dynamics of formation and decay of coherence in a ... · dynamics of formation and decay of...

Dynamics of formation and decay

of coherence in a polariton BECof coherence in a polariton BECElena del Valle8 September 2008

SEMICUAM (up to this summer)SEMICUAM (up to this summer)

Experiments: Daniele Sanvitto, Alberto Amo, Dario Ballarini, Lola Martín, Luis Viña

&Theory: Fabrice Laussy, Carlos Tejedor

Outline

1. Introduction on polaritons and polariton BEC 2. Time-Polarization resolved pulsed experiment in CdTe

µµµµ.c.: dynamics of coherence in BEC3. Reproduction of dynamics with a theoretical model4. Conclusions

Excitons

( ) [ ]�� ++++++ +=��

�

�

��

�

�

+++=⊕ k

kkukkk

lk

kkkkkkkkcavkkkexckpol qqEppExcxcgccxxH

RWAH��

int

,,ˆ ωω

Cavity Photons

Dispersion

Excitons Cavity Photons

(Planar Distributed

Bragg Reflector)

2/1 Ban << ⊕

Hopfield Transformation

Strong Coupling Regime Upper polaritons

( )22,,

, 421

kkcavkexckqp

k gE +±+= δωω

exckcavkk ,, ωωδ −≡Cavity

Polariton

00 ==kδ

Bare excitons

Bare

photons

Lower polaritons

• Bosonic behaviour: Bose statistcs

final state stimulation

• Spin degree of freedom ( ) : transferred to light polarization (right/left circular)

Characteristics

↓↑• Spin degree of freedom ( ) : transferred to light polarization (right/left circular)

• 2D particles with light mass:

Berezinskii-Kosterlitz-Thoules phase transition

BEC/lasing at room temperature

↓↑

• Relaxation processes: PL-PL scattering, phonon emission, electron-PL… allows for efficient population transfer towards ground state

( ) ∞ →−

= →→−

0

0

,0

1

1 ETT

TKE

BEC

Ben µ

µ

knn kETT BEC ∀>> → →→ ,0

, 0µ

• Spontaneous Symmetry Breaking with an order parameter: the condensate wave function

• Ground state macro population

of weakly-interacting bosons

BEC of polaritons


• Ideally, with no decoherence, at thermal equilibrium, BEC is a macroscopic coherent state

• BEC of two spin components � Spontaneous appearance of polarization

↑↓Ψ

( ) ( )trtr ,,ˆ00

�� Ψ=Ψ

( )( ) 102 ==τgα φα ien=

↑↓Ψ0

arXiv:0808.1674 (2008)

( ) ∞ →−

= →→−

0

0

,0

1

1 ETT

TKE

BEC

Ben µ

µ


, 0µ



weakly-interacting bosons

BEC of polaritons



• BEC of two spin components � Appearance of linear polarization

↑↓Ψ

( ) ( )trtr ,,ˆ00

�� Ψ=Ψ

( )( ) 102 ==τgα φα ien=F. P. Laussy et al.,

Phys. Rev. B 73, 035315 (2006)I. A. Shelykh et al.,

Phys. Rev. Lett. 97, 066402 (2006)↑↓Ψ0Phys. Rev. Lett. 97, 066402 (2006)

0>−

= ⇔

totl I

IID �

Degree of linear polarization

( ) ∞ →−

= →→−

0

0

,0

1

1 ETT

TKE

BEC

Ben µ

µ


, 0µ



weakly-interacting bosons

BEC of polaritons



• BEC of two spin components � Appearance of linear polarization

( ) ( )trtr ,,ˆ00

�� Ψ=Ψ

( )( ) 102 ==τgα φα ien=F. P. Laussy et al.,

Phys. Rev. B 73, 035315 (2006)I. A. Shelykh et al.,

Phys. Rev. Lett. 97, 066402 (2006)0>−

= ⇔ IID �

• Pinning of the linear polarization axis:

cavity/well asymmetry

exciton anisotropic localization

Phys. Rev. Lett. 97, 066402 (2006)

J. Kasprzak et al., Nature 73, 035315 (2006)

0>−

= ⇔

totl I

IID �

Outline



Pulsed non-resonant excitation

CdTe semiconductor microcavity

(R. André, U. Grenoble)Excitation:

circularly polarised2ps pulsed laser

at 0<t

00 ==kδmeVRabi 24=

at 0<t

More details on experiments: Daniele Sanvitto’s talk on Wednesday


Excitation:

circularly polarised ps pulsed laser

at 0<t


(R. André, U. Grenoble)

00 ==kδmeVRabi 24=

at

Relaxation: PL-PL interactions, phonon emission

Lower-polariton formation

0<t



Excitation:

circularly polarised ps pulsed laser

at 0<t


(R. André, U. Grenoble)

00 ==kδmeVRabi 24=

at

Relaxation: PL-PL interactions, phonon emission

Lower-polariton formation

0<t

Spectrometer Streak Camera


Ground state emission

Analysed in time (from ):

• Blueshift and intensity

• Linewidth of spectral lineshape

• Direction/degree of linear polarization

0=t

Ground state energy blueshift

We observe in strong coupling:

(a) Blueshifted emission thatrecovers monotonicallythe bare lower polariton, withthe total branch populationdecay

(b) This is not affected by theground state populationdynamics of formation-decay

1meV

Ground state energy blueshift

We observe in strong coupling:

(a) Blueshifted emission thatrecovers monotonicallythe bare lower polariton, withthe total branch populationdecay

(b) This is not affected by theground state population

Transition at around 60ps?

1meV

dynamics of formation-decay

Total ground state population

0n

0n

0

Pulse arrival

Transition?

( )pstimeExponential

increase

(logaritmic scale)

Energy linewidth of ground state emission

( )meV0κ

( )0

00

0 1

1

n

nWk

kk

+

Γ++≈� →

κ5.25max0 ≈n

Line narrowing

0n

0κ

and recovering

(logaritmic scale)

meV896.00lim0 =Γ≈κ

bare polaritonlifetime of 0.73ps

Transition around

pstcoh 61≈

( )pstime

10 ≈nD. Porras et al.,

Phys. Rev. B 67, 161310(R) (2003)

Degree of linear polarization of ground state emission

When fitting procedure breaks, it can lead to

�

><

10

lDlD

0n

0κ

lD

(we consider 0) �>1

Transition

pstcoh 61≈

( )pstime


lD

0n

0κ

lD

Spontaneous

formation

Transition

pstcoh 61≈

( )pstime

formation

(logaritmic scale)


↓+↓↑

+↑

↓+↑

++

+

+⇔

+⇔

+⇔

+⇔⇔

+=

+=

+

−=

−=

pppp

pp

cccc

cc

cccc

cccc

I

IID

leftleftrightright

leftright

totl

]Re[2]Re[2

��

��

Intensity of light emitted � mean number of cavity photons � polariton spin at k=0

=lD


pure/mixed linearly pol. state1

�eliptically polarised state

partially polarised light

=lD

0 Pure circularly polarised

unpolarised light

mixed state state of circular pol.

�


↓+↓↑

+↑

↓+↑

++

+

+⇔

+⇔

+⇔

+⇔⇔

+=

+=

+

−=

−=

pppp

pp

cccc

cc

cccc

cccc

I

IID

leftleftrightright

leftright

totl

]Re[2]Re[2

��

��


=lD


pure/mixed linearly pol. state with 1

�eliptically polarised state

partially polarised light

pure coherent states φβαβα ienn == ,, ( ) ↓↑

+=

nnCosDl

2φ

0,, ≠↓↑↓↑ mmnn ρ

=lD

0 Pure circularly polarised

unpolarised light

mixed state state of circular pol. ( polaritons: )

� pure coherent states

Cothermal state

φβαβα ienn ↓↑↓↑ == ,,

( )�↓↑

↓↑↓↑↓↑=nn

nnnnnnP,

,,,ρ↓↑

( )↓↑ +

=nn

CosDl φ

Cothermal state

Interpolates between thermal mixture and a coherent pure stateF. P. Laussy et al.,

Phys. Rev. B 73, 035315 (2006)

( )

( ) =

=�ρ

nnP

nnnP

nth

nthth

( )!

=

=

−

αα φ

nn

enP

enncohn

Poisson

icoh

coh

Adequate in the ( ) ( )( )( )

0220

122

1

=−===

+= +

χχτg

nn

nP nth

thth

( )( )( )

110

!2

===

χτg

nPoissonAdequate in the presence of decoherence

n n

Cothermal state

Interpolates between thermal mixture and a coherent pure stateF. P. Laussy et al.,

Phys. Rev. B 73, 035315 (2006)

( )

( ) =

=�ρ

nnP

nnnP

nth

nthth

( )!

=

=

−

αα φ

nn

enP

enncohn

Poisson

icoh

coh

Adequate in the ( ) ( )( )( )

0220

122

1

=−===

+= +

χχτg

nn

nP nth

thth

( )( )( )

110

!2

===

χτg

nPoisson

Convolution of the Glauber’s P probability distribution representations:

( ) �= � Pdd ** , ααααααρ

Adequate in the presence of decoherence

n n

thcoh nnn +=0

( ) ( ) ( )[ ]( )[ ]

( )( )( )��

�

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

1

01

0

011coth

0

0

nn

nenP nn

nn

n

0nncoh=χ

( )

( ) �

�

�

=

=

−−

⊄�

thi nen

th

en

P

Pdd

/*coth

*coth

*coth

2coth1

1,

,

φα

παα

ααααααρ

Coherent and thermal fractions: Second order coherence degree:

10 ≤≤ χ

Degree of linear polarization of cothermal states

Supposing that in the ground state each spin polaritons are in similar cothermal states

n

�

=≈

==≈

=≈

↓↑

↓↑

↓↑

2

2

0

0

coh

coh

n

nn

nnn

αα

χχχ↓↑ ⊗= cothcoth0 ρρρ

F. P. Laussy et al., Phys. Rev. B 73, 035315 (2006)

χ≈+

=−

=↓

+↓↑

+↑

↓+↑⇔

pppp

pp

I

IID

totl

]Re[2�


lD

0n

0κ

lD

75.0max ≈lD

Transition

pstcoh 61≈

( )pstime


lDCoherence formation

Coherence decay

Transition

pstcoh 61≈

( )pstime

Coherence disappears

pstdecoh 290≈

Coherent and thermal fractions of ground state emission

Coherence formation

Coherence decay

0n

cohn

thn

Transition

pstcoh 61≈

( )pstime

Coherence disappears

pstdecoh 290≈

Evolution of particle probability distribution

in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

Thermal

( )0nP

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

Growing coherence

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

Maximum coherence

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

Decaying coherence

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

lD≈χ( ) ( ) ( )[ ]

( )[ ]( )

( )( )��

��

�

−+−−

−+

−= +

−+−

χχχ

χχχ

χ

111/

Laguerre11

11

011coth

0

0

nn

nenP nn

nn

n

( )pstime

( )0nP

Back to thermal

( )[ ] ( )( )�� −+−+ + χχ 1111 0

10

coth nnnn

( )0nP

0n


in the ground state

( )pstime

( )0nP

0n

( )pstime

Polarization of ground state emission

Polar plot of emitted intensity (through polarizer):

To follow the evolution of linear polarization

( )pstime

To follow the evolution of linear polarization


( )pstime


( )pstime

Pinning


( )pstime


( )pstime

Maximum degree


( )pstime

Histeresis with 0n

( )0nDl

0nn

D cohl ≈

0


t

Transition

pstcoh 61≈

0n

Histeresis with 0n

( )0nDl

0nn

D cohl ≈

( )00 nκ

0

( )0

00

0 1

1

n

nWk

kk

+

Γ++≈� →

κ


t

Transition

pstcoh 61≈

D. Porras et al., Phys. Rev. B 67, 161310(R) (2003)

Interactions start to broaden at

0n

meVVn 001.00int0 ≈≈κ

Outline



Reproducing the coherence and ground state population dynamics

Pulsed coherent excitation

Last steps of each spin relaxation chain:

(two is enough for qualitative description)

1

Effective pulsed incoherent excitation

( )tW1

0

0Γ

01→W

1

1Γ10→W from upper levels

( )tW1

1001 →→ > WWphoton emission

(but it could be PL-PL interactions---evaporative cooling)

Formalism: density matrix, master eq. and Lindblad terms

W

1

Γ10→W

( )tW1

0

0Γ

01→W 1Γ

• harmonic oscillators10 , aa

• Incoherent dynamics with a master eq.

( diagonal but not separable )

• We follow coherence through the statistics (cothermal):

• Linewidth extracted from spectrum

(quantum regression th.)

ρ( ) ( ) �� ==

11

1010100 ,,,nn

nnnnnnPnP ρ

( ) ( ) ( )�∞

+ +≈0

00Re, ωτττω ietatadtS

10 ρρρ ⊗≠

( ) ( )

( )111111

1 22

)(aaaaaa

tWdt

td +++

Γ

−−≈ ρρρρ

Formalism: density matrix, master eq. and Lindblad terms

W

1

Γ10→W

( )tW1

( )

( ) ( ) ( )( ) ( )( )[ ]( ) ( ) ( ) ( ) ( ) ( )[ ]

( )0000000

10101010101010

10101010101001

1111111

22

22

22

22

aaaaaa

aaaaaaaaaaaaW

aaaaaaaaaaaaW

aaaaaa

+++

+++++++++→

+++++++++→

+++

−−Γ+

−−+

−−+

−−Γ+

ρρρ

ρρρ

ρρρ

ρρρ0

0Γ

01→W 1Γ

• harmonic oscillators10 , aa

• Incoherent dynamics with a master eq.

( diagonal but not separable )

• We follow coherence through the statistics (cothermal):

• Linewidth extracted from spectrum

(quantum regression th.)

ρ( ) ( ) �� ==

11

1010100 ,,,nn

nnnnnnPnP ρ

( ) ( ) ( )�∞

+ +≈0

00Re, ωτττω ietatadtS

10 ρρρ ⊗≠

( ) 101010 ,,, nnnndtd

nnPdtd ≈= ρ

Growing Poissonian distribution

W

1

Γ10→W

( )tW1

( )( ) ( ) ( )( )[ ] ( )

( ) ( ) ( )( ) ( )( ) ( )

( ) ( )1000

101001

100110

10111011

100101001011111

101010

,11

1,11

1,11

1,11,)(

,11)(1

,,,

nnPn

nnPWnn

nnPWnn

nnPnnnPtWn

nnPWnnWnnntWn

nnnndt

nnPdt

+Γ++−++++−++

+Γ++−+Γ+++++Γ++−

≈=

→

→

→→

ρ

0

0Γ

01→W 1Γ

( ) 101010 ,,, nnnndtd

nnPdtd ≈= ρ


W

1

Γ10→W

( )tW1

effΓ( )

( ) ( ) ( )( )[ ] ( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )1000

101001

100110

10111011

100101001011111

101010

,11

1,11

1,11

1,11,)(

,11)(1

,,,

nnPn

nnPWnn

nnPWnn

nnPnnnPtWn

nnPWnnWnnntWn

nnnndt

nnPdt

+Γ++−++++−++

+Γ++−+Γ+++++Γ++−

≈=

→

→

→→

ρ

0

0Γ

01→W 1Γ

( )κ t

( )( ) 2

00

00002 0,aa

aaaatg

+

++

==τ

( )22 g−=χor fitting with cothermal( )0nP

( )( )

( ) 22

0

0

2

21,

ωκ

κ

πω

+��

��

�≈

t

t

tS

ω0κ

( )0

01010 1

1

n

Wn

+Γ++

≈ →κ


0n

0κ

lD

0n

cohn

thn

( )pst

l thn

Conclusions

• We analyzed experimental results on the dynamics of formation and decay of coherence of polaritons through the degree of linear decay of coherence of polaritons through the degree of linear polarization.

• With a description in terms of cothermal states for both spins, wecould extract the coherent/thermal fraction and analyse the particleprobability distribution.

• A model of incoherent transfer of population from one excited level tothe ground state reproduces qualitatively the coherence dynamics.

• As a consequence, we report BEC of polaritons created under pulsedexcitations, with a fraction of ~75% of the particles in a condensedstate formed out of an incoherent bath.

Dynamics of formation and decay

of coherence in a polariton BECof coherence in a polariton BECElena del Valle8 September 2008

dynamics of formation and decay of coherence in a ... · dynamics of formation and decay of...

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