Quantum Fluids of Light

L ^ABORATOIRE K ^ASTLER B ^ROSSEL

R

S

Quantum Fluids of Light

Author:

Titus FANZ

Supervisor:

Alberto BRAMATI

Master LUMI in the

Quantum Optics Group Sorbonne University

2017

T µ

N =X

k

nk=X

k

1 e ^{(✏k µ)} 1

nk k = _kB¹_T

✏k = ^¯h_2m^2k2 µ <0 nk <0

P

k! lim

V!1 V (2⇡)^d

R1 0 d^dk n= ^N_V > nc µ= 0

n0 k= 0 n=n0+

Z

k6=0

d^dk (2⇡)^d

1 e ^{(✏k µ)} 1.

n0

c

ei =_4n⁰

i 0 ni

i 2 {a, b}

✓ k = (k_k, k_z)

k_z = 2⇡nc 0

✓

E (k_k =¯hc nc

s✓2⇡nc 0

◆2

+⇣ k_k⌘2

⇡hc

0 +¯h²k²_k 2m^⇤ m^⇤ = ^n2ch

0c

⌧c =Lef f

nc

⇡ F

c Lef f = c

⇣1 + 2_n^nanb

b na

⌘ F

Incident light

| {z }

| {z } | {z }

optical cavity of length

c=m_2nc layera

with thickness ea=m_4na

layerb with thickness

eb=m_4nb

0

ea eb

k = (k_k,k_z) ✓

E_exc^2D(k^exc_k ) =Eexc+

⇣¯hk_k^exc⌘2

2m^⇤_exc

Eexc k_k^exc

m^⇤_exc

k^exc=k = Enc

¯

hc sin✓^c,

8

E ✓^c

Hˆ =X

k

✓

Eexc(k)ˆb^†_kˆbk+E (k)ˆa^†_kˆak+¯h⌦R

2

⇣ˆa^†_kˆbk+ ˆb^†_kˆak

⌘◆,

ˆbk ˆb^†_k ˆak ˆa^†_k

k = |k_k| ⌦R

ˆ

⇢^(LP)_k ˆ

⇢^{(U P}_k ⁾

!

=

✓ Ck Xk

Xk Ck

◆ ✓ˆak

ˆbk

◆ .

ˆ

⇢^(LP)_k ⇢ˆ^{(U P}_k ⁾

E(U P)=1 2

✓

Eexc(k) +E (k) +q

2

k+ (¯h⌦R)²

◆ ,

E_(LP)= 1 2

✓

Eexc(k) +E (k) q

2k+ (¯h⌦R)²

◆ .

InGaAs GaAs GaAs

Incident light

| {z }

| {z } | {z }

optical cavity of length

c=m_2nc layera

with thickness ea=m_4na

layerb with thickness

eb=m_4nb

| {z } | {z }

Bragg mirror

| {z }

Polariton

0

ea eb

k = (k_k,k_z)

✓

ˆb^†_k+qˆbk⁰ qˆbkˆbk⁰

ˆ

a^†_k+qˆb^†_k0 qˆbkˆbk⁰

ˆ

⇢^{(U P)}

Hˆ = ˆHlin+ ˆHint=X

k

2

4E_k⇢ˆ^†_k⇢ˆk+1 2

X

k⁰,q

V_k,k^{pol pol}0,q ⇢ˆ^†_{k q}⇢ˆ^†_k0+q⇢ˆk⇢ˆk⁰

3 5.

ˆ

⇢k= 1 V

Z

r (r)eˆ ^k·r/¯h

q

V_k,k^{pol pol}0,q =V_q^{pol pol}= Z

rV^{pol pol}(r)e ^k·r/¯h,

Hˆ = Z

r

✓¯h²

2mrˆ^†rˆ + ˆ^†Vext(r) ˆ

◆

+1 2

ZZ

r r⁰⇣

ˆ^†(r) ˆ^†(r⁰)V^{pol pol}(r r⁰) ˆ (r⁰) ˆ (r)⌘ .

i¯h@ (r, t)ˆ

@t =h

(r, t),ˆ Hˆi

=

 ¯h²r²

2m +Vext(r, t) + Z

r⁰ˆ^†(r⁰, t)V^{pol pol}(r⁰ r) ˆ (r⁰, t) ˆ (r, t).

ˆ =a0⇢ˆ0+X

i6=0

ai⇢ˆ0⇡a0⇢ˆ0⇡ ⁰.

Pthr

Pthr Pthr a

±23

k_k= 0

b a

V^{pol pol}(r⁰ r) =g (r⁰ r)

t ⁰(r, t) =

 ¯h²r²

2m +Vext(r, t) +g| ⁰(r, t)|^{2 0}(r, t).

!L

=Elas ELP = 0 Elas= ¯h!L

ELP

t ⁰(r, t) =

 ¯h²r²

2m +Vext(r, t) +g| ⁰(r, t)|² i¯h

2 ⁰(r, t) +FL(r, t).

FL(r, t)

kLP ⌘

(3)

!0

0 =@_z²E(r, z) +r²?E(r, t) +!²₀ c²

⇣✏+ ✏(r, z) + ³⁾|E(r, z)|²⌘

E(r, z).

E(r, z) (r, z) = (x, y, z)

✏ ✏(r, z)

E(r, z) = E(r, z)e ^ik0z

i@zE(r, z) = r²?E(r, t) k²₀ 2✏

⇣ ✏(r, z) + ³⁾|E(r, z)|²⌘ E(r, z).

z

Vext(r, t) ˆ= ^k0_2✏^✏(r,z)

(3)

g= ⁽³⁾₂^k0

T =

2.17K

v

p E0+✏(p) E0

✏(p

E⁰=E0+✏(p) p·v+1 2Mv².

E =✏(p) p·v<0 ,v > vc=✏(p)

p .

v < vc= lim

p!0

✏(p) p .

0

(r, t) =h

0(r) +Ae^{i(l·r !t)}B^⇤e ^{i(l·r !t)}i e ^iµth¯.

A B^⇤

e^±!t

¯

h!_±=±

s✓¯h²k² 2m

◆²

+¯h²k² m gn0.

cs= @!

@k ^k=0= rgn0

m .

v = ^hk¯_m⁰ cs

v > cs

I III IV VI

Pthr I IV

Pthr III VI

k_k

II V

I III IV VI

Pthr I IV

Pthr III VI

k_k

II V

(r, t) =p

n0(r, t)e^i✓(r,t).

@n

@t +r·(n0(r, t)v(r, t)) = 0 v(r, t) = _m^¯hr✓(r, t)

r ⇥v= _m^h¯r ⇥ r✓= 0

0(r= (r, )) =p

n0(r)e^{.✓( )}

v(r) = ¯h mr

@✓

@ u = ¯h mrlu✓.

✓

✓ 2⇡

I

C

v s= ¯h ml.

1

r n0(r)^r!!⁰0

l² E_kin^l =

ZZ 1

2mv²(r)nl(r) ²r/l²..

N l = 1

l=N

Epair / ZZ

(v1(r) +v2(r))^{2 2}r )Epair =E_kin^l1 +E_kin^l2 + ⇤l1l2.

✓ k_k

= 0,5,5,10,15,21

3⇡ N

l = 1

l = N l = 1

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