1. Polariton spin transport in a microcavity channel:
A mean-field modeling
M.Yu.Petrov1 and A.V.Kavokin1,2
1Spin Optics Laboratory, Saint Petersburg State University, Russia
2Physics and Astronomy School, University of Southampton, UK
SOLAB seminar, Apr. 17, 2012

2. Outline
• Motivation for experimentalists
• Mean-field model and its numerical implementation
• Results of modeling
• Interference of two flows in a channel
• Spin-polarized polariton transport in a channel
• Interpretation in terms of spin conductivity
• Summary and further steps

3. Motivation for experimentalists
pump-L pump-R
Quantum Well
Bragg
Mirrors

4. Mean-field model
↵2 = 0.1↵1
8 ⇣ 2
⌘
<iù @ ù iù
@t + (x, t) = r2 + ↵1 | 2
+ (x, t)| + ↵2 | (x, t)|2 + (x, t) + p+ (x, t)
⇣ 2m 2⌧ ⌘
:iù @ ù2 iù
@t
(x, t) = 2m
r2 + ↵1 | (x, t)|2 + ↵2 | + (x, t)|
2
2⌧
(x, t) + p (x, t)
⌧ ' 10ps
5
m ' 3 ⇥ 10 m0 (x xL )2
L
p± (x, t) =p± e x2 eiwL t+ikL ·x +
(x xR )2
R
p± e x2 eiwR t+ikR ·x

5. Numerical implementation
• Spatial discretization by using Finite Element Method
• Quadratic elements
• Grid size: ∆x<0.25µm @ Lch~100µm
• Time discretization
u0 = f (t, u), u(t0 ) = u0 ;
• 5-step Backward Differentiation Formula s
X
ak un+k = h f (tn+s , un+s );
k=1
• Max time step: ∆t<100fs @ τ=10ps tn = t0 + nh.
• Implementation using Comsol (2D and 1D) and a private software (1D)

6. Interference of two pump pulses

7. Interference of two pump pulses (2)
current density
iù ⇤ ⇤
j= ( ±r ± ±r ±)
2m

8. Interference of two pump pulses (2)
current density
iù ⇤ ⇤
j= ( ±r ± ±r ±)
2m

9. Spin-polarized polariton transport

10. Spin-polarized polariton transport

11. Spin-polarized polariton transport
2 2
| +| | |
⇢c =
| + |2 +| |2

12. Conductivity tensor
@ ⇣ ù2 2 iù ⌘
2 2
iù ± = r + ↵1 | ±| + ↵2 | ⌥| ± + P±
@t 2m 2⌧
p
± = n± ei'±
@n±
Imaginary part of GP eq. decomposition gives: + div j = 0
@t
ù h n+ r'+ i ⇣ ⌘h + +
µR µL
i
j= n r' = ++ +
µR µL
m +
(x x0 )2
± (x, t) = ± (x)e
iµ± t/ù
P± = p0 e 2
ei!± t/ù+ik± ·x
⇣ ù2 2 iù ⌘
2 2 0
µ± ± (x) = r + ↵1 | ±| + ↵2 | ⌥| ± (x) + P± (x)
2m 2⌧

13. Spin-current density with diferent pump-pulses
h
ù n+ r'+ i ⇣ ⌘h + + i
µR µL
j= n r' = ++ +
µR µL
m +

14. Summary and further steps
• A mean-field model describing polariton spin transport based on coupled
Gross-Pitaevskii equations is developed
• Numerical implementation of the model demonstrates interference of two
flows stimulated by CW excitation near both boundaries of a channel
• If pumps are cross-polarized the effect can be emphasized by detection of
circular polarization degree
• Further steps
• Possibility of experimental observation