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Journal of Physics Condensed Matter

Strong-coupling BCS models of Josephson qubits

Robert Alicki, W. Miklaszewski

Published Dec 10, 2012• 1 citation
Publisher

Abstract

The strong-coupling version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions--charge qubit, flux qubit and phase qubit. Applied to Josephson qubits it yields a more complicated structure of the lowest-lying energy levels than that obtained from phenomenological models based on quantization of the Kirchhoff equations. In particular, highly degenerate levels emerge, which act as probability sinks for the qubit. The alternative formulae concerning spectra of superconducting qubits are presented and compared with the experimental data. In contrast to the existing theories those formulae contain microscopic parameters of the model. In particular, for the first time, the density of Cooper pairs at zero temperature is estimated for an Al-based flux qubit. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.

DOI
10.1088/0953-8984/25/3/035701

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The strong-coupling version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions--charge qubit, flux qubit and phase qubit. Applied to Josephson qubits it yields a more complicated structure of the lowest-lying energy levels than that obtained from phenomenological models based on quantization of the Kirchhoff equations. In particular, highly degenerate levels emerge, which act as probability sinks for the qubit. The alternative formulae concerning spectra of superconducting qubits are presented and compared with the experimental data. In contrast to the existing theories those formulae contain microscopic parameters of the model. In particular, for the first time, the density of Cooper pairs at zero temperature is estimated for an Al-based flux qubit. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
Flux qubits shed a new light on BCS theory and high-TcT_cTc​ superconductivity
A simple microscopic model of a small superconducting loop interrupted by Josephson junction (flux qubit) allows to compute from the experimental data of Wal et.al \cite{Wal} an important parameter - the density of Cooper pairs at zero temperature. This density is determined by the cut-off energy in the BCS model and agrees with the original BCS suggestion but is lower by two orders of magnitude than the value accepted in the modern literature. The immediate consequences of this result are: the validity of the strong coupling BCS model, a plausible picture of electrons recombination into Cooper pairs, and a much weaker condition for the appearance of high-temperature superconductivity. Another consequence is that the popular interpretation of Josephson qubits as macroscopic quantum systems is replaced by a picture of qubit states being superpositions of the ground state and the state containing only a single excited Cooper pair.
The standard phenomenological Hamiltonian of a small superconducting Josephson junction in the charge regime (Cooper Pair Box) produces a model of the effective charge qubit with possible applications to quantum information processing. In this note a new model based on the BCS Hamiltonian with individual tunneling yields an effective multi-level picture with a highly degenerated level placed between the ground state and the excited state. Unlike in the standard approach, the excited Cooper pairs play here an important role. For such a system coupled to a zero temperature bath the additional levels act as a probability sink. In contrast to the standard large-spin model the coupling to phonons can be an effective source of dissipation. This model provides also alternative explanations of various effects observed in experiments and sheds new light on the issue of Josephson junctions as macroscopic quantum systems.
Abstract We consider a specific instance of a superconducting circuit, the so-called charge-qubit, consisting of a capacitor and a Josephson junction that we describe by means of the BCS microscopic model in terms of two tunnelling superconducting systems in the strong-coupling quasi-spin formulation. Then, by means of collective observables we derive the Hamiltonian governing the quantum behaviour of the circuit in the limit of a large numberNof quasi-spins. Our approach relies on suitablequantum fluctuations, i.e. on collective quasi-spin operators, different from mean-field observables, that retain a quantum character in the large-Nlimit. These collective operators generate the Heisenberg algebra on the circle and we show that their dynamics reproduces the phenomenological one generated by the charge qubit Hamiltonian obtained by quantizing the macroscopic classical Hamiltonian of the circuit. The microscopic derivation of the emergent, large-Nbehaviour provides a rigorous setting to investigate more in detail both general quantum circuits and quantum macroscopic scenarios; in particular, in the specific case of charge-qubits, it allows to explicitly obtain the temperature dependence of the critical Josephson current in the strong coupling regime, a result not accessible using standard approximation techniques.
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.
Within a semiclassical approximation and at the weak-coupling limit, where the charging energy dominates the Josephson coupling energy, the dynamics of small-capacitance superconducting tunnel junctions can be described by a stochastic master equation for the probability distribution of junction charges. We develop a method of solving this equation and obtain an analytic solution when Cooper-pair, quasiparticle, and Zener tunneling are all present under an arbitrary Ohmic conductance at zero temperature. We use this solution to systematically investigate the static and dynamic properties of small Josephson junctions. The charge distributions, current-voltage characteristics, and voltage spectra are analytically calculated and numerically evaluated for wide ranges of parameters. The separate contributions from quasiparticle and Cooper-pair tunneling to voltage spectra are evaluated and compared based on their respective sum rules. The linewidths of the resonance peaks for single-electron tunneling and Bloch oscillations are also discussed.