As a newly developed ultrafast optical spectroscopy, terahertz two-dimensional coherent spectroscopy (2DCS) has become a promising method to characterize the physical properties of optical excitations in various materials. It utilizes two or more THz pulses to detect nonlinear responses of materials, thereby incorporating multiple frequency variables. Experimentally, it has uncovered a host of interesting phenomena in quantum wells, electronic glasses, and superconductors. In this work, we theoretically investigate the 2DCS of the Josephson plasmon mode in layered high-temperature superconductors.

Layered high-temperature superconductors consist of alternating superconducting and insulating layers. The adjacent superconducting-insulating-superconducting layers form a Josephson junction. Consequently, the layered high-temperature superconductors possess a nonlinear mode, known as Josephson plasmon mode. The Josephson plasmon mode arises from Josephson tunneling of Cooper pairs across the neighboring superconducting layers separated by insulating block layers.

In this work, we compute the 2DCS of the Josephson plasmon mode and analyze the features therein. We expect that our findings will provide the theoretical basis for the future 2DCS experiments on layered high-temperature superconductors.

We begin with a semiclassical effective model of layered high-temperature superconductors. We derive and solve numerically the equation of motions of the Josephson mode coupled to the electrical field. Fourier transforming the time-domain data yields 2DCS. Meanwhile, assuming that the electrical field is weak, we are able to solve the equation of motion analytically by using the perturbation theory. Finally, we associate the peaks in the two-dimensional spectrum with the Liouville paths and clarify the response process in detail.

Figure 3 shows the 2DCS of the Josephson plasmon mode. The numerical results [Figs. 3(a) and 3(b)] are found to be consistent with the analytical results [Figs. 3(c) and 3(d)]. In the time domain [Figs. 3(a) and 3(c)], the signals decay exponentially with increasing t1 or t2. The decay rate is proportional to the resistance of the Josephson junctions in layered high-temperature superconductors, γ. Meanwhile, the signal oscillates with the frequency of the Josephson plasmon mode,ω. In the frequency domain [Figs. 3(b) and 3(d)], the 2DCS exhibits eight peaks. In Fig. 3(b), the blue circles label the pump-probe peaks. The red circles label the photon echo peaks, which can help to separate the homogeneous broadening and inhomogeneous broadening. The green circles label the dephasing peaks. The orange circles label the two-quantum (2Q) peaks, which correspond to the simultaneous excitation of two plasmon mode quanta.

We have investigated the 2DCS of Josephson plasmon mode in layered high-temperature superconductors. Our findings reveal that the 2DCS contains various peaks, including pump-probe peaks, photon echo peaks, dephasing peaks, and 2Q peaks. We have also clarified their origins by associating these peaks to the various optical transition processes.

We envision that our model may be extended and improved in various aspects. Firstly, by adding the spatial degrees of freedom, the corresponding equation of motions would become sine-Gordan equation. Sine-Gordan equation hosts a specific excitation: soliton. Soliton has been detected in experiments but its 2DCS signature is still unclear and requires further investigation. Secondly, in deriving the equation of motion, the resistance is introduced phenomenologically. It would be interesting to derive the dissipative term from the first principles.