Interacting exciton-polaritons in cylindric micropillars Yury S. Krivosenko1Ivan V. Iorsh1and Ivan A. Shelykh1a Department of Physics and Engineering ITMO University St. Petersburg 197101 Russia

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Interacting exciton-polaritons in cylindric micropillars

Yury S. Krivosenko,1Ivan V. Iorsh,1and Ivan A. Shelykh1, a)

Department of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia

(*Electronic mail: y.krivosenko@gmail.com)

We present a quantitative microscopic analysis of the formation of exciton-polaritons, the composite particles pos-

sessing light and material components, polariton-polariton interactions, and resonant pumping dynamics in cylindrical

semiconductor micropillars. We discuss how the redistribution effect can be used in devices generating photons with

non zero orbital angular momenta.

I. INTRODUCTION

Cavity polaritons, also known as exciton-polaritons1–3 are

composite quasiparticles consisting of excitonic and a pho-

tonic components. Conventional geometry where polaritons

can be routinely observed is a planar semiconductor micro-

cavity, which consists of a semiconductor quantum well, sand-

wiched between a pair of distributed Bragg reﬂectors (DBRs)

and placed in the position where photonic mode of such

Fabry-Perot cavity has an antinode.

The physics of polaritons attracted substantial interest of

the researches working in the domains of photonics and con-

densed matter physics. This is mainly due to the unique

properties of polaritons, related to their composite nature.

The combination of extremely low effective mass, inherited

from the photonic component, with giant nonlinear optical

response, stemming from the excitonic part, enables the ob-

servation of a set of intriguing collective phenomena at sur-

prisingly high temperatures2,4,5. Examples include polariton

BEC and polariton lasing6–9, formation of topological de-

fects such as solitons10–13, quantized vortices14,15 and vortex

lattices16,17, skyrmions and merons18,19, shock waves20 and

many others. Moreover, polariton systems can form a ba-

sis for creation of nanophotonic devices of the next genera-

tion, including optical logic gates and all-optical integrated

circuits21–24.

The visible current trend in polaritonics is the shift towards

quantum applications25–27. Geometries with cavity polari-

tons were suggested as candidates for creation of the sources

of single photons28–32 and entangled photon pairs33–35, the

corresponding onset of polaritonic quantum correlations was

reported36–39. Moreover, the possibility to use polariton sys-

tems in quantum computing40,41 and as quantum and neuro-

morphic simulators was actively discussed41–46.

For applications of polaritonics at the quantum level two in-

gredients are necessary: spatial conﬁnement of the polaritons

and pronounced nonlinearity. The ﬁrst can be achieved by

etching of a planar microcavity, which allows to get individual

polariton pillars47–52, systems of several coupled pillars form-

ing so-called polariton molecules53 or periodically arranged

arrays of the pillars forming polariton superlattices43,54–56.

As for polariton nonlinearity, it is provided by the excitonic

fraction, and the main contribution is given by the exchange

a)Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Ice-

land

interaction of electrons and holes forming the excitons57–61.

The quantitative analysis of the polariton nonlinearities in mi-

cropillars is thus an important problem, which we solve in the

current work.

We consider in detail the formation of exciton-polaritons in

cylindric micropillars, develop a microscopic theory for the

calculation of the matrix elements of polariton-polariton in-

teraction in this geometry, and analyze nonlinear dynamics

of such a system subject to a partially coherent pumping, in-

vestigating, in particular, the redistribution of energy between

polaritonic states with different angular momenta.

The paper is organised as follows. In sections II A, II B,

and II C we brieﬂy describe the formation of the photonic

cavity modes, quantum-well excitonic modes (accounting for

exciton-exciton interaction), and polaritonic modes, respec-

tively. In section II D we consider the dynamics of the co-

herently pumped polaritons. Section III contains numeric re-

sults and their discussion. Conclusions are summarized in sec-

tion IV. Appendices contain the technical details of the calcu-

lations.

II. THEORY

The system we consider is schematically shown in Figure 1

(left panel). It consists of a semiconductor micropillar with

cylindrical symmetry (grey cylinder) ﬂanked by a pair of di-

electric Bragg mirrors (DBRs, the orange planes), and con-

taining a quantum well (QW, the green circle in the middle)

placed in the position of an antinode of an electric ﬁeld of the

optical cavity mode. The energy of the cavity mode is brought

close to the resonance to the energy of the excitonic transition,

favoring the formation of the hybrid polariton modes.

As polaritons contain an excitonic fraction, they interact

with each other. The scheme of the polariton-polariton inter-

action is shown in the right panel of Figure 1. As the system

we consider has cylindrical symmetry, polaritons are char-

acterized by their projections of angular momenta (λj) onto

the structure growth axis. States with different angular mo-

menta have different proﬁles of their wave functions, shown

schematically by the purple surfaces. Polaritons can exchange

their angular momenta due to the exciton-exciton scattering,

which, obviously, conserve the total angular momentum.

The Hamiltonian of the interacting excitons coupled to a

photonic cavity mode is:

Htot =b

Hx+b

Hp+b

Hxp +b

Hxx,(1)

arXiv:2210.09258v1 [cond-mat.mes-hall] 17 Oct 2022

lower DBR

z= 0

QW plane

upper DBR

|λ1i

|λ2i

|λ1+∆λi

|λ2−∆λi

Hxx

QW top view

FIG. 1. Left panel: Sketch of the considered geometry, consisting

of a semiconductor micropillar (grey cylinder) ﬂanked by a pair of

dielectric Bragg mirrors (DBRs, the orange planes), and containing

a quantum well (QW, the green circle in the middle) placed in the

position of an antinode of an electric ﬁeld of the optical cavity mode.

The energy of the cavity mode is brought close to the resonance to

the energy of the excitonic transition, giving rise to the formation

of the polaritons. Right panel: Top view of the system illustrating

the scheme of the exciton-exciton scattering. Excitons in the states

|λ1i,|λ2i,|λ1+∆λi, and |λ2−∆λi(the red rounds) are coupled via

interaction Hamiltonian b

Hxx (the white round). Excitonic wave func-

tions λ1=λ2=∆λ=1 and ν1,2,3,4=1 are schematically shown by pur-

ple surfaces next to the states markers.

where, b

Hxand b

Hpare Hamiltonians of free excitons and pho-

tons, b

Hxp and b

Hxx are Hamiltonians of photon-exciton and

exciton-exciton interaction, respectively.

In second quantization representation, they can be written

as:

Hx=∑

λ,ν

hΩλ ν b

X†

λ ν b

Xλ ν =∑

hΩκb

X†

κb

Xκ,(2a)

Hp=∑

l,n

hωln b

a†

ln b

aln =∑

hωkb

a†

ak,(2b)

Hxp =∑

k,κ

hgkκb

X†

κb

ak+h.c.,(2c)

Hxx =1

2∑

κ1κ2κ3κ4

Vκ1κ2κ3κ4b

X†

κ1b

X†

κ2b

Xκ3b

Xκ4.(2d)

Here, b

X(†)

λ ν and b

a(†)

ln are the λ ν-excitonic and ln-photonic mode

annihilation (creation) operators, respectively, and ¯

hΩλ ν and

hωln are the corresponding eigenenergies. The indices λ

and ν(land n) are the excitonic (photonic) angular and

radial quantum numbers, respectively. In equation (2c),

hgkκ≡¯

hglnλ ν represent the amplitudes of the interaction be-

tween λ ν-excitonic and ln-photonic modes. To spare the no-

tations, we hereafter denote the pairs of the excitonic, λ(j)ν(j),

and photonic, l(j)n(j), indices as κ(j)and k(j), respectively.

A. Photonic modes

To analyze the photonic modes in the cylindric symmetry

microcavity, we follow the scheme of the reference 62. We

consider a microcavity placed between a pair of DBRs and

containing a cylindric micropillar of a radius Rmade of a

semiconductor with high-frequency dielectric permittivity ε1.

The effective inter-DBRs distance is denoted as L. Within

the conventional cylindric coordinates frame (r,z, and θ), the

z-component of the ﬁeld is sought as a product of Bessel’s

functions (BFs) of either the ﬁrst kind (core region, argument

β1r) or the modiﬁed BF (air region, argument β2r), the com-

plex exponential of (ilθ), and cosæz. This introduces the ra-

dial, angular, and zdependences, respectively. The value æis

chosen to be π/Lwhich corresponds to the fundamental (in

z-direction) photonic mode and obeys the zero boundary con-

ditions at the cylinder bases (z=±L/2).

The use of the continuity condition for the ﬁelds tangential

components at the core-to-vacuum boundary (r=R) leads to

the characteristic equation (see (A2)) and the photonic target

function (PTF):

Fptf(¯

hω) = u4v4(η1+η2)k2

1η1+k2

2η2

−l2æ2u2+v22,(3a)

where

η1=J0

l(u)

uJl(u),η2=K0

l(v)

vKl(v),u=β1R,v=β2R.(3b)

The PTFs are displayed in Fig. 2 as vertically oriented curves.

Their roots (photonic modes eigenenergies) are marked by the

round grey symbols.

The core-region (index 1), air-region (index 2) quasimo-

menta β1,2, and frequency ωare interconnected by

β2

1=ω2ε1

c2−æ2,β2

2=æ2−ω2ε2

c2.(3c)

The system of eqs. (3) is invariant under the l→ −lsubstitu-

tion and, hence, is solved only for l>0. The procedure results

in a set of frequencies numbered by the index nfor each l:ωln,

n=1,2,...N(|l|).

All the details of the computational procedure are presented

in Appendix A.

B. Excitonic modes

Considering the excitonic part, we use the following as-

sumptions:

RaBohr au.c.,(4)

where aBohr is the two-dimensional exciton Bohr radius, and

au.c. is the characteristic size of the micropillar material unit

cell. These conditions allows to neglect the internal structure

of excitons when considering their conﬁnement inside a pillar.

Moreover, the excitonic density nXis supposed to be small

enough to treat them as bosons, i.e. the following condition is

satisﬁed:

nXa2

Bohr 1.(5)

The dielectric-to-vacuum boundary is assumed to be an inﬁ-

nite barrier.

Inside the cavity, we can factorize the excitonic wave func-

tion as

φκ(r,ρ) = Φλ ν (r)χ1s(ρ),(6)

where Φλ ν (r)describes the exciton centre-of-mass motion,

and χ1s(ρ)corresponds to the relative motion of an electron

and a hole, and for conventional semiconductor materials can

be well approximated by the two-dimensional hydrogen-like

atom wave function (in this paper, we consider only the 1s

excitonic ground state):

Φλ ν (r) = Jλ(αλ ν r)expiλ θ

J|λ|+1(xλ ν )R√π,(7a)

χ1s(ρ) = exp(−ρ/aBohr)

aBohr r2

π.(7b)

The quantum numbers λand νare thus associated with the

exciton centre-of-mass motion, xλ ν is the νth root of BF Jλ,

and αλ ν =xλ ν /R.

The excitonic energies are expressed as:

hΩλ ν =¯

2m∗

e+m∗

hxλ ν

R2−2¯

µa2

Bohr

+Eg(8)

with m∗

eand m∗

hbeing the electron and hole effective masses,

respectively, µ– the effective reduced mass of an electron-

hole pair, and Eg– the semiconductor energy band gap. The

second term in (8) is the 1s-state eigenenergy.

For realistic cavity parameters, we got that the ﬁrst term, re-

sponsible for the dimensional quantization energy of the mo-

tion of the center of the mass is orders of magnitude smaller,

then last two terms, so we can neglect the excitonic dispersion

and take the energies of all excitonic states to be the same,

hΩλ ν =¯

hΩx=const,(9)

which is in agreement with the reference 63.

The matrix element of the exciton-exciton scattering

|κ3,κ4i → |κ1,κ2iin the Born approximation can be written

Vκ1κ2κ3κ4=hκ1,κ2|Vint,xx|κ3,κ4i,(10a)

where |κi,κjiis the two-exciton wave function antisymmetric

with respect the electron-electron and hole-hole exchange:

|κi,κji=1

2·φκi(e1,h1)φκj(e2,h2) + φκi(e2,h2)φκj(e1,h1)

−φκi(e1,h2)φκj(e2,h1)−φκi(e2,h1)φκj(e1,h2)(10b)

with e1(2)and h1(2)presenting the electron and hole coordi-

nates, respectively.

Within the same notations, the matrix elements of the

exciton-exciton interaction potential, Vint,xx, can be presented

as64

Vint,xx =VC(e1,h2)−VC(e1,e2) +VC(e2,h1)−VC(h1,h2).

(10c)

Here, VC(r1,r2) = −e2/4πε0εst|r1−r2|is Coulomb poten-

tial, εst is the micropillar static dielectric permittivity, e

is the elementary charge. The matrix element (10a) can

be expressed as the sum Vdir

κ1κ2κ3κ4+Vxx

κ1κ2κ3κ4+Vee

κ1κ2κ3κ4+

Vhh

κ1κ2κ3κ4that reveals the four channels (direct, excitons ex-

change, electron-electron, and hole-hole exchange ones, re-

spectively).

The two latter terms jointly form the fermion-fermion

exchange channel, Vfx

κ1κ2κ3κ4, which gives the major

contribution64 and is only retained in our further considera-

tion. The corresponding interaction matrix element is

Vκ1κ2κ3κ4=−Z···Zd8τφ∗

κ1(e1,h1)φ∗

κ2(e2,h2)Vint,xx

×φκ3(e1,h2)φκ4(e2,h1) + φκ3(e2,h1)φκ4(e1,h2),(10d)

where d8τdenotes the coordinate phase space volume differ-

ential. The elements Vdir,xx,fx

κ1κ2κ3κ4respect the angular momentum

conservation law, i.e. Vdir,xx,fx

κ1κ2κ3κ4=0 if λ1+λ26=λ3+l4, which

can be schematically written down in terms of the transferred

angular momentum ∆λin the scattering channel:

|λ1i+|λ2i ↔ |λ1+∆λi+|λ2−∆λi.(11)

C. The Hamiltonian in the polariton basis

Within the dipole approximation, the operator of exciton-

photon interaction can be put down as

Vint,ln =−(dcv ·Eln),(12)

where ln denotes the photonic mode, dcv is the dipolar matrix

element between valence and conduction bands. For the case

of GaAs material (studied here), dcv can be expressed as65

dcv =dcv

√2(ex+iey) = dcv

√2(er+ieθ)eiθ=d0

cv eiθ.(13)

Then, the exciton-photon interaction matrix element can be

presented as the excitonic radiative linewidth multiplied by

the overlap of the excitonic and photonic modes wave func-

tions. Following the deduction of reference 63, we eliminate

the extra phase factor eiθin (13) and arrive at the following

expression for the interaction term:

hglnλ ν =−χ∗

1s(0)ZZ

Φ∗

λ ν (r)d0

cv(r)·Eln(r)z=0d2r(14)

Utilizing the expression for the planar 2D microcavity Rabi

splitting (¯

hΩ2D, see Appendix B), we recast the matrix ele-

ment as

hglnλ ν =−¯

hΩ2D

2rε0ε1L

2¯

hΩxZZ

Φ∗

λ ν (r)Eln,||(r)d2r,(15)

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Interactingexciton-polaritonsincylindricmicropillarsYuryS.Krivosenko,1IvanV.Iorsh,1andIvanA.Shelykh1,a)DepartmentofPhysicsandEngineering,ITMOUniversity,St.Petersburg197101,Russia(*Electronicmail:y.krivosenko@gmail.com)Wepresentaquantitativemicroscopicanalysisoftheformationofexciton-polaritons,thecom...

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Interacting exciton-polaritons in cylindric micropillars Yury S. Krivosenko1Ivan V. Iorsh1and Ivan A. Shelykh1a Department of Physics and Engineering ITMO University St. Petersburg 197101 Russia

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