Quantization of Foster mesoscopic circuit and DC–pumped Josephson parametric amplifier from fractal measure arguments

Abstract

A quantum theory of the mesoscopic LC-circuit based on the product-like fractal measure which was introduced by Li and Ostoja-Starzewski is proposed. On the basis of the theory, the Schrödinger equation and the energy spectrum for the quantum LC circuit were derived. By introducing special forms of position-dependent LC-electric components, the associated creation and annihilation operators were obtained and analyzed. The quantization of the DC-driven Josephson circuit and its parametric amplifier were studied in details. The main outcome of this study concerns the finite form of the energy expectation value at very high temperature in contrast to the results obtained in literature which is time-dependent. Further details were analyzed and discussed. • The mesoscopic circuit and its quantum structure were explored using the concept of fractal product measure. • The theory is characterized by position-dependent electronic circuit's components. • The quantization of the DC-driven Josephson circuit and its parametric amplifier were constructed. • At very high temperature, a finite value of the energy expectation value was obtained.