Double nuclear spin relaxation in hybrid quantum Hall systems

Letter

Double nuclear spin relaxation in hybrid quantum Hall systems

M. H. Fauzi, William J. Munro, Kae Nemoto, and Y. Hirayama

Phys. Rev. B 104, L121402 – Published 10 September 2021

Abstract

Recent advances in quantum engineering have given us the ability to design hybrid systems with novel properties normally not present in the regime they operate in. The coupling of spin ensembles and magnons to microwave resonators has for instance lead to a much richer understanding of collective effects in these systems and their potential quantum applications. We can also hybridize electron and nuclear spin ensembles together in the solid-state regime to investigate collective effects normally only observed in the atomic, molecular, and optical world. Here we explore in the solid state regime the dynamics of a double domain nuclear spin ensemble coupled to the Nambu-Goldstone boson in GaAs semiconductors and show it exhibits both collective and individual relaxation (thermalization) on very different time scales. Further the collective relaxation of the nuclear spin ensemble is what one would expect from superradiant decay. This opens up the possibility for the exploration of novel collective behavior in solid state systems where the natural energies associated with those spins are much less than the thermal energy.

Received 30 August 2020
Revised 31 August 2021
Accepted 2 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L121402

Physics Subject Headings (PhySH)

Quantum Hall effect Quantum transport

Two-dimensional electron system

Master equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. H. Fauzi^1,2,*, William J. Munro^4,3, Kae Nemoto^3,†, and Y. Hirayama ^1,5,6,‡

¹Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan
²Research Center for Physics, National Research and Innovation Agency, South Tangerang City, Banten 15314, Indonesia
³National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁴NTT Basic Research Laboratories & NTT Research Center for Theoretical Quantum Physics, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa, 243-0198, Japan
⁵Department of Physics, Tohoku University, Sendai 980-8578, Japan
⁶Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai 980-8577, Japan

^*moha065@lipi.go.id
^†nemoto@nii.ac.jp
^‡yoshiro.hirayama.d6@tohoku.ac.jp

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Issue

Vol. 104, Iss. 12 — 15 September 2021

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Images

Figure 1
(a) Schematic illustration of our hybrid quantum systems device used to investigate the behavior of nuclear spins coupled with an electronic reservoir of electrons in QH states through the hyperfine interaction. Here two identical 20-nm wide GaAs quantum wells, indicated by the two red lines, are in close proximity to create a strongly coupled bilayer two-dimensional electron gas system. The electronic states in both layers are electrostatically controlled by applying a gate bias between the top gate ( $V_{TG}$ ) and bottom gate ( $V_{BG}$ ). The inset depicts our QH system coupled through the hyperfine interaction to the NG boson. The nuclear spins are forming the double-spin domains A and B (two independent ensembles) with the NG boson operating as a reservoir. Domain A is described by the upward red arrows while domain B with the downward blue arrows. The number of spins in each domain is approximately the same. (b) A color map of longitudinal resistance as a function of top and bottom gate measured at a field of $5.85 T$ and a lattice temperature of 50 mK. We highlight the data around total filling factor $ν_{tot} = 2$ where three different electron spin configurations (FM, CAF, and SS) along the dotted white line can be realized depending on the charge imbalance ( $δ n$ ) between the two layers. Among those three possible spin states at $ν_{tot} = 2$ , the CAF phase houses the NG boson mode.
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Figure 2
Plot of the initialization of the double-nuclear-spin domain (a) indicating the number of polarized nuclear spins (measured as a hyperfine field change in millitesla) vs the time $τ_{p}$ the source-drain current $I_{sd}$ is applied. At $τ_{p} = 50 s$ , the number of up nuclear spin polarization $J_{↑}^{z}$ (down nuclear spin polarization $J_{↓}^{z}$ ) was $- 170 mT$ ( $+ 43 mT$ ), respectively. For $τ_{p} = 400 s$ , they increased to $- 353 mT$ ( $+ 215 mT$ ), a factor of 2 (5) polarization improvement, respectively. As a calibration a fully polarized sample would result in a hyperfine field near $5.3 T$ [19] meaning we can estimate the maximum percentage of spins in the up (down) domain at approximately $6.6 %$ ( $4 %$ ) respectively. In (b), we depict the total nuclear spin polarization for varying pumping time $τ_{p}$ followed by a 5-ms interaction the NG boson with a subsequent total magnetization measurement (in terms of the hyperfine field strength). For ease of comparison, the initial polarization in (a) is replotted. In (c), we show the nuclear spin relaxation dynamics vs the Dwell time to the NG boson for pumping times $τ_{p} = 100 s$ ( $400 s$ ) respectively with the solid lines providing a visual guide. It is clear that at long times the double-nuclear-spin domains reach the same thermal equilibrium. Our minimum dwell time measurement is 5 ms restricting our ability to explore the really short time dynamics.
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Figure 3
Simulation of the total magnetization of the double domain nuclear spin vs the interaction (dwell) time, nothing that we have an arbitrary scaling on the $y$ axis. The results are presented for a) $N = 10^{12}$ , b) $N = 6 \times 10^{12}$ . Here we have used the parameters $ω_{ns} / 2 π = 10$ MHz, $T = 50$ mK, $T_{2}^{*} \sim 1$ ms, and $T_{1} \sim 40$ s. This implies that $\bar{n} \sim 110$ which is much less than $N$ . Two distinct regions are shown in the main figure: White (red) where individual spin dephasing (thermalization) are the dominant behaviours. The yellow colored inset shows the short time behavior where collective thermalization effects dominate. Within the inset the black (red-dashed) curves correspond to thermal mean photon number $\bar{n} \sim 110$ (zero temperature $\bar{n} \sim 0$ ). Little difference is seen for the two $\bar{n}$ as both are much less than $N$ .
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