Interaction of vector Bose gases with fermionic superfluids

We study the effects of vector Bose gases, which are described by massive vector Bose fields, on fermionic superfluids in the low-energy region. It is demonstrated that the vector Bose gases can give rise to the Meissner effect, the flux quantization, and the Josephson effect, which are similar to those of the electromagnetic field in fermionic superfluids. However, unlike the electromagnetic field where the Meissner effect is unrelated to mass and the quantized flux is a constant, the Meissner effect of the vector Bose gases can be related to mass via a classical kinetic energy, and the value of the quantized flux is tunable. Our analyses also show that, although the origin model violates gauge invariance, the vector Bose gases in fact are gauge-invariant Maxwell-Chern-Simons systems under a Gauss constraint. It is also proposed that these effects can be observed by using the spin-1 cold atomic Bose gases in experiments.