Polariton Bose-Einstein condensation: A theoretical...

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008 Polariton Bose-Einstein condensation: A theoretical overview Institute of Theoretical Physics Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland Vincenzo Savona Outlook Lecture 1: Overview of symmetry breaking field theory of a Bose condensate Lecture 2: The gas of interacting polaritons: BEC-relevant aspects Lecture 3: Hartree-Fock-Popov theory of a polariton condensate + Boltzmann kinetics

Transcript of Polariton Bose-Einstein condensation: A theoretical...

Page 1: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Polariton Bose-Einstein condensation:A theoretical overview

Institute of Theoretical Physics

Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland

Vincenzo Savona

Outlook

Lecture 1: Overview of symmetry breaking field theory

of a Bose condensate

Lecture 2: The gas of interacting polaritons: BEC-relevant aspects

Lecture 3: Hartree-Fock-Popov theory of a polariton condensate

+ Boltzmann kinetics

Page 2: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Bose-Einstein 1925

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Bose-Einstein 1925

The idea is that something happens, similar to the isothermal compression of a vapor across the saturation volume. A separation occurs; one part “condenses”, the remaining part stays a “saturated ideal gas”.

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas

Nk = nB(εk) =1

eβ(εk−µ) − 1

εk =2k2

2mβ =

1

kBT

k

Nk = N → µ

Statistical distribution of Bose particles

with

Total number of particles

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas: condensation

0 20 40 60 80 100

10-2

10-1

100

101

102

103

104

j

n(Ej)

103

104

105

-4

-3

-2

-1

0

N

µ (meV)

1000N =

k

Nk = N → µ

k

Nk

Nk =1

eβ(εk−µ) − 1

Box of size L

k =π

L(nx, ny, nz)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas: condensation

0 20 40 60 80 100

10-2

10-1

100

101

102

103

104

j

n(Ej)

103

104

105

-4

-3

-2

-1

0

N

µ (meV)

8000N =

k

Nk

k

Nk = N → µ

Nk =1

eβ(εk−µ) − 1

Box of size L

k =π

L(nx, ny, nz)

Page 7: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas: condensation

0 20 40 60 80 100

10-2

10-1

100

101

102

103

104

j

n(Ej)

103

104

105

-4

-3

-2

-1

0

N

µ (meV)

45000N =

k

Nk

k

Nk = N → µ

Nk =1

eβ(εk−µ) − 1

Box of size L

k =π

L(nx, ny, nz)

Page 8: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas: condensation

0 20 40 60 80 100

10-2

10-1

100

101

102

103

104

j

n(Ej)

103

104

105

-4

-3

-2

-1

0

N

µ (meV)

64000N =

k

Nk

k

Nk = N → µ

Nk =1

eβ(εk−µ) − 1

Box of size L

k =π

L(nx, ny, nz)

Page 9: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Ideal Bose gas: phase transition

kBTc =2π2

m

( n

2.612

)2/3

N

Ld=

1

Ld

k=0

Nk →∫ ′

dkkd−1nB(ε(k))

=

(2m

2

) d2∫ ′

dEE( d2−1)nB(E)

N(k) = N0δ(k) +V

(2π)31

eβE(k) − 1

L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford Univ. Press, 2003

Density of noncondensed particles

Phase transition in 3-D

Macroscopic occupation of ground state

nλ3T = 2.612

λT =√2π2/mkBT

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BEC of an atomic Bose gas

M. H. Anderson, et al., Science 269, 198 (1995)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

n(r, r′) = N0φ∗0(r)φ0(r

′) +∑

j =0

Njφ∗j (r)φj(r

′)

= n0(r, r′) + n(r, r′)

n(r, r′) = 〈ψ†(r)ψ(r′)〉

ψ(r) = φ0(r)a0 +∑

j =0

φj(r)aj

∫dr′n(r, r′)φj(r

′) = Njφj(r)

One-body density matrix

Ideal gas: diagonal form on the basis of single-particle states

Expansion of field operator in the eigenstate basis

Diagonalization of the density matrix of an interacting gas

L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford Univ. Press, 2003

A. J. Leggett, Rev. Mod Phys. 73, 307 (2001)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Off-diagonal long-range order

φj(r) =1

L3/2eikj·r

00

r

g(1

) (r)

n0

n

cT T>

cT T<

O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956)

L→∞

n(|r− r′|) = n0 +∑

k =0

nkeik·(r′−r)

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ODLRO in an atomic Bose condensate

I. Bloch, T. W. Hänsch and T. Esslinger, Nature 403, 166 (2000)

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Symmetry-breaking assumption

ψ(r) =√N0φ0(r) + ψ(r)

〈ψ(r)〉 = 0[ψ(r), ψ†(r′)] = δ(r− r′)

φ0 =1√Veiθ

[a0, a†0] = 1

〈a†0a0〉 = N0 1

a0 √N0

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

V. A. Zagrebnov and J. B. Bru, Phys. Rep. 350, 291 (2001)

Fluctuation field obeys Bose commutation rules

Why should it work?

Phase is arbitrary: U(1) gauge symmetry broken

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Symmetry-breaking Hamiltonian of the excitation field

K = H − µN

=

∫drψ†(r)

(−

2

2m∇2 − µ

)ψ(r)

+

∫drdr′ψ†(r)ψ†(r′)v(r− r′)ψ(r′)ψ(r)

ψ(r) =

√N0

V+

1√V

k=0

akeik·r

K = E0 − µN0 +1

V

k=0

(εk − µ)a†kak +

7∑

j=1

Vj

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

Grand-canonical Hamiltonian of an interacting Bose system

Symmetry-breaking assumption for a spatially uniform system

Symmetry-breaking Hamiltonian

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Symmetry-breaking Hamiltonian of the excitations field

v(q) =

∫drv(r)eiq·r

E0 =1

2n20V v(k = 0)

V1 =1

2n0∑

k =0

v(k)aka−k

V2 =1

2n0∑

k =0

v(k)a†ka†−k

V3 = n0∑

k =0

v(k)a†kak

V4 = n0∑

k =0

v(0)a†kak

V5 =

√n0V

k,q

v(q)a†k+qakaq

V6 =

√n0V

k,q

v(q)a†ka†qak+q

V7 =1

2V

k,k′,q

v(q)a†k+qa†k′−qak′ ak

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Finite-temperature Green’s function formalism

G(rτ, r′τ ′) = −TrρTτ [ψ(r, τ)ψ†(r′, τ ′)]

= −〈Tτ [ψ(r, τ)ψ†(r′, τ ′)]〉

τ = it

A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems, McGraw-Hill, 1971G. D. Mahan, Many particle physics, Plenum Press, 1981

“Imaginary time” formalism

G(k, iωn) =1

(2π)3

∫dreik·r

∫ β

0

eiωnτG(r, τ)

Momentum and Matsubara frequency representation (uniform system)

ωn =2nπ

β

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Dyson-Belaev equations

G11(p) = G(0)(p) +G(0)(p)Σ11(p)G11(p) +G(0)(p)Σ21(p)G21(p)

G12(p) = G(0)(p)Σ12(p)G11(−p) +G(0)(p)Σ11(p)G12(p)

G21(p) = G(0)(−p)Σ21(p)G11(p) +G(0)(−p)Σ11(−p)G21(p)

G(0)(p) ≡ G(0)(k, iωn) =1

iωn − εk + µ

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

Anomalous Green’s functions

G12(k, iωn) = −〈Tτ [a−k(τ)ak(τ′)]〉

G21(k, iωn) = −〈Tτ [a†k(τ)a

†−k(τ

′)]〉

Dyson-Belaev equations in terms of the self-energy

Free-particle Green’s function

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Dyson-Belaev equations

G11(p) =iωn + εk − µ+Σ11(−p)

[iωn + εk − µ− Σ11(−p)][iωn − εk + µ− Σ11(p)] + (Σ12(p))2

G12(p) =−Σ12(p)

[iωn + εk − µ− Σ11(−p)][iωn − εk + µ− Σ11(p)] + (Σ12(p))2.

11Σ12Σ

11Σ21Σ

=

= +

+ +G11

G12 G(0)

G(0) G(0)

G(0)

G(0) G11

G12

G21

G22

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

Formal solution of the Dyson-Belaev equations

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Hugenholtz-Pines theorem

n = n0 + n

n =∑

k=0

nk ,

(∂Ω(T, V, µ,N0)

∂N0

)

T,V,µ

= 0 N0 = N0(µ)

µ = Σ11(0, 0)− Σ12(0, 0)

condensate,µ

excitations, Ek

gap

condensate,µ

excitations, Ek

P. C. Hohenberg and P. C. Martin, Annals of Physics 34, 291 (1965)

We no longer have a simple relation for the chemical potential

First method: minimize the grand-canonical thermodynamic potential

Second method:require a gapless excitation spectrum

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S-wave scattering amplitude

r

v(r

)

0

T 2B(k,k′,q) ∼ 4π2a

m+O(ka2) , (k = k′ → 0)

T = + vv

T+ vv=

v + v v v + · · ·k′ − qk′

k+qk

v(k)→ g =4π2a

m

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

S. A. Morgan et al., Phys. Rev. A 65, 022706 (2002)

Hard-core potential: at large momentum all orders contribute to the scattering amplitude

Two-body T-matrix in the long-wavelength limit

Replace the bare potential with effective contact interaction

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Bogoliubov approximation for the self-energy

11Σ

12Σ

v(0) ++++

====

====

v( )k

v( )k

k

kk−−−−

k

Σ11(k) = n0[v(0) + v(k)]

Σ12(k) = n0v(k)

µ = Σ11(0, 0)− Σ12(0, 0) = gn0

Σ11 = 2gn0

Σ12 = gn0

In the s-wave scattering length limit

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Bogoliubov Hamiltonian

H =g

2VN2 +

g2

2n2∑

k=0

m

p2+∑

k=0

εka

†kak +

gN

2

[2a†kak +

(a†ka

†−k + h.c.

)]

ak = ukαk + v−kα†−k [αk, α

†k′ ] = δkk′

|uk|2 − |vk|2 = 1

H = U0 +∑

k =0

Ekα†kαk

U0 =g

2VN2 +

g2

2n2∑

k =0

m

p2+

1

2

k=0

(Ek − gn− εk)

Quadratic Hamiltonian: diagonalize using a linear transformation

Ideal gas of Bogoliubov collective excitations

L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford Univ. Press, 2003

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Bogoliubov collective excitation spectrum

uk, v−k =

[εk + gn

2Ek± 1

2

]1/2

( )E k

k

ck

kE

εk

Ek =√ε2k + 2εkng

Page 25: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

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Measured excitation spectrum of a Bose gas

J. Steinhauer et al., Phys. Rev. Lett. 88, 120407 (2002)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Quantum fluctuations at T=0

Nk ≡ 〈α†kαk〉 = nB(Ek)

Nk = 〈a†kak〉 =(|uk|2 + |vk|2

)Nk + |vk|2

N0 = N −∑

k=0

Nk

→ N

(1− 8

√na3

)(T = 0) .

Thermal distribution of Bogoliubov collective excitations

Population of unperturbed single-particle states

Population of the condensate

Quantum fluctuations: even at T=0 the condensate is depleted!

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Quantum vs thermal fluctuations

Page 28: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Popov approximation for the self-energy

11Σ

12Σ

v(0) ++++

====

====

v( )k

v( )k

k

kk−−−−

k

(((( ))))(0)

nG q,iω

++++++++v(0)

v( )k q−−−−

(((( ))))(0)

nG q,iω

Σ11(k) = n0[v(0) + v(k)]− 1

βV

q,iωn

G(0)(q, iωn)(v(0) + v(k− q))

= n0[v(0) + v(k)] +

∫d3q

(2π)3(v(0) + v(k− q))eβ(εq−µ

(0)) − 1

Σ12(k) = n0v(k)H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Popov self-energy in the s-wave scattering length limit

Σ11 = 2g(n0 + n(0))

Σ12 = gn0

n(0) =

∫d3k

(2π)31

eβ(εk−µ(0)) − 1.

µ = Σ11 − Σ12 = g(2n(0) + n0) .

Hugenholtz-Pines theorem in the Popov approximation

Remember that in the Bogoliubov approximation we had

µ = Σ11 − Σ12 = gn0

where

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Self consistent equations of the Popov model

n = −∫

dr3G11(r, τ = 0)

=

∫d3k

(2π)3

(εk + gn02Ek

coth

(βEk

2

)− 1

2

)

n = n0 + n

Ek =√ε2k + 2εkn0g

Excitation spectrum

Total density

Density of excitations

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Result of the Popov model

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T/Tc

n0/n

0 0.01 0.020.997

0.998

0.999

1

na1/3=0.0073

Ideal gas

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Experiments of polariton BEC

J. Kasprzak, et al., Nature 443, 409 (2006)

Groups of B. Deveaud (Lausanne) and Le Si Dang (Grenoble)

H. Deng, et al., Science 298, 199 (2002)

H. Deng, et al., PRL 97, 146402 (2006)

Group of Y. Yamamoto (Stanford)

R. Balili, et al., Science 316, 1007 (2007)

Group of D. Snoke (Pittsburgh)

Group of N. Grandjean (Lausanne)

Group of M. S. Skolnick

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Polaritons in a two-dimensional semiconductor microcavity

0 0.002 0.004 0.006 0.008 0.011496

1498

1500

1502

1504

1506

1508

1510

wave vector (nm-1

)e

ne

rgy (

me

V)

mirror A

mirror B

spacer

0 0.002 0.004 0.006 0.008 0.011496

1498

1500

1502

1504

1506

1508

1510

wave vector (nm-1

)e

ne

rgy (

me

V)

0 0.002 0.004 0.006 0.008 0.011496

1498

1500

1502

1504

1506

1508

1510

wave vector (nm-1

)e

ne

rgy (

me

V)

QW

Page 34: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

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Are excitons really bosons?

† † †1e hm m

M M

b c dV

ϕ∗

+ −= ∑k q

k q k qq

( ) ( )2 2

2

2

eE

M rϕ ϕ

ε

− ∇ + =

r r

Exciton creation operator Hydrogen-like equation for e-h relative motion

† † †1 1,

e e h hm m m m

M M M M

b b c c d dV V

δ ϕ ϕ ϕ ϕ∗ ∗′ ′ ′ ′+ + − −

′ ′+ + − − = − − ∑ ∑k k kk k q k q k q k q

q k q k q k q kq q

Commutator

( )†, D

Bb b O naδ′ ′ = − k k kk

Expectation value (on, e.g., a thermal distribution)

For excitons: B. Laikhtman, “Are excitons really Bosons?”, J. Phys.: Cond. Mat. 19, 295214 (2007)

For atoms, see e.g. works by M. D. Girardeau

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Effective bosonic Hamiltonian for excitons and photons

Can we map the composite fermion problem onto an effective Bose Hamiltonian with interactions?

X-X interaction:

X- oscillator strength saturation:

( ) † †1ˆ , ,2

x xH v b b b b

A′ ′+ −

′= ∑ k q k q k k

kk q

k k q

( ) † †1ˆ , , . .s s

H v c b b b h cA

′ ′+ −′

′= +∑ k q k q k k

kk q

k k q

X-ph linear interaction:

Ω

( ) ( )† † † †ˆ x c

linH b b c c c b b cε ε= + + Ω +∑ ∑k k k k k k k k k k

k k

ˆ ˆ ˆ ˆlin x s

H H H H= + +

†,c c δ′ ′ = k k kkphotons

†,b b δ′ ′ = k k kkexcitons Rochat, et al., PRB 61, 13 856 (2000),

Ben-Tabou de-Leon, et al., PRB 63 125306 (2001),

Okumura, et al., PRB 65, 035105 (2001),

Ch. Schindler and R. Zimmermann, PRB (2008), to appear.

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Models of effective interaction

vx(0, 0, 0) = vx = 6Eba20

vs(0, 0, 0) = vs = −ΩR

ns

C. Ciuti, et al., PRB 58, 7926 (1998); G. Rochat, et al., PRB 61, 13 856 (2000)

Φ(re1, rh1, re2, rh2) = ψk(re1, rh1)ψk(re2, rh2)

+ ψk(re2, rh2)ψk(re1, rh1)

− ψk(re1, rh2)ψk(re2, rh1)

− ψk(re2, rh1)ψk(re1, rh2)

Hartree-Fock, or “rigid exciton” approximation

Ben-Tabou de-Leon, et al., PRB 63 125306 (2001),

Okumura, et al., PRB 65, 035105 (2001),Ch. Schindler and R. Zimmermann, PRB (2008), to appear.

Better approximations (Heitler-London, etc.) account for Van der Waals effects

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Effect of the two interaction terms

b†kbk′ ∼ δk,k′nxk

Hpol + Hs ∼ Ω(1− nx

ns

)∑

k

(c†kbk + h.c.)

Mean-field limit

k

en

erg

y

Hx ∼ vxnx∑

k

b†kbk

k

en

erg

y

nx

nx

nx

nx

Mean-field expression for the Hamiltonians

Saturation X-X interaction

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Effective interaction for N quantum wells

HN = Hc0 + HN

0 + HNR + HN

x + HNs

HN0 =

N∑

j=1

H(j)0 =

k

εxk

N∑

j=1

b(j)†k b

(j)k

HNR = ΩR

k

N∑

j=1

(b(j)†k ck + h.c.)

HNx =

1

2A

k,k′,q

vx(k,k′,q)

N∑

j=1

b(j)†k+qb

(j)†k′−qb

(j)k′ b

(j)k

HNs =

1

A

k,k′,q

vs(k,k′,q)

N∑

j=1

(c†k+qb(j)†k′−qb

(j)k′ b

(j)k + h.c.)

Assume N QWs at the antinodes of the cavity field

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Effective interaction for N quantum wells

Bk =N∑

j=1

β(j)k b

(j)k =

1√N

N∑

j=1

b(j)k

HB = Hc0 + HB

0 + HBR + HB

x + HBs

HB0 =

k

ExkB

†kBk

HBR = ΩB

R

k

(B†kck + h.c.)

HBx =

1

2A

k,k′,q

vBx (k,k′,q)B†

k+qB†k′−qBk′Bk

HBs =

1

A

k,k′,q

vBs (k,k′,q)(c†k+qB

†k′−qBk′Bk + h.c.)

Totally symmetric linear superposition of exciton modes

Effective Hamiltonian for the totally symmetric mode

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Effective interaction for N quantum wells

Require that for an arbitrary state

〈Ψ | HB | Ψ〉 = 〈Ψ | HN | Ψ〉

Exk = εxk

ΩBR =

√NΩR

vBx (k,k′,q) =

1

Nvx(k,k

′,q)

vBs (k,k′,q) =

1√N

vs(k,k′,q)

One obtains

In particular

nBs = Nns

Higher saturation density, lower effective interactions: more dilute gas

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Is the polariton gas a “quantum” gas?

-5 0 5

momentum (µm-1)

5

010M m−=

2

2 2E ck

n

π

λ

= +

k

2

ph

T

B

E

k Tλ λ

π=

But…

Thermal

wavelength

not of the form

2 2

2

k

m

Polariton effective mass is not a kinetic mass. It is enforced by Maxwell equations

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What is a “dilute” Bose gas?

T ∼ 2

2mn2/3 U ∼ gn

U

T 1

8π(na3)1/3 1

Good criterion for perturbation theory: potential energy smaller than kinetic energy

In 3-D:

We find the usual diluteness criterion:

g =4π2

ma

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Dilute Bose gas in 2-D

g 4π2

m

1

ln(µm4π2 a

2)

T ∼ 2

2mn U ∼ gn

U

T=

ln(µm4π2 a

2)

In 2-D: For a review: A. Posazhennikova, Rev. Mod. Phys. 78, 1111 (2006)

The ratio depends only very weakly on density (and is never very small)

2-D scattering lengt, see e.g.: D. S. Petrov et al., Phys. Rev. A 64, 012706 (2001)

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V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Diluteness criterion for 2-D polaritons

T ∼ 2

2mn

g 4π2

mx

1

ln(µmx

4π2 a2)

k

en

erg

y

Polariton mass Exciton mass

U

T=

ln(µmx

4π2 a2) m

mx 1

The ratio is now governed by the exciton-polariton mass ratio

T-matrix interaction constant comes from a sum over large momenta: it

is governed by the exciton mass

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Literature (non exhaustive) on the theory of polariton BEC

Including spin and polarization effects

D. Sarchi and V. Savona, PRB 77, 045304 (2008)

BCS-like approach for strongly localized excitons

Kinetics and nonequilibrium properties

Statistics and correlation functions

2-D physics and disorder

K. V. Kavokin et al., PRL 92, 017401 (2004)I. A. Shelykh et al., PRL 97, 066402 (2006)

Y. G. Rubo, PRL 99, 106401 (2007)

J. Keeling et al., Semicond. Sci. Technol. 22, R1 (2007)

F. Marchetti et al., PRB 76, 115326 (2007)

D. Sarchi and V. Savona, PRB 75, 115326 (2007)

M. H. Szymanska et al., PRL 96, 230602 (2006)M. Wouters and I. Carusotto, PRL 99, 140402 (2007)

F. P. Laussy et al., PRL 93 016402 (2004)

P. Schwendimann and A. Quattropani, PRB 74, 045324 (2006)P. Schwendimann and A. Quattropani, PRB 77, 085317 (2008)

J. Keeling, PRB 74, 155325 (2006)

A. Kavokin et al., Phys. Lett. A 306, 187 (2003)G. Malpuech et al., PRL 98, 206402 (2007)

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( ) ( ) ( )( ) ( ) ( )ˆ ;X C X C X C P X CψΨ = Φ + ⇒ Φ = Φ +Φr r r

Generalized symmetry breaking ansatz

( )( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

† †

† †

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ . .

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ . .

X C

kin kin X C

R X CA

X X X XA

RC X X X

Asat

H N H H N N

d h c

vd

d h cn

µ µ− = + − +

+ Ω Ψ Ψ +

+ Ψ Ψ Ψ Ψ

Ω− Ψ Ψ Ψ Ψ +

r r r

r r r r r

r r r r r

X-X interaction:

X- oscillator str. saturation:

Symmetry-breaking theory of a polariton condensate

Linear X-photon coupling

Grand canonical polariton Hamiltonian

D. Sarchi and V. Savona, PRB 77, 045304 (2008)

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Generalized Dyson-Beliaev formalism for excitation field

0 0G G G G= + Σ

11 12 11 12

21 22 21 22

11 12 11 12

21 22 21 22

XX XX XC XC

XX XX XC XC

CX CX CC CC

CX CX CC CC

g g g g

g g g gG

g g g g

g g g g

=

( ) ( ) ( ) ( )( )0

0 0 0 0

tX X C C

G g k g k g k g k= − −1

gχξjl (k, iωn) = −∫ β

0

dτeiωnτ 〈Ojχ (k, τ) O

lξ (k, 0)

†〉τ,β

O1ξ(k) = Oξ(k)

O2ξ(k) = O†

ξ(−k)Ox = b, Oc = c

Four-component field: four by four Green’s tensor and self-energy tensor

( )( )

0

1X ( C )

X ( C )

n k X C

g ki iω ε γ µ

± ≡± − − +

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Hartree-Fock-Popov aproximation

11

χξΣ

12

χξΣ

11gυς

11gυς

υΦ ςΦ

υΦ ςΦ

ςΦυΦ

++++

++++

====

====

( ), , , ,x cχ ξ υ ς =

++++

(((( ))))1 χξδ−−−− ++++

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Popov self-energy

Σ Σ Σ Σ

Σ Σ Σ Σ Σ = Σ Σ Σ Σ

11 12 11 12

21 22 21 22

11 12

21 22

0 0

0 0

XX XX XC XC

XX XX XC XC

Popov CX CX

CX CX

Σxx11 = Σxx

22 = 2

[vxnxx −

ΩR

ns(ncx + nxc)

]

Σxx12 = (Σxx

21 )∗= vxΦ

2x − 2

ΩR

nsΦxΦc

Σxc11 = Σxc

22 = ΩR

(1− 2

nxxns

)

Σxc12 = (Σxc

21)∗= −ΩR

nsΦ2x

Σcxjl = Σxc

jl

Σccjl = 0

nxc = Φ∗xΦc +∑

k=0

〈b†kck〉

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Generalized Hugenholtz-Pines relation

Φx = X0Φ

Φc = C0Φ

|X0|2 + |C0|2 = 1

n0 = |Φ|2

Elp = µ

E

(X0

C0

)=

[(εx0 00 εc0

)+ (Σ11 − Σ12)

](X0

C0

)

Exciton and photon components of the condensate field

Φx(c)(t) = e−iEt Φx(c)(0)

Time evolution

Generalized Hugenholtz-Pines relation

Chemical potential from the lowest eigenvalue (lower polariton)

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Polariton excitation spectrum

E(k)

Xu

Xv

Cu

Cv

=

εxk − µ+Σxx11 Σxx

12 Σxc11 Σxc

12

−Σxx21 −(εxk − µ+Σxx

22 )∗ −Σxc

21 −Σxc22

Σcx11 Σcx

12 εck − µ 0−Σxc

21 −Σcx11 0 −(εck − µ)∗

Xu

Xv

Cu

Cv

|Xju|2 − |Xj

v |2 + |Cju|2 − |Cj

v |2 = 1

πlp(up)k = X lp(up)

u (k)bk +X lp(up)v (k)b†−k + Clp(up)

u (k)bk + C lp(up)v (k)c†−k

≡ ulp(up)(k)pk + vlp(up)(−k)∗p†−k

N jk ≡ 〈π

j†k πjk〉 =

1

eβEj

k − 1j = lp, up

Equation for the collective excitation spectrum

Operators of collective excitations

Thermal distribution of collective excitations

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Excitation spectrum

0 2 4 6 8 10-40

-20

0

20

40

Ek-E

0 (

me

V)

k|| (µµµµm

-1)

Photon dispersion

Exciton dispersion

Normal modes for npol=100 µµµµm-2

Normal modes for npol=10 µµµµm-2

Measured in Yamamoto’s experiment:

See Tuesday and Wednesday lectures

Bogolubov linear spectrum of excitations

0 1 2 3

-4

0

4

Ek-E

0 (

me

V)

k|| (µµµµm

-1)

D. Sarchi and V. Savona, PRB 77, 045304 (2008)

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Energy shifts

1010

-2

10-1

1

0.0

0.4

0.8

12.4

12.7

13.0

Eu

p (meV

)E

X (

meV

)

np (µµµµm

-2)

0 50 10010.7

11.1

11.5

25.1

25.5

25.9

(b)

(a)E

lp (

meV

)0

0

0

Elp (

meV

)0

ΣΣ ΣΣ X

C (meV

)1

1

D. Sarchi and V. Savona, PRB 77, 045304 (2008)

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Measured energy shifts vs pump power

Upper polariton Lower polariton

J. Kasprzak, et al., Nature 443, 409 (2006)

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Off-diagonal long-range order

( ) ( )( ) ( ) ( ) ( )† †

1 0

ˆ ˆ 0 0r rng r

n n n

ψ ψΨ Ψ= = +

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

gp

(1)

gc

(1)

np = 20 µµµµm

-2

np = 10 µµµµm

-2

g(1

) (r)

r (µm)

np = 7 µµµµm

-2

J. Kasprzak, et al., Nature 443, 409 (2006)

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Phase diagram

10 15 20 25 30

100

101

102

HFP A=100 µm2

HFP A=1000 µm2

HFP A=1 cm2

N-superfluid

np (

µµ µµm

-2)

T (K)

23CN mµ −≈

20pol

T K=

In good agreement with experiments

D. Sarchi and V. Savona, PRB 77, 045304 (2008)

A=100

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Polariton kinetics

-15 -10 -5 0 5 10 15-5

-4

-3

-2

-1

0

1

2

3

4

5

in-plane momentum (µm-1

)

en

erg

y (

me

V)

Electron-hole continuum

formation

relaxation

Radiative

recombination

excitation

k −k q

( ),z

qq

k +k q

Phonon emission Phonon absorption

Scattering with thermal bath of phonons at L

T T=

Polariton-polariton

scattering

collisions

form 1 psτ <<

relax 100 1000 psτ ÷∼

r ad 1 10 psτ ÷∼1

coll nτ −∝

No equilibrium at low

Quasi-equilibrium at high : n

npol L

T T≠

′k ′ −k q

k +k q

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Number-conserving Bogolubov approach to kinetics

One- and two-body density matrix

kinetic theory

Account for collective excitationspectrum in an adiabatic limit

Solve assuming steady-state pump

and finite lifetimes

Account for both polariton-polaritonand polariton-phonon scattering

D. Sarchi and V. Savona, phys. stat. sol. (b) 243, 2317 (2006)

D. Sarchi and V. Savona, Phys. Rev. B 75, 115326 (2007)

Excitation

Relaxation

coherent scattering quantum fluctuations(see e.g.: J. Leggett, Rev. Mod. Phys., 73, 307 (2001))

† †ˆ ˆk k km a a ψ ψ −=

k-k

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Parameters as in experiment by Kasprzak, et al.

Increasing excitation

intensity f

0 1 2 3 410

-2

10-1

100

101

f=10 pol µµµµm-2ps

-1

f=12 pol µµµµm-2ps

-1

f=15 pol µµµµm-2ps

-1

f=30 pol µµµµm-2ps

-1

f=50 pol µµµµm-2ps

-1

BE fit

de

ns

itie

s N

k/A

(µµ µµm

-2)

energy (meV)

J. Kasprzak, et al., Nature 443, 409 (2006)

Polariton stationary distribution from kinetic model

D. Sarchi and V. Savona, Phys. Rev. B 75, 115326 (2007)

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0 1 2 310

0

102

104

kinetic model

equilibrium Popov

BE distribution

po

pu

lati

on

s N

k

Ek-E

0 (meV)

CdTe

Population distribution: Equilibrium vs non-equilibrium

Quantum fluctuationscondensate depletion

long-wavelength excitations

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Spatial correlation function

J. Kasprzak, et al., Nature 443, 409 (2006)

See also: R. Balili, et al., Science 316, 1007 (2007)

(25% long-range correlation)

Deviation from the ideal case:

1) Non-thermal dependence on r

2) Long-range order partially destroyed

ExperimentTheory:

0 2 4 60.0

0.2

0.4

0.6

0.8

1.0

1.2 kinetic model

Popov

g(1

) (r)

r(µµµµm)

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Boltzmann equation for polariton-polariton and polariton-phonon sc.

Nk = Fk − γkNk

+∑

k′,q,q′

[W ppqq′→kk′NqNq′(1 +Nk)(1 +Nk′)

− W ppkk′→qq′NkNk′(1 +Nq)(1 +Nq′)

]

+∑

k′

[W phk′→kNk′(1 +Nk)

− W phk→k′Nk(1 +Nk′)

]

-0.2-0.1

00.1

0.20.3

-0.2-0.1

00.1

0.20.3

Assume isotropic distributions

Nk → N(Ek)

T. D. Doan et al., PRB 72, 085301 (2005)F. Tassone et al., PRB 59, 10830 (1999)

I. Shelykh et al., PRB 70, 115301 (2004)

Including spin kinetics

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Polariton distribution under steady-state pump

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Polariton thermalization vs polariton lifetime

0 5 10 15 20 25 3010

-1

100

101

102

103

kinetic model

equilibrium Popov

nQ

W( µµ µµ

m-2)

ττττpol

(ps)

0 5 10 15 20 25 30

5

10

15

20

25

30 kinetic model

equilibrium Popov

Teff (

K)

D. Sarchi and V. Savona, Solid State Commun. 144, 371 (2007)

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Polariton thermalization vs polariton splitting

4 6 8 10 12 14 16

10-1

100

101

102

103

nQ

W (

µµ µµm

-2)

2ΩΩΩΩR

(meV)

kinetic model (ττττpol

=3 ps)

equilibrium Popov (T=10 K)

equilibrium Popov (T=20 K)

4 6 8 10 12 14 16

15

17

19

(b)

Teff (

K)

(a)

D. Sarchi and V. Savona, Solid State Commun. 144, 371 (2007)

Page 66: Polariton Bose-Einstein condensation: A theoretical overviewstatic.sif.it/SIF/resources/public/files/va2008/Savona.pdf · Polariton Bose-Einstein condensation: A theoretical overview

V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008

Final considerations

Polaritons in semiconductor microcavities are a good physical realization of a

weakly interacting Bose gas

We have presented the basic theory of a polariton condensate at thermal equilibrium and of the polariton BEC kinetics

Theory explains all existing experimental data in terms of polariton BEC physics

Kinetic theory indicates that full thermalization is not reached in present experiments

Polaritons have a great potential as the ideal system for studying 2-D Bose physics

Theory should progress towards a better modeling of nonequilibrium properties

and of 2-D Bose physics (in view of future high-quality samples)