Abstract
We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [Phys. Rev. Lett. $\mathbf{113}:157001$] is shortly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantisation techniques in terms of discrete mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.
Highlights
Parametric amplification arises as a result of nonlinear optics
After an introduction to the relevant terminology and the classical theory of the coupled-mode equations of Josephson traveling-wave parametric amplifiers, we derived the mesoscopic quantum Hamiltonian up to first nonlinear order describing the process using discrete-mode operators. We found that such a description is possible, even when taking into account dispersion effects in the transmission line, and showed that the classical coupled-mode equations can be derived from this Hamiltonian
Such a Hamiltonian description of traveling-wave parametric amplifiers (TWPAs) is necessary for treating the amplifier as a quantum device
Summary
Parametric amplification arises as a result of nonlinear optics. In the case where a nonlinear medium is traversed by a (weak) signal and a strong pump, a wave-mixing interaction causes the signal to be amplified. A quantum theory allows one to calculate averages, standard deviations, and higher-order moments of measurement operators, and takes into account the effects of commutation relations, whereas a classical theory only allows averages to be calculated straightforwardly and the effect of noncommuting operators is neglected Some authors consider such a Hamiltonian description impossible due to the difficulties of quantum mechanics in describing dispersion ([20] and references therein), an important characteristic in TWPAs. in the case of a TWPA based on Josephson junctions, such a description appears to be possible.
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