Characterization of strong light-matter coupling in semiconductor quantum-dot microcavities via photon-statistics spectroscopy

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by  L. Schneebeli

Characterization of strong light-matter coupling in semiconductor quantum-dot
microcavities via photon-statistics spectroscopy
L. Schneebeli, M. Kira, and S.W. Koch
∗
Department of Physics and Material Sciences Center, Philipps-University, 35032 Marburg, Germany
(Dated: February 2, 2008)
It isshown that spectrally resolved photon-statistics measurementsof theresonance fluorescence
from realistic semiconductor quantum-dotsystemsallow forhigh contrast identification ofthetwo-
photon strong-coupling states. Using a microscopic theory, the second-rung resonance is analyzed
and optimum excitation conditions are determined. The computed photon-statistics spectrum dis-
playsgigantic, experimentallyrobustresonancesattheenergeticpositionsofthesecond-rungemis-
sion.
8
0 PACSnumbers: 42.50.Pq,42.50.Ar,78.67.Hc,78.90.+t
0
2
Eventhoughlight-quantizationeffects,suchassqueez- ofthe second-rungobservationsinstandardsemiconduc-
n
a ing[1],antibunching[2],andentanglement[3],havebeen tor experiments that monitor PL after carriercapture of
J observedunderawiderangeofconditions,onecanrarely electrons,initiallycreatedtothewettinglayer. However,

3 detect the discrete energy levels of the different photon- we demonstrate that it should be completely feasible to
2 number states directly as resonances. Under strong- observe the second rung in resonance fluorescence mea-

couplingconditions betweena two-levelresonanceatthe surementswheretheQD-cavitysystemisresonantlylaser
]
h energy ~ω and the quantized resonant light field, the re- pumped while the re-emitted light spectrum is recorded.
p sultingdressed-stateresonancesappearat~ωn g√n+1 Our results clearly show that the spectrum of the two-
t- wherenistheoccupationofthephoton-numbe±rstate n photoncorrelations,g(2),containsagiganticresonanceat
n | i
and g is the light-matter coupling constant [4]. The re- theenergeticpositionofthesecondrung. Duetoitslarge
a
sulting discrete energystructure resembles a ladder with value,thisg(2) resonanceremainsvisibleevenwhenscat-
u
q n-dependent rungs. The first rung shows the so-called teringbecomesappreciablemakingthisphoton-statistics
[ vacuum Rabi splitting 2g, the second rung shows the spectroscopy a viable experimental scheme to detect the

splitting2g√2,andsoon. Forsufficientlysmalldamping second rung in realistic QD systems.
1
v and dephasing, the different rungs are clearly visible as The first QD strong-coupling experiments [10, 11, 12]
4 discrete resonances, e.g., in transmission spectra. were performed with QDs grown on top of a quantum
0
The firstobservationsof strongcoupling [5,6]andthe well(QW)whichactsastheso-calledwettinglayer(WL).
6
second rung [7] were reported for atoms in high-quality Due to the three-dimensional confinement, the QD elec-
3
. cavities. Besidesthedemonstrationofthediscretenature tronsoccupyadiscretesetofstateswhichcoupletoeach
1
oflight, this researchhas leadto majoradvances,e.g. in otherandtoacontinuumofWLstatesviatheCoulomb,
0
8 utilizingentanglementeffectsasabasisforquantum-logic phonon,andlight-matterinteractions,yieldingacompli-
0 applications[8,9]. Theclearidentificationofthesecond- cated many-body problem [14, 15, 16, 17]. For strong-
: rung resonance can be considered as the critical step if coupling investigations, one typically studies strongly
v
i one wants to use other materials, suchas solid-state sys- confining dots where only one discrete electron and hole
X
tems for strong-coupling applications. A promising can- level–constitutinganeffectivetwo-levelsystem–couples
r didate are semiconductor quantum dots (QD) in micro- tothecavityresonance. Thisscenariosimplifiestheanal-
a
cavities. Recentexperiments[10,11,12,13]havealready ysis considerably because the remaining WL electrons
demonstrated that one can see clear vacuum Rabi split- and the phonon interactioneffectively act as noise reser-
ting effects in the photoluminescence (PL) of such sys- voirsto the isolatedtwo-levelsystem. As in the realsys-
tems. However, despite continuing attempts, the second tems, we assume that a variable number of dots is posi-
rung resonance has not yet been observed. The most tionedinsidethecavity. Thedifferentdotsarelabeledby
likely reason for this failure is that dephasing and the jandthecorrespondingelectronsandholesaredescribed
other broadening effects in the real systems smear out viaFermionicoperatorseˆ ,hˆ ,witheigenenergyEe,Eh,
j j
the expected discrete features. respectively. The light field corresponding to a mode
InthisLetter,weproposeanovelschemetoobservethe function uq(r) is quantized by introducing the Bosonic
second-rungstrongcouplingeffectsevenundertherealis- operatorBˆq whereqisthewavevectoroflightoutsidethe
tic broadeningconditions of the currently available sam- cavity. ThesystemHamiltonianfortheinvestigateddots
ples. In our approach, we use a fully microscopic theory and its interaction with light follows then from [17, 18]
teoviddeetnecremfionretthheeepxraecstenccoendoiftitohnestsoecoobntda-irnuunngarmesboignuaonuces Hˆ = Pj(cid:16)Eeeˆ†jeˆj +Ehhˆ†jˆhj(cid:17) + Pq~ωq(cid:16)Bˆq†Bˆq+ 21(cid:17) +
in QD microcavities. Our results confirm the difficulty Pqj(cid:16)Fq⋆Bˆq†hˆjeˆj +h.c.(cid:17), where ~ωq = ~|q|c is the pho-
2
ton energy and = id u (r ) defines the coupling QDposition. ThecouplingtotheWLcontinuumandthe
q q q j
F − E
strength via the dipole-matrix element d, the vacuum- phononsintroducesdephasingforpolarization-dependent
field amplitude , and the position of the dot r . The quantities, included phenomenologically via the dephas-
q j
E
used multi-mode description of the light field allows us ing constant γ . Otherwise, we explicitly include effects
P
to flexibly describe the coupling between an external up to three-particle correlations, denoted as ∆ 3 . We
pump field to the internal cavity field, propagation ef- observe that the squeezing signal ∆ BˆBˆ couplehsito the
h i
fects, as well as the finite linewidth of the cavity mode correlateddestruction of both a photon and an electron-
without additional phenomenological parameters. To hole pair, ∆ BˆPˆ . This correlation is created sponta-
h i
get a physically correct form for the light modes, we neously via the nonlinear polarization source P2. Af-
solve u (r) for a cavity between distributed Bragg re- ter its creation,∆ BˆPˆ produces non-vanishing ∆ BˆBˆ .
q
flector (DBR) mirrors. Since the resulting u (r)2 pro- The generated sqhueeziing signatures are coupledhbacik
q
duces nearly a Lorentzian resonance [19],| this |model to ∆ BˆPˆ via the stimulated contribution ∆ BˆBˆ in
h i P h i
is well-suited for other types of cavities after we ad- Eq. (2), which together with the photon-density correla-
just the cavity resonance ~ωc and its half width γcav tions, ∆ Bˆfˆeh , eventually produce the different rungs.
h i
to match the studied experimental cavity. In this con- Equations (1)–(2) are structurally similar to those ob-
nQec=tio~nω,ci/t(2isγccaovn).veFnuiernthtetromdoertee,rtmhienevatchueuqmuaRliatbyifsapcltiotr- taiWneedefvoarlu∆ahtBeˆq†oBˆuqr′ithaneodry∆huBsˆiq†nPˆgjit.he parameters of the
ting 2g = 2d N u (r )2 can also be adjusted three different, recently published experimental config-
Ecq dotPq| q j | urations in which vacuum Rabi splitting has been re-
to the experiment having N dots within the cavity.
dot
Since Hˆ involves infinitely many light modes, it is ported. TheQD-pillarinvestigations[10,22]havendot =
1.3 109cm 2 withinDBRmirrorsyieldinga qualityfac-
not feasible to solve the corresponding QD emission by tor·of Q =−2.4 104 with cavity frequency ~ω = 1.33
computing the wave-function or density-matrix dynam- · c
eV. The effective cavity area is S = 3.0µm2 yielding
ics. Therefore, we evaluate the time-evolution of the rel-
N =39andg =20GHz. InanotherQD-crystalexper-
evant expectation values directly. Due to the quantum- dot
optical light-matter coupling within Hˆ, the correspond- iampehnott[o1n1i]c,cnrdyostt=al6p.r0o·v1id0i9ncgmQ−2=Q2D.2s w10er4e, ~pωlac=ed1w.0itehVin,
ing Heisenberg equations of motion formally produce an · c
S = 10µm2, N = 600, and g = 22 GHz. In the QD-
infinite hierarchy of equations. We systematically trun- dot
diskexample[23],n =1010cm 2 QDswerepositioned
cate this hierarchy with the so-called cluster-expansion dot −
within a microdisk giving ~ω = 1.0 eV, Q = 4 105,
approach [18, 20]. At the lowest level, we find the c ·
S = 2.5µm2, N = 250, and g = 11 GHz. The dot
usualMaxwell-Blochequations[18,21]forcoherentlight dot
Bˆ , QD polarization P = hˆ eˆ , as well as electron dipole moment is d=5.3 ˚Ae in all these systems.
q j j j
h i h i In all cases, we carefully match the system parame-
(hole) occupations fje = heˆ†jeˆji (fjh = hhˆ†jhˆji). We as- ters with our microscopic model. As an initial condition
sume that the pump pulse is a classical field defined by
for the resonance-fluorescence computations, we assume
Eˆ(r) = i u (r) Bˆ +c.c. positioned initially out-
h i Pq Eq q h qi that the QDs are initially unexcited. We then solve the
side the cavity. As it propagatesinside the cavityit gen-
full set of equations including the three-particle corre-
erates the QD polarization and densities.
lations to determine the re-emission following the exci-
At the same time, the light-matter coupling induces
tation. Especially, we monitor the photoluminescence
quantum-optical correlations. In particular, we concen- spectrum I(ω ) Bˆ Bˆ in directions different than
trateontheanalysisofsqueezinggenerationdescribedby q ≡ h q† qi
the excitation together with the two-photon correlation
tBhˆeqBdˆyq′namicBˆsqofBtˆhqe′ twfool-lopwhointognfrcoomrrelations∆hBˆqBˆq′i≡ sthpeecptrroubmabgi(l2i)t(yωoqf)d≡etheBcˆtq†iBnˆgq†BtˆwqoBˆpqih/ohtBoˆnq†Bsˆwqii2t,hdmetoemrmenintuinmg
h i−h ih i
∂ q at the same time. We have verified with an indepen-
i~∂t∆hBˆqBˆq′i = ~(ωq+ωq′)∆hBˆqBˆq′i (1) dent single-mode analysis that the four-photon quantity
+ X∆h(cid:16)Fq⋆′Bˆq+Fq⋆Bˆq′(cid:17)Pˆji, ctraunnbcaetaiocncu.ratelydescribed by its singlet-doublet-triplet
j
At the level of the Jaynes-Cummings model [4], the
∂
i~ ∆ Bˆ Pˆ = ~ω +Eeh iγ ∆ Bˆ Pˆ (2) second-rungwavefunction follows from φ 1 up
∂t h q ji (cid:0) q − P(cid:1) h q ji | ±i∼| i| i±
2 down where up (down )refertotheexcited(unex-
+ (cid:0)1−fje−fjh(cid:1)XFq′∆hBˆqBˆq′i |citie|d) doit/atom.| Tihu|s, oneican access this state either
q′ by having the system originally in the excited state and
+ Ω ∆ Bˆ fˆeh ⋆P2+T [∆ 3 ]. providing sufficient occupation of the 1 photon state
j h q j i−Fq j h i or when the system is originally unexcit|edi and the light
HerewehavedefinedEeh Ee+Eh,thepolarizationop- has a strong occupation of the state 2 . If carriers are
eratorPˆj ≡hˆjeˆj,thedens≡ityoperatorfˆjeh =eˆ†jeˆj+hˆ†jhˆj, created nonresonantly to the wetting|laiyer, the carrier-
as wellas the classicalRabi energyΩ =d Eˆ(r ) atthe capture processes will bring the dot toward the excited
j j
h i
3
dition guarantees the selective excitation at the second
rung with a probability determined by the two-photon
state occupation P2 = |α2|4 e−|α|2 within the external
pump. We also notice that the second-rungemission en-
ergy, ~ω = ~ω +( ∆2+8g2 ∆2+4g2)/2, and
the optim2nudm pumcp enpergy ~ω are±gepnerally nondegener-
ate.
To illustrate how the resonant second-rung pumping
works, we consider the excitation of the QD-disk sys-
tem (∆ = 0) with coherent light energetically centered
at ~ω = ~ω + g . Figure 1(a) shows the pump spec-
c √2
trum (shaded area) together with the resulting emission
I(ω) for the low γ = 0.06 GHz dephasing (solid line)
P
and the elevated γ =0.4 GHz dephasing (dashed line).
P
Besides the usual vacuum-Rabi peaks at ~ω g we ob-
c
±
serve for low γ that the fluorescent light contains clear
P
second-rungemissionpeaksat~ω =~ω + √2 1 g.
2nd c
(cid:0) ± (cid:1)
Thisfeaturesareclearsignaturesofstrongcouplingeven
FIG. 1: Resonance fluorescence spectra for a QD-disk sys- though they gradually vanish for increasing γ . If the
P
tem (∆ = 0). (a) Spectrum of the resonant second-rung same analysis is repeated for non-resonant excitation,
pump(shadedarea)andresultingemissionforlowdephasing
I(ω) shows just the vacuum-Rabi peaks, regardless of
γP =0.06 GHz (solid line) and elevated dephasing γP =0.4 γ . Thus, it is clearly easier to demonstrate true strong
GHz (dashed line). The vertical lines mark the second-rung P
coupling using the resonant second-rung pumping. To
resonances. (b) The second-rung emission intensity as func-
tion of pump frequency. The inset shows the second-rung verify that the emerging resonances result from the sec-
emission intensity and the two-photon state occupation P ond rung, we performed additional test by changing the
2
(shaded area) in the pump field as function of pump inten- frequency and intensity of the pump. Figure 1(b) shows
sity. I I(ω )atthelowersecond-rungpeakasfunction
2nd 2nd
of pu≡mping ~ω. Indeed, I peaks exactly at the opti-
2nd
mumpumpingfrequency~ω =~ω +g/√2. Furthermore,
c
state up with a characteristic time τcapt. At the same theinsetshowsthatI2ndscaleswiththetwo-photonstate
| i
time,the exciteddotemits lightintothecavitymode. A occupationP2 inthepumpfield,asthecoherenceparam-
typical time scale to create 1 out of the vacuum is de- eter α is increased. Thus, it is very important to prop-
fined by half of the Rabi per|ioid, givingτ ~. Thus, erly adjust not only the excitation frequency but also
Rabi ≡ g
thecarriercapturehastosupportboththedotandcavity theexcitationintensitysuchthatthetwo-photonstateis
excitation,whichdecayswiththelifetimeτ . Toclearly sufficiently occupied.
cav
see the second-rung from PL alone, one thus must have Even though the second-rung pumping induces true
τcapt τRabi and τcapt τcav, whichis difficult to fulfill strong coupling effects in the QD system, the realistic
≪ ≪
in currentexperiments since the typicalcapture times of dephasing of γ =0.4 GHz washesout the most intrigu-
P
τcapt = 50 ps [24] exceed both the Rabi (11 ps) and the ing features in the standard experiments. Thus, we an-
cavity lifetime (25 ps) [11]. Further problematic aspects alyze next the g(2) correlations and show that they can
are the dephasing and the incomplete carriercapture re- serveasmorerobustsignatures. InFig.2(a)-(c),thesolid
sulting in fe,h <1. linepresentsthecomputedg(2) spectrumforthedifferent
The second-rung state can also be reached by bring- QD-cavity systems after resonant second-rung pumping
ing the cavity directly into the Fock state 2 while the (shadedarea). The energeticpositionofthe secondrung
| i
dotremainsunexcited(down ). Aresonantexcitationof (upper NMC peak) is marked by the solid (dashed) ver-
| i
thecavity-dotsystemwithacoherentlaser,describedby tical line. Our results verify that all QD-cavity systems
|αi = P∞n=0αn/√n!exp(cid:2)−|α|2/2(cid:3)|ni, always produces yield g(2) resonances with gigantic values close to 103 at
someoccupationat 2 . Inparticular,the 2 component the second-rung energy.
| i | i
oftheexternallightisselectivelyconvertedtothesecond- This strongly enhanced g(2) follows from the funda-
rung state φ+ if its energy Epump = 2~ω matches the mental properties of the resonant second-rung pumping
energy of t|he diressed dot-cavity state E = 2~ω +
dress c which exclusively enables the Fock-state 2 to interact
√2g,givingthecondition~ω =~ωc+√g2 forzerocavity- with the QD. Since the cavity initially is i|nithe vacuum
dot detuning ∆ Eeh ~ω . The transfer of other n state, the addition of this Fock state essentially creates
c
≡ − | i
statesissuppressed. Fornon-zero∆,theoptimumenergy cavity light into state 0 +√P 2 , which is a squeezed
2
is defined by ~ω =~ω +(∆+ ∆2+8g2)/4. This con- statewithanappreciab|lyi smallP| .iThesameconclusion
c p 2
4
for a large dephasing γ = 2.3 GHz. We notice that the
p
g(2) signalremainsunchangedinthelow-intensityregime
but decreases for too strong excitation. Thus, the reso-
nant second-rung pumping has to be performed in the
low-intensity regime where g(2)(ω ) approaches a con-
2nd
stant value. The calculations in Fig.2(a)-(d) were done
in this stable regime, as indicated by the vertical line in
Fig.2(e).
Insummary,ourproposedmethodofphoton-statistics
spectroscopy identifies a way for the experimental ver-
ification of strong-coupling resonances in the resonance
fluorescence of semiconductor QD microcavities. Based
on a consistent microscopic analysis, we predict that es-
pecially the two-photon strong-coupling state should be
clearlyvisibleasapronouncedresonanceintheg(2)spec-
trum. These measurements allow for clear identification
oftruestrongcouplingsituationsinrealisticQDsystems.
Acknowledgements: This work is supported by the
FIG.2: Photon-statisticsspectra(solidline)for(a)QD-pillar,
Quantum Optics in Semiconductors DFG Research
(b) QD-crystal, and (c) QD-disk system, at the time when
Group. We thank H.M. Gibbs and G. Khitrova for valu-
the pump (shaded area) has its maximum. In all systems,
able discussions.
∆ = 0 and γP = 0.23 GHz. (d) The corresponding g(2) at
the second rung as function of dephasing for the QD-pillar
(shaded),QD-crystal(solid), and QD-disk (dashed) systems.
The horizontal line at g(2) = 3 indicates the visibility limit.
(e)Thepump-intensitydependenceofg(2) atthesecondrung
∗
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