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Influence of exciton-exciton correlations on the polarization characteristics of the

polariton amplification in semiconductor microcavities

S. Schumacher, N. H. Kwong, and R. BinderCollege of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

(Dated: February 1, 2008)

Based on a microscopic many-particle theory we investigate the influence of excitonic correlationson the vectorial polarization state characteristics of the parametric amplification of polaritons insemiconductor microcavities. We study a microcavity with perfect in-plane isotropy. A linearstability analysis of the cavity polariton dynamics shows that in the co-linear (TE-TE or TM-TM) pump-probe polarization state configuration, excitonic correlations diminish the parametricscattering process whereas it is enhanced by excitonic correlations in the cross-linear (TE-TM orTM-TE) configuration. Without any free parameters, our microscopic theory gives a quantitativeunderstanding how many-particle effects can lead to a rotation or change of the outgoing (amplified)probe signal’s vectorial polarization state relative to the incoming one’s.

PACS numbers: 71.35.-y, 71.36.+c, 42.65.Sf, 42.65.-k

I. INTRODUCTION

In the past decade the parametric amplification of po-laritons in planar semiconductor microcavities has beenthe subject of intense experimental and theoretical re-search, see, e.g., Refs. 1,2,3,4,5,6 or the reviews given inRefs. 7,8,9. In a typical pump-probe setup in a co-circularpolarization configuration the amplification of a weakprobe pulse has mainly been attributed to four-wave mix-ing (FWM) processes mediated by the repulsive Coulombinteraction of the exciton constituent of the polaritonsexcited on the lower polariton branch (LPB).3,4,6,10 Fora specific pump in-plane momentum (defining the so-called “magic angle”), energy and momentum conserva-tion is best fulfilled for the FWM processes and thusa pronounced angular dependence of this amplificationis observed.1,3 Since in the strong coupling regime theLPB is spectrally well below the two-exciton scatteringcontinuum, the influence of excitonic correlations in thescattering processes of polaritons on the LPB is stronglysuppressed [compared to the situation in a single quan-tum well (QW) without the strong coupling to a confinedphoton cavity mode11]. However, even for co-circularpump-probe excitation these correlations must be con-sidered for a complete understanding of the experimentalresults.5,10,12

Whereas for co-circular pump-probe excitation onlyexciton-exciton scattering in the electron-spin tripletchannel plays a role, for excitations in other vectorialpolarization state configurations, excitonic scattering inthe electron-spin singlet channel is also expected to con-tribute to the amplification mechanism. In the lattercase, a change (in the following loosely referred to as‘rotation’) of the vectorial polarization state of the am-plified probe signal compared to the incoming one’s canbe attributed to this coupling of the two spin subsys-tems excited with right (+) and left (−) circularly po-larized light, respectively.13,14,15,16,17,18,19,20,21 However,different effects can overshadow rotations in the vec-torial polarization state that are caused by the spin-

dependent many-particle interactions that mediate theamplification process, e.g., a splitting of the TE andTM cavity modes22,23 (longitudinal-transverse splitting),or an in-plane anisotropy of the embedded QW or thecavity21,24. Furthermore, for not linearly polarized pumpexcitation an imbalance in the polariton densities in thetwo spin subsystems (+ and −) will also lead to a rota-tion in the vectorial polarization state of the amplifiedsignal.17,21 These different mechanisms have previouslybeen investigated16,17,18,19,21 based on models describingthe effective polariton dynamics in the cavity. In thesemodels that describe the system dynamics at the po-lariton quasi-particle level, the spin-dependent polariton-polariton scattering matrix elements are included as in-put parameters for the theory. With a reasonable choiceof the parameter set, good agreement with experimentalresults showing rotations in the probe’s vectorial polar-ization state has been obtained.16,18,19,21

In contrast to these previous studies,16,17,18,19,21 weemploy a microscopic theory that calculates, from afew material parameters, the scattering matrices driv-ing the polariton amplification in the different vectorialpolarization state channels25. No additional assump-tions for the effective polariton-polariton interaction areneeded, which is directly included in our theory via thefrequency dependent and complex exciton-exciton scat-tering matrices.10 Our theoretical analysis incorporates:(i) the well-established microscopic many-particle the-ory for the optically-induced QW polarization dynam-ics based on the dynamics-controlled truncation (DCT)formalism26,27, and (ii) the self-consistent coupling of thisdynamics to the dynamics of the optical fields in the cav-ity modes10,28,29 including all vectorial polarization statechannels. The theory consistently includes all coherentthird order (χ(3)) nonlinearities and the resulting equa-tions of motion are solved in a self-consistent fashion inthe optical fields which includes a certain class of higher-order nonlinearities.30,31,32 Correlations involving morethan two excitons and those involving incoherent exci-tons are neglected. These effects are not expected to

2

qualitatively alter the presented results for the consid-ered coherent exciton densities of ∼ 1010 cm−2, especiallyfor excitation well below the exciton resonance.

Based on this theory we introduce a linear stabilityanalysis (LSA) of the cavity polariton dynamics as a gen-eral and powerful tool to study the role of spin-dependentpolariton-polariton scattering (including time-retardedquantum correlations). For steady-state pump excita-tion and as long as depletion of the pump from scatteringinto probe and FWM signals can be neglected, the LSAgives comprehensive information about growth and/ordecay (in the following only referred to as ‘growth’) ratesfor probe and FWM intensities in all vectorial polariza-tion states. The growth rates determine the exponentialgrowth of components in different vectorial polarizationstates over time, and determine together with the initialconditions the ratio of these components after a givengrowth duration, uniquely determining the final vectorialpolarization state. Although the results are obtained forstrict steady-state pump excitation, as discussed belowthey can to a large extent be carried over to the analysisof pump-probe experiments with finite pulse lengths.11

We use this theory to investigate a microcavity sys-tem with perfect in-plane isotropy. As an intrinsic ef-fect that is not caused by structural imperfection, weinclude a splitting of the TE and TM cavity modes asshown in Fig. 1(a). This way we study a system whereall vectorial polarization state rotations of the amplifiedprobe signal can unambiguously be traced back to intrin-sic phenomena always present in planar semiconductormicrocavities, the TE-TM cavity-mode splitting and thespin-dependent polariton-polariton scattering mediatingthe amplification.

For this system we analyze results that show how fora linearly polarized pump many-particle correlations andthe TE-TM cavity-mode splitting lead to different growthrates of the linearly polarized components (TE,TM)in the probe pulse. This difference can lead to rota-tion in the vectorial polarization state of the amplifiedprobe compared to the incoming one’s.16,17,18,19,21 Start-ing from the equations governing the cavity-polariton dy-namics, we take advantage of our theoretical approach toisolate and discuss the frequency-dependent scatteringmatrices that give rise to this difference in the vectorialpolarization state channels. We show that in the studiedregime, close to the amplification threshold, even wherethe correlation contribution in the spin-singlet channelis weak15,20, these correlations can give rise to an al-most complete vectorial polarization state rotation intothe “preferred” cross-linear (TE-TM or TM-TE) pump-probe configuration for the pump and outgoing probepulses. For sufficiently long amplification duration, thisresult can become virtually independent of the inputprobe’s vectorial polarization state as long as it contains asmall component polarized perpendicular to the linearlypolarized pump.

-5 0 5

-5

0

5

TM: TE:

cavity

UPB(a)

k [106m-1]

frequ

ency

[meV

]

xk

LPB

2 1 0 -1

-10

0

10ImT

++

ImT

+- R

eT+-

(b)

[ax20 Ex

b]

-2x 0 [m

eV]

Re

T++

FIG. 1: (color online) (a) Shown is the dependence of thecavity polariton modes on the magnitude k of the in-planemomentum. Depicted are the lower (LPB) and upper (UPB)polariton branches for TE (solid) and TM (dashed) cavitymodes, and the bare cavity and exciton (dotted) dispersions.For details on the modeling of the cavity modes see Sec. II.(b) Real and imaginary parts of the two-exciton scattering

matrices T (Ω) in the co-circular (++) and counter-circular(+−) polarization state channels. (a,b) Results are shown fortypical GaAs parameters33 as used throughout this work.

II. THE THEORETICAL MODEL

We use a microscopic many-particle theory to describethe coherent QW response to the light field confined inthe cavity. Based on the dynamics-controlled trunca-tion (DCT) approach26,27 all coherent optically-inducedthird order nonlinearities, i.e., phase-space-filling (PSF),excitonic mean-field (Hartree-Fock) Coulomb interactionand two-exciton correlations are included on a micro-scopic level. We use a two-band model (including spin-degenerate conduction and heavy-hole valence band) todescribe the optically induced polarization in the GaAsQW.33 Since we are mainly interested in pump excitationin the LPB, i.e., energetically below the bare exciton res-onance (cf. Fig. 1), we account for the dominant contri-butions to the QW response by evaluating the opticallyinduced QW polarization in the 1s heavy-hole excitonbasis.29,31,32,34

We start from the coupled equations of motion forthe field Ek in the cavity modes with in-plane mo-mentum k (treated in quasi-mode approximation35) andthe optically induced interband polarization amplitudepk in the embedded QW. We formulate our theory inthe TE-TM basis for the optical fields in the cavity,Ek = ETE

k eTE + ETMk eTM (see Fig. 2 for the excitation

geometry), where the field components in the TE modeETE

k eTE (also called s-polarized) are characterized by anelectric field vector with in-plane (in the plane of the

3

z

x

j

plane ofincidence

quantum-wellplane

k

TEe

y

h

TMe

in-plane

in-plane

FIG. 2: (color online) Schematic of the excitation geometry.The plane of incidence is spanned by the wave vector of theincoming light field and the z axis. All lines in this planeare solid. The quantum-well plane is the x-y plane. All linesin this plane are broken. The figure shows the basis vectorsein-planeTE and e

in-planeTM that span the projection of the TE-TM

basis on the x-y plane. The polar angle ϑ and azimuthal angleϕk are also shown. For more explanation see text in Sec. II.

QW) component perpendicular to the in-plane momen-tum k, and field components in the TM mode ETM

k eTM

(also called p-polarized) by an electric field vector within-plane component parallel k. In this basis it is most in-tuitive to include different cavity-mode exciton couplingsfor the TE and TM modes in the theory: the in-planecomponent of the fields in the TE mode does not dependon the polar angle of incidence ϑ (cf. Fig. 2) and thus thecoupling strength to the excitonic dipole in the quantum-

well plane does not depend on ϑ. A different result isfound for fields in the TM mode where the magnitudeof the in-plane component depends on the polar angle ofincidence ϑ. Since the z component of fields in the TMmode does not couple to the excitonic dipole for exci-tation of heavy-hole excitons in the QW36, the effectivecoupling constant of excitons and fields in the TM modedecreases with the polar angle like ∼ cosϑ. Additionally,we include a slightly different polar angular dependence

of the bare TE and TM cavity dispersions ωTMTE

k that ingeneral depend on the specific materials and design ofthe cavity.22 The resulting cavity dispersions are shownin Fig. 1(a) and the parameters will be given later in thissection. In order not to complicate the structure of thenonlinear terms in the equations of motion for the polar-ization amplitudes, we use the usual Cartesian basis (X-Ybasis) in the QW plane (x − y plane) to decompose thepolarization into its components as pk = pX

k ex + pYk ey.

In the X-Y basis, the projection of the TE-TM basis vec-

tors on the x − y plane, ein-planeTE and e

in-planeTM , rotates

with the in-plane component k of the momentum of theincident wave. The azimuthal angle between k and thex axis is denoted by ϕk (cf. Fig. 2). Simple geometricconsiderations lead to the azimuthal angular dependen-cies that appear in the terms coupling the equations ofmotion for field and polarization amplitude componentsin the different bases:

i~ETMTE

k =(

~ωTMTE

k − iγc

)

ETMTE

k

− VTMTE

k

[

pXY

k cosϕk ± pYX

k sin ϕk

]

+ i~tcETMTE

k,inc , (1)

i~pXY

k =(

~ωxk − iγx

)

pXY

k −(

VTMTE

k ETMTE

k cosϕk ∓ VTETM

k ETETM

k sin ϕk

)

+A∑

k′k′′

(

pXY∗

k′+k′′−kpXY

k′ + pYX∗

k′+k′′−kpYX

k′

)(

VTMTE

k′′ ETMTE

k′′ cosϕk′′ ∓ VTETM

k′′ ETETM

k′′ sinϕk′′

)

+A∑

k′k′′

(

pYX∗

k′+k′′−kpXY

k′ − pXY∗

k′+k′′−kpYX

k′

)(

VTETM

k′′ ETETM

k′′ cosϕk′′ ± VTMTE

k′′ ETMTE

k′′ sinϕk′′

)

+1

2

∑

k′k′′

pXY∗

k′+k′′−k

∫ ∞

−∞

d t′[

(

T ++(t − t′) + T +−(t − t′))

pXY

k′(t′)p

XY

k′′(t′) −

(

T ++(t − t′) − T +−(t − t′))

pYX

k′(t′)p

YX

k′′(t′)]

+∑

k′k′′

pYX∗

k′+k′′−k

∫ ∞

−∞

d t′[

T ++(t − t′)pXY

k′′(t′)p

YX

k′(t′)

]

. (2)

These equations constitute the generalization of theequations given in Ref. 10, now including all vectorialpolarization states. As discussed above, the polarizationamplitudes are given in the X-Y basis, while the fieldsare given in the TE-TM basis.37 The meaning of thesymbols in Eq. (2) are to be discussed in the remain-

der of this paragraph, along with the used parametersand approximations. Unless otherwise noted, the timeargument in Eqs. (1) and (2) is t. tc is the couplingconstant of the cavity mode to the external light fields

ETMTE

k,inc, and the dephasing constant γc describes optical

losses from the cavity to the outside world.10 The depen-

4

dence of the bare cavity modes ωTMTE

k and the dependence

of the exciton-cavity mode coupling VTMTE

k on the polarangle ϑ is modeled on a phenomenological level alongthe guidelines given in Ref. 22. We approximate the barecavity dispersions with ωTM

k = ω0

cos ϑ+ 100 meV · sin2 θ

and ωTEk = ω0

cos ϑwith sinϑ = |k|c0

ωnbg. This way a TE-

TM cavity-mode splitting from the mismatch of thecenter of the stopband of the cavity mirrors and theFabry-Perot frequency of the cavity is phenomenologi-cally included.22 The cavity-mode exciton couplings areV TM

k = V TM0 cosϑ and V TE

k = V TE0 , respectively, with

V TM0 = V TE

0 = 5.2 meV. With ~ωTMTE

0 = εx0 we assume

zero cavity-mode exciton splitting for k = 0. The cho-sen parameters33 give a reasonable magnitude and polarangular dependence of the TE-TM mode splitting in theLPB, comparable to the results in, e.g., Ref. 23. Sincethe presented results do not crucially depend on the de-tails of the cavity-mode splitting, no further insight isexpected from a more elaborate treatment. In Eq. (2),~ωx

k is the 1s heavy-hole exciton in-plane dispersion, γx

a phenomenological dephasing constant of the excitonicpolarization amplitude, and A is related to the excitonic

PSF constant APSF by A = APSF

φ∗

1s(0) , with φ1s(r) being

the two-dimensional QW exciton wavefunction. With-out loss of generality, in the following the quantities φ1s,

VTMTE

0 , and tc are chosen to be real-valued. The parametervalues are listed in Ref. 33. Although in this paper weinvestigate a spatially isotropic microcavity-system, spa-tial anisotropy can easily be included in the theory viaωx

k → ωxk to model an anisotropic dispersion of the QW

excitons and via ωTMTE

k → ωTMTE

k to model anisotropy of thecavity modes. The two-exciton scattering matrices (T-matrices) T in the co-circular (++) and counter-circular(+−) polarization state channels include a two-excitondephasing rate 2γ and are given by T ++ = T + andT +− = (T + + T −)/2, with the T-matrices T + and T −

in the electron-spin triplet and singlet channel, respec-tively, as defined in Eq. (32) of Ref. 25. The frequencydependence of real and imaginary parts of T ++ and T +−

is shown in Fig. 1(b). We neglect the momentum depen-dence of the T-matrices for scattering processes involv-ing two excitons with different in-plane momenta. Cal-culations in a different context have shown that this isjustified in a good approximation for the small opticalmomenta contributing here.38 We also neglect possiblecorrections to the excitonic T-matrices from the couplingto the photons in the cavity modes. This is supportedby experimental observations that indicate that even inthe strong coupling regime the biexciton binding energyis not significantly affected by the coupling to the cavitymodes.39 Also, good theory-experiment agreement hasbeen achieved in Refs. 28,29,40 using a theory based onpure exciton-exciton scattering matrices.

To simulate a typical pump-probe setup, we start fromEqs. (1) and (2) and chose a finite in-plane momentumkp for the pump propagating along a given axis, here,

without loss of generality the x axis, i.e., ϕkp= 0. We

“detect” the probe in normal incidence with in-plane mo-mentum k = 0 where in the past the strongest amplifica-tion has been observed.10,12,16,41 This fixes the in-planemomentum of the background-free FWM signal to 2kp.We go beyond an evaluation of the theory on a strictχ(3) level by self-consistently calculating the resulting ex-citonic and biexcitonic polarization amplitude dynamicsup to arbitrary order in the pump field30,31 and we lin-earize the equations of motion in the weak probe field.Only via this self-consistent solution the coupling of theprobe signal to the background-free FWM signal is in-cluded in the theory which provides the basic feedbackmechanism that leads to the unstable behavior in, e.g.,Refs. 1,2,3,4,5,7,8. We limit our analysis to coherent ex-citon densities that are low enough [<∼ 2× 1010 cm−2, cf.Figs. 3 and 4] so that neglect of higher than two-excitonCoulomb correlations can be justified.

III. LINEAR STABILITY ANALYSIS

In this section we introduce the linear stability analysis(LSA) used in the remainder of this paper. To analyzethe stability of the pump-probe dynamics the LSA is donewithout an incoming probe field and for a linearly po-larized monochromatic continuous wave (cw) pump field

ETMTE

kp(t) = E

TMTE

kp(ωp)e

−iωpt inducing the pump polariza-

tion amplitude pXY

kp(t) = p

XY

kp(ωp)e

−iωpt, with ˙pXY

kp= 0 and

˙ETMTE

kp= 0 (ωp is the pump frequency). The pump po-

larization amplitude is a solution of the cubic nonlinearpump equation following from Eqs. (1) and (2) for unidi-rectional light propagation and is determined as outlinedin Appendix A. The resulting coherent exciton density

|pXY

kp|2 for excitation with a linearly polarized pump of

fixed intensity is shown in Figs. 3 and 4 as a functionof the magnitude of the pump in-plane momentum |kp|and pump detuning ∆ε from the bare exciton resonancefor excitation in the TE or TM mode, respectively. Nobistable behavior of the pump-induced exciton densityin Figs. 3 and 4 is found (this follows from the solu-tion of the nonlinear pump equation as outlined in Ap-pendix A). However, for other values of cavity or QW pa-rameters and a different pump intensity, bistability mayoccur, which would complicate our discussion41,42. Thepump densities shown have their maxima close to the lin-ear polariton dispersions (included as the dashed lines)and decrease along the LPB with increasing in-plane mo-mentum because of the decreased coupling of the (mostlyexciton-like) large-momentum polariton states to the in-coming field. Furthermore, a reduced exciton densityis found for excitation of the UPB caused by strongexcitation-induced dephasing (EID) for pump excitationin the two-exciton scattering continuum (the spectral re-gion where the two-exciton scattering matrices shown inFig. 1 exhibit a large imaginary part). For the stability

5

analysis we evaluate the memory integrals in Eq. (2) con-tributing to the probe and FWM directions in a Markovapproximation for the two-exciton scattering continua inthe T-matrices T ++ and T +−. Our Markov approxi-mation is effected by taking p0(t

′) ≈ p0(t)eiωp(t−t′) and

p2kp(t′) ≈ p2kp

(t)eiωp(t−t′) where the probe and FWMpolarization amplitudes p0 and p2kp

appear under thetime-retarded integrals in Eq. (2) together with the con-tinuum contributions in T ++ and T +−. In contrast, weinclude the bound biexciton state exactly via the time-dependent amplitudes bkp

(t) and b3kp(t). For this we

separate the bound biexciton contributions T +−xx from the

correlation kernels T +− in Eq. (2) as T +− = T +−cont+T +−

xx .The bound biexciton contributions to Eq. (2) can be ex-actly included via the equations of motion [cf. Eqs. (11)and (12) of Ref. 25] for the biexciton amplitudes bkp

(t)and b3kp

(t) which are labeled according to the total in-plane momentum of their source terms: ∼ p0pkp

and∼ p2kp

pkp, respectively. This way we include quantum

memory effects related to the excitation of bound biexci-tons, which were previously shown to play an importantrole in the study of FWM instabilities in single semicon-ductor QWs11. Since, for the chosen excitation geometrywith finite pump in-plane momentum kp, the probe andFWM signals do not oscillate at the pump frequency ωp,the Markov approximation for the two-exciton scatteringcontinuum may not be as justified as it is for the singleQW system investigated in Ref. 11. However, close to orin the unstable regime, those wave mixing processes thatdescribe the pairwise scattering of pump polaritons intothe probe and FWM directions play the dominant role inthe probe and FWM dynamics. For monochromatic cwpump excitation these terms are of purely Markovian na-ture and hence the T-matrices in these terms contributeexactly at frequency Ω = 2ωp; no approximation is re-quired. And indeed, for the results discussed here, noteven from the excitation of the bound biexciton statehave we found a sizable contribution from non-Markovian(quantum memory) effects. For monochromatic cw pump

excitation and with the ansatz ETMTE

0 (t) = ETMTE

0 (t)e−iωpt,

ETMTE

2kp(t) = E

TMTE

2kp(t)e−iωpt, p

XY

0 (t) = pXY

0 (t)e−iωpt, pXY

2kp(t) =

pXY

2kp(t)e−iωpt and bkp

(t) = bkp(t)e−i2ωpt, b3kp

(t) =

b3kp(t)e−i2ωpt the coupled probe and FWM dynamics can

be written in the form

~ ˙p(t) = Mp(t) . (3)

The vector

p(t) = [ETE0 (t), ETE∗

2kp(t), ETM

0 (t), ETM∗2kp

(t),

pY0 (t), pY∗

2kp(t), pX

0 (t), pX∗2kp

(t), bkp(t), b∗3kp

(t)]T ,

groups field and polarization amplitude variables to-gether. M is a time-independent matrix where all systemparameters and the steady-state pump polarization am-plitude (Figs. 3 and 4) and the corresponding pump fieldin the cavity modes parametrically enter the analysis.For excitation with a linearly polarized pump eitherexciting the TE or the TM mode, the matrix M is block-diagonal for the components of probe and FWM parallel(co-linear configuration) and perpendicular (cross-linearconfiguration) to the pump’s vectorial polarization state.Then for each pump polarization state (TE or TM,respectively), the 10 × 10 matrix M can be decomposedinto the 6 × 6 block M

σs,σp

‖ with σs = σp and the 4 × 4

block Mσs,σp

⊥ with σs 6= σp which describe the dynam-ics of the coupled variables in the vectors pσs

‖ (t) =

[E σs

0 (t), E σs∗2kp

(t), pσs

0 (t), pσs∗2kp

(t), bkp(t), b∗3kp

(t)]T

with σs = σp for the co-linear configurations and

pσs

⊥ (t) = [E σs

0 (t), E σs∗2kp

(t), pσs

0 (t), pσs∗2kp

(t)]T with σs 6= σp

for the cross-linear configurations, respectively. Theindices σs and σp relate to the cavity modes TE andTM, or to the corresponding excitonic polarizationamplitude components X and Y that are excited by thefields in these modes. Note, that for the above-describedexcitation situation (pump pulse propagating along thex axis and linearly polarized excitation in TE or TMmode) the x component of the polarization amplitudeis exclusively excited by fields in the TM mode and they component by fields in the TE mode. The matricesM

σs,σp

‖ and Mσs,σp

⊥ can be derived from Eqs. (1) and

(2) and take the following form:

Mσs,σp

⊥ =

hσs

0 0 iV σs

0 00 hσs∗

2kp0 1

iV σs∗

2kp

Vσs,σp

0,eff 0 aσp

0,⊥ bσp

⊥

0 Vσs,σp∗

2kp,eff bσp∗⊥ a

σp∗2kp,⊥

, (4)

Mσs,σp

‖ =

hσs

0 0 iV σs

0 0 0 00 hσs∗

2kp0 1

iV σs∗

2kp0 0

Vσs,σp

0,eff 0 aσp

0,‖ bσp

‖ C σp 0

0 Vσs,σp∗

2kp,eff bσp∗

‖ aσp∗

2kp,‖ 0 C σp∗

0 0 − 12C σp∗ 0 B0,kp

00 0 0 − 1

2C σp 0 B∗kp,2kp

. (5)

6

The time-independent coefficients are defined as:

hσs

k =1

i(~ω σs

k − ~ωp − iγc) ,

Vσs,σp

k,eff = iV σs

k

(

1 − A|pσp

kp|2

)

,

aσp

k,i =1

i

[

− ∆εk − iγx + AVσp

kpp

σp∗kp

Eσp

kp

+(

T ++(2ωp) + δi,‖T+−cont(2ωp)

)

|pσp

kp|2

]

,

bσp

i =1

i

[

(−1)δi,⊥AVσp

kpp

σp

kpE

σp

kp

+1

2

(

(−1)δi,⊥ T ++(2ωp) + T +−(2ωp))

pσp

2

kp

]

,

Bk1,k2=

1

i(−∆εk1

− ∆εk2− 2iγx − Exx

b ) ,

C σp =1

i(Cxxp

σp∗kp

) ,

Cxx =([

∑

q

W−†xx (q, 0)ζ(q)

][

∑

q′

ζ†(q′)W−xx(q′, 0)

])12

.

T ++(Ω) and T +−(Ω) are the Fourier transformed cor-relation kernels as shown in Fig. 1(b) and defined byEqs. (27) and (32) of Ref. 25. The coupling strengthCxx ≈ 0.54 Ex

b ax0 of the excitonic polarization amplitudes

to the bound biexciton amplitude is given by the biexci-ton ground state wave function ζ(q) in the electron spinsinglet configuration and the corresponding two-excitonCoulomb interaction matrix element W−

xx(q, 0), both asdefined in Eqs. (14) and (24) of Ref. 25.

For steady-state monochromatic pump excitation, thelinear stability analysis formulated in this section givesus comprehensive information about growth (real-partof the eigenvalues of M) and frequency (imaginary-partof the eigenvalues of M) of the polariton modes in theprobe and FWM directions, after the initial external driv-ing pulse (seed) in the probe direction is gone. Infor-mation how the different eigenmodes contribute to thepolarization amplitudes and fields with different in-planemomenta (0 or 2kp) can be obtained from the eigenvec-tors of the matrices M . If at least one of the eigenvaluesλi of the matrices M fulfills Reλi > 0, the system is

unstable. An arbitrarily small seed of pXY

0 or pXY

2kp(X or

Y, depending on which subspace shows an unstable dy-namics) or in the corresponding cavity modes would growexponentially until the matrix M ceases to describe thesystem correctly.

In addition to the strict steady-state analysis (regard-ing the pump excitation), the general information ob-tained from the stability analysis can – to a large extent

0 1 2 3 4 5

-5

0

5

TE

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

00.30.50.81.11.31.61.82.1

coherent steady-state exciton density [1010cm-2]

TE

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

-0.50-0.42-0.35-0.27-0.19-0.12-0.040.040.12

TETE

2 3-4

-2

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

-0.50-0.42-0.35-0.27-0.19-0.12-0.040.040.12

TETM

2 3-4

-2

FIG. 3: (color online) Top: Coherent steady-state excitondensity |pY

kp|2 for a fixed pump intensity of a linearly TE po-

larized pump as a function of the magnitude kp of the pumpin-plane momentum and the pump detuning ∆ε = ~ωp − εx

0

from the bare exciton resonance εx0 . The linear polariton dis-

persions are included as the dashed lines. Middle and lowerfigures show the real part of the eigenvalue of MTE,TE

‖ and

MTE,TM

⊥ with the largest real part in meV (maximum growthrate if larger than zero). The linear polariton dispersions areincluded as the dashed lines and the insets show the samedata around the “magic angle” for pump excitation on theLPB.

– be carried over to the discussion of pump-probe experi-ments with finite pulse lengths. From the linear stabilityanalysis reasonable predictions for growth rates of probeand FWM signals and thus for polarization rotations canbe made as long as no external probe pulse significantlydrives the probe polariton dynamics during the period of

7

0 1 2 3 4 5

-5

0

5

TM

TM

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

00.30.50.81.11.31.61.82.1

coherent steady-state exciton density [1010cm-2]

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

-0.50-0.42-0.35-0.27-0.19-0.12-0.040.040.12

TMTM

2 3-4

-2

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV]

-0.50-0.42-0.35-0.27-0.19-0.12-0.040.040.12

TMTE

2 3-4

-2

FIG. 4: (color online) Same as Fig. 3 but for linearly TMpolarized pump.

amplification. Furthermore, for interpretation of pulsedexperiments based on the linear stability analysis, thepump pulse must be spectrally sufficiently narrow andpump and probe must have significant temporal overlap.After a sufficiently long period of time that particulareigenmode of M corresponding to the eigenvalue withthe largest real part will dominate the overall outgoingsignal in probe and FWM directions. In a pump-probeexperiment an incoming probe pulse in this particularmode will be most efficiently amplified, or for steady-state pump excitation without an incoming probe, fluc-tuations in this mode (serving as a seed) will grow mostefficiently over time and dominate the signal in probe andFWM direction after a sufficiently long growth period.

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV] -0.50

-0.42-0.35-0.27-0.19-0.12-0.04-0.02

TETE

2 3-4

-2

0 2 4 6

-5

0

5

10

pump in-plane momentum kp [106m-1]

pum

p de

tuni

ng [m

eV] -0.50

-0.42-0.35-0.27-0.19-0.12-0.04-0.02

TETM

2 3-4

-2

FIG. 5: (color online) Same as the two lower panels in Fig. 3but without exciton-exciton scattering in the +− channel(T +− ≡ 0).

IV. RESULTS AND DISCUSSION

Without excitonic correlations (and neglecting the TE-TM-splitting of the cavity modes) the equations of mo-tion for the two circular polarization state channels, +and −, are decoupled. In this case, for excitation with alinearly polarized pump, where equal densities of polari-tons are excited in these two different polarization statechannels, the incoming (seed in our linear stability anal-ysis) and outgoing probe are always in the same vectorialpolarization state, in the stable (all Reλ < 0) as well asin the unstable (at least one Reλ > 0) regime. Neglect-ing the TE-TM cavity-mode splitting, for the pump ina linear polarization state, only the spin-dependent exci-tonic correlations can give rise to a rotation of the outgo-ing probe signal’s vectorial polarization state relative tothe incoming one’s. The actual fraction of polaritons thatis scattered into the probe and FWM directions in a vec-torial polarization state perpendicular or parallel to thepump’s vectorial polarization state, respectively, stronglydepends on the excitonic correlations in the +− polar-ization state channel (included in T +−).9 As discussed inthe previous section, for a linearly polarized (either TE orTM) pump, all the eigenmodes [eigenvectors of the ma-trix M in Eq. (3)] for the probe and FWM dynamics areeither polarized parallel (co-linear configuration) or per-pendicular (cross-linear configuration) to the pump, evenwhen both excitonic correlations and cavity-mode split-

8

ting are included. However, either excitonic correlationsalone or the cavity-mode splitting alone (when the po-lariton scattering is mediated by Hartree-Fock Coulombinteraction) is sufficient to give different probe and FWMdynamics in the two different polarization state configu-rations.

Since the above-listed effects lead to a difference in thegrowth rates of the modes polarized parallel or perpen-dicular to the pump, for an arbitrarily polarized probethe two different vectorial polarization state components(parallel or perpendicular) will grow differently over time.The stronger the amplification of the probe and thelonger the amplification duration, the more the fractionsof the probe in those modes that exhibit the fastest ex-ponential growth, will dominate the outgoing probe andFWM signals. Thus, for strong amplification, the growthrates of the fastest growing modes in the two polarizationstates (parallel and perpendicular) ultimately determinethe rotation in the probe’s vectorial polarization state.Studying these growth rates also answers the questionabout the preferred mode for the growth of probe fluc-tuations over time, when no incoming probe is present.Figures 3 and 4 show these growth rates – the real partof the eigenvalue of M with the largest real part in eachcase – for a fixed pump intensity for the different po-larization state configurations (co-linear or cross-linear)for pump excitation of TE (Fig. 3) and TM (Fig. 4)mode, respectively. The results show that pumping ei-ther TE or TM mode does not significantly influence theoverall result regarding the maximum growth rates inthe co-linear or cross-linear configuration; merely a smallchange in the optimum pump momentum and frequencyis observed. However, a significant difference between co-linear and cross-linear configuration is found for pump-ing close to the inflection point of the LPB (the so-called“magic angle”) where phase-matching is best fulfilled sothat triply-resonant (resonant for the pump excitationand at an angle so that the dispersions in probe andFWM directions allow for phase-matched scattering ofpairs of pump-excited polaritons into these two direc-tions) amplification of the polaritons can occur. For thechosen intensity, close to the instability threshold, andfor pump excitation under the “magic angle” we are ina regime where instability (Reλ > 0) and correspond-ing exponential signal growth is only found in the cross-linear configurations (TE-TM and TM-TE) while all themodes in the co-linear configurations (TE-TE and TM-TM) are exponentially decaying. In this regime, for anarbitrarily polarized probe, only that component polar-ized perpendicular to the pump is exponentially grow-ing over time and thus only this component experiencessignificant amplification. To isolate the mechanism thatleads to this striking difference in the polarization stateconfigurations, Fig. 5 shows results for both configura-tions in Fig. 3 but without taking into account the cor-relations in the +− channel (T +− ≡ 0). Without thesecorrelations the results for the two configurations almostlook alike; we find only a small difference in the growth

rates caused by the TE-TM cavity-mode splitting. Notethat without the correlations in the +− channel in bothconfigurations the instability threshold is not reached forthe same pump intensity as used in Figs. 3 and 4.

In the co-circular (++) excitation configuration it wasfound earlier10 that excitonic correlations in the ++channel may considerably reduce the maximum growthrates in the polariton amplification. In the ++ channelthe driving mechanism for the instabilities, the phase-conjugate feedback, is weakened by the two-exciton cor-relations. Additionally, but for large negative detuningless important, correlations in the ++ channel give rise topump-induced EID, also reducing the exponential growthrate over time. Figures 3 and 4 show that the correla-tions in the +− channel enhance the growth rate in thecross-linear configuration and diminish it in the co-linearconfiguration, compared to the results shown in Fig. 5where these correlations are absent.

In the following we will interpret these results interms of the exciton-exciton scattering matrices shownin Fig. 1(b) for the different polarization state channelsof polariton-polariton scattering. For this discussion weignore the small PSF nonlinearities that contribute tothe probe and FWM dynamics, and concentrate on thenonlinearities in the excitonic polarization amplitudes inEq. (2) that contribute to the probe and FWM dynamicsand therefore determine the amplification process. Beingsufficient for a qualitative understanding we discuss allcontributions in Markov approximation. Three differentterms have to be analyzed that enter the matrices M inEqs. (4) and (5):

(i) Excitation-induced dephasing for the excitonic com-ponent of the polaritons (entering M via a

σp

k,i):

Im

T ++(2ωp) + δi,‖T+−(2ωp)

|pσp

k |2 . (6)

(ii) Nonlinear shifts to the effective exciton resonances inprobe and FWM direction (entering M via a

σp

k,i):

Re

T ++(2ωp) + δi,‖T+−(2ωp)

|pσp

k |2 . (7)

(iii) The phase-conjugate oscillation feedback for the ex-citonic constituents of the polaritons that drives the in-stability (entering M via b

σp

i ):

1

2

(

(−1)δi,⊥ T ++(2ωp) + T +−(2ωp))

pσp

2

k . (8)

The index i ∈ ⊥, ‖ labels the co- and cross-linear polar-ization state configurations, respectively. The maximumgrowth rates in Figs. 3 to 5 are obtained when the pump istuned about 3 meV below the exciton resonance and closeto the “magic angle”. For this pump detuning and inMarkov approximation the two-exciton scattering matri-ces shown in Fig. 1(b) contribute at ~Ω−2εx ≈ −6 meV.

For this detuning the ImT ++ is much smaller than

ImT +− which according to Eq. (6) leads to a muchlarger EID in the co-linear (‖) configuration. According

9

to Eq. (7) a partial cancelation of the nonlinear energyshifts from contributions in the ++ and +− channels isfound in the co-linear (‖) configuration only. This, how-ever, only slightly modifies the effective resonance fre-quencies (polariton dispersions) and thus slightly changesthe optimum pump momentum and frequency. Most im-portant for the stimulated amplification process is thedifference in the polarization state configurations thatcan be seen in Eq. (8). Whereas for the co-linear (‖)configuration the sum of the scattering matrices in the++ and +− polarization state channels determines thestrength of the phase-conjugate feedback driving the in-stability, for the cross-linear (⊥) configuration the dif-ference of the two is relevant. Compared to the resultswithout correlations in the +− channel, the differencein sign in the real parts of these two contributions [cf.Fig. 1(b)] leads to strong cancelation in Eq. (8) for theco-linear configuration and to strong enhancement in thecross-linear configuration. As previously pointed out inRef. 18 this leads to an imbalance in the pairwise scat-tering of polaritons into the probe and FWM directionswith polarization parallel or perpendicular to the pump.From our results we conclude that this does not necessar-ily lead to an overall rotation of the vectorial polarizationstate by 90 in contrast to the conclusions in Ref. 21.Based on our explicit treatment of the bound biexciton,we also find that in the studied system the scatteringmatrix element T +− that drives the amplification pro-cess by scattering of polaritons with opposite spins is notsmall compared to T ++. We note that this is in contrastto the case reported in Ref. 19. Our results indicate thatin the cw regime for a linearly polarized pump and closeto threshold, spontaneous fluctuations preferably growin the cross-linear polarization configuration. This is inagreement with recent observations for a slightly differentsystem and excitation geometry.24

The above discussion has a very general character.The actual cancelation of the different contributions inEqs. (6)-(8) depends quantitatively on parameters suchas the coupling strength of the cavity-modes to the exci-tonic polarization amplitudes. The general trend followsfrom the frequency dependence of the two-exciton scat-tering matrices as shown in Fig. 1(b): The stronger thecavity-mode exciton coupling (shifting the “magic angle”to larger negative detuning) the less pronounced the roleof T +− will be. However, especially close to thresholdeven a small difference in growth rates can be crucial, andin this regime even a small T +− contribution can play amajor role for the analysis of vectorial polarization staterotations. Although for the system studied here it wasfound to be almost insignificant, the overall role of theTE-TM cavity-mode splitting can depend on parametersand excitation conditions, too. Furthermore the impor-tance of a TE-TM cavity-mode splitting can be differentin other systems such as the quasi-one-dimensional mi-crocavity system studied in Ref. 20.

Although not relevant for the amplification of polari-tons on the LPB, we finally note that Figs. 3 to 5 show

strong pump-induced EID for excitation on the UPB. Al-though the two-exciton scattering matrices evaluated inthe 1s approximation are not quantitatively accurate inthis energy region, even in a theory including only coher-ent excitations this additional EID would likely inhibitthe observation of any instability in the UPB (previouslydiscussed, e.g., in Ref. 43) in analogy to the situation forpositive pump detuning in a single QW11.

V. CONCLUSIONS

Based on a microscopic many-particle theory we haveinvestigated the influence of excitonic correlations on thevectorial polarization state characteristics of the para-metric amplification of polaritons in semiconductor mi-crocavities. By means of a linear stability analysis ithas been analyzed how a linearly polarized pump can in-duce a polarization state anisotropy in an otherwise per-fectly isotropic microcavity system. For the discussionof this effect we take advantage of our theoretical ap-proach which – in contrast to previous models16,17,18,19

– is based on microscopically calculated exciton-excitonscattering matrix elements. Accounting for all coherentcorrelations between two excitons, these matrix elementsdetermine the nonlinear cavity-polariton dynamics in theprobe and FWM directions in the amplification regime.

A previous study10 found that excitonic correlationsweaken the polariton amplification for co-circular pump-probe excitation. We confirm these findings and addi-tionally investigate the effects of correlations on the po-lariton amplification in linearly polarized pump-probeconfigurations. We find that scattering contributionsof the excitonic components of polaritons with oppositespins can strongly diminish the driving force for the am-plification, the phase-conjugate coupling, in the co-linear(TE-TE or TM-TM) pump-probe polarization configura-tion and strongly enhance it in the cross-linear (TE-TMor TM-TE) configuration. In the spectral region whereinstability occurs the scattering of polaritons with oppo-site spins is dominated by the virtual formation of boundbiexcitons.

In or close to the unstable regime this polarizationstate anisotropy has the potential to alter the polariza-tion state of an amplified probe pulse compared to the in-cident probe. If the incoming probe has both componentspolarized parallel and perpendicular to the pump’s po-larization state, in general the maximum growth rate forthese two components over time is not the same. Then af-ter a certain growth period the amplification of these twopolarization state components will be different, and hencethe vectorial polarization state of the outgoing probe isrotated compared to the incoming one’s. Since the probecomponent polarized perpendicular to the pump’s polar-ization vector grows faster over time than the parallelcomponent, the probe is always rotated toward the “pre-ferred” cross-linear configuration. However, the overallrotation in the vectorial polarization state depends on

10

both the duration of amplification and the difference inthe growth rates in the two polarization state channels.

Finally, we note that in a situation where steady-statepump excitation brings only the cross-linear configura-tion above the unstable amplification threshold, withoutan incoming probe, only spontaneous fluctuations in thecross-linear polarization state channel will be amplifiedand thus observed as a finite signal in probe and FWMdirection.

Acknowledgments

This work has been financially supported by ONR,DARPA, JSOP. S. Schumacher gratefully acknowledgesfinancial support by the Deutsche Forschungsgemein-schaft (DFG, project No. SCHU 1980/3-1).

APPENDIX A: NONLINEAR PUMP EQUATION

The linear stability analysis of probe and FWM dy-namics in Sec. III is done for monochromatic cw pumpexcitation. The stationary pump field inside the cav-

ity, ETMTE

kp, and pump-induced polarization amplitude in

the QW, pXY

kp, which enter the matrix M in Eq. (3) are

needed to analyze the probe and FWM dynamics. In thiswork we have considered pump excitation with a linearlypolarized pump in the TM or TE cavity mode with in-plane momentum kp along a certain axis, here, withoutloss of generality the x axis, i.e., ϕkp

= 0. Seeking asteady-state solution for the pump-induced polarization

amplitude pXY

kpwe use the ansatz E

TMTE

kp,inc = ETMTE

kp,ince−iωpt,

ETMTE

kp= E

TMTE

kpe−iωpt for the incoming pump field and

the field in the excited cavity mode, respectively, and

pXY

kp= p

XY

kpe−iωpt for the polarization amplitude, with

˙ETMTE

kp,inc = ˙ETMTE

kp= ˙p

XY

kp= 0. Then from Eq. (1) it fol-

lows that the electric field in the cavity mode for in-planemomentum kp is given by:

ETMTE

kp=

−VTMTE

kpp

XY

kp+ i~tcE

TMTE

kp,inc

~ωp − ~ωTMTE

kp+ iγc

. (A1)

For unidirectional light propagation, i.e., by removing allsums in Eq. (2), taking all field and polarization ampli-tude variables at momentum kp, and replacing the pumpfield by Eq. (A1), a cubic equation of the form

0 = a0 + a1

∣

∣pXY

kp

∣

∣

2+ a2

∣

∣pXY

kp

∣

∣

4+

∣

∣pXY

kp

∣

∣

6(A2)

can be derived. This equation determines the monochro-matic solutions for each pump frequency ωp and incom-

ing pump intensity (∼∣

∣ETMTE

kp,inc

∣

∣

2). The coefficients in

Eq. (A2) are

a0 = −1

|T |2

~2t2cV

TMTE

kp

2

|ETMTE

kp,inc|2

(~ωp − ~ωTMTE

kp)2 + γ2

c

,

a1 =1

|T |2

(

|ε|2 +2~

2t2cVTMTE

kp

2

A|ETMTE

kp,inc|2

(~ωp − ~ωTMTE

kp)2 + γ2

c

)

,

and

a2 =2

|T |2

(

ReεReT + ImεImT

−~

2t2cVTMTE

kp

2

A2|ETMTE

kp,inc|2

(~ωp − ~ωTMTE

kp)2 + γ2

c

)

,

with the definitions

ε = ~ωxkp

− ~ωp − iγx +V

TMTE

kp

2

~ωp − ~ωTMTE

kp+ iγc

,

T =1

2(T ++ + T +−) −

VTMTE

kp

2

A

~ωp − ~ωTMTE

kp+ iγc

.

Being a cubic equation in |pXY

kp|2, depending on the coef-

ficients a0, a1, a2, Eq. (A2) can have either one or threereal-valued solutions. These solutions are given, e.g., inRef. 44.

Equation (A2) only determines the magnitude of the

pump-induced polarization amplitude |pXY

kp| and contains

no information about the different phases of the polariza-tion amplitude and the incoming field. We define a phase

φ according to ETMTE

kp,inc ·pXY

kp= |E

TMTE

kp,inc||pXY

kp|eiφ. This phase

φ is required to determine the field in the cavity modefrom Eq. (A1) that is coupled to the polarization ampli-tude. The field enters the linear stability analysis via thePSF terms in M . If we choose the solution of Eq. (A2)

to be real, pXY

kp= |p

XY

kp|, then the incoming field inducing

this polarization amplitude is ETMTE

kp,inc = |ETMTE

kp,inc|eiφ. The

phase of the incoming field can be obtained from Eq. (2)and is given by

eiφ =

(

ε|pXY

kp| + T |p

XY

kp|3

)

(~ωp − ~ωTMTE

kp+ iγc)

i~tcVTMTE

kp

(

1 − A|pXY

kp|2

)∣

∣ETMTE

kp,inc

∣

∣

,

which then also determines the phase of the field in thecavity mode given by Eq. (A1).

11

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33 The background refractive index, effective electron mass,and effective hole mass are nbg = 3.61, me = 0.067 m0,and mh = 0.1 m0, respectively. The corresponding excitonbinding energy is Ex

b ≈ 13meV and the bulk Bohr-radiusis ax

0 ≈ 170 A. For the evaluation of the theory also εxk =

1.4965 eV, APSF = 4ax0

√2π/7, Exx

b ≈ 1.84 meV, and γx =0.25 meV are required. The cavity parameters are γc =

0.25 meV, and VTETM

0 = 5.2 meV.34 M. E. Donovan, A. Schulzgen, J. Lee, P.-A. Blanche,

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38 N. H. Kwong, S. Schumacher, and R. Binder, to be sub-mitted (2007).

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