# Influence of Anharmonic and Frustration Effects on Josephson Phase Qubit Characteristics

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{c}superconductors, the current-phase relation includes the second term [3,5,6],

_{c}superconductors and multiband behavior of the superconducting state in new iron-based compounds [3,7].

## 2. Results

_{i}are the masses of the carriers in different bands, (i = 1–3); α

_{i}= γ

_{i}(T − T

_{ci}) which are linearly dependent on temperature T; β

_{i}and γ

_{i}are constants; ε

_{ij}= ε

_{ji}and ε

_{1}

^{ij}

_{=}ε

_{1}

^{ji}mean the interaction between superconducting gaps and their gradients, respectively; H is the external magnetic field applied to example; and Φ

_{0}is the magnetic flux quantum. In the case of single- and two-band junctions, for the phase differences $\varphi $ of gap parameters, we have the effective critical current, as presented in Ref. [14].

_{ij}= ε

_{ji}= ε > 0, the phase differences in frustration states are given as $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}2\pi /3\\ -2\pi /3\end{array}\right)$ and $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}-2\pi /3\\ 2\pi /3\end{array}\right)$ [15]. In other possible frustration states, we have $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}0\\ \pi \end{array}\right);\left(\begin{array}{l}\pi \\ 0\end{array}\right)$ and $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}\pi \\ \pi \end{array}\right)$. From the expression for the potential energy of single-/three-band junctions $U(\varphi )$, we can get effective critical current

_{c}

_{2}and I

_{c}

_{3}in Equation (12) were replaced by the places. The frustration case $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}0\\ \pi \end{array}\right)$ corresponds to the effective critical current

## 3. Discussion

_{c}

_{3}/I

_{c}

_{1}. At high values of I

_{c}

_{2}/I

_{c}

_{1}= 1, we have the behavior similar to the single-band/two-band case with the opposite phase difference $\varphi =\pi $. The ratio $\mathrm{\Delta}E/\mathrm{\Delta}{E}_{0}$ reveals a minimum in the case of low values of the ratio I

_{c}

_{2}/I

_{c}= 0, 0.5. In the frustration case, $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}0\\ \pi \end{array}\right)$ and $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}\pi \\ \pi \end{array}\right)$, using the effective critical current (see Equations (13) and (14)), has a form similar to Figure 4 in the case of a single-/two-band structure.

^{_}/two-band junctions with positive interband interaction parameters. In Ref. [22], single-/three-band junction was investigated, and it described the effects of the asymmetric critical current, Shapiro steps. The effect of the asymmetrical critical current has been observed in the edge-type junction between PbIn and many-band Co-doped BaFe

_{2}As

_{2}thin film, as presented in Ref. [23]. In such junctions, a critical voltage I

_{c}R

_{N}of about 12 $\mathsf{\mu}\mathrm{V}$. In Ref. [24], the junction between PbIn and the Ba

_{1−x}K

_{x}(FeAs)

_{2}x = 0.29 and 0.49 was realized. In this study, it was studied experimentally as a PbIn/BaK(FeAs)

_{2}point-contact junction. It was also theoretically shown that the three-band superconducting state scenario provides better results for the treatment of the observed data. In papers [25,26], Nb/BaNa(FeAs)

_{2}junctions were reported with very a small critical voltage I

_{c}R

_{N}, approximately 3 $\mathsf{\mu}\mathrm{V}$. This fact can be explained by the cancellation of the opposite supercurrents in the frustrated state of multiband iron-based superconductors. The reduction of the Josephson plasma frequency in such three-band structures was also obtained by the theoretical investigation in paper [27]. We hope that the obtained theoretical results for changing $\mathrm{\Delta}E/\mathrm{\Delta}{E}_{0}$ phase qubits will be observed experimentally.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Profile of potential energy $U(\varphi )=-{E}_{J}\left\{\mathrm{cos}\varphi +{i}_{b}\varphi \right\}$ of the Josephson phase qubit.

**Figure 3.**Changing the energy difference between levels $\mathrm{\Delta}E/\mathrm{\Delta}{E}_{0}$ versus anharmonicity.

**Figure 4.**Changing the energy difference between levels $\mathrm{\Delta}E/\mathrm{\Delta}{E}_{0}$ in qubit based on the single-/two-band junction versus ${I}_{c2}/{I}_{c1}$.

**Figure 5.**Changing the energy difference between levels in qubit based on single-/three-band junction $\mathrm{\Delta}E/\mathrm{\Delta}{E}_{0}$ versus ${I}_{c3}/{I}_{c1}$ for ${I}_{c2}/{I}_{c1}$ = 0, 0.5, 1 (from top to bottom).

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**MDPI and ACS Style**

Askerzade, I.N.
Influence of Anharmonic and Frustration Effects on Josephson Phase Qubit Characteristics. *Condens. Matter* **2023**, *8*, 20.
https://doi.org/10.3390/condmat8010020

**AMA Style**

Askerzade IN.
Influence of Anharmonic and Frustration Effects on Josephson Phase Qubit Characteristics. *Condensed Matter*. 2023; 8(1):20.
https://doi.org/10.3390/condmat8010020

**Chicago/Turabian Style**

Askerzade, Iman N.
2023. "Influence of Anharmonic and Frustration Effects on Josephson Phase Qubit Characteristics" *Condensed Matter* 8, no. 1: 20.
https://doi.org/10.3390/condmat8010020