Table Of ContentMultiple-pulse lasing from an optically induced harmonic confinement in a highly
photoexcited microcavity
Wei Xie,1, Feng-Kuo Hsu,1, Yi-Shan Lee,2 Sheng-Di Lin,2 and Chih Wei Lai1,
∗ ∗ †
1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
2Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan
Wereporttheobservationofmacroscopicharmonicstatesinanopticallyinducedconfinementina
highlyphotoexcitedsemiconductormicrocavityatroomtemperature. Thespatiallyphotomodulated
refractive index changes result in the visualization of harmonic states in a micrometer-scale optical
potentialatquantizedenergiesupto4meVevenintheweak-couplingplasmalimit. Wecharacterize
5 thetimeevolutionoftheharmonicstatesdirectlyfromtheconsequentpulseradiationandidentify
1 sequential multiple 10 ps pulse lasing with different emitting angles and frequencies.
∼
0
2
Opticaltrappingofatomsisessentialtorealizeatomic coupling to the cavity light field. Multiple-pulse las-
l
u Bose-Einstein condensates [1]. Similarly, the 2D spatial ing from discrete states in a harmonic confinement (po-
J
confinementofexcitonsorexciton-polaritonsbyinhomo- tential) manifests as a result of the time- and energy-
1
geneousstrains[2],naturaldefectsandpotentialfluctua- dependent competition between the gain and reservoir
2
tions[3–6],nanofabrication[7–13],andopticalpotentials carrier relaxation.
] [14–17] in a semiconductor heterostructure/microcavity The Fabry–P´erot microcavity sample consists of a λ
ll is considered conducive for the formation and control GaAs cavity layer containing three sets of three In-
a
of the condensates of these quasiparticles. These dy-
h GaAs/GaAs QWs embedded within GaAs/AlAs dis-
namic condensates can form a meta-stable state in a
- tributed Bragg reflectors (DBRs) (see also Supplemen-
s finite-momentumstate[8,18]andmultiplespatialmodes
e tal Material). The sample is nonresonantly excited by
m [19]. Theopticalvisualizationofmacroscopicinteracting a 2-ps pulse pump laser at E = 1.58 eV at room tem-
p
quantum states in solid-state systems has been demon-
. perature. The pump energy is about 250 meV above the
t strated in condensates of excitons (bound electron-hole
ma pairs) and kindred quasiparticles. Such light–matter hy- cQaWvitybarnesdongaanpc(eE(g(cid:48)E≈ 1.13.340e–V1).4a1nedV1)7.0TmheeVhigahb-odveenstihtye
c
- brid condensates are typically formed at cryogenic tem- e-h plasmasofadensi≈ty 1 5 1012cm−2 perQWper
d peraturesinaphotoexciteddensitymuchbelowtheMott ≈ − ×
pulse are formed momentarily after non-resonant pulse
n transition where constituent quasiparticles can be con-
excitationasaresultofrapid(<10ps)energydissipation
o
sidered boson-like. Optically defined potentials enabled
c through optical phonons. The radiative recombination
bytheeffectiverepulsiveinteractionsofpolaritonsresult
[ rate of these e-h carriers in the reservoir is suppressed
in further real-time manipulation of these light-matter
because the cavity resonance E is detuned to 70–80
3 fluids in a steady state [15–17]. On the other hand, re- c ∼
v meV above the QW bandgap Eg(cid:48), i.e., the e1hh1 tran-
laxationoscillationswith 10–50psperiodcausedbythe
0 ∼ sition between the first quantized electron and (heavy-
4 interplay betweenreservoirfeedingand Bosestimulation )hole states in a QW. Below the lasing threshold, the
0 have been reported in localized condensates in natural
high-density e-h plasmas in the microcavity are subject
0 potential fluctuations [5].
to non-radiative loss with a long decay time (>500 ps)
0
. In this study, we report the optical visualization of (Supplemental Fig. S1). Therefore, the chemical poten-
2
tial of the e-h plasmas (µ ) appears to be stationary
0 dynamic quantized states in a highly photoexcited mi- 0
within 100 ps after pulse excitation. The bare cav-
5 crocavity at room temperature. Photoexcitation creates
∼
1 mainly free pairs of electrons and holes in a III-V-based ity resonance Ec of the sample studied here is close to
: E , the e2hh2 transition between the second quantized
v semiconductor quantum well (QW) as a result of ther- g(cid:48)(cid:48)
electron and hole levels of InGaAs/GaAs QWs.
i mal ionization at high temperature [20, 21]. In the
X
highly photoexcited microcavity studied here, the non- When µ advances toward E , the average refractive
0 c
r
a linearly photomodulated refractive index results in a siz- index near Ec can be considerably modified, and this
able effective cavity resonance shift, which enabls opti- results in a sizable blueshift of the effective cavity reso-
callyinducedconfinement. Inanopticalconfinementini- nance(E )andconsequentemissionenergy(Supplemen-
c(cid:48)
tiated by spatially modulated nonresonant ps pulse ex- tal Fig. S2a). In general, E increases with the photoex-
c(cid:48)
citation, sequential multiple-pulse lasing commences at cited density and can be seen as an effective potential V
several quantized energy levels. The transverse optical forcehps(SupplementalMaterialsectionS3). Therefore,
modes in a spatially modulated refractive index can also a quasi-stationaryconfining potentialV(r) forcehps can
be understood as an optically induced potential for a be established by a ring-shaped spatial distribution of
fictitious quasiparticle (see Supplemental Material), e.g., photoexcited carrier density. In our experiments, we use
correlated e-h pairs (cehp) resulting from the effective a double-hump-shaped beam to fixate the orientation of
2
r-space k-space
8 a 1 1.41 c d 1.41 e f 1
-088 b Pump 0 (ev)Energy (ev)Energy EE32 000...631466
E 2.3 meV
0 1 0.04
1.40 1.40
PL P = 0.8 P P = 1.3 P P = 0.8 P P = 1.3 P 0
-8 th th th th
-8 0 8 -4 0 4 -4 0 4 -4 0 4 -4 0 4
Position ((cid:80)m) Position ((cid:80)m) Position ((cid:80)m) k ((cid:80)m-1) k ((cid:80)m-1)
|| ||
FIG.1. Visualization of the macroscopic harmonic states. (a)Intensityimageofthering-shapedpumplaserbeam. (b)
Photoluminescence(PL)imageunderapumpfluxofabout1.3P ,wherethethresholdpumpfluxP =1.8 108 photonsper
th th
×
pulse. Thewhitedashedlinerepresentstheintensitypeakofthepump. PLemergesatthecenterwithaminimaloverlapwith
theannularpumplaserbeam. (c–d)r-spaceimagingspectraatP =0.8P and1.3P . Theblackdashedlinerepresentsthe
th th
harmonic confining potential V(x), whereas the white lines represent the spatial probability distributions of the lowest three
states of a corresponding harmonic oscillator. (e–f) k-space imaging spectra. The energy splitting is (cid:126)ω 2 meV, consistent
with the quantized energy of a quantum oscillator for a particle with mass m∗ =3 10−5 me, as determi≈ned by the E vs. k
dispersion (doted grey line). The quantized modes spectrally blue shift about 1 m×eV from P = 0.8 to 1.3 P , whereas th(cid:107)e
th
quantized energy splitting remains the same. The potential and spectral shifts are due to a density-dependent increase in the
chemical potential of the high-density e-h plasma in the reservoir.
the optically defined potential (Fig. 1a–b). 1.412
aD b c 1
Real-space(r-space)imagingspectraprovidedirectvi- V) 0.64
sualization of the optical potential. Fig. 1c–d shows the (ey1.408 0.36
g
r-spaceimagingspectraofanarrowcross-sectionalstripe ner1.404 0.16
E 0.04
acrossthetrap. Anearlyparabolicpotentialwellof 10
∼ 1.400 0.8 P 0.4 P 1.0 P 0
meV across 3 µm is revealed. Such a quasi-1D harmonic th th th
-4 0 4 -4 0 4 -4 0 4
potential, V(x) = 1/2 αx2, is spontaneously formed un-
1.412
d e f
der 2 ps pulse excitation. The resultant radiation ap-
V)
pearsin-betweenthetwohumps,afewmicrometersaway (ey1.408
from the pump spot. The r-space intensity profiles agree nerg1.404
withtheprobabilitydistributionsofthequantizedstates E
of a harmonic oscillator (harmonic states). The devia- 1.400 1.1 P 0.5 P 1.1 P
th th th
tions of the actual potential from a perfectly harmonic -4 0 4 -4 0 4 -4 0 4
k ((cid:80)m-1) k ((cid:80)m-1) k ((cid:80)m-1)
trap result in slightly asymmetric luminescence inten- || || ||
sity distributions. These standing wave patterns form
in the self-induced harmonic optical potential when a FIG. 2. Quantized states in optically controlled con-
fining potentials. K-space imaging spectra under below-
macroscopiccoherentstateemerges. Atacriticaldensity,
threshold (a–b) and above-threshold (d–e) for two double-
the occupation number of cehps (n ) in these quantized
i hump-shaped pump beams with peak-to-peak distances of 5
harmonic states approaches unity when the conversion
µmand3µm,respectively. Forcomparison,thek-spaceimag-
from the reservoir (Neh) overcomes the decay of the har- ingspectraunderaflat-toppumpbeamareshownin(c)and
monic states. Consequently, when a stimulated process (f). Thecorrespondingr-spaceimagesandspectraareshown
( N n ) prevails, n and the resultant radiation in- in Supplemental Fig. S1.
eh i i
∝
creasenonlinearlybyafewordersofmagnitudewiththe
increasing N .
eh
Anevenmoreregularpatternappearsink-spaceimag- of the pump beam (Fig. 2). The energy quantization
ing spectra (Fig. 1e–f). The E vs. k dispersion mea- ((cid:126)ω) varies with α/m∗, while the emission patterns
suredbelowthethresholdallowsthedi(cid:107)rectmeasurement evolve into the probability distributions for a particle
(cid:112)
of the effective mass m∗ = 3 105me, where me is the with an effective mass m∗ in V(x) (Fig. 2 and Supple-
electron rest mass. Moreover×, the strength of the opti- mental Fig. S3).
callydefinedharmonicpotentialα canbetunedthrough Temporally,theseharmonicstatesemergesequentially
a variation in the spatial distance between two humps anddisplaydistinctdensity-dependentdynamics. InFig.
3
a P = 1 P P = 1.1 P P = 1.2 P P = 1.6 P
0 th th th th
s)
p
e ( 50
m 1
Ti
0.5
0
100
-2 0 2 -2 0 2 -2 0 2 -2 0 2
k ((cid:80)m-1)
b ||
1
1.410
V) 0.25
e
y (
g
ner1.406 0
E
1.402
0 50 100 50 100 50 100 50 100
Time (ps)
FIG. 3. Dynamics. (a) Time-dependent luminescence in k-space at P =1.0, 1.1, 1.2 and 1.6P . The E , E and E states
th 3 2 1
appear sequentially with the increasing pump flux. The rise times decrease with the increasing pump flux for all states. (b)
Time-dependent spectra in r-space. The false color represents normalized intensities.
3a,westudythetimeevolutionsofthreeharmonicstates nitude across a threshold and then reach a plateau at
in k-space. The corresponding time-integrated imaging a saturation density (Fig. 4a). On the other hand, the
spectra in k-space are shown in Supplemental Fig. S4. emission energy of these three states increases to a con-
The energy relaxation of these three states is revealed in stant with the increasing pump flux (Fig. 4b). The en-
the time-resolved spectra in Fig. 3b. At a critical pho- ergyspacing(cid:126)ω onlyincreasesslightlywithdensity. The
toexcited density, the high-energy E state arises 25 spectral linewidths increase slightly for the E state but
3 1
∼
ps after pulse excitation and lasts for 20 ps. The cor- by about a factor of 10 for the E state. Next, we study
3
∼
responding pump flux is defined as the threshold (P ). the density-dependent dynamics. Fig. 4c shows the rise
th
With the increasing pump flux, the E state emerges timesandpulsedurationsforE ,E ,andE . Theprod-
2 3 2 1
∼
25 ps after E at 1.1 P , whereas the ground E state uct of the variances of the spectral linewidth (∆E) and
3 th 1
appear50psafterE at1.2P . Intheopticallyinduced the pulse duration (∆t) is found to be close to that of a
2 th
harmonic confinement studied here, the effective cavity transform-limited pulse: (cid:38) 4(cid:126) and 1(cid:126) for the E3/E1
≈
resonance E decreases with time as a result of the de- and E states, respectively. These harmonic states are
c(cid:48) 2
cay of reservoir carriers. However, the conversion from macroscopically coherent states with finite phase and in-
the reservoir to specific confined E state can be efficient tensity fluctuations induced by interactions.
i
only when E is resonant with E . Such a temporally
c(cid:48) i
decrease in E results in a time- and energy-dependent We use a rate-equation model to describe the dy-
c(cid:48)
conversion efficiency for these confined harmonic states namic formation of quantized states in an optically
(Fig. S1)andconsequentmultiple-pulselasingabovethe defined harmonic confinment (Supplemental Material
critical density threshold. Sec. S4Supplemental Material). This phenomenolog-
ical model reproduces qualitatively the dynamics and
We further analyze the emission flux and energy of integrated emission flux of the harmonic states when
these harmonic states by using time-integrated spectra photoexcited density is varied (Supplemental Figs. S5,
measured with increasing photoexcited density (Fig. 4 S6, and S7). Non-equilibrium polariton condensates
and Supplemental Fig. S2). Far below the threshold have been modeled by a modified Gross–Pitaevskii (GP)
(P < 0.4 P ), emission is dominated by luminescence (or complex Ginzberg–Landau [cGL]) equation that ac-
th
from GaAs spacer layers. When the pump flux is in- counts for the finite lifetime of polaritons [22, 23]. How-
creased, the emissions from the cehp states become in- ever, cGL-type equations are inapplicable for the multi-
creasingly dominant, and the E state eventually lases ple dynamic states examined in this study. Additionally,
3
at the threshold. The emission fluxes of all three states the formation of a BEC-like condensate that underpins
increase nonlinearly by more than two orders of mag- the GP- or cGL-type equation is not necessarily justified
4
1 0 10 bodyeffects,suchasscreening,bandgaprenormalization,
) a and phase-space filling, are all considered. For example,
its 1 0 9 one can consider the formation of index-guided multiple
n
. u transverse modes as a result of an optically-induced re-
(arb 1 0 8 fractive index reduction (∆nc(x)). The cavity resonance
x shift (δE) can be estimated from δE /E = ∆n /n ,
n Flu 1 0 7 EE 3 where nc is the effective refractive incdexcaver−agedcovecr
sio 1 0 6 E 2 the longitudinal cavity photon mode which spans over
is 1 µm in the growth direction. Therefore, a cavity res-
m ∼
E 1 0 5 onance shift δE 10 meV (Figs. 1 and 2) corresponds
0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0 2 .2 ∼
1 .4 1 2 to ∆nc/nc 1%. Such a significant refractive index
| | ∼
b is probable with resonance-enhance optical nonlinearity
1 .4 1 0
) fromthee-h correlation[30–33]orcarrier-inducedchange
(eV1 .4 0 8 in refractive index at a high carrier density (of ∼ 1019
ak cm−3 or more) [34–36]. To uncover the microscopic for-
e1 .4 0 6
P mation mechanisms of such a sizable spatial modulation
y
rg1 .4 0 4 in refractive index or equivalent effective harmonic con-
e
n finement under pulse excitation, further characterizing
E1 .4 0 2
the photoexcited density distribution with the use of
1 .4 0 0 otherultrafastspectroscopictechniques,suchasapump-
0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0 2 .2
probe spectroscopy, is necessary.
1 2 0
c We identify sequential multiple 10 ps pulse lasing in
1 0 0 ∼
anopticallyinducedharmonicconfinementinasemicon-
)
s 8 0 ductor microcavity at room temperature. Laser radia-
e (p tion emerges at the quantized states of an optically in-
6 0
im duced harmonic confinement. The lasing frequency, rise
T
e 4 0 time, pulse width, and radiation angle can be controlled
is
R through a variation in the photoexcited density or opti-
2 0
cal pump spot dimensions in real time. The sample has
0
a composition structure similar to those used for stud-
0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0 2 .2
P u m p F lu x (P ) ies of polariton condensates/lasers and the widely used
th
VCSELs. In the highly photoexcited microcavity stud-
ied here, harmonic confinement can be optically induced
FIG. 4. Density dependence. (a) Temporally and spec-
trallyintegratedemissionfluxvs. pumpflux. Allthreemodes and controlled as a result of photomodulated refractive
displaynon-linearincreasesinintensitybymorethantwoor- index changes even in the weak-coupling plasma limit.
dersofmagnitude,andsaturateat1.1,1.2and1.3Pth,respec- Our demonstration of macroscopic harmonic states in
tively. (b) Peak energy (solid shapes) and linewidths 2∆E a room-temperature optically induced harmonic confine-
(error bars) vs. pump flux. These states spectrally blue shift
ment improves our understanding of nonlinear laser dy-
by1to4meV.Thespectrallinewidths(∆E)andpulsewidths
namicsandshouldstimulatestudiesonemergentordered
(∆t) are reciprocal with a product of ∆E ∆t 4(cid:126) ((cid:126)) for
× ≈ states near the Fermi edge of a high-density e-h plasma
E and E (E ), which is closed to the uncertainty (Fourier-
3 1 2
transform) limit. (c) Rise time vs. pump flux for the three in a semiconductor cavity [37–39].
states E1 (blue), E2 (red) and E3 (black). The error bar WethankChengChin,MarkDykman,BrageGolding,
represents 2∆t. Peter B. Littlewood, John A. McGuire, David W. Snoke
and Carlo Piermarocchi for the discussions. This work
wassupportedbyNSFgrantDMR-0955944andJ.Cowen
in our room-temperature experiments. endowment at Michigan State University.
Transverse light-field patterns and confined optical
modes have been identified in nonlinear optical sys-
tems [24], vertical-cavity surface-emitting lasers (VC-
SELs) [25, 26], and microscale photonic structures [27].
In principle, the multiple transverse mode lasing in a ∗ These authors contributed equally to this work.
high-density e-h-plasma described in the present study † [email protected]
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Supplemental Material for
Multiple-pulse lasing from an optically induced harmonic confinement in a highly
photoexcited microcavity
Wei Xie,1 Feng-Kuo Hsu,1 Yi-Shan Lee,2 Sheng-Di Lin,2 and Chih-Wei Lai1
1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
2Department of Electronics Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan
S.1. MATERIALS AND METHODS
Sample. The microcavity sample is grown on a semi-insulating (100)-GaAs substrate with the molecular beam
epitaxy method. The top (bottom) DBR consists of 17 (20) pairs of GaAs(61-nm)/AlAs (78-nm) �/4 layers. The
central cavity layer consists of three sets of three In Ga As/GaAs (6 nm/12 nm) quantum wells each, positioned
0.15 0.85
at the anti-nodes of the cavity light field. The structure is undoped and contains a � GaAs cavity sandwiched by
DBRs. The bare cavity resonance E 1.40 eV at room temperature. The QW bandgap (E 1.33 eV) is tuned
c ⇡ g0 ⇠
througharapidthermalannealingprocess(at1010 C-1090 Cfor5-10s),inwhichtheInGaAsQWbandgapblueshifts
� �
because of the di↵usion of gallium ions into the MQW layers. With increasing photoexcited density, the chemical
potentialµ ofthee-h plasmacanincreaseupto 80meVaboveE . ThecavityqualityfactorQisabout4000–7000,
0 ⇠ g0
corresponding to a cavity photon lifetime of 2 ps.
⇠
Optical excitation. The front surface of the sample is positioned at the focal plane of a high-numerical-aperture
objective(N.A.=0.42,50 ,e↵ectivefocallength: 4mm). Theholographicbeamshapingisachievedwithareflected
⇥
Fourier transform imaging system that consists of a 2D phase-only spatial light modulator (SLM), a 3 telescope, a
⇥
Faraday rotator, a polarizing beam splitter, and the objective. The light fields at the SLM and the sample surface
formaFouriertransformpair. The2DSLM(1920 1080pixels, pixelpitch=8µm)enablesustogeneratearbitrary
⇥
pump geometries with a 2 µm spatial resolution at the sample surface by using computer-generated phase patterns.
⇡
Thepolarizationpropertiesofpumpandluminescencearecontrolled/analyzedwithliquid-crystaldevices. Thepump
flux is varied by more than two orders of magnitude with a liquid-crystal-based attenuator.
Thermal management. Atahighpumpflux,thesteady-stateincidentpowertransmittedtothesampleatthe76
MHz repetition rate of the laser would exceed 50 mW, and significant thermal heating results. Thermal heating can
inhibitlaseractionandleadtospectrallybroadredshiftedluminescence. Tosuppressthermalheating, wetemporally
modulate the 2 ps 76 MHz pump laser pulse train with a duty cycle (on/o↵ ratio) < 0.5% by using a double-pass
acousto-optic modulator system. The time-averaged power is limited to below 1 mW for all experiments.
R-space and k-space imaging spectroscopy. We measure the angular, spectral, and temporal properties of
luminescence in the reflection geometry. The angle-resolved (k-space) luminescence images and spectra are measured
throughaFouriertransformopticalsystem. Aremovablef =200mmlensenablestheprojectionofeitherthek-space
or r-space luminescence on to the entrance plane or slit of the spectrometer. Luminescence is collected through the
objective, separated from the reflected specular and scattered pump laser light with a notch filter, and then directed
to an imaging spectrometer. A lasing mode with a spatial diameter 8 µm is isolated for measurements through a
⇡
pinhole positioned at the conjugate image plane of the microcavity sample surface. The spectral resolution is 0.1
⇡
nm (150 µeV), which is determined by the dispersion of the grating (1200 grooves/mm) and the entrance slit width
(100–200 µm). The spatial (angular) resolution is 0.3 µm (6 mrad) per CCD pixel.
⇡
2
S.2. SUPPLEMENTARY FIGURES
a 1.0 b 1.0
0.8 0.8
m. Intensity00..46 m. Intensity00..46
Nor0.2 Nor0.2
0.0 0.0
0 50 100 150 200 -5.0 -2.5 0.0 2.5 5.0
Time (ps) Position (µm)
c 1.411 d 1.0
0 1 E
Energy (eV) 1111....444400007395 EE23 orm. Intensity000...468 EE321
E N
1.401 1 0.2
0.0
0 50 100 150 200 0 50 100 150 200
Time (ps) Time (ps)
Fig. S1. Below-threshold dynamics. (a) Time-dependent spectrally and spatially integrated luminescence at P = 0.9 P .
th
The luminescence decays with a > 500 ps time constant largely because of the nonradiative loss. (b) Spatial luminescence
distribution integrated up to a delay of 50 ps after pulse excitation, revealing a double-hump-shaped profile. (c) Time-
dependent luminescence spectra at P = 0.9 P . The false color represents the square root of the luminescence intensity. (d)
th
Time-dependentluminescenceintensityfortheE (black),E (red)andE (blue)harmonicstates. Theintensityisintegrated
3 2 1
over a spectral range of 1 meV centering the energies, as indicated by the dashed lines in (c).
1.415 1.415
a b (cid:86)(cid:14)(cid:18)(cid:86)(cid:14) c (cid:86)(cid:14)(cid:18)(cid:86)(cid:16)
1
n
o
si1.410 1.410 0.64
s
mi
Peak of E1.405 nergy (eV) 1.405 00..1366
al E
ctr1.400 1.400 0.04
e
p
S 0
1.395
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.395
0.2 0.6 1.0 1.4 1.8 2.2 0.2 0.6 1.0 1.4 1.8 2.2
Photoexcited Density (D) Pump Flux (P )
th
Fig. S2. Pump-flux-dependent resonance energy and polarized luminescence spectra. (a) Resonance energy (the
energy of emission mode at k 0 in K-space spectral mapping) versus photoexcited density under a tightly-focused circular
pumping beam (radius 1 µmk ⇡). The spectral peak can be considered as the e↵ective cavity resonance and displays a sizable
⇠
spectral blueshift with increasing the photoexcited density. Such an energy shift with photoexcited density is the basis of
optically induced harmonic confinement (potential) using a spatially modulated pump beam profile. D 4 1012cm�2 per
⇡ ⇥
quantumwellperpulse. (b,c)Normalizedspectraintegratedoverk-spaceasafunctionofpumpfluxinthedouble-hump-shaped
pumping case: Co-circularly (b) and cross-circularly (c) polarized component. Quantized harmonic states are formed above
0.8P . Below the threshold, the luminescence from all states is unpolarized. Above the threshold, the E state is highly
th 3
⇠
circularly polarized, where as the E state has a diminishing circular polarization. The fourth quantized state (E ) is weakly
1 4
confined.
3
5 a 5 b 10 c Density
m) 3.5x1012
(cid:80)
(on 0 0 0
siti
o
P
-5 -5 -10 0
-5 0 5 -5 0 5 -10 0 10
1.412
d e f 1
V)1.408 0.64
e
(y 0.36
erg1.404 0.16
n
E 0.04
1.400
0.8 P 0.4 P 1.0 P 0
th th th
-5 0 5 -5 0 5 -5 0 5
1.412 g h 1 i 0.4 P
V)1.408 units)0.5 S0.H5O Ptthh
(eergy1.404 de (arb. 1 E1
n u
E1.400 1.1 P 0.5 P mplit0.5 E
th th A 0 2
-5 0 5 -5 0 5 -5 0 5
Position ((cid:80)m) Position ((cid:80)m) Position ((cid:80)m)
Fig. S3. Quantized states vs. the trap size. (a–c) Optical images of pump beam profiles at the front sample surface for
twodouble-hump-shapedandoneflat-topfocalspots. Thefalsecolorrepresentsthephotoexciteddensity(cm�2)perquantum
well per pulse at 0.8 P for (a) and at 0.4 P for (b). P = 1.8 108 per pulse, as defined in the main text. The lasing
th th th
⇥
thresholds are 0.8 P for (a) and 0.4 P for (b). (d–f) Real-space imaging spectra measured below the threshold. The black
th th
dashedlineshowsthee↵ectiveharmonicpotential,whereasthewhitelinesrepresentthespatialprobabilitydistributionsofthe
lowest few states of a corresponding quantum oscillator. The quantized energy splittings for (d) and (e) are 2.2 and 4.0 meV,
respectively. The false color represents the square root of intensity. (g–h) Real-space imaging spectra under above-threshold
pump. (i) The measured wave functions of the two lowest states E and E at 0.4 P (solid blue lines) and 0.5 P (solid
2 1 th th
red lines) as shown in (e) and (h). The ideal single-particle wave functions of the corresponding quantum simple harmonic
oscillator (SHO) are represented by the dashed lines.
1.41
P = 0.9 P P = 1 P P = 1.2 P P = 1.3 P
th th th th 1
)
V
e
(
y
g
r
e
n
E
1.40
0
-4 0 4 -4 0 4 -4 0 4 -4 0 4
k(µm-1)
||
Fig. S4. K-space imaging spectra. The energy vs. in-plane momentum k dispersion corresponding to the data set in the
main paper with the increasing pump flux from 0.9 to 1.3 P . The quantized||E , E , and E states lase in sequence with the
th 3 2 1
increasing pump flux. E -state lasing commences first, followed by E -state lasing at about 1.2 P , and then E -state lasing
3 2 th 1
appears at 1.3 P and dominates over the other two states at a higher pump flux.
th
4
S.3. OPTICALLY INDUCED POTENTIAL AND REFRACTIVE INDEX CHANGES
The transverse optical modes confined by a spatially modulated refractive index can also be understood as an
optically induced potential for a fictitious quasiparticle.
Considering the electromagnetic fields in a planar Fabry-P´erot cavity with a medium of a 2D spatially modulated
e↵ective refractive index profile n(r), we can express the Helmholtz equation for the electric field in the following
simplified form:
@2F(r,',z)
+n2(r) k2 F(r,',z)=0 (1)
@r2 · 0 ·
The electric field F in the cavity can be decomposed into lateral and transverse components, and approximated as
F(r,',z) (r) exp[ in(r)k (r) z)],wherek2 =k2(r)+k2(r),andk (r)andk (r)aretransverseandlongitudinal
wavenumb/ers in t·he va�cuum, rezspec·tively. For s0implizcity, wekconsider thke 1D casezand reduce Eq. 1 to:
@2F(x,z)
+n2(x) [k2 k2(x)] F(x,z)=0 (2)
@x2 · 0 � z ·
Applying the e↵ective-mass approximation for small k , we express the in-plane dispersion E (k ) in terms of an
c
e↵Eecqt.ive2mcaansstmhe⇤nfobretmheapexpceidtattoiona:tEimc(ek-ikn)d=epEenc0de+ntES||kc=hrd(ni2n~mgke⇤z)r2e+qu(ant2~imok⇤kn)2b,ywrheeprleacmin⇤g⌘th~e22Ene2clk0ez2ct=rikc~nfi22ecklzd.amplitude
with the probability wavefunction � of a hypothetical quasi-particle with an e↵ective mass m in the presence of a
⇤
potential V(x), thus:
~2 @2�(x)
� +[V(x) E ] �(x)=0 (3)
2m · @x2 � c ·
⇤
Replacing the e↵ective mass for an optical cavity mode with longitudinal wavenumber k , we obtain
z
@2�(x) n2k
z
+ [E V(x)] �(x)=0 (4)
@x2 ~c · c� ·
Comparing Eqs. 2 and 4 gives
V(x)=~c kz(x) =Ec0(x) (5)
·
Therefore, E (x) and E correspond to the potential V(x) and the kinetic energy of a quasi-particle of mass m .
c0 ⇤
Considering E (x) k~c , where L is the e↵ective cavity length, we can relate the potential to the spatially
c0 / n(x)Lc c
modulated refractive index �n(x) as follows: V(x)=V [1+ �V(x)]= Ec0 E (1 �n(x)), where n¯ and V¯
n¯ · V 1+�n(x)/n¯ ⇡ c0 � n¯
are the refractive index and potential at x=0. Thus, the spatial modulation of the potential (�V(x)) and refractive
index (�n(x)) have a one-to-one correspondence: �V(x) �n(x).
V¯ ⇡� n¯
S.4. THEORETICAL MODELING
As discussed in section S.3, the spatial modulation of the refractive index can be considered a spatially dependent
e↵ective potential for a quasiparticle with an e↵ective mass associated with the bare cavity resonance and mean
refractive index. The multiple transverse optical modes are equivalent to harmonic states in harmonic confinement
(potential). In this section, we use a set of rate equations that consider the formation of macroscopic harmonic states
via the stimulation of a fictitious quasiparticle (correlated e-h pairs [cehps]) in the presence of an e↵ective harmonic
potential. We consider the temporal evolution of the reservoir distribution N (x,t) and quantized sates n (t). The
R i
e-h pairs photoexcited nonresonantly by a 2 ps pulse laser cool down rapidly (<5 ps) to the band edge. The cooled
carriers in the reservoir N (x,t) are subject to nonradiative loss (� ) and thus result in a slow decrease ( 0.1
R nr
⇠
meV/ps) in the chemical potential µ of the reservoir carriers. In this model, the bare cavity resonance E is fixed.
0 c
By contrast, the e↵ective cavity resonance E (t) blueshifts with the increasing photoexcited density (Fig. S2) and
c0
redshifts with time as a result of the temporal decay of reservoir carriers (Fig. S5). Standing waves (macroscopic
coherent states of cehps) are formed in an optically induced harmonic confinement (potential V(x) x2) associated
/
with photomodulated refractive index changes initiated by a double-humped pump beam profile. Such a harmonic
potentialisquasi-stationarywithin 100psafterthepulseexcitationasaresultof nsdecayofthereservoircarriers.
⇠ ⇠
5
A fraction of the reservoir carriers [N (x,t) = �N (x,t)] near the e↵ective cavity resonance E are considered as
eh R c0
cehps that can be scattered into the harmonic states via a spontaneous or stimulated process. Those cehps with
energy nearly in resonance with a specific E state can be spontaneously scattered into the E state at a rate W .
i i s
Above the threshold, thepopulationofE statesincreases nonlinearlyasa resultofstimulation ( W N n ). The
i ss eh i
/
consequent leakage photons from the decay of these confined cehps at a rate associated with the cavity photon decay
rate � are then measured experimentally. The dynamics of the quantum oscillator of cehps can then be described by
c
the following set of coupled rate equations:
3
dN (x,t)
R
=G (x,t)P � N (x,t) ⌘ ⇣ N (x,t)[W n (t)+W ], (6)
p nr R i i eh ss i s
dt � �
i=1
X
dn (t)
i
= dx⌘ ⇣ N (x,t)[W n (t)+W ] � n (t). (7)
i i eh ss i s c i
dt �
Z
The spatial generation rate G(x,t) follows the 2 ps Gaussian temporal and double-hump-shaped beam profile
(Fig. S5c and Fig. S6c). P is the pump flux. ⇣ is a scaling factor associated with � for a specific E state. ⌘ =
i i i
exp (E E )2/(2 �E2) is the coupling e�ciency of the reservoir cehps to the E state, where �E is the spectral
� c0 � i i i i
linewidth of the E state.
i
⇥ ⇤
The fitting parameters are as follows: � = 0.0125, � = 1/700 ps 1,� = 0.5 ps 1, W = 1.0 10 4 ps 1, W
nr � c � s � � ss
⇥
= 1.0 10 2 ps 1, ⇣ = 0.55, ⇣ = 0.67, ⇣ = 0.80, �E =�E = 1 meV, E = 1.401 eV, E = 1.403 eV, and E =
� � 1 2 3 i 1 2 3
⇥
1.405eV.Inthissimulation,weneglectthespindegreeoffreedomandonlyconsiderthedensityregimeabove0.8Nth
R
and E (x,t) E < 15 meV (Nth 500, corresponding to the occupation number at P = P ). We approximate
| c0 � i| R ⇠ th
E (x,t)= E +g N (x,t), where E and g are determined by the following initial conditions: E (x=0,t=0) =
c0 c0 eh c0 c0
1.4 eV and E (x= 3µm,t=0) = 1.407 eV.
c0 ±
This model reproduces the time evolution of the quantized states with increasing photoexcited density (Fig. S6).
Belowthethreshold,thescatteringofthereservoircehpsintotheharmonicstatesoccursthroughaspontaneousdecay
process. The radiative quantum e�ciency is low (<10 4) because of the dominant non-radiative recombination loss.
�
Above the threshold, the quantum e�ciency increases by two to three orders of magnitude as a result of stimulation.
Thismodelreproducesqualitativelytherisetimesandthetime-integratedemissionfluxesofthemultiplepulselasing
from the quantized states when the pump flux is varied (Fig. S7).
We note a few shortcomings of this simplified model: (1) The model does not consider the local energy shift and
the evolution of potentials induced by the interactions of the cehps because the density-dependent E (N ,N ,n ) is
c0 R eh i
not well determined. As a result, the density-dependent energy shifts and linewidths cannot be reproduced. (2) The
model does not consider carrier di↵usion, which a↵ects dynamics at a density far above the threshold. (3) Phase and
intensity fluctuations are neglected.
