Photonic Crystal Architecture for Room-Temperature Equilibrium Bose-Einstein Condensation of Exciton Polaritons

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Photonic Crystal Architecture for Room-Temperature Equilibrium Bose-Einstein Condensation of Exciton Polaritons

Jian-Hua Jiang and Sajeev John

Phys. Rev. X 4, 031025 – Published 13 August 2014

Abstract

We describe photonic crystal microcavities with very strong light-matter interaction to realize room-temperature, equilibrium, exciton-polariton Bose-Einstein condensation (BEC). This goal is achieved through a careful balance between strong light trapping in a photonic band gap (PBG) and large exciton density enabled by a multiple quantum-well (QW) structure with a moderate dielectric constant. This approach enables the formation of a long-lived, dense $10 - μ m - 1 - cm$ - scale cloud of exciton polaritons with vacuum Rabi splitting that is roughly 7% of the bare exciton-recombination energy. We introduce a woodpile photonic crystal made of ${Cd}_{0.6} {Mg}_{0.4} Te$ with a 3D PBG of 9.2% (gap-to-central-frequency ratio) that strongly focuses a planar guided optical field on CdTe QWs in the cavity. For 3-nm QWs with 5-nm barrier width, the exciton-photon coupling can be as large as $ℏ Ω = 55 meV$ (i.e., a vacuum Rabi splitting of $2 ℏ Ω = 110 meV$ ). The exciton-recombination energy of 1.65 eV corresponds to an optical wavelength of 750 nm. For $N = 106$ QWs embedded in the cavity, the collective exciton-photon coupling per QW ( $ℏ Ω / \sqrt{N} = 5.4 meV$ ) is much larger than the state-of-the-art value of 3.3 meV, for the CdTe Fabry-Pérot microcavity. The maximum BEC temperature is limited by the depth of the dispersion minimum for the lower polariton branch, over which the polariton has a small effective mass of approximately $1 0^{- 5} m_{0}$ , where $m_{0}$ is the electron mass in vacuum. By detuning the bare exciton-recombination energy above the planar guided optical mode, a larger dispersion depth is achieved, enabling room-temperature BEC. The BEC transition temperature ranges as high as 500 K when the polariton density per QW is increased to $(11 a_{B})^{- 2}$ , where $a_{B} ≃ 3.5 nm$ is the exciton Bohr radius and the exciton-cavity detuning is increased to 30 meV. A high-quality PBG can suppress exciton radiative decay and enhance the polariton lifetime to beyond 150 ps at room temperature, sufficient for thermal equilibrium BEC.

Received 9 April 2014

DOI:https://doi.org/10.1103/PhysRevX.4.031025

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Jian-Hua Jiang¹ and Sajeev John^1,2

¹Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
²Department of Physics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Popular Summary

Light-matter interactions give rise to the colors that we see in daily life. If an atom is placed in a material that traps light, the photons will be emitted, reflected, and reabsorbed, again and again, if the material itself does not absorb the light. Quantum mechanics tells us that in this case, a single photon and the atomic excitation form a quantum superposition. In the past decades, such a “polariton” has been realized in semiconductor microcavities. Three-dimensional photonic band-gap microcavities enable complete trapping of light propagating in all directions and focus light to nearly 20 times the intensity of a Fabry-Pérot microcavity. We describe such a microcavity using CdTe (a material that balances optical and electronic properties) and achieve room-temperature, equilibrium Bose-Einstein condensation.

In a Fabry-Pérot microcavity, Bragg mirrors focus light on a quantum well, where excitons reside, to enhance the exciton-photon interaction. The strong light-matter interaction induces a sharp minimum in the energy-momentum dispersion of the polariton, and it behaves like a particle with a mass 9 orders of magnitude smaller than an atom. This interaction has been utilized to realize polariton Bose-Einstein condensation, for a trillionth of a second, at low temperature. However, light leaks out from the Fabry-Pérot microcavity in directions parallel to the Bragg mirrors, resulting in a short polariton lifetime and the nonequilibrium nature of the polariton condensate. We describe a new microcavity using CdTe, based on a woodpile architecture; CdTe was chosen for its balance of light-confining ability and high exciton density. We find that with the stronger and complete light trapping by the photonic band gap, the exciton-photon interaction is considerably enhanced, reaching a collective vacuum Rabi splitting over 100 meV. This state enables room-temperature, equilibrium Bose-Einstein condensation of a cloud of long-lived polaritons. The transition temperature to the Bose-Einstein state even increases to about 500 K when the polariton density per quantum well is increased.

Our finding that equilibrium Bose-Einstein condensates can be achieved at room temperature opens up broad applications in which it is impractical or difficult to maintain cryogenic temperatures. Long-lived polariton condensates represent a coherent light source with novel quantum correlations between photons, useful, for instance, in a futuristic quantum internet.

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Vol. 4, Iss. 3 — July - September 2014

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Figure 1
Photonic crystal architecture and photonic band dispersion. (a) A slab layer sandwiched between two woodpile photonic crystals induces 2D photonic bands (“guided modes”) in the 3D PBG with $δ ω / ω_{c} = 9.2 %$ , where $δ ω$ is the band-gap width and $ω_{c}$ is the PBG central frequency. (b) QWs are grown in the slab and the eight nearby layers of rods. Typically, $a ≃ 360 nm$ , $w = 0.25 a ≃ 90 nm$ , $h = 0.3 a ≃ 110 nm$ , and $b = 0.06 a ≃ 20 nm$ . There are about 100 QWs with the same width. There are two QWs in the slab layer and about 12 QWs in each rod. The QW widths are chosen to be 3, 4, 5, and 6 nm for comparison, while the width of the barrier layers between QWs is fixed to be 5 nm. QWs are made of CdTe (dielectric constant 8.5), while the barriers and photonic crystals are made of ${Cd}_{0.6} {Mg}_{0.4} Te$ (dielectric constant 7.5). In the regions with QWs, the dielectric constant is taken to be 8 (average of the two materials). (c),(d) Dispersion of the 2D guided modes (dashed green and solid red curves) and the bulk 3D photonic bands (shaded regions). The first Brillouin zone of the 2D guided modes is $- \frac{π}{a} \leq q_{x}, q_{y} < \frac{π}{a}$ . The lowest 2D photonic band [the solid curves in (c) and (d)] is coupled with excitons [dotted curve in (d)] in QWs. The lower and upper polariton branches [solid blue curves in (d)] are depicted for exciton recombination in resonance with the lowest 2D guided mode. In this case, each QW width is 4 nm, the exciton-recombination energy is $E_{X 0} = 1.59 eV$ , and the photonic lattice constant is chosen to be $a = 360 nm$ . The exciton-photon-coupling strength and the on-resonance lower-polariton dispersion depth is 48 meV. Larger dispersion depth and higher BEC temperature are achieved by detuning the exciton-recombination energy above the lowest 2D guided mode.
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Figure 2
Dispersion of confined photonic and polaritonic bands in 2D wave-vector space. (a) Dispersion of the lowest confined photonic band in the first Brillouin zone $- \frac{π}{a} \leq q_{x}, q_{y} < \frac{π}{a}$ . The lattice constant is $a = 365 nm$ . (b) Dispersion of the lower polariton branch in the same wave-vector region. The QW width is 4 nm. The lower band edge (1.56 eV) of guided 2D modes within the 3D PBG is 30 meV below the exciton-recombination energy ( $E_{X 0} = 1.59 eV$ ). The exciton-photon coupling of 48 meV involves the collective response of 100 QWs.
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Figure 3
Very strong light-matter interaction in the photonic band-gap 2D guided mode. (a) Exciton-photon collective interaction strength $ℏ Ω$ as a function of QW width for a fixed barrier width of 5 nm. The horizontal axis spans from $N = 106$ to 83 QWs. Inset: Exciton-recombination energy $E_{X 0}$ . The lowest 2D photonic band edge $ℏ ω_{0}$ is kept in resonance with the exciton-recombination energy $E_{X 0}$ as the QW width is varied. The barrier layers between QWs are made of ${Cd}_{0.6} {Mg}_{0.4} Te$ . (b) As $E_{X 0}$ varies with the QW width, the photonic crystal lattice constant $a$ is adjusted (red curve) to keep $ℏ ω_{0} = E_{X 0}$ . Since $a$ changes, the total number $N$ of QWs (blue curve) in the slab and rods also changes. (c) Exciton-binding energy and Bohr radius for different QW widths calculated using an effective mass approximation [45] (see the Appendix). (d) Average electric field-intensity $S^{- 1} \int d \vec{ρ} | \vec{E} (\vec{ρ}, z) |^{2}$ profile at the 2D guided mode edge in the PBG cavity (red curve) compared to that in the ${Cd}_{0.4} {Mg}_{0.6} Te − {Cd}_{0.8} {Mn}_{0.2} Te$ Fabry-Pérot cavity [44] (green curve) along the $z$ direction. The photons in both cavities have a band-edge energy of 1.59 eV. The photonic field intensity in the PBG cavity is multiplied by a factor of 0.1 in the figure. The Fabry-Pérot cavity is a $λ / 2$ cavity with $λ = 780 nm$ . The lattice constant of the photonic crystal is $a = 360 nm$ . (e) Field-intensity $| \vec{E} |^{2}$ profile in the $x − y$ plane (in one unit cell of photonic crystal) at $z = 0$ . Here, the averaged intensity is 1. The $x$ and $y$ coordinates are in units of the photonic lattice constant $a$ .
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Figure 4
Polariton population on the ground state and other low-energy states for the exciton-box trap with $D = 10 μ m$ , an exciton-photon coupling of $ℏ Ω = 48 meV$ , and an exciton-recombination energy of $E_{X 0} = 1.59 eV$ . The QW width is 4 nm, and the barrier-layer width is 5 nm. The detuning is (a) $Δ = 30 meV$ and (b) $Δ = 40 meV$ . (a) Population of the ground state (GS) vs polariton density (solid curve). The rapid, nonlinear increase of the GS population indicates the BEC transition. The GS population fraction (dashed curve), i.e., the GS population divided by the total population, is of order unity for densities beyond the BEC transition. (b) GS population fraction (dashed curves) and the population fraction on the low-energy discrete states (solid curves) below the continuum exciton-recombination energy $E_{X 0}$ . (There are more than 1000 such states.) These low-energy states are photonlike states with very small effective mass, whereas states higher than $E_{X 0}$ are excitonlike. When a large population fraction resides in the photonlike states, the BEC threshold density is reduced. For $T = 100 K$ (blue curves), populated polariton states are mostly photonlike. The threshold density is much lower than for $T = 300 K$ (brown curves), where the population fraction on low-energy states is negligible. (c) The dispersion of polaritons (solid red curves), bare excitons (dashed green curve), and pure photons (dash-dotted blue curve) for detuning $Δ = 40 meV$ with $q_{y} = π / a = 8.6 μ m^{- 1}$ for $a = 370 nm$ . The dispersion depth of the lower polariton branch $V$ is also depicted. (The dotted black line denotes the lower polariton band edge.) Inset: Excitonic fraction of the lower polariton branch as a function of detuning $Δ$ . (d) The dispersion depth $V$ (solid red curve) and polariton effective mass $m_{pl}$ (dashed green curve) of the lower polariton branch vs the detuning $Δ$ .
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Figure 5
Polariton BEC transition temperature $T_{c}$ at fixed densities. (a) Transition temperature $T_{c}$ with detuning $Δ = 30 meV$ for different QW widths (with QWs filling an entire active region) at fixed polariton densities: $1.5 \times 1 0^{4} μ m^{- 2}$ (curve with solid circles) and $1.9 \times 1 0^{3} μ m^{- 2}$ (curve with solid squares). For the higher total density, the exciton density in each QW is below $(9 a_{B})^{- 2}$ . For the lower total density, the exciton density in each QW is below $(26 a_{B})^{- 2}$ . (b) Dependence of the transition temperature $T_{c}$ on the detuning $Δ$ and the exciton-trap size $D$ when the exciton-photon coupling is $ℏ Ω = 48 meV$ (QW width 4 nm and barrier width 5 nm) and the polariton density is $3 \times 1 0^{3} μ m^{- 2}$ . (c) Dependence of the transition temperature $T_{c}$ on the detuning $Δ$ and the exciton-photon coupling $ℏ Ω$ when the exciton-trap size is $D = 10 μ m$ and the polariton density is $3 \times 1 0^{3} μ m^{- 2}$ .
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Figure 6
BEC transition temperature $T_{c}$ as a function of total polariton density and (a) box-trap size, (b) exciton-photon detuning, and (c) exciton-photon-coupling strength. (d) Root-mean-square deviation of the vacuum Rabi splitting (diamonds) and lower-polariton dispersion depth (circles) as a function of exciton inhomogeneous broadening $δ E_{X 0}$ . In (a) and (b), the exciton-photon coupling is $ℏ Ω = 48 meV$ (QW width 4 nm and barrier width 5 nm). For (a) and (c), the detuning is 30 meV. For (b) and (c), the exciton-box-trap size is $10 μ m$ . The exciton-recombination energy is $E_{X 0} = 1.59 eV$ . For (d), the detuning is $Δ = 0$ . The dashed curve in (d) depicts the value of $δ E_{X 0}$ itself for reference.
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