Dynamical Condensation of Exciton-Polaritons

kξ=1

-15o 15o

1.613 eV

8 m

eV

kξ=1-15o 15o

1.613 eV

S. Utsunomiya, C.W. Lai, G. Roumpos and Y.YamamotoStanford University, National Institute of Informatics

A. Loeffler, S. Hoefling, and A. ForchelTechnische Physik, Universität Wurzburg

International School of Physics “Enrico Fermi”:Quantum Coherence in Solid State PhysicsVarenna (Italy) July 1 – 11, 2008

1

Lecture 3 Lecture 3 BogoliubovBogoliubov excitation and excitation and superfluiditysuperfluidity

2

Outline

Bose-Einstein condensation and superfluidityBogoliubov theory for weakly interacting Bose gasInteraction energy of exciton-polaritonsEnergy-momentum dispersion relation: revisited - Condensates and excitation spectrum - Sound velocity - Free particle energyQuantum depletion vs. thermal depletion

Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -Future prospects

3

Interpretation of superfluid He interms of BEC of ideal gas (1938)

Phenomenological model based onstrongly interacting systems (1941)

London, Tisza

L. D. Landau

Discovery of superfluid 4He (1937)

P.L. Kapitsa

Excitation spectrum of superfluid He

BEC of non-interacting ideal gas (1925)

A. Einstein

Atomic theory of Landau’s two fluid model（1955）

R. P. FeynmanL.D.Landa

u

P.L. Kapitsa

A. Einstein

R.D. Feynman

Bose-Einstein Condensation and Superfluidity

4

Weakly interacting Bose gas : Bogoliubov transformation (1947)

• Diagonalization of the Hamiltonianof interacting bosonic particles

Diagonalized Hamiltonian

p1 p2

p2-qp1+q

Mutual interactionof condensate

Bogoliubov dispersion law for elementary excitations

N.N. Bogoliubov

Diagonalization using the linear transformation

Energy of acondensate Energy of excitations

0)(g ˆˆˆˆ2

1ˆˆ

2 ,,

2

21

2121>+= !! +

"+

+

+

qpp

ppqpqp

p

pp aaaaV

gaa

m

pH

,ˆˆˆ ,ˆˆˆ **

pppppppp bvbuabvbua pp !

+++

! !! +=+=

gnm

pp +=

2)(

2

!cpp =)(!

linearat small k

quadraticat large k

kinetic term interacting term

5

70 years after Einstein

E. Cornell

C.E. Wieman

W. Ketterle

Bogoliubov excitation spectrum

Bose-Einstein Condensation of Neutral Atoms(1995)

6

Outline

Bose-Einstein condensation and superfluidity

Bogoliubov theory for weakly interacting Bose gas

Interaction energy of exciton-polaritonsEnergy-momentum dispersion relation: revisited - Condensates and excitation spectrum

- Sound velocity

- Free particle energy

Quantum depletion vs. thermal depletion

Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -

Future prospects

7

Polariton-Polariton Interaction: Recap

s

Bexc

n

nE=!"h

• M. Kuwata-Gonokami et al., Phys. Rev. Lett. 79, 1341 (1997)• S. Schmitt-Rink, et al., Phys. Rev. B 32, 6601 (1985)

Exciton energy blue shift due to fermionic exchange interaction

Dipole moment reduction due to phase space filling and fermionic exchange interactionbased on Pauli’s exclusion principle based on Coulomb interaction

where

•C. Ciuti et al., Phys. Rev. B58, 7926 (1998)•P.R. Eastham et al., Phys. Rev. B68, 075324 (2003)

'0

sn

ngg !=" where

22*2.2 Xa

SNn

B

QW

s

!=

22*4

'

Xa

SNn

B

QW

s

!=

22*2.2 Xa

SNn

B

QW

s

!=

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

ExcitonLP UP

smaller spitting

Blue shift

Δωexc

g!

exc!"h

g!

Origin of strong mutual interaction because oflarge Bohr radius (~100A) .

22 )(42

)(2

phexkphexLPUP gE !!!! "+±+=

hh

!"

#$%

&'+'+= ]4/1[

2

1 2

0

2gX

exciton fraction at k=0

exciton binding energy

initial state

final state

Total number of polaritons

Inte

ract

ion

ener

gy (µ

eV)

10

100

1000

104

105

Numericalsimulation

TheoryExperiment

Interaction energy vs. total number of polaritons

gndN

dEnU

NV

gE

==

=

0

2

0

)(

2(Total energy of the condensate)

(Blue shift of the photon from the condensate at k=0 LP)

(S:fixed)

(nonlinear increase of S isincluded in Gross-Pitaevskiiequation)

4

)()(

42

1 22

0

22

0exc

excggg

!!

"#"+"+#

"++"=

hh

8

9

Outline

Bose-Einstein condensation and superfluidity

Bogoliubov theory for weakly interacting Bose gas

Interaction energy of exciton-polaritons

Energy-momentum dispersion relation: revisited - Condensates and excitation spectrum - Sound velocity - Free particle energyQuantum depletion vs. thermal depletion

Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -

Future prospects

10-15o 15o-15o 15o -15o 15o

1.613 eV1.613 eV

kξ=1

-15o 15o

1.613 eVCo-polarization detection

8 m

eV

kξ=1-15o 15o

1.613 eVLinear scale

Cross-polarization detection Mixed-polarization detection

Dispersion relation for untrapped condensatewith circularly polarized pump wave

11

-15o 15o

1.614 eV

-15o 15o

1.614 eV

kξ=1

-15o 15o-15o 15o

1.614 eV

8 m

eV1.614 eV

-15o 15o

P=1.2PthP=0.05Pth

Dispersion relation for trapped condensatewith different pump levels

P=6PthP=4Pth

12

Bogoliubov dispersion for trapped and untrappedcondensates

Linear dispersion at low momentum regime c=dE(p)/dp~108 cm/s(c~1cm/s for atomic BEC, c~104cm/s for 4He)

Nature Physics (in press)

Trapped

2

1

0

-1

E/U

-1.0 0.0 1.0

k!

BogoliubovA B

C D

quadratic

2

1

0

-1

E/µ

-1 0 1

k!

Untrapped

E: d=7um, Δ=3.3 (meV), P=2.9PthF: d=7um, Δ=2.9 (meV), P=3.8PthG: d=8um, Δ=1.6 (meV), P=2PthH: d=8um, Δ=2.5 (meV), P=3.8Pth

A: Δ=1.41 (meV), P=4PthB: Δ=0.82 (meV), P=8PthC: Δ=4.2 (meV), P=4PthD: Δ=-0.23 (meV), P=24Pth

Sound Velocity vs. Pump Power / Interaction Energy

13

Far Blue Detuning Zero Detuning

14

Energy shift Ek(P>Pth)-Ek(P<<Pth) vs. Interaction energy Ufor free particle regime (at kξ=1)

Ek(P>Pth)-Ek(PaPth)=2U

Interaction energy U (meV)

A B

C D

E B-E

LP (m

eV)

Interaction energy U (meV)

E F

G H

E B-E

LP (m

eV)

0.1

1

0.1 1

0.1

1

0.1 1

Trapped system Untrapped system

EK(P>Pth)-EK(P<<Pth)=2U

Energy shift for condensate particleU=gn0

Energy shift for non-condensed particleU=gn0

( )!!"

#+

#+++ ++++=

0

2

0

2

ˆˆˆˆˆˆ22

1

2ˆˆ

2

ˆ

p

pppppppp

p

aaaaaagnNV

gaa

m

pH

15

Quantum depletion vs. thermal depletion

Quantum depletion 　　　　　　　 (proportional to 1/k)

Thermal depletion 　　　　　　 (proportional to 1/k2)

pppppppp bbubbuvn !

+

!!

+

! ++= ˆˆ||ˆˆ|||| 222

!!"

#$$%

&

'=

+

1)](exp[

1ˆˆp

bb pp()

00ˆˆˆˆ aaaa

kk

+

!

+

k

mUnk

h2!

2)( k

Tmkn

B

k

h!

=0 at T=0

phk baa ˆˆˆ0

+

phonon

Quantumdepletion

Thermaldepletion

Quantum depletion Thermal depletion

16

Outline

Bose-Einstein condensation and superfluidity

Bogoliubov theory for weakly interacting Bose gas

Interaction energy of exciton-polaritons

Energy-momentum dispersion relation: revisited - Condensates and excitation spectrum

- Sound velocity

- Free particle energy

Quantum depletion vs. thermal depletion

Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -Future prospects

17

Cold atoms in 3D Optical lattice Experimental evidence forquantum phase transition from

BEC, superfluid to Mott insulator

M. Greiner et al., Nature 419, 6901 (2002)

Bose – Hubbard Hamiltonian

M.P.A. Fisher et al., PRB 40, 546 (1989)D. Jaksch et al., PRL 81, 3108 (1998)

Exciton-polaritons in 2D lattice 2D physicsMass can be varied over four orders of magnitudeOptical input and readout

Quantum EmulationQuantum Emulation- - SuperfluidSuperfluid to Mott Insulator Phase Transition in 3D-Optical Lattice - to Mott Insulator Phase Transition in 3D-Optical Lattice -

180 1 2 30

50

100

X/a

δE0 (

µeV

)

Au/Ti

12 mm

SEM

1.4µm line & space

Theory & Independent Measurements under a uniform layer: ~200µeVLimited by imaging resolution

Spatial modulation of LP energy

akzθ

substrate

k

DBR

DBR

λ/2

AlA

s ca

vity

3 st

acks

of 4

GaA

s Q

Ws

GaAlAsAlAs

Au/Ti strips

k||=k×sin(θ)

Au/Ti

Air Air

GaAlAs

GaAs

E

cavity only

k||

QW exciton

LPU0

cavity + Aucavityphoton

0

~2U0

a = 2.8 µm

LP energy can be spatially modulated by periodic metallicLP energy can be spatially modulated by periodic metallicfilmsfilms

19

on top of metal gate

bright = gap region bright = metal region bright = gap region

above thresholdbelow threshold well above threshold

Real Space and Momentum Space Distributions in 1DReal Space and Momentum Space Distributions in 1DLatticeLattice

19

20

“0-state” (s-wave)

π-state (s-wave)

”π-state” (p-wave)

Periodic potential

Nature 450, 529 (2007)

Below threshold

Above threshold

Anti-Phased p-wave and In-Phased s-wave inAnti-Phased p-wave and In-Phased s-wave inOne-Dimensional One-Dimensional Exciton-PolaritonExciton-Polariton Condensate Array Condensate Array

(π state) (0 state)

Excited state condensation Ground state condensation

Diffraction Patterns of the Diffraction Patterns of the ““Zero-StateZero-State”” and and ””ππ-State-State””

21

22

n2

Pump: n3

Γ3<< Γ1, Γ2

Γ2

Γ1

Γ31<< Γ32Γ21n1

π-state

0-state

Γ32

0 20 40 60 80 1000

2

4

6

8

10

PL

In

ten

sit

y (

arb

. u

nit

s)

Time (ps)

zero-state

!-state

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

zero-state

Inte

ns

ity

(a

rb.

un

its

)

Time (ps)

!-state

MetastableMetastable ππ-State-State (p-wave) Forms before Stable Zero-State (s-wave)(p-wave) Forms before Stable Zero-State (s-wave)

23

Future Prospect

• Quantum states of the ground state (phase-locking, quantum correlation)• Superfluidity (second sound wave, quantized vortices)• Exotic excitation spectrum (maxon, roton, negative-energy branch)•Orbital physics (p-wave, d-wave and f-wave superfluid states)

Physics of Bose-Einstein Condensation

Quantum emulation of many body systems

• Quantum annealing machine (bosonic stimulated cooling)Ground state searching

• Polariton BEC as laser without inversion• Electrically driven polariton laser

New coherent light source

BEC-BKT cross-over, BEC-BCS cross-over

Generation of single photons (repulsive interaction) andentangled photon-pairs (attractive interaction) from MI state

Phys. Rev. A 77, 031803(R) (2008).

Q/104

Goal: Construction of Bose-Hubbard phase diagram and validationcheck with cold-atom-based quantum emulator

PolaritonicPolaritonic Quantum Emulator: Quantum Emulator: SuperfluidSuperfluid to Mott- Insulator to Mott- InsulatorQuantum Phase TransitionQuantum Phase Transition

24

arXiv:0804.1829 (2008)

Generation of Indistinguishable Single Photons andGeneration of Indistinguishable Single Photons andPolarization Entangled Photon PairsPolarization Entangled Photon Pairs

25