We propose a frequency-mixed effect of Terahertz (THz) and Gigahertz (GHz) electromagnetic waves in the cryogenic system of electrons floating on liquid helium surface. The THz wave is near-resonant with the transition frequency between the lowest two levels of surface state electrons. The GHz wave does not excite the transitions but generates a GHz-varying Stark effect with the symmetry-breaking eigenstates of electrons on liquid helium. We show an effective coupling between the inputting THz and GHz waves, which appears at the critical point that the detuning between electrons and THz wave is equal to the frequency of GHz wave. By this coupling, the THz and GHz waves cooperatively excite electrons and generate the low-frequency ac currents along the perpendicular direction of liquid helium surface to be experimentally detected by the image-charge approach [Phys. Rev. Lett. 123, 086801 (2019)]. This offers an alternative approach for THz detections. IntroductionElectrons floating on liquid helium surface have been considered as an alternative physical system to implement quantum computation [1][2][3][4][5][6][7][8][9][10][11]. Sommer first noticed that there is a barrier at the surface of liquid helium that prevents electrons from entering its interior [12]. Therefore, in addition to the attractive force of image charge in liquid helium, the surface electron forms a one-dimensional (1D) hydrogen atom above liquid helium surface [13][14][15]. The two lowest levels of the 1D hydrogen atoms have been proposed to encode the qubits, which have good scalability due to the strong Coulomb interactions between electrons [1]. However, the single electron detection is still very difficult in experiments. Recently, E. Kawakami et.al. have used the image-charge approach to experimentally detect the Rydberg states of surface electrons on liquid helium [16]. Compared with the conventional microwave absorption method, this method may be more effective to detect the quantum state of surface electrons [17,18].The transition frequency between the lowest two levels of surface electrons is in the regime of Terahertz (THz) and therefore has perhaps applications in the fields of THz detection and generation. The THz spectrum is between the usual microwave and infrared regions, and has many practical applications [19], such as THz wireless communication [20], astronomical observation [21], Earth's atmosphere monitoring [22], and the food safety [23]. Specially, in biological and medical fields [24,25], THz imaging have smaller damages in body than X-ray, but higher resolution than mechanical supersonic waves.In this paper, we show a frequency-mixed effect of the THz and the Gigahertz (GHz) electromagnetic waves in the system of electrons floating on liquid helium. This effect could be applied to detect THz radiation. In our mixer, the inputting THz wave near-resonantly excites the transition between the lowest two levels of surface state electrons. The GHz wave does not excite the transitions but generates a GHz-varying Stark effect. T...