98) $I=0.8\,\mu A$.  Estimate of self-induced flux. By using estimate for self-inductance of the thin-film ring as   \[L=\mu_0 R\left[\ln\left(\frac{8R}{w}\right)-1/2\right]\approx 6\,\text{pH}\]  we can estimate  \[\Phi=4.88\times 10^{-18}\text{Wb} =2.4\,m\Phi_0\]  So we can neglect by the self-induced flux.  \paragraph{Coupling estimates.}  %Oscillation amplitude during linescans was 25~\AA, which corresponds to $6.3\,m\Phi_0$ .   %It corresponds to $1.38 \cdot 2.5 \times 10^{-3} \mu m\, 1.82 \Phi_0/\mu m =6.3\times 10^{-3}$ $\Phi_0$.   In order to compare measured strength of the signals to the model we need to estimate a product  \[\frac{dP^{eq}}{dh}\cdot \alpha \cdot\Delta I\cdot \frac{\delta h}{k x_0}\]  Coupling $\alpha$ is the proportionality coefficient between the bending force acting on cantilever $F_x$, and the current in the ring $I$, so that $F_x=\alpha \cdot I$.   Apparently, coupling $\alpha$ depends on the mutual position of the MFM tip and the ring.   Lets notice, that amount of work that the magnetic field of the MFM tip performs on ring is $\delta A_{tip}=-\frac 1 c I \delta \Phi$.   On the other hand, the mechanical work that the magnetic force acting on MFM tip performs is $ \delta A_{ring}=F_x \delta x$, where $\delta x$ is the displacement of the MFM tip.  Since $\delta A_{ring}=-\delta A_{tip}$ , we find   \[F_x=\frac 1c I \frac{d\Phi}{dx}, \quad \text{and} \quad\alpha=\frac {\Phi_0}{c} \frac{dh}{dx}\]  The position of the tip was converted to flux through the ring by interpolating positions of the peaks, see figure~\ref{3um_positionToFlux}.  The flux dependance on position around studied transition (marked by red circle) is nearly linear.  Fit gives position-to-flux conversion $d \Phi/dx=1.82 \,\Phi_0/\mu m$.  Combining all, we end up with:  \begin{align*}  \frac{dP^{eq}}{dh}\cdot \alpha \cdot\Delta I\cdot \frac{\delta h}{k x_0}&=  \frac 1 2 \frac{\epsilon_0}{k_B T} \cdot\frac {\Phi_0}{c} \frac{dh}{dx} \cdot \frac{c}{\Phi_0} 2\epsilon_0 \cdot \frac{dh}{dx} \frac 1 k  =\frac{\epsilon_0^2}{k_B T} \left(\frac{dh}{dx}\right)^2 \frac 1 k \\  \end{align*}  \paragraph{Further fit discussion.}  In our case LAMH theory should give an lower estimate of $\nu_r$, because we do not take into account the slight suppression of the order parameter under the MFM tip.  It should be noted that our measurement does not give an answer where the phase slip actually happen.  It is possible that if the effect of the tip is strong, it creates the most favorable spot for PS.   Alternatively, because of the relative roughness of the walls and small local variations of critical temperature, PSs can be happening in some other spot of the ring.  Without a set of measurements of PS rates at different height it is impossible to say.  \subsection {Ring with crossbar}  Sample 2 has a shape of a ring with crossbar.   Inner diameter $D_{in}=3.56\mu m$, outer diameter $D_{out}=4.04\mu m$, wall width $w=220-230\, \text{nm}$, film thickness $d= 45\, \text{nm}$.   Superconducting transition temperature is $T_c=1.199$~K.   For this sample coherence length is estimated to be $\xi(0) =125 \pm 2 \, \text{nm}$.  \paragraph{Fluxoid states of ring with a crossbar. Different types of transitions}  The fluxoid state of the ring with crossbar can be described by a pair of winding numbers $\{ n_1, n_2 \}$ for top and bottom halves of the ring.  It is analogous to the honeycomb stability diagram of double quantum dot \cite{Wiel2002}.  The region of stability of each of the fluxoid states is a hexagon in $\phi_1$, $\phi_2$ coordinates (Fig.~\ref{crossbarSimulations}).  MFM scanning effectively creates non-linear mapping between ($\phi_1$, $\phi_2$) and (x,y) coordinates ( again look at figure.~\ref{crossbarSimulations}).  Images of Sample 2 at different tip-sample separations are shown on Fig.~\ref{4umCB_heights}.  By taking the transition lines that are furthers form the center of the structure as those that corresponds to first vortex entering the structure, we can enumerate all fluxoid states, as it is shown on figure~\ref{4umCB_markedStates}.  \textcolor[rgb]{1,0.41,0.13}{(Here I will add the simulated patterns from with the tip model.)}  The energy of the ring with crossbar can be expressed as  \[E\propto (\phi_1-n_1)^2+(\phi_2-n_2)^2+\alpha(\phi_1-n_1+\phi_2-n_2)^2 \]  Parameter $\alpha$ can be estimated from the shape of the honeycomb cells $\alpha=1.024$, which is different from estimate from geometry of device $\alpha=0.63$.  Let us notice, that three distinct types of transitions between fluxoid states are possible: i)fluxoid transitions when the winding number in top half-ring changes by one $\{n_1,n_2\}\to \{n_1\pm 1,n_2\}$; ii) winding number in the bottom half-ring changes by one $\{n_1,n_2\}\to \{n_1,n_2\pm 1\}$; iii) $\{n_1\pm 1,n_2\}\to \{n_1,n_2\mp 1\}$ winding number in one half increases, and in the other half decreases by one.   These three cases corespondent to phase slips that happen in different 'wires' of the ring: top, bottom and crossbar.  Strictly speaking, the transitions can also happen by co-tunneling events.  For example, for the transition between halves, it can happen by one 'vortex' leaving the ring from the top half and another vortex entering the ring from the bottom half.  We studied transitions of three different types between states $\{5,4\}$, $\{5,5\}$ and $\{6,4\}$ (Fig.~\ref{4umCB_posTri}).  Temperature dependance of in-phase and out-of-phase components of the signal are shown on figure~\ref{4umCB_tempTri}.  The transition for the top half is at resonance around 1.175~K, the transition for the bottom half around 1.173~K, and the transitions between halves around 1.170~K.  It can be seen that for all three transitions signal behavior is consistent with stochastic resonance model.  By taking the 'ratio' of frequency shift and damping scans with appropriate scaling we can find phase slip rate image of the transitions \ref{trijunction}.  Such tomography scan of phase slip rate has a bandwidth about three orders of magnitude, which is limited by frequency and damping noise and errors of background subtraction.  It gives a good way to demonstrate that three transitions are qualitatively distinct and how they evolve with temperature.   \subsection{Imaging constrictions}  (Comment: the data with constriction that I took last time doesn't look convincing enough to include it in this shorter paper.  On the other hand presenting an image like this significantly enhance s the paper, because it shows how we can image the effect of the local magnetic field on sample.  It fully utilize the advantage of this being a scanning microscopy technique - that we can image some interesting effects.  To demonstrate this we can use one of the images of a ring with constriction taken earlier.  For example \ref{RingConstriction}.  The purpose of this image is to merely to qualitatively showcase the capability to image the effect on the barrier.  We could even use qualitative colorscale legend High/Low phase slip rate.  Ideally, if we could get a better image in the next run, but even if it will not happen  (Note: I propose to drop this part and corresponding figures in current version of the paper)  Sample 3 is a loop of square shape, that is 10 $\mu m$ on a side, with a constriction at the center of one of sides.  The narrowest part of the constriction is 90 nm wide and have length 680 nm. The total length with tapered parts is 1.58~$\mu m$.  The width of the wire is 220-230~nm.  Color-coded images of the phase slip rates of constriction area are shown on Fig.~\ref{constrColor}.  The amplitude of the in-phase and out-of-phase parts of signal extracted from image is shown on Fig.~\ref{constrProfiles}  \section{Further discussion and conclusion}  Discussion of the analogy and relation between flux and charge.   There is an analogy between SC ring and a quantum dot.   On the other hand flux couples to phase which is tight with particle number by uncertainty relationship.  Vortices correspond to electrons, kinetic energy of the ring to electrostatic charging energy, flux to electrostatic potential, e-EFM to $\Phi_0$-MFM.   In superconducting devices important interplay is interplay between Josephson energy, charging energy and $k_B T$.  Our ring with crossbar is similar to double quantum dot.  Peaks in frequency are similar to Coulomb blockade.   What if we go further? Are there any new ideas we can get from pushing this analogy further?  We have demonstrated that MFM can be used to map the boundaries between fluxoid/vortex states.  It utilizes spatially localized field of MFM tip, that allows to access fluxoid/vortex states that are not accessible with external homogenous field that can be applied with superconducting magnate in the cryostat.   Localized field of the tip also allows to apply differential flux modulation between different parts of studied structures.   When the combination of the magnetic field of the tip at a given position and homogenous external field biases the studied superconducting structure to the point where two fluxoid/vortex states are degenerate, flux modulation leads to switching between this states.  As a consequence, supercurrents switch too, which creates an enhances back action on cantilever, that might result in resonant frequency shift and enhanced damping.   Spatial mapping is extremely helpful in distinguishing fluxid sates and ascribing physical meaning to individual transitions.  Once the fluxoid sates of the structure are mapped, this map can be used to drive the system between them in a desired way, using scan as a road-map for the movements of the tip.  Discussion of the applicability of vortex manipulation technique to other experiments such as braiding Majorana fermions.  We also demonstrated that the cantilever interaction with thermally activated phase slips is well described by stochastic resonance model.  \appendix  \section{Critical temperatures, coherence length and penetration measurements of devices}  Critical temperature of structures was measured by parking MFM tip at considerable distance above one the walls and monitoring frequency shift while sweeping the temperature.  We checked that the tip-sample separation does not affect the measured transition temperature (at smaller values measured $T_c$ becomes lower by several mK).  Frequency shift, which is expected to be proportion to the supercurrent ($\propto 1/\lambda^2$) is fitted with function $(1-(T/T_c)^3)$.   It is not clear why this particular temperature dependance works so well.   For comparison, Gorter-Casimir two fluid dependance $1/\lambda^2\propto 1-(T/T_c)^4$, and for \textit{s}-wave $1/\lambda^2~\propto~1~-~(T/T_c)^2$.  To estimate $\xi (T)$ from critical temperature suppression we use the result from section 4.6 'PARALLEL CRITICAL FIELD OF THIN FILMS' p.130 in Tinkham's book \cite{Tinkham1996}.  \begin{align*}  & H_{c\Vert}=2\sqrt{6} H_c\frac{\lambda}{d},\\  &\text{by replacing}\quad H_c=\frac {\Phi_0}{2\sqrt{2} \pi \xi\lambda},\quad \xi(T)=\xi(0)/\sqrt{1-T/T_c},\quad  H_{c\Vert} (T)=H_{c\Vert} (0) \sqrt{1-T/T_c}\\   &\text{we obtain for $T_c(H)$ suppression:}\\  & T_c(H)=T_c-\frac{T_c} {H_{c\Vert}^2 (0)}{H_{c\Vert} ^2(T)}  \end{align*}  The results of analysis measurements for samples 1-3 are shown in Tab.~\ref{tab1}.  %%%%%%%%%% TABLE %%%%%%%%%%%%%%  \begin {table}  %\caption {Table Title}% \label{tab:title}   \begin{center}  \caption{Sample parameters}  \begin{tabular}[t]{|c|c|c|c|c|}  \hline  Sample & $T_c$, (K) & $H_{c\Vert}$, (Oe)& $\xi(0)$, (nm) & $\lambda (0)$, (nm)\\  \hline  3 $\mu m$ ring & 1.163 & $496\pm 6$ &$108 \pm 8$ & \\  \hline  4 $\mu m$ ring crossbar & 1.199 & $457\pm 9$ &$125$ & \\  \hline  constriction & 1.140 & & & \\  \hline  \end{tabular}  \label{tab1}  \end{center}  \end {table}  \section {Superconducting ring theory and main formulas}  \subsubsection{Fluxoid states, occupation probabilities, currents, and phase slip barriers in superconducting ring }  We use notation $h=\Phi/\Phi_0$ for flux in units of $\Phi_0$.  For a ring with thin walls $S=w\cdot d \ll\lambda^2$.  \[E_n=- \frac{H_c^2}{8 \pi} V \cdot \left(1-\frac{\xi^2}{R^2} (h-n)^2\right)\]  For the case $\xi\ll R$  \begin{align*}  %&j_s=-\frac{c}{4 \pi \lambda^2} A \qquad\text{London equation}\\  %&\lambda^2=\frac{m c^2}{4 \pi n e^2}\\  %&A=\frac{\Phi_0 h}{2 \pi R}\\  %&W=n \frac m 2 (\frac{j_s}{n e})^2=\frac{m j_s^2}{2ne^2}=\frac{2\pi \lambda^2}{c^2} j_s^2\\  &E_n= \frac{H_c^2}{8 \pi} V\cdot 2\frac{\xi^2}{R^2} (h-n)^2=\frac{\Phi_0^2}{16\pi^2 \lambda^2} \frac {S} {R} h^2\\  %&E=S (2 \pi R) (2\pi/c^2 ) (\frac{c}{4 \pi}) ^2 \frac1 {\lambda^2} (\frac{\Phi_0 h}{2 \pi R})^2= \frac{\Phi_0^2}{16\pi^2 \lambda^2} \frac {S} {R} h^2\\  &E_n=\epsilon_0 (h-n)^2,\quad \epsilon_0=\frac{\Phi_0^2}{16\pi^2 \lambda^2} \frac {S} {R} \\  &I_n=\frac {c}{\Phi_0}\frac{dE_n}{dh}=\frac {c}{\Phi_0} 2 \epsilon _0 (h-n)\\  %&L_k=\frac{ 8\pi^2 \lambda^2}{c^2} \frac R S \qquad \text{Kinetic inductance}\\  \end{align*}  $E_1(h)$ $E_2(h)$, $E_b(h)$ are the energies of the first and the second fluxoid states ($n_1=0$, $n_2=1$) and the energy of the barrier respectively( here in $k_B T$ units).  \begin{align*}  &\Delta E =E_2-E_1\\  &P_1=\frac{\exp(-E_1)}{\exp(-E_1)+\exp(-E_2)}=\frac{1}{1+\exp(-\Delta E)}\\  &\frac{dP_1}{dh}=\frac{1}{4}\frac{d\Delta E}{dh}\frac{1}{(\cosh (-\Delta E/2)))^2}=-\frac{1} {2}\frac{ \epsilon_0 }{\cosh (\epsilon_0(h-1/2))^2}  \end{align*}  Given relaxation rates for both states we can write the equation for probability dynamics:  \begin{align*}  &\tau_1^{-1}=\Omega \exp(-E_b+E_1), \qquad \tau_2^{-1}=\Omega \exp(-E_b+E_2)\\  &dP_1=-\frac{P_1}{\tau_1}dt+\frac{1-P_1}{\tau_2}dt=-P_1(\tau_1^{-1}+\tau_2^{-1}) dt +\tau_2^{-1} dt, \quad \text{ denoting effective   relaxation rate $\omega^*=\tau_1^{-1}+\tau_2^{-1}$}\\  &dP_1=-P_1\omega^* dt +\tau_2^{-1} dt\\  &\nu_r=\tau_1^{-1}+\tau_2^{-1}= 2\Omega \exp(-E_b)\exp\left( \frac{E_1+E_2}{2} \right) \cosh \left( \frac{E_2-E_1}{2} \right)  \end{align*}  For $h=1/2$ :  \begin{align*}  &\frac{dP_1}{dh}=-1/2\epsilon_0\quad \text{(in $k_B T$ units)} \\  &\nu_r=2\Omega \exp(-E_b+E_1)  \end{align*}  \section{Superconducting ring with crossbar}  Derivation of the fluxoid states of the ring with crossbar  \begin{align*}  j_1 S=j S +j_2 S\quad \text{ current conservation}\\  j_1=\Lambda \frac{\phi_1}{L_1};\quad j_2=\Lambda \frac{\phi_2}{L_2};\quad j=\Lambda \frac{\phi}{l}\\  E=L_1 \left(\frac{n_1}{L_1}\right)^2+L_2 \left(\frac{n_1}{L_2}\right)^2+l \left(\frac{n_1}{l}\right)^2\\  \left\{  \begin{aligned}  &\varphi_1+\varphi=2\pi (\phi_1-n_1)\\  &\varphi_2+\varphi=2\pi (\phi_2-n_2) \\  &\left(\frac{\varphi_1}{L}\right) S=\left(\frac{\varphi}{l}\right) s+\left(\frac{\varphi_2}{L}\right) S  \end{aligned}\right.\\  \varphi=\underbrace{\frac{S l}{sL}}_{=\alpha} (\varphi_1-\varphi_2)\\  \left\{  \begin{aligned}  &\varphi_1=2 \pi \frac{\alpha}{1+2\alpha} ((\phi_1-n_1)+(\phi_2-n_2))+2 \pi \frac{1}{1+2\alpha}(\phi_1-n_1)\\  &\varphi_2=2 \pi \frac{\alpha}{1+2\alpha} ((\phi_1-n_1)+(\phi_2-n_2))+2 \pi \frac{1}{1+2\alpha}(\phi_2-n_2)\\  &\varphi=2 \pi \frac{\alpha}{1+2\alpha} ((\phi_1-n_1)-(\phi_2-n_2))  \end{aligned}\right.\\  E=\frac{S}{L} \frac{(2\pi)^2}{1+2\alpha} \{\phi_1^2+\phi_2^2+\alpha (\phi_1+\phi_2)^2\}=\\  =\frac{S}{L} \frac{(2\pi)^2}{1+2\alpha} \{ (\alpha+1/2)(\phi_1+\phi_2)^2+1/2(\phi_1-\phi_2)^2\}  \end{align*}  Several limiting cases for coupling $\alpha$:  1. Two isolated rings $\alpha \to 0$ ($s$ - fixed, $l\to 0$)  $E_{two}=(2\pi)^2\frac{S}{L} (\phi_1^2+\phi_2^2)$  2. One ring $\alpha \to \infty$ ($s\to 0$, $l$ - fixed)  $E_{one}=(2\pi)^2\frac{S}{2L} (\phi_1+\phi_2)^2$  3. Intermediate case $\alpha=1$: $E\propto \phi_1^2+\phi_1 \phi_2+\phi_2^2$  Let's consider the transitions between adjacent fluxoid states:  \begin{align*}  &\{n_1,n_2\}\to \{n_1\pm 1,n_2\} \\  &(\phi_1\pm 1)^2+\phi_2^2+\alpha (\phi_1\pm 1+\phi_2)^2=\phi_1^2+\phi_2^2+\alpha (\phi_1+\phi_2)^2\\  &(\phi_1\pm 1/2)(\pm 1)=-\alpha (\phi_1 +\phi_2 \pm 1/2)(\pm 1)\\  &(1+\alpha)\phi_1+\alpha \phi_2= \mp 1/2(1+\alpha)\\  &\{n_1,n_2\}\to \{n_1,n_2 \pm 1\}\\  &\alpha \phi_1+(1+\alpha) \phi_2= \mp 1/2(1+\alpha)\\  &\{n_1,n_2\}\to \{n_1\pm 1,n_2\mp1\}\\  &(\phi_1\pm 1)^2+(\phi_2\mp1)^2+\alpha (\phi_1+\phi_2)^2=\phi_1^2+\phi_2^2+\alpha (\phi_1+\phi_2)^2\\  &\phi_1-\phi_2\pm 1=0\\  &\left\{  \begin{aligned}  &\left|\phi_1+\frac{\alpha}{1+\alpha} \phi_2\right|< 1/2\\  &\left|\frac{\alpha}{1+\alpha}\phi_1+ \phi_2\right|< 1/2\\  &|\phi_1-\phi_2|< 1  \end{aligned}  \right.  \end{align*}  Positions of the corners of the diamond centered at (0, 0):   $\frac 1 2 \frac{1}{1+2\alpha} \times\{ ( -1-3\alpha, 1+\alpha),   ( -1-\alpha, 1+3\alpha),(1+\alpha,1+\alpha),\\  ( 1+3\alpha, -1-\alpha),( 1+\alpha, -1-3\alpha),( -1-\alpha, -1-\alpha) \}$