PHYSICAL REVIEW B 86, 045315 (2012)

Cavity quantum electrodynamics with a single quantum dot coupled to a photonic molecule

Arka Majumdar,* Armand Rundquist, Michal Bajcsy, and Jelena VuckovicE. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA

(Received 30 January 2012; published 18 July 2012)

We demonstrate the effects of cavity quantum electrodynamics for a quantum dot coupled to a photonicmolecule consisting of a pair of coupled photonic crystal cavities. We show anticrossing between the quantumdot and the two supermodes of the photonic molecule, signifying achievement of the strong coupling regime.From the anticrossing data, we estimate the contributions of both mode-coupling and intrinsic detuning to thetotal detuning between the supermodes. Finally, we also show signatures of off-resonant cavity-cavity interactionin the photonic molecule.

DOI: 10.1103/PhysRevB.86.045315 PACS number(s): 42.50.Pq, 42.70.Qs

I. INTRODUCTION

A single quantum emitter coupled to a resonator constitutesa cavity quantum electrodynamic (cQED) system for studyingstrong light-matter interaction. With an eye toward achievinga scalable quantum computer1 and more recently a photonicquantum simulator2–4 significant progress has been achievedin optical cQED as well as its microwave counterpart circuitQED. One such cQED system consisting of a single quantumdot (QD) coupled to a photonic crystal (PC) cavity hasrecently emerged as an attractive candidate for integratednanophotonic quantum information processing devices.5 Thissolid-state cQED system is of considerable interest to thequantum optics community for the generation of nonclassicalstates of light,6,7 for its application to all-optical8,9 andelectro-optical switching,10 and due to unusual effects likethe off-resonant dot-cavity interaction due to electron-phononcoupling.11 However, all of the cQED effects demonstrated sofar in this system involve a single cavity. Although numeroustheoretical proposals employing multiple cavities coupled tosingle quantum emitters exist in the cQED and circuit-QEDliterature,2–4,12 experimental development in this directionis rather limited. Most of these proposals, for example,observing the quantum phase transition of light, require anonlinearity in each cavity, which is a formidable task withcurrent technology. However, several proposals involving asingle QD coupled to multiple cavities predict novel quantumphenomena, for example, generation of bound photon-atomstates13 or sub-Poissonian light generation in a pair of coupledcavities or in a photonic molecule containing a single QD.14,15

This photonic molecule, coupled to a single QD, formsthe first step toward building an integrated cavity networkwith coupled QDs. Photonic molecules made of PC cavitieswere studied previously16,17 to observe mode-splitting due tocoupling between the cavities. In those studies, a high densityof QDs was used merely as an internal light source to generatephotoluminescence (PL) under above-band excitation and noquantum properties of the system were studied. In anotherexperiment, a photonic molecule consisting of two micropostcavities was used along with a single QD to generate entangledphotons via exciton-biexciton decay, but the QD-cavity systemwas in the weak coupling regime and the Purcell enhancementwas the only cQED effect observed.18 In addition to thephotonic molecule, fabrication of large-scale coupled cavities(without any quantum emitters) have been reported both in

photonic crystal platform19 and in circuit QED,20 revealingthe widespread interest in exploring the scalability of thesesystems.

In this paper, we demonstrate strong coupling of a photonicmolecule with a single QD. We note that even though thephotonic molecule described in this paper is based on photoniccrystal optical microcavities, one can also employ FabryPerot cavities, DBR microcavities, microposts or microwavecavities20,21 for this task. However, to best of our knowledge,coupling multiple cavities to single quantum emitters hasnot been shown in any other cQED system. We show clearanticrossing between the QD and two supermodes formedin the photonic molecule. In general, the exact couplingstrength between two cavities in a photonic molecule isdifficult to calculate, as the observed separation betweenthe two modes has contributions both from the cavity couplingstrength as well as from the mismatch between the two cavitiesdue to fabrication imperfections. However, by monitoring theinteraction between a single QD and the photonic moleculewe can exactly calculate the coupling strength between thecavities and separate the contribution of the bare detuning dueto cavity mismatch. In fact, without any coupling between twocavities, one cannot have strong coupling of the QD with bothof the observed modes. Hence, the observed anticrossing of theQD with both modes clearly indicates coupling between thecavities. Apart from the strong coupling, we also demonstrateoff-resonant phonon-mediated interaction between the twocavity modes, a recently found effect in solid-state cavitysystems.22

II. THEORY

Let us consider a photonic molecule consisting of twocavities with annihilation operators for their bare (uncoupled)modes denoted by a and b, respectively. We assume that a QDis placed in and resonantly coupled to the cavity described byoperator a. The Hamiltonian describing such a system is

H = �ob†b + J (a†b + ab†) + g(a†σ + aσ †), (1)

where �o is the detuning between the two bare cavity modes; Jand g are, respectively, the intercavity and dot-cavity couplingstrength; σ is the QD lowering operator; and the resonancefrequency ω0 of the cavity with annihilation operator a isassumed to be zero. We now transform this Hamiltonian bymapping the cavity modes a and b to the bosonic modes α and

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MAJUMDAR, RUNDQUIST, BAJCSY, AND VUCKOVIC PHYSICAL REVIEW B 86, 045315 (2012)

β introduced as a = cos(θ )α + sin(θ )β and b = sin(θ )α −cos(θ )β. We note that this mapping maintains the appropriatecommutation relations between operators a and b. Underthese transformations we can decouple the two cavity modes(α and β) for the following choice of θ :

tan(2θ ) = − 2J

�o

. (2)

Under this condition the transformed Hamiltonian becomes

H = α†α[�o sin2(θ ) + J sin(2θ )] + g cos(θ )(α†σ + ασ †)

+β†β[�o cos2(θ ) − J sin(2θ )] + g sin(θ )(β†σ + βσ †).

Therefore, a QD coupled to a photonic molecule has exactlythe same eigenstructure as two detuned cavities with the QDcoupled to both of them (from the equivalence of the twoexpressions above for the Hamiltonian H). The supermodesof the transformed Hamiltonian α and β will be separated by� = √

�2o + 4J 2 [obtained by subtracting the terms multiply-

ing α†α and β†β, under the conditions of Eq. (2)] and the in-teraction strength between the QD and the supermodes will beg1 = g cos(θ ) and g2 = g sin(θ ). Hence, g =

√g2

1 + g22 , θ =

arctan(g2/g1), and so tan(2θ ) = 2g1g2/(g21 − g2

2) = −2J/�o.If the two cavities are not coupled (J = 0 and θ = 0), we canstill observe two different cavity modes in the experimentdue to �o, the intrinsic detuning between two bare cavities.However, if we tune the QD across the two cavities in thiscase, we will observe QD-cavity interaction only with onecavity mode [in this case α, as the term coupling β tothe QD in the transformed Hamiltonian will vanish, as aresult of sin(θ ) = 0]. In other words, in this case the QDis spatially located in only one cavity and cannot interactwith the other, spatially distant and decoupled cavity. Figure 1shows the numerically calculated cavity transmission spectra(proportional to 〈a†a〉 + 〈b†b〉) when the QD is tuned acrossthe two cavity resonances. When the two cavities are coupled(J �= 0), we observe anticrossing between each cavity modeand the QD [Fig. 1(a)]. However, only one anticrossing isobserved when the cavities are not coupled [Fig. 1(b)].

III. CHARACTERIZATION OF PHOTONIC MOLECULE

The actual experiments are performed with self-assembledInAs QDs embedded in GaAs, and the whole system iskept at cryogenic temperatures (∼10–25 K) in a helium-flowcryostat. The cavities used are linear three-hole defect GaAsPC cavities coupled via spatial proximity. The photonic crystalis fabricated from a 160-nm-thick GaAs membrane, grown bymolecular beam epitaxy on top of a GaAs (100) wafer. Alow-density layer of InAs QDs is grown in the center of themembrane (80 nm beneath the surface). The GaAs membranesits on a 918-nm sacrificial layer of Al0.8Ga0.2As. Underthe sacrificial layer, a 10-period distributed Bragg reflector,consisting of a quarter-wave AlAs/GaAs stack, is used toincrease collection into the objective lens. The photonic crystalwas fabricated using electron beam lithography, dry plasmaetching, and wet etching of the sacrificial layer in dilutedhydrofluoric acid, as described previously.11,23

We fabricated two different types of coupled cavities: inone case, the two cavities are offset at a 60◦ angle [inset ofFig. 2(a)] and in the other the two cavities are laterally coupled

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FIG. 1. (Color online) Numerically calculated cavity transmis-sion spectra when the QD resonance is tuned across the two cavityresonances. (a) Anticrossing is observed between the quantum dotand both cavity modes when the two cavities are coupled (couplingrate between the two cavities is J/2π = 80 GHz). (b) When the twocavities are not coupled (J = 0), we observe anticrossing in onlyone cavity. Parameters used for the simulation: cavity decay rateκ/2π = 20 GHz (for both cavities); QD dipole decay rate γ /2π = 1GHz; dot-cavity coupling rate of g/2π = 10 GHz; intrinsic detuningbetween the bare cavity modes �o/2π = 40 GHz for (a) and120 GHz for (b). The plots are vertically offset for clarity. The hori-zontal axis corresponds to the detuning of the probe laser frequencyωp from the cavity a resonance ω0 in units of cavity field decay rate.

[inset of Fig. 2(b)]. In the first case the coupling between thecavities is stronger as the overlap between the electromagneticfields confined in the cavities is larger along the 60◦ angle.Figures 2(a) and 2(b) show the typical PL spectra of thesetwo different types of coupled cavities for different spacing

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FIG. 2. Photoluminescence spectra of the coupled cavities fordifferent hole spacings between two cavities: (a) the cavities areseparated at an angle of 60◦ [see the inset for a scanning electronmicrograph (SEM)]; (b) the cavities are laterally separated (see theinset for SEM). A decrease in the wavelength separation between twocavity modes is observed with increasing spatial separation betweenthe cavities (i.e., with increasing number of holes inserted in betweenthe two cavities). A much larger separation is observed in (a) when thecavities are coupled at an angle compared to the lateral coupling (b).

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FIG. 3. (Color online) Normalized PL intensity plotted whenwe tune the QD across the cavity resonance by temperature:(a) before nitrogen deposition (i.e., the QD is temperature tunedacross the longer wavelength resonance) and (b) after nitrogendeposition (which red-shifts the cavity resonances and allows us totemperature tune the QD across the shorter wavelength resonance).Clear anticrossings between the QD and the cavity are observed forboth supermodes. In both cases, the temperature is increased frombottom to top (the plots are vertically offset for clarity). In the insetthe resonances of the two anticrossing peaks (as extracted from curvefitting) are plotted. Clear anticrossing is observed in both cases.

between the cavities. A clear decrease in the frequency sepa-ration between the cavities is observed with increasing spatialseparation. Note that the consistency of this trend betweendifferent fabrication runs already indicates that this frequencyseparation cannot be purely due to the fabrication-relatedintrinsic detuning between the two cavities. Nevertheless, itis very difficult to quantify how much of the separation is dueto coupling (J ) and how much is due to intrinsic detuning(�o) of the cavity resonances. However, we will show that byobserving the anticrossing between the QD and the two modeswe can conclusively determine both J and �o.

IV. STRONG COUPLING WITH A SINGLE QD

First, we investigate the strong coupling between a singleQD and the photonic molecule in PL. For this particularexperiment, we used a photonic molecule consisting of cavitiesseparated by four holes along the 60◦ angle. In practice itis not trivial to tune the QD over such a long wavelengthrange as required by the observed separation of the twocavity peaks. Hence, we use two different tuning techniques:We tune the cavity modes by depositing nitrogen on thecavity,24 and then tune the QD resonance across the cavityresonance by changing the temperature of the system. ForFig. 3(a) there is no nitrogen deposition and the temperatureis changed from 20 K to 30 K to tune the QD across thelonger-wavelength cavity mode. For Fig. 3(b) we deposited

nitrogen (to red-shift cavities) and changed the temperaturefrom 20 K to 40 K, showing the anticrossing between theQD and the shorter-wavelength cavity mode. We note thatall these temperatures are measured at the cold-finger of thecryostat. We observe less shift of the cavity resonance withtemperature after nitrogen deposition, most likely becausethe nitrogen starts evaporating with increasing temperaturecausing the cavity to blue-shift, thus compensating for thered-shift caused by increased temperature. We fit the PLspectra to find the resonances of coupled QD-cavity systemand plot the resonances in the inset of Figs. 3(a) and 3(b).We observe clear anticrossings for both coupled modes. Thenitrogen and the temperature tuning do not cause a significantchange in the coupling and the detuning between the cavities,as confirmed in the experiments described below. We note thatthere is another QD (shown by the red arrow) in Fig. 3(a)weakly coupled to the cavity as no anticrossing between thatQD and the cavity is observed. Within the cavity region, wetypically have about four QDs that are spectrally distributedover a wide range. Spatially, they are also distributed withinthe cavity and, thus, have different coupling strengths tothe cavity mode. Tuning the system over a large range ofwavelengths (as we do in this experiment, with temperatureand nitrogen tuning) can bring into resonance some of theseother QDs, as shown by this additional peak. However, nosignature of coupling between this QD and the QD understudy for anticrossing is observed, and the presence of thisadditional weakly coupled dot does not change the conclusionsof the experiment. Finally, the additional QD is not visible inFig. 3(b), as it is off-resonant from both cavity modes.

We perform curve fitting for the PL spectra when theQD is resonant to the cavity super-modes and estimate thesystem parameters [Figs. 4(a) and 4(b)]. The supermode atshorter (longer) wavelength is denoted as sm1 (sm2). Asthe detuning between the supermodes is much larger than thevacuum Rabi spitting caused by the QD, we can assume thatwhen the QD is resonant to sm1(2), its interaction with sm2(1)is negligible. Therefore, we can fit the PL spectra of sm1

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FIG. 4. (Color online) QD-photonic molecule spectrum, (a) whenthe QD is resonant with super-mode sm2 and (b) when the QDis resonant with supermode sm1. From the fit we extract thesystem parameters (see text). Numerically simulated (c) second-orderautocorrelation g2(0) and (d) transmission from cavity b, as a functionof laser frequency, with the experimental system parameters that wereextracted from the fits.

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FIG. 5. (Color online) Off-resonant interaction between twocoupled cavities and a QD. We scan the laser across both coupledmodes and observe emission from the off-resonant supermode, underexcitation of the other supermode. A close-up spectrum for eachresonance shows the relative position of the laser and the cavitymodes.

(sm2) modes exhibiting Rabi splitting individually. For sm1,we extract from the fit the field decay rate κ1/2π = 22.4 GHzand the QD-field interaction strength g1/2π = 14.2 GHz[Fig. 4(b)]; for sm2, κ2/2π = 16.7 GHz and g2/2π =23.7 GHz [Fig. 4(a)]. We note that we can achieve very highquality factors (∼ 7000–10 000) of the coupled cavity modesas seen from the extracted κ values. We also estimate the totaldetuning between two observed modes as �/2π ∼ 246.5 and253.5 GHz before and after nitrogen tuning. This minimaldifference in � resulting from the nitrogen tuning does notimpact our further analysis, and we take � to be the averageof these two values. The change in the cavity field decay ratesarising from the nitrogen deposition is also minimal. Fromthese data, we use the relations 2g1g2/(g2

1 − g22) = −2J/�o

and � = √4J 2 + �2

o (previously derived in the paper) toobtain J/2π ≈ 110 GHz and �o/2π ≈ 118 GHz.

We now numerically simulate the performance of such aQD-photonic molecule for generation of sub-Poissonian lightusing the quantum optical master equation approach.25 Twobare cavity modes are separated by �o/2π = 118 GHz; aQD is resonant and strongly coupled to one of the modes(a) with interaction strength g/2π = 27.6 GHz (as discussedearlier, g =

√g2

1 + g22 , where g1 and g2 are the two values

of QD-cavity interaction strengths obtained by fitting the PLspectra); mode b is the empty cavity. The mode b is drivenand the second order autocorrelation g2(0) = 〈b†b†bb〉

〈b†b〉2 of the

transmitted light through cavity b is calculated.15 We alsoassume the two cavities to have the same cavity decay rate,which is an average of the cavity decay rates measured from thetwo supermodes. Note, however, that having slightly differentdecay rates does not significantly affect the performance of thesystem. The numerically simulated cavity b transmission andg2(0) of the transmitted light is shown in Figs. 4(c) and 4(d).From these simulated data, we observe that with our system

parameters we should be able to achieve strongly sub-Poissonian light with g2(0) ∼ 0.03. Unfortunately, in practiceit is very difficult to drive only one cavity mode without affect-ing the other mode due to the spatial proximity of two cavities.This individual addressability is critical for good performanceof the system15 and to retain such a capability in a photonicmolecule the cavities should be coupled via a waveguide.26

V. OFF-RESONANT COUPLING

Finally, as a further demonstration of cQED effects in thissystem, we report off-resonant interaction between the coupledcavities and the QD, similar to the observations in a singlelinear three-hole defect cavity22 and a nano-beam cavity.27

This experiment was performed on a different QD-photonicmolecule system than the one where we observed strongcoupling. Figure 5 shows the spectra indicating off-resonantcoupling between the cavities and the QD. Under resonantexcitation of the supermode at longer wavelength (sm2), wesee pronounced emission from both sm1 and a nearby QD.Similarly, under resonant excitation of sm1, we see emissionfrom sm2, although the emission is much weaker. We excludethe presence of any nonlinear optical processes by performinga laser-power-dependent study of the cavity emission, whichshows a linear dependence of the cavity emission on the laserpower (not shown here).

VI. CONCLUSION

In summary, we demonstrated strong coupling of a singleQD to a photonic molecule in a photonic crystal platform.Clear anticrossings between the QD and both supermodesof the photonic molecule were observed, showing conclusiveevidence of intercavity coupling. From the anticrossing datawe were able to separate the contributions of the intercavitycoupling and intrinsic detuning to the cavity mode splitting. Wehave also reported observation of off-resonant cavity-cavityand cavity-QD interaction in this type of system. Such asystem could be employed for nonclassical light generation (astheoretically studied in this article) and represents a buildingblock for an integrated nanophotonic network in a solid-statecQED platform.

ACKNOWLEDGMENTS

The authors acknowledge financial support provided by theOffice of Naval Research (PECASE Award; No: N00014-08-1-0561), DARPA (Award No: N66001-12-1-4011), NSF (DMR-0757112), and Army Research Office (W911NF-08-1-0399).A.R. is also supported by a Stanford Graduate Fellowship.We acknowledge Dr. Hyochul Kim and Dr. Pierre Petroff forproviding the quantum dot sample. This work was performedin part at the Stanford Nanofabrication Facility of NNIN,supported by the National Science Foundation.

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