Spontaneous Polarization, Spin Current and Quantum Vortices in Exciton-Polariton Condensates

2.7 MB·

by  Bo Xiong

Spontaneous Polarization, Spin Current and Quantum Vortices in Exciton-Polariton
Condensates
Bo Xiong1,2
1Department of Physics, Nanchang University, 330031 Nanchang, China
2Skolkovo Institute of Science and Technology, Novaya Street 100, Skolkovo 143025, Russian Federation
(Dated: January 10, 2017)
We show that how to support propagation of spin degree in a spin-symmetric exciton-polariton
condensates in a semiconductor microcavity. Due to the stimulated spin-dependent scattering be-
tween hot excitons and condensates, exciton polaritons form a circular polarized condensate with
7 spontaneous breaking the spin rotation symmetry. The spin antiferromagnetic state are developed
1 evidently from the density and spin flow pumped by localized laser source. The low energy spin
0 current is identified where the steady state is characterized by the oscillating spin pattern. We
2 predictviasimulationthatitisverypromisingtodynamicalcreationoffractionalizedhalf-quantum
n vortices induced by effective non-abelian gauge potential within currently experiment procedure.
a
J PACSnumbers: 72.25.Rb,75.30.Ds,72.70.+m,71.36.+c

8

I. INTRODUCTION. superconductors Sr RuO [46] or spinor atomic Bose-
] 2 4
l Einstein condensates [47–50], described as having two or
l
a moresuperfluidcondensateswithdifferentspinstatesand
h Recently,insemiconductormicrocavitieswithquantum
the HQV is then a vortex in just one of them [51–55].
-wells sandwiched between highly reflective mirrors, the
s In this paper, we study spontaneous polarization un-
strong coupling is achieved between excitons and pho-
e
der nonlocal spin injection in a weakly interacting gas of
mtons [1–4]. Such coherent light-matter particles called
exciton-polaritoncondensates. Wefindadramaticallyen-
exciton-polaritons obey the Bose-Einstein statistics and
. hanced spin-polarized signal at the appropriate pumping
tthus condense at critical temperatures ranging from tens
a regime when taking into account incoherent hot exciton
mKelvin [5–7] till several hundreds Kelvin [8, 9], which ex-
reservoirscatteredintocoherentstates. Thecoherentspin
ceedsbymanyordersofmagnitudetheBose-Einsteincon-
- antiferromagneticstateisalsoidentifiedwhichrevealsspin
ddensation temperature in atomic gases. Recently, electri-
injection and spin current that can be manipulated by
ncallypumpedpolaritonlaserorcondensationwasrealized
obasedonamicrocavitycontainingmultiplequantumwells spin rotation symmetric pumping source, and may open
c[10,11]. Consideringthehightransitiontemperaturesand up new ways of thinking about spintronic devices. Fi-
[ nally, we find direct ways for the dynamic generation of
high tunability from pumping source, semiconductor mi-
fractionalized HQV around which spin currents circulate
1crocavities are perfectly suited for studies of macroscopi-
vcally collective phenomenon and have initiated the fasci- as a result of Berry’s phases induced by spin rotations.
6
natingresearchonthepolaritonquantumhydrodynamics.
0
0 Thepolaritonshavetwoallowedspinprojectionsonthe
2structure growth axis, ±1, corresponding to right- and II. PHYSICAL BACKGROUND.
0left- circular polarizations of photons. In diverse semi-
1.conductor materials like GaAs/GaAlAs [12], Si [13], or- In the absence of external magnetic field the “spin-up”
0ganicsingle-crystalmicrocavitySiNx/SiO2 [14]andsoon, and “spin-down” states σ = ± of noninteracting polari-
7spininjectionanddetectionhasbeensuccessfullyrealized tons,ortheirlinearlypolarizedsuperpositions,aredegen-
1which is one of the key ingredients for functional spin- erate corresponding to the right (σ ) and left (σ ) circu-
+ −
v:tronics devices. A number of prominent spin-related phe- lar polarizations of external photons. The spinor nature
inomena both in interacting and in noninteracting polari- ofexcitonpolaritonscanthereforebemanifestedsincethe
X
ton systems have already been predicted and observed in spin of the exciton polaritons are essentially free in semi-
rthe microcavities, such as, spontaneous polarization [15– conductor microcavities. To illustrate the fully degener-
a
19], polarization multistability [20–22], optical spin Hall ate spinor nature, and as a first step, to reach a good
effect [23–29] and topological insulator [30, 31]. approximation of this case, the Zeeman energy must be
Spin degrees of freedom in two-dimensional exciton- much smaller than the interaction energy, thus we shall
polaritons superfluid can drastically change elementary consider only the case of zero magnetic field. Since the
topological vortices referred to as half-quantum vortices interaction between two exciton polaritons depends on
(HQV) [32–36] which are characterized by a half-integer their total spins (singlet or triplet), the spin states of the
value of vorticity in contrast to the regular quantum vor- exciton polaritons can be changed after the scattering.
tex [37, 38, 40, 41] where the vorticity takes only integer The spin-dependent interactions cause the polariton spin
values. UsuallyHQVcarryonlyonehalf-integertopologi- states exchange. Moreover, additional mixing may comes
calchargeoriginatingfromthe superfluid currentpropor- from the longitudinal-transverse (LT) splitting of polari-
tional to ∇θ and due to the π spin disclinations super- tons(referredtoastheMaiallemechanism)[56]andfrom
imposed with half-vortices as a result of Berry’s phases structural anisotropies [57].
induced by spin rotations [42]. Similar half vortices have The low energy dynamics of the system is therefore de-
been discussed in A phase of 3He [43–45] or in triplet scribed by a pairwise interaction that is rotationally in-
2
variant in the spin space and preserves the spin of the representsthesumofthefieldresponsibleprovidedbythe
individualexcitonpolaritons. Thegeneralformofthisin- spindependentandindependentpolariton-polaritoninter-
teractionisVˆ (r −r )=δ(r −r )(cid:88)2f g ·Pˆ where actionsandpolariton-hotexcitoninteraction(LTsplitting
1 2 1 2 F F
gF =4π(cid:126)2aF/M,M isthemassofexcitoFn=0polaritons,PˆF HVeLrTy idsiffaesrseunmtefdrotmo btehonseegliisgoilbalteedinohrigchlodseednssityystreemgi,mteh)e.
is the projection operator which projects the pair 1 and 2 dynamic of spin pattern in such open-dissipative system
into a total spin F state, and aF is the s-wave scattering is crucially determined by the pump source. We will go
length in the total spin F channel. For exciton polaritons into further details in the following.
of f = 1 bosons, we have Vˆ = g ·Pˆ +g ·Pˆ . In terms
0 0 2 2
of nonlinear optics, the coupling coefficients of polariza-
tion independent c and so-called linear-circular dichro-
0 III. THEORETICAL MODEL.
ism c can be estimated through the matrix elements of
2
thepolariton-polaritonscatteringinthesingletandtriplet
In the following, we study the propagation of polarized
configurations.
polariton in the a planar microcavity and generation of
ItisconvenienttowritetheBosecondensateΨ (r)≡<
ψˆ (r)>asΨ (r)=(cid:112)n(r)ζ (r),wheren(r)istheadensity, spin injection, spin current and the observability of the
a a a HQV, in realistic structures. The equation of motion for
andζ isanormalizedspinorζ+·ζ =1. Itisobviousthat
a
all spinors related to each other by gauge transformation
eiθ andspinrotationsU(α,β,γ)=e−iSxαe−iSyβe−iSzγ are
degenerate, where (α,β,γ) are the Euler angles.
Thenon-equilibriumdynamicsofpolaritoncondensates
is described by a Gross-Pitaevskii type equation for the
coherent polariton field, which should be coupled to a
reservoir of hot excitons that are excited by the nonreso-
nant exciting pump. The model is, however, generalized
to take into account the polarization degree of freedom
of hot exciton. In this approach, instead of polarization
independentscattering,wemusttakeintoaccountdichro-
ismscatteringbetweenhotexcitonandcoherentpolariton
field.
Let us turn to the pseudospin representation, then the
→−
local spin density s at the position r and time t is
→−s (r,t) = Ψ†(r,t)→−ˆsΨ(r,t), where →−ˆs = ((cid:126)/2)→−σˆ with →−σˆ Figure 1: (Color online) The spontaneously circular polar-
ization of spinor condensate non-resonantly pumped by lin-
beingthePaulimatrices. Theusualdefinitionofthefree- early polarized laser. (a) Proposed scheme to experimentally
(cid:104) (cid:105)
particle probability current J = Re Ψ†(r,t)PˆIˆΨ(r,t) , stimulating spontaneous circular polarization by nonpolarized
n m laser beam. (b) Spinor is polarized when the laser power is
where Iˆ is the identity, and probability spin current larger than one threshold value, however, unpolarized after
J→− = Re(cid:104)Ψ†(r,t)Pˆ→−s Ψ(r,t)(cid:105). The emergent magnetic laser power is above second threshold value. (c) Density dis-
s m tribution of hot exciton (left picture which is the same profile
monopoles are defined by analogy with Maxwell’s equa-
→− for both components) and spinor polariton (middle and right
tion as ∇· s characterized by a divergent in-plane pseu-
picturesforeachcomponents)inrealspace. Here,simulations
dospin pattern which is are present in other systems intheabsenceofdisorderforfourpumpingpointswithasmall
,suchasmagneticallyfrustratedmaterialsorspin-ice[58– radius1.54µm. Thevaluesoftheparametersusedinthesim-
64]such as magnetic nanowires [65] and atomic spinor ulations are shown in the paper.
Bose-Einsteincondensates[66,67]. Thedynamicsofeach
spininamagneticfieldisgovernedbytheprecessionequa-
tion ∂ S=H×S/(cid:126). The total effective magnetic field H the spinor polariton wave function reads [68–71]
t
i(cid:126)∂ ψ (r)=(cid:26)− (cid:126)2 ∇2+ i(cid:126)(cid:0)g n +h n +β |ψ |2+f |ψ |2−γ (cid:1)+V (r)(cid:27)ψ (r)
t ± 2m 2 2 R± 2 R∓ 2 ± 2 ∓ C ext ±
+(cid:8)(cid:126)(cid:0)β |ψ |2+f |ψ |2(cid:1)+V (r)(cid:9)ψ (r), (1)
1 ± 1 ∓ R ±
where ψ represents the condensed field, with σ =± rep- density of the incoherent hot exciton reservoir. And here,
σ
resenting the spin state of polaritons with effective mass V (r) = (cid:126)[g n +h n +ΩP (r)] represents the
R 1 R± 1 R∓ ±
m. γ represents the coherent polariton decay rate. β reservoir produces spin-conserved and spin-exchange in-
C 1
andf isthespin-conservedandspin-exchangepolariton- teractions where P (r) is the spatially dependent pump-
1 ±
polariton interaction strength, respectively. n is the ing rate and g , h , Ω > 0 are phenomenological coef-
Rσ 1 1
3
under a spatially homogeneous pumping and in the ab-
sence of any external potential, Eqs. 1 and 2 admit
analytical stationary spinor configuration. Below the
pumpingthreshold, thecondensateremainsunpopulated,
while the reservoir grows linearly with the pump inten-
sity n = W/Γ. At the threshold pump intensity
R±
Wth, the stimulated emission rate exactly compensates
the losses g n +h n =γ and condensate becomes
2 R± 2 R∓ C
populated dynamically. We notice that threshold pump
intensity becomes Wth = Γγ /(g +h ). Above the
C 2 2
threshold, the reservoir density is homogeneous n =
R±
W/(cid:0)Γ+g |ψ |2+h |ψ |2(cid:1), from this, we obtain
2 ± 2 ∓
W (g −h )
Z ∼− 2 2 Z , (3)
Figure 2: (Color online) The spontaneously circular polariza- R Γ2+Γ(g +h )n C
2 2 c
tion of spinor condensate non-resonantly pumped by 6 and 8
linearlypolarizedlaser,respectively. Toppanel: densitydistri- here, we have defined reservoir polarization Z = n −
R R+
butionofhotexciton(leftpicturewhichisthesameprofilefor n , condensate polarization Z = |ψ |2 − |ψ |2 and
R− C + −
bothcomponents)andspinorpolariton(middleandrightpic- condensate total density n = |ψ |2 +|ψ |2. As long as
turesforeachcomponents)inrealspacefor6pumpingpoints. c + −
g (cid:54)= h , condensate polarization is directly proportional
Bottompanel: distributionofmagneticpolarizationalongthe 2 2
to the reservoir polarization. From the Eqs. 1, we find
Zaxisfor6pumpingpoints(leftpicture)and8pumpingpoints
that the condensate density grows as
(right picture, inset shows density distribution of two compo-
nent polariton). The values of the parameters used in the (cid:0)W −Wth(cid:1) 1
simulations are the same as those in the Fig. 1.
n ∼ · , (4)
c γC 1− 1(cid:16) W + β2+f2 Γ (cid:17)
2 Wth g2+h2γC
ficients to be determined experimentally. V (r) repre-
ext and condensate polarization
sents the static disorder potential typical in semiconduc-
tor microcavities, which is chosen as the same for both M Z =0. (5)
C C
component polaritons. g n and h n is related the
2 R± 2 R∓
condensation rate, representing the process where hot ex- where
citons with same spin or cross spin are stimulated growth (cid:18) WΓ2γ2 (cid:19)
of condensate, respectively. [72]. β2 and f2 are the same- MC = 4Wg2h2+Γ2(β2−f2)− Wth·WCth .
spin and cross-spin nonradiative loss rates, respectively.
The equation 1 for the condensate is coupled to a rate
The Eq. 5 has solution for the magnetization of conden-
equation for the evolution of the incoherent hot exciton
sate Z = 0 except very stringent condition M = 0.
C C
density n which is given by the rate equation:
Rσ Specially, if assuming cross-spin radiative and nonradia-
∂ n =−Γn −(cid:2)g |ψ |2+h |ψ |2(cid:3)n +P , (2) tive loss rates is negligible. we find laser power should
t R± R± 2 ± 2 ∓ R± ± satisfy
where the reservoir relaxation rate Γ is much faster Γ (cid:29) γ2 g2
γ under the Gaussian pump laser P = W which is as- W = C = 2 , (6)
suCmednonpolarized(correspondingto±linearorhorizontal β2(Wth)2 β2Γ2
polarization through the paper), giving a sufficient large
then, from necessary condition W > Wth, in order to
occupation in momentum space of incoherent hot exci-
spontaneous magnetization, we find
ton. The stimulated emission of the hot exciton reservoir
intothetwo-componentcondensatemodeistakenintoac- g3
count by the term (cid:2)g |ψ |2+h |ψ |2(cid:3)n . The spatial 2 >1,
2 ± 2 ∓ R± β γ Γ3
2 C
diffusion rate of reservoir density has been neglected. In
the following, we solve the coupled Eqs. 1 and 2 numeri- should be complied.
callystartingfromasmallrandominitialcondition,then, If assuming condensate wave function takes the form
the time evolution of the system can be calculated until a ψ±(r) = (cid:80)ψk±ω±ei(k±·r−ω±t) ∼ ψ0±ei(k±·r−ω±t), we
steady state is reached which is independent of the initial find spectrum of spinor condensate is given by
noise.
(cid:126)k2
ω = ± +Ω˜ W, (7)
± 2m ±
IV. STEADY STATE.
where
(β +f )n ±(β −f )Z
A. Spatially homogeneous system Ω˜ =Ω+ 1 1 c 1 1 C
± 2W
2(g +h )Γ+G·n ±H ·Z
Letusbeginwithsomeanalyticalconsiderationonfind- + 1 1 c C,
ing spinor condensate. In the homogeneous case, i.e., 2[Γ2+Γ(g2+h2)nc+A]
4
here, wave vector k and frequency ω remains so far Different from the single component condensate, now in
± ±
undetermined, and coefficience G=g g +g h +g h + Eq. 10, the quantum pressure terms are not only origi-
1 2 1 2 2 1 √
h h , H = (g h +g h −g g −h h ). However, from nated from density ∇2 ρ but also from the spinor ∇2ζ
1 2 1 2 2 1 1 2 1 2 √
Eq. 7, we find energy difference between two component and even spin-density coupling ∇ ρ·∇ζ. In particular,
condensate is in Eq. 11, besides the current divergence term liking in
single component condensate, the terms associated with
(cid:126)(cid:0)k2 −k2(cid:1)
ω+−ω− = +2m − +∆Ω˜, (8) c∇o√upρli·nkg b(ert)weoernspsuinpeprrfleusisdurceurkren(tr)an·d∇dζenarseityapppreesasruerde.
C C
We can make local density approximation (LDA) and lo-
here,
cal spin approximation (LSA) if the spatial variation of
∆Ω˜ ∼Z {(β −f )/W thelaserpumpprofileW (r)issmoothenough, wherethe
C 1 1
quantum pressure term in Eq. 10 and 11 are neglected.
−(g −h )(g −h )/(cid:2)Γ2+Γ(g +h )n +A(cid:3)(cid:9),
1 1 2 2 2 2 c Under these LDA and LSA, similar to the homogeneous
case,thecondensatedensityprofileandcondensatepolar-
where A is high order term of density and polarization
A=(g +h )2(cid:0)n2+Z2(cid:1)/4whichcanbedominantterm izationisgivenbytheEq. 4andEq. 5, respectively, with
2 2 c C the local value of the laser pump power W (r).
for the larger condensate density and polarization. Inter-
Under the Gaussian laser pump profile, we can look
estingly, we can see that energy gap is polarization Z
C forcylindricallysymmetricstationarysolutionswherethe
dependence and especially, which does not depend on the
condensate frequency ω is determined by the boundary
laser power when β (cid:39)f or large enough laser power. ±
1 1 condition that the local condensate density wave vector
vanishes k (r=r )=0 at the center of the each pump-
C p
ing spot, i.e.,
B. Local density and spin approximation
Ω˜ ·W
In the presence of an inhomogeneous laser pump W (r) ω± = ±γ , (13)
(or multiple pump W (r)), much richer phenomena will C
i
be represented in our system, like, spin domain, magnetic where
monopole, half vortex and so on. In this case, we can
look for stationary spinor polariton wave function of the Ω˜ =Ω+ (β1+f1)ρ±(β1−f1)ρSZ
following form ± 2W
2(g +h )Γ+[G·ρ±H ·ρS ]
Ψ=(cid:18) ψ+ (cid:19)=(cid:112)ρ(r)ζ(r)e−i(φ(r)−ω±t), (9) + 2[1Γ2+1Γ(g2+h2)ρ+B·ρ2Z] , (14)
ψ
−
from here, we can find frequency difference between two
where ρ(r) and φ(r) are the local density and phase of component condensate as
the condensate wave function, and ζ(r) is spinor func-
tion. We are going to assume that the local pump im- ∆Ω˜ ·W
posesaboundaryconditionforthespinor-functionateach ω+−ω− = γ , (15)
C
pumping spot r : lim ζ(r) = λ, lim k (r) = 0,
p r→rp r→rp C
here, we have defined local condensate density wave vec- here,
tor k (r)=∇ φ(r). In what follows, we use the dimen-
C r
sionless form of the model obtained by using the scaling ∆Ω˜ =Ω˜ −Ω˜
+ −
units of time, energy, and length: T = 1/γ , E = (cid:126)γ ,
C C =ρ(r )S (r ){(β −f )/W
L=(cid:112)(cid:126)/mγC. Inserting spinor form Eq. 9 into the Eqs. −(g −p hZ)(gp−h1)/(cid:2)Γ12+Γ(g +h )ρ+Aρ2(cid:3)(cid:9),
of motion 1 and 2, one obtains the following set of condi- 1 1 2 2 2 2
tions after considering stationary solution:
here, we have defined condensate polarization S (r ) =
Z p
1(cid:18)∇2√ρ ∇2ζ ∇√ρ·∇ζ (cid:19) |ζ+(rp)|2 −|ζ−(rp)|2 and coefficience of density square
ω± =−2 √ρ + ζ ± +2 √ρζ ± −kC2 term B = (g2+h2)2(cid:0)1+SZ2(cid:1)/4, which has maximal
± ± value(g +h )2/2forthetotalpolarization±1. Interest-
+ 1 (cid:0)β |ζ |2ρ+f |ζ |2ρ+g n +h n (cid:1)+ ΩW, ingly, w2e can2see that energy gap is polarization S (r )
γ 1 ± 1 ∓ 1 R± 1 R∓ γ Z p
C C dependence and especially, which does not depend on the
(10)
laser power when β (cid:39)f or large enough laser power.
1 1
Localcondensatedensitywavevectork (r)isreaching
and C
maximalvaluewiththecondensatedensitydecreasedand
1(cid:16)g n +h n +β |ζ |2ρ+f |ζ |2ρ−γ (cid:17) spin polarized away from the pumping center. Polaritons
2 2 R± 2 R∓ 2 ± 2 ∓ C condenseatthelaserspotpositionhasalargeblueshifted
√
1 ∇ ρ·k (r) k (r)·∇ζ energyduetotheirinteractionswithuncondensedhotex-
+2∇·kC(r)+ √ρC + C ζ ± =0, (11) citons, thus within a short time, these interaction energy
±
will lead to the motion of polariton initially localized at
and pumping point. In particular, spontaneous polarization
would happen because polarization may lower the fre-
Γn +(cid:0)g |ζ (r)|2+h |ζ (r)|2(cid:1)ρ(r)n =W(r). (12) quencysignificantlyunderthelaserpowerislargeenough
R± 2 ± 2 ∓ R±
5
as seen from Eq. 14, especially, spin domain, spin current
and topological defect may be formed under appropriate
condition.
In the following section, through extensive numerical
simulationsoftheEq. 1coupledtothereservoirevolution
Eq. 2, the robustness of above analytical considerations
and, in particular, the dynamical formation of spin do-
main, spin current and half vortex for the typical values
of the experimental parameters have been verified for a
wide range of pump parameters.
V. NUMERICAL RESULTS FOR
SPONTANEOUS POLARIZATION.
Equations can be solved numerically with the initial
Figure 3: (Color online) Normalized average density current
condition n (x,y,t) ≈ 0, ψ (x,y,t) ≈ 0. The param-
Rσ σ J ofacoherentpolaritoncondensatewhichisnon-resonantly
eters of the pump are chosen in a way to compare with nx
excitedwith6pointslinearlypolarizedlaser. Theinsetsshows
the experimentally observation [27, 27–29, 36] and con- thetotaldensityprofilebeforeandaftershiftingpositionoftwo
dition. In our calculations we used the following param- linearly polarized lasers (see the schematic picture) along the
eters, typical for state-of-the-art GaAs-based microcavi- xdirection,andalsoshowsJ underdecreasingthepumping
nx
ties: the polariton mass is set to m = 10−4 m where power to 80%.
e
m is the free electron mass. The decay rates are cho-
e
sen as γ = 0.152 ps−1 and Γ = 3.0γ . The interaction
C C
strengthsaresetto(cid:126)β =40µeVµm2,f =−0.1β ,g = polarized polaritons ballistically fly away from the laser
1 1 1 1
2β , h = −0.2β , and condensation rate to (cid:126)g = 0.16 spot due to their interactions converted into kinetic en-
1 1 1 2
meV µm2, (cid:126)h = 0.016 meV µm2, and condensation loss ergy of coherent polariton.
2
rate −(cid:126)β = 0.16 meV µm2, (cid:126)f = 0.016 meV µm2. The Fig. 2 show the density distribution of the calculated
2 2
pump intensity was chosen to match the experimentally incoherent hot exciton and coherent polariton condensate
measuredblueshiftofthepolaritoncondensate,andpump forsixandeightunpolarizedpumpingpoints,respectively.
profile is Gaussian shape: As is shown, while incoherent hot excitons experience a
limited diffusion in such case, however, the neighbouring
polarization of condensed polaritons are polarized with
W(r)= πww02(cid:88)ni=1e−(x−xi)w2−12(y−yi)2, panredcissepliyn-odpeppoesnitdeenptolparriezsastuiroentdeertmersm. inWedebsyhtohwesdethnesitsyz-
1 distribution which clearly has the opposite circular po-
here, for a typical case, w = 1.0, |x | = |y | = 1.5, and larization for two neighbouring site. This steady state
1 i i
w is tuned accordingly. is characterized by the magnetic domain wall formation
0
corresponds to a vanishing total magnetization. As we
As expected, the dynamics of spinor condensate tends
have emphasized above that total effective external field
to a dynamically stable steady state with a sponta-
is fundamentally generated and controlled by the spatial-
neously circular polarization under increasing laser power
dependent pump source which provide the coupling po-
as shown in Fig. 1(b). Threshold laser power for sponta-
tential for the dynamics of particle density and spin den-
neouslycircularpolarizationislargerthanthatofstarting
sity of exciton-polariton condensates. Thus, geometrical
condensation which can be understood from the Eqs. 5
dependent interactions or geometrical effective magnetic
and 6. The coherent polarized polaritons ballistically fly
fieldmayleadtodifferentand,perhaps,morecomplicated
away from the laser spot due to their interactions con-
magneticstructures. Thus,inthefollowing,wewillstudy
verted into kinetic energy of coherent polariton. In par-
how to dynamically generate the spin current, fractional-
ticular, the circular polarization rapidly saturates with
ized HQV and magnetic charge (or magnetic monopole)
increasing the pumping power and may lead to an almost
via dynamical tuning pumping (or geometrical) source.
full polarization of spinor condensate [17–19]. Surpris-
ingly, almost full circular polarization will finally change
backtothelinearpolarizationwithfurtherincreasingthe
laser power (i.e., the density of condensate exceeding a VI. DENSITY CURRENT, SPIN CURRENT,
TOPOLOGICAL CHARGE.
thresholdvalue). Thequantumpressureterms(especially,
spin-dependent pressure terms) in Eqs. 10 and 11 has to
betakenintoaccounttounderstandthispolarizationsat- A. Density current
urates and going back to the linear polarization. In Fig.
1(c), densityprofilesofincoherenthotexcitonandpolari- Physically, condensed fluid is a rather long-range co-
ton condensate represent linear and circular polarization, operative phenomenon and is characterized by a special
respectively, foronesetofparameterschoseninthespon- long-range correlation between the particles involving the
taneouslycircularpolarizationregime. Asisshown,while coherent ordering of the momenta. The density-density,
unpolarized hot excitons experience a limited diffusion, density-current and current-current correlation function
6
in some cases implies that the liquid has net surface cur-
rentsandanetorbitalangularmomentum. Itistherefore
important to evaluate and generate the density current
and spin current. We find that, by suddenly shifting a
distance of pumping laser, the density and spin current
can be dynamically generated, and integer vortices can
be created as well. In particular, if shifted the pumping
laseriscircularpolarized, fractionalizedHQVcanbesuc-
cessfullycreatedwhichcanbedetectedexperimentallyby
means of polarization-resolved interferometry, real-space
spectroscopy, and phase imaging technique [34].
To explain these features, we have performed simula-
tionsunderasuddenlyshiftingthepositionoftwomiddle
pumpinglaseractingonpolaritons. Fig. 3givesanexam-
ple of the outcome for a given realization of six linearly
polarized pumping laser, and shows the normalized av-
erage polariton density current as function of time. As Figure 4: (Color online) Normalized average spin current
(cid:10) (cid:11)
is shown in the Fig. 3, after switching on the pumping Jsx,x of a coherent polariton condensate which is non-
lasers, polariton experiences a large oscillation of density resonantly excited with 6 points linearly polarized laser. The
insets shows density profile of the spin current J at the fi-
current within a short time, then, such density current sx,x
nalstageaftershiftingpositionoftwolinearlypolarizedlasers
oscillation decays very quickly and completely decreases
along the x direction, and that for normalized average spin
to zero at 60 unit of time. The reason of large oscillation (cid:10) (cid:11)
current along the y direction J .
current in the early process is due to the large overlap of sx,y
pumpedhotexcitonandpolaritonleadingtothelargere-
pulsiveforceactingonthecondensedpolariton. However,
with the diffusion of polariton under the repulsive force,
a steady state with zero averaged current can be reached
finally. Here, zero density current means polariton con-
densate reaches a balanced configuration in momentum
space.
Next, in order to generate net density current, we sud-
denly shifting the two pumping lasers in the middle site
at time 1050 (see the schematic picture of inset in the
Fig. 3), a persistent current (about -0.15 in amplitude)
with non-decay small oscillation is created successfully.
Here, non-zero current can be understood as a new con-
figuration in momentum space of condensed polariton af-
ter non-adiabatic modifying interaction energy between
polariton and hot exciton by shifting pumping lasers. Es-
pecially, the small and fast oscillation in the density cur- Figure 5: (Color online) Phase profile of each circular com-
rent can be understood as generating surface oscillation ponentofcondensedpolaritonwhichisnon-resonantlyexcited
mode which is moved back and forth due to it’s confined with 6 points linearly polarized laser. The up row and down
rowcorrespondtothephasemapafterandbeforeshiftingpo-
by the pumping laser. Interestingly, such surface oscil-
sition of two linearly polarized lasers along the x direction,
lation mode can be suppressed completely by decreasing
respectively.
the pumping laser power (as is shown in the inset of Fig.
3, where pumping laser power has been decreased to 80
percent.). Fundamentally, generation of net surface cur-
magnetic field which can be utilized to generate polariza-
rentscanbethoughtasgeneratinglong-rangedensitycor-
tion patterns as well as spin-polarized vortices.
relation in separated region by shifting a distance of the (cid:10) (cid:11)
Figure 4 shows average spin current J of a coher-
pumping lasers. sx,x
entpolaritoncondensateasfunctionoftimewhichisnon-
resonantly excited with 6 points linearly polarized lasers.
Similar to the behavior of average density current shown
B. Spin current, topological defect. in Figures 3, persistent spin current is quickly developed
aftersuddenlyshiftingthetwopumpinglasersinthemid-
In the strong coupling regime of semiconductor micro- dle site at time 1050. Here, the spin current is induced
cavities, due to their strong optical nonlinear response, bytheeffectivenon-abeliangaugepotentialwhichisorig-
spin polarization properties, and fast spin dynamics, po- inated from spin-dependent interaction between hot exci-
lariton condensates are excellent candidates for design- tonandcoherentcondensate. SimilartotheFigure3, the
ing novel spin-based devices. Here, we show the coher- small and fast oscillation in the spin current is generated
ent transport of the spin vector of propagating polariton which can be understood as generating surface oscillation
condensates. Theobservednondissipativelong-rangespin modeofspinorfunctionduetoit’sconfinedbythepump-
transportiscausedbyexcitingdensity-dependenteffective ing laser. Especially, average spin current moving along
7
ponent is still zero.
Now, the question is how to generate stable HQV in
our studied system. In order to realize that, we now shift
the position of one circularly polarized pumping lasers in
the middle site (one linearly polarized laser is just the
superposition of two circularly polarized lasers equally),
andthenseehowtheHQVisformeddynamically[32–36].
Very interestingly, steady exotic HQV can be successfully
generated as is shown in 6 which has significant features
that one component has vortices, but second component
doesn’t exist vortices. In particular, there are always two
opposite circulation of HQV generated for one of compo-
nent due to keep total angular momentum zero.
Generally, what kinds of stable topological defects are
developed depending on the dynamics of gauge poten-
Figure 6: (Color online) Phase profile of each circular com- tial together with vector field, such as Maxwell-Chern-
ponentofcondensedpolaritonwhichisnon-resonantlyexcited
Simons-vector Higgs model for the the superconductivity
with 6 points linearly polarized laser. The up row and down
of Sr RuO [46]. In our studied non-equilibrium exciton-
rowcorrespondtothephasemapafterandbeforeshiftingpo- 2 4
polaritons liquid, incoherent hot exciton with pumping
sition of two linearly polarized lasers along the x direction,
source and spin-dependent dissipation play important
respectively.
roles in topological excitation and make the dynamics of
gaugepotentialmorecontrollablecomparingwithconven-
the y direction (cid:10)J (cid:11) expects large oscillation across the tional solid state system and ultracold atoms. HQV is
zero value equallysxa,ys is shown in the inset which means analogous to the multicomponent quantum Hall system
the stronger spin-dependent interaction and polarization [73], which is the two-dimensional electron system in an
reversedalongtheydirectionwithtimeevolution. Thelo- external magnetic field violating parity and time reversal
cal spin current can be positive or negative value depend- symmetry. Here we have mainly focused on dynamical
ing on the effective local gauge connection as is shown in creation of density, spin current and HQV induced by the
the density of s . effective non-abelian gauge potential at relatively short
x,x
Comparison to solid-state systems, the spatial- time; these phenomena can be conveniently probed by
dependent pump sources inducing spin-dependent stimu- real-space spectroscopy, and phase imaging [34].
lationanddissipationforthecondensedpolaritonprovide
averypromisingwaytogenerateandcontrolthespincur-
rentandspinstructureinthepresenceofdifferentkindsof
VII. CONCLUSIONS.
effectivegaugefields(likeDresselhausandRashbafields),
therefore open very broad possibilities for the studies of
spinor quantum fluids and accessibility to their diverse In conclusion, we have demonstrated a practical way
quantum phases. In particular, such flexible and efficient to control spin polarization, induce density and spin
way to control external pumping laser may generate fas- current, and creating fractionalized vortices in exciton-
cinating topological defects dynamically which properties polaritonssemiconductormicrocavities. Forthepolariton
dependonthewayhowtomanipulatethepumpinglaser. lifetime, the spin localization, spin polarization and ex-
Figures 5 shows the phase profile of each circular com- otic vortices can be readily excited in photoluminescence
ponent of condensed polariton under shifting the linearly experiments and detectable by the time-resolved micro-
polarized lasers in the middle site. In such case, shifted photoluminescence spectroscopy [37, 74] or spin noise
laser is linearly polarized which means each circular com- spectroscopy [75, 76]. Our results are of particular sig-
ponentofcondensedpolaritonexperienceaequallypump- nificanceforcreatingtheseexcitationsinexperimentsand
ing source mainly originated from V (r) in the Eq. 1. forexploringnovelphenomenaassociatedwiththem. This
R
Thus, as expected, after some time evolution, the nor- noticeably spin amplification and spin transport could of-
malintegerquantumvortexforeachcomponenthasbeen fer a promising way to optimize spin signals in future de-
generated and stable robustly as is shown in the upper vices with using polariton condensates.
row of 5 which is similar to the single component case We are grateful to N. Berloff for discussions. The fi-
[37, 38, 40, 41]. As is shown, two separated integer vor- nancial support from the early development program of
ticeswithoppositecirculationhasbeengeneratedforeach NanChangUniversityandSkoltech-MITNextGeneration
component, thus total angular momentum for each com- Program is gratefully acknowledged.
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