Coherent Three-Level Mixing in an Electronic Quantum Dot

C. Payette, G. Yu, J. A. Gupta, D. G. Austing, S. V. Nair, B. Partoens, S. Amaha, and S. Tarucha

Phys. Rev. Lett. 102, 026808 – Published 16 January 2009

Abstract

We observe magnetic-field-induced level mixing and quantum superposition phenomena between three approaching single-particle states in a quantum dot probed via the ground state of an adjacent quantum dot by single-electron resonant tunneling. The mixing is attributed to anisotropy and anharmonicity in realistic dot confining potentials. The pronounced anticrossing and transfer of strengths (both enhancement and suppression) between resonances can be understood with a simple coherent level mixing model. Superposition can lead to the formation of a dark state by complete cancellation of an otherwise strong resonance, an effect resembling coherent population trapping in a three-level-system of quantum and atom optics.

Received 15 August 2008

DOI:https://doi.org/10.1103/PhysRevLett.102.026808

Authors & Affiliations

C. Payette^1,2, G. Yu¹, J. A. Gupta¹, D. G. Austing^1,2,*, S. V. Nair³, B. Partoens⁴, S. Amaha⁵, and S. Tarucha^5,6

¹Institute for Microstructural Sciences M50, NRC, Montreal Road, Ottawa, Ontario, K1A 0R6, Canada
²Department of Physics, McGill University, 3600 rue University, Montréal, Quebec, H3A 2T8, Canada
³Center for Advanced Nanotechnology, University of Toronto, 170 College Street, Toronto, Ontario, M5S 3E3, Canada
⁴Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
⁵Quantum Spin Information Project, ICORP, JST, Atsugi-shi, Kanagawa 243-0198, Japan
⁶Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

^*guy.austing@nrc-cnrc.gc.ca

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Vol. 102, Iss. 2 — 16 January 2009

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Figure 1
Measurement principle: energy separation between the $1 s$ state in the emitter dot and mixed states ( $a$ , $b$ , $c$ ) in the collector dot is a function of ( $V_{s d}$ , $V_{g}$ ) ( $1 s$ - $b$ resonance condition shown). Empty nonresonant states below ( $a$ , $b$ , $c$ ) omitted. Micrograph is of mesa similar to measured mesa. Lowest three exact three-level-crossings are circled in the Fock-Darwin spectrum for circular parabolic dot. Energy of states is with respect to energy of the $1 s$ state. States that merge into the same Landau level at high $B$ field are colored the same.Reuse & Permissions
Figure 2
(a) Differential conductance resonance position versus $B$ field maps out the high energy spectrum of the probed dot at 0.3 K [15, 16]. (b) Calculated spectrum with confinement strengths $ℏ ω_{x} = 6.1 meV$ and $ℏ ω_{y} = 4.6 meV$ ( $δ = ω_{x} / ω_{y} \sim 4 / 3$ ) reproduces a well the measured spectrum except in the vicinity of level crossings. Energy of the ( $n_{x}$ , $n_{y}$ ) state is with respect to the energy of the (0,0) ground state ( $n_{x}$ and $n_{y}$ are $x$ - and $y$ - quantum numbers [13]). Relevant states are labeled with atomic orbital-like notation for circular dot. (c) Selected current traces (vertical sections) through the $γ$ and $τ$ crossings from the boxed regions in panel (a) [nonresonant component not subtracted]. (d) Basic types of three-level crossings. Dimensionless couplings are set to $C_{12} = C_{23} = C_{13} = 0$ ; $C_{12} = 1$ , $C_{23} = C_{13} = 0$ ; $C_{12} = C_{23} = 1$ , $C_{13} = 0$ ; and $C_{12} = - 1$ , $C_{23} = C_{13} = 1$ .Reuse & Permissions
Figure 3
Three-level-crossing $γ$ . (a) Energy level (differential conductance resonance) positions. (b) Extracted branch currents (resonance current with nonresonant component subtracted). (c) Fit of energy level positions: couplings (in meV) $C_{12} = 0.30$ , $C_{23} = 0.29$ , and $C_{13} = 0.03$ . Lines estimating position of uncoupled basis levels provide a guide to the eye. (d) Fit of currents. Resonances through uncoupled basis states have amplitudes (in $p A^{1 / 2}$ ) $s_{1} = - 0.4$ , $s_{2} = 0.804$ , and $s_{3} = 0.219$ at 1.8 T, and $s_{1} = - 0.213$ , $s_{2} = 1.22$ , and $s_{3} = 0.209$ at 2.5 T. (e) Reconstructed eigenvectors. $Δ B = 0 T$ is at $\sim 2.12 T$ .Reuse & Permissions
Figure 4
Calculations reproducing correct general behavior at $γ$ crossing in Fig. 3. (a) Energy level position, and (b) overlap integral squared, proportional to resonance current in coherent tunneling picture, $| ⟨ g | ψ_{j} ⟩ |^{2}$ , for each branch. Parameters: Probed dot’s $V_{eff} (x, y) = 1 / 2 [(4 / 3) x^{2} + y^{2} + 0.12 x^{2} y + 0.05 x y^{2} + 0.05 y^{3} + 0.01 x^{6} + 0.01 y^{6}$ ], $δ = 4 / 3$ and $ℏ ω_{y} = 3.8 meV$ . Emitter dot potential is the same but 3rd and 6th degree terms are excluded ( $1 s$ -like prober state is least sensitive to addition of perturbation terms).Reuse & Permissions

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