Transcript No Slide Title

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?
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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)
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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
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Absorption and
re-emission
Stimulated
emission
Stimulated Emission > Absorption
 Laser
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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?
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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 = 4ù a3 r 3
3
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B ohr s
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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
–
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[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 ]
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•
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
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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)  f a  a 0
k ck vk
( fk is Fourier transform of
k
hydrogenic wavefunction)
This is not a boson!
k+q
evk
õ
k
e
P
y
k þ k a ck a 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  v a a

k ck vk
0 ;
uk  vk  1
2
2
[Keldysh and Kopaev 1964]
k
k
v , ku
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
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Mean field theory of excitonic insulator

 BCS   uk  vk a a

ck vk
0
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 );
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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
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Ek
k
Correlation energy
Absorption
m
k
k
Emission
kf
k
m
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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
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•
•
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//
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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)
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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  bibi  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
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P
y
i
< ayibi >
PR Eastham, 2000, 2001
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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)
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Excitation spectrum with inhomogeneous broadening
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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
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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
(wm/g
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MH Szymanska 2002
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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
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Small decoherence -- BEC of polaritons
• order parameter mixed exciton/photon
• excitation spectrum has a gap
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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
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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….
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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
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