Different terrestrial terahertz applications would benefit from large-format arrays, operating in compact and inexpensive cryocoolers at liquid helium temperature with sensitivity, limited by the 300-K background radiation only. A voltage-biased Transition-Edge Sensor (TES) as a THz detector can have sufficient sensitivity and has a number of advantages important for real applications as linearity of response, high dynamic range and a simple calibration, however it requires a low-noise current readout. Usually, a current amplifier based on Superconducting Quantum-Interference Device (SQUID) is used for readout, but the scalability of this approach is limited due to complexity of the operation and fabrication. Recently, it has been shown that instead of SQUID it is possible to use a current sensor, which is based on the nonlinearity of the kinetic inductance of a current-carrying superconducting stripe. Embedding the stripe into a microwave high-Q superconducting resonator allows for reaching sufficient current sensitivity. More important, it is possible with the resonator approach to scale up to large arrays using Frequency-Division Multiplexing (FDM) in GHz range. Here, we demonstrate the operation of a voltage-biased TES with a microwave kinetic-inductance current amplifier at 4.2 K. We measured the expected intrinsic Noise-Equivalent Power NEP ~$5times 10^{-14} ; rm W/Hz^{1/2}$ and confirmed that a sufficient sensitivity of the readout can be reached in conjunction with a real TES operation. The construction of an array with the improved sensitivity ~ $10^{-15}; rm W/Hz^{1/2}$ at 4.2 K could be realized using a combination of the new current amplifier and already existing TES detectors with improved thermal isolation.
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