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Home , Bose gas

condensation, beyond the weakly interacting Bose .

Jonathan Keeling1 N. G. Berloff1, P. R. Eastham1, P. B. Littlewood1, F. M. Marchetti2, M. H. Szyma´nska1 1University of Cambridge, 2University of Oxford

July 24th 2007

J. Keeling, FOPS, July 2007 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas

• Three dimensional

• Non-interacting

• Structureless bosons

• Infinite lifetime

• Solved 80 years ago

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas Polariton Condensate

• Three dimensional • Two dimensions

• Non-interacting bosons

• Structureless bosons

• Infinite lifetime

• Solved 80 years ago

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas Polariton Condensate

• Three dimensional • Two dimensions

• Non-interacting bosons • Coulomb/saturation interactions

• Structureless bosons

• Infinite lifetime

• Solved 80 years ago

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas Polariton Condensate

• Three dimensional • Two dimensions

• Non-interacting bosons • Coulomb/saturation interactions

• Structureless bosons • Tc comparable to ΩR, Ry

• Infinite lifetime

• Solved 80 years ago

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas Polariton Condensate

• Three dimensional • Two dimensions

• Non-interacting bosons • Coulomb/saturation interactions

• Structureless bosons • Tc comparable to ΩR, Ry

• Infinite lifetime • Pumped, decaying

• Solved 80 years ago

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Polariton condensation vs W.I.D.B.G

Non-interacting Bose gas Polariton Condensate

• Three dimensional • Two dimensions

• Non-interacting bosons • Coulomb/saturation interactions

• Structureless bosons • Tc comparable to ΩR, Ry

• Infinite lifetime • Pumped, decaying

• Solved 80 years ago • Open questions remain

J. Keeling, FOPS, July 2007 1 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift ◦ Equilibrium diagram

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift ◦ Equilibrium phase diagram

• Pumping and decay

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift ◦ Equilibrium phase diagram

• Pumping and decay ◦ Relevant energy scales

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift ◦ Equilibrium phase diagram

• Pumping and decay ◦ Relevant energy scales ◦ Lineshape, linewidth

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. Overview

• Polariton internal structure ◦ Polariton blueshift ◦ Equilibrium phase diagram

• Pumping and decay ◦ Relevant energy scales ◦ Lineshape, linewidth ◦ Density profile

J. Keeling, FOPS, July 2007 2 Polariton condensation, beyond the weakly interacting Bose gas. in a disordered Quantum well

Cavity

J. Keeling, FOPS, July 2007 3 Polariton condensation, beyond the weakly interacting Bose gas. Excitons in a disordered Quantum well

Centre-of-mass wavefunction satisfies:

· 2 ¸ ∇R − + V (R) Φα(R) = ²αΦα(R) Cavity Quantum Wells 2mX

J. Keeling, FOPS, July 2007 3 Polariton condensation, beyond the weakly interacting Bose gas. Excitons in a disordered Quantum well

Centre-of-mass wavefunction satisfies:

· 2 ¸ ∇R − + V (R) Φα(R) = ²αΦα(R) Cavity Quantum Wells 2mX

V (R) smoothed by Bohr radius

J. Keeling, FOPS, July 2007 3 Polariton condensation, beyond the weakly interacting Bose gas. Excitons in a disordered Quantum well

Centre-of-mass wavefunction satisfies:

· 2 ¸ ∇R − + V (R) Φα(R) = ²αΦα(R) Cavity Quantum Wells 2mX

V (R) smoothed by exciton Bohr radius

Want distribution of: Energies, ²α

J. Keeling, FOPS, July 2007 3 Polariton condensation, beyond the weakly interacting Bose gas. Excitons in a disordered Quantum well

Centre-of-mass wavefunction satisfies:

· 2 ¸ ∇R − + V (R) Φα(R) = ²αΦα(R) Cavity Quantum Wells 2mX

V (R) smoothed by exciton Bohr radius

Want distribution of: Energies, ²α Oscillator strengths: gα,p ∝ ψ1s(0)Φα,p

J. Keeling, FOPS, July 2007 3 Polariton condensation, beyond the weakly interacting Bose gas. Exciton energies and oscillator strengths

DoS(ε) [arb. units] 2 | p=0 α

|g 2 g (ε,0)

-3 -2 -1 0 1 2 ε -Ex [meV]

[FMM, JK, MHS, PBL cond-mat/0608096]

J. Keeling, FOPS, July 2007 4 Polariton condensation, beyond the weakly interacting Bose gas. Exciton energies and oscillator strengths

θ[degrees] 0 50 60 70 80 2 ε=p2/2m 1 g2(ε,p) ε DoS( ) 0 [arb. units] 2 [mev] | x -1 p=0 - E α ε

|g 2 g (ε,0) -2 -3 -3 -2 -1 0 1 2 0 1 2 3 4 5 6 7 ε p/105 cm-1 -Ex [meV]

[FMM, JK, MHS, PBL cond-mat/0608096]

J. Keeling, FOPS, July 2007 4 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions ◦ WIDBG model: Average interaction

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions ◦ WIDBG model: Average interaction ◦ Our model:

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect)

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position Represent sites as two-level systems (spins):

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position Represent sites as two-level systems (spins):

X † H = ωkψkψk k

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position Represent sites as two-level systems (spins): " # X X † z H = ωkψkψk + ²αSα k α

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Polariton model

• Couple excitons to photons

• Consider exciton-exciton interactions Energy ◦ WIDBG model: Average interaction ◦ Our model: ∗ Strong on-site interaction ∗ Weak inter-site (neglect) Position Represent sites as two-level systems (spins): " # X X X † z 1 + H = ωkψkψk + ²αSα + √ gα,kψkSα + H.c. k α Area k

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 5 Polariton condensation, beyond the weakly interacting Bose gas. Blueshift

10

1

0.1 Energy [meV] 0.01 threshold mean−field δ =5.3meV, kBTeff=17K 0.001 107 108 109 1010 n [cm−2]

J. Keeling, FOPS, July 2007 6 Polariton condensation, beyond the weakly interacting Bose gas. Blueshift

Clean limit: 10 2 2 δELP 'RyXaXn + ΩRaXn 1

0.1 Energy [meV] 0.01 threshold mean−field δ =5.3meV, kBTeff=17K 0.001 107 108 109 1010 n [cm−2]

J. Keeling, FOPS, July 2007 6 Polariton condensation, beyond the weakly interacting Bose gas. Blueshift

Clean limit: 10 2 2 δELP 'RyXaXn + ΩRaXn 1

0.1 Here: 2 2 ΩRaX → ΩRξ Energy [meV] 0.01 threshold mean−field δ =5.3meV, kBTeff=17K 0.001 107 108 109 1010 n [cm−2]

J. Keeling, FOPS, July 2007 6 Polariton condensation, beyond the weakly interacting Bose gas. Blueshift

Clean limit: 10 2 2 δELP 'RyXaXn + ΩRaXn 1

0.1 Here: 2 2 ΩRaX → ΩRξ Energy [meV] 0.01 threshold mean−field For CdTe, ×100 larger δ =5.3meV, kBTeff=17K 0.001 107 108 109 1010 n [cm−2]

J. Keeling, FOPS, July 2007 6 Polariton condensation, beyond the weakly interacting Bose gas. Blueshift

Clean limit: 10 2 2 δELP 'RyXaXn + ΩRaXn 1

0.1 Here: 2 2 ΩRaX → ΩRξ Energy [meV] 0.01 threshold mean−field For CdTe, ×100 larger δ =5.3meV, kBTeff=17K 0.001 Upper polariton: 107 108 109 1010 n [cm−2] 2 2 δEUP 'RyXaXn − ΩRaXn

J. Keeling, FOPS, July 2007 6 Polariton condensation, beyond the weakly interacting Bose gas. Phase diagram

40 non-condensed

30 condensed T

B 20 k

10

0 9 9 0 1×10 2×10 n [cm-2]

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 7 Polariton condensation, beyond the weakly interacting Bose gas. Phase diagram

40 non-condensed 40 30 condensed 30 UP 20 T

B 20 k 10

Energy [meV] Exciton 10 0

-10 LP 0 0 10 20 30 40 0 9 9 1×10 2×10 Angle [degrees] n [cm-2]

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 7 Polariton condensation, beyond the weakly interacting Bose gas. Phase diagram

40 non-condensed 40 30 condensed 30 100 UP Photon 20 T

B 20 k 10 10

Energy [meV] Exciton 10 0

-10 1 7 8 9 10 LP 10 10 10 10 0 0 10 20 30 40 0 9 9 1×10 2×10 Angle [degrees] n [cm-2]

[JK, FMM, MHS, PBL Semicond. Sci. Technol. 22 R1 (2007)]

J. Keeling, FOPS, July 2007 7 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Atoms

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Polaritons 5ps 0.5ps Atoms

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Polaritons 5ps 0.5ps Atoms 10s 10ms

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Linewidth Polaritons 5ps 0.5ps 0.5meV Atoms 10s 10ms

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Linewidth Temperature Polaritons 5ps 0.5ps 0.5meV 20K 2meV Atoms 10s 10ms

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Linewidth Temperature Polaritons 5ps 0.5ps 0.5meV 20K 2meV Atoms 10s 10ms 2.5 × 10−13meV

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Linewidth Temperature Polaritons 5ps 0.5ps 0.5meV 20K 2meV Atoms 10s 10ms 2.5 × 10−13meV 10−8K 10−9meV

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Pumping and decay; energy scales

Lifetime Thermalisation Linewidth Temperature Polaritons 5ps 0.5ps 0.5meV 20K 2meV Atoms 10s 10ms 2.5 × 10−13meV 10−8K 10−9meV Pumping Bath

      2 γ=πΓ Np

  System     g Excitons Cavity mode

κ=πζ2Nζ Bulk photon modes

J. Keeling, FOPS, July 2007 8 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges Gap equation/Ginzburg-Landau:

0 = G−1(ω, k)

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges Gap equation/Ginzburg-Landau:

−1 ∗ 0 = G (ω, k) ' (ω−ωk)

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges Gap equation/Ginzburg-Landau:

−1 ∗ 0 = G (ω, k) ' (ω−ωk)−iα

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges Gap equation/Ginzburg-Landau:

−1 ∗ 0 = G (ω, k) ' (ω−ωk)−iα(µeff−ω)

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

Approaching transition, susceptibility diverges Gap equation/Ginzburg-Landau:

−1 ∗ 0 = G (ω, k) ' (ω−ωk)−iα(µeff−ω)

Linewidth vanishes at transition

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

κ/g=0.02 γ/g=0.3 Approaching transition, susceptibility γ/g=0.5

-1 diverges 10 Gap equation/Ginzburg-Landau:

−1 ∗ -2 0 = G (ω, k) ' (ω−ωk)−iα(µeff−ω) 10 Homogeneous LP linewidth Linewidth vanishes at transition -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 µ Bath , B/g

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Linewidth on approaching transition

2 10 κ/g=0.02 γ/g=0.3 γ/g=0.5 Approaching transition, susceptibility 1 10 diverges -1 10 =0) p 0 Gap equation/Ginzburg-Landau: 10

-1 −1 ∗ 10 -2 0 = G (ω, k) ' (ω−ωk)−iα(µeff−ω) 10 -2 Integrated PL ( Homogeneous LP linewidth 10

Linewidth vanishes at transition -3 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 10 µ Bath chemical potential, B/g

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 9 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become:

ω = c|k|

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become: p ω = − ix ± c2k2 − x2

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become: 0 p momentum ω = − ix ± c2k2 − x2 energy or decay rate Real part (energy) Imaginary part (decay rate)

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become: 0 p momentum ω = − ix ± c2k2 − x2 energy or decay rate Real part (energy) Imaginary part (decay rate)

" # † hψ (r, t)ψ(0, 0)i ' ρ0 exp −

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become: 0 p momentum ω = − ix ± c2k2 − x2 energy or decay rate Real part (energy) Imaginary part (decay rate)

" ( # † η ln(r/ξ) r → ∞, t ' 0 hψ (r, t)ψ(0, 0)i ' ρ0 exp −

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Excitations of decaying condensate; power spectrum

When condensed, poles become: 0 p momentum ω = − ix ± c2k2 − x2 energy or decay rate Real part (energy) Imaginary part (decay rate)

" ( # † η ln(r/ξ) r → ∞, t ' 0 hψ (r, t)ψ(0, 0)i ' ρ0 exp − η 2 2 2 ln(c t/xξ ) r ' 0, t → ∞

J. Keeling, FOPS, July 2007 10 Polariton condensation, beyond the weakly interacting Bose gas. Phase boundary; effect of dephasing

Mean-field theory of pumped decaying system; self-consistent distribution:

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 11 Polariton condensation, beyond the weakly interacting Bose gas. Phase boundary; effect of dephasing

Mean-field theory of pumped decaying system; self-consistent distribution:

κ/g=0.02

0.2

0.1 Temperature/g γ/g=0.10 γ/g=0.20

0 -0.45 -0.4 -0.35 -0.3 Bath chemical potential

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 11 Polariton condensation, beyond the weakly interacting Bose gas. Phase boundary; effect of dephasing

Mean-field theory of pumped decaying system; self-consistent distribution:

κ/g=0.02

0.2 0.2

0.1 0.1 Temperature/g γ/g=0.10 γ/g=0.20

0 0 -0.45 -0.4 -0.35 -0.3 0 0.1 0.2 Bath chemical potential Density

[MHS, JK, PBL Phys. Rev. B 75 195331 (2007)]

J. Keeling, FOPS, July 2007 11 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation:

[JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 ψ t 2m

[JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 + i(γ − Γ|ψ|2 ) ψ t 2m

[JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 + i(γ − Γ|ψ|2 − κ) ψ t 2m

[JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 + i(γ − Γ|ψ|2 − κ) ψ = µ ψ t 2m eff

[JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 + i(γ − Γ|ψ|2 − κ) ψ = µ ψ t 2m eff 2 | ψ

| 30

25

20

Consider V (r) = mω2r2/2 Density, 15 γ−κ =0.7 10 hω¯

5

0 0 2 4 6 8 Radius [JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Effect of pumping on density profile

Gross-Pitaevskii equation: " # ¯h2∇2 i¯h∂ ψ = − + V (r) + U|ψ|2 + i(γ − Γ|ψ|2 − κ) ψ = µ ψ t 2m eff 2 | ψ

| 30 γ−κ 25 hω¯ =2.2

20

Consider V (r) = mω2r2/2 Density, 15 γ−κ =0.7 10 hω¯

5

0 0 2 4 6 8 Radius [JK, NGB arXiv:0706.3686v1 [cond-mat.other]]

J. Keeling, FOPS, July 2007 12 Polariton condensation, beyond the weakly interacting Bose gas. Summary

Differences between polariton condensate and non-interacting Bose gas allow investigation of interesting physics

• Internal structure/disorder

◦ Critical temperature comparable to ΩR, Ry ◦ Exciton disorder affects effective interactions

• Non-equilibrium ◦ Diffusive spectrum / Lineshape ◦ Phase boundary ◦ Persistent supercurrents and density profile

J. Keeling, FOPS, July 2007 13 Polariton condensation, beyond the weakly interacting Bose gas. Supplementary Slides

J. Keeling, FOPS, July 2007 14 Polariton condensation, beyond the weakly interacting Bose gas. Density scales/exciton localisation

θ[degrees] 0 50 60 70 80 40 2 non-condensed ε=p2/2m 1 g2(ε,p)

0

condensed [mev] 30 x -1 - E ε -2

-3 T 0 1 2 3 4 5 6 7 B 20 5 -1

k p/10 cm

5 -1 p=6.3×10 cm ε) 10 DoS(

0 1 2 [arb. units] 2 | p α 2 |g g (ε,p) 0 9 9 0 1×10 2×10 -2 n [cm ] -3 -2 -1 0 1 2 ε -Ex [meV]

J. Keeling, FOPS, July 2007 15 Polariton condensation, beyond the weakly interacting Bose gas. Limit of validity

occupation DoS(ε)/ν Model neglects: 1 • Inter-site Coulomb

• (High energy) multiple occupancy Limits to low density: How low? Limited to states below band-edge 0 -12 -10 -8 -6µ -4 -2 0 2 4 ε -Ex [meV]

J. Keeling, FOPS, July 2007 16 Polariton condensation, beyond the weakly interacting Bose gas. Lineshape in the normal state

-1.5 -1 -0.5 0 0.5 1 3 1 R Im[K1 ] 2 R Re[K1 ] 1 K -iK1 0 0 3

Luminescence Absorption -1 2 Spectral weight γ/g=0.20, µeff Distribution, FS γ/g=0.20, ω* eff -2 γ/g=0.15, µ γ ω 1 Energy of zero (units g) /g=0.15, * γ/g=0.10, µeff Intensity (a.u.) γ/g=0.10, ω* -3 0 -0.6 -0.5 -0.4 -0.3 -1.5 -1 -0.5 0 0.5 1 Bath chemical potential, µ /g Energy (units of g) B

J. Keeling, FOPS, July 2007 17 Polariton condensation, beyond the weakly interacting Bose gas. Finite size, condensate linewidth

Correlations, hψ†(t, r)ψ(0, r)i ∝ exp(−f(t, r, r), with: Z nXmax dω C|ϕ (r)|2(1 − eiωt) f(t, r, r) = − n , 2π [ω2 − (n∆)2]2 + 4ω2x2 n

√x/t E p max p ∆ ¿ x/t ¿ Emax f(t, r, r) ∼ 1 + ln(Emax t/x) ∆ Energy √x/t E p max ¡ ¢ ¡ ¢ πC t x/t ¿ ∆ ¿ Emax f(t, r, r) ∼ 2x 2x ∆ Energy E p max x/t ¿ Emax ¿ ∆ f(t, r, r) ∼ 1 ∆ Energy

J. Keeling, FOPS, July 2007 18 Polariton condensation, beyond the weakly interacting Bose gas. Pumped, Decaying GPE

Rescaling: £ ¤ µψ˜ = −∇2 + r2 + |ψ|2 + i(α − σ|ψ|2) ψ √ iφ Then, writing: ψ = ρe µ/˜ µ˜TF 1.25

1.2

1.15 ∇ · [ρ∇φ] = (α − σρ) 1.1 √ 1.05 2 1 2 2 ∇ ρ µ = |∇φ| + r + ρ − √ 0.95 ρ 0 1 2 3 Pumping, α

J. Keeling, FOPS, July 2007 19