POLARITON CONDENSATES IN MICROSTRUCTURES Jacqueline Bloch Laboratoire de Photonique et de Nanostructures LPN/CNRS Marcoussis, France [email protected] http://www.lpn.cnrs.fr/fr/GOSS/CFMC.php

(a) 3 µm 25 µm

20 20µm µm

LPN/CNRS –Marcoussis- France

Institut Pascal -Clermont Ferrant-France

Esther Wertz PhD 2007-2010 Dimitrii Tanese PhD 2010-2013 Vera Sala PhD 2010-2013 Lydie Ferrier Post-doc 2008-2010 Marco Abbarchi Post-doc 2011-2012

Robert Johne (PhD 2007-2010) Hugo Flayac (PhD 2009-2012)

Hai Son Nguyen Post-doc 2011-2014 Thibaut Jacqmin Post-doc (2012-2014) Chris Sturm Post-doc 2011-2013 Marta Galbiati Master2 2010

Guillaume Malpuech Dimitri Solnishkov INO-CNR BEC Center, Trento, Italy Iacoppo Carusotto

Jacqueline Bloch Alberto Amo Aristide Lemaître Elisabeth Galopin Isabelle Sagnes Pascale Senellart

Department of Physics, Technion, Haifa Israël Eric Akkermans Evgueni Gurevich

Boson fluids: Cavityquantum polaritons in 2Dpolaritons Polariton BEC

Quasi long-range order phases

Real space

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

Quantised vortices

Momentum space

Lai et al., Nature 450,50 529µm (2007) Kim et al. Nature Phys. (2011)

Lagoudakis et al., Nature Phys. 4, 706 (2008), Lagoudakis et al., Science 326, 974 (2009) Sanvitto et al., Nature Phys. 6, 527 (2010) Krizhanovskii et al., PRL 104, 126402 (2010) Roumpos et al., Nature Phys. 7, 129 (2010) Nardin et al., Nature Phys. 7, 635 (2011) Sanvitto et al., Nature Phot. 5, 610 (2011)

Superfluidity

Dark solitons

Bright solitons

obstacle

A. Amo et al., Nature Phys. 5, 805 (2009)

FLOW

FLOW

A. Amo et al., Science 332, 1167 (2011) R. Hivet et al., Nat. Phys. 8, 724 (2012)

Sich et al., Nature Phot. 6, 50 (2012)

Cavity polaritons in microstructures Use of nanotechnology to engineer the potential landscape: * A quantum simulator * Toward polaritonic circuits

(a) 3 µm 25 µm

20 20µm µm

Outline 1. Introduction to cavity polaritons 2. Polariton gases in 1D potential - Spontaneous coherence, ballistic propagation - Resonant tunneling 3. Polaritons in modulated potentials - Localization in a 1D periodic potential - 1D Quasi-periodic potentials - Spin-Orbit coupling in a benzene photonic molecule - Honeycomb lattices : Dirac cones and flat bands Perspectives

Microcavity polaritons GaAs/AlGaAs based structures

Angle θ (º)

θ

-20

Top DBR

Bottom DBR

10

Bragg mirror GaAs/AlAs Cavity Bragg mirror GaAs/AlAs

20

Photon

-2

Optical cavity

TEM, G. Patriarche, LPN

0

Emission energy (eV)

5K

-10

0

kin-plane (µm-1)

2

Microcavity polaritons GaAs/AlGaAs based structures

Angle θ (º)

θ

-20

Emission energy (eV)

5K

Top DBR Quantum Wells Bottom DBR

Quantum well exciton AlGaAs

AlGaAs

GaAs

e-

Exciton

h+

QW

-10

0

10

20

Photon Exciton

 
  
   0  2 -2

0

kin-plane (µm-1)

2

Microcavity polaritons GaAs/AlGaAs based structures

Angle θ (º)

θ

-20

Emission energy (eV)

5K

Top DBR Quantum Wells Bottom DBR

Weisbuch, et al., PRL 69, 3314 (1992)

Microcavity polaritons : mixed exciton-photon states AlGaAs

Cavity

-10

0

Upper polariton

20

Photon Exciton

~ 5meV Lower polariton -2

0

kin-plane (µm-1)

AlGaAs

GaAs

10

e-

polariton h+

QW

Courtesy D.Sanvitto

2

Microcavity polaritons Angle θ (º)

θ

-20

-10

5K

Top DBR Quantum Wells Bottom DBR

Microcavity polaritons : mixed exciton-photon states

Properties Bosons Photonic component Short lifetime (~1-50 ps) Excitonic component

0

10

20

Upper polariton

Emission energy (eV)

GaAs/AlGaAs based structures

Photon Exciton

~ 15meV Lower polariton -2

0

kin-plane (µm-1)

2

pol = X k exc + Ck phot

low mass (10-5 me) condensation at high temperature escape out of the cavity optical access repulsive interactions

Polariton spin Spin : electron : +- ½ heavy hole : +- 3/2

Exciton : Jz = +- 1 e

h e

h

e

h e

h

Jz =+- 2 Photon have an angular momentum : +- 1

Only J=1 excitons are coupled to light Upper Polariton

Polaritons have two spin projections: jz = +1 jz = -1

Photon

Exc Jz = +- 1 Jz = +- 2

Lower Polariton

σ+ σ−

1

2

pseudospin

Polariton spin Spin : electron : +- ½ heavy hole : +- 3/2

Exciton : Jz = +- 1 e

h e

h

e

h e

h

Jz =+- 2 Photon have an angular momentum : +- 1

Only J=1 excitons are coupled to light Polaritons have two spin projections: jz = +1 jz = -1

σ+ σ−

1

2

pseudospin

One-to-one relationship between pseudospin state and polarisation degree

Probing polariton states E

Far field imaging: k space ky (µm-1)

z

-0.5

émission(θ θ)

0.5

(µm-1)

Energy

kx

0.0

θ

θ (d)

kx (µm-1) R. Houdré et al., Phys. Rev. Lett. 73, 2043 (1994)

0

k//

k// = ω/c sin(θ)

1 2 3 k// (104 cm-1) 4 -1 k ( 10 cm )

Probing polariton states Far field imaging: k space ky (µm-1)

Impossible d’afficher l’image.

-0.5

0.0

1

0 10 µm

Density

0.5

kx (µm-1) Energy

Real space imaging Real space imaging

(d)

Interference with a reference beam

Coherence map g(1)

Phase dislocations - vortices - solitons

kx (µm-1) R. Houdré et al., Phys. Rev. Lett. 73, 2043 (1994)

Resonant injection of polaritons

Polariton condensation under non resonant excitation Polariton density

Reservoir

1-50 ps Nature 443, 409 (2006)

When f exceeds unity :

Γrelaxation ∝ (f+1)

Bosonic stimulation

Massive occupation of the lowest energy states See also LPN, Stanford, Pittsburgh, Cambridge, Southampton, Lecce …

Polariton condensation under non resonant excitation Polariton density

Reservoir

1-50 ps Nature 443, 409 (2006)

When- fDissipative exceeds unity : Γrelaxation ∝ (f+1) Bosonic stimulation system - Coexistence of the polariton condensate with Massive the lowest energy states theoccupation reservoir ofofuncondensed excitons See also LPN, Stanford, Pittsburgh, Cambridge, Southampton, Lecce …

Ginzburg Landau equation for the polariton wavefunction Polariton wavefunction: Kinetic energy

Confinement potential

Interactions with the excitonic reservoir

 ∂ψ(x,t)  h2∇2 h 2 ih = − * +Vext(x) −i [γC − RnR(x,t)] + gC ψ(x,t) + gRnR(x,t)ψ(x,t) ∂t 2  2m  Decay

Populating term

Excitonic reservoir

Interactions within the condensate

 , 

 ,    ,      ,    M. Wouters et al., Phys. Rev. B 77, 115340 (2008)

Polariton gases in 1D microcavities

E. Wertz et al., Nat. Phys. 6, 860 (2010) L. Ferrier et al., Phys. Rev. Lett. 106, 126401 (2011) D. Tanese et al., Phys. Rev. Lett. 108, 36405 (2012) E. Wertz et al., Phys. Rev. Lett. 109, 216404 (2012) H.S. Nguyen et al., Phys. Rev. Lett. 110, 236601 (2013) Viewpoint: L. Pilozzi, Physics 6, 64 (2013)

Polariton in 1D cavities 1D polariton sub-bands

Fabrication : E-beam lithography Dry Etching

Polariton lateral confinement Discretisation of kx:

kx= nπ/d n=1,2,3,…

J. Bloch et al.. Superlatt. and Microst. 22, 371 (1998). T. Gutbrod et al. Phys. Rev. B 57, 950 (1998). A. Kuther et al., Phys. Rev. B 58, 15744 (1998)

Spreading of the polariton gas T = 10 K

2

Peak Intensity (a. u.)

10

1

10

0

10

-1

10

-2

10 0.1

1 P/Pth

Polariton gas

E. Wertz et al., Nature Physics 6, 860 (2010)

10

Spreading of the polariton gas

Excitonic reservoir

 ∂ψ(x,t)  h2∇2 h 2 ih = − * +Vext(x) −i [γC − RnR(x,t)] + gC ψ(x,t) + gRnR(x,t)ψ(x,t) ∂t 2  2m  Pump

Local blueshift

Polariton gases in 1D cavities Pump

Eb

Pump

Spreading of the polariton gas T = 10 K

2

Peak Intensity (a. u.)

10

1

10

0

10

-1

10

-2

10 0.1

1 P/Pth

Polariton gas Interference

Coherent ballistic propagation over E. Wertz et al., Nature Physics 6, 860 (2010)

more than 200 µm (entire wire sample)

10

Resonant tunneling of polaritons

H.S. Nguyen et al., Phys. Rev. Lett. 110, 236601 (2013) Viewpoint: L. Pilozzi, Physics 6, 64 (2013)

1D cavity with double potential barrier Luminescence of the island

3µm

6µm

2µm

2µm

1µm

1.2 meV

0.5 meV

10 K

RTD operation

θ1 Polaritons Flow

Non-resonant

P1 : Polariton injection at energy E1 and angle θ1

Pc: excitonic reservoir injection

Polaritons Flow

10 K

Local blueshift

RTD demonstration resonant p1, E1

non resonant p2< pth

p2 (mW) 0,0

0,5

1,0

1,5

2,0

Theory for cold atoms : T. Paul et al., Phys. Rev. Lett. 94, 020404 (2005) Photonic crystal single cavity : K. Nozaki et al., Nat. Phot. 6, 248 (2012)

Transmission

0,06 0,04 0,02 0,00 0 100 200 300 Blueshift of Mode 1TM (µeV) x

x

x

y

y

y

1D cavity with double potential barrier Theory for cold atoms : T. Paul et al., Phys. Rev. Lett. 94, 020404 (2005)

resonant p1,

E1

Photonic crystal single cavity : K. Nozaki et al., Nat. Phot. 6, 248 (2012)

Experiment

Simulation

PInc = 40 mW

PInc = 5 mW Transmission

0,15

0,10

27 µeVΓ

0,05

0,00 1573,8

1573,9

EInc (meV)

Linear regime

1573,7

1573,8

EInc (meV)

High density: interactions

H.S. Nguyen et al., Phys. Rev. Lett. 110, 236601 (2013)

Optical bistability

Towards the quantum regime

U ≥γ pol-pol interaction

Polariton blockade

radiative decay

Single photon emission

Single polariton current

A. Verger et al., Phys. Rev. B 73, 193306 (2006) I. Carusotto et al., Phys. Rev. Lett. 103, 033601 (2009)

Condensation in a periodic potential

Period = 2.1 µm

Metallic pattern : Lai et al., Nature 450, 529 (2007) SAW :E. A. Cerda-Méndez, PRL (2010).

Period = 2.7 µm

Condensation in a periodic potential Low power

High power

GAP

Log (l)

GAP

Condensation in a highly localized gap state D. Tanese et al. Nature Comm. 4, 1749 (2013)

Condensation in a periodic potential Pump

Energy

excitonic reservoir

condensate y

 h2∇2  2  − + Vper (r ) + gC ψ (r ) + gresnRes (r ) ψ (r) = µψ (r )  2m 

D. Tanese et al., Nat. Commun. 4, 1749 (2013)

Condensation in a periodic potential Pump

Energy

excitonic reservoir

y

 h2∇2  2  − + Vper (r ) + gC ψ (r ) + gresnRes (r ) ψ (r) = µψ (r )  2m  Excitonic potential

cavity

localised state

D. Tanese et al., Nat. Commun. 4, 1749 (2013)

Condensation in a periodic potential

Energy

Pump

condensate y

 h2∇2  2  − + Vper (r ) + gC ψ (r ) + gresnRes (r ) ψ (r) = µψ (r )  2m  sech2(x)

Excitonic potential

cavity

Condensate interactions

localised state

GAP SOLITON

D. Tanese et al., Nat. Commun. 4, 1749 (2013)

Condensation in a periodic potential Pump

Energy

excitonic reservoir

condensate y

 h 2∇ 2  2  − + V per ( r ) + g C ψ ( r ) + g res nRe s ( r ) ψ ( r ) = µψ ( r )  2m 

D. Tanese et al., Nat. Commun. 4, 1749 (2013)

ps pulsed excitation sech2(x)

Time (ps)

reservoir localised

GAP SOLITON

Polaritons in 1D quasi-periodic potential FIBONACCI sequence

Collaboration with E. Akkermans, Technion, Israel

S12 =…..B A A B A B A A B A A B A B A A B A .

- Extended localized modes - Fractal energy spectrum: gap labelling - log periodic oscillation of the DOS

Polaritons in 1D quasi-periodic potential Direct imaging of the Fibonacci modes in real space

Theory : effective 1D Schrödinger equation

S13: 233 letters

Polaritons in 1D quasi-periodic potential Direct imaging of the Fibonacci modes in reciprocal space

S13: 233 letters Gap labeling theorem :

Similar approach in 2D Penrose tilted quasicrystal: J.M. Gambaudo and P. Vignolo, arXiv:1309.6420

Polaritons in 1D quasi-periodic potential Direct measure of the spectral Density of States Gap labeling theorem :

Polaritons in 1D quasi-periodic potential Direct measure of the spectral Density of States Gap labeling theorem : Log-periodic oscillations of the IDOS :

Coupling 0D polaritons

4 µm

Polaritons in micropillars Micropillars: photonic atoms 2 → 10µm

kx = pxπ/Lx ky = pyπ/Ly

Two coupled micropillars

d Energy

2J 2 µm

Michaelis de Vasconcellos et al., APL 99, 101103 (2011)

10 µm

Galbiati et al., PRL 108, 126403 (2012)

Optical modes of a photonic molecule

Polariton in honeycomb lattices

See also Yamamoto’s group N. Y. Kim et al 2013 New J. Phys. 15 035032 (2013) M. C. Rechtsman et al., Nature Photonics 7, 153 (2013)

Polariton in honeycomb lattices: Dirac cones

T. Jacqmin et al., PRL to appear (arXiv:1310.8105)

Polariton in honeycomb lattices: a Flat band

T. Jacqmin et al., PRL to appear (arXiv:1310.8105)

Polariton in honeycomb lattices: Dirac cones

T. Jacqmin et al., PRL to appear (arXiv:1310.8105)

Polariton condensation in a honeycomb lattice

Spontaneous spatial coherence over more than 30 micropillars

Polariton in honeycomb lattices: a Flat band

Flat band

C.Wu et al., PRL 99, 070401 (2007)

Polariton in honeycomb lattices: a Flat band Tight binding approach

t// Flat band

2D Schrödinger equation

t┴

t┴=0

Polariton in honeycomb lattices: a Flat band Real space imaging of the flatband

Flat band

- Kagome Lattice frustrated phase - Flat band : particles with infinite mass Strong effects of interactions S. D. Huber and E. Altman, Phys. Rev. B 82, 184502 (2010). C. Wu et al., Phys. Rev. Letters 99, 070401 (2007).

T. Jacqmin et al., PRL to appear (arXiv:1310.8105)

Cavity polaritons: perspectives Quantum simulator of complex systems -

Honeycomb lattices: klein tunneling, Berry phase, edge states Flat bands : strongly correlated states Artificial Gauge field Black hole analogue

ERC Starting Grant Honeypol PI : A. Amo (2013-2018) Polariton devices - Optical gates, Optical memories, Routers

Quantum optics - Polariton blockade - Non classical polariton current