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Bose condensation of excitons, optical coherence and lasers Exciton trap, Butov et al Rb atom condensate, JILA, Colorado Blue Laser, Nakamura, Nichia, Japan Polariton trap, Baumberg et al., Southampton Josephson array, Mooij group, Delft All(?) these objects possess coherent fields. Is there any relationship? 5/24/2017 BC 1 Bose condensation of excitons, optical coherence and lasers The people who did the work….. Paul Eastham Marzena Szymanska Peter Littlewood Cavendish Laboratory University of Cambridge [email protected] Also thanks to Gavin Brown, Alexei Ivanov, Francesca Marchetti, Ben Simons PR Eastham and PB Littlewood, Phys. Rev. B 64, 235101 (2001) M Szymanska and PB Littlewood, PhD thesis cond-mat/0204294 & 0204307 and preprint cond-mat/0204271 Useful background reading: Bose-Einstein Condensation, ed Griffin, Snoke, and Stringari, CUP, (1995) PBLittlewood and XJZhu, Physica Scripta T68, 56 (1996) 5/24/2017 BC 2 Outline • Characteristics of a Bose condensate • Excitons, and why they might be candidates for BEC How do you make a BEC wavefunction based on pairs of fermions? • Some recent experiments • Excitons usually have microscopic dipoles, and hence can decay directly into photons What happens to the photons if the “matter” field is coherent? • Two level systems interacting via photons How do you couple to the environment ? • Decoherence phenomena and the relationship to lasers + - + - + Excitons are the solid state analogue of positronium + - Photon + - Combined excitation is called a polariton These photons have a fixed phase relationship Emission 5/24/2017 Absorption and re-emission Stimulated emission Stimulated Emission > Absorption Laser BC 3 Systems with macroscopic phase coherence • • Dilute atomic gases Superfluid 4He • • Superconductors Lasers – Is this an example of BEC? – Photon condensate?? Photons are massless • Ketterle group, MIT – Non-equilibrium, coupled to the environment, strongly “decohered” Excitons and polaritons – Observation? 5/24/2017 BC 4 Excitons in semiconductors Conduction band (empty) Recombination (slow) Thermal equilibration (fast) H [Ti e Ti h ] [Vijee Vijhh Vijeh ] i i, j 2 i p Ti 2m Optical excitation of electron-hole pairs Valence band (filled) Vij e2 e ri rj At high density - an electron-hole plasma At low density - excitons Exciton - bound electron-hole pair (analogue of hydrogen, positronium) In GaAs, m* ~ 0.1 me , e = 13 Rydberg = 5 meV (13.6 eV for Hydrogen) Bohr radius = 7 nm (0.05 nm for Hydrogen) Measure density in terms of a dimensionless parameter rs - average spacing between excitons in units of aBohr 1=n = 4ùa3 r 3 3 5/24/2017 BC Bohr s 5 Some experimental systems Must have long-lived excitons to allow thermalisation • Cu2O - long-lived optically excited excitons (dipole-forbidden) – anomalous transport [Fortin et al PRL 70, 2951 (1993)] & luminescence [Lin and Wolfe, PRL 71, 1222 (1993)] – dominated by Auger recombination [O’Hara and Wolfe PRB 62 12909 (2000)] • Biexcitons in CuCl - analogue of H2 – • Chase et al, PRL 42, 1231 (1979); Hasuo et al PRL 70, 1303 (1992) Double quantum well - keep electrons and holes physically apart – Conduction band Optical excitation in double wells [Fukuzawa et al, PRL 64, 3066 (1990); Kash et al, PRL 66, 2247 (1990)] – Indirect G-X exciton at GaAs/AlAs interface [Butov et al, PRL 73, 301 (1992), PRL (2001), Nature (2002)] – Valence band Separately gated electron and hole layers [Sivan et al, PRL 68, 1196 (1992)] – Type II quantum wells (artificial 2D semimetal) [Lakrimi et al, PRL 79, 3034 (1997)] • Position Optical microcavities – 5/24/2017 [Pau et al, PRA 54, 1789 (1996); Senellart and Bloch, PRL 82, 1233 (1999); Le Si Dang et al. PRL 81, 3920 (1998); Stevenson et al., PRL 2000 ] BC 6 • Bose-Einstein condensation R D (ï ) Macroscopic ground state occupation n = dï eì ( ï à ö) à 1 finit e as ö ! Density of states D(e) e 0 Thermal occupation T T>T0 e m/kT T<T0 T0 n(e) k B T0 = 1:3 r 2s Ryd: / n 2=3 m • Macroscopic phase coherence Condensate described by macroscopic wave function y eif which arises from interactions between particles y -> y eif Genuine symmetry breaking, distinct from BEC • Superfluidity Implies linear Goldstone mode in an infinite system with dispersion w = vs k and hence a superfluid stiffness vs 5/24/2017 BC 7 Excitonic insulator A dilute Bose gas should condense - generalisation to dense electronhole system is usually called an excitonic insulator eck Single exciton wavefunction (q 0) fk ack avk 0 ( fk is Fourier transform of k hydrogenic wavefunction) This is not a boson! k+q evk eõ k P y k þ ka cka vk j0 > ? hP i y þ a a k k ck vk N j0 > ? Coherent wavefunction for condensate in analogy to BCS theory of superconductivity BCS uk vk a avk 0 ; ck uk vk 1 2 2 [Keldysh and Kopaev 1964] k v , ku k variational solutions of H = K.E. + Coulomb interaction Same wavefunction can describe a Bose condensate of excitons at low density, as well as two overlapping Fermi liquids of electrons and holes at high density 5/24/2017 BC 8 Mean field theory of excitonic insulator BCS uk vk a avk 0 ck k BCS-like instability of Fermi surfaces Bose condensation of excitons 1 Low density nk ~ n1/2fk Efc m Efv High density nk nk ~ Q(k-kf) 0 kf Special features: order parameter; gap ack avk uk vk ( k / 2 Ek ); 5/24/2017 BC Ek (e k m ) 2 2k 9 Excitation spectra +(-)Ek is energy to add (remove) particle-hole pair from condensate (total momentum zero) Ek Band energy (e k m ) 2 2k Chemical potential (<0 for bound exciton) Low density m<0 Chemical potential below band edge High density m>0 No bound exciton below band edge Ek -m m 5/24/2017 Ek k Correlation energy Absorption m k k Emission kf k m BC 10 Coherent light emission and superfluidity • Order parameter <ck+vk>eimt oscillating dipole in III-V semiconductor at optical frequency - an antenna • Not a structureless boson -- coherent light emission from the condensate Phase fixing and superfluidity Order parameter has a phase eif collective mode which produces a superfluid stiffness When coupled to photon mode in a cavity, phase entrained to cavity mode -- coherent but not superfluid Same physics arises in electron-hole layers with tunnelling between layers Only if the electrons and holes are not allowed to recombine does the condensate have a gapless phase mode (Biexcitons OK [Kohn and Sherrington 1968]) Excitonic insulator is just a species of commensurate charge density wave Must go back to a model that has coupled photons and excitons right from the start - polaritons 5/24/2017 BC 11 • • Correct linear excitations about the ground state are mixed modes of excitonic polarisation and light - polaritons Optical microcavities allow one to confine the optical modes and control the interactions with the electronic polarisation – – – – small spheres of e.g. glass planar microcavities in semiconductors excitons may be localised - e.g. as 2-level systems in rare earth ions in glass RF coupled Josephson junctions in a microwave cavity Photon Upper polariton Exciton Lower polariton Frequency Optical microcavities and polaritons Wavevector k// 5/24/2017 BC 12 Microcavity polaritons A simplified model - the excitons are localised and replaced by 2-level systems and coupled to a single optical mode in the microcavity Density of states Conduction band Energy w N Localized excitons Cavity mode of light Valence band ( H e i bi bi ai ai 2-level system b a i wy y photon g Dipole coupling (bi aiy y ai bi ) N i Fermionic representation - ai creates valence hole, b+i creates conduction electron on site i Photon mode couples equally to large number N of excitons since l >> aBohr R.H. Dicke, Phys.Rev.93,99 (1954) K.Hepp and E.Lieb, Ann.Phys.(NY) 76, 360 (1973) 5/24/2017 BC 13 Localized excitons in a microcavity - the Dicke model ( H e i bi bi ai ai wy y i g N ( b a y y ai bi ) i i i Excitation number (excitons + photons) conserved ( L y y 12 bibi ai ai i Variational wavefunction (BCS-like) is exact in the limit N , L/N const. (easiest to show with coherent state path integral and 1/N expansion) l , u, v e ly v b i i ui ai 0 i Two coupled order parameters Coherent photon field Exciton condensate Excitation spectrum has a gap 5/24/2017 P y i < ayibi > PR Eastham, 2000, 2001 BC 14 Condensation in the Dicke model (g/T = 2) Increasing excitation density úex = hL i N à 1 2 Upper polariton Excitation energies (condensed state) Lower polariton Chemical potential (condensed state) Chemical potential (normal state) 5/24/2017 BC 15 Excitation spectrum with inhomogeneous broadening 5/24/2017 BC 16 Decoherence Collisions and other decoherence processes Decay of cavity photon field Pumping of excitons Decay, pumping, and collisions may introduce “decoherence” loosely, lifetimes for the elementary excitations include this by coupling to bosonic “baths” of other excitations ck õ õ P P in analogy to superconductivity, the external fields may couple in a way that is “pair-breaking” or “non-pair-breaking” y y y à (b b a a )(c + ck ) i i i ;k i i k non-pairbreaking (inhomogeneous distribution of levels) y y y (b b + a a )(c + ck ) i i i ;k i i k pairbreaking disorder • Conventional theory of the laser assumes that the external fields give rise to rapid decay of the excitonic polarisation - incorrect if the exciton and photon are strongly coupled • Correct theory is familiar from superconductivity - Abrikosov-Gorkov theory of superconductors with magnetic impurities • Here consider the limit where photon escape rate is vanishingly small - maintain thermal equilibrium 5/24/2017 BC 17 Phase diagram of Dicke model with pairbreaking Pairbreaking characterised by a single parameter g l2N Dotted lines - non-pairbreaking disorder Solid lines - pairbreaking disorder 0 (wm/g 5/24/2017 MH Szymanska 2002 BC 18 This “laser” is indeed an example of BEC Matter/Light ratio 0 0.5 í =g 1.0 1.5 Large decoherence -- “laser” • order parameter nearly photon like • electronic excitations have short lifetime • excitation spectrum gapless real lasers are often far from equilibrium, however í =g 5/24/2017 Small decoherence -- BEC of polaritons • order parameter mixed exciton/photon • excitation spectrum has a gap BC 19 Physical systems • • • • • • • Rough estimate of g ~ 1/T2 , which grows ~ linearly with nexciton in the normal state Need ~ g exp (-1/N(0)V) > g , where N(0) set by inhomogeneous broadening (assuming low nexciton limit) CdTe quantum wells g = 29 meV ; at n = 0.05, measured 1/T2 = 2.5 meV ; – no inhomogeneous broadening = 20 meV – with s = 6 meV reduced = 2 meV Organics? g ~ 80 meV ; T2 ?? Solid state laser materials? Dilute atomic gases (microwave cavity) ? dephasing rate ~ 10-3 x dipole coupling (reduce further by starting with atomic BEC) Undoubtedly more serious problem is to have a thermalised equilibrium system; high quality cavities; more theory needed to model non-equilibrium effects 5/24/2017 BC 20 Conclusions • Exciton condensation How do you make a BEC wavefunction based on pairs of fermions? BCS But be careful, because the order parameter phase corresponds to an “internal degree of freedom” (dipole) which couples to light. If we deal with a closed system - a cavity where the photons don’t escape - then the modes are gapped, and it is not a superfluid. • Coupled excitons and photons - polaritons What happens to the light field if the “matter” field is coherent? Still BCS Two order parameters have phases that are entrained. In the low density regime, this “looks like” BEC of polaritons. • Open systems How do you treat coupling to the environment? BCS + pairbreaking (AG) Weak pairbreaking, gap is robust, and BEC persists. Strong pairbreaking, gap closes, order parameter becomes almost entirely photon-like No fundamental distinction between BEC of polaritons and a laser. ? Not yet any experimental evidence for spectral structure that would indicate proximity to the BEC limit…. 5/24/2017 BC 21 Current projects • Quantum Monte Carlo calculations of T=0 phase diagram of excitonic matter – exciton, biexciton solids and liquids, plasma phases, role of me/mh Gavin Brown (PhD Student), Richard Needs, Cambridge • Mesoscopic physics of (nearly) coherent exciton and exciton-polariton systems – – – – effect of disorder, pairbreaking perturbations non-equilibrium pumping and dynamics luminescence spectra finite-dimensional, multimode and fluctuation effects Paul Eastham, Francesca Marchetti, Marzena Szymanska, Ben Simons (Cambridge) • Semiconductor microcavities – condensate, laser, or parametric oscillator? Paul Eastham (Cambridge), Jeremy Baumberg (Southampton), David Whittaker (Sheffield) • Resonant acousto-optics Alex Ivanov (Cardiff) + incipient experimental programme in Cardiff/Cambridge 5/24/2017 BC 22