We are developing a new cryogenic neutrino detector: electron bubble chamber, using liquid helium as the detecting medium, for the detection of low energy p-p reaction neutrinos (<420 keV), from the Sun. The program focuses in particular on the interactions of neutrinos scattering off atomic electrons in the detecting medium of liquid helium, resulting in recoil electrons which can be measured. We designed and constructed a small test chamber with 1.5 L active volume to start the detector R&D, and performed experimental proofs of the operation principle. The test chamber is a stainless steel cylinder equipped with five optical windows and ten high voltage cables. To shield the liquid helium chamber against the external heat loads, the chamber is made of double-walled jacket cooled by a pumped helium bath and is built into a LN2/LHe cryostat, equipped with 80 K and 4 K radiation shields. A needle valve for vapor helium cooling was used to provide a 1.7-4.5 K low temperature environments. The cryogenic test chamber has been successfully operated to test the performance of Gas Electron Multipliers (GEMs) in He and He+H2 at temperatures in the range of 3-293 K. This paper will give an introduction on the cryogenic solar neutrino detector using electron bubbles in liquid helium, then present the cryogenic design and operation of liquid helium in the small test chamber. The general principles of a full-scale electron bubble detector for the detection of low energy solar neutrinos are also proposed.