Abstract Body

Josephson current has a problem called the cosine-term problem, which has been known since the 1970s. The dissipative Josephson current has a term that depends on the phase difference (cosine term) and a term that does not. The cosine-term problem is an inconsistency between theories and experiments, in which theories show that these two terms have the same sign, but experiments give results with opposite signs.[1] Recently, in connection with the superconducting qubit, a positive-sign cosine term has been obtained in an experiment at the extremely low-temperature and low-frequency limit. However, since the finite value of the original dissipative Josephson current is caused by thermally excited quasiparticles, the author considers that the cosine-term problem in temperature ranges other than extremely low temperatures is still unsolved. Several theories have been suggested so far regarding the cosine-term problem, but all of these considered only the sign of the cosine-term at the limit of zero frequency. In actual experiments, the frequency dependence of the damping rate at the resonant frequency of the Josephson plasmon has been observed, so calculations must be performed at a finite frequency. Furthermore, the frequency dependence does not arise only from the term that depends on the phase difference, but it is necessary to examine the frequency dependence of the entire dissipative current. Recently, the author showed that when the electronic state is in a non-equilibrium steady state due to an external field at the resonant frequency, the effective temperature changes due to the phase difference, and the damping rate of Josephson plasmon behaves differently from previous theories.[2]

In this symposium, in addition to this change in electronic state, we show that collective excitation modes also depend on the phase difference, which results in the dependence of the damping rate of Josephson plasmon on the phase difference that is consistent with experiments. In general, phase fluctuations and amplitude fluctuations exist in superconductors, and in simple cases, these two modes are uncoupled, but when the particle-hole symmetry is broken, these two fluctuations couple. In Josephson junctions, the amplitude mode also exists, but when the phase difference is zero, they do not couple with Josephson plasmons, which is a phase difference fluctuation. On the other hand, when a finite phase difference occurs due to the presence of a dc current, these two modes couple due to the breaking of time-reversal symmetry. As a result, Josephson plasmons are affected by the amplitude mode and has a dependence on the phase difference. Figure 1 shows the frequency dependence of the impedance of a Josephson junction. The numbers shown in the figure are values of cosine of the phase difference; as the phase difference increases, the resonance frequency decreases and the peak position shifts to lower energy. The inset shows the damping rate of the Josephson plasmons at this resonance frequency, which becomes smaller as the phase difference decreases. This result is consistent with that of experiments.

References

[1] D. N. Langenberg, Rev. Phys. Appl. (Paris) 9, 35 (1974).
[2] T. Jujo, J. Phys. Soc. Jpn. 93, 094701 (2024).

Acknowledgment

The numerical computation in this work was carried out at the Yukawa Institute Computer Facility.

Figure 1. The curves show the frequency dependence of impedance. The values of the lines indicate the cosine of the phase difference. The inset shows the value of the damping rate of the Josephson plasmons at the resonance frequency, and the value of the cosine ranges from 0.05 to 0.95. The temperature is at T/Tc=0.6.

Keywords: Josephson plasmons, Amplitude fluctuation, Damping rate, Cosine-term problem