Polariton Bose-Einstein condensation: A theoretical...
Transcript of Polariton Bose-Einstein condensation: A theoretical...
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Bose-Einstein 1925
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”.
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
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)
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)
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)
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)
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
BEC of an atomic Bose gas
M. H. Anderson, et al., Science 269, 198 (1995)
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)
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
ODLRO in an atomic Bose condensate
I. Bloch, T. W. Hänsch and T. Esslinger, Nature 403, 166 (2000)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
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
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Measured excitation spectrum of a Bose gas
J. Steinhauer et al., Phys. Rev. Lett. 88, 120407 (2002)
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
3π
√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!
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Quantum vs thermal fluctuations
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)
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)
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)
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
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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.
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
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Dilute Bose gas in 2-D
g 4π2
m
1
ln(µm4π2 a
2)
T ∼ 2
2mn U ∼ gn
U
T=
8π
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)
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=
8π
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
( ) ( ) ( )( ) ( ) ( )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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
± ≡± − − +
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Hartree-Fock-Popov aproximation
11
χξΣ
12
χξΣ
11gυς
11gυς
υΦ ςΦ
υΦ ςΦ
ςΦυΦ
++++
++++
====
====
( ), , , ,x cχ ξ υ ς =
++++
(((( ))))1 χξδ−−−− ++++
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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〉
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Measured energy shifts vs pump power
Upper polariton Lower polariton
J. Kasprzak, et al., Nature 443, 409 (2006)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
Polariton distribution under steady-state pump
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
V. Savona, Int. School of Physics “E. Fermi”, Varenna, July 2008
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)
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)