Indistinguishable single photons with flexible electronic triggering

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by  Adetunmise C. Dada

Indistinguishable single photons with flexible electronic triggering
Adetunmise C. Dada,1,∗ Ted S. Santana,1 Ralph N. E. Malein,1 Antonios Koutroumanis,1
Yong Ma,1,† Joanna M. Zajac,1 Ju Y. Lim,2,‡ Jin D. Song,2 and Brian D. Gerardot1
1Institute for Photonics and Quantum Sciences, SUPA,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
2Center for Opto-Electronic Convergence Systems,
Korea Institute of Science and Technology, Seoul, Korea
A key ingredient for quantum photonic technologies is an on-demand source of
indistinguishable single photons. State-of-the-art indistinguishable-single-photon
sources typically employ resonant excitation pulses with fixed repetition rates,
creating a string of single photons with predetermined arrival times. However, in
future applications, an independent electronic signal from a larger quantum circuit or
6 network will trigger the generation of an indistinguishable photon. Further, operating
1
the photon source up to the limit imposed by its lifetime is desirable. Here we report
0
ontheapplicationofatrueon-demandapproachinwhichwecanelectronicallytrigger
2
the precise arrival time of a single photon as well as control the excitation pulse

y duration, based on resonance fluorescence from a single InAs/GaAs quantum dot.
a We investigate in detail the effect of finite duration of an excitation π pulse on the
M degree of photon antibunching. Finally, we demonstrate that highly indistinguishable
single photons can be generated using this on-demand approach, enabling maximum

6 flexibility for future applications.

]
h
p I. INTRODUCTION Here, we apply a flexible scheme for pulsed RF which
-
t triggers the generation of highly indistinguishable single
n
Single photons remain prime candidates for realising photons such that true on-demand operation is achieved
a
u scalable schemes of quantum communication [1] and lin- via real-time electronic control. Our system uses a GHz-
q ear optical quantum computing [2, 3]. The performance bandwidth electro-optic modulator (EOM) to modulate
[ of such schemes rely critically on the indistinguishability the output of a tunable continuous-wave (CW) laser for

ofthesinglephotons[4],inparticularforkeyapplications resonant excitation of a QD emitting at ∼ 960nm. In
2
v suchasquantumrepeaters[5]andbosonsampling[6,7]. turn, the EOM is driven by a fast programmable elec-
7 Ofthevarioustypesofsinglephotonsources[8,9],semi- tronic pulse-pattern generator (PPG). Such flexibility
6 conductor quantum dot (QD) systems are particularly will greatly benefit practical applications of single pho-
6 promisingforgeneratingindistinguishablesinglephotons tons in quantum technologies.
1
because they offer a robust platform in which a single Key performance measures for an on-demand single
0
quantumsystemcanbeembeddedwithinsemiconductor photonsourceinclude: theefficiency,definedastheprob-
.
1 devices and designed into bright single- and entangled- ability to detect a photon for a given electronic trig-
0 photonsources. Theidealsingle-photonsourceforquan- ger; the purity, defined by the degree of antibunching
6
tum information processing (QIP) applications is one as quantified by the second-order correlation function at
1
which generates a pure single photon Fock state on de- zero delay; the degree of indistinguishability between in-
:
v mand, i.e., in response to an independent trigger signal dividual photons as measured, e.g., by the Hong–Ou–
Xi from a user. Pulsed resonance fluorescence (RF) has Mandel (HOM)-type two-photon interference (TPI) vis-
been identified as the optimal way to deterministically ibility [16]; and crucially, the ability to determine or
r
a generate high-quality photons with minimal dephasing. adjust, on demand, the timing and sequence of trigger
However, good quality pulsed RF systems have so far pulses.
utilised pulsed excitation generated by lasers with fixed Considerable effort has been made towards realizing
repetition rates (∼ 80MHz) [10–15]. While this type of on-demand triggering of single photon generation by di-
triggering could be said to be deterministic, it is not on- rectly driving a QD electrically. GHz-bandwidth elec-
demand since a user in this case has limited control over trical pulses (with pulse width w > 270ps) have been
the excitation pulse arrival time and duration. usedtorapidlymodulatetheQDemissioninresonantor
non-resonantexcitation[17–19]. Unfortunatelythesingle
photon purity in such hybrid schemes is less than ideal.
∗ [email protected]
SimilareffectshavebeenobservedwhenusinganEOM
† Current address: Chongqing Institute of Green and Intelligent
to generate optical trigger pulses for a single-photon
Technology, Chinese Academy of Sciences, Chongqing, China
400714 source (e.g., Ref. [20], w = 500ps) where significant
‡ Current address: Korea Photonics Technology Institute, overlap between quantum dot RF pulses results in a
Gwangju61007,Korea quasi-CW stream of RF photons. EOM-generated op-
2
FIG. 1. (a) Flexibly-triggered generation of resonance fluorescence from a quantum dot. We modulate the CW
laser output using a 20Gb/s electro-optic modulator (EOM) driven by a pulse-pattern generator (PPG) capable of custom
pulse patterns at up to a frequency of f =3.35GHz. A modulator bias controller (MBC) optoelectronic circuit maintains the
high extinction ratio of the excitation pulses at >30dB using a dual feedback system for increased dynamic range. BS: beam
splitter; PC: polarization controller; VOA: variable optical attenuator; LP: linear polarizer; SPAD: single-photon avalanche
diode. (b), (c) Time-resolved QD resonance fluorescence under 100-ps π-pulse excitation. We overlay pulsed RF
on a real-time measurement of the 100-ps excitation pulse (with spectral FWHM ∼ 5.4µeV, see Supplemental Document)
obtainedbytappingoffsomeofthepowerfromtheEOMoutput[see(a)]. Afitofasingleexponentialfunctiontotheexciton
decay yields lifetimes of TX1− = 0.79(1)ns and TX0 = 0.78(2)ns for X1− and X0 respectively. The V-type energy structure
1 1
of X0 leads to quantum beats between excited states which are directly detected here in the pulsed RF transient decay. (d)
Direct observation of Rabi oscillations in the charged exciton. A fit of the theoretical excited state population (see
Supplemental Document) to the Rabi oscillations gives a dephasing time T =(2.1±0.2)T .
2 1
tical pulses have been used for direct detection of Rabi mined by high background counts and the widths of
oscillations in QD excitons (w =2ns) [4], as well as fast the excitation pulses. We use an EOM to demonstrate
triggeringofsingle-photongenerationwithalargemulti- narrower optical-excitation-pulse widths and low back-
photon contribution in the emission due to large trigger ground counts in the on-demand single-photon emission
pulsewidths(w >300ps)[22]. EOMshavealsobeenap- from a QD, better highlighting the potential of the flex-
plied for triggered photon generation from a QD also us- ible triggering for high-quality indistinguishable single-
ing optical pulses with w ≥400ps specifically applied to photon generation. We also exploit the flexibility of our
QD spin manipulation and quantum teleportation [23]. setup for a detailed experimental study of the effect of
Efforts have also been made to synchronously modu- finite duration of excitation π pulses on the degree of
late QD photoluminescence generated using pulsed opti- photon antibunching.
calpumpingwiththegoalofwaveformshapingandtem-
poral matching [24], as well as improved single-photon
generation by filtering out multi-photon events and the II. METHODS
incoherentportionofthephotonwavepackets [25]. The
usesofEOMsformodulationofsingle-photonwavepack- A. Sample details. Our experiments were performed
ets generated in pulsed mode by non-QD sources have on self-assembled InGaAs quantum dots embedded in a
also been demonstrated [26, 27]. GaAs Schottky diode for deterministic charge-state con-
In all these works, on-demand operation and pure sin- trol. A broadband planar cavity antenna is used to en-
gle photon generation of the sources have been under- hancethephotonextractionefficiency[28]. TheQDsare
3
(b)
1.0 X1- 20 MHz (c) X1- 0.8
(a) es 0.5 )0.10 (e) X0
Pulsed SPAD 1 denc 10..00 40 MHz 2g(τ0.05 0.6 X1-
RF nci 0.5 (0)
5B0S:50 d Coi 10..00 80 MHz 0.00 (d) X0 2G 0.4
SPAD 2 se 0.5 )0.10
ali 0.0 2(τ 0.2
m 1.0 160 MHz g
r 0.05
No 0.5
0 1000 2000
0.0 0.00
-100 -50 0 50 100 -4 -2 0 2 4 Pulse width w (ps)
τ (ns)
τ (ns)
FIG. 2. Pulsed antibunching of on-demand triggered resonance-fluorescence photons. (a) Measurement setup (b)
Demonstrationofflexibletriggeringofsingle-photongenerationwithexamplesatvariousfrequencies. Allmeasurementshavea
180-sintegrationtime. (c)and(d)showzoomsintothetime-zeropeaksrevealingidealantibunchingsmearedoutbyjitterinour
detection system (FWHM∼150ps). The data points represent raw experimental data, while the solid coloured (g2(0)(cid:39)0.05)
and black (g2(0) = 0.0) lines respectively represent the results of quantum numerical simulation of the master equation (see
Section SII of the Supplemental Document for details) with and without convolution with the instrument response function
(IRF) of our detection system (FWHM∼ 150ps). The pulsed antibunching is limited by the effect of the finite width of our
excitation 100-ps pulses (the limit of our pulse generator) giving G2 (0) (cid:39) 0.1 and g2 (0) (cid:39) 0.05. (e) G2(0) as a function
exp exp
excitation pulse width under π-pulse excitation. Measurements were performed on both neutral and charged exciton states.
The solid lines are linear fits to the experimental data.

pulsed RF measurements on both the neutral exciton
5
1 4 0 (X0) and charged exciton states (X1−) of a quantum
) dot. Oursetupfortriggeringsingle-photongenerationon
z) 4 C W s a tu ra tio n c o u n ts : X 1- X 0 1 2 0 Hz demandisillustratedinFig.1(a). ForRF,weuseacross-
H
s (M PS uinlsg eled--pRh Fo tco on u cnotsu :n ts : X 1- X 1- X 0 X 0 1 0 0 ns (M peaorlapriozlaatriioznertsecahrneiqpuleacinedwhinichthoertehxocgiotnatailolynoarniedntceodllleinc--
d count rate 0 .43 1 0 tes at 1st le teeCixxo.ctniiPntaacurttmiiloosnsen-dolrafastaterioricgsopgnoheffoormtcoagonlersmenieitnchrraatohntsieco1ocn0po7.elliteWnoctCeseudWgpelpniogrepehrseatsrta[er2te0ioso]ou,nnr.waonitpth--
cte 0 .3 t ra tical excitation trigger signals using a programmable
ete 0 .2 5 oun pulse-patterngenerator(PPG)whichproduceselectronic
D 0 .1 C pulses with widths of down to w = 100ps at up to 3.35
GHz (period T (cid:39) 300ps). Notably, much faster pulses
0 .0 0 (w < 30ps) can be achieved in the future with com-
2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0
merciallyavailableelectronicpulsegenerators. ThePPG
T r ig g e r fr e q u e n c y f ( M H z )
drivesa20-Gb/sEOMwhichinturnmodulatestheout-
putofaresonantCWlasertoobtainopticalpulsesprac-
FIG. 3. Count rates as a function of trigger pulse
ticallyidenticaltothedrivingelectronicpulseswithtypi-
frequency. Rawexperimentalcountratesonthedetectorare
calextinctionratiosinexcessof30dB.Thisextinctionra-
plotted for both X0 and X1−, as well as single photon count
tio is actively maintained by a modulator bias controller
rates which are calculated from corresponding multiphoton
probabilities [G2(0)]. CW saturation counts are also shown (MBC) optoelectronic circuit through optical feedback.
for comparison. We are able to vary pulse widths and repetition rates of
the trigger pulses with high precision, and also obtain
optical pulses with user-defined bit-cycle data patterns.
D. Efficiencies. The efficiency of our microscope and
at an antinode of a fifth-order planar cavity on top of
detectors are as follows: coupling of far-field radiation
a Au layer which functions simultaneously as a cavity
into single-mode fiber: ∼ 31.4%; linear polarizer: 43%;
mirror and Schottky gate. Simulations predict a photon beam splitter surfaces: (96%)4; SPAD at λ ∼ 950nm:
extractionefficiencyof∼27%intothefirstobjectivelens
∼30%. Thecombinationgives∼3.5%. Thecombination
from this device.
gives ∼3.5%. The measured total efficiency of detecting
B. Resonance fluorescence system. We perform a single photon per trigger pulse is ∼ 0.36%. Based on
4
this, we determine the photon extraction efficiency from up to 5.5K are measured at 160MHz with a 256-ps time
the sample into the first lens to be ∼10.4%. bin size in 180-s acquisitions. Beyond ∼ 160MHz, the
pulses in the RF autocorrelation function begin to over-
lap. Thislimitisimposedbytheexcitonlifetime. Single-
III. RESULTS photoncountratesareobtainedusingtheemissionprob-
ability of more than one photon in a pulse, as obtained
from the corresponding values of G2(0). From the count
Fig. 1(a) illustrates our basic excitation and measure-
rates, we calculate the overall efficiency, i.e. probability
ment setup. In Fig. 1(b) and (c), we respectively show
todetectapuresinglephotonstatepertriggerπ-pulseto
time-resolved RF from the charged exciton state X1−
be0.36±0.01%. Basedonthecombinedefficiencyofthe
and X0 following excitation with a π pulse (w =100ps),
collection optics and detectors (∼ 3.5%, see Methods),
whichgivesexcitonlifetimesofTX1− =0.79±0.01nsand
1 we determine an extraction efficiency of 10.4±0.7% into
TX0 = 0.78±0.02ns . The V-type energy structure of
1 thefirstlenswhilethesimulatedextractionefficiencyfor
X0 leads to quantum beats between excited states (e.g.,
oursampleis∼27%fora0.68-NAobjectivelens(asused
see [29]) which are directly detected here in the pulsed
in our experiment) [28].
RF transient decay. The beat frequency corresponds to
thefine-structuresplitting(duetoelectron-holeexchange To reveal the nature of the non-ideal raw antibunch-
interaction) of δ = 3.3GHz for this QD. In Fig. 1(d), ing measured in our pulsed experiments and verify the
0
we demonstrate direct measurement of X1− Rabi oscil- truequalityofoursingle-photonsource,weperformhigh
lations using 2-ns pulses from which we extract a de- timing resolution (jitter ∼ 150ps) measurements of the
phasing time of T =1.66±0.18ns, using the lifetime of intensity autocorrelation. Figs 2 (c) and (d) show zooms
2
T =0.79±0.01ns obtained from the measured X1− de- into the small time-zero peaks for X1− and X0, which
1
cay with 100-ps pulses. This is consistent with the case both reveal characteristic central dips. At zero delay, we
of no pure dephasing where T = 2T , confirming the see clear antibunching within the small peak with a van-
2 1
absence of excitation induced dephasing effects [30, 31]. ishing raw multiphoton probability of g2(0) = 0.05. To
The first peak in the RF counts corresponds to a pulse provide further insight, we use numerical simulations of
area of π. We confirm the π-pulse area/power both us- the master equation for both X0 and X1− (at a mag-
ingthedirectmeasurementandbyconventionalmethods netic field of Bext = 0), as a V-type atomic system and
(e.g., as used in Ref. [10]). a two-level system, respectively (see Section SII of the
Antibunching and efficiency. For our main au- Supplemental Document for details). The small peaks
tocorrelation measurements, we excite the quantum dot surroundingτ =0alsomanifestinthesimulationresults,
with 100-ps π pulses. Fig. 2(a) shows a schematic of the in good agreement with the experimental data as shown
Hanbury Brown and Twiss (HBT)-type set-up used in in Figs 2 (c) and (d). The underlying mechanism of this
our antibunching measurements. In what follows, while non-ideality is a small probability to re-excite the sys-
we will use g2(τ) to represent the autocorrelation func- tem (following a first photon emission event) within the
tion of the continuous time delay τ, G2(τ ) denotes the pulse duration. The re-excitation probability increases
n
pulsed-mode autocorrelation function of the discretised with pulse width, as confirmed in the simulation and ex-
timedelayτ =nT obtainedbyintegratingthenth pulse perimentally [see Fig. 2(e)]. Importantly, the g2(0) is
n
ing2(τ),whereT =1/f isthepulseperiod. InFig.2(b), always zero at the middle of the time-zero peak, indi-
we demonstrate antibunching at various trigger frequen- cating that arbitrarily low g2(0) values can be achieved
cies as seen in the intensity-correlation histograms for with shorter excitation pulses. Taking the IRF of our
the RF emission from the QD under pulsed excitation. detection system into account, we estimate perfect an-
Pulsed second-order correlation at zero delay G2(0) are tibunching (g2(0) = 0.0). We conclude that, although
calculatedbyintegratingphotoncountsinthezero-time- more than one photon may be emitted during the 100-
delay peak and dividing by the average of the adjacent ps-long excitation pulse with a small probability, these
peaksoverarangeof∼650nsaroundthetime-zeropeak, photons are not emitted at the same time.
withstandarddeviationsobtainedfrompropagatedPois- Pulsed two-photon interference. For TPI mea-
soniancountingstatisticsoftherawcounts. With100-ps surements, we send the QD photons into a HOM-type
π pulses,weobtainrawexperimentalvaluesG2(0)∼0.1, setup [see Fig. 4(a)] which consists of an unbalanced
and g2(0) ∼ 0.05, as shown in Figs 2 (c-e). An increase Mach-Zender (MZ) interferometer with delay of ∆t =
inpulsewidthsleadstoworsepulsedantibunchingG2(0) 49.70ns and polarization control in each arm to enable
[Fig. 2(e)], while g2(0) values are unaffected. measurements with parallel ((cid:107)) and orthogonal (⊥) po-
We demonstrate the flexibility of the system and how larizations of interfering photons. The beamsplitters in
itmaybeexploitedto,e.g.,maximizesingle-photonrates the MZ setup have nearly perfect 50:50 splitting ratios.
by performing autocorrelation measurements at varying We filter out the zero-phonon line from the most of the
repetition rates of 20MHz to 160MHz and detect up to phonon sideband using a grating-based spectral filter
∼ 0.45MHz of single photon counts (see Fig. 3). Also (bandwidth∆f =12GHzandefficiencyη =22%). Due
f
shownarethedetectedcountsratesatsaturationinCW to the flexibility of trigger pulse generation, we are able
mode for each charge state. Peak coindicence counts of to precisely match the repetition period of the pulses to
5
FIG. 4. Demonstration of indistinguishablility of single photons triggered on demand. (a) Hong-Ou-Mandel
(HOM)-type two-photon interference (TPI) results. The flexibility of our approach allows us to set the pulse period to match
the delay in our HOM setup (∆t = 49.7ns). (b) TPI visibility versus period. The measurements were perfomed on a
neutral exciton line for X0 using various pulse periods and hence delays between interfering photons with π-pulse excitation.
TPI autocorrelation at zero relative delay. (c) shows results for X0 photons and (d) for the charged exciton (X1−)
both at B = 0T. Measurements were performed using 100-ps-wide excitation pulses. The measurements plotted in grey
ext
are with orthogonal polarizations of interfering photons. (c) and (d) are measured with most of the phonon band filtered out
using a grating-based spectral filter. The X0 photons show TPI visibiliities of v = 0.76±0.06 as raw experimental data and
v=0.96±0.07 when corrected only for multiphoton emission (G2 corrected).
∆t (see Fig. 4(b)) to obtain pulsed autocorrelation at a
1E+00 T = 250 ps
relativedelayT−∆t=0,showninFigs4(c)and(d). The 1
TPI visibilitiy is defined as v =[G2(0)−G2(0)]/G2(0). T = 800 ps
⊥ (cid:107) ⊥ 1
For the X0 and X1−, we measure raw visibilities of 0)
v =0.76±0.06 and 0.28±0.03 respectively. The raw in- 2G( 1E-01
distinguishability of the X0 photons is limited primarily d
by the multiphoton probability of G2(0) = 0.10±0.01, e
at
which is in turn limited by the excitation pulse width ul
as described above. When this is corrected for (by us- m 1E-02
ing G(cid:48)2(0) = G2(0)−G2(0)), we obtain a TPI visibil- Si
(cid:107) (cid:107)
ities of v = 0.96 ± 0.06 and 0.47 ± 0.03 respectively
for X0 and X1− without accounting for any other ex-
1E-03
perimental imperfections. The reduced visibility of the 1 10 100 1000
X1− (Bext = 0T) is understood to be due to detuned Trigger pulse width w (ps)
Raman-scatteredphotonswhicharedistinguishablefrom
both the elastic and incoherent components of the reso- FIG. 5. Simulated G2(0) as a function of excitation
nance fluorescence due to nuclear spin fluctuations (fur- pulse width under 0.81π-pulse excitation. Simulation
therdetailsareprovidedinRef.[32]). Westressthatthe of a two-level system with lifetimes T = 800ps and 250ps
1
Raman-scattered photons result in a total linewidth of using a Gaussian (temporal) 0.81π-excitation pulse profiles
less than 1GHz which is not filtered out by the 12-GHz- withvaryingwidths. Weuse0.81πforthesimulatedGaussian
bandwidth spectral filter. pulsesbecausewitha100-pswidth,theygivethesameG2(0)
as the asymmetric 100-ps π pulses used in the experiment.
6
IV. DISCUSSION durationsislikelytobeattractive. Anotheradvantageof
this approach is the possibility to specifically tailor the
For on-demand single photon sources to underpin pumppulsesforquantumcontrolprocessessuchasstim-
scalable and efficient linear-optical quantum comput- ulated Raman adiabatic passage [40] in quantum dots
ing and networking, stringent criteria must be satis- exhibiting spin-Lambda systems [41, 42]. Finally, we
fied [33, 34]. Our experimental results provide insight note that the flexible technique presented here enables
into the prospect of realizing the G2(0) requirements us- an excitation repetition rate up to the limit of that im-
ing resonance-fluorescence-generated single photons. A posed by T1, offering a significant boost in count rates
crucial result is the effect of the pulse width relative to for real applications. While overall system efficiencies
T on G2(0). Typically, Purcell enhancement is consid- need to be improved to realize an ideal single-photon
1
ered desirable to reduce the impact of dephasing mecha- source, recent developments in QIP protocols have made
nisms[35–37]andenableincreasedclockrates. However, efficiency requirements considerably less stringent (e.g.,
inpulsedRFafasterT alsoincreasestheprobabilityfor in [33] efficient linear optical quantum computation is
1
re-excitation given a certain excitation pulse width. We possible with an overall efficiency of 2/3) even as high
illustrate this trade-off using a numerical simulation for quality indistiguishablility, antibunching, and brightness
G2(0) as a function of pulse width (Gaussian profile) for arenowsimultaneouslybeingachieved(e.g.,see[13–15]).
T = 250ps and 800ps (Fig. 5). We see that in both The approach we demonstrate here is an important step
1
cases vanishing G2(0) can be obtained for ultra-short towards combining these key performance features with
pulse widths, but practically the minimization of G2(0) true on-demand operation.
is best achieved with larger T values. This is impor-
1 Funding and Acknowledgements. The authors
tant for prospective applications (such as linear-optical
would like acknowledge the financial support for
QIP) of single photons generated using pulsed resonance
this work from the Engineering and Physical Sci-
fluorescence.
ences Research Council (EPSRC) (EP/G03673X/1,
We have demonstrated flexible electronic triggering of
EP/I023186/1, EP/K015338/1) and the European Re-
on-demand single indistinguishable photons. This sys-
searchCouncil(ERC)(307392). B.D.Gacknowledgesthe
tem offers several intriguing advantages for future appli-
Royal Society for support via a University Research Fel-
cations. Whereas ultra-short excitation pulses lead to
lowship. The KIST authors acknowledge support from
excitation induced dephasing (EID) [31], coherent con-
KIST’s flagship program and GRL.
trol with longer pulse durations is expected to minimize
EID[38,39]. Hence, forsomecoherentcontrolandread-
out schemes the flexibility of electronically tunable pulse
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8
Supplementary Information for:
“Indistinguishable single photons with flexible electronic triggering”
SI. NUMERICAL SIMULATION FOR PULSED RESONANCE FLUORESCENCE
Ournumericalsimulationisbasedonthemasterequationmethod(e.g.,see[1–3]). WeusethewellknownHamilto-
nians describing a driven (1) two-level atom for the X1− and (2) V-type three level atom for X0 state of the InGaAs
quantum dot (see Fig. S1).
X0 X1-
|+⟩
|↑↓⇑⟩ |↑↓⇓⟩ ≡|e⟩
δ
|−⟩ 0
ω
ω
|0⟩ |↑⟩ |↓⟩ ≡|g⟩
FIG. S1. Level diagrams for X0 and X1− at an external magnetic field of B = 0. The X1− is modelled as a
ext
two-level system (due to the degeneracy of the excited and ground state levels), and the X0 as a V-type three-level atomic
system. δ is the electron-hole exchange interaction energy, the red arrows represent the driving field and ω its frequency.
0
The Hamiltonian for the two-level system is
∆ Ω
Hˆ = (|e(cid:105)(cid:104)e|−|g(cid:105)(cid:104)g|)+ (|g(cid:105)(cid:104)e|+|e(cid:105)(cid:104)g|), (S1)
X1− 2 2
while for X0, we have
∆ ∆
Hˆ = (|−(cid:105)(cid:104)−|−|0(cid:105)(cid:104)0|)+( +δ )|+(cid:105)(cid:104)+|+
X0 2 2 0
Ω
[cosθ(|−(cid:105)(cid:104)0|+|0(cid:105)(cid:104)−|)+sinθ(|+(cid:105)(cid:104)0|+|0(cid:105)(cid:104)+|)], (S2)
2
under the rotating-wave approximation. ∆ is the detuning between the driving field and the transition from |0(cid:105)
(|g(cid:105)) to |−(cid:105) (|e(cid:105)), δ the electron-hole interaction energy (corresponding to a fine-structure splitting of 13µeV for the
0
QD reported in Figs 1-4 of the manuscript). Ω represents the Rabi frequency, θ is a constant determined by the
polarization angle of the (linearly polarized) driving field.
Spontaneous decay and dephasing are included in the time evolution of the system via the Lindblad terms of the
masterequation. TheLindbladoperatoractingonthedensitymatrixρ(foragivencollapseoperatorCˆ)isdefinedas
1
L(Cˆ)ρ=CˆρCˆ†− (Cˆ†Cˆρ+ρCˆ†Cˆ) (S3)
2
The master equation can then be written as
ρ˙ =−i[ρ,Hˆ]+(cid:88)Γ L(σ )ρ. (S4)
(cid:126) ij ij
ij
Here, σ = |i(cid:105)(cid:104)j|, Γ is the decay rate from state |j(cid:105) to |i(cid:105) and Γ represents the pure dephasing rate of state |i(cid:105),
ij ij ii
where i,j = e,g,0,+,−. For example, Γ = 1/T , Γ = Γ = 1/T −1/2T = 0 for T = 2T where T is the
eg 1 ee gg 2 1 2 1 1
exciton lifetime, and 1/T is the total dephasing rate.
2
UsingtheRunge-Kuttafourth-ordermethod,weobtainnumericalsolutionsofthemasterequationandcalculatethe
autocorrelation function from the obtained density matrix elements. We simulate pulsed excitation by incorporating
the corresponding temporal pulse intensity profile and pattern into the driving field Ω(t) as I(t)∝Ω(t)2.
While we use Gaussian temporal pulse profiles to obtain the simulation results in the trend shown in Fig. 5 of the
primarymanuscript,fortheresultsshowninFig.2(c)and(d),weusea∼100-ps-widelognormaltemporalprofilewhich
is obtained by fitting to the intensity profile measured from the EOM output. This closely matches the (asymmetric)
intensity profile of our excitation pulses.
9
SII. MODULATION SETUP
Weuse az-cut electro-optic modulator having adual bias-port (forcoarse-and fine-tuning)with V =2Vto cause
π
the phase shift required to change from minimum to maximum transmission. The EOM is driven by a PPG with a
12.5 Gb/s pulse amplifier at the output to ensure full modulation depth. As mentioned in the main manuscript, the
high modulation extinction ratio is achieved and actively maintained by a modulator bias controller which uses two
photodetectors, with different sensitivities. The lower(higher)-sensitivity detector is used in the coarse(fine) tuning
of the modulator bias. Two 95:5 beam splitters are used to tap off light from the EOM output for coupling to the
respective photo-detectors.
SIII. SPECTRAL PROFILE OF EXCITATION PULSES
Wecharacterisethespectralprofileofthe100-pspulsesfromtheoutputoftheelectro-opticmodulator(EOM)used
in the experiment with a Fabry-Perot scanning interferometer having 27-MHz resolution and a free spectral range of
5.5GHz. There is a broadening of the modulated light in the spectral domain as compared with the continuous-wave
(CW) light, with longer pulses exhibiting narrower spectral widths as expected. Fig. S2 compares the measured
frequency-domain profile of the 100-ps pulses used for the two-photon interference and antibunching measurements
reported in the manuscript with those of 1-ns and 4-ns pulses.
w = 100 ps, FWHM: 1.31 GHz
1.2
w = 1 ns, FWHM: 0.68 GHz
w = 4 ns, FWHM: 0.23 GHz
1.0
y
sit
n
e0.8
nt
d i
ze0.6
ali
m
or0.4
N
0.2
0.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Frequency detuning (GHz)
FIG.S2. Highresolutionspectrumofexcitationpulsesatdifferentpulsewidthsw. Thepointsandlinesrespectively
represent experimental data and fits of Gaussian functions to the data, from which we obtain a full-width-at-half-maximum
(FWHM) of 1.3±0.1 GHz (5.4±0.4µeV) for the 100-ps pulses.
SIV. RF SPECTRUM/FILTERING IN TPI MEASUREMENTS
As mentioned in the main manuscript, the RF photons were spectrally filtered using a 12-GHz-bandwidth grating-
based filter for the TPI experiment. We show typical spectral data for a 100-ps pulsed RF from a quantum dot in
our sample in Fig. S3 to compare the unfiltered RF spectral widths with the filter bandwidth. We show the pulsed
RF spectrum on a Log scale to highlight the phonon sideband. In this QD the phonon sideband to zero-phonon line
ratio is ∼1:9.
SV. FITTING FUNCTION FOR RABI-OSCILLATIONS MEASUREMENT
The function fitted to the Rabi oscillation data describing the probability of being in the excited state is obtained
by solving the density-matrix equations for a resonant driving field, including the effects of decoherence due to
10
FIG. S3. High-power spectrum of RF photons under 100-ps pulsed excitation . Typical spectral data for a 100-ps
pulsed RF from a single quantum dot (λ ∼ 960nm) in our sample with no filtering plotted on log scale to show the phonon
sideband and showing the bandwidth of the filter used in the Hong-Ou-Mandel-type TPI measurements. This spectral data
was acquired using a 35-µeV resolution grating spectrometer. The linewidth of the zero-phonon line shown here is limited by
spectrometer resolution. We note that >87% of the total signal spectra lies within the pass band of the filter.
FIG. S4. X1− Rabi oscillations measurement using (a) 2-ns at ∼ 188.5 nw peak power and (b) 5-ns excitation pulses
at ∼ 55.5nw peak power. The red line represents a fit of the theoretical excited state population to the Rabi oscillations
when the shape of the excitation pulse is taken into account, i.e., using Eq. S7 in Eq. S5 for the fit gives a dephasing time
T =(2.02±0.27)T .
2 1
spontaneous emission and pure dephasing, which is a well known result (e.g., it is the same result used to fit Rabi-
oscillation data as reported in Ref. [4]), i.e.,
Ω2/2
ρ (t)=
22 Ω2+1/(T T )
1 2
(cid:26) (cid:104) (cid:105) 1 (cid:27)
× 1− cos(ξt)+ 1/T1+1/T2 sin(ξt) e−2(1/T1+1/T2)t . (S5)
2ξ