Heat transport through a Josephson junction

We discuss heat transport through a Josephson tunnel junction under various bias conditions. We first derive the formula for the cooling power of the junction valid for arbitrary time dependence of the Josephson phase. Combining it with the classical equation of motion for the phase, we find the time-averaged cooling power as a function of bias current or bias voltage. We also find the noise of the heat current and, more generally, the full counting statistics of the heat transport through the junction. We separately consider the metastable superconducting branch of the current-voltage characteristics allowing quantum fluctuations of the phase in this case. This regime is experimentally attractive since the junction has low power dissipation, low impedance, and therefore may be used as a sensitive detector.

Figure 1

A Josephson junction between two superconductors with the gaps ${\Delta }_1,{\Delta }_2$ and temperatures $T_1,T_2$. The junction is shunted by a capacitor $C$, a resistor $R_S$, and the whole circuit is biased by a current $I_x$. The dynamics of this circuit is described by RCSJ model (7).

Figure 2

The quasiparticle cooling power $P_{qp}\left(V\right)$ (solid line) and the absolute value of the amplitude $|P_{\cos }\left(V\right)|$ (dotted line). The parameters are chosen as follows: ${\Delta }_1=100$ $\mu$eV, ${\Delta }_2=200$ $\mu$eV, $R=1$ k$\Omega$, $T_1=T_2=230$ mK.

Figure 3

The Josephson component of the cooling power $P_{\sin }\left(V\right)$ for the same parameters as in Fig. 2.

Figure 4

The components of the heat current noise $S_P^{qp}\left(V\right)$ (solid line) and the $|S_P^{\cos }\left(V\right)|$ (dotted line). The parameters of the junction are the same as in Fig. 2.

Figure 5

Time-averaged current $\langle I\rangle$ (61) (top panel) and cooling power $\langle P\rangle$ (62) (lower panel) at different temperatures. The parameters are the same as in Fig. 2, i.e., ${\Delta }_1\left(0\right)=100$ $\mu$eV, ${\Delta }_2\left(0\right)=200$ $\mu$eV, $R=1$ k$\Omega$. The temperature dependence of the energy gaps ${\Delta }_1\left(T\right),{\Delta }_2\left(T\right)$ is modeled by Bardeen-Cooper-Schrieffer theory. Besides that we have put $T_1=T_2=T^{*}$ and chosen $R_S=100$ $\Omega$. The horizontal axis in both plots shows the average voltage drop across the junction $\langle V\rangle$ given by Eq. (B2).

Figure 6

The average cooling power $\langle P^{\left(1\right)}\rangle$ (45) and (64) of a resistively shunted Josephson junction at low bias and in the absence of the noise. Top panel: $\langle P^{\left(1\right)}\rangle$ as a function of the bias current at $T_1=T_2$, $T_1>T_2$ and $T_1<T_2$. Lower panel: $\langle P^{\left(1\right)}\rangle$ as a function of the time-averaged voltage $\langle V\rangle =R_S\sqrt{I_x^2-I_C^2}$; for comparison we have also shown the quasiparticle cooling power $P_{qp}$ (42) of a voltage biased Josephson junction. The parameters are chosen as follows: ${\Delta }_1=100$ $\mu$eV, ${\Delta }_2=200$ $\mu$eV, $R=1$ k$\Omega$, $R_S=0.1$ k$\Omega$. For these parameters we find $I_C=212$ nA and $I_C{R}_S=21.2$ $\mu$V.

Figure 7

Time-averaged cooling power $\langle P\rangle$ (73), (74), (81) at $T_1=T_2=230$ mK and at various values of the noise temperature $T^{*}$. Other parameters are the same as in Fig. 6. Top panel: Cooling power as a function of the bias current, inset: zero-bias cooling power versus the environment temperature $T^{*}$. Bottom panel: Cooling power as a function of the average voltage for three different values of $T^{*}$.