Stochastic Optimization of Bose-Einstein Condensation Using a
Bose-Einstein Condensation of Light
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Transcript of Bose-Einstein Condensation of Light
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Martin Weitz
Institut für Angewandte Physik der Universität Bonn
Bose-Einstein Condensation of Light
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Bose-Einsteincondensation of atoms: matter waves in lockstep
T~100nK
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Outline of Talk
-cold atomic gases
- thermodynamics of a two-dimensional photon gas in a dye-filled optical microcavity
- Bose-Einstein condensation of photons
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Cold Atomic Gases: Temperature Scale
1000 K
1 mK
1 K
1 K
1 nK
Bose-Einstein-condensate(T~100nK, n~1014cm-3)
liquid helium
laser-cooled atoms
room temperaturesurface of sun
T
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From Thermal Gas to Bose-Einstein-Condensate
classical gas
T<Tc
matter waves overlap → BEC
T<<Tc
pure Bose-Einstein-condensate
cold gas, but T>Tc
dB = h/mv
atoms show wave propertiesT1
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T>>Tc
TTc
T<<Tc
BEC of rubidium atoms @ 180nK
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photons ?
Ground State of Bosonic Ensembles (3D-Regime)
Bose-Einstein-condensate
atoms
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Earlier Work related towards a Photon BEC
- Proposal for a photon BEC in Compton scattering off a thermal electron gas
Zel‘dovich and Levich, 1969
photons
scattered photons:shock wave structure
electrons in thermal equilibium (within plasma)
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… Earlier Work
photon-photon scattering(four-wave mixing)
- Exciton-polariton condensates
strong coupling (‚half matter, half light‘); in equilibrium for condensed part
- Proposal for photon fluid in nonlinear resonator
R. Chiao
atom
Yamamoto, Deveaud-Pledran, Littlewood, Snoke, …
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Bonn 2D-Photon Gas Experimental Scheme
- use curved-mirror microresonator to modify photon dispersion
d=n/2
(for n=1)
- thermal equilibrium of photon gas by scattering off dye molecules…
dyemolecule
n=1 n=2(transversal) wavevector
energy
photon in free space
min. (‚cutoff‘)energy of light in resonator
cutoff
J. Klaers et al.
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I (a.
u.)
wavelength (nm)
absorption () fluorescence f()
quantum 0.97
Kennard-Stepanov theory:
Spectrum of Perylene-Dimide Molecule (PDI)
Tkf
B
exp
)()(
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Photon Gas Thermalization: Background
Collisionally induced thermalization in dye medium
Kennard 1912, Stepanov 1956
Tkf
B
exp
)()(+ -´
T: (internal rovibrational) temperatureof dye solution
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Model for Photon Thermalization
multiple absorption and emission processes by dye molecules in resonator
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Planck Blackbody Radiation
thermalexcitation
photonemission
EkBT=1/40 eV
New Scheme
photonemission
Ecutoffphotonabsorption
thermal excitation suppressed3610
Tke
B
cutoffby
=2.1eV
‚white-wall box‘ for photons
photon average number conserved
Photon Number Variation during Thermalization?
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Photon Trapping versus Atom Trapping
(transversal) wavevector
ener
gy
free photon
photonin cavity kz (fixed)
kr
k
- quadratic photon dispersion
z
rzrz k
kkckkcE2
222
eff
reff m
kcm2
)( 22
2/ cckm cutoffzeff with
In paraxial approximation (kz>>kr):
cutoff
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..Photon versus Atom trapping
0 x
Vtrap
- trapping potential from mirror curvature
resonator
222
2
21
2)( rm
mkcmE eff
eff
reff
2cm cutoffeff System formally equivalent to 2D-gas of massive bosons with
BEC expected for 770003
22
TkNN Bc
(T=300K,=2·4·1010 Hz,meff6.7·10-36 kg10-10·mRb)
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BEC versus Lasing
Optical laser Photon BEC
thermodynamicstate:
far from equilibrium thermal equilibrium
gain/thermalisationmedium:
three or more levels,inversion (or quantumcoherence, high couplingeff. to single cavity mode)
non-inverted two-level system sufficient(many transversal cavity modes)
phase transitioncondition:
for the lasing mode:gain (stim. emission) > loss
22
3
TkNN Bc
(or )122 dBDn
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Two-Dimensional Photon Gas in Dye-Filled Optical Resonator
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Experimental Setup: 2D Photon Gas
pumping laser(532 nm)
d 7/2
spectrometerdye-filledmicrocavity
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pumpingirradiation
resonator axis
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Spectrum of Thermal Photon Gas in Cavity
evidence for thermalized two-dimensional photon gas with ∫0!
J. Klaers, F. Vewinger, M. Weitz, Nature Phys. 6, 512 (2010)
residual pumpradiation
1)300/)exp((
Kkuu
Btr
tr
= -6.76(17)kBT
from Pout=50(5) nW
= -7.16(17)kBT
trucavity cutoff
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Spectra for Different Cavity Cutoff Frequencies
optically dense regime,
thermalization of photon gas
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dyemolecule
‘ ‘‘
… Reabsorption: Required for Photon Thermalization
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Snapshot: Thermalization of Photon Gas in Dye Microcavity
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Thermalization – Photon Diffusion towards Center
x
thermalizedphoton gas
pumping
phot
ontr
appi
ngpo
tent
ial
from
mirr
orcu
rvat
ure
0
Experimental data
trapcenter
pump beam position xexc (m)
xm
ax(m
)
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Photon Gas at Criticality
N<<NcN>Nc
Rh6G, duty cycle 1:16000, 0.5s pulses
BEC!
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ground state:filament off
Cooling(or increase of ndb
2)
belowthreshold
Bose-Einsteincondensate
Bose-Einstein condensate of Light
Light Bulb
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Spectra for Densities around Photonic BEC Threshold
J. Klaers, J. Schmitt, F. Vewinger, M. Weitz, Nature 468, 545 (2010)
22
3
TkN Bc
Pc,exp1.55(60) W
Nc,exp (6.3±2,4)·104
Theory: 7.7·104
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Spatial Intensity Distribution around BEC Threshold
for atoms: geff,2D 10-1 - 10-2 (Dalibard,Phillips)
mode diameter increase could be explained by photon mean field interactionwith geff,2D 7·10-4 (too small for Kosterlitz-Thouless physics) BEC expected
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optical path length difference: 15 mm
Michelson Interference Pattern above Photon BEC Threshold
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Phase Transition Onset versus Resonator Geometry
Rc
TkNP Brt
cc
2
2
)(12
expected critical optical power:
- Variation of mirror radius R
n=7, R variabletheory
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.. Phase Transition Onset
- Variation of resonator length
Rc
TkP Bc
22
)(12
n
R=1m, n variable
(n-4.77)-1 Vdye-1 pumping
beam
dye
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De Martini+Jacobovitz, PRL 60, 1711 (1988)
threshold lowerfor a small mirrorspacing!
Published Results for the Phase Transition of a Microlaser
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.. Phase Transition Onset
- Variation of resonator length
Rc
TkP Bc
22
)(12
n
R=1m, n variable
(n-4.77)-1 Vdye-1
microlaser
photon BEC
usuallaser
pumpingbeam
dye
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Condensation for Off-Center Pumping
Rea
bsor
ptio
ncutoff
pump spot
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+
photon
molecule inground state
molecule in electronicallyexcited state
average photon number is fixed, but fluctuations of the photon numberaround the average value can occur
Effective Particle Exchange of Photons with Dye Excitations
for a large number M of dye molecule, the photon gas is well decribedby a grandcanonical model
exchange
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MexcN
photons
molecules inelectronicallyexcited state
microcanonical ensemble
(similar: canonical ensemble)For large reservoir size: grandcanonical ensemble
N
atom number fixed
usual BECs
Ensembles for Bose-Einstein Condensation
Photon BEC with dye reservoir
Poissonian particle statisticsin condensed state, g(2)(0) =1
enhanced particle fluctuations incondensed state, g(2)(0) =2
general theory grandcanonical BEC fluctuations: Fujiwara et al. (1970), Ziff et al. (1977), Holthaus (1998)
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Expected Phase Diagram
thermal gas
PoissonianBEC regime
J. Klaers et al., PRL 108, 160403 (2012), see also: D. Sobyanin, PRE 85, 061120 (2012)
fluctuatingregime
Poissonianregime
fluctuating BECregime
normalized cavity detuning /kBTc
T/T
c
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0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30
g(2) (
)
delay [ns]
Measured Photon BEC Intensity Correlation vs. Delay Time
preliminary data,
grandcanonicalregime
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excitation fluorescence
|e>
|g>rubi
dium
ener
gy
distance to colliding atom
Laser Cooling by Collisional Redistribution
Cooling cycle ExperimentRb + 200 bar argon filled cell
cooling inside cell: 4100C -1200C
proposal: Berman+Stenholm, 1978
U. Vogl and M. Weitz, Nature 461, 70 (2009);A. Sass, U. Vogl, M. Weitz, to be published
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Conclusions
- thermal 2D-photon gas with nonvanishing chemical potential (average particle number conserved)
- Bose-Einstein condensation of photons. Signatures:
Bose-Einstein distributedphoton energies
phase transitionabsolute value+scaling
condensation foroff-center pumping
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Outlook
- photon thermalization: concentration of diffuse sunlight
- photon BEC: new states of light
- light sources in new wavelength regimes, coherent UV sources
V
position
thermalizedgas of light atroom temperature
possible application:lithography
(some) future directions:
• canonical vs. grandcanonical photon ensemble regimes• study of quantum manybody states in periodic potentials
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J. KlaersS. KlingA. SassH. BrammerJ. SchmittT. DammC. GrossertJ. UlitzschM. LederD. DungT. BurgermeisterS. HüweP. MoroshkinF. Vewinger
M. Weitz
Quantum optics group, IAP Bonn: