Cooper quartets in interacting hybrid superconducting systems

Open Access

Cooper quartets in interacting hybrid superconducting systems

Luca Chirolli, Alessandro Braggio, and Francesco Giazotto

Phys. Rev. Research 6, 033171 – Published 14 August 2024

Abstract

Cooper quartets represent exotic fermion aggregates describing correlated matter at the basis of charge- $4 e$ superconductivity and offer a platform for studying four-body interactions, of interest for topologically protected quantum computing, nuclear matter simulations, and more general strongly correlated matter. Focusing on solid-state systems, we show how to quantum design Cooper quartets in a double-dot system coupled to ordinary superconducting leads through the introduction of an attractive interdot interaction. A fundamentally novel, maximally correlated double-dot ground state, in the form of a superposition of vacuum $| 0 〉$ and four-electron state $| 4 e 〉$ , emerges as a narrow resonance in a many-body quartet correlator that is accompanied by negligible pair correlations and features a rich phenomenology. The system represents an instance of correlated Andreev matter and the results open the way to the exploration of interaction effects in hybrid superconducting devices, and the study of novel correlated states of matter with ingredients available in a quantum solid-state laboratory.

Received 4 March 2024
Revised 18 May 2024
Accepted 28 June 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.033171

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Fermions Josephson effect Proximity effect Superconducting order parameter

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Luca Chirolli ^1,2, Alessandro Braggio ¹, and Francesco Giazotto ¹

¹NEST, Istituto Nanoscienze CNR, and Scuola Normale Superiore, I-56127 Pisa, Italy
²Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE

Article Text

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Issue

Vol. 6, Iss. 3 — August - October 2024

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Figure 1
(a) Schematics of the double-dot system with on-site interaction $U$ , interdot density-density interaction $W$ , tunnel-coupled to a superconductor $S$ through rates $γ_{S}$ and $γ_{S}^{'}$ . (b) Scheme of energy levels and their coupling when the resonance condition is met, $ε = - U / 2 - W + δ ε$ , up to a small detuning $δ ε$ . (c) Phase diagram of the quartet correlator $Q$ at resonance $ε = - U / 2 - W$ for $γ_{S}^{'} = γ_{S}$ : for $- W > U > 0$ the ground state has a value $Q = - 1 / 2$ compatible with a quartet state $| ϕ_{Q}^{-} 〉$ , whereas for $U < W < 0$ the ground state has a value $Q = 1 / 2$ associated to the quartet state $| ϕ_{Q}^{+} 〉$ . (d) Quartet resonance as a function of $δ ε = ε + U / 2 + W$ along the dashed gray line in (c), where $U - W = 3 γ_{S}$ , showing the dependence of the correlator $Q$ on the detuning of the vacuum and four-electron states.
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Figure 2
(a) Schematics of the three-terminal structure composed by two quantum dots, 1 and 2, coupled to a common superconducting lead $S_{0}$ at the left and to superconducting leads, $S_{1}$ and $S_{2}$ , at the right. The latter are biased at phases $φ_{1}$ and $φ_{2}$ with respect to $S_{0}$ , respectively. (b) Contour plot of the ground state energy of the system in (a) with respect to the phases $φ_{1}$ and $φ_{2}$ , showing two minima at approximately $φ_{1} = φ_{2} ≃ \pm π / 2$ , for $γ_{S}^{'} = γ_{S, l} = γ_{S, r} = γ_{S}, ε = - U / 2 - W, U / γ_{S} = 10$ and $W / γ_{S} = - 12$ . (c) Plot of the ground state and the first excited states as a function of the phases $φ_{1}$ and $φ_{2}$ , representing a correlated Andreev matter and realizing a quartet Andreev qubit. (d) Total current $I_{1} + I_{2}$ as a function of $φ_{1} = φ_{2} = φ$ along the white line in (c), shown for the case $γ_{S, l} = γ_{S, r}$ , as in (b) and (c), and for $γ_{S, l} = 0$ for which the current is exactly $π$ periodic.
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Figure 3
(a) Current $I_{1} (φ_{1}, φ_{2} = 0)$ and (b) $I_{2} (φ_{1}, φ_{2} = 0)$ as a function of the phase $φ_{1}$ and the detuning $δ ε$ , obtained by keeping $φ_{2} = 0$ . (c) Cuts of $I_{1} (φ_{1}, φ_{2} = 0)$ in (a) at three different values of $δ ε$ shown in the legend of (d). (d) Cuts of $I_{2} (φ_{1}, φ_{2} = 0)$ at the values of $δ ε$ specified in the legend. The parameters for the plots are the same as in Fig. 2.
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Figure 4
(a) Schematics of a possible implementation of the system. The double quantum dot is formed on a carbon nanotube in tunnel contact with superconducting leads $S_{0}, S_{1}$ , and $S_{2}$ . Gates $V_{g 1}$ and $V_{g 2}$ allow for alignment of the dots levels. A second carbon nanotube, on the right side is suspended between contacts $S_{1}$ and $S_{2}$ but not in tunnel contact. Its vibrating lowest energy flexural mode couples to the charge on the quantum dots and provided a mechanism for strong interdot density-density interaction. (b) Modified setup that allows the pinning of the phase $φ_{2} = 2 e Φ_{x} / h$ and the resulting current $I_{1}$ shown in Fig. 3. (c) Quartet correlator $Q$ away from resonance, with $δ ε / γ_{S} = 0.1$ and (d) $δ ε / γ_{S} = 1$ , with $γ_{S}^{'} = γ_{S}$ .
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