Quantum and classical multiple-scattering effects in the spin dynamics of cavity polaritons

Abstract

The transport properties of exciton-polaritons are studied with allowance for their polarization. Both classical multiple scattering effects and quantum effects such as weak localization are taken into account in the framework of a generalized kinetic equation. The longitudinal-transverse (TE-TM) splitting of polariton states which plays a role that is analogous to the spin-orbit splitting in electron systems is taken into account. The developed formalism is applied to calculate the particle and spin diffusion coefficients of exciton-polaritons, spin relaxation rates, and the polarization conversion efficiency under the conditions of the optical spin Hall effect. In contrast to the electron systems, a strong spin splitting does not lead to the antilocalization behavior of the particle diffusion coefficient, while quantum corrections to spin diffusion and polarization conversion can be both negative and positive depending on the spin splitting value.

Full text

arXiv:0711.4702v1 [cond-mat.mes-hall] 29 Nov 2007
Quantum and classical multiple scattering effects in spin dynamics of cavity polaritons
M. M. Glazov∗and L. E. Golub
A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
The transport properties of exciton-polaritons are studied with allowance for their polarization.
Both classical multiple scattering effects and quantum effects such as weak localization are taken
into account in the framework of a generalized kinetic equation. The longitudinal-transverse (TE-
TM) splitting of polariton states which plays role analogous to the spin-orbit splitting in electron
systems is taken into account. The developed formalism is applied to calculate the particle and
spin diffusion coefficients of exciton-polaritons, spin relaxation rates and the polarization conversion
efficiency under the conditions of the optical spin Hall effect. In contrast to the electron systems,
strong spin splitting does not lead to the antilocalization behavior of the particle diffusion coefficient,
while quantum corrections to spin diffusion and polarization conversion can be both negative and
positive depending on the spin splitting value.
PACS numbers: 73.20.Fz, 72.25.Fe, 71.36.+c, 72.25.Rb, 78.35.+c
I. INTRODUCTION
Spin dynamics of charge carriers and their complexes
attracts lately an increasing interest. The issues of spin
coherence generation, detection and manipulation be-
came topical during last years.1
The spin properties of cavity polaritons are of special
interest both from the fundamental point of view and
due to possible future device applications.2The quantum
microcavity is the quantum well embedded between two
highly reflective Bragg mirrors. In such structures the
strong coupling between a cavity photon and a quantum
well exciton takes place, which leads to the formation of
new quasi-particles: exciton-polaritons. These half-light
half-matter particles exhibit both photonic and excitonic
properties. Their polarization (or spin) dynamics is ex-
tensively studied both experimentally and theoretically,
see Ref. 3 and references therein.
The polarization eigenmodes of the quantum microcav-
ity are so-called TE- and TM-modes where the electric
or magnetic field vector is oriented perpendicularly to
the polariton wave vector, respectively. They are split-
ted by the longitudinal-transverse (also known as TE-
TM) splitting4which plays a role similar to the spin
splitting of electron states in quantum wells.5It leads
to D’yakonov-Perel’-like spin relaxation in the collision-
dominated regime and to spin precession in the collision-
free regime.6,7 One of the brightest manifestations of the
polariton spin splitting is the polarization conversion or
the optical spin Hall effect: under the Rayleigh scatter-
ing of linearly polarized polaritons the scattered particles
obtain a certain degree of circular polarization.5,8 The
angular distribution of the circular polarization demon-
strates the second angular harmonics thus reflecting the
symmetry of the longitudinal-transverse splitting.
Spin splitting can strongly modulate the interference
phenomena inherent to the quantum particles. It is well
∗Electronic address: glazov@coherent.ioffe.ru
known that the spin-orbit interaction modifies quantum
corrections to electron diffusion coefficient or conductiv-
ity and electron spin relaxation times.9,10,11 Although the
quantum interference of excitons has a long history,12,13
it became topical only recently with the development of
microcavities.14,15
Here we analyse the spin-dependent interference effects
in quantum microcavities. We focus on weak localization
effects in exciton-polariton diffusion and spin diffusion,
effects of interference in spin relaxation and in polar-
ization conversion. Our results can be summarized as
follows:
1. The quantum correction to the polariton diffu-
sion coefficient is negative despite the value of the
longitudinal-transverse splitting. It is in sharp con-
trast with the case of electrons, where the suf-
ficiently strong spin-orbit interaction changes the
sign of quantum correction to the diffusion coeffi-
cient.
2. The relaxation of the circular polarization degree
of exciton-polaritons is enhanced by the quantum
interference effects while the relaxation of linear
polarization can either speed up or slow down de-
pending on the value of the longitudinal-transverse
splitting.
3. The efficiency of the polarization conversion can ei-
ther be increased or decreased by the interference
effects depending on the value of the longitudinal-
transverse splitting and relation between the scat-
tering time and the radiative lifetime of polaritons.
The paper is organized as follows: in Sec. II we present
the model based on the kinetic equation for exciton-
polariton spin density matrix. The quantum corrections
to the collision integral describing the effects of coherent
scattering are introduced and calculated in the frame-
work of Green’s function technique. Section III is devoted
to the calculation of quantum corrections to the particle
and spin diffusion coefficients of exciton-polaritons. The

2
interference effects on exciton-polariton spin relaxation
times are discussed in Sec. IV. The multiple scattering
effects and quantum interference effects in the polariza-
tion conversion are discussed in Sec. V.
II. THEORY
Below we present the kinetic theory of the spin dynam-
ics of exciton-polaritons with allowance for the interfer-
ence effects.
A. Model
We consider exciton-polaritons formed from the heavy-
hole quantum well excitons. Their spin projection on the
structure growth axis zcan take values ±1 or ±2. The
latter states are optically inactive and do not participate
to the light-matter coupling while the former ones con-
stitute the radiative doublet. It is convenient to describe
this doublet as a pseudospin 1/2 state,16 the pseudospin z
component describes the emission intensity in the circular
polarization, and the in-plane components correspond to
the linear polarization: namely, sxcomponent is propor-
tional to the intensity measured in the given axes x−y
while sycomponent corresponds to the intensity mea-
sured in the axes x′−y′rotated by π/4 with respect to
x−ycoordinate frame.
The (pseudo)spin dynamics of exciton-polaritons is
most conveniently described within the spin density ma-
trix approach. It can be represented in a form
ρk=fk+sk·σ,
where fkis the particle distribution function and skis
the average spin in the given state k. Under the con-
ditions of Rayleigh scattering experiments the monoen-
ergetic distribution of the polaritons is excited and the
processes of energy relaxation can be neglected.2It means
that the absolute values of polariton wave vectors k0are
conserved. We consider the situation where the mean
free path of the particles lis large enough, k0l≫1. In
this regime the dynamics of scattered particles can be de-
scribed in the framework of the classical kinetic equation,
and the quantum effects can be incorporated as correc-
tions to the collision integral.17
In the steady-state regime the kinetic equations writes:
fk
τ0
+Q{fk}=gk,(1)
sk
τ0
+sk×Ωk+Q{sk}=gk.(2)
Here τ0is the polariton lifetime, Q{fk}and Q{sk}are
the collision integrals, and gk,gkare the components of
the generation density matrix γk= (gk+gk·σ) describing
the particle and spin generation rates, respectively.
The quantity Ωkin Eq. (2) is the pseudo-spin pre-
cession frequency related to the longitudinal-transverse
splitting of polariton modes equal to ~|Ωk|.4It can be
written as
Ωk= Ω(k)[cos 2ϕk,sin 2ϕk,0],(3)
where Ω(k) is some function of the wave vector absolute
value k,ϕkis the angular coordinate of k. It depends
strongly on the microcavity parameters. Note, that the
components of Ωkare described by the second angular
harmonics of the wave vector angle, because the pseu-
dospin flip is accompanied with the change of the polari-
ton spin by two. It is strongly different from the case
of two-dimensional electrons, where the spin splitting is
described by the first and third angular harmonics.
We assume that the scattering of exciton-polaritons is
caused by a short-range disorder, i.e. the scattering cross-
section is angular independent. This condition can be vi-
olated in real structures,18 but our goal is to consider the
simplest case which allows an analytical solution. Fur-
thermore, we assume that the polaritons are described
by a parabolic dispersion, Ek=~2k2/2m, with an effec-
tive mass m, and introduce the density of states per spin
D=m/2π~2.
In typical microcavities under the conditions of
Rayleigh scattering the kinetic energy of polaritons, Ek,
is of the order of several meV, while the longitudinal-
transverse splitting is of the order of tenths of meV.
Therefore the effect of the polarization on the orbital
dynamics of polaritons can be neglected. At the same
time, the spin precession frequency, inverse lifetime and
the scattering rate can be comparable.
The collision integrals entering Eqs. (1), (2) can be
written as a sum of the classical contribution
Qcl{fk}=QX
k′
(fk−fk′)δ(Ek−Ek′),(4)
Qcl{sk}=QX
k′
(sk−sk′)δ(Ek−Ek′),
and the quantum corrections. Here Qis the elastic scat-
tering constant, QD=τ−1
1with τ1being the momentum
scattering time.
The quantum corrections to the collision integrals can
be most conveniently found in Green’s function tech-
nique. Various contributions to the scattering cross-
sections are exemplified in Fig. 1. Solid lines are retarded
and advanced exciton-polariton Green’s functions which,
with allowance for polariton spin and the longitudinal-
transverse splitting have a form of 2 ×2 matrices and
read19
GR,A(k, ω ) = [~ω−Ek−~(σ·Ωk)/2±i~/2τ]−1,(5)
where
τ−1=τ−1
0+τ−1
1.
Note that the radiative lifetime of exciton-polaritons
plays role of the phase relaxation time in the theory of

3
(a) (b) (c) (d)
FIG. 1: Examples of irreducible diagrams which contribute
to the collision integral. (a) Single scattering, (b) coherent
backscattering, (c) and (d) coherent scattering by an arbitrary
angle.
electron weak localization.10 The dashed line in Fig. 1 is
the scattering amplitude which reads ~Q/2π=~3/mτ1.
The diagram Fig. 1(a) shows the single scattering pro-
cess. The diagram in Fig. 1(b) describes the interference
of polariton which passes the same configuration of scat-
terers in the clockwise and counterclockwise directions
and propagates exactly backwards. Diagrams Fig 1(c)
and (d) describe the same interference but accompanied
by the scattering by an arbitrary angle. The corrections
to the collision integral can be expressed in the terms of
the Cooperon operator Cαβ
δγ (q) which is the sum of di-
agrams depicted in Fig. 1(b) with any number of lines
N>3.(1) The Cooperon satisfies the equation
Cαβ
δγ (q) = [P3(q)]αβ
δγ +X
β′γ′Cαβ′
δγ′(q)Pβ′β
γ′γ(q),(6)
with
Pαβ
δγ (q) = ~Q
2πX
k′GR
αβ (k′, E0)GA
δγ (q−k′, E0).(7)
Here it is assumed that the polaritons have the same en-
ergy E0determined by the excitation. As a result the
distribution functions can be reduced to the angular de-
pendent parts only
fk=f(ϕk)δ(Ek−E0),
sk=s(ϕk)δ(Ek−E0).
The shape of the coherent backscattering cone ob-
tained by the summation of the diagrams in Fig. 1(b)
is given in the limit of Ωτ= 0 by the following expres-
sion12
I1(k) = I0(k)τ
τ0
1
p(τ1/τ)2+ (k+k0)2l2−1.(8)
Here I0(k) describes the classical intensity distribution.
In the case of exact backscattering k=−k0and τ0≫τ1
1The diagram with two intersecting impurity lines is dominated
by the pairs of impurities separated by ∼k−1
0and can not be
treated within our approximation k0l≫1.
the coherent backscattering intensity is I1(k) = I0(k). If
τ0≪τ1the coherent backscattering is negligible I1(k)≪
I0(k). The backscattering cone angular width is small,
of the order of (k0l)−1≪1 and the details of its shape
are irrelevant for our consideration. Thus, the coherent
backscattering processes simply couple states with kand
−k.
The quantum effects are most pronounced in the mul-
tiple scattering regime where τ0≫τ. This assumption
is used hereafter in the description of the interference
phenomena. Furthermore, in the limit ln(τ0/τ1)≫1 the
sum of the diagrams Fig. 1(c) and (d) weakly depends on
the scattering angle. Even if this condition is violated the
classical effects of finite correlation length of the potential
leading to the angular dependent scattering cross-section
may be more important. Therefore their angular depen-
dence is neglected, and the quantum corrections to the
collision integral can be written as11,17
Qqnt{fk}=−W0Zdϕ′
2π[f(ϕ−π)−f(ϕ′)]δ(Ek−E0),(9)
Qqnt{sk}=−ˆ
WZdϕ′
2π[s(ϕ−π)−s(ϕ′)]δ(Ek−E0).
Here the quantities W0,ˆ
Wdescribe spin-dependent re-
turn probabilities10
W0=l
k0τX
αβ X
qCαβ
βα (q),(10)
(ˆ
W)ij =l
k0τX
αβγ δ X
q
σi
γα Cαβ
δγ (q)σj
βδ ,
where σifor i=x, y, z are the Pauli matrices.
We consider an isotropic spin splitting, i.e. Ωkis inde-
pendent of the angles of the wave vector kand equals to
Ω(k), see Eq. (3). Thus, for the cylindrical symmetry of
the problem under study, the only non-zero components
of ˆ
Ware
W⊥=Wxx =Wyy , Wk=Wzz .
Thus, the decription of the spin dynamics of the exciton-
polaritons is reduced to the solution of the kinetic equa-
tions with the collision terms in the form of Eqs. (4), (9).
Latter depend on the spin-dependent return probabilities
W0,Wkand W⊥which can be found straightforwardly
from the Cooperon operator Cαβ
δγ (q) Eq. (6).
B. Cooperon
In order to find the Cooperon we follow the procedure
outlined in Refs. 20,21 and make use of the fact that the
operator Pcan be presented as
P=τ
τ1
2π
Z
0
dϕk
2π[1 −iτL·Ωk+ iτvk·q]−1,(11)

4
where
Lαγ,β δ =σαβ −σδγ
2
is an operator of the difference of spins of two inter-
fering particles and vkis the velocity operator. This
result can be contrasted with the situation realized for
two-dimensional electrons where the spin splitting is odd
function of kand, thus, the total spin of interfering par-
ticles enters into the definition of P.
Nevertheless, in our treatment we use the representa-
tion of the total spin of the interfering particles: αγ →
Sms, where S= 0,1 is the absolute value of the total spin
S, and msis its projection onto the zaxis (|ms| ≤ S).
The pair of particles with S= 0 is in the singlet state
while S= 1 corresponds to the triplet one.
Since Pand, hence, the Cooperon are determined
by the operator L, two independent contributions to
Cooperon can be separated. Namely, the matrix Pcan
be block-diagonalized, one of the blocks corresponds to
the pair in the triplet state with ms= 0 and another is
a 3 ×3 matrix corresponding to two triplet states with
|ms|= 1 and a to singlet. The part with (S, ms) = (1,0)
reads
P=1
p(τ1/τ)2+ (ql)2,(12)
and the corresponding Cooperon is given by
C0=P3
1−P.(13)
The operator Pin the basis of three other states,
(S, ms) = (1,1); (0,0); (1,−1), has the following form
P1=

P−S0ie−2iϕqR e−4iϕqS4
ie2iϕqR P −2S0−ie−2iϕqR
e4iϕqS4−ie2iϕqR P −S0

,(14)
where ϕqis the angular coordinate of q,
Sm=
∞
Z
0
dx exp (−x)Jm(xql) sin2xΩτ
2,(15)
R=1
√2
∞
Z
0
dx exp (−x)J2(xql) sin (xΩτ),
and Jm(y) are the Bessel functions. (2) The Cooperon
corresponding to these three states is given by
C1=P3
1[I− P1]−1,(16)
where Iis the 3 ×3 unit matrix.
2The integrals (15) can be calculated analytically but the expres-
sions are very cumbersome, therefore we leave them in integral
forms.
C. Spin-dependent return probabilities
Using Eqs. (10) and rewriting the expressions for W0
and Wk,⊥in the same basis as C0,C1, we have
W0=l
k0τX
q{Tr [E1C1] + C0},(17)
Wk=l
k0τX
q{Tr [C1]−C0},
W⊥=l
k0τX
q{Tr [E2C1] + C0},
where E1is the matrix with the diagonal (1,−1,1) and
other elements being zero, and E2is the matrix with
unit anti-diagonal elements and other elements being
zero. Note that W0can be recast in a conventional
form as a difference of triplet and singlet contributions
to C.10,19,20,21
The asymptotic values for the components of ˆ
Wtensor
can be obtained analytically in the case ln (τ0/τ1)≫1,
Ωτ1≪1. They read:
W0=1
2πτ k0l2 ln Ts⊥
τ+ ln τ0
τ−ln Tsk
τ,(18)
Wk=1
2πτ k0l2 ln Ts⊥
τ+ ln Tsk
τ−ln τ0
τ,
W⊥=1
2πτ k0lln Tsk
τ+ ln τ0
τ.
Here the lifetimes are introduced for spin components
parallel and perpendicular to the growth axis z:
1
Tsk
= Ω2τ+1
τ0
,1
Ts⊥
=Ω2τ
2+1
τ0
.(19)
In this limit, the spin-dependent return probabilities
contain logarithmic factors, which is a specific feature of
two-dimensional systems where the return probability is
proportional to the logarithm of the ratio of longest and
shortest allowed travel times. The lower boundary for the
travel time is obviously ∼τwhile the upper boundary
depends on the particle lifetime τ0and spin lifetimes Tsk,
Ts⊥. The expression for the particle return probability
W0differs from the corresponding transport coefficient
derived for electrons in Ref. 19. It reflects the specifics
of the even in kspin splitting in polariton systems.
In the regime of Ωτ≫1 the leading logarithmic con-
tributions to ˆ
Whave the following form:
W0=W⊥≈1
2πτ k0l0.06 + ln τ0
τ,(20)
Wk≈1
2πτ k0l0.06 −ln τ0
τ.
Clearly, the main (logarithmically large) correction is
determined by the (S, ms) = (1,0) contribution to the

5
0.1 1 10 100
-0.5
0.0
0.5
1.0
1.5
Return proba bility
W
k
0
l
Electrons
Spin splitting
Ω τ
W
0
W
⊥
W
||
(b)
0.1 1 10 100
-0.5
0.0
0.5
1.0
1.5
(a)
Return proba bility
W
•
k
0
l
Spin splitting
Ω τ
W
0
W
⊥
W
||
Exciton-polaritons
FIG. 2: (Color online) Components of the tensor ˆ
Wplotted as the functions of the product Ωτ, where Ω = Ω(k0). The radiative
lifetime τ0= 100τ1.
Cooperon. Other states give small additional term. It is
worth noting that at Ωτ→ ∞ quantitites W0, W⊥have
the same sign as at Ω →0, while Wkchanges its sign as
compared with the case Ωτ≪1.
The dependence of W0,Wkand W⊥on spin splitting
value in a wide range of Ωτvariation is presented in Fig. 2
at a fixed ratio τ0/τ1= 100. Panel (a) of Fig. 2 shows
the non-zero components of ˆ
Win the case of exciton-
polaritons (i.e. where the spin splitting contains second
angular harmonics). The case of electrons where the spin
splitting contains first harmonics is presented for com-
parison in Fig. 2(b). Although the overall behavior of Wk,
W⊥,W0is similar in both cases, there is a strong qual-
itative difference. Namely, for electrons W0changes its
sign with an increase of Ωτwhich is a direct consequence
of the antilocalization phenomena: the sufficiently strong
spin-orbit interaction leads to the phase πacquired by an
electron on the closed trajectories, and the probability
for electron to avoid its initial point increases.21 The real
spin of polaritons is 1 therefore the aquired phase is 2π,
and the quantity W0does not change its sign as a func-
tion of Ωτ. Mathematically, it is a direct consequence of
the k-even spin splitting: the interference is governed by
the difference spin Lof the particles contrary to the case
of odd spin splitting where the interference is controlled
by the total spin of the particles S. Therefore, in the case
of electrons the Cooperon can be separated in the singlet
and triplet parts with respect to the total spin Swhich
enter to W0with different signs.9On the contrary, in the
case of polaritons the part corresponding to the triplet
with zero projection of the total spin (S, ms) = (1,0)
is separated from the Cooperon, therefore W0keeps its
sign, while Wkdemonstrates the antilocalization behav-
ior. Therefore, the antilocalization of polaritons does not
occur.
Figure 3 shows the dependence of W0on the ratio
0.01 0.1 0.3
0.0
0.5
1.0
W
||
W
W
0
W k
0
l
1
/
0
= 1
0.01 0.1 0.3
0.0
0.5
1.0
1.5
Return prob ability
W
0
k
0
l
1
/
0
= 0
= 1
= 10
FIG. 3: (Color online) The dependence of W0on the ratio
τ1/τ0calculated for the different values of Ωτ. The inset
shows the dependence of all components of ˆ
Wtensor at fixed
Ωτ= 1.
τ1/τ0calculated for exciton-polaritons. The curves cor-
respond to the different values of Ωτ. It is seen that
W0decreases monotonously with an increase of the ra-
diative rate. An inset to the figure shows W0,Wkand
W⊥calculated as functions of τ1/τ0at fixed Ωτ= 1. In
agreement with Eqs. (17) and (18) only Wkbehaves non-
monotonously with the decrease of the polariton lifetime.
Its dependence on τ1/τ0is determined by the competi-
tion of two contributions to Cooperon: Ω-independent
C0and Ω-dependent C1. They enter into Eq. (17) with
the opposite signs and lead to non-monotonic behavior.
Cooperons C0and C1contribute to W0and W⊥with the
same signs and therefore these components of the tensor
ˆ
Ware monotonous functions of τ1/τ0.

6
III. QUANTUM CORRECTIONS TO THE
PARTICLE AND SPIN DIFFUSION
COEFFICIENTS
The diffusion coefficient Dnfor the quantity n(r, t)
where ncan be either the particle density or one of the
spin density components sx, syand szcan be defined
from the Fick’s law
Dn∇n+jn(r, t) = 0,(21)
where jnis the flux density of the corresponding quan-
tity, ris the coordinate, tis time.
Let us first consider the particle diffusion coefficient
D0. We assume that the distribution function of polari-
tons can be recast as f(r,k, t) = n(r, t)f(ϕk)δ(Ek−~ω)
and that n(r, t) weakly depends on the coordinate and
on time. The kinetic equation with the collision integrals
(4), (9) reads
∂n
∂t f(ϕk) + n
τ0
f(ϕk) + ∂n
∂r·~k
mf(ϕk) + n
τ1hf(ϕk)−¯
fi
−nW0hf(ϕk−π)−¯
fi= 0.(22)
Here ¯
f= (2π)−1R2π
0f(ϕ)dϕis the average number of
particles. Obviously f(ϕ) contains only zeroth and first
harmonics in ϕk. The particle current thus reads
jn(r, t) = n(r, t)
2π
Z
0
dϕk
2π
~k
mf(ϕk),(23)
and Eq. (23) together with Eq. (21) yields for the parti-
cle diffusion coefficient
D0=Dcl(1 −W0τ1),(24)
where Dcl =~2k2τ1/2m2is the classical diffusion coeffi-
cient.
Analogously one can derive the diffusion coefficients
for the spin components sz(k) and sx,y (⊥):
Dk,⊥=Dcl(1 −Wk,⊥τ1).(25)
From Eqs. (24), (25) one can see that the quantum
corrections to the particle and spin diffusion coefficients
are determined by the respective components of the ten-
sor ˆ
W. It follows from the previous section that W0is
positive for all values of Ωτand τ1/τ0. Therefore the
backscattering is enhanced and the quantum interference
leads to the decrease of the polariton diffusion coefficient,
cf.(24). The same holds for the in-plane spin diffusion
coefficient, D⊥, which is determined by W⊥>0. The
quantum correction to the longitudinal spin diffusion co-
efficient, Dk, is determined by Wkwhich can be either
positive or negative depending on Ωτand τ1/τ . In par-
ticular, Wkis negative for large Ωτand τ0/τ1, and quan-
tum effects will lead to an increase of the longitudinal
spin diffusion coefficient. For small values of Ωτlongi-
tudinal diffusion coefficient is decreased by the quantum
interference.
Quantum corrections to diffusion coefficients can be
extracted from the dependence of these coefficients on ex-
ternal perturbations which introduce an extra dephasing
of the particles.10 These effects can be incorporated into
the effective lifetime τ0. For instance, the temperature
variation modifies the rates of inelastic processes. Since
exciton is a neutral particle, the magnetic flux through
the trajectory is proportional to the electron-hole separa-
tion. Hence the magnetic field affects the interference at
lB∼aBwhere lBis the magnetic length and aBis Bohr
radius.22 However, such a field is not classically weak,
therefore strong diamagnetic effects are dominant, there-
fore the weak localization corrections to magnetodiffusion
can be hardy separated.
In microcavities τ0can be efficiently varied with inci-
dence angle of light proportionally to the photonic frac-
tion in polariton. It is seen for Fig. 3 that the presence of
the spin-splitting does not lead to the non-monotonous
dependence of W0(and, therefore, of the quantum cor-
rection to the diffusion constant) on the lifetime. On the
contrary, the antilocalization behaviour can be observed
in Dk
IV. QUANTUM CORRECTIONS TO THE SPIN
RELAXATION RATES
The spin dynamics is known to be non-exponential
with allowance for the quantum interference effects: weak
localization leads to the appearance of the long-living
tails in spin polarization.11,13 It is convenient to deter-
mine the tensor of spin relaxation rates ˆ
Γ from the bal-
ance equation for the total spin in the system at the
steady-state spin pumping
τ−1
0+ˆ
Γ¯
s=g.(26)
Here ¯
s= (2π)−1R2π
0s(ϕ)dϕis the angular average of the
spin distribution, and the spin generation is assumed to
be isotropic
gk=gδ(Ek−E0).
We represent the distribution functions separating
terms of zeroth and first order in quantum corrections:
f(ϕ) = f0(ϕ) + f1(ϕ),s(ϕ) = s0(ϕ) + s1(ϕ).(27)
Here the upper index refers to the order in W0,ˆ
W.
In order to solve the kinetic equations (1), (2) we note
that the solution of the following equation
s(ϕ)
τ+s(ϕ)×Ω(ϕ) = F(ϕ),(28)

7
where Ω(ϕ) = Ωkwrites
s(ϕ) = τ
1 + Ω2τ2×
F(ϕ) + τΩ(ϕ)×F(ϕ) + τ2Ω(ϕ)(Ω(ϕ)·F(ϕ)).(29)
Neglecting the weak-localization effects (i.e. putting
W0, Wk, W⊥= 0) one arrives to Eq. (29) for the spin
distribution function s0with
F=s0
τ1
+g.(30)
It is enough to consider the cases where gis directed
along the growth axis of the sample, gkz, or lies in the
plane of the structure, because tensor ˆ
Γ for cylindrical
symmetry has only two independent components,
Γk= Γzz ,Γ⊥= Γxx = Γyy ,
describing the longitudinal and transverse relaxation
rates, respectively.
If gkzthe self-consistency equation for ¯
sgives s0
z=
gzTsk, and hence
s0(ϕ) = [g+τΩ(ϕ)×g]Tsk.(31)
Therefore, the quantum correction for z-component of
spin reads
s1
z=−gzT2
sk(Ωτ)2W⊥,(32)
and according to Eq. (26) the longitudinal spin relaxation
rate is given by
Γk= Ω2τ(1 + τW⊥).(33)
Since W⊥is positive the z-component spin relaxation is
enhanced by the quantum interference effects. This equa-
tion shows the correction to τ1in the longitudinal spin
relaxation rate has an inverse sign as compared to the
correction to the spin diffusion coefficient Eqs. (25). It
is due to the fact that the spin relaxation rate is gov-
erned by the relaxation of the second harmonic of the
spin distribution function16 whereas the spin diffusion is
determined by the relaxation of the first harmonic.
The calculation of the transverse relaxation rate yields
Γ⊥=Ω2τ/2
1 + (Ωτ)2/21 + τ Wk−τW⊥(Ωτ)2/2
1 + (Ωτ)2/2.(34)
This equation shows that at small Ωτthe quantum cor-
rections to Γ⊥are determined by Wk>0 and the spin
relaxation is enhanced by an interference. In contrast, at
large Ωτ1the quantity Wkbecomes negative and slows
dows spin relaxation. Besides, the second contribution
proportional to W⊥(Ωτ1)2>0 becomes even more im-
portant and suppresses spin relaxation as well.
V. OPTICAL SPIN HALL EFFECT
Under the conditions of the optical spin Hall effect the
TE- or TM-eigenstate of the quantum microcavity with
the wave vector k0is excited and the circular polarization
in the scattered state kis observed.5In this case the
generation rate can be represented as
gk=gδk,k0,gk=gδk,k0,(35)
where the Kronecker δ-symbol used here is defined as
δk,k0=2π
Dδ(Ek−Ek0)δ(ϕk−ϕk0).
We first consider classical multiple scattering effects in
the optical spin Hall effect regime and demonstrate that
the spin relaxation decreases the polarization degree as
compared with the single scattering regime considered in
Ref. 5. Further, we calculate the quantum corrections to
polarization conversion and demonstrate that they can
either increase of decrease conversion efficiency.
A. Classical effects
Under the condition Eq. (35), the solution of Eq. (1)
for f0(ϕ) reads
f0(ϕ) = gτ
Dτ0
τ1
+ 2πδ(ϕ−ϕk0).(36)
The solution of Eq. (2) for s(ϕ) can be written in a
straightforward way as well using Eqs. (28), (29). In
what follows we concentrate on the important case of ex-
citation of the pure TE- or TM-state, i.e. Ωk0kgkx.
Thus, s0=guτ/D, and
s0
x(ϕ) = gτ
D2πδ(ϕ) + (1 + Ω2
xτ2)νu,(37)
s0
y(ϕ) = gτ
DΩxΩyτ2νu,
s0
z(ϕ) = −gτ
DΩyτνu,
where we introduced
u=1 + (Ωτ)2−ν(1 + Ω2τ2/2)−1, ν =τ
τ1
.
The angular distribution of the scattered polaritons is
symmetric with respect to the rotation by an angle πbe-
cause the longitudinal-transverse spitting contains even
harmonics of the wave vector angle ϕk. Moreover, sz
pseudospin component appears to be proportional to the
y-component of the splitting, Eq. (37), (see also Ref. 5),
therefore the maxima of circular polarization appear at
scattering angles equal to π/4, 3π/4, 5π/4 or 7π/4. The
sign of polarization is opposite in the adjacent maxima.
Figure 4 shows the circular polarization degree of the po-
laritons at the scattering by π/4. The dependence of the

8
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
Circul ar pol arizati on deg ree
FIG. 4: (Color online) The absolute value of the circular po-
larization degree of polaritons as a function of Ωτcalculated
for the different values of τ0/τ1. The scattering angle is π/4.
An inset shows a contour plot of the angular distribution of
circular polarization degree, and arrow denotes the excitation
point in the k-space.
polarization degree on Ωτis non-monotonous, its maxi-
mum shifts to the lower values of Ωτwith decrease of the
scattering time τ1. The maximum value of the polariza-
tion is observed for τ1/τ0→ ∞, i.e. in the regime of a
single scattering described in Ref. 5.
Multiple scattering leads to spin relaxation of polari-
tons. At τ0≫τ1, (Ω2τ1)−1one can see that the factor
ureduces to 2/(Ωτ1)2which is nothing but the ratio of
the classical value of the transverse spin relaxation time
τ⊥and the scattering time τ1. The circular polarization
degree in the scattered state
ρ0
c(ϕ) = s0
z(ϕ)
f0(ϕ)= Ωy(ϕ)τ1
τ⊥
τ0
is the smaller the shorter spin relaxation time.
B. Quantum effects
The quantum corrections to the particle and spin dis-
tribution functions in the lowest order in W0,ˆ
Wcan be
found similarly to Sec. IV. The distribution functions
are represented as the sum of zeroth order contributions
and the first order corrections, see Eq. (27). Functions
f0and s0are given by Eqs. (36), (37). Thus, for f1(ϕ)
we have
f1(ϕ) = τW0
gτ
D[2πδ(ϕ−π)−1] (38)
The first term in the brackets describes the coherent
backscattering, i.e. an increase by interference effects
of the number of the particles scattered into the oppo-
site from the source direction. The second term describes
the coherent scattering by an arbitrary angle. The total
number of particles is conserved, R2π
0f1(ϕ)dϕ= 0.
The solution procedure for s1(ϕ) is analogous to the
outlined above. We introduce the auxiliary function
F1(ϕ) = s1
τ1
+ˆ
Whs0(ϕ−π)−s0i,(39)
and the kinetic equation for s1reduces to Eq. (28) with
F1instead of Fand s1instead of s. Thus, s1is given
by Eq. (29). One needs to average the solution over ϕto
obtain self-consistency equations for the components of
s1and then find s1(ϕ). The result reads
s1
x(ϕ) = gτ
D2πτ W⊥δ(ϕ−π) +
u2
4A0+u2
4A2Ω2
xτ2,(40)
s1
y(ϕ) = gτ
D
u2
4
ΩxΩyτ2
2C0,
s1
z(ϕ) = −gτ
D
u2
4
Ωyτ
2B0,
Here the following quantities are introduced:
A0= 2τWkτ2(ν−2)νΩ2τ2+ 4τ W⊥(1 −ν)(ν−1−Ω2τ2),
A2= 4τWk(1 −ν)ν−2τW⊥p,
C0= 8τWk(1 −ν)ν−4τW⊥p,
B0= 8τWk(1 −ν)ν+ 8τ W⊥(1 −ν)(ν−1−Ω2τ2),
p= Ω2τ2(2 + ν2−4ν) + 2 + 4ν2−6ν.
Quantum interference leads to the appearance of extra
backscattered polaritons. The total number of backscat-
tered particles is proportional to W0, Eq.(38). This in-
crease is compensated by the coherent scattering by an
arbitrary angle which leads to the decrease of the num-
ber of scattered polaritons in all other directions than
exactly backwards. An increase of the spin splitting at
fixed τ1and τ0leads to the decrease of W0therefore the
number of backscattered particles decreases.
The coherent backscattering is also pronounced in the
xpseudospin component. In this case the overall inten-
sity of the backscattering peak is proportional to W⊥.
The cross-linear polarization and the circular polariza-
tion appear proportionally to ΩxΩyand Ωy, therefore
both of them vanish for ϕ=π, i.e. for the detection in
the backscattering geometry. However, the coherent scat-
tering by an arbitrary angle leads to the modification of
the polarization conversion efficiency. The circular polar-
ization degree can be written as ρc(ϕ) = ρ0
c(ϕ) + ρ1
c(ϕ),
where ρ0
cis the classical value of the circular polarization
degree and the quantum correction ρ1
c(ϕ) is given by
ρ1
c(ϕ) = ρ0
c(ϕ)s1
z
s0
z−f1
f0.(41)

9
0.01 0.1 1 10 100
-0.02
-0.01
0.00
0.5
1.0
1.5
Total correction
s
1
z
/
s
0
z
f
1
/
f
0
Quantum correction
(
1
c
/
0
c
)
•
k
0
l
Ω τ
FIG. 5: (Color online) Quantum corrections to the circular
polarization degree observed under optical spin Hall effect
conditions. Solid curve shows the total correction, dash-
dotted one shows first contribution, s1
z/s0
z, in Eq. (41),
the dashed curve shows second term, f1/f0, in Eq. (41),
τ0/τ1= 100.
The relative value of the polarization conversion effi-
ciency ρ1
c/ρ0
cas a function of Ωτis plotted in Fig. 5.
From Eq. (41) it is clear that there are two contri-
butions to the quantum correction to the polarization
degree: first one arises due to the modification of spin
distribution (s1
z) while second one is determined by the
change of the number of particles in a given state. The
latter correction is always positive because the coherent
scattering by an arbitrary angle reduces the number of
particles in a given state, see Eq. (38) and dashed curve
in Fig. 5. The former one can be either positive or nega-
tive depending on the values of Ωτand τ0/τ1, see Fig. 5.
For instance, if Ωτ≪1 and τ0≫τ1both Wk,W⊥are
positive and B0>0. Therefore, quantum corrections in
this regime increase the polarization as compared with
the classical result. On the contrary, if Ωτ≫1 both
Wkand W⊥(ν−1−Ω2τ2)≈ −W⊥Ω2τ2are negative
and z-pseudospin component is decreased. Thus, inter-
ference of polaritons can either increase of decrease the
polarization conversion efficiency.
Physically, it can be interpreted as follows. The effi-
ciency of the polarization conversion is strongly sensitive
to the spin relaxation times. The correction to the trans-
verse relaxation time can either be positive or negative
depending on the sign of Wkand the value W⊥(Ωτ)2,
therefore the x-pseudospin component can be preserved
better or worse depending on the value of Ωτ. There-
fore generated circular polarization and cross-linear po-
larization may either increase or decrease as a result of
quantum interference.
VI. CONCLUSIONS
To summarize, we have studied in detail the spin-
dependent quantum interference and classical multiple
scattering effects in dynamics of exciton-polaritons. We
have derived the quantum corrections to the collision
integral of exciton-polaritons in the leading order in
(k0l)−1. The quantum corrections are strongly sensitive
to the value of the spin splitting of exciton-polaritons.
Contrary to the case of electrons where the strong spin
splitting can lead to the anti-localization, the quantum
correction to the polariton diffusion coefficient is nega-
tive. The quantum correction to spin z-component diffu-
sion coefficient changes its sign from negative to positive
with the increase of the spin splitting while the correc-
tion to the diffusion coefficient of the in-plane spin com-
ponents is negative. The relaxation of the longitudingal
spin component is accelerated by the quantum interfer-
ence effects and the relaxation rate of the transverse spin
components can increase or decrease depending on the
spin splitting value. The polarization conversion effi-
ciency in the regime of the optical spin Hall effect can
also be larger or smaller than the value predicted by the
classical theory depending on the relations between the
lifetime of polaritons, their scattering time and the value
of the spin splitting.
Acknowledgments
The discussions with A.V. Kavokin, A.N. Poddubny
and I.A. Shelykh are gratefully acknowledged. This
work was partially supported by RFBR, ‘Dynasty’
Foundation-ICFPM and RSSF.
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