Interacting Exciton-Polaritons in Cylindric Micropillars

Yury S. Krivosenko, Ivan V. Iorsh, and Ivan A. Shelykh

Phys. Rev. Applied 19, 044008 – Published 4 April 2023

Abstract

We present a quantitative microscopic analysis of the formation of exciton-polaritons, the composite particles possessing light and material components, polariton-polariton interactions, and resonant pumping dynamics in cylindrical semiconductor micropillars. We obtain that, due to the exciton-exciton interaction, the polaritonic states’ populations become redistributed among pumped and pristine states. We discuss how the redistribution effect can be used in devices generating photons with nonzero orbital angular momenta.

Received 18 October 2022
Revised 9 February 2023
Accepted 23 February 2023

DOI:https://doi.org/10.1103/PhysRevApplied.19.044008

Physics Subject Headings (PhySH)

Exciton polariton Integrated optics Optical quantum information processing

Quantum cavities Quantum wells Semiconductors

Monte Carlo methods Second quantization

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Yury S. Krivosenko ^1,*, Ivan V. Iorsh¹, and Ivan A. Shelykh^1,2

¹School of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia
²Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland

^*y.krivosenko@gmail.com

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Vol. 19, Iss. 4 — April 2023

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Images

Figure 1
Left panel: sketch of the considered geometry, consisting of a semiconductor micropillar (gray cylinder) flanked by a pair of dielectric Bragg mirrors (DBRs, the orange planes), and containing a quantum well (QW, the green circle in the middle) placed in the position of an antinode of an electric field of the optical cavity mode. The energy of the cavity mode is brought close to the resonance to the energy of the excitonic transition, giving rise to the formation of the polaritons. Right panel: top view of the system illustrating the scheme of the exciton-exciton scattering. Excitons in the states $| λ_{1} ⟩$ , $| λ_{2} ⟩$ , $| λ_{1} + Δ λ ⟩$ , and $| λ_{2} - Δ λ ⟩$ (the red rounds) are coupled via interaction Hamiltonian ${\hat{H}}_{x x}^{}$ (the white round). Excitonic wave functions $λ_{1} = λ_{2} = Δ λ = 1$ and $ν_{1, 2, 3, 4} = 1$ are schematically shown by purple surfaces next to the states’ markers.
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Figure 2
Photonic target functions (A2) and hybridization schemes of exciton-polariton formation for $l = 0, \pm 1, \pm 2$ angular quantum numbers. The vertical line at each PTF part shows the zero-PTF axis, the gray rounds and x markers—the PTF roots and lower photonic cutoff energies, respectively, the horizontal lines—the energy levels of photons (left), excitons (right), and polaritons (mid). The slant connection-line transparencies indicate the corresponding Hopfield coefficients (HCs) absolute values (the lower the value of the HC, the more the transparency). The open gray rectangles highlight the areas shown in Fig. 3. The micropillar is made of $GaAs$ and has the radius $R = 1.5 μ m$ , its effective height ( $134.5 nm$ ). The two-dimensional (2D) Rabi splitting is $3.2 meV$ .
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Figure 3
The lower polaritons’ formation scheme (for the states around the excitonic resonance, the magnified section of Fig. 2). The states $P A_{1 l}$ , $P B_{1 l}$ , and $P C_{1 l}$ are the lowest polaritonic states with $l = 0$ , $\pm 1$ , and $\pm 2$ , respectively. The $P B_{1 u}$ state is the counterpart to the $P B_{1 l}$ one: photonic mode $B_{1}$ splits into them almost equally. The percentage fractions present the absolute values of the corresponding Hopfield coefficients, squared.
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Figure 4
(a) The real parts of exciton-exciton, $V_{κ_{1} κ_{2} κ_{3} κ_{4}}$ , and (b) the absolute values of polariton-polariton, $W_{ϰ_{1} ϰ_{2} ϰ_{3} ϰ_{4}}$ , interaction matrix elements, and (c) their difference, $| V | - | W |$ . All the $V s$ and $W s$ are measured in $μ eV$ . The states examined are $κ_{1} = ϰ_{1} = (1, n_{1})$ , $κ_{2} = ϰ_{2} = (1, n_{2})$ , $κ_{3} = ϰ_{3} = (2, n_{2})$ , $κ_{4} = ϰ_{4} = (0, n_{1})$ , with integers $n_{1}, n_{2} = 1, \dots 5, 11, \dots 15$ . The essence of the color schemes used for all the panels is common: the associated absolute values grow with the increase in the color saturation. The horizontal color bars show the scales.
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Figure 5
(a)–(c) The population dynamics of the system consistent of the five polaritonic states: ${| c_{0} (t) |}^{2}$ (blue lines), ${| c_{1} (t) |}^{2} + {| c_{- 1} (t) |}^{2}$ (red lines), and ${| c_{2} (t) |}^{2} + {| c_{- 2} (t) |}^{2}$ (green lines), where the indices stand for the angular momenta. The lowest polaritonic modes ( $n_{l} = 1$ ) are taken for all $l$ . The time axis is broken to show both the transient period and the periodic behavior. (d) The phase trajectory (the limit cycle here; black solid line) for the later times [right parts in (c)]. The states with (a) $l = \pm 1, \pm 2$ , and (b, c, d) $l = 0, \pm 1$ are coherently pumped, which is marked by (p) in the legends. In the case (c), the $| 1 ⟩$ and $| - 1 ⟩$ states are pumped in counterphase. The semitransparent red curves in (d) are the trajectory projections onto the coordinate planes.
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