Polariton formation in a microcavity embedding a charged

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Polariton formation in a microcavity embedding a charged quantum well Maarten Baeten, Michiel Wouters

Polariton formation in a microcavity embedding a charged quantum well Maarten Baeten, Michiel Wouters TQC, Universiteit Antwerpen, Belgium ICSCE-7 April 21 – April 25, Hakone, Japan Theory of Quantum and Complex systems

Microcavity exciton-polaritons: hybrid photon-exciton modes Ø Microcavity: photon confinement Ø Undoped quantum well: dipole

Microcavity exciton-polaritons: hybrid photon-exciton modes Ø Microcavity: photon confinement Ø Undoped quantum well: dipole carrying excitation, i. e. exciton Eigenmodes of the coupled system: exciton-polaritons Polariton dispersion Many properties have been studied: § BEC and superfluid properties: I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013) Kasprzak, J. , et al. , 2006, Nature (London) 443, 409

Doped semiconductor: many-body effects • Pauli blocking: quantum statistics • Screening of the Coulomb

Doped semiconductor: many-body effects • Pauli blocking: quantum statistics • Screening of the Coulomb interaction Conduction band Fermi level • Mahan (Fermi-edge) singularity: absorption diverges at Fermi edge Valence band • Anderson orthogonality catastrophe: P. Anderson, Phys. Rev. Lett. 18, 1049 (1967) Experiment: Skolnick et al. , Phys. Rev. Lett. 58, 2130 (1987) Theory: G. D. Mahan, Phys. Rev. 153, 882 (1967)

OUTLOOK • `Fermi edge’ polaritons: experimental results • Our theory: Ø 2 DEG optical

OUTLOOK • `Fermi edge’ polaritons: experimental results • Our theory: Ø 2 DEG optical susceptibility: Combescot and Nozières full numerics • Results: linear polariton properties Ø Lineshapes Ø Rabi splitting Ø Effective mass • Conclusion and outlook ky kx kz

Fermi edge polaritons: A. Gabbay et al. , Phys. Rev. Lett. 99, 157402 (2007)

Fermi edge polaritons: A. Gabbay et al. , Phys. Rev. Lett. 99, 157402 (2007) • • • Highly quantum degenerate electron gas No exp. observation of bound excitonic states However, still polariton formation reflection photoluminescence Modification of bare interband matrix element to incorporate effect of FES: ky kx kz A. Gabbay et al. , Phys. Rev. Lett. 99, 157402 (2007) N. S. Averkiev et al. , Phys. Rev. B 76, 045320 (2007)

Theory: 2 DEG optical susceptibility - physical picture Optical interband transition Quiet Fermi sea

Theory: 2 DEG optical susceptibility - physical picture Optical interband transition Quiet Fermi sea Create electron in bound state Create electron in scattering state Perturbed Fermi sea Absorption + + Broadening: Anderson orthogonality Catastrophe (AOC) Energy

2 DEG optical susceptibility: Combescot and Nozières Ø non-interacting electron gas Ø Scattering potential:

2 DEG optical susceptibility: Combescot and Nozières Ø non-interacting electron gas Ø Scattering potential: infinite heavy valence band hole = external single particle potential for the electrons Ground state shift: N electrons adjust their SPW `clear’ the bound state level M. Combescot and P. Nozières, Le Journal de Physique. 32, 913 (1971)

2 DEG optical susceptibility F(t) • Transient behaviour of electrons in presence of the

2 DEG optical susceptibility F(t) • Transient behaviour of electrons in presence of the scattering potential (infinite heavy valence band hole) • Adjustment of SPW leads to decay of overlap in time • CN: Exact result in terms of SPW Optical susceptibility: In the long time limit, CN analytically showed AOC Scattering phase shift at Fermi surface determines exponents

Model: separable potential in k-space • Solid: full numerics • Dashed: • First threshold:

Model: separable potential in k-space • Solid: full numerics • Dashed: • First threshold: singularity in Fourier domain • Delta peak in absorption at zero density (exciton) • Non-zero electron density: asymmetric broadened lineshape: Fermi-edge singularity.

Polariton formation: photon spectral function • Why? Direct access to photonic component of the

Polariton formation: photon spectral function • Why? Direct access to photonic component of the polaritons due to leaky mirrors Photon propagator at normal incidence: CN theory

Rabi splitting : photon in resonance with the singular threshold • High densities: decrease

Rabi splitting : photon in resonance with the singular threshold • High densities: decrease of Rabi splitting for increasing electron density: AOC reduces e-h overlap • Small increase at low densities for large LM-coupling: short time behaviour of optical response becomes important • Full numerics ; asymptotic long times

Photonic Hopfield coefficients (spectral weights) • CLP: integral around the lower polariton energy :

Photonic Hopfield coefficients (spectral weights) • CLP: integral around the lower polariton energy : spectral function sum rule • § § Fermi energy as detuning parameter Cavity mode fixed Dashed line: resonance CLP > 50% at resonance § Photon in resonance for all densities § CLP > 50% § Only for small LM at zero density: 50% mixture of both at resonance D. S. Citrin, and J. B. Khurgin, Phys. Rev. B, 68, 205325 (2003)

Lower polariton effective mass • LP acquires heavier mass for increasing electron density •

Lower polariton effective mass • LP acquires heavier mass for increasing electron density • Faster convergence towards bare photon mass for negative detuning • Two-level system: dashed lines : detuning at normal incidence

Summary § Numerical calculation of 2 DEG optical susceptibility § Effects of the 2

Summary § Numerical calculation of 2 DEG optical susceptibility § Effects of the 2 DEG on the polariton properties: Ø Upper polariton inherits asymetric lineshape from 2 DEG susceptibility Ø Heavier lower polariton mass as compared to bare quantum well Ø Rabi splitting decreases for increasing electron density § Role of the electron spin polarization as compared to Fermi sea spin? § Hole recoil? M. Baeten and M. Wouters, ar. Xiv: 1404. 2048