Analytical theory of DC SQUIDS operating in the presence of thermal fluctuations

A comprehensive analytical theory of symmetric DC SQUIDs is presented taking into account the effects of thermal fluctuations. The SQUID has a reduced inductance β < 1/π where β = 2LIc/Φ0, L is the loop inductance, Φ0is the flux quantum, and Icis the critical current of the identical Josephson junctions which are assumed to be overdamped. The analysis, based on the two dimensional Fokker–Planck equation, has been successfully performed in first order approximation with β considered a small parameter. All important SQUID characteristics (circulating current, current-voltage curves, transfer function, and energy sensitivity) are obtained. In the limit βΓ ≪ 1( Γ = 2πkBT/IcΦ0is the noise parameter, kBis the Boltzmann constant, and T is the absolute temperature) the theory reproduces the results of numerical simulations performed for the case of small thermal fluctuations. It was found that for Γ < 1 the SQUID energy sensitivity is optimum when β is higher than 1/π, i.e., outside the range for which the present analysis is valid. However, for Γ ≥ 1 the energy sensitivity has a minimum at L = LF, where LF = (Φ0/2π)2/kB, and therefore, in this case, the optimal reduced DC SQUID inductance is βopt= 1/πΓ, i.e., within the range for which the present analysis is valid. In contrast to the case of an RF SQUID, for a DC SQUID the transfer function decreases not only with increasing L/LFbut also with increasing Γ (as 1/Γ). As a consequence, the energy sensitivity of a DC SQUID with β < 1/π degrades more rapidly (as Γ4) with the increase of Γ than that of an RF SQUID does (as Γ2).