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arXiv:0910.3622 · Physica Scripta

Electronic structure of superposition states in flux qubits

Jan Ivar Korsbakken, Frank K. Wilhelm, K Birgitta Whaley

Published Dec 1, 2009• 19 citations
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Abstract

Flux qubits, small superconducting loops interrupted by Josephson junctions, are successful realizations of quantum coherence for macroscopic variables. Superconductivity in these loops is carried by $\sim 10^6$ -- $10^{10}$ electrons, which has been interpreted as suggesting that coherent superpositions of such current states are macroscopic superpositions analogous to Schrödinger's cat. We provide a full microscopic analysis of such qubits, from which the macroscopic quantum description can be derived. This reveals that the number of microscopic constituents participating in superposition states for experimentally accessible flux qubits is surprisingly but not trivially small. The combination of this relatively small size with large differences between macroscopic observables in the two branches is seen to result from the Fermi statistics of the electrons and the large disparity between the values of superfluid and Fermi velocity in these systems.

DOI
10.1088/0031-8949/2009/t137/014022

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Flux qubits, small superconducting loops interrupted by Josephson junctions, are successful realizations of quantum coherence for macroscopic variables. Superconductivity in these loops is carried by ∼106\sim 10^6∼106 -- 101010^{10}1010 electrons, which has been interpreted as suggesting that coherent superpositions of such current states are macroscopic superpositions analogous to Schrödinger's cat. We provide a full microscopic analysis of such qubits, from which the macroscopic quantum description can be derived. This reveals that the number of microscopic constituents participating in superposition states for experimentally accessible flux qubits is surprisingly but not trivially small. The combination of this relatively small size with large differences between macroscopic observables in the two branches is seen to result from the Fermi statistics of the electrons and the large disparity between the values of superfluid and Fermi velocity in these systems.
The question as to whether or not quantum mechanics is applicable to the macroscopic scale has motivated efforts to generate superposition states of macroscopic numbers of particles and to determine their effective size. Superpositions of circulating current states in flux qubits constitute candidate states that have been argued to be at least mesoscopic. We present a microscopic analysis that reveals the number of electrons participating in these superpositions to be surprisingly but not trivially small, even though differences in macroscopic observables are large.
The consequences of the transition to a quantum description of magnetic flux motion in the superconducting ring closed by an ScS type Josephson junction are considered. Here we review the principal results regarding macroscopic quantum tunneling (MQT) of Bose condensate consisting of a macroscopically large number of Cooper electron pairs. These phenomena are illustrated by the original data obtained from the study of MQT and coherent states in a modified flux qubit with energy level depletion ΔE01 ≈ 2·10–23 J (ΔE01/h ≈ 30 GHz). State superposition properties in a two-well potential and the issues associated with quantum measurements of local curvature of qubits’ superposition energy levels are analyzed.
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.
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.
From a physicist's standpoint, the most interesting part of quantum computing research may well be the possibility to probe the boundary between the quantum and the classical worlds. The more macroscopic are the structures involved, the better. So far, the most "macroscopic" qubit prototypes that have been studied in the laboratory are certain kinds of superconducting qubits. To get a feeling for how macroscopic these systems can be, the states of flux qubits which are brought in a quantum superposition corresponds to currents composed of as much as 10^5-10^6 electrons flowing in opposite directions in a superconducting loop. Non-superconducting qubits realized so far are all essentially microscopic.
Abstract. Superconducting circuits with Josephson tunnel junctions are interesting systems for research on quantummechanical behavior of macroscopic degrees of freedom. A particular realization is a small superconducting loop containing three Josephson junctions. Close to magnetic frustration 1=2, the physics of this system corresponds to a double well, whose minima correspond to persistent currents of opposite sign. We present DC measurements of the flux indicating a smooth transition close to the degeneracy point even at very low temperatures. Furthermore, microwavespectroscopy experiments allow for the excitation to the next excited state. The dependence of the energy of the resonance on the applied flux clearly indicates the nature of these states as tunneling-splitted superpositions of flux states. We theoretically analyze the system using a generalized master-equation formulation of the spin-boson model. We address the nature of the measuring process by a switching DC SQUID and the possible interpretation of the spectroscopy data in terms of quantum coherence. We discuss these aspects in the context of further applications as a quantum bit.