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Low Temperature Physics

Superposition of states in flux qubits with a Josephson junction of the ScS type (Review Article)

V. I. Shnyrkov, A. A. Soroka, A. M. Korolev, Oleg G. Turutanov

Published Apr 1, 2012• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.3699781

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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.
In this paper we discuss solid-state nanoelectronic realizations of Josephson flux qubits with large tunneling \namplitude between the two macroscopic states. The latter can be controlled via the height and form of \nthe potential barrier, which is determined by quantum-state engineering of the flux qubit circuit. The simplest \ncircuit of the flux qubit is a superconducting loop interrupted by a Josephson nanoscale tunnel junction. \nThe tunneling amplitude between two macroscopically different states can be essentially increased, by \nengineering of the qubit circuit, if tunnel junction is replaced by a ScS contact. However, only Josephson \ntunnel junctions are particularly suitable for large-scale integration circuits and quantum detectors with preset- \nday technology. To overcome this difficulty we consider here the flux qubit with high-level energy separation \nbetween «ground» and «excited» states, which consists of a superconducting loop with two lowcapacitance \nJosephson tunnel junctions in series. We demonstrate that for real parameters of resonant superposition \nbetween the two macroscopic states the tunneling amplitude can reach values greater than 1 K. Analytical \nresults for the tunneling amplitude obtained within semiclassical approximation by instanton technique \nshow good correlation with a numerical solution.
In this paper we discuss solid-state nanoelectronic realizations of Josephson flux qubits with large tunneling amplitude between the two macroscopic states. The latter can be controlled via the height and form of the potential barrier, which is determined by quantum-state engineering of the flux qubit circuit. The simplest circuit of the flux qubit is a superconducting loop interrupted by a Josephson nanoscale tunnel junction. The tunneling amplitude between two macroscopically different states can be increased substantially by engineering of the qubit circuit if the tunnel junction is replaced by a ScS contact. However, only Josephson tunnel junctions are particularly suitable for large-scale integration circuits and quantum detectors with present-day technology. To overcome this difficulty we consider here a flux qubit with high energy-level separation between the “ground” and “excited” states, consisting of a superconducting loop with two low-capacitance Josephson tunnel junctions in series. We demonstrate that for real parameters of resonant superposition between the two macroscopic states the tunneling amplitude can reach values greater than 1K. Analytical results for the tunneling amplitude obtained within the semiclassical approximation by the instanton technique show good correlation with a numerical solution.
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.
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.
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.
We report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on a Josephson phase escape for switching from the superconducting (the zero-voltage) state to a resistive one. Above the crossover temperature Tcr{T}_{cr}Tcr​ the thermal fluctuations of the Josephson phase induce such a switching, and as T<TcrT<{T}_{cr}T<Tcr​ the regime of the MQT occurs. In the absence of two-level systems (TLSs) a magnetic field applied parallel to the junction plane results in a smooth reduction of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$, where $\mathrm{\ensuremath{\Phi}}$ is an applied magnetic flux. As the TLSs are present in Josephson junctions we obtain a resonant enhancement of the MQT. This phenomenon manifests itself by a narrow peak in the dependence of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$ occurring in the intermediate range of $\mathrm{\ensuremath{\Phi}}$, i.e., $0<\mathrm{\ensuremath{\Phi}}<{\ensuremath{\phi}}_{0}$ (${\ensuremath{\phi}}_{0}$ is the magnetic flux quantum). We explain this effect quantitatively by a strong resonant suppression of the potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.