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