^{1,a)}, A. A. Soroka

^{2}, A. M. Korolev

^{3}and O. G. Turutanov

^{4}

### 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 Δ*E* _{01} ≈ 2·10^{–23} J (Δ*E* _{01}/*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.

I. INTRODUCTION

II. MACROSCOPIC QUANTUM TUNNELING

III. SUPERPOSITION OF TWO INDEPENDENT STATES OF MAGNETIC FLUX IN SUPERCONDUCTORS

IV. THE MODIFIED SCHEME OF SIGNAL AMPLIFICATION DURING THE CONTINUOUS FUZZY MEASUREMENT OF QUANTUM SUPERPOSITION STATES OF A FLUX QUBIT

V. CONCLUSION

### Key Topics

- Qubits
- 86.0
- Josephson junctions
- 35.0
- Superconductivity
- 26.0
- Amplifiers
- 23.0
- Quantum measurement theory
- 21.0

## Figures

The design of an RF SQUID with a pure ScS type contact: *1*–Nb (99.999%) needle, *2*–resonant circuit coil, *3*–body, *4*–Nb membrane (a). Micrograph of the membrane surface (mosaic crystal) (b). Potential *U* of the superconducting ring, closed by pure ScS type contact, depending on the magnetic flux Φ in the ring when external flux is Φ_{ e } = 0.52Φ_{0}; parameter β_{ L } = 0.8, contact capacitance *C* = 3.77 fF. Magnetic flux, represented by the square of the wave function |Ψ_{1}|^{2}, tunnels into the right well, then the state relaxes to a lower energy level (MQT phenomenon, the process is shown by dashed lines with arrows) (c).

The design of an RF SQUID with a pure ScS type contact: *1*–Nb (99.999%) needle, *2*–resonant circuit coil, *3*–body, *4*–Nb membrane (a). Micrograph of the membrane surface (mosaic crystal) (b). Potential *U* of the superconducting ring, closed by pure ScS type contact, depending on the magnetic flux Φ in the ring when external flux is Φ_{ e } = 0.52Φ_{0}; parameter β_{ L } = 0.8, contact capacitance *C* = 3.77 fF. Magnetic flux, represented by the square of the wave function |Ψ_{1}|^{2}, tunnels into the right well, then the state relaxes to a lower energy level (MQT phenomenon, the process is shown by dashed lines with arrows) (c).

The current-voltage characteristics *VT*(*I*0) of an RF SQUID and their derivatives *dV _{T} */

*dI*

_{0}(

*V*) at

_{T}*T*= 0.5 K. An RF SQUID with an oxidized ScS type contact (a). An RF SQUID with a pure ScS type contact (b). Additional peaks on the derivative of HF CVC SQUID, shown with arrows, correspond to the macroscopic resonance tunneling around degeneracy of energy levels (see Ref. 56) of a quantum oscillator.

The current-voltage characteristics *VT*(*I*0) of an RF SQUID and their derivatives *dV _{T} */

*dI*

_{0}(

*V*) at

_{T}*T*= 0.5 K. An RF SQUID with an oxidized ScS type contact (a). An RF SQUID with a pure ScS type contact (b). Additional peaks on the derivative of HF CVC SQUID, shown with arrows, correspond to the macroscopic resonance tunneling around degeneracy of energy levels (see Ref. 56) of a quantum oscillator.

Superposition of states in a flux qubit, calculated for a three-atom point contact. The calculation was performed for parameters *C* = 3.77 fF, β_{ L } = 0.8. Potential *U*/*k _{B} *(

*f*), expressed in units of temperature, for Φ

_{ e }= Φ

_{0}/2 with tunneling splitting of the energy levels Δ

*E*

_{01}/

*k*= 1.52 K; the square of the wave function for the ground state is shown schematically (a). Dependences

_{B}*E*

_{i}/

*k*(

_{B}*f*) of the energy levels

_{e}*E*

_{0},

*E*

_{1}, and

*E*

_{2}on the external magnetic flux, expressed in units of temperature (b). Effective quantum inductance as a function of the reduced external magnetic flux (

*LL*

_{Q}^{–1})

_{eff}(

*f*) for different values of noise variance σ; parameter σ

_{e}^{1/2}for curves

*1*–

*4*equals, respectively, 0, 0.005, 0.01, and 0.02 (c). Family of HF CVCs

*V*(

_{T}_{ I }0) near low currents of excitation

*I*

_{0}for Φ

_{ dc }= Φ

_{0}/2. Parameter σ

^{1/2}for curves

*1*–

*3*equals, respectively, 0, 0.01, and 0.02. Curve

*4*corresponds to the values Φ

_{ dc }= Φ

_{0}, σ

^{1/2}= 0 (d).

Superposition of states in a flux qubit, calculated for a three-atom point contact. The calculation was performed for parameters *C* = 3.77 fF, β_{ L } = 0.8. Potential *U*/*k _{B} *(

*f*), expressed in units of temperature, for Φ

_{ e }= Φ

_{0}/2 with tunneling splitting of the energy levels Δ

*E*

_{01}/

*k*= 1.52 K; the square of the wave function for the ground state is shown schematically (a). Dependences

_{B}*E*

_{i}/

*k*(

_{B}*f*) of the energy levels

_{e}*E*

_{0},

*E*

_{1}, and

*E*

_{2}on the external magnetic flux, expressed in units of temperature (b). Effective quantum inductance as a function of the reduced external magnetic flux (

*LL*

_{Q}^{–1})

_{eff}(

*f*) for different values of noise variance σ; parameter σ

_{e}^{1/2}for curves

*1*–

*4*equals, respectively, 0, 0.005, 0.01, and 0.02 (c). Family of HF CVCs

*V*(

_{T}_{ I }0) near low currents of excitation

*I*

_{0}for Φ

_{ dc }= Φ

_{0}/2. Parameter σ

^{1/2}for curves

*1*–

*3*equals, respectively, 0, 0.01, and 0.02. Curve

*4*corresponds to the values Φ

_{ dc }= Φ

_{0}, σ

^{1/2}= 0 (d).

The superposition of flux qubit states, designed for a three-atom point contact of increased capacity *C* = 9.42 fF. Potential *U*/*k _{B} *(

*f*), expressed in temperature units, for Φ

_{ e }= Φ

_{0}/2 and β

_{ L }= 0.8 with a tunnel splitting Δ

*E*

_{01}= 0.36 K; the square of the wave function for the ground level is shown schematically (a). Dependences

*E*

_{i}/

*k*(

_{B}*f*) of the energy levels

_{e}*E*

_{0},

*E*

_{1}, and

*E*

_{2}on the external magnetic flux, expressed in units of temperature (b). Effective quantum inductance as a function of the reduced external magnetic flux (

*LL*

_{Q}^{–1})

_{eff}(

*f*) for different values of noise variance σ; parameter σ

_{e}^{1/2}for curves

*1*–

*4*equals, respectively, 0, 0.005, 0.01, and 0.02 (c). Family of HF CVCs

*V*(

_{T}*I*) near low currents of excitation for Φ

_{P}_{ dc }= Φ

_{0}/2; parameter σ

^{1/2}for curves

*1*–

*3*equals, respectively, 0, 0.01, and 0.02. Curve

*4*corresponds to the values Φ

_{ dc }= Φ

_{0}, σ

^{1/2}= 0 (d).

The superposition of flux qubit states, designed for a three-atom point contact of increased capacity *C* = 9.42 fF. Potential *U*/*k _{B} *(

*f*), expressed in temperature units, for Φ

_{ e }= Φ

_{0}/2 and β

_{ L }= 0.8 with a tunnel splitting Δ

*E*

_{01}= 0.36 K; the square of the wave function for the ground level is shown schematically (a). Dependences

*E*

_{i}/

*k*(

_{B}*f*) of the energy levels

_{e}*E*

_{0},

*E*

_{1}, and

*E*

_{2}on the external magnetic flux, expressed in units of temperature (b). Effective quantum inductance as a function of the reduced external magnetic flux (

*LL*

_{Q}^{–1})

_{eff}(

*f*) for different values of noise variance σ; parameter σ

_{e}^{1/2}for curves

*1*–

*4*equals, respectively, 0, 0.005, 0.01, and 0.02 (c). Family of HF CVCs

*V*(

_{T}*I*) near low currents of excitation for Φ

_{P}_{ dc }= Φ

_{0}/2; parameter σ

^{1/2}for curves

*1*–

*3*equals, respectively, 0, 0.01, and 0.02. Curve

*4*corresponds to the values Φ

_{ dc }= Φ

_{0}, σ

^{1/2}= 0 (d).

A family of current-voltage characteristics *V _{T} *(

*I*

_{0}) of a flux qutrit with a superposition step. The family parameter is the value of the external magnetic flux ΔΦ

_{ e }≈ Φ

_{0}/10. The superposition step has a periodic dependence on Φ

_{ e }, while the period is Φ

_{0}, and is observed around the symmetrical three-well potential. The slope of this step is partly due to the noise temperature of the resonant circuit.

A family of current-voltage characteristics *V _{T} *(

*I*

_{0}) of a flux qutrit with a superposition step. The family parameter is the value of the external magnetic flux ΔΦ

_{ e }≈ Φ

_{0}/10. The superposition step has a periodic dependence on Φ

_{ e }, while the period is Φ

_{0}, and is observed around the symmetrical three-well potential. The slope of this step is partly due to the noise temperature of the resonant circuit.

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