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arXiv:2504.13779 · Physical Review Letters

Quantum Theory of the Josephson Junction between Finite Islands

Thomas J. Maldonado, Alejandro W. Rodríguez, Hakan E. Türeci

Published Aug 8, 2025• 1 citation
arXivHTMLPDF

Abstract

Superconducting circuits comprising Josephson junctions have spurred significant research activity due to their promise to realize scalable quantum computers. Effective Hamiltonians for these systems have traditionally been derived assuming the junction connects superconducting islands of infinite size. We derive a quantized Hamiltonian for a Josephson junction between finite-sized islands and predict measurable corrections to the qubit frequency and charge susceptibility to test the theory.

DOI
10.1103/mwvt-wlkj

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Superconducting circuits comprising Josephson junctions have spurred significant research activity due to their promise to realize scalable quantum computers. Effective Hamiltonians for these systems have traditionally been derived assuming the junction connects superconducting islands of infinite size. We derive a quantized Hamiltonian for a Josephson junction between finite-sized islands and predict measurable corrections to the qubit frequency and charge susceptibility to test the theory.
The quantized Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through canonical quantization of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well-known quantized Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong-coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean-field limit and predict further finite-size terms in addition to the basic equation.
We derive a mesoscopic theory of the Josephson junction from nonrelativistic scalar electrodynamics. Our theory reproduces the Josephson relations with the canonical current phase relation acquiring a weak second harmonic term, and it improves the standard lumped-element descriptions employed in circuit quantum electrodynamics by providing spatial resolution of the superconducting order parameter and electromagnetic field. By providing an ab initio derivation of the charge qubit Hamiltonian that relates traditionally free qubit parameters to geometric and material properties, we progress toward the quantum engineering of superconducting circuits at the subnanometer scale.
Recently, a new theory of superconductivity has been put forward that attributes the origin of superconductivity to the appearance of a non-trivial Berry connection from many-electron wave functions. This theory reproduces the major results of the BCS theory with conserving the particle number, and predicts the single-electron supercurrent tunneling across the Josephson junction with keeping the correct Josephson relation. We re-examine the quantization of superconducting qubit circuits by taking into account the above development, and show that the dynamical variables used in the standard theory, the flux nodes relating to the voltage, should be replaced by those relating to the electromagnetic vector potential. The fact that the Josephson junction tunneling allows the single-electron supercurrent tunneling is the reason for the existence of excited single electrons in superconducting qubits with Josephson junctions. We predict that it will be avoided by weakening the coupling between two superconductors in the Josephson junction.
This thesis presents experimental research on Josephson junction devices that behave quantum mechanically. The devices are formed by micrometer-sized superconducting islands, that are interconnected by a Josephson tunnel junction: a thin insulating layer between two superconductors. With current microfabrication technology it is possible to make very clean and well-defined junctions. The research that is presented in this thesis aimed at investigating whether Josephson junction circuits can have quantum coherent dynamics with a long decoherence time.
This thesis describes a number of experiments with superconducting cir- cuits containing small Josephson junctions. The circuits are made out of aluminum islands which are interconnected with a very thin insulating alu- minum oxide layer. The connections form a Josephson junction. The current trough the junction is related to the superconducting phase difference across the junction and described by the well-known Josephson relation. The char- acteristic energy scale for this effect is the Josephson energy, which is a measure for the strength of the coupling of the islands. Due to the parallel plate geometry of the junction a capacitance is formed. This capacitance leads to a charging effect. The characteristic energy scale for this effect is given by the charging energy. With the shadow evaporation technique it is possible to make the Josephson junction area very small and the two characteristic energy scales of the same order of magnitude. Charge and phase are conju- gate variables. Therefore a circuit containing such small Josephson junctions exhibits quantum effects. The microfabrication technique used in this the- sis allows fabrication of solid circuits with controlled quantum mechanical behavior.
In superconducting quantum circuits, Josephson junctions are a key component to provide nonlinearity and enable superconducting qubits. Fabricating junctions with precise critical currents is crucial to defining qubit frequency. Here we suppress the critical current variation of Josephson junctions by varying the critical current density Jc and junction areas. The relative standard deviation of the critical current Ic is controlled to be below 0.8%. This uniform fabrication is promising for realizing scalable superconducting quantum computation.