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arXiv:2108.05468

Berry connection from many-body wave functions and superconductivity: Circuit quantization for superconducting qubits and absence of a dissipative quantum phase transition in Josephson junctions

Hiroyasu Koizumi

Published Aug 11, 2021• 0 citations
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Abstract

The new superconductivity theory that attributes the $U(1)$ superconductivity phase to a Berry phase arising from many-body wave functions is applied to the circuit quantization for superconducting qubits. The phase-charge duality required for the occurrence of superconductivity in the standard theory becomes irrelevant in the new theory, and the absence of a dissipative quantum phase transition in Josephson junctions is explained. It is shown that a charge-decaying term leads to the compact phase description, and the appearance of Shapiro steps is explained without introducing normal current.

DOI
10.48550/arxiv.2108.05468

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The new superconductivity theory that attributes the U(1)U(1)U(1) superconductivity phase to a Berry phase arising from many-body wave functions is applied to the circuit quantization for superconducting qubits. The phase-charge duality required for the occurrence of superconductivity in the standard theory becomes irrelevant in the new theory, and the absence of a dissipative quantum phase transition in Josephson junctions is explained. It is shown that a charge-decaying term leads to the compact phase description, and the appearance of Shapiro steps is explained without introducing normal current.
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.
Neglected U(1)U(1)U(1) phase in the Schroedinger representation of quantum mechanics and particle number conserving formalisms for superconductivity
Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U(1)U(1)U(1) phase neglected by Dirac in the Schroedinger representation of quantum mechanics. The neglected phase gives rise to a U(1)U(1)U(1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected U(1)U(1)U(1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the U(1)U(1)U(1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does one agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exit in the new one in accordance with the recent experimental result.
Abstract Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U (1) phase neglected by Dirac in the Schrödinger representation of quantum mechanics. The neglected phase gives rise to a U (1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected U (1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the U (1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does not agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exist in the new one in accordance with a recent experiment (Murani et al 2020 Phys. Rev. X 10 021003).
Large superconducting quantum circuits have a number of important applications in quantum computing. Accurately predicting the performance of these devices from first principles is challenging, as it requires solving the many-body Schrödinger equation. This work introduces a new, general ab-initio method for analyzing large quantum circuits based on lattice field theory, a tool commonly applied in nuclear and particle physics. This method is competitive with state-of-the-art techniques such as tensor networks, but avoids introducing systematic errors due to truncation of the infinite-dimensional Hilbert space associated with superconducting phases. The approach is applied to fluxonium, a specific many-component superconducting qubit with favorable qualities for quantum computation. A systematic study of the influence of impedance on fluxonium is conducted that parallels previous experimental studies, and ground capacitance effects are explored. The qubit frequency and charge noise dephasing rate are extracted from statistical analyses of charge noise, where thousands of instantiations of charge disorder in the Josephson junction array of a fixed fluxonium qubit are explicitly averaged over at the microscopic level. This is difficult to achieve with any other existing method.
Superconducting circuit quantization conventionally starts from classical Euler-Lagrange circuit equations of motion. Invoking the correspondence principle yields a canonically quantized description of circuit dynamics over a bosonic Hilbert space. This approach has been very successful for describing experiments, but implicitly starts from the classical Ginsberg-Landau mean-field theory for the circuit. Here, we employ a different approach that starts from a microscopic fermionic Hamiltonian for interacting electrons, whose ground space is described by the Bardeen-Cooper-Schrieffer (BCS) many-body wave function that underpins conventional superconductivity. We introduce the BCS ground space as a subspace of the full fermionic Hilbert space, and show that projecting the electronic Hamiltonian onto this subspace yields the standard Hamiltonian terms for Josephson junctions, capacitors, and inductors, from which standard quantized circuit models follow. This approach does not impose a spontaneously broken symmetry so that it consistently describes quantized circuits that support superpositions of phases, and the canonical commutation relations between phase and charge are derived from the underlying fermionic commutation properties, rather than imposed. By expanding the projective subspace, this approach can be extended to describe phenomena outside the BCS ground space, including quasiparticle excitations.
Abstract Quantum circuit theory is a powerful tool to describe superconducting circuits. In its language, quantum phase slips (QPSs) are considered to be the exact dual to the Josephson effect. This duality renders the integration of QPS junctions into a unified theoretical framework challenging. As we argue, different existing formalisms may be inconsistent, and the correct inclusion of time-dependent flux driving requires introducing a large number of auxiliary, nonphysical degrees of freedom. We resolve these issues by describing QPS junctions as inductive rather than capacitive elements, and reducing the Hilbert space to account for a compact superconducting phase. Our treatment provides an approach to circuit quantization exclusively in terms of node-flux-node variables, and eliminates spurious degrees of freedom. Finally, the inductive treatment reveals the possibility of a voltage-dependent renormalization of the QPS amplitude, by accounting for spatial variations of the electric field built up across the junction.