Fractional transconductance via nonadiabatic topological Cooper pair pumping

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Fractional transconductance via nonadiabatic topological Cooper pair pumping

Hannes Weisbrich, Raffael L. Klees, Oded Zilberberg, and Wolfgang Belzig

Phys. Rev. Research 5, 043045 – Published 16 October 2023

Abstract

Many robust physical phenomena in quantum physics are based on topological invariants arising due to intriguing geometrical properties of quantum states. Prime examples are the integer and fractional quantum Hall effects that demonstrate quantized Hall conductances, associated with topology both in the single particle and the strongly correlated many-body limit. Interestingly, the topology of the integer effect can be realized in superconducting multiterminal systems, but a proposal for the more complex fractional counterpart is lacking. In this work, we theoretically demonstrate how to achieve fractional quantized transconductance in an engineered chain of Josephson junctions. Crucially, similar to the stabilization of the conductance plateaus in Hall systems by disorder, we obtain stable transconductance plateaus as a result of nonadiabatic Landau-Zener transitions. We furthermore show that the fractional plateaus are robust to disorder and study the optimal operation regime to observe these effects. Our proposal paves the way for quantum simulation of exotic many-body out-of-equilibrium states in Josephson junction systems.

Received 26 April 2023
Revised 18 August 2023
Accepted 25 August 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.043045

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)

Fractional quantum Hall effect Quantum transport Topological superconductors

Josephson junctions Superconductors Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hannes Weisbrich ¹, Raffael L. Klees ², Oded Zilberberg ¹, and Wolfgang Belzig ^1,*

¹Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany
²Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, D-97074 Würzburg, Germany

^*Corresponding author: wolfgang.belzig@uni-konstanz.de

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Vol. 5, Iss. 4 — October - December 2023

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Images

Figure 1
(a) Array of Josephson junctions (blue boxes) arranged as a chain of superconducting loops with fluxes $φ_{n} = {\dot{φ}}_{x} t + n φ_{off}$ and applied voltage bias $V_{y} = ℏ {\dot{φ}}_{y} / (2 e)$ . (b) Transconductance ${〈 G_{y x} 〉}_{T} = {〈 I 〉}_{T} / V_{x}$ in units of $G_{0} = 4 e^{2} / h$ as a function of $φ_{off}$ averaged over the time $T = 200 h / (2 e V_{y})$ for a chain of length $N = 18$ with $2 E_{J_{1}} = E_{J_{2}}, 2 e V_{y} = 0.01 E_{J_{1}}$ , and ${\dot{φ}}_{x} = - 2 π {\dot{φ}}_{y}$ , cf. Eq. (2). (c) Local probability density of the excess Cooper pair $ρ (n) = {| 〈 n | ψ_{1}^{0} 〉 |}^{2}$ of the ground state for different offset fluxes $φ_{off}$ at different times $t_{1} < t_{2}$ given by $φ_{x} (t_{1})$ and $φ_{x} (t_{2})$ , respectively.
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Figure 2
(a) Ground-state band and first excited states [cf. Eq. (1)] for $φ_{off} = 2 π ν / N$ with $ν = 2, N = 12, E_{J_{2}} = 2 E_{J_{1}}$ , and $φ_{x} = 0$ . There is an energy gap $E_{inter}$ between the ground-state band and other excited states. By changing $φ_{off}$ from $ν = 2$ to $ν = 3$ , one of the excited states is pushed toward the ground-state band [see arrow and panel (c)]. (b) Zoom in on the ground-state band for $ν = 2$ as a function of $φ_{x}$ and $φ_{y}$ with the same parameters as in panel (a). The adiabatic state evolution is indicated by the black arrow for finite voltages $V_{ℓ} = ℏ {\dot{φ}}_{ℓ} / (2 e)$ . (c) Same as panel (b) for $ν = 3$ . An additional energy level moved from the excited states toward the ground-state band leading to a $6 π$ periodicity of the whole band. The width of the lowest band $E_{width}$ increases with increasing $ν$ . (d) Applying incommensurate voltages $V_{x}$ and $V_{y}$ leads to a nonperiodic time evolution $(φ_{x} (t), φ_{y} (t))$ for $t \in [0, T]$ that covers the whole torus for long times $T \to \infty$ .
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Figure 3
(a) Lowest band for $φ_{off} = 3.9 π / N, N = 12, E_{J_{2}} = 2 E_{J_{1}}$ , and $φ_{x} = π / 2$ . At $φ_{off} \neq 2 π ν / N$ , small energy gaps $E_{intra}$ open between the individual states within the ground-state band. As long as $2 e V ≫ E_{intra}$ with $V \equiv \sqrt{V_{x}^{2} + V_{y}^{2}}$ , the state evolution will diabatically jump over these gaps (marked by dots) and maintain the $2 π ν$ periodicity [provided that $2 e V ≪ E_{inter}$ , cf. Fig. 2]. (b) Transconductance ${〈 G_{y x} 〉}_{T} = {〈 I 〉}_{T} / V_{x}$ in units of $G_{0} = 4 e^{2} / h$ as a function of the applied voltage $V = \sqrt{V_{x}^{2} + V_{y}^{2}}$ averaged over the time $T = 200 h / (2 {e V}_{y})$ for $φ_{off} = 3.36 π / N, N = 12, 2 E_{J_{1}} = E_{J_{2}}$ , and $V_{x} = - π V_{y}$ . We observe a quantized plateau only in the region where the condition $E_{intra} (φ_{off}) ≪ 2 e V ≪ E_{inter} (φ_{off})$ is fulfilled, while the fractional quantization vanishes for too large (small) voltages $2 e V \sim E_{inter} (2 e V \sim E_{intra})$ . (c) and (d), Standard deviation of the transconductance $δ G_{ν}$ [see Fig. 1] with respect to the quantized values $G_{0} / ν$ as a function of the disorder strength $δ$ . $δ G_{ν}$ is determined in the range of $φ_{off}$ where quantized plateaus would appear in the case without disorder. The transconductance is averaged over the time $T = 200 h / (2 {e V}_{y})$ for $N = 18, 2 E_{J_{1}} = E_{J_{2}}$ , and $V_{x} = - 2 π V_{y}$ , as in Fig. 1.
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