Two-Color Photon Correlations of the Light Scattered by a Quantum Dot

3.6 MB·English

by  M. Peiris

Two-Color Photon Correlations of the Light Scattered by a Quantum Dot
M. Peiris,1 B. Petrak,1 K. Konthasinghe,1 Y. Yu,2,3 Z. C. Niu,2,3 and A. Muller1,∗
1Physics Department, University of South Florida, Tampa, Florida
2State Key Laboratory of Superlattices and Microstructures,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China
3Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, China
(Dated: January 6, 2015)
Two-colorsecond-ordercorrelationsofthelightscatterednear-resonantlybyaquantumdotwere
measured by means of spectrally-filtered coincidence detection. The effects of filter frequency and
bandwidthwerestudiedundermonochromaticlaserexcitation,andacompletetwo-photonspectrum
was reconstructed. In contrast to the ordinary one-photon spectrum, the two-photon spectrum
is asymmetric with laser detuning and exhibits a rich structure associated with both real and
5 virtualtwo-photontransitionsdownthe“dressedstates”ladder. Photonpairsgeneratedviavirtual
1 transitionsarefoundtoviolatetheCauchy-Schwartzinequalitybyafactorof60. Ourexperiments
0 arewelldescribedbythetheoreticalexpressionsobtainedbydelValleet al. viatime-andnormally-
2 ordered correlation functions.

n
PACSnumbers: 78.67.Hc,78.47.–p,78.55.Cr
a
J

5
Quantum particles generated in pairs are an essential tecting a photon of frequency ω (obtained experimen-

resource in quantum information science and a unique tally by recording the transmission of light through a
]
h testbed for the investigation of quantum mechanical frequency-tunablefilter),theTPSmeasurestheprobabil-
p paradoxes[1]. Therearecurrentlytwomajorapproaches ityofdetectingonephotonoffrequencyω attimeT and
1 1
-
for generating pairs of photons which are correlated another of frequency ω at time T . Experimental mea-
t 2 2
n strongly enough to violate classical inequalities such as surement of the TPS requires frequency-resolved coinci-
a
theCauchy-SchwartzorBell’sinequalities[2,3]. Onere- dence detection, such as is obtained by placing tunable
u
lies on nonlinear parametric processes like spontaneous filters in front of each detector of a Hanbury-Brown and
q
[ down-conversion [4] or four-wave mixing [5]. The other Twiss (HBT) setup and histogramming photon arrival

uses multilevel atomic cascades, as in the pioneering ex- times [24]. The theoretical framework for the TPS cal-
1
periments of Aspect et al. [6], or more recently, in biex- culation rigorously yields the frequency and bandwidth
v
citonic decays of quantum dots (QDs) [7–9]. dependent two-color correlations which are required to
8
9 In light of the complexities associated with obtaining determine whether or not classical inequalities are vio-
8 pairedphotonsinamultilevelsystem,itisnaturaltoask lated [10].
0 under which conditions a two-level system may generate
We report here the measurement of the TPS of the
0 photon pairs correlated strongly enough to violate clas-
light near-resonantly scattered by a QD exposed to a
.
1 sical inequalities [10]. It is well-known that spectrally strong monochromatic laser. The 2D TPS maps reveal
0 filtering the light scattered by strongly driven two-level
intricate and unexpected features of two-photon cascade
5 atoms yields correlated pair emission [11–21]. In QDs
emission such as transitions proceeding via virtual inter-
1 this cascaded emission has been investigated recently
: mediate states showing particularly strong correlations,
v by using a Michelson interferometer to separate differ- significantly violating the Cauchy-Schwartz inequality.
i ent components of the spectral Mollow triplet [22]. In
X Furthermore, we evidence the asymmetric nature of the
this way the sequential emission of photons in pairs em-
TPS under laser detuning as well as the effect of filter
r anating from the two Mollow triplet sidebands has been
a bandwidth on the correlations at a level at which Rabi
demonstrated.
oscillations can be resolved. The TPS measurement pro-
In general, scattering of photon pairs from a strongly
videsnewopportunitiesforthecharacterizationofquan-
driven two-level system may occur via numerous path-
tumopticpathwayswhichcouldhelpimproveourunder-
ways, and any photon with one color and emission time
standing of a variety of systems [25].
may be correlated, to some degree, to another photon
with possibly different color and emission time. For We used InAs QDs grown by molecular beam epitaxy
the purpose of describing such correlations, del Valle et (see Ref. [26] for sample details) in a cryogenic or-
al. have introduced a “two-photon spectrum” (TPS) as thogonal excitation/detection setup, at a cryostat base
an extension of the ordinary one-photon spectrum [23]. temperature of 5 K [Fig. 1(a)]. With this geometry
While the latter simply measures the probability of de- the light scattered by a single QD exposed to a tun-
able continuous-wave laser is collected efficiently, free of
unwantedbackgroundlaserscattering[27]. Weareinter-
estedinthesituationinwhichtwofilters,tunablebothin
∗ [email protected] theirresonancefrequencies,ω andω ,aswellasintheir
1 2
2
(d)
(a) Fiber Correlation
Ω Ω Ω
Electronics 2π =1.0 GHz 2π =1.6 GHz 2π =2.2 GHz
Detectors
Exp.
Filter 1
(ω1,Γ1) Filter 2 Hz)-2 ω,0)2
(ω,Γ) ωπ )/2(GL0 ω(2) g(,1Γ,Γ
2 2 ω-2 3.8
(
Cryostat Mirrors 2
≈5K QD
Theo. Dvi
z)-2 Ai Ci Aiii 1
GaAs ELaxcseitration ωωπ (-)/2(GH2L0 1 0.15.5 1 DviiiACiv Dii DiBv DDiiiiCiii Dv 1 1.5 1.5
2 iv C A 0.3
nits) (b) (c) Dvii ii ii 0.5
u Ω
b. 2π= -2 0 2 -2 0 2 -2 0 2
y (ar 1.3 GHz ×8 (e) (ω1-ωL)/2π (GHz) (ω1-ωL)/2π (GHz) (ω1-ωL)/2π (GHz)
nsit 3 Ω (-3,3) (f) Ai Aii Aiii Aiv B Ci Cii Ciii Civ Di DiiDiiiDiv Dv DviDviiDviii
ght inte ωωτ,,)12 2π =2.2 GHz 12
ered li ×8 (2) g(Γ,Γ1 (0,0) 12
catt (-2.2,-2.2)
S -1.5
-1.5 0 1.5 -1.5 0 1.5 -10 0 10 1
Relative Frequency (GHz) Time τ (ns) 2
FIG. 1. (a) Experimental setup. (b) Unfiltered Mollow triplet for Ω/2π=1.3 GHz. (c) From top to bottom: isolated central
peak, blue and red sidebands, respectively (Γ/2π=0.5 GHz). (d) Experimental and theoretical TPS for the Rabi frequencies
indicated, with filter bandwidths of Γ /2π = Γ /2π=0.5 GHz. (e) Experimental correlations for Ω/2π=2.2 GHz labeled by
1 2
(ω1−ωL,ω2−ωL) in units of GHz. (f) Dressed-state diagram illustrating the transitions labelled in the middle theory panel of
2π 2π
part (d).
bandwidths, Γ and Γ , are placed before the detectors will be denoted by ω . We define the laser detuning as
1 2 L
of a HBT setup [28] receiving the scattered light. Figure δ = ω −ω . Figure 1(d) shows the results of the TPS
L 0
1(b) displays the scattered light (one-photon) spectrum measurement under coincidence detection, i.e., for a cor-
for a single QD obtained using a high resolution (≈ 20 relation time τ = T − T = 0, and with δ = 0 and
2 1
MHz)scanningFabry-PerotinterferometerataRabifre- Γ/2π=0.5 GHz. Only the Rabi frequency was changed
quencyofΩ/2π=1.3GHz. Thefunctionalityofthefilters between each panel (increasing from left to right). The
isillustratedinFig. 1(c). Eachfiltercouldbetunedcon- theoretical one-photon spectrum is plotted above the
tinuously; in particular it could select the Mollow triplet TPSforeaseofcomparisonandidentificationofspectral
red or blue sidebands, its central peak, as shown, or any features. Figure 1(e) shows several sample raw correla-
other frequency window. The filters’ long-term stability tion functions from which the images were constructed.
has been verified in separate measurements. Further details are provided in the form of supplemental
material.
In order to record the TPS associated with the QD
scattered light, photon arrival times were histogrammed For a qualitative understanding we treat the
for a matrix of filter frequencies (ω ,ω ), using a fixed resonantly-driven QD as an ideal two-level system with
1 2
filter bandwidth Γ = Γ = Γ. The QD resonance fre- ground state |g(cid:105), excited state |e(cid:105), and radiative de-
1 2
quency will be denoted by ω while the laser frequency cay rate κ. By diagonalizing the Hamiltonian of the
0
3
coupled system (QD+laser field), the “dressed states”, ments.
|1(cid:105)=c|g(cid:105)−s|e(cid:105)and|2(cid:105)=s|g(cid:105)+c|e(cid:105),areobtainedforeach Ascanbeseen,allmajorfeaturesarepredictedwellby
rung of the dressed states “ladder” [11, 19]. The ampli- Eq. (1). Theoreticalcontourshavebeenoverlaidontothe
(cid:112)
tudes of the eigenstates are given by c= (Ω(cid:48)+δ)/2Ω(cid:48) experimental measurements for highlighting purposes.
and s = (cid:112)(Ω(cid:48)−δ)/2Ω(cid:48), with Ω(cid:48)2 = Ω2 +δ2 [19]. The Crucially,thevirtualtransitionsDi-Dviii ofFig. 1(f)can
dressed states picture is ideally suited for visualizing the be reproduced. These have been first identified through
two-photon cascade [Fig. 1(f)]. In this picture, the low- the inspection of the Jaynes-Cummings model in Ref.
est coincidence rate, i.e., highest degree of photon anti [25], where they were termed “leapfrog transitions”.
bunching,ispredictedtooccurnear(ω −ω ,ω −ω )= The detuning provides another interesting control pa-
1 L 2 L
(±Ω,±Ω) [labelled A and A ], due to disconnected de-
i ii
caypaths. Thesecorrespondtofilteringoflikesidebands.
Ontheotherhand,ahighcoincidencerateisexpectedto (a) (c)
occurnear(ω1−ωL,ω2−ωL)=(±Ω,∓Ω)[Aiii andAiv]. 2Ωπ = 1.6 GHz 2δπ= 1.0 GHz
These are associated with the cascaded alternating side- 1
band emission [19]. Other notable pathways are those τ)
atwosistyohiceifiladltteeudrnincwogirtrthehlefialtcteeedrnitsnrtgaalttihlsitenieccsean(nBtdr)a,olanlniendeo,tfhwtohhseiechsaisdisneobtceairanftedersde, -2 0) ωω(2) g(,,on LLΓ,Γ 0 Unfiltered
w[1h2i]c.h interfere to yield partial anti bunching (Ci-Civ) GHz) ωω,,12 uncti 1
[(1a8tT,th1he9e,9f2e2ian]t,tuearrnessdecadtireoesnccsrlieobaferdtlhyeasbdeeoanvsheiendatrghereidwTleiPnllSesdo)o.fcFHuiomgw.een1vt(eedrd), ωωπ (-)/2(2L02 (2) g(Γ,Γ8.8 er correlation f 10 Γ1./82 Gπ H=z
theydonotformadominantpattern. Rather,theregions d
with the highest coincidence rates are generally located d-or Γ/2π =
along anti-diagonal lines defined by ω1+ω2−2ωL =±Ω econ 0 1.0 GHz
banudncωhi1n+g)ωo2bs−er2vωedL o=n 0th,ewseithlinsetsroantgpmoinaxtsimnaea(rpwhohtiochn Hz)-2 1 alized s 1
G m Γ/2π =
piω.he1.,o2,tto−hnrωoduLegch≈ayasn−, t3inhΩte/erc2ma,s−ecdΩaid/ae2te,pΩsrto/a2cte,ee3dwΩsh/v2ic.ihaiaFsovnriorttthuaeansleesittgawetnoe--, ωωπ -)/2(2L0 Nor 10 0.5 GHz
state of the system. Figure 1(f) depicts several of these (
2 Γ/2π =
transitions (Di-Dviii). 0.4 0.05 GHz
For the purpose of quantitatively describing the
-2 0 2 -5 0 5
TPS measurements of Fig. 1(d), del Valle et al.
(ω-ω)/2π (GHz) Time τ (ns)
have introduced the quantity S(2) (ω ,T ,ω ,T ) = (b) 1 L
(cid:82)(Γ21π(cid:82)Γ−)T22∞2(cid:82)d(cid:82)t−T(cid:48)2∞d1t(cid:48)3det−(cid:48)1dΓ2t2(cid:48)4(eT−2−Γ2t1(cid:48)2()Te1−−Γt2(cid:48)12)(eT−2−Γ2t1(cid:48)3Γ()T1eΓ1i−2ω2t(cid:48)4()t1(cid:48)3e−iωt1(cid:48)21()t×(cid:48)4−2t(cid:48)1)2× Ωτ+,) 180 τ(2)g()15 2δπ= 2 Ω1π.0 =1 .06. 5GHz 0 -0.5 -1.5 GHz
(cid:104)T−[a†(t(cid:48)1)a†(t(cid:48)2)]T+[a(t(cid:48)3)a(t(cid:48)4)](cid:105), where T− and T+ Ωω,L 6 0 Ti1m0e τ (ns)20 30
are time-ordering operators [23], and a† and a are the ω-L
photon creation and annihilation operators, respectively. (2) g(Γ,Γ 4 -0.45 - 1.5G0Hz
After normalizing by the one-photon time-dependent 0
2 0.70
power spectra SΓ(11)(ω1,T1) and SΓ(12)(ω2,T2), the time 2δπ = 1.65
and frequency resolved physical TPS is given by
0 40 80 120
Time τ (ns)
S(2) (ω ,T ,ω ,T ) (cid:12)(cid:12)
g(2) (ω ,ω ,τ)= Γ1,Γ2 1 1 2 2 (cid:12) . FIG.2. (a)ExperimentalandtheoreticalTPSunderdetun-
Γ1,Γ2 1 2 SΓ(11)(ω1,T1)SΓ(12)(ω2,T2)(cid:12)(cid:12)T2−T1=τ i(nbglaδc/k2)πan=d1t.h0eGorHetzi,cwalit(hreΩd)/2pπho=to1n.6cGorHrezl.at(ibo)nsEbxpetewriemenenrteadl
(1)
and blue sidebands (Ω/2π = 1.6 GHz) for different laser de-
In the supplemental material of Ref. [23] it is shown
tuning (δ/2π= 1.65, 1.40, 0.85, 0.70, 0.30, 0, -0.15, -0.45,
howthisquantitycanbecalculatedinageneralcontext.
-0.75,-1.20and-1.50GHz),offsetforclaritywiththeshaded
Here we have evaluated it numerically for the case of a
region indicating the zero. Inset: corresponding photon cor-
radiatively-broadened two-level system (radiative decay relation measurements on both Mollow triplet sidebands for
rate of κ/2π=0.2 GHz). It is represented below the ex- a range of detunings as indicated. (c) Experimental (black)
perimentalTPSinFig. 1(d). Aconvolutionwiththede- and theoretical (red) photon correlations on the central Mol-
tectors’ instrument response (Γ−1=350 ps) function has lowtripletpeakforarangeoffilterbandwidths(Γ =Γ =Γ)
det 1 2
also been applied for accurate comparison with experi- as indicated.
4
rameterfortheinvestigationoftheTPS.InFig. 2(a)the (a) (b)
TPShasbeenmeasuredinthepresenceofalaserdetun- 2Ωπ =1.0 GHz 2Ωπ= 1.6 GHz 2δπ= 1.0 GHz
ing δ/2π=1.0 GHz. Despite the moderate magnitude of
this detuning, a mirror asymmetry in the TPS is read-
ily seen relative to the central anti-diagonal, in contrast
Exp.
tothestrictsymmetryoftheone-photonspectrumunder
detuning. Again,thetheoreticalcalculationbasedonEq. Hz)-2 -2
G
(wp1lii)tTthu[hbdeeoexltspatsoceermarimndmdeentasutp.nooifnftgFhaeiglsd.or2ec(soasne)td]ropslrtsoavttheidse.essuUcpnleodrseperoaslagitrrigeoeenmpaeomnst-- ωωπ (-)/2(2L02 ωω R (,,0)12Γ,Γ 02 ωω R (,,0)12Γ,Γ
15 90
itive detuning (|c| (cid:29) |s|) the branching ratios are such
that transitioning into state |1(cid:105) is favored over transi-
tioning into state |2(cid:105). Thus in steady-state the system
is found predominantly in state |1(cid:105) and the emission of Hz)-2 1 -2 1
G
a photon at the blue sideband is likely to be followed π (
by the emission of a photon at the red sideband [Aiii ω)/2L0 0
in Fig. 1(f)]. This asymmetric time sequence is clearly ω-2 0.4 0.1
visible in the data of Fig. 2(b). Likewise, when the de- (2 2
tuningisnegativethentheredsidebandphotonemission Theo.
is likely to be followed by a blue sideband photon emis- -2 0 2 -2 0 2
(ω-ω)/2π (GHz) (ω-ω)/2π (GHz)
sion [Aiv in Fig. 1(f)]. On resonance Aiii and Aiv are 1 L 1 L
equally likely to occur and thus the resulting correlation
FIG.3. (a)MapoftheCauchy-SchwartzcriteriaR forwith
function is symmetric, with a dip at τ = 0 due to their
nolaserdetuning. (b)Sameasin(a)butwithalaserdetuning
interference[12]. Notethattheeffectofspectraldiffusion
of δ/2π=1.0 GHz.
[29] has been accounted for in the theory via an averag-
ing over a 1 GHz wide distribution of random detunings
as described in Ref. [26]. We have also verified that our
bunchingcanbeunderstoodasbeingtheresultofacon-
resultsagreewiththosereportedinRef. [22],forthecor-
structive multiphoton interference process originating in
relations observed when the Mollow triplet is stripped of
photonindistinguishability[30,31]. HadtheTPSofFigs.
its central peak. This case is shown in the inset of Fig.
1,2beenrecordedwithafilterbandwidthmuchlessthan
2(b), where one sideband was selected by each filter and
the emitter bandwidth κ, then this indistinguishability
the two signals were subsequently recombined at a beam
bunchingwouldhaveappearedasasharpcenter-diagonal
splitter before histogramming.
line [25].
Wealsoexploredtheeffectofreducedfilterbandwidth Lastlyweexaminetheconditionsunderwhichthescat-
on the two-photon correlations. Here we fixed the fre- tered photon pairs violate the Cauchy-Schwartz inequal-
quency of each filter to select the central peak from the ity. In quantum optics, for two electromagnetic modes,
Mollow triplet. The correlations were then recorded for this inequality is written in the form of the ratio R of
increasingly smaller filter bandwidths, as shown in Fig. thesquareofcross-correlationsovertheproductofauto-
2(c). Here the detector time resolution (Γ−1 ≈ 80 ps) correlations between the modes at τ = 0, i.e, using the
det
was such that Rabi oscillations are seen when no filter notation of Eq. (1) [10],
is present [uppermost trace in Fig. 2(c)]. A deconvo-
lution with the detectors’ instrument response function R (ω ,ω ,0)=
Γ1,Γ2 1 2
has been applied to the data. The red continuous traces (cid:104) (cid:105)2 (cid:104) (cid:105)
in Fig. 2(c) correspond to the theoretical correlations gΓ(21),Γ2(ω1,ω2,0) / gΓ(21),Γ1(ω1,ω1,0)gΓ(22),Γ2(ω2,ω2,0) .
[Eq. (1)]. These follow closely the experimental data (2)
as the filter bandwidth is reduced. Note that the effect
of the reduced filter bandwidth is not equivalent to a In Fig. 3 the experimentally obtained R is plotted as a
slower detector response (which would result simply in a function of the filter frequencies. The theoretical value
removal of high-frequency components of the Rabi oscil- isrepresentedinthebottompanels. Onthechosencolor
lations and a “flattening” of the correlations). Here the scale, green indicates a violation (R > 1). As can be
reduced filter bandwidth eventually leads to increased seen,significantviolationsoccuronlyforpairsemittedin
coincidence rates, an effect well-known from the quan- theMollowtripletsidebandtails, unlessalaserdetuning
tum treatment of filter transfer functions [20, 21]. In is present [Fig. 3(b)], in which case a violation is also
particular, regardless of the specific photon generation observedforfilterssetexactlyatoppositesidebands. The
mechanism, a filter with a bandwidth smaller than the largest experimental value seen is R≈60.
light’sbandwidthunavoidablyintroducesquantumnoise In conclusion, we have measured the two-photon spec-
[16], i.e., it will cause photon bunching to occur. This trum of the light resonantly scattered by a QD. Its
5
distinctive features, such as the visualization of virtual atomdressedbyacavityvacuumfield[35],aswellascou-
transitions, asymmetry under laser detuning, and filter pled quantum systems. Finally, our results also suggest
bandwidth-dependent effects have been identified. Al- that a demonstration of Bell’s inequalities, as predicted
thougharefinedtheoreticaldescriptionmaybeobtained theoretically [10], should be possible experimentally.
by including, e.g., phonon-scattering—known to play an
importantroleinQDexcitondephasing[32–34]—ourob- The authors acknowledge financial support from the
servations agree well with the theoretical model of del National Science Foundation (NSF grant No. 1254324)
Valleetal. [23]. TheTPSmeasurementmayprovidenew and the National Natural Science Foundation of China
insightsintomorecomplexsystems,suchasthetwo-level (Grant No. 90921015).
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