Metastability and quantum coherence assisted sensing in interacting parallel quantum dots

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Metastability and quantum coherence assisted sensing in interacting parallel quantum dots

Stephanie Matern, Katarzyna Macieszczak, Simon Wozny, and Martin Leijnse

Phys. Rev. B 107, 125424 – Published 30 March 2023

Abstract

We study the transient dynamics subject to quantum coherence effects of two interacting parallel quantum dots weakly coupled to macroscopic leads. The stationary particle current of this quantum system is sensitive to perturbations much smaller than any other energy scale, specifically compared to the system-lead coupling and the temperature. We show that this is due to the presence of a parity-like symmetry in the dynamics, as a consequence of which two distinct stationary states arise. In the presence of small perturbations breaking this symmetry, the system exhibits metastability with two metastable phases that can be approximated by a combination of states corresponding to stationary states in the unperturbed limit. Furthermore, the long-time dynamics can be described as classical dynamics between those phases, leading to a unique stationary state. In particular, the competition of those two metastable phases explains the sensitive behavior of the stationary current towards small perturbations. We show that this behavior bears the potential of utilizing the parallel dots as a charge sensor, which makes use of quantum coherence effects to achieve a signal to noise ratio that is not limited by the temperature. As a consequence, the parallel dots outperform an analogous single-dot charge sensor for a wide range of temperatures.

Received 20 December 2022
Revised 8 March 2023
Accepted 17 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.125424

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. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum interference effects Quantum transport

Double quantum dots

Quantum master equation

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

Authors & Affiliations

Stephanie Matern ¹, Katarzyna Macieszczak ^2,3, Simon Wozny ¹, and Martin Leijnse ¹

¹NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
²Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
³TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Ave., Cambridge CB3 0HE, United Kingdom

Article Text

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Issue

Vol. 107, Iss. 12 — 15 March 2023

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Images

Figure 1
(a) The parallel dots with energies $ε_{i}$ are coupled to macroscopic leads with the tunneling rates $Γ_{j s}$ between dot $j$ and lead $s$ , where $j = 1, 2$ and $s = L, R$ . A bias is applied across the left and right leads, with temperatures $T_{L}$ and $T_{R}$ and chemical potentials $μ_{L}$ and $μ_{R}$ , respectively. (b) $d I / d V_{B}$ as a function of the bias voltage $V_{B}$ and the gate voltage $V_{G}$ . In Figs. 1, 2, 3, 4, and 6, we consider perturbations away from the balanced setup, that is Eqs. (22) and (23). Here, $δ ε = δ Γ = 0.04 Γ$ and $U = 250 Γ, T_{L} = T_{R} = 10 Γ$ . The star marks the voltage parameters $V_{G} = 0 Γ$ and $V_{B} = 30 Γ$ used in other figures.
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Figure 2
(a) $Re (λ_{3}) / λ_{2}$ as a function of $δ ε, δ Γ$ . We (somewhat arbitrarily) define the parameter regime where metastability occurs by $Re (λ_{3}) / λ_{2} > 20$ . (b) Stationary probabilities of Eq. (25) (purple line) well approximate the numerical decomposition into metastable phases ${\tilde{p}}_{1}$ (black dotted line) for constant $δ Γ / Γ = 0.04$ [shown as the white dashed line in (a)], see Appendix pp2-s3. (c) The transient current for $δ ε / Γ = 0.04$ (purple line) and $δ ε / Γ = 0.35$ (orange line). For a small perturbation $δ ε$ , the transient dynamics shows a plateau of approximately constant current corresponding to Eq. (32) (gray line), which at later times follows the evolution of Eq. (25) (purple dashed line). In both cases, $δ Γ / Γ = 0.04$ , and the initial state is the fully mixed state $ρ_{PD} (0) = 1 / 4$ in the local basis. Other parameters are chosen as in Fig. 1.
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Figure 3
Real (top panels) and imaginary part (bottom panels) of the spectrum of $L$ as function of the perturbations $δ Γ, δ ε$ with the Lamb shift $H_{LS}$ included (purple) and excluded (orange). The solid lines correspond to purely real eigenvalues while the dashed lines represent eigenvalue branches with a nonvanishing imaginary part. (a) Spectrum for varying $δ ε / Γ$ and constant $δ Γ / Γ = 10^{- 8}$ . (b) Spectrum for varying $δ Γ / Γ$ and constant $δ ε / Γ = 10^{- 6}$ ; all other parameters are chosen as in Fig. 1.
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Figure 4
(a) The stationary current, (b) absolute value of the signal rate $| \partial_{δ ε} I |$ , and (c) fluctuation rate as functions of $δ Γ$ and $δ ε$ . All other parameters are chosen as indicated by the marker in Fig. 1. The black dashed line in (a)–(c) marks the approximate border where metastability occurs and the solid black contour marks where $- λ_{2} / Γ > 10^{3}$ . (d) Temperature dependence of $σ_{δ ε}^{2}$ of the parallel dots (PD) for $δ ε = δ Γ = 0.04 Γ$ (purple, marker 1), $δ ε = 0.35, δ Γ = 0.04 Γ$ (orange, marker 2). These points are indicated by stars in (a)–(c). The error $σ_{ε}^{2}$ for a single quantum dot (SD) in the ideal configuration is plotted in black.
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Figure 5
(a) Differential conductance and (b) current of the parallel dots described by Pauli rate equation in the local basis. All other parameters are chosen as in Fig. 1. (c) Spectrum of the Liouvillian [as in Fig. 3, purple solid lines indicating purely real $λ_{1}, λ_{2}, λ_{5}, λ_{6}$ , while purple dashed lines the complex eigenvalues $λ_{3}$ and $λ_{4}$ ] and of the Pauli rate equation for the local basis (black dotted lines). For large $δ ε$ , the coherences in the evolution are eliminated and the rate equation becomes exact.
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Figure 6
(a) Stationary current, (b) noise, (c) signal rate, and (d) the error with the same parameters as Fig. 1 as functions of $V_{B}$ and $V_{G}$ . The structure of the error in (d) within the Coulomb diamond is due to the current, noise, and sensitivity not being exactly zero, but exponentially suppressed in this region. The star indicates the position in the stability diagram used for Fig. 4.
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