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, Russia2Physics and Astronomy School, University of Southampton, UK

SOLAB seminar, Apr. 17, 2012

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

Motivation for experimentalists

pump-L pump-R

BraggMirrors

Quantum Well

Mean-field model

↵2 = �0.1↵1

⌧ ' 10ps

8<:iù @

@t +(x, t) =⇣� ù2

2mr2 +↵1| +(x, t)|2 +↵2| �(x, t)|2 � iù2⌧

⌘ +(x, t)+ p+(x, t)

iù @@t �(x, t) =

⇣� ù2

2mr2 +↵1| �(x, t)|2 +↵2| +(x, t)|2 � iù2⌧

⌘ �(x, t)+ p�(x, t)

m ' 3⇥ 10�5m0

p±(x, t) =pL±e�(x�x

L

)

2

�x

2e

iw

L

t+ikL

·x+

p

R

±e� (x�x

R

)

2

�x

2e

iw

R

t+ikR

·x

Numerical implementation

• Spatial discretization by using Finite Element Method

• Quadratic elements

• Grid size: ∆x<0.25µm @ Lch~100µm

• Time discretization

• 5-step Backward Differentiation Formula

• Max time step: ∆t<100fs @ τ=10ps

• Implementation using Comsol (2D and 1D) and a private software (1D)

u0 = f(t,u), u(t0) = u0;sX

k=1

akun+k = h�f(tn+s , un+s);

tn = t0 +nh.

Interference of two pump pulses

Interference of two pump pulses (2)current density

j = � iù2m

( ⇤±r ± � ±r ⇤±)

Interference of two pump pulses (2)current density

j = � iù2m

( ⇤±r ± � ±r ⇤±)

Spin-polarized polariton transport

Spin-polarized polariton transport

Spin-polarized polariton transport

⇢c =| +|2 � | �|2| +|2 + | �|2

Conductivity tensor

j = ùm

hn+r'+n�r'�

i=⇣�++ �+���+ ���

⌘hµ+R�µ+Lµ�R�µ�L

i

± =pn±ei'±

@n±@t

+ div j = 0

iù @@t ± =

⇣� ù

2

2mr2 +↵1| ±|2 +↵2| ⌥|2 �

iù2⌧

⌘ ± + P±

±(x, t) = ±(x)eiµ±t/ù P± = p0e�(x�x0)2

�2 ei!±t/ù+ik±·x

Imaginary part of GP eq. decomposition gives:

�µ± ±(x) =⇣� ù

2

2mr2 +↵1| ±|2 +↵2| ⌥|2 �

iù2⌧

⌘ ±(x)+ P 0±(x)

Spin-current density with diferent pump-pulses

j = ùm

hn+r'+n�r'�

i=⇣�++ �+���+ ���

⌘hµ+R�µ+Lµ�R�µ�L

i

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