Signal and noise theory for a dc SQUID amplifier

A theory is presented for the gain and noise in an amplifier based on a dc SQUID. In the lumped circuit approximation, the total inductance of the input circuit L/sub T/ is coupled to the SQUID inductance L via a mutual inductance M/sub i/ = α/sub e/(LL/sub T/)/sup 1/2/ and is in series with a voltage source with a resistance R/sub i/ and a capicitance C/sub i/. The results are expressed in terms of parameters for a SQUID with reduced inductance (1-α/sub e/2)L. The voltage gain of the amplifier at frequency ω/2π is M/sub i/V/sub phi//sup r//Z/sub T/((ω), while the total voltage noise at the output of the SQUID is V/sub N//sup r/(ω)+M/sub i/2 V/sub phi//sup r/J/sub N//sup r/(ω)(R/sub i/+1/jωC/sub i/)/L/sub T/Z/sub T/( (ω). Here, Z/sub T/((ω) = Z/sub T/(ω)-J/sub Phi//sup r/M/sub i/2(R/sub i/+ 1/jωC/sub i/)/L/sub T/, where Z/sub T/(ω) is the total impedance of the unloaded input circuit, V/sub Phi//sup r/ and J/sub Phi//sup r/ are the flux-to-voltage and flux-to-circulating current transfer functions of the reduced SQUID, and V/sub N//sup r/(ω) and J/sub N//sup r/(ω) are the noise voltage and noise current of the reduced SQUID