Quantum noise in SQUIDs

The sensitivity limitations due to thermal, quantum and shot noise are calculated for dc and ac SQUIDs. Two approximations are used for the Josephson junctions: i. RSJ model with the equilibrium source of thermal-and-quantum fluctuations and ii. microscopic theory of the tunnel junctions with small capacitance (the latter theory takes into account the shot noise as well). The ultimate "output" sensitivity ε_{V}=<deltaPhimin{x}max{2}>/2LDeltaf is shown to be substantially less than the Planck\´s constant h, for the both types of SQUIDs. This result does not contradict to the uncertainty principle, because the "total" sensitivity ε=(ε_{V}ε_{I}-εmin{IV}max{2})^{1/2} rather than ε_{V} is an adequate figure of merit of SQUID as a linear amplifier. The total sensitivity is shown to be, in fact, limited by the quantum fluctuations, ε>h/2. These fluctuations originate in the shunting resistances of the dc SQUID, while in the ac SQUID they are due to quantum noise of the following stage of amplification. Nevertheless, even the total sensitivity ε can be made less than h/2, if the "degenerate" type of SQUID is used, being sensitive to the only one of the quadrature components of the input signal.