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AIP conference proceedings

Squeezed States in Josephson Junctions

Francesco Antonino Raffa

Published Jan 1, 2004• 0 citations
Publisher

Abstract

We consider the superconductive regime of a Josephson junction with an external biasing circuit. After recalling that the eigenvalue equation of the junction Hamiltonian HJ is a special case of the Mathieu equation, we diagonalize HJ in a suitable Fock space and prove that this leads quite naturally from the use of the two‐boson algebra. We utilize then the normalized even and odd coherent states of the two‐boson algebra to investigate the squeezing properties inherent in the nonlinearity of the junction with respect to the Cooper pair number and phase operators, N̂ and ϑ̂. This task is accomplished by utilizing the relationship between N̂, ϑ̂ and the generators of the dynamical quantum algebra of the junction.

DOI
10.1063/1.1834454

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We consider the superconductive regime of a Josephson junction with an external biasing circuit. After recalling that the eigenvalue equation of the junction Hamiltonian HJ is a special case of the Mathieu equation, we diagonalize HJ in a suitable Fock space and prove that this leads quite naturally from the use of the two‐boson algebra. We utilize then the normalized even and odd coherent states of the two‐boson algebra to investigate the squeezing properties inherent in the nonlinearity of the junction with respect to the Cooper pair number and phase operators, N̂ and ϑ̂. This task is accomplished by utilizing the relationship between N̂, ϑ̂ and the generators of the dynamical quantum algebra of the junction.
Our investigation concerns the class of Josephson-like systems, sharing the same nonlinear Hamiltonian. Among the latter a Josephson junction with an external biasing circuit is considered. We diagonalize the fully nonlinear Hamiltonian (in the superconductive regime of the junction) in the Fock space of the TBHA (two-boson Heisenberg algebra) and prove that such algebra leads quite naturally to the theoretical realization of codewords and logical operators: the codewords are defined as the even and odd coherent states of the TBHA, while the logical operators are expressed in terms of operators in the same algebra. Our theoretical construction corresponds to a continuous variable quantum computation scheme; the continuous variables are identified in terms of the physical operators of the junction. The link between this scheme and the technique of fermionization of bosonic systems is also discussed.
We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos ϑ-like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h (1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications.
Based on Feynman's explanation about the Cooper pair that "a bound pairs act as a Bose particle" and the bosonic operator Hamiltonian of Josephson junction (H.-Y. Fan, Int. J. Mod. Phys. B17 (2003) 2599) as well as the entangled state representation, we establish a possible number–phase squeezing mechanism for the Cooper-pair and the phase difference between the two sides of the junction. We find that when an extra energy (e.g. microwave radiation) is provided to the junction, then this squeezing mechanism can happen.
We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos -like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h(1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications
In this paper, we consider a system consisting of two capacitively coupled superconducting islands via Josephson junctions. We show that it can be reduced to two coupling harmonic oscillators under certain conditions, and can be solved exactly in terms of a displacing transformation, a beam-splitter-like transformation, and a squeezing transformation. It is found that the system evolves by a rotated-squeezed-coherent state when the system is initially in a coherent state. Quantum dynamics of the Cooper pairs in the two superconducting islands are investigated. It is shown that the number of the Cooper pairs in the two islands evolves periodically.