Cavity Quantum Electrodynamics with a Single Quantum Dot Coupled to a Photonic Molecule

2.1 MB·English

by  Arka Majumdar

Cavity Quantum Electrodynamics with a Single Quantum Dot Coupled to a Photonic
Molecule
Arka Majumdar,∗ Armand Rundquist, Michal Bajcsy, and Jelena Vuˇckovi´c
E.L.Ginzton Laboratory,
Stanford University, Stanford, CA, 94305
We demonstrate the effects of cavity quantum electrodynamics for a quantum dot coupled to a
photonic molecule, consisting of a pair of coupled photonic crystal cavities. We show anti-crossing
betweenthequantumdotandthetwosuper-modesofthephotonicmolecule,signifyingachievement
of the strong coupling regime. From the anti-crossing data, we estimate the contributions of both
2
mode-coupling and intrinsic detuning to the total detuning between the super-modes. Finally, we
1
also show signatures of off-resonant cavity-cavity interaction in the photonic molecule.
0
2

n Asinglequantumdot(QD)coupledtoaphotoniccrys- coupling strength as well as from the mismatch between
a
tal (PC) cavity is an important building block for inte- the two cavities due to fabrication imperfections. How-
J
gratednanophotonicquantuminformationprocessingde- ever, by monitoring the interaction between a single QD
0
vices[1]. Thissolid-statecavityquantumelectrodynamic and the photonic molecule we can exactly calculate the
3
(cQED) system is of considerable interest to the quan- coupling strength between the cavities and separate the

] tumopticscommunityforthegenerationofnon-classical contributionofthebaredetuningduetocavitymismatch.
l statesoflight[2,3],foritsapplicationtoall-optical[4,5] In fact, without any coupling between two cavities, one
l
a and electro-optical switching [6], and due to unusual ef- cannot have strong coupling of the QD with both of the
h
fects like the off-resonant dot-cavity interaction due to observed modes. Hence, the observed anti-crossing of
-
s electron-phonon coupling [7]. However, all of the cQED the QD with both modes clearly indicates coupling be-
e
effects demonstrated so far in this system involve a sin- tween the cavities. Apart from the strong coupling, we
m
gle cavity. Although numerous theoretical proposals em- also demonstrate off-resonant phonon-mediated interac-
t. ploying multiple cavities coupled to quantum dots exist tionbetweenthetwocavitymodes,arecentlyfoundeffect
a
intheliterature[8–10],experimentaldevelopmentinthis in solid-state cavity systems.
m
directionisratherlimited. Recentlyithasbeenreported Let us consider a photonic molecule consisting of two
- thatstronglysub-Poissonianlightcanbegeneratedfrom cavities with annihilation operators for their bare (un-
d
apairofcoupledcavitiescontainingasingleQD[11,12]. coupled) modes denoted by a and b, respectively. We
n
o This double cavity, also called a photonic molecule, cou- assume that a QD is placed in and resonantly coupled
c pled to a single QD forms the first step towards building to the cavity described by operator a. The Hamiltonian
[ an integrated cavity network with coupled QDs. Pho- describing such a system is:

tonic molecules made of PC cavities were studied previ-
1
v ously [13, 14] to observe mode-splitting due to coupling H=∆ob†b+J(a†b+ab†)+g(a†σ+aσ†) (1)
4 between the cavities. In those studies, a high density of
4 QDswasusedmerelyasaninternallightsourcetogener- where ∆o is the detuning between the two bare cavity
2 atephotoluminescence(PL)underabove-bandexcitation modes; J and g are, respectively, the inter-cavity and
6 and no quantum properties of the system were studied. dot-cavity coupling strength; σ is the QD lowering oper-
.
1 Inanotherexperiment,aphotonicmoleculeconsistingof ator; and the resonance frequency ω0 of the cavity with
0 two micropost cavities was used along with a single QD annihilation operator a is assumed to be zero. We now
2 to generate entangled photons via exciton-biexciton de- transformthisHamiltonianbymappingthecavitymodes
1 cay, but the QD-cavity system was in the weak coupling a and b to the bosonic modes α and β introduced as
:
v regime and the Purcell enhancement was the only cQED a=cos(θ)α+sin(θ)β andb=sin(θ)α−cos(θ)β. Wenote
i effect observed [15]. that this mapping maintains the appropriate commuta-
X
tion relations between operators a and b. Under these
r In this paper, we demonstrate strong coupling of a transformations we can decouple the two cavity modes
a
photonicmoleculewith asingleQD.Weshow clearanti- (α and β) for the following choice of θ:
crossing between the QD and two super-modes formed
in the photonic molecule. In general, the exact coupling 2J
tan(2θ)=− (2)
strength between two cavities in a photonic molecule is ∆
o
difficult to calculate, as the observed separation between
Under this condition the transformed Hamiltonian be-
the two modes has contributions both from the cavity
comes:
H = α†α(∆ sin2(θ)+Jsin(2θ))+gcos(θ)(α†σ+ασ†)
o
∗Electronicaddress: [email protected] + β†β(∆ocos2(θ)−Jsin(2θ))+gsin(θ)(β†σ+βσ†)
2
Therefore, a QD coupled to a photonic molecule has ex- (a) 7 (b) 7
actly the same eigen-structure as two detuned cavities
with the QD coupled to both of them (from the equiv- t 6 6
u
alence of the two expressions above for the Hamiltonian p
t
H). The super-modes of the transformed Hamiltonian α u 5 5
O
and β will be separated by ∆ = (cid:112)∆2+4J2 (obtained y
bysubtractingthetermsmultiplyingα†oαandβ†β,under vit 4 4
a
the conditions of Eq.2) and the interaction strength be- C
d 3 3
tween the QD and the super-modes will be g =gcos(θ) e
1 z
and g2 = gsin(θ). If the two cavities are not coupled ali 2 2
(J = 0 and θ = 0), we can still observe two different m
r
cavity modes in the experiment due to ∆o, the intrinsic No 1 1
detuning between two bare cavities. However, if we tune
the QD across the two cavities in this case, we will ob- 0 0
serve QD-cavity interaction only with one cavity mode -5 0 5 10 -5 0 5 10
(in this case α, as the term coupling β to the QD in (ωp-ω0)/κ (ωp-ω0)/κ
the transformed Hamiltonian will vanish, as a result of
sin(θ) = 0). In other words, in this case the QD is spa-
FIG. 1: (color online) Numerically calculated cavity trans-
tiallylocatedinonlyonecavityandcannotinteractwith
mission spectra when the QD resonance is tuned across the
the other, spatially distant and decoupled cavity. Fig.
two cavity resonances. (a) Anticrossing is observed between
1 shows the numerically calculated cavity transmission the quantum dot and both cavity modes when the two cav-
spectra (proportional to (cid:104)a†a(cid:105)+(cid:104)b†b(cid:105)) when the QD is ities are coupled (coupling rate between the two cavities is
tuned across the two cavity resonances. When the two J/2π=80 GHz). (b) When the two cavities are not coupled
cavitiesarecoupled(J (cid:54)=0),weobserveanti-crossingbe- (J =0), we observe anti-crossing in only one cavity. Param-
tweeneachcavitymodeandtheQD(Fig. 1a). However, eters used for the simulation: cavity decay rate κ/2π = 20
only one anti-crossing is observed when the cavities are GHz (for both cavities); QD dipole decay rate γ/2π=1 GHz;
dot-cavitycouplingrateofg/2π=10GHz;intrinsicdetuning
not coupled (Fig. 1b).
between the bare cavity modes ∆ /2π =40 GHz for (a) and
o
The actual experiments are performed with self-
120 GHz for (b). The plots are vertically offset for clarity.
assembled InAs QDs embedded in GaAs, and the whole
Thehorizontalaxiscorrespondstothedetuningoftheprobe
system is kept at cryogenic temperatures (∼ 10−25 K) laser frequency ω from the cavity a resonance ω in units of
p 0
in a helium-flow cryostat. The cavities used are linear cavity field decay rate.
three hole defect GaAs PC cavities coupled via spatial
proximity. The photonic crystal is fabricated from a 160
nmthickGaAsmembrane,grownbymolecularbeamepi- cannot be purely due to the fabrication-related intrinsic
taxy on top of a GaAs (100) wafer. A low density layer detuning between the two cavities. Nevertheless, it is
of InAs QDs is grown in the center of the membrane (80 very difficult to quantify how much of the separation is
nm beneath the surface). The GaAs membrane sits on duetocoupling(J),andhowmuchisduetointrinsicde-
a 918 nm sacrificial layer of Al0.8Ga0.2As. Under the tuning (∆o) of the cavity resonances. However, we will
sacrificial layer, a 10-period distributed Bragg reflector, showthatbyobservingtheanti-crossingbetweentheQD
consisting of a quarter-wave AlAs/GaAs stack, is used and the two modes we can conclusively determine both
to increase collection into the objective lens. The pho- J and ∆ .
o
toniccrystalwasfabricatedusingelectronbeamlithogra- First,weinvestigatethestrongcouplingbetweenasin-
phy,dryplasmaetching,andwetetchingofthesacrificial gleQDandthephotonicmolecule. Forthisparticularex-
layerindilutedhydrofluoricacid,asdescribedpreviously periment, weusedaphotonicmoleculeconsistingofcav-
[7, 16]. itiesseparatedby4holesalongthe30o angle. Inpractice
We fabricated two different types of coupled cavities: it is not trivial to tune the QD over such a long wave-
in one case, the two cavities are offset at a 30o angle length range as required by the observed separation of
(inset of Fig. 2a) and in the other the two cavities are the two cavity peaks. Hence we use two different tuning
laterally coupled (inset of Fig. 2b). In the first case the techniques: we tune the cavity modes by depositing ni-
coupling between the cavities is stronger as the overlap trogenonthecavity[17],andthentunetheQDresonance
between the electromagnetic fields confined in the cavi- acrossthecavityresonancebychangingthetemperature
ties is larger along the 30o angle. Figs. 2a,b show the of the system. We observe clear anti-crossings for both
typicalPLspectraofthesetwodifferenttypesofcoupled themodesasshowninFigs. 3a,b. Fig. 3aisobtainedby
cavitiesfordifferentspacingbetweenthecavities. Aclear temperature-tuningtheQDacrossthelonger-wavelength
decreaseinthefrequencyseparationbetweenthecavities cavitymodebeforenitrogendeposition. Wethenperform
is observed with increasing spatial separation. Note that thenitrogendepositiontored-shiftthecavityresonances,
the consistency of this trend between different fabrica- and repeat the temperature tuning. Fig. 3b shows the
tionrunsalreadyindicatesthatthisfrequencyseparation anti-crossingbetweentheQDandtheshorter-wavelength
3
(a) (b)
6 holes y
y t
sit 1µm 1µm nsi
n e
e 5 holes t
t n
n
L I 4 holes L I
P P
d d
e 3 holes e
z z
ali 2 holes ali
m m
or or
N
N
1 hole 1 hole
935 940 945 920 925 930
λ (nm) λ (nm)
923.6 924 923.6 924
(a) (b)
λ (nm) λ (nm)
FIG.2: Photoluminescencespectraofthecoupledcavitiesfor
different hole spacings between two cavities: (a) the cavities
are separated at an angle of 30o (see the inset for a scanning FIG. 3: (color online) Normalized PL intensity plotted when
electronmicrograph(SEM));(b)thecavitiesarelaterallysep- we tune the QD across the cavity resonance by temperature:
arated(seetheinsetforSEM).Adecreaseinthewavelength (a) before nitrogen deposition (i.e., the QD is temperature
separationbetweentwocavitymodesisobservedwithincreas- tuned across the longer wavelength resonance), and (b) after
ingspatialseparationbetweenthecavities(i.e.,withincreas- nitrogen deposition (which red-shifts the cavity resonances
ing number of holes inserted in between the two cavities). A andallowsustotemperaturetunetheQDacrosstheshorter
much larger separation is observed in (a) when the cavities wavelength resonance). Clear anti-crossings between the QD
arecoupledatananglecomparedtothelateralcoupling(b). and the cavity are observed for both super-modes. In both
cases, the temperature is increased from top to bottom (the
plots are vertically offset for clarity).
cavity mode. The nitrogen and the temperature tuning
donotcauseasignificantchangeinthecouplingandthe
to obtain: J/2π ≈110 GHz and ∆ /2π ≈118 GHz.
detuningbetweenthecavities,asconfirmedintheexper- o
We now numerically simulate the performance of such
iments described below.
aQD-photonicmoleculeforgenerationofsub-Poissonian
We perform curve-fitting for the PL spectra when the
light using the quantum optical master equation ap-
QD is resonant to the cavity super-modes and estimate
proach [18]. Two bare cavity modes are separated by
the system parameters (Figs. 4a,b). The super-mode
∆ /2π = 118 GHz; a QD is resonant and strongly cou-
at shorter (longer) wavelength is denoted as sm1 (sm2). o
pled to one of the modes (a) with interaction strength
Asthedetuningbetweenthesuper-modesismuchlarger
(cid:112)
than the vacuum Rabi spitting caused by the QD, we g/2π =27.6GHz(g = g12+g22,whereg1andg2arethe
can assume that when the QD is resonant to sm1(2), its twovaluesofQD-cavityinteractionstrengthsobtainedby
interaction with sm2(1) is negligible. Therefore, we can fitting the PL spectra); mode b is the empty cavity. The
fit the PL spectra of sm1 (sm2) modes exhibiting Rabi mode b is driven and the second order autocorrelation
splitting individually. For sm1, we extract from the fit g2(0) = (cid:104)b†b†bb(cid:105) of the transmitted light through cavity
(cid:104)b†b(cid:105)2
the field decay rate κ /2π =16.7 GHz and the QD-field b is calculated [12]. We also assume the two cavities to
1
interaction strength g /2π = 23.7 GHz (Fig. 4a); for have the same cavity decay rate, which is an average
1
sm2,κ /2π =22.4GHzandg /2π =14.2GHz(Fig. 4b). of the cavity decay rates measured from the two super-
2 2
We note that we can achieve very high quality factors modes. Note however that, having slightly different de-
(∼ 7,000−10,000) of the coupled cavity modes as seen cay rates does not significantly affect the performance of
from the extracted κ values. We also estimate the total thesystem. Thenumericallysimulatedcavitybtransmis-
detuningbetweentwoobservedmodesas∆/2π =0.7and sion and g2(0) of the transmitted light is shown in Figs.
0.72GHzbeforeandafternitrogentuning. Thisminimal 4c,d. We note that with our system parameters we can
difference in ∆ resulting from the nitrogen tuning does achieve strongly sub-Poissonian light with g2(0) ∼ 0.03.
notimpactourfurtheranalysis, andwetake∆tobethe Unfortunately,inpracticeitisverydifficulttodriveonly
average of these two values. The change in the cavity one cavity mode without affecting the other mode due
field decay rates arising from the nitrogen deposition is to the spatial proximity of two cavities. This individual
also minimal. From these data, we use the relations θ = addressabilityiscriticalforgoodperformanceofthesys-
(cid:112)
arctan(g /g ), tan(2θ) = −2J/∆ and ∆ = 4J2+∆2 tem [12] and to retain such a capability in a photonic
2 1 o o
4
tal platform. Clear anti-crossings between the QD and
1200 3
both super-modes of the photonic molecule were ob-
0)
1000 2g( served, showing conclusive evidence of inter-cavity cou-
1
pling. From the anti-crossing data we were able to sep-
800
-5(ωp-ω00)/κ5 ainrtartienstihcedceotunntrinibguttoionthseocfatvhiteyinmtoerd-ecasvpiltiytticnogu.pWlineghaanvde
600 n1.5 (c)
400 nsmissio 1 aalnsdo craevpiotyrt-eQdDoibnsteerrvaacttiioonn ionf tohffis-rteysopneaonftscyasvteitmy.-caSvuicthy
200 Cavity Tra0.05 aerasytisotnem(ascotuhledorbeeticeamllpylosytueddiefodrinnotnh-icslaasrstiiccalel),ligahntdgreenp--
923 924 923 924 925 -10 0 10 20 resents a building block for an integrated nanophotonic
λ (nm) λ (nm) (ωp-ω0)/κ
(a) (b) (d) network in a solid-state cQED platform.
The authors acknowledge financial support provided
FIG. 4: (color online) QD-photonic molecule spectrum, (a) by the Office of Naval Research (PECASE Award; No:
whentheQDisresonantwithsuper-modesm1and(b)when
N00014-08-1-0561), DARPA (Award No: N66001-12-1-
the QD is resonant with super-mode sm2. From the fit we
extract the system parameters (see text). Numerically sim-
ulated (c) second order autocorrelation g2(0) and (d) trans-
mission from cavity b, as a function of laser frequency, with sm1
g
theexperimentalsystemparametersthatwereextractedfrom n
ni
the fits. u
T
er
s
a
L
molecule the cavities should be coupled via a waveguide
[19]. sm2
Finally, as a further demonstration of cQED effects in
928.6 929.8
this system, we report off-resonant interaction between 928.6 929 929.4 929.8 λ (nm)
the coupled cavities and the QD, similar to the obser- λ (nm)
vations in a single linear three hole defect cavity [20]
and a nano-beam cavity [21]. This experiment was per- FIG. 5: (color online) Off-resonant interaction between two
formedonadifferentQD-photonicmoleculesystemthan coupledcavitiesandaQD.Wescanthelaseracrossbothcou-
theonewhereweobservedstrongcoupling. Fig. 5shows pledmodes,andobserveemissionfromtheoff-resonantsuper-
the spectra indicating off-resonant coupling between the mode, under excitation of the other super-mode. A close-up
spectrumforeachresonanceshowstherelativepositionofthe
cavitiesandtheQD.Underresonantexcitationofthesu-
laser and the cavity modes.
permode at longer wavelength (sm2), we see pronounced
emissionfrombothsm1andanearbyQD.Similarly, un-
derresonantexcitationofsm1,weseeemissionfromsm2,
although the emission is much weaker. We exclude the 4011), NSF (DMR-0757112) and Army Research Office
presence of any nonlinear optical processes by perform- (W911NF-08-1-0399). A.R. is also supported by a Stan-
ingalaser-powerdependentstudyofthecavityemission, ford Graduate Fellowship. We acknowledge Dr. Hyochul
which shows a linear dependence of the cavity emission Kim and Dr. Pierre Petroff for providing the quantum
on the laser power (not shown here). dotsample. ThisworkwasperformedinpartattheStan-
In summary, we demonstrated strong coupling of a fordNanofabricationFacilityofNNIN,supportedbythe
single QD to a photonic molecule in a photonic crys- National Science Foundation.
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