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Verification of very strong coupling in a semiconductor optical microcavity

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2015 New J. Phys. 17 023064

(http://iopscience.iop.org/1367-2630/17/2/023064)

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New J. Phys. 17 (2015)023064 doi:10.1088/1367-2630/17/2/023064

PAPER

Veri

cation of very strong coupling in a semiconductor opticalmicrocavity

Ming-JayYang1,NaYoungKim2, YoshihisaYamamoto2,3 andNeilNa1

1 Institute of PhotonicsTechnologies, National Tsing-HuaUniversity, Hsinchu 30013,Taiwan2 Edward L. Ginzton Laboratory, Stanford University, Stanford, California94305, USA3 National Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

E-mail:[email protected]

Keywords: exciton-polariton, quantum well,microcavity

AbstractWe study the very strong coupling effect in a semiconductor optical microcavity in which the exciton

radius is dramatically modied due to the presence of strong photonexciton coupling by means of

nonlinear numerical optimization.To experimentally verify the features of very strong coupling and

distinguish them from those of strong coupling, we propose two schemes and show that our proposals

canprovide unequivocal experimental proof of the existence of very strong coupling.

Introduction

A semiconductor optical microcavity (MC) providesa unique platform for studying the coupling betweenMC

photons and quantum well (QW) excitons. If the photonexcitoncoupling constant is smaller than the

linewidths of MC photons and QW excitons, a weak coupling occurs and manifests itself as an enhancement ofexcitondecay rate [1]. When the coupling constant is larger than the linewidths of MCphotons and QW

excitons, new eigenmodes of the strongly-coupled cavity quantum electrodynamics (QED) system are formed

and often referred to as exciton-polaritons [2]. In thiseld, numerous experimental demonstrations of the

coherent properties of exciton-polaritons have been made, such as BoseEinstein condensation [3],

superuidity [4] and vortices [5]. If the couplingconstant can be further enlarged tobe of the same order of

magnitude as the excitonbinding energy, the system enters the so-called verystrongcoupling (VSC) regime [6].

In the VSC regime, it is theoretically predicted that the properties of exciton-polaritons are strongly modied,

including exciton radius reduction/enlargement, unequal photonand exciton fractionsat resonance, and strong

asymmetric Rabi splitting betweenthe upper polariton (UP)and lower polariton (LP) [6, 7]. In addition,

because the effect of VSC increases the thermal dissociationtemperature and nonlinear saturationdensity of

LPs, it may be possible to construct a room temperature, ultralow threshold polariton laser using a GaAs-basedoptical MC ifVSC can be achieved [7]. Compared to the cases of using wide bandgap semiconductors such as

GaN, ZnO etc as MC material, this approach should feature signicantlyless inhomogeneous broadening and

may be applicable to optical communication systems at near-infrared wavelengths. Nevertheless, despite the fact

that a Rabi splitting larger than the excitonbinding energy has beenobserved experimentally [8], the existence of

VSC remainselusive for various reasons, which will be discussed later.

In this paper,we rstreview the analytical variational approach used in [6, 7], and then expand the

theoretical conceptby utilizinga nonlinear numerical optimizationtechnique. Sucha technique allowsus to

produce the exact shape of the excitonwavefunction in the relative coordinate for both UP and LP energy

branches, evenwhen an arbitrary potential seenby the excitonwavefunctionin the relative coordinate is added.

By means of this technique, we can further examine the problem whenan external in-plane dc electricaleldis

applied to the exciton-polaritons, in which the excitonic dipole moment and the coupling constant are reduced

so thatthe bare cavity energy is revealed. The value of the recoveredbare cavity energy depends highly onwhether photons and excitons are strongly coupled or verystrongly coupledto eachother, which serves as an

experimental proof of the existence of VSC.In addition, we examine the problem whenan external vertical dc

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magnetic eld is applied to the exciton-polaritons. Due to the dependence of excitonradius on the coupling

constant in the VSC regime, the UP and LP manifest themselves in different diamagnetic energy blue-shifts that

serve as another experimental proof of the existence of VSC. Detailed analyses on these twophenomena in a

GaAs-based optical MC are presented with realistic experimental parameters.

Formalism

The exciton-polariton state can be written as = + K K,pol ex where the Hopeld coefcients

andare for QW exciton state ex and MC photonstate K K, respectively.K is the transverse wavevector

and K is the longitudinal wavevector. The system Hamiltonian can be written as

=

+

+

+ +

+

( )H

M

e

ra a

ep

m Va a

2 2 4

2 . (1)r R

sK K

eK K

2 2 2 2 2

0

0

The rst twoterms are the kinetic energies including the relative motionand the center-of-mass motionof

an exciton. The third term isthe Coulomb interactionbetweenan electron and a hole, andthe fourth term is the

MC photonenergy = +( )c K Kz2 2 1/2

whereKzis the discretized wavevector due to an optical connementin the zdirection. The last term is the coupling betweenthe QW excitonand the MC photonand is proportional

to the coupling constant = ( )g e f m L4 e z2 2 2D 01/2

considering a two-dimensional (2D) exciton. The 2D

exciton oscillator strengthf2D is linearly proportional to |(0)|2, i.e.,the probability oflocating an electron and a

hole in the same position.Lzis the cavity effective length in thezdirection. For most of the MCsin the literature,

gis muchsmaller than the excitonbinding energyEB, sothe photonexciton coupling is treated as a

perturbation tothe total Hamiltonian. However, inthe case ofmany QWs embedded in a high quality DBR MC,

gandEB can be at the same order of magnitude which invalidates this assumption. The photonexciton coupling

and the electronhole Coulomb interaction should be then treated on an equal footing, and the inuence of this

VSC effect was previously investigated by using a variational method withand as variationalparameters, and

= ( ) ( )r a r a( ) (2 ) exp1/2 0 0 as a trial exciton wavefunction [6, 7]. and a0 are the exciton radiusreduction factor and the 2D excitonBohr radius respectively. It was predicted that the exciton wavefunction in

the relative coordinate experiences a substantial radial change for both polariton energy branches in the regime

of VSC, and the resulting coupling constantgbecomes nonlinear and can be written as

=g g

(0)

(0). (2)0

0

The subscript0 means is taken as unity, i.e., whenthe coupling constant is much smaller than the exciton

binding energy. Anotherway tounderstand the effect of VSCis towrite down the 2 by 2 matrix

=

+ H

E g

g Eg

E

(3)

B 0

0 B02

B

which describes the effective system Hamiltonian [7] assuming the exciton ground state wavefunction is

Gaussian. is the photonexcitonenergy detuning. It is seenthat the effect of VSC can be alternatively

interpreted as an effective renormalization of the bare cavityenergy byg02/EB. In addition, such an effect wears

off when g E0 B and (3) reduces to the usual strongcoupling (SC) Hamiltonian. Ingure 1,weplottheUPandLP energiesEpol considering variousg0 and using (3). It shows that for a specic detuning, the LP(UP) energy

decreases (increases) with increasingcoupling constant as expected. Nevertheless, compared to the conventional

Rabi splitting whereEpol changes linearly withg0, here the change isnonlinearand a signicant asymmetrical

splitting can be observed.

In orderto handle a larger classof problems such as studying an exciton

polaritonunder an external electricor magneticeld, we extend the previous variational formalismto include the spatial prole of exciton

wavefunction as variational parameters,so that the system energy functional can be written as

2

New J. Phys. 17 (2015) 023064 M-J Yang etal


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=

=

+

+

+ +

=

( )

E H

r e

rU r

r r E g

f r

( )2 4

( )

( )d 2 (0)

(0)( ( ), ). (4)

r

s

pol pol

22 2 2

0

2 2B 0

0

U(r) is an externalpotential exerted on the excitonwavefunction. Note that =K 0 has been assumed. Theminimizationof this energy functional will deliver the ground state exciton wavefunction, Hopeld coefcients,

and polaritonenergy. Numerically, the QW excitonwavefunctionisrst discretized spatially as a 2D matrix

x y a( , ) (5 )

n

n

i j

m m m n

1,1 1,2 1,

2,1 2,2 2,

,

,1 ,2 ,

with eachmatrix element representing the magnitude of excitonwavefunctionat a specic spatial grid inside thesimulation space. Its second-order derivative is thenconstructed by anite difference approach

+ + + + +( )x yh

b( , ) 1

4 (5 )i j i j i j i j i j 2

2 1, 1, , , 1 , 1

where (i,j)and(m,n) stand for the label andthe number of grid points in thexandydirections. = h L n( 1) isthe distance betweenthe grid points assuming =m n, and L isthe size of simulation space. By using (5a) and(5b), we may rewrite (4) as

= ( )E f , , , , , , . (6)i j m n1,1 1,2 , ,

The energy functional now becomes a function with (nm + 1) variables, and the problem is transformed intonding a vector that minimizes the objective functionfsubjectto the constraints of = 1

m ni j1,1

,,

2and

+ = 1.2 2 Thenonlinearity of both the objective function and the constraints render the minimizationanonlinear optimizationproblem (NLP) of the general form [9, 10]:

Figure 1.Polaritonenergy plottedas a function ofcoupling constantand energydetuningfor (a)LP and(b) UP.Notethat thewhitedashedlinein (b)indicates theboundary( =E 0)pol betweenbound excitons andfree electronhole pairs.

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= = = +

f x

c x i p

g x i p l

min ( )

subjectto: ( ) 0 1,...,

( ) 0 1,..., (7)

x R

i

i

s

Note that if =l 0the problem is referred to as unconstrained optimizationproblem. In our numerical

optimization = +s nm 1, and since the number of grid points involves a rather large scale, we apply the widely

usedsequential quadratic programming (SQP) [11

13] algorithm to save the computational time. It beginswitha given guessxkand then iterates through the quadratic programming to generate a next approximated solution

xk+1. The successive iteration is performed to construct a sequence of solutionsbelonging to a specicsetthat

converges to an optimum solution.We implement the iterationprocess by means of the Optimization Toolbox

in the MATLAB software package [14] to solve our NLP. Byloading the system energy function together with its

constraints into the SQP programwhere the 1s 2D excitonwavefunction is given as an initial guess, the solution

corresponding to the exciton wavefunction(r) and the exciton fractionofthe LP isobtained. The UP

solution can then be retrieved by implementing the parity condition of Hopeld coefcients (i.e. for UPand

are switched and have opposite signs) in the system energy functional and repeating the SQP programgiven the

LP solution as an initial guess.

Ingure 2(a), we plot the energy-detuning relations for both the UP and LP using the analytical solutionsof

Hamiltonian (3) and the numerical solutionsof Hamiltonian (4) assuming no externalpotential presents. Note

that the yellow region indicating the bound exciton no longer existsso any calculated UP state that penetratesthis region should be ignored. It can be seenthat the numerical results match the analytical results well, which

attests to the applicability of our numerical technique. Note that the differences in analytical and numerical

results are more apparent for the UP due tothe fact the UP solution isiterated from the LP solution, and the

accuracy can be further improved by for exampleincreasing the number of grid points. Ingure 2(b),Epol at

resonance for differentg0 is calculated to show the strongasymmetric Rabi splittingwhich nonlinearly depends

ong0. In gure 2(c), the LP exciton (photon) fraction at resonance is shown tobe signicantlysmaller (larger)

than50%at a largeg0. Note that in bothgures 2(b) and (c), again the analytical and numerical resultsagree well

with eachother. In all above calculations, = =m n 89 and =L a10 0 are used.

Recovering thebare cavityenergy

From the experimental point of view,it isimpossible to distinguish the effect ofVSC from SC by simply

measuring the UP and LP energy-detuning relations. Thisis due tothe fact that although the UP and LP as well as

the QW exciton energies can be determined by the energy-resolved photoluminescence measurements at a

verticalangle (for the UP and LP) as well as at a large oblique angle (for the QW exciton), the MCphotonenergy

is nevertheless treated as atting parameter. As a consequence, the ambiguity of whether one should use the

black dashed line or the purple dashed line ingure 2(a)to represent the MC photonenergy results inthe

uncertainty of whetherthe VSC model or the SC model should apply to the experimental system. We believe

such a problem prevents a direct vericationof the existence of VSC experimentally to date.

In viewing (3), ifg0 canbe reduced in a controllable fashion so that the bare cavity energyEB + is

recovered by the energy-resolved photoluminescence measurement at a vertical angle, the ambiguity oftting

the MCphoto energy with the blackdashed line (VSC model) or the purple dashed line (SCmodel) ingure 2(a)

can be resolved. This can be realized by applying an in-plane electriceld tothe QWs so that the excitonic

oscillator strength is bleached by separating the relative distance between an electron and a hole. The physical

properties of such a system can be calculated by our numerical technique introduced previously with the

externalpotentialU(r)in(4)givenas eFr. F is the magnitude of an in-plane electriceld. In our calculations,

an electricaleld is applied laterally in theydirection and the dimensionlesseld strength = f F F0 where=F E ea0 B 0is introduced. The proposed experimental structure is shown ingure 3(a), in which the

semiconductor opticalMC is etched to dispose electrical contacts in close proximity to the QW layer for

applying an in-plane electriceld.To implement this structure, one thing that should be considered

experimentally is how to precisely control the DBR etchdepth so that the in-plane electricaleldcanbe

efciently applied to the QW layers at the strongest cavityeld antinodes. Moreover, the etched sidewalls may

introduce undesired recombination centers that quench the LP signals so the area of etched mesa should be

designed tobe sufciently larger than that ofthe LP or an efcient sidewall passivation should be applied.

We calculate the spatial pro

le of the excitonwavefunctionin the relative coordinate under the in

uence ofan in-plane electriceld ingures 3(b) and (c),and the energy-detuning relations are calculated ingure 4.For

the LP (UP) that isphoton(exciton)-like, the dc Stark effect onthe spatial prole is less obvious (obvious). Both

UP and LP excitonwavefunctions are stretched by the in-plane electricelds as shown ingures 3(b) and (c),

and the reduction of oscillator strength can be observed ingure 4. Note that the stuck of the exciton

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wavefunctionat = y a5 0 ingures 3(b) and (c) is due to the null-wavefunction boundary condition. In

gure 4(a) where =f 1, the energy branches that are exciton-like ( 0 for LP; 0 for UP) shiftdownward because of the dc Starkeffect, similar to the energy red-shift ofa hydrogen atom in a static electric

eld. Ingure 4(b) where =f 2, the LP becomes a completeat line correspondingto the red-shiftedQW

excitonenergies and the UP unveils the bare cavity energies. The proposed experimental scheme shall provide a

direct verication of the existence of VSC.

Vericationof the reduced (enlarged) radiusof the LP(UP) inVSC

The modicationof excitonradius is the origin of the VSC phenomena discussed above. Here we propose

another experimental scheme to verify the existence of the VSC by measuring the diamagnetic energy shifts in

the presence of a verticalmagneticeld.The diamagnetic energy shift is frequently usedto determine the size ofan exciton ina semiconductor and can be expressed as [1517]

Figure 2. (a)Polariton energyplotted asa functionof energydetuning given =g E1.2 .0 B TheMC photonand theQW excitonenergy-detuning relations are indicted by the blackdashed-dottedline andthe blackdashed linerespectively. The renormalized MC photonenergy-detuningrelationis indicatedby thepurple dashedline.Notethatthe equal photonexcitonfractionpoint i.e. =2 2 isshifted to= E1.44 ,B whichis indicatedby theinterception ofthe blackdashed-dotted line andthe purpledashed line.(b) Polaritonenergycalculated with SC andVSCmodels.(c) Exciton fractionof LPplotted asa functionof coupling constant.Notethat allenergiesarein theunit ofexciton binding energyEB (10 meV).

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= H

eB r

r

rB

8

, (8)

s

diam

222

2

2

2

1

2

where the diamagnetic coefcient 2 isabout 13eV T2 fora1s2DexcitoninaGaAsQW[17],i.e.,a

diamagnetic energy blue-shift of about 1.3 meV can be achievedfor a 10 T verticalmagneticeld.IntheSC

regime, arigid

excitonmodel is assumed so that the exciton radius is independent of photon

exciton coupling.Accordingly, the net diamagnetic energy blue-shift, dened as the difference betweenUP and LP energy blue-

shifts, should be zeroat the resonance condition(equal photonexciton fraction) anda small number at a

detuned condition (unequal photonexcitonfraction). Nevertheless in the VSC regime, a exible exciton

model is assumed so that a considerable net diamagnetic energy blue-shift shall present due to the signicant

Figure 3. (a)Schematic constructionof ourdevice where multiple QWsare inserted inthe cavityregion andthe DBR region. The(b)LP and(c) UPexciton wavefunctions under theinuence ofan electricaleldappliedlaterally intheydirection with =f 1 given= EB.

Figure 4.Theenergy-detuningrelations ofUP andLP underthe inuence ofan electriceldappliedlaterally intheydirection with(a) =f 1 and(b) =f 2 given =g E1.20 B.

6



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difference in UP and LP excitonradii, which are enlarged and reduced respectively through the photonexciton

coupling. The apparentnet diamagnetic energy shift serves as a direct verication of the existence of VSC, which

can be experimentally measuredthrough the photoluminescence signals from UPsand LPsin a vertical magnetic

eld.

We solve the NLP with diamagnetic interaction added and determine the exact exciton wavefunctions and

the correspondingenergy shifts. Ingures 5(a)(c), the exciton wavefunctions are presented respectively for the

1s 2D exciton(without applying a magneticeld) as well as LPs and UPs (applying a magneticeld). In

gure 6(a), we plot the numerical calculations of the diamagnetic energy shiftsfor UPs and LPs using our

numerical technique. A diamagnetic energy blue-shift of about 2 meV and 0.2 meV can be observed for UPs and

LPs respectively when =g E1.2o B and =B 10T, which are accessible experimentally. An even larger energyblue-shift can be observed when =B 15T. It isapparent that in the VSCregime, the signicant energy blue-shiftfor the UP originates from the enlarged exciton radius, a phenomenon that isabsent in the SC regime. In

gure 6(b), we calculate the net diamagnetic energy shiftsEUPELP for different coupling constants based on

the VSC modelsolved by our numericaltechnique, and based onthe SC modelsolved byan analytical

calculation [18]. For the SC model (solid lines),the net diamagnetic energy shift at a given coupling constant

vanishes at resonance,and remainsas a small value in the small detuning region. For the VSCmodel (dashedlines), the net diamagnetic energy shift up to 2.5 meV can be observed at resonance and increases/decreases

continuouslywith negative/positive detunings. Thus the nature of photonexcitoncoupling can be clearly

resolved on the level of the wavefunction by the proposed experiment, which provides another direct

vericationof the existence of VSC. Note that the above calculated diamagnetic energy shifts agree reasonably

Figure 5. (a)1s 2Dexciton wavefunction. Theexcitonwavefunction for(b) LPand (c)UP underthe inuence ofa vertical magneticeld =B 10T.

Figure 6. (a)The diamagnetic energyshifts ofboth theUP andLP asa functionof detuning when =g E1.2o B inthepresenceofavertical magneticeld. (b)The netdiamagneticenergy shiftsas a functionof detuning fordifferentg0 calculatedusing boththeanalyticalSC model(solid lines) and numerical VSC model(dashedlines) when =B 10T. Note forVSC thenet diamagnetic energyshift isa nite positive value reectingthe enlarged exciton radiusfor theUP andthe reduced exciton radiusfor theLP. When

=g E0.5 ,o B thedetuningandthe netdiamagnetic energyshift foran equalphotonexciton fraction is 0.25EB and0.016EB respectively,indicted bythe twopurple dashedlines.

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well with the results of usingrst-order perturbationtheory [15, 16] assuming the cyclotron energy is small

compared to the excitonbinding energy.

Conclusion

To summarize, the VSC effect in semiconductor cavity QED systems where the QW exciton radius is

dramatically modied has been veried by means of a nonlinear numerical optimizationtechnique.Furthermore, we propose two experimental schemes, one by reducing the oscillator strength of QW exciton

with an in-plane electriceld to recover the bare cavity energy,and the other by inducing the diamagnetic energy

shiftsin both UP and LP energy branches with a vertical magneticeld to compare their energy difference, and

theyexperimentally provide unequivocal proof of the existence of VSC. Our work offers further insight into the

verystronglightmatter interaction in semiconductor optical MCs, and the quest of developing a room-

temperature GaAs polariton laser.

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verification of very strong soupling in a semiconductor optical microcavity

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