The Stationary SQUID

Abstract

In the customary mode of operation of a SQUID, the electromagnetic field in the SQUID is an oscillatory function of time. In this situation, electromagnetic radiation is emitted and couples to the sample. This is a back action that can alter the state that we intend to measure. A circuit that could perform as a stationary SQUID consists of a loop of superconducting material that encloses the magnetic flux, connected to a superconducting and to a normal electrode. This circuit does not contain Josephson junctions, or any other miniature feature. We study the evolution of the order parameter and of the electrochemical potential in this circuit; they converge to a stationary regime, and the voltage between the electrodes depends on the enclosed flux. We obtain expressions for the power dissipation and for the heat transported by the electric current; the validity of these expressions does not rely on a particular evolution model for the order parameter. We evaluate the influence of fluctuations. For a SQUID perimeter of the order of 1$\mu$m and temperature $0.9T_c$, we obtain a flux resolution of the order of $10^{-5}\Phi _0/$Hz$^{1/2}$; the resolution is expected to improve as the temperature is lowered.

Appendix A: Energy Balance in a Wire

Our analysis is an adaptation of Schmid’s seminal work [45]. Schmid’s analysis was intended to deal with the case of TDGL, but we will keep our expressions in a form that is valid for whatever evolution of the order parameter.

We initially use the gauge in which the scalar potential $\varphi$ is zero. In the units defined in Eq. (1), the Ginzburg–Landau energy of a segment of 1D wire of length dx is

$$\begin{aligned} F_\mathrm{GL}=\frac{u}{2} (-\epsilon |\psi |^2+\frac{1}{2}|\psi |^4+|{\mathcal D}\psi |^2)w\hbox {d}x. \end{aligned}$$

(10)

The unit of energy is $t_0\varphi _0j_0\xi ^2 (0)=2\sigma \hbar k_BT_c\xi (0)/\pi e^2$.

Taking the derivative with respect to time we obtain

$$\begin{aligned} \hbox {d}F_\mathrm{GL}/\hbox {d}t = (u/2) [(\partial _t\psi ^*)(-\epsilon +|\psi |^2\,+\,iA{\mathcal D})\psi \,+(\partial _{tx}^2\psi ^*){\mathcal D}\psi +i(\partial _tA)\psi ^*{\mathcal D}\psi ]w\hbox {d}x\,+\,\mathrm{cc}, \end{aligned}$$

(11)

and substituting $w(\partial _{tx}^2\psi ^*){\mathcal D}\psi =\partial _x[w(\partial _t\psi ^*){\mathcal D}\psi ]-(\partial _t\psi ^*)[(\partial _xw){\mathcal D}\psi +w\partial _x({\mathcal D}\psi )]$, this becomes

(12)

We now switch to the gauge in which A does not depend on time. $\partial _tA$ is substituted by $\partial _x\varphi$ and $\partial _t\psi$ by $\partial _t\psi +i\varphi$. We obtain

$$\begin{aligned} \hbox {d}F_\mathrm{GL}/\hbox {d}t=u\{\mathrm{Re}[(\partial _t\psi +i\varphi \psi )^*\left( -\epsilon +|\psi |^2-{\mathcal D}^2-(w'/w){\mathcal D}\right) \psi ]w \nonumber \\- {} (\partial _x\varphi )\mathrm{Im}(\psi ^*{\mathcal D}\psi )w +\partial _x\mathrm{Re}[w(\partial _t\psi +i\varphi \psi )^*{\mathcal D}\psi ]\} \hbox {d}x. \end{aligned}$$

(13)

In addition to the Ginzburg–Landau energy, we take into account the electrochemical energy. Assuming electroneutrality, its rate of increment is $wj\partial _x\varphi \, \hbox {d}x$. Adding this quantity to Eq. (13) and using Eq. (3), the rate of energy change of the segment becomes

$$\begin{aligned} \hbox {d}F_\mathrm{total}/\hbox {d}t= & {} \{u\mathrm{Re}[(\partial _t\psi +i\varphi \psi )^*(-\,\epsilon +|\psi |^2-{\mathcal D}^2-(w'/w){\mathcal D})\psi ]w \nonumber \\- & {} (\partial _x\varphi )^2 w +u\partial _x\mathrm{Re}[w(\partial _t\psi +i\varphi \psi )^*{\mathcal D}\psi ]\} \hbox {d}x. \end{aligned}$$

(14)

We finally note that if the volume density of power dissipation is W, and the energy flow is $I_\mathrm{E}$, then this rate of change is $\hbox {d}F_\mathrm{total}/\hbox {d}t=-(Ww+\partial _xI_\mathrm{E})\hbox {d}x$. From here and from Eq. (14) we identify

$$\begin{aligned} W=(\partial _x\varphi )^2+u\mathrm{Re}[(\partial _t\psi +i\varphi \psi )^*\left( \epsilon -|\psi |^2+{\mathcal D}^2+(w'/w){\mathcal D}\right) \psi ] \,, \end{aligned}$$

(15)

and

$$\begin{aligned} I_\mathrm{E}= -\,uw\mathrm{Re}[(\partial _t\psi +i\varphi \psi )^*{\mathcal D}\psi ]. \end{aligned}$$

(16)

The first term in W is the Joule dissipation, and the second term is due to the relaxation of the order parameter. In the limiting case of TDGL, $\partial _t \psi =[\epsilon -|\psi |^2- i\varphi +{\mathcal D}^2+(w'/w){\mathcal D}]\psi$ and we recover the result in [45].

Since $\psi$ and $\varphi$ are continuous, it follows from Eq. (16) that the constraint $\sum _{n=1}^3\pm \,{\mathcal D}\psi _n=0$ not only ensures charge conservation at the contacts between the ring and the connectors, but also energy conservation.