Abstract
Superconducting circuits for quantum information processing are often described theoretically in terms of a discrete charge, or equivalently, a compact phase/flux, at each node in the circuit. Here we revisit the consequences of lifting this assumption for transmon and Cooper-pair-box circuits, which are constituted from a Josephson junction and a capacitor, treating both the superconducting phase and charge as noncompact variables. The periodic Josephson potential gives rise to a Bloch band structure, characterised by the Bloch quasicharge. We analyse the possibility of creating superpositions of different quasicharge states by transiently shunting inductive elements across the circuit, and suggest a choice of eigenstates in the lowest Bloch band of the spectrum that may support an inherently robust qubit encoding.
Highlights
In 1985, Likharev and Zorin [1] addressed the question of whether the superconducting phase at a circuit node, φ = 2π / 0, is a compact variable, φ ∈
We revisit the consequences of lifting this assumption for transmon and Cooper-pair box circuits, which are constituted from a Josephson junction and a capacitor, treating both the superconducting phase and charge as noncompact variables
We review a convenient basis proposed by Zak [8], and use this to reanalyze the circuit for a transmon/Cooper-pair box (CPB)
Summary
If the Hamiltonian for the device is invariant under a subset of phase translations (e.g., the 2π -periodic phase-translation symmetry of a Josephson junction), it is reasonably common to identify the phase with that of rotor [2], so that it is compact, and the states |φ φand |φ + 2π φare identical (the subscript φdenotes the phase basis) It follows that the Hilbert space in the charge representation is discrete. The Bloch bands include the pendulum spectrum as a discrete subset, corresponding to the Bloch eigenvalues at the center of the Brillouin zone (i.e., the point, in band-structure nomenclature) It follows that the noncompact periodic system includes the pendulum as a subspace, but its larger Hilbert space admits richer physics.
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