AMPLITUDE AND PHASE DYNAMICS OF SPIN-DEGENERATEDPOLARITON CONDENSATE IN SEMICONDUCTOR MICROCAVITIES MAI HOANG LINH PHI, CAO HUY THIEN Ho Chi Minh City Institude of Physics, Vietnam Acade
Trang 1AMPLITUDE AND PHASE DYNAMICS OF SPIN-DEGENERATED
POLARITON CONDENSATE IN SEMICONDUCTOR
MICROCAVITIES
MAI HOANG LINH PHI, CAO HUY THIEN
Ho Chi Minh City Institude of Physics, Vietnam Academy of Science and Technology, 1
Mac Dinh Chi District 1 Ho Chi Minh City Viet Nam
Abstract The complex Gross-Pitaevskii equations for amplitude and phase of spin-degenerated polaritons condensate in semiconductor microcavities are built within spinor polariton model These equations have been obtained by adiabatic elimination the corresponding three-point po-larizations with p-p scattering as the dominant mechanism If neglecting scattering terms, our results is agreement with calculation of M.O.Borgh et al [9] Together with the solutions the Boltz-mann equations for the excited states, our result will form the basis for a complete evaluation of the linewidth and the second-order correlation function.
I INTRODUCTION Bose-Einstein condensation of polaritons in semiconductor microcavities have been recently shown in many experiment [1], [2] Many theoretical calculations have been caried out to confirm existence of the condensated state of microcavity polariton The result
by H.Haug et al for polariton distribution function and chemical potential by using the semiclassical Boltzmann equation [3] is good quantitative agreement with experimental observation by Yamamoto et al [1] Recently, the interest in the condensed microcavity polaritons shifted toward detailed studies of the dynamics of their coherence properties The coherence has been found by a Schawlow-Townes decrease in the emission linewidth, but only at slightly higher values of the condensate populations, the linewidth is observed
to go through a sharp minimum, after that, the linewidth increase again The analytical formula and numerical results for linewidth have been evaluated completely by H.Haug
et al by using quantum Langevin equation for the coherent condensate amplitude [4] Another important test of the coherence properties of the condensate is the second-order correlation function This function have been measured by Deng et al for GaAs mc’s [7] and Kasprazk et al in CdTe mc’s [8], and have been calculated by many different methods [5], [4], [6] Coherence of condensed microcavity polaritons has been continued studying for spinor polariton model A method to derive analytical formula for the linewidth and the second-order correlation function is building condensed amplitude and phase equations
In this paper, spinor polariton model is studied, with p-p scattering as the domi-nant mechanism We build the complex Gross-Piteavskii equations for the ground-state operators The corresponding three-point polarizations have been eliminated adiabati-cally to obtain the complex Gross-Pitaevskii equations for amplitude and phase of spin-degenerated polaritons condensate in semiconductor microcavities
Trang 2II DERIVATION OF THE COMPLEX GROSS-PITEAVSKII FOR THE
GROUND-STATE OPERATORS Hamiltonian of spinor polariton in semiconductor microcavity:
~
k,s
e~kb+~
k,sb~k,s+1
2 X
~ k
Ω~kb~+
k,1b~k,2+ H.c
+1
4
"
X
~
k, ~ k 0 ,~ q,s
V (~k + ~q, ~k0− ~q, ~k, ~k0)b+~
k,sb+~
k 0 ,sbk~0 −~ q,sb~k+~q,s
~k, ~k0 ,~ q,s6=s 0
U (~k + ~q, ~k0− ~q, ~k, ~k0)
b+~
k,sb+~
k 0 ,s 0bk~0 −~ q,s 0b~k+~q,s+ b~+
k,sb+~
k 0 ,s 0bk~0 −~ q,sb~k+~q,s0
+ H.c
#
Where e~k is energy of the lower p branch Ω~k is complex energy, related to the
TE-TM splitting The matrix elements V ~k + ~q, ~k0− ~q, ~k, ~k0
, U~k + ~q, ~k0− ~q, ~k, ~k0
describe scattering of ps in the relative triplet and singlet configuration, respectively They are given by:
V ~k, ~k0, ~k0− ~q, ~k + ~q
= 6EBa2Bu~k+~quk~0 −~ quk~0u~k
U~k, ~k0, ~k0− ~q, ~k + ~q = −αV ~k, ~k0, ~k0− ~q, ~k + ~q, α > 0
Where EB and ab are the binding energy and Borh radius of exciton in 2D, respectively The Heisenberg equations for the ground-state operators (with i, j are spin indexes (i 6= j))
db0,i
i
~e0b0,i− i
2~Ω0b0,j
−i
~ X
~k
V (~k, 0, ~k, 0)n~k,ib0,i− i
2~
X
~k6=~q,~q6=0
V (~k − ~q, ~q, ~k, 0)b~+
k,ib~k−~q,ib~q,i
−i
~ X
~k
U (~k, 0, ~k, 0)n~k,jb0,i− i
~ X
~k,~q6=0
U (~k − ~q, ~k, ~q, 0)b~+
k,jb~k−~q,jb~q,i (1)
Adiabatic elimination the corresponding three-point polarizations, we have the complex Gross-Piteavskii for the ground-state operators:
db0,i
i
~e0b0,i− i
2~Ω0b0,j
−i
~ X
~ k
V (~k, 0, ~k, 0)n~k,ib0,i− i
~ X
~ k
U (~k, 0, ~k, 0)n~k,jb0,i
+ i 2~2
X
~ k6=~ q,~ q6=0
| V (~k, ~k − ~q, ~q, 0) |2
n~k−~q,in~q,i− n~k,i1 + n~k−~q,i
− n~k,in~q,i
P
Ωi
b0,i
Trang 3+ π 2~2
X
~k6=~q,~q6=0
| V (~k, ~k − ~q, ~q, 0) |2
n~k−~q,in~q,i− n~k,i1 + n~k−~q,i− n~k,in~q,iδ (Ωi) b0,i + i
~2 X
~ q6=0
| U (~k, ~k − ~q, ~q, 0) |2
n~k−~q,jn~q,i− n~k,j1 + n~k−~q,j− n~k,jn~q,i P
Ω0i
b0,i
+π
~2 X
~ q6=0
| U (~k, ~k − ~q, ~q, 0) |2
n~k−~q,jn~q,i− n~k,j1 + n~k−~q,j− n~k,jn~q,iδ Ω0i b0,i (2) where
~
"
e~k− e~− e~k−~q+ e0
~
k 0
h
U (~k, ~k0, ~k, ~k0) − U (~q, ~k0, ~q, ~k0) − U (~k − ~q, ~k0, ~k − ~q, ~k0)ink~0 ,j
#
(3)
Ω0i = 1
~
"
e~k− e~− e~k−~q+ e0
~
k 0
V (~k, ~k0, ~k, ~k0) − V (~k − ~q, ~k0, ~k − ~q, ~k0)nk~0 ,j− V (~q, ~k0, ~q, ~k0)nk~0 i
# (4)
III AMPLITUDE AND PHASE DYNAMICS OF SPIN-DEGENERATED
POLARITON CONDENSATE Use the decomposition of the condensate amplitude:
b0,1 = ρ0,1e−i(φ+θ2); b0,2= ρ0,1e−i(φ−θ2) (5)
To discuss the dynamics, we using:
R = ρ
2 0,1+ ρ20,2
ρ20,1− ρ2
0,2
We take amplitude and phase equations of spin-degenerated polariton condensate:
˙
θ = − Ω0z cos θ
~p(R2− z2) +
2
~V (0, 0, 0, 0) (1 + α) z +1
~ X
~k6=0
V (~k, 0, ~k, 0) (1 + α)
n~k,1− n~k,2
Trang 4− 1
2~2
X
~
q6=0
| V (~q, 0, ~q, 0) |2
n2~q,1 P
Ω1
− n2q,2~ P
Ω2
+ 1
2~2
X
~k6=0
| V (~q, ~q, 0, 0) |2 P
Ω1
+ P
Ω1
+ 2
n~q,1 P
Ω1
+ n~q,2 P
Ω2
z
+ 1
2~2
X
~k6=0
| V (~q, ~q, 0, 0) |2 P
Ω1
− P
Ω1
+ 2
n~q,1 P
Ω1
− n~q,2 P
Ω2
R
− 1
2~2
X
~k6=~q,~q6=0,~k6=0
| V (~k, ~k − ~q, ~q, 0) |2
n~k−~q,1n~q,1− n~k,11 + n~k−~q,1
− n~k,1n~q,1
P
Ω1
−n~k−~q,2n~q,2− n~k,21 + n~k−~q,2− n~k,2n~q,2 P
Ω2
−1
~2
X
~
q6=0
| U (~k, ~k − ~q, ~q, 0) |2
n~k−~q,2n~q,1− n~k,21 + n~k−~q,2− n~k,2n~q,1 P
Ω01
−n~k−~q,1n~q,2− n~k,11 + n~k−~q,1− n~k,1n~q,2 P
Ω02
(7)
˙
~
p
(R2− z2) sin θ
+ π
2~2
X
~
q6=0
| V (~q, ~q, 0, 0) |2
n2~q,1δ (Ω1) + n2~q,2δ (Ω2)
−2h(δ (Ω1) + δ (Ω2)) + 2 n~q,2δ (Ω2) + n~q,1δ (Ω1)i
R
z
+ π
2~2
X
~
q6=0
| V (~q, ~q, 0, 0) |2 n2~q,1δ (Ω1) − n2~q,2(Ω2) R
+ π
2~2
X
~
q6=0
| V (~q, ~q, 0, 0) |2
h
2 n~q,2δ (Ω2) − n~q,1δ (Ω1) + (δ (Ω2) − δ (Ω1))
i
R2
+ π
2~2
X
~
q6=0
| V (~q, ~q, 0, 0) |2 h2 n~q,2δ (Ω2) − n~q,1δ (Ω1) + δ (Ω2) − δ (Ω1)i
z2
+ π
2~2
X
~k6=~q,~q6=0,~k6=0
| V (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,1, n~q,1− n~k,11 + n~k−~q,1
− n~k,1n~q,1
δ (Ω1)
−n~k−~q,2, n~q,2− n~k,21 + n~k−~q,2− n~k,2n~q,2δ (Ω2)iR
Trang 5+ π
2~2
X
~k6=~q,~q6=0,~k6=0
| V (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,1, n~q,1− n~k,11 + n~k−~q,1− n~k,1n~q,1δ (Ω1) +
n~k−~q,2, n~q,2− n~k,21 + n~k−~q,2
− n~k,2n~q,2
δ (Ω2)
i z +π
~2
X
~ q6=0
| U (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,2, n~q,1− n~k,21 + n~k−~q,2− n~k,2n~q,1δ Ω01
−n~k−~q,1, n~q,2− n~k,11 + n~k−~q,1− n~k,1n~q,2δ Ω02i
R +π
~2
X
~ q6=0
| U (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,2, n~q,1− n~k,21 + n~k−~q,2
− n~k,2n~q,1
δ Ω01 +
n~k−~q,1, n~q,2− n~k,11 + n~k−~q,1
− n~k,1n~q,2
δ Ω02
i
˙
2~2
X
~ q6=0
| V (~q, ~q, 0, 0) |2 hδ (Ω1) + δ (Ω2) + 2 n~q,1δ (Ω1) + n~q,2(Ω2)i
n2~q,1δ (Ω1) + n2~q,2δ (Ω2)
δ (Ω1) + δ (Ω2) + 2 n~q,1δ (Ω1) + n~q,2(Ω2) R − R
2− z2
+ π
2~2
X
~ q6=0
| V (~q, ~q, 0, 0) |2 h2 (δ (Ω2) − δ (Ω1)) + 4 n~q,2δ (Ω2) − n~q,1δ (Ω1)i
Rz
+ π
2~2
X
~ q6=0
| V (~q, ~q, 0, 0) |2 n2~q,1δ (Ω1) − n2~q,2δ (Ω2) z
+ π
2~2
X
~k6=~q,~q6=0,~k6=0
| V (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,1, n~q,1− n~k,11 + n~k−~q,1− n~k,1n~q,1δ (Ω1) +n~k−~q,2, n~q,2− n~k,21 + n~k−~q,2− n~k,2n~q,2δ (Ω2)iR + π
2~2
X
~k6=~q,~q6=0,~k6=0
| V (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,1, n~q,1− n~k,11 + n~k−~q,1
− n~k,1n~q,1
δ (Ω1)
−n~k−~q,2, n~q,2− n~k,21 + n~k−~q,2
− n~k,2n~q,2
δ (Ω2)
i z +π
~2
X
~ q6=0
| U (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,2, n~q,1− n~k,21 + n~k−~q,2− n~k,2n~q,1δ Ω01 +n~k−~q,1, n~q,2− n~k,11 + n~k−~q,1− n~k,1n~q,2δ Ω02i
R
Trang 6~2 X
~ q6=0
| U (~k − ~q, ~k, ~q, 0) |2
h
n~k−~q,2, n~q,1− n~k,21 + n~k−~q,2− n~k,2n~q,1δ Ω01
−n~k−~q,1, n~q,2− n~k,11 + n~k−~q,1
− n~k,1n~q,2
δ Ω02
i
IV CONCLUSION Although we haven’t take solutions for the complex Gross-Pitaevskii equations, our result, if neglecting scattering terms, is agreement with caculation of M O Borgh et al [9] Our result, together with the solutions the Boltzmann equations for the excited states, will form the basis for a complete evaluation of the linewidth and the second-order correlation function
ACKNOWLEDGMENT
We gratefully acknowledge the financial support of the National Foundation for Science and Technology Development (NAFOSTED)
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Received 30-09-2011