Spin purity of the quantum dot confined electron and hole in an external magnetic field

Letter

Spin purity of the quantum dot confined electron and hole in an external magnetic field

Dan Cogan, Zu-En Su, Oded Kenneth, and David Gershoni

Phys. Rev. B 105, L041407 – Published 14 January 2022

Abstract

We investigate experimentally and theoretically the temporal evolution of the spin of the conduction band electron and that of the valence band heavy hole, both confined in the same semiconductor quantum dot. We use all-optical pulse techniques to perform complete tomographic measurements of the spin as a function of time after its initialization and study the total spin purity (coherence), measured here. In the important limit of a weak externally applied magnetic field, comparable in strength to the Overhauser field due to fluctuations in the surrounding nuclei spins, the measured spin purity performs complex temporal oscillations. We use a central-spin model encompassing the spin' s Zeeman and the hyperfine interactions to reproduce the measured results quantitatively. Our studies are essential for designing and optimizing quantum-dot-spin-based entangled multiphoton sources that set stringent limitations on the magnitude of the externally applied field.

Received 30 August 2021
Revised 2 January 2022
Accepted 5 January 2022

DOI:https://doi.org/10.1103/PhysRevB.105.L041407

Physics Subject Headings (PhySH)

Excitons Mesoscopics Quantum communication Quantum memories Quantum optics Spin dynamics Spintronics Trions Zeeman effect

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Dan Cogan , Zu-En Su, Oded Kenneth, and David Gershoni ^*

The Physics Department and the Solid State Institute, Technion–Israel Institute of Technology, Haifa 3200003, Israel

^*dg@physics.technion.ac.il

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Issue

Vol. 105, Iss. 4 — 15 January 2022

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Images

Figure 1
(a) The QD sample. The 12-ps laser $π$ pulses (marked in red) resonantly excite the central spin. The excited spin then returns to its ground state by emitting single photons (pink circles). The spin evolves under the joint influence of the external and nuclear magnetic fields. Green arrows represent the randomly distributed nuclear spins. (b) The polarization selection rules for the hole-trion optical transition. $\pm Z$ is the polarization of the exciting laser and the emitted photons. (c) The experimental setup for initializing and probing the central-spin evolution. Liquid-crystal variable retarders (LCVRs) and polarizing beam splitters (PBSs) are used to project the photon polarization. By selecting the polarization of the exciting pulses and that of the projected photons, while setting the time difference between the pulses, one can fully investigate the electron and hole spin evolution.
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Figure 2
Measurements of the trion spin evolution. Polarization-sensitive time-resolved photoluminescence (PL) emission from the positive trion transition for a Zeeman-hyperfine ratio [Eq. (1)] of (a) $r = 0$ , and (b) $r = 11$ . The inset in (a) describes the pulse (red arrows) and detection (magenta markings) sequence used for these measurements. The trion spin is alternately initialized to spin down and up to prevent buildup of the Overhauser field in the sample. (c)–(g) Time-resolved degree of circular polarization $[D_{cp}$ , Eq. (3)] of the PL emission for various $r$ parameters. Red marks represent the measured data and the overlaid blue lines represent our central-spin model best fits [29]. From these multiple fits, we deduce the electron's $g$ factor $g_{e}$ , the interaction energy with the nuclear environment $γ_{e}$ , and the trion's radiative time $τ_{photon}$ , as summarized in Table 1.
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Figure 3
The heavy-hole spin evolution. (a)–(d) Full tomography of the HH spin [40] at time $t = 1$ ns after its initialization to $+ Z$ for a Zeeman-hyperfine ratio $r = 30$ [Eq. (1)]. (a) and (b) are polarization-sensitive time-resolved intensity correlation measurements of the trion emission after the second pulse. The insets describe the polarized pulse sequence (red arrows) and the resulting emission detection (magenta). (c) and (d) describe the degree of circular polarization $[D_{cp}$ , see Eq. (3)] of the two-photon correlations, deduced from the measurements in (a) and (b), respectively. The black lines in (a)–(d) describe the best fitted model [see Supplemental Material (SM) 29] from which the HH spin state $[S_{x}, S_{y}, S_{z}]$ at the second pulse time is extracted [40]. Similar spin tomography measurements are performed for various pulse separation times ( $t$ ) and for different ratios $r$ . (e)–(h) Full tomography of the HH spin evolution is presented for various ratios $r$ . The red, green, and blue marks describe the HH projections $S_{x} (t), S_{y} (t)$ , and $S_{z} (t)$ , respectively, while the black marks describe the spin purity $| S | = \sqrt{S_{x}^{2} + S_{y}^{2} + S_{z}^{2}}$ . The bars represent the measurement uncertainties and the color matched solid lines represent the model fit to the measured data [29]. From these fits, we extract the HH in-plane $g$ factor $g_{h}$ , and the hyperfine-interaction energies $γ_{h_{p}}$ and $γ_{h_{z}}$ (Table 1).
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Figure 4
Model calculated central-spin purity $| S (t) | = \sqrt{S_{x}^{2} (t) + S_{y}^{2} (t) + S_{z}^{2} (t)}$ vs time for (a) the electron, and (b) the heavy hole. The spin purity is calculated using the central-spin model (see SM and Table 1). The spin is initialized in the $z$ direction $[S_{z} (0) = 1]$ and the external field is applied in the $x$ direction. The colored lines represent the purity vs time from initialization for various Zeeman-hyperfine ratios $r$ [Eq. (1)]. The shaded areas represent the uncertainties in the measured values of Table 1. The calculations result from Fig. 2 for the electron and Fig. 3 for the hole. The dashed black line represents the analytic solution $| S (t) | = e^{- \frac{1}{2} {(\frac{γ_{x}}{ℏ} t)}^{2}}$ describing the purity in the limit $r ≫ 1$ , where the spin dephasing time $T_{2}^{*} = \sqrt{2} ℏ / γ_{x}$ is unambiguously defined. In the intermediate regime $r \sim 1, T_{2}^{*}$ is not very well defined and can be regarded as the time it takes the spin purity to decay to 37% of its initial value.
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