nonequilibrium theory of exciton-polariton condensatesnonequilibrium theory of exciton-polariton...

Nonequilibrium theory of exciton-polariton condensates

Michiel Wouters

Collaboration with

• Iacopo Carusotto (Trento, theory)

• Vincenzo Savona (EPFL theory)

• Cristiano Ciuti (Paris 7, theory)

• Konstantinos Lagoudakis, Barbara Pietka, Augustin Baas, Benoit Deveaud (EPFL experiment)

• Maxime Richard, Le Si Dang (Grenoble, experiment)

Outline

• Introduction• Mean field model• Elementary excitations• Spectral shape of inhomogeneous condensates• Vortices• Synchronization• Beyond mean field• Fluctuations

Planar LaserPolariton

condensate/laser

Coherent field

Interaction energy

Decay rate

Relation with Laser and BEC

|light>

< 10-5 meV

1 meV

Atomic BEC

|matter>

1 kHz

10-4 kHz

|light>+|matter>

1 meV

1 meV

The mean field model

relaxation from free carriers

phonon emission pol-polscattering

condensate

reservoir

pump

Generalized Gross-Pitaevskii equation:

Rate-diffusion equation:

• Inspired by model for atom laser by Kneer et al. [PRA 58, 4841 (1998)].• Related to the Complex Ginzburg-Landau equationfrom non-equilibrium pattern formation [M.C. Cross and P.C. Hohenberg, RMP 1993], used by Cambridge group for polaritons [J. Keeling and N. Berloff, arXiv:0706.3686].

[M.W. and I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007)]

Steady state

P/Pth

|ψ0|2

1

(P-Pth)/γ

20

~0

||

2

)(

ψμ

μμ

ψψ μ

g

ng

et

thRT

ti T

=

+=

= −

0)( RR ntn =0Rn

condensate interactions only

Elementary excitation spectrum

• Goldstone mode diffusive at small k

• recovers spectrum of equilibriumcondensate at large k:

ωbog(k)>>Γ.

• form also found by Littlewoodgroup with Keldysh Green function technique

[M.Szymanska et al. PRL 2006] → general, model independent

( ) ( ) 42

22 Γ−±Γ

−=± kik bogωω

Negative frequencies

• Bogoliubov theory predicts negative frequencies in the elementary excitation spectrum.

• Should be visible for polaritons• Not seen so far• Try with FWM?

Transmission vs FWM

Outline


Inhomogeneous system

Large pump spotExperiment

M. Richard et al. PRB 72, 201301 (2005)

Small pump spot

M. Richard et al. PRL 94, 187401 (2005)

Generalized Gross-Pitaevskii equation

M.W, I. Carusotto and C. Ciuti, Phys. Rev. B 77, 115340 (2008).

Inhomogeneous system: intuitive picture

• Blue shifts/quantum pressurecreate antitrapping potential

• Potential energy converted in kinetic energy

• Depending on spot size:– small spot : more polaritons with large kinetic energy

= on dispersion, finite k– large spot : more polaritons with large potential energy

= blue shifted from dispersion, small k

Outline


Vortices

Some correlation measurement images show a dislocation in the fringe pattern

Singularity in relative phase φ(x)-φ(-x)

2π phase winding

K. Lagoudakis et al. arXiv:0801.1916

Unlikely explanations

• Thermally activated (BKT physics): these vortices would move and be not visible after averaging over many experiments, but give a decrease of fringe contrast.

• Kibble-Zurek (defects formed when going through the phase transition): could be pinned to disorder, but why always the same sign?

Phase singularities in gGPE

•No rotation of the sample!•Not at equilibrium (GP equation has real ground state)•Not found with flat external potential•Interplay between pumping-losses-potential•Also found in Complex Ginzburg Landau equation

[J. Keeling and N. Berloff, PRL08]

A simple configuration

‘energy conservation’

forbids this solution.

‘small’ potential wellIlluminated with bigExcitation laser

?

A state with a vortex is formed instead

Outline


One frequency ?

• Equilibrium: chemical potential μ is constant.• What in nonequilibrium polariton condensates?• Experiment:

Depends on the position on the sample

A. Baas et al. Phys. Rev. Lett. 100, 170401 (2008).

Theory for 2 coupled wells

J

Δε

P P

M. Wouters, Phys. Rev. B 77, 121302(R) (2008)

Josephson Oscillations in the desynchronized state

Spatially overlapping states

x

E

P.R. Eastham PRB 78 035319 (2008).

Increasing pumping

P

Outline


Beyond mean field

• Occupation of excited states• Decay of first order spatial correlation function (BKT??)• Second order correlation function (density fluctuations)

Include noise term in Gross-Pitaevskii equation by interpretation as stochastic motion equation for Wigner distribution function

Why ?

How ?

( ) xRRR dWnRtxgngnRi

mdttxid

4)(),(2)(

22),( 2

22 γψψγψ ++⎥⎦

⎤⎢⎣

⎡++−+

∇−=h

dWx is a noise term that comes from the quantum nature of the field ψMain physical process: shot noise of gain and losses

Link with Boltzmann (I)

Incoherent limit of GP equation = Boltzmann without ‘+1’

),(22

),( 222

txgngm

txdtdi R ψψψ ⎥

⎦

⎤⎢⎣

⎡++

∇−=h

If ',* )(),'(),( kkkntktk δψψ = it follows that

( ) ( )kkkkkkkkkkkk

kkkkkkkkkk nnnnnngdtkdn

−++−−+−+ +−−−+= ∑ 2121212121

21212121

22)( εεεδεεεδπ

GP equation

Simplest model for occupation of excited states = Boltzmann

can (almost) be reduced to Boltzmann if we

See e.g. Y. Kagan in Bose-Einstein condensation, A. Griffin, D. Snoke and S. Stringari (eds.), 1994

Link with Boltzmann (II)

Boltzmann with ‘+1’

In the incoherent limit, it follows that

Boltzmanncolkkoutkkin

k InRnRdt

dn++−+= ))(()1)(( γεε

Include gain/losses/fluctuations:

( ) xRoutRinRRoutRin dWnRnRtxgngnRnRi

mdttxid

4)()(),(2)()(

22),( 2

22 γψψγψ +++⎥⎦

⎤⎢⎣

⎡++−−+

∇−=h

∑ −−+−−+−21

431432])[(

2...

2

kkkkkkkkkk nnnn

g εεεεδπ

Quantum ≠ Classical

Spurious

, but OK if i) nΔV >>1ii) γ >> g/ΔV

Preliminary numerical results

Built up of long range spatial coherence for increasing pump intensity

Conclusions Perspectives

simple and generic model for nonequilibrium condensationelementary excitations: diffusive Goldstone modeexplanation for different condensate states for small and large pump spotsinterpretation of experimentally observed vortices due to driving, dissipation and external potentialpicture of synchronizationformalism for fluctuations

understand fluctuations study time dependence of condensate formation…

Nonequilibrium theory of �exciton-polariton condensatesCollaboration withOutlineRelation with Laser and BECThe mean field modelSteady stateElementary excitation spectrumNegative frequenciesTransmission vs FWMOutlineInhomogeneous systemInhomogeneous system: intuitive pictureOutlineVorticesUnlikely explanationsPhase singularities in gGPEA simple configurationOutlineOne frequency ?Theory for 2 coupled wellsSpatially overlapping statesOutlineBeyond mean fieldLink with Boltzmann (I)Link with Boltzmann (II)Preliminary numerical resultsConclusions Perspectives

nonequilibrium theory of exciton-polariton condensatesnonequilibrium theory of exciton-polariton...

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