Tuning the Photon Statistics of a Strongly Coupled Nanophotonic System
Constantin Dory, Kevin A. Fischer, Kai Müller, Konstantinos G. Lagoudakis, Tomas Sarmiento, Armand Rundquist, Jingyuan L. Zhang, Yousif Kelaita, Neil V. Sapra, Jelena Vučković
aa r X i v : . [ phy s i c s . op ti c s ] M a r Tuning the Photon Statistics of a Strongly Coupled Nanophotonic System
Constantin Dory, ∗ Kevin A. Fischer, ∗ Kai M¨uller, ∗ Konstantinos G. Lagoudakis, Tomas Sarmiento, Armand Rundquist, Jingyuan L. Zhang, Yousif Kelaita, Neil V. Sapra, and Jelena Vuˇckovi´c † E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA (Dated: March 23, 2017)We investigate the dynamics of single- and multi-photon emission from detuned strongly coupledsystems based on the quantum-dot-photonic-crystal resonator platform. Transmitting light throughsuch systems can generate a range of non-classical states of light with tunable photon countingstatistics due to the nonlinear ladder of hybridized light-matter states. By controlling the detuningbetween emitter and resonator, the transmission can be tuned to strongly enhance either single-or two-photon emission processes. Despite the strongly-dissipative nature of these systems, wefind that by utilizing a self-homodyne interference technique combined with frequency-filtering weare able to find a strong two-photon component of the emission in the multi-photon regime. Inorder to explain our correlation measurements, we propose rate equation models that capture thedominant processes of emission both in the single- and multi-photon regimes. These models are thensupported by quantum-optical simulations that fully capture the frequency filtering of emission fromour solid-state system.
I. INTRODUCTION
The generation of nonclassical states of light for ap-plications such as quantum computing [1], quantum keydistribution [2, 3], or quantum lithography and metrol-ogy [4] has been extensively investigated for quantumdots (QDs) [5, 6] due to their large optical dipole mo-ment, discrete optical transitions, and nearly Fouriertransform limited linewidth. However, any potentialquantum light source must be efficiently integrated witha resonator for effective use in a practical quantum net-work. Towards this goal, it was demonstrated thatstrongly coupled QD-photonic crystal resonator systemsare capable of high-fidelity single-photon generation withsuperior generation rates [7–9]. Importantly, such sys-tems are promising for on-chip geometries, since theycan be integrated into optical circuits due to their ef-ficient coupling to waveguides [10]. Nevertheless, theirpromising potential for multi-photon generation has yetto be experimentally investigated.Strongly coupled QD-nanocavity systems have longbeen seen as a versatile platform for the genera-tion of nonclassical light. The enhancement of thelight-matter interaction due to the presence of a res-onator leads to a nonlinear ladder of hybridized po-laritonic states. This enables transmitted light with asub-Poissonian photocount distribution [7–9, 11] in the ∗ These authors contributed equally † Correspondence to: [email protected] regime known as photon-blockade and it also enablestransmitted light with a super-Poissonian photocountdistribution [7, 8, 11–13] in the photon-induced tunnel-ing regime. However, the highly-dissipative nature ofnanophotonic systems has so far obscured the generationof multi-photon pulses with n photons, where n > II. STRONGLY COUPLED NANOPHOTONICSYSTEMS
The sample under investigation consists of a singleInAs QD strongly coupled to a photonic crystal L3 cav- -2 -1 0 1 2 3 40.20.61.01.41.8 -4 -2 0 2 430 32 34 36 38922.2922.0921.8921.6921.4 Temperature (K) W a v e l e ng t h ( n m ) g ( ) D ( ) Laser detuning (g) (b)(a)
Polariton Cavity QDLP2UP2UP1LP1 E n e r gy ( a r b . un it s ) Detuning (g)
Ground state (c)
FIG. 1.
Strongly coupled QD-cavity system: ( a ) Spec-tra of the strongly coupled QD-photonic crystal cavity sys-tem detected in cross-polarized reflectivity measurements, ex-hibiting an anticrossing - the signature of strong coupling.The QD resonance is tuned through the cavity resonance bychanging the sample temperature. ( b ) Calculated energy levelstructure of the first two rungs of a strongly coupled Jaynes-Cummings system. Each rung n consists of an upper polari-ton and a lower polariton UP n and LP n , respectively. ( c )Measured degree of second-order coherence as a function ofthe laser detuning for a QD-cavity detuning of ∆ = 3 g . Thepolaritonic frequencies are indicated by solid black lines, thebare cavity frequency a dashed red line and the bare QD fre-quency by a dotted blue line. ity [16]. The strong coupling between QD and cavitycan be observed in cross-polarized reflectivity measure-ments [17]. Because the in- and out-coupled light modesare orthogonally polarized, reflectivity is mathematicallyequivalent to a transmission experiment and we hence-forth refer to the process as transmission. By chang-ing the sample temperature, we can control the QD-cavity detuning ∆, tune the QD through the cavity res-onance and observe a distinct anticrossing (figure 1(a)).This anticrossing results from the strong coupling be-tween the QD and cavity. The energy level structure of astrongly coupled system can be described by the Jaynes-Cummings (JC) Hamiltonian H JC = (∆ + ω C ) σ † σ + ω C a † a + g (cid:0) a † σ + aσ † (cid:1) , (1)with ω C denoting the cavity frequency, σ the quantumdot lowering operator, ∆ the detuning between quantumemitter and cavity, a the cavity mode operator and g theemitter-cavity field coupling strength. Including dissipa- tion, the complex eigenenergies E n ± of the system are [18]: E n ± = nω C + ∆2 − i (2 n − κ + γ ± s(cid:0) √ ng (cid:1) + (cid:18) − ∆2 − i κ − γ (cid:19) , (2)where n corresponds to the rung of the system (num-ber of excitations) and κ and γ are the cavity and QDenergy decay rates, respectively. The resulting lowestenergy levels are depicted in figure 1(b) as a functionof ∆. They consist of pairs of anticrossing branches, theupper polaritons (UP n ) and the lower polaritons (LP n ).When transitioning through the anticrossing, the polari-tons change their character from QD-like/cavity-like tocavity-like/QD-like. Fitting the data results in values of g = 12 . · π GHz and κ = 18 . · π GHz in energydecay rates. Importantly, our system satisfies the strongcoupling condition, which occurs if the coupling strengthovercomes the losses of the system (cid:0) g > κ − γ (cid:1) [19]. Fur-thermore, this type of nanophotonic system operates inthe good emitter limit, where κ is much larger than γ .However, in photonic crystal cavity-QD based systems, γ can be neglected relative to the other rate, since forQDs in bulk the radiative lifetime is about 1 ns and evenfurther lengthened by the photonic band gap. Instead,the lifetimes of the far detuned polaritons are dominatedby phonon bath-induced dephasing processes.Specifically in QD-cavity systems, phonon assisted-population transfer between polaritonic branches is im-portant and well studied [20–23]. It is very efficient forstrongly coupled systems [8] and for the system investi-gated here, we find transfer rates of Γ nr ∼ · π GHzfor detunings in the range of 0 − g .As discussed in the introduction, our QD-cavity plat-form can produce a wide variety of nonclassical lightstatistics. To visualize this capability, we present in fig-ure 1(c) the laser detuning-dependent measured degreeof second order coherence g (2) D (0) for ∆ = 3 g , obtainedusing cross-polarized reflectivity [12] and a Hanbury-Brown-Twiss (HBT) type measurement. Note that dueto the extremely fast emission rates of nanophotonic sys-tems, all correlation experiments presented throughoutthis paper are performed in the pulsed regime, wherewe measure the degree of total second-order coherence g (2) D (0) ≡ h m ( m − i / h m i , with m signifying the num-ber of detections [24, 25]. A super-Poissonian photondistribution can be found at laser detunings of 1 − . g ,known as photon-induced tunneling regime, while a sub-Poissonian photon distribution can be found at detun-ings of 3 − g , known as photon-blockade regime. In thefollowing sections, we investigate both regimes in moredetail and in particular examine the interplay betweenphonon effects and frequency filtering. III. PHOTON-BLOCKADE REGIME
First, we discuss in greater detail the generation ofsingle photons in the photon-blockade regime and pro-vide insight into phonon-assisted processes. In contrastto our prior photon blockade work, we now considerphonon assisted coupling between dressed ladder eigen-states, which impacts the properties of the single photongeneration. As schematically illustrated in figure 2(b),photon-blockade occurs if the laser is resonant with thefirst rung of the JC-ladder (solid blue upward arrow)but not resonant with higher rungs of the ladder dueto its anharmonicity. In this configuration, only singlephotons can be transmitted. Due to the fast dissipationrates of nanophotonic systems, a detuning between QDand cavity of a few g has been shown to be essential forhigh-fidelity single photon generation [7]; based on thisstudy, we have chosen an optimal detuning of 3 . g . Thepulse length of the excitation laser has to be chosen tobe significantly smaller than the state lifetime to mini-mize re-excitation during the presence of the excitationpulse [7, 8]. At the same time, the pulse needs to bespectrally narrow to avoid unnecessary overlap with sub-sequent climbs up the ladder. We determined an optimalcompromise at 25 ps which is smaller than the state life-time of 48.5 ps at this detuning. A typical transmissionspectrum obtained in this configuration with an excita-tion pulse area of π [26] is presented in figure 2(a). Thesignal is composed of three contributions: emission fromthe resonantly excited UP1 (schematically illustrated bya solid blue arrow in figure 2(b)), phonon-assisted emis-sion from LP1 (schematically illustrated a dashed redarrow in figure 2(b)) and coherent scattering of the exci-tation laser. We note here that in this configuration, co-herent scattering of the excitation laser would normallydominate the signal. However, in our case it is largelysuppressed due to a self-homodyne suppression (SHS) ef-fect that results from interference of light scattering fromthe cavity and continuum modes of the photonic crys-tal [15]. The relative intensities of UP1 and LP1 aremainly given by the ratio of the (detuning dependent)radiative transition rate Γ rUP and the phonon-assistedtransfer rate Γ nr (illustrated by a curved orange arrowin figure 2b).We now investigate the quantum character of the emis-sion through measurements of the degree of second-ordercoherence. The result of a measurement without spec-tral filtering (similar to our previous bockade results)is presented in figure 2(c) and results in a value of g (2) D (0) = 0 . ± . . · π GHzas a spectral filter. The result for filtering on theUP1 emission (indicated in blue on the left in fig- -4 -2 0 2 4-60 -30 0 30 6005001000150020002500 (c) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)921.2 921.6 922.0 I n t e n s it y ( a r b . un it s ) Wavelength (nm) (a) -60 -30 0 30 600100200300 (d) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)-60 -30 0 30 6004008001200 (e) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) -60 -30 0 30 600100200300 (f) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) (b) E n e r gy ( a r b . un it s ) Detuning (g) X FIG. 2.
Photon-blockade regime: ( a ) Spectrum of thestrongly coupled system at ∆ = 3 . g and resonant excita-tion of UP1 with a 25 ps long pulse. ( b ) Illustration ofthe JC-ladder. The excitation laser is resonant with UP1(depicted with a solid blue upward arrow) but not resonantwith higher climbs up the ladder. Following excitation ofUP1, possible recombination channels are from UP1 to theground state (solid blue downward arrow) or from LP1 tothe ground state (dashed red arrow) via a phonon-assistedpopulation transfer from UP1 to LP1 (curved orange ar-row). ( c-f ) Correlation measurement of the ( c ) unfilteredsignal, revealing g (2) D (0) = 0 . ± . d ) filteredemission from UP1 (indicated in blue on the left in ( a )),revealing g (2) D (0) = 0 . ± . e ) filtered emissionfrom LP1 (indicated in red on the right in ( a )), revealing g (2) D (0) = 0 . ± .
010 and ( f ) cross-correlation betweenUP1 and LP1, revealing g (2) D (0) = 0 . ± . ure 2(a)) is presented in figure 2(d) and shows valuesof g (2) D (0) = 0 . ± . g (2) D (0) = 0 . ± . g (2) D (0) is lowest in this case.In order to determine the relationship between photonsemitted at the LP1 and UP1 frequencies, we performedcross-correlation measurements using two spectrometersas spectral filters in front of the two detectors of ourHBT setup. The result of a cross-correlation measure-ment between UP1 and LP1 is presented in figure 2(f)and shows clear antibunching with a measured degree ofsecond-order coherence of g (2) D (0) = 0 . ± . g (2) D (0) to coherent laser scattering. Most importantly, itdemonstrates that after exciting UP1, emission of a sin-gle photon occurs either at the UP1 energy or phonon-mediated via LP1. We also note here that measurementswith longer pulses (see supplementary material) showedqualitatively the same behavior but with higher values of g (2) D (0) due to an enhanced probability of re-excitation.To support our interpretation of the data, we devel-oped a model based on the measured lifetimes of UP1 andLP1. In our model, the system is initialized to the excitedstate UP1 and then decays via two independent channels,which thus would have zero self- or cross-correlation be-tween intensities. In rate equation form, the model isgiven by: ddt (cid:16) P UP ( t ) P LP ( t ) (cid:17) = (cid:16) − ( Γ rUP +Γ nrf ) Γ nrr Γ nrf − (Γ rLP +Γ nrr ) (cid:17) · (cid:16) P UP ( t ) P LP ( t ) (cid:17) , (3)where P UP ( t ) and P LP ( t ) are the population of UP1and LP1, respectively. The rates used in the model arethe radiative recombination rate Γ rUP from UP1 andΓ rLP from LP1 and the phonon-assisted transfer ratesfrom UP1 to LP1 (Γ nrf ) and vice versa (Γ nrr ). Using thismodel and the measured rates (see supplemental mate-rial for details), we calculate that 52 . . IV. PHOTON-INDUCED TUNNELING REGIME
We now turn our attention to multi-photon emission[27]. We again investigate frequency filtered photonstatistics from a detuned strongly coupled system, but with the laser tuned to a multi-photon resonance ofthe Jaynes-Cummings system [28]. As discussed above,photon-induced tunneling describes the enhanced prob-ability of a multi-photon transmission for an excitationlaser tuned in between the polaritons of the first rung.This configuration is schematically illustrated by thesolid green arrows in figure 3(b) and known to result insuper-Poissonian counting statistics of the transmittedlight [7, 8, 12, 13]. Although two-, three- and higher n -photon resonances are located quite close in frequency,we expect to observe mainly effects from two-photon ex-citation. Because n -photon resonance transition ratesscale with the n-th order of the laser power, we expectthat for the relatively low powers used in the experiment,the two-photon resonance will completely dominate theemission statistics.In general, for nanophotonic systems in the photon-tunneling regime, the transmission is dominated by co-herent scattering of the laser and the probability P ( n ) ofobtaining n photons in a transmitted pulse only slightlydeviates from a coherent beam. Especially in the de-tuned tunneling regime, this coherent scattering com-pletely dominates the emission even for arbitrarily lowpowers. As can be seen from the gray dashed line in fig-ure 3(a), the cavity strongly reflects the laser pulse as toobscure the interesting quantum light emission. There-fore to observe non-trivial emission statistics it is criticalto employ a self-homodyning interference in order to re-move the unwanted coherently scattered light [15].Utilizing this SHS effect, a typical spectrum obtainedat a detuning of ∆ = 5 . g with a pulse length of25 ps and the excitation laser in resonance with the sec-ond rung (for two-photon excitation) is presented in fig-ure 3(a) as gray circles. The data is fitted with a quan-tum optical model (solid black line), that we will discusslater. Similar to the photon-blockade case, the emis-sion contains three components: emission at the UP1energy, emission at the LP1 energy and coherent scat-tering from the laser. Here, the detuning is chosen largeenough to separate the three components while keeping itsmall enough to lend oscillator strength from the cavityto enable multi-photon excitation of higher rungs.Now, we discuss a multi-photon tunneling process thatis schematically illustrated in figure 3(b): After populat-ing UP2 via a two-photon excitation (two solid greenarrows) the system mainly relaxes into UP1 emitting aphoton at the LP1 frequency with the rate κ . From thereit decays either through emission of a photon at the UP1frequency with the rate Γ rUP or phonon-assisted at theLP1 frequency with the rate Γ nrf .Next, we test if this model is consistent with cor-relation measurements. The results of unfiltered andfrequency-filtered measurements are presented in fig-ures 3(c-e) whereby the filtering frequency is indicatedby the colors in figure 3(a). The measured valuesfor unfiltered, filtered on UP1 and filtered on LP1 are g (2) D (0) = 1 . ± . g (2) D (0) = 0 . ± .
028 and g (2) D (0) = 1 . ± . -9 -6 -3 0 3 6 9921.3 921.6 921.9 I n t e n s it y ( a r b . un it s ) Wavelength (nm)(a) (b) E n e r gy ( a r b . un it s ) Detuning (g)-60 -30 0 30 600200040006000(c) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) -60 -30 0 30 600100200300400(d) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)-60 -30 0 30 600100200300400 (f) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) (e) g ( ) D ( ) Filter position (nm)
FIG. 3.
Photon-induced tunneling regime: ( a ) Spec-trum of the strongly coupled system at ∆ = 5 . g for excitingUP2 with a 25 ps long pulse (gray circles) fitted with a quan-tum optical model with (solid black line) and without SHS(gray dashed line). ( b ) Illustration of the JC-ladder. Theresonant two-photon excitation of UP2 is depicted with twosolid green arrows. The most likely relaxation channels arefrom UP2 to UP1 (upper dashed red arrow), from UP1 to theground state (solid blue arrow) or from LP1 to the groundstate (lower dashed red arrow) via a phonon-assisted popula-tion transfer from UP1 to LP1 (curved orange arrow). Cor-relation measurements of ( c ) the unfiltered signal, revealing g (2) D (0) = 1 . ± . d ) the filtered emission UP1,revealing g (2) D (0) = 0 . ± .
028 and ( e ) the emissionfrom the transitions from UP2 to UP1 and from LP1 to theground state, revealing g (2) D (0) = 1 . ± . f ) Simu-lated pulse-wise second-order coherence versus the position ofthe frequency filter and taking into account the experimentalparameters (laser pulse approximately tuned to two-photonresonance). The dotted blue (dashed red) line represents thefrequency of UP1 (LP1) and black circles represent measuredvalues. small bunching value for the unfiltered measurement isconsistent with literature and our prior work [7, 8, 11]and with the measurement presented in figure 1(c). Thefact that we observe strong antibunching filtered on theUP1 frequency is consistent with our proposed model,where independent of which rung is excited, only onephoton can be emitted at the UP1 frequency per exci-tation cycle. The increase in g (2) D (0) relative to the casefor frequency-filtered photon-blockade results from leak-age of the coherently scattered laser component into thedetection channel due to its spectral proximity. Impor-tantly, the frequency filtered measurement at the cavityfrequency shows strongly enhanced bunching relative tothe unfiltered case.Here, we propose a second-order scattering process tointerpret these results and use a rate equation model toanalyze the dynamics. The system is initialized to the ex-cited state UP2 and then decays via different channels.With this model we can calculate the population of UP2,LP2, UP1 and LP1, in our notation labeled P UP ( t ), P LP ( t ), P UP ( t ) and P LP ( t ), respectively. This allowsus to calculate the radiative emission that occurs fromeach polariton. The rate equation model is given by: ddt P UP ( t ) P LP ( t ) P UP ( t ) P LP ( t ) ! = Γ · P UP ( t ) P LP ( t ) P UP ( t ) P LP ( t ) ! , (4)with Γ representing the following rates:Γ = − ( Γ rQ +Γ nrf ) Γ nrr Γ rC nrf − (Γ rC +Γ nrr ) Γ rQ · Γ rC − ( Γ rC +Γ rQ +Γ nrf ) · Γ nrr nrf − rC +Γ nrr ) , where Γ rQ and Γ rC , are the radiative recombination ratesof UP1 and LP1, respectively. The model also includesnonradiative phonon-assisted transfer rates from UP n toLP n ( n Γ nrf ) and vice versa ( n Γ nrr ).Using the measured rates at this detuning [8], we com-pare the emission intensities at the frequencies of UP1and LP1 estimated from the rate equation model withthe ones fitted from the spectrum in figure 3(a). Fromthe model we calculate that 88 . . . . g (2) D (0) values as presented in figure 3(f).The simulated second-order coherence fits almost per-fectly to the measured data at the UP1 (dotted blue line)and the LP1 frequency (dashed red line). We again em-phasize that the self-homodyne suppression effect is ofparamount importance for these experiments. WithoutSHS, the coherently scattered laser light would dominatethe spectrum, leading to a Poissonian photon distribu-tion, as illustrated by the gray dashed line in figure 3(a).Finding a model that describes the system’s be-havior well allows us to make an important in-sight into the multi-photon emission from our sys-tem. Unlike in previous studies of photon-inducedtunneling where g (3) D (0) > g (2) D (0) [14], our fil-tered emission both strongly bunches in second-order( g (2) D (0) = 1 . ± . g (3) D (0) = 0 . ± . V. CONCLUSION
In this article, we provided further insight into the dy-namics of strongly coupled QD-photonic crystal cavitysystems for nonclassical light generation. By modifyingthe excitation laser detuning, we showed that the emit-ted photon distribution can be tuned from sub- to super-Poissonian.In the photon-blockade regime we addressed the firstpolaritonic rung with resonant laser pulses and foundtwo decay channels: direct recombination from UP1 andphonon-assisted emission from LP1. In cross-correlationmeasurements we found strong antibunching, demon-strating for the first time that the system only emits onephoton at a time through any of its decay channels.In the photon-tunneling regime we excited the sec-ond polaritonic rung resonantly and generated photonswith a super-Poissonian distribution. We presented amodel, where the emission from the system is explainedthrough the subsequent emission of two photons either with or without a phonon-mediated population trans-fer. This finding was supported through quantum op-tical simulations that showed excellent agreement. Fur-thermore, we calculated a third-order coherence valueof g (3) D (0) = 0 . ± . ACKNOWLEDGMENTS
We acknowledge support from the Air Force Officeof Scientific Research (AFOSR) MURI Center for Mul-tifunctional Light-Matter Interfaces Based on Atomsand Solids (FA9550-12-1-0025), the Army Research Of-fice (ARO) (W911NF1310309) and the National ScienceFoundation (NSF) Division of Materials Research (DMR)(1503759). CD acknowledges support from the AndreasBechtolsheim Stanford Graduate Fellowship. KM ac-knowledges support from the Alexander von HumboldtFoundation. KAF acknowledges support from the LuStanford Graduate Fellowship and the National DefenseScience and Engineering Graduate Fellowship. JLZ ac-knowledges support from the Stanford Graduate Fellow-ship. YK acknowledges support from the Art and MaryFong Stanford Graduate Fellowship and the National De-fense Science and Engineering Graduate Fellowship.
Appendices
A. APPENDIX A: PHOTON-BLOCKADEREGIME
In the following, we demonstrate that the excita-tion pulse length has a strong influence on the single-photon character of the system’s emission. In particu-lar the pulse length must be carefully chosen to avoidre-excitation that leads to a decreased fidelity of singlephoton generation.With resonant excitation of UP1, we acquire the emis-sion of the strongly coupled system at ∆ = 3 . g asshown in figure 4(a). Note that due to the nonlinear JCladder, the excitation laser is resonant with UP1 (solidblue upward arrow), while it is not resonant with LP1 orthe second rung via two-photon-excitation. With this ex-citation scheme we can detect emission from UP1 (solidblue downward arrow) and emission from LP1 (dashedred arrow) after a phonon-assisted population transferfrom UP1 (curved orange arrow). During these exper-iments we use a laser pulse length of 80 ps and expectthe second-order coherence to be imperfect due to re-excitation, since the polariton lifetime is roughly 49 ps.Now, we detect the unfiltered signal with a fiber-coupled Hanbury-Brown and Twiss setup to measure thedegree of second-order coherence. In figures 4(c-f), wepresent the measured degree of second-order coherenceof the unfiltered signal ( g (2) D (0) = 0 . ± . g (2) D (0) = 0 . ± . g (2) D (0) = 0 . ± . g (2) D (0) = 0 . ± . B. APPENDIX B: PHOTON-INDUCEDTUNNELING REGIME
In the photon-induced tunneling regime we excite thesecond polaritonic rung with an excitation pulse length of80 ps, while the polariton’s lifetime is 43 ps at ∆ = 2 . g .The spectrum of the system is shown in figure 5(a)and the contributing decay channels are colored in thesame way, as shown in figure 5(b). Here, the two-photonexcitation of UP2 is depicted with two solid green arrows.The most likely recombination channel from UP2 is thedecay through UP2 to UP1 (upper dashed red arrow).UP1 allows for immediate emission (solid blue arrow)or for a phonon-assisted population transfer from UP1to LP1 (curved orange arrow) and subsequent emissionfrom LP1 (lower dashed red arrow).Figures 5(c-e) show the results of the second-order coherence measurements of the unfiltered signal( g (2) D (0) = 1 . ± . g (2) D (0) = 0 . ± . g (2) D (0) = 2 . ± . . g leads to shorter lifetimesof UP1, making the phonon-assisted population transferless prominent. This leads to a mainly cascaded emis-sion of two photons at different frequencies. The higherbunching values compared to figure 3 in the main textresult from less emission at LP1, which leaves a higherproportion of vacuum state and increases the bunchingvalue at the LP1 frequency. The shorter lifetime alsoleads to re-excitation and thus an increased second-ordercoherence value at UP1. -4 -2 0 2 4-60 -30 0 30 6005001000150020002500 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)921.2 921.6 922.0 I n t e n s it y ( a r b . un it s ) Wavelength (nm) -60 -30 0 30 60050100150200250 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)-60 -30 0 30 60050100150200 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) -60 -30 0 30 600100200300 (a) g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) E n e r gy ( a r b . un it s ) Detuning (g) (b) X (c) (d)(e) (f) FIG. 4.
Photon-blockade regime: ( a ) Spectrum of thestrongly coupled system at ∆ = 3 . g and resonant excitationof UP1 with an 80 ps long pulse. The emission from UP1 is il-lustrated in blue on the left and the emission from LP1 in redon the right. ( b ) Illustration of the JC-ladder. The excitationlaser is resonant with UP1 (depicted with a solid blue upwardarrow) but not resonant with higher climbs up the ladder.Following excitation of UP1, possible recombination channelsare from UP1 to the ground state (solid blue downward ar-row) or from LP1 to the ground state (dashed red arrow) via aphonon-assisted population transfer from UP1 to LP1 (curvedorange arrow). ( c-f ) Correlation measurement of the ( c ) un-filtered signal, revealing g (2) D (0) = 0 . ± . d ) filteredemission from UP1, revealing g (2) D (0) = 0 . ± . e )filtered emission from LP1, revealing g (2) D (0) = 0 . ± . f ) cross-correlation between UP1 and LP1, revealing g (2) D (0) = 0 . ± . I n t e n s it y ( a r b . un it s ) Wavelength (nm) E n e r gy ( a r b . un it s ) Detuning (g)(a) (b)(c) (d)(e) -60 -30 0 30 6005001000150020002500 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns) -60 -30 0 30 60050100150200 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)-60 -30 0 30 60050100150200 g (2)D (0) I n t e n s it y ( c oun t s ) Time delay (ns)
FIG. 5.
Photon-induced tunneling regime: ( a ) Spectrumof the strongly coupled system at ∆ = 2 . g for exciting UP2with an 80 ps long pulse. ( b ) Illustration of the JC-ladder.The resonant two-photon excitation of UP2 is depicted withtwo solid green arrows. The most likely recombination chan-nels are from UP2 to UP1 (upper dashed red arrow), fromUP1 to the ground state (solid blue arrow) or from LP1 to theground state (lower dashed red arrow) via a phonon-assistedpopulation transfer from UP1 to LP1 (curved orange arrow).Correlation measurements of ( c ) the unfiltered signal, reveal-ing g (2) D (0) = 1 . ± . d ) the filtered emission UP1,revealing g (2) D (0) = 0 . ± .
013 and ( e ) the emission fromthe transitions from UP2 to UP1 and from LP1 to the groundstate, revealing g (2) D (0) = 2 . ± . C. APPENDIX C: RATE EQUATION MODEL
Throughout the main text we compare the results offits to the measured spectra with rate equation mod-els. Figure 6(a) shows the spectra of the strongly cou-pled system in the photon blockade regime at a detuningof ∆ = 3 . g . The fit (solid black line) to the dataconsists of two Lorentzian and one Gaussian lineshapes.The Lorentzians correspond to emission from the system,while the Gaussian simply corresponds to reflected laserlight. With a rate equation model consisting of two statesand four rates, we calculate the time-resolved populationof UP1 (dotted blue line) and LP1 (dashed red line) asshown in figure 6(c). In our model we start with a fullypopulated state UP1. As can be seen in figure 6(c), itspopulation immediately decays, while the population ofLP1 first needs to build up via a phonon-assisted transferprocess. From these results, we calculate the integratedphotoluminescence intensity as shown in figure 6(e). Atthis detuning of ∆ = 3 . g the constellation of radia-tive recombination rates and phonon-assisted populationtransfer results in comparable emission from UP1 andLP1.In the photon-induced tunneling regime at a detuningof ∆ = 5 . g we fit the spectrum with two Lorentzianand one Gaussian lineshapes as shown in figure 6(b).The Lorentzians correspond to emission from the sys-tem, while the Gaussian simply corresponds to reflectedlaser light. With the model stated in the main text,we calculate the time-resolved population of UP1 (dot-ted blue line), LP1 (dashed red line), UP2 (dot-dashedgreen line) and LP2 (solid orange line) in figure 6(d).Here, we can see that the fast radiative decay rate ofUP2 results in a quick decay. Due to this fast decay,the phonon-assisted population transfer to LP2 is minor,which results in a weakly populated LP2. As shown infigure 6(d), UP2 mainly decays to UP1, from where thephonon-assisted population transfer also populates LP1.The integrated photoluminescence intensities at the UP1(dotted blue line) and the LP1 (dashed red line) frequen-cies are shown in figure 6(f). At this comparably largedetuning of ∆ = 5 . g , the radiative recombination rateof UP1 is long compared to the phonon-assisted trans-fer rate. This leads to strongly enhanced emission fromLP1 compared to UP1. Note here that the radiative re-combination rate of LP1 is extremely short, so the reversephonon-assisted transfer rate from LP1 to UP1 plays onlya minor role. D. APPENDIX D: SAMPLE FABRICATION
We use the same sample fabrication as in our previouswork [7] and have reproduced the details from the sup-plemental material.The molecular beam epitaxy-grownstructure consists of an ∼
900 nm thick Al . Ga . Assacrificial layer followed by a 145 nm thick GaAs layercontaining a single layer of InAs QDs. Our growth con- P L I n t e n s it y Time (ps)
UP LP P opu l a ti on Time (ps)
UP1 LP1 P L I n t e n s it y Time (ps)
UP LP P opu l a ti on Time (ps)
UP1 LP1 UP2 LP2 (a) (b)(c) (d)(e) (f)921.2 921.6 922.0 I n t e n s it y ( a r b . un it s ) Wavelength (nm) 921.2 921.6 922.0 I n t e n s it y ( a r b . un it s ) Wavelength (nm)
FIG. 6.
Rate equation model:
Spectrum of the stronglycoupled system at ( a ) ∆ = 3 . g for exciting UP1 with a25 ps long pulse and ( b ) ∆ = 5 . g for exciting UP2 witha 25 ps long pulse. The data (gray circles) is fitted with afunction built of Lorentzian and Gaussian lineshapes (solidblack line). The Lorentzian at UP1 (LP1) frequency is shownas a dotted blue (dashed red) line. Calculated time-resolvedpopulation of ( c ) UP1 (dotted blue line) and LP1 (dashed redline) in the photon blockade and ( d ) UP1 (dotted blue line),LP1 (dashed red line), UP2 (dot-dashed green line) and LP2(solid orange line) in the photon induced tunneling regime.Integrated photoluminescence intensity at the frequency ofUP1 (dotted blue line) and LP1 (dashed red line) in ( e ) thephoton blockade and ( f ) the photon induced tunneling regime. ditions result in a typical QD density of (60 − µ m − .Using 100 keV e-beam lithography with ZEP resist, fol-lowed by reactive ion etching and HF removal of the sac-rificial layer, we define the photonic crystal cavity. Thephotonic crystal lattice constant was a = 246 nm andthe hole radius r ∼
60 nm. The cavity fabricated is a lin-ear three-hole defect (L3) cavity. To improve the cavityquality factor, holes adjacent to the cavity were shifted.
E. APPENDIX E: OPTICAL SPECTROSCOPY
We use the same optical spectroscopy techniques asin our previous work [7, 8, 26] and have reproduced thedetails from the supplemental material. All optical mea-surements were performed with a liquid helium flow cryo-stat at temperatures in the range of 20 −
30 K. Forexcitation and detection, a microscope objective with anumerical aperture of NA = 0 .
75 was used. Cross-polarized measurements were performed using a polar-izing beam splitter. To further enhance the extinctionratio, additional thin film linear polarizers were placedin the excitation/detection pathways and a single modefibre was used to spatially filter the detection signal. Fur-thermore, two waveplates were placed between the beam-splitter and microscope objective: a half-wave plate to ro-tate the polarization relative to the cavity and a quarter-wave plate to correct for birefringence of the optics andsample itself. Photons are detected after spectral filteringwith an Hanbury-Brown and Twiss setup.The thin film polarizers and polarizing beamsplittersallow us to achieve an extinction ratio of 10 − betweenexcitation and detection path on bulk. This suppressionratio is large enough that light geometrically rotated bythe high NA objective plays little role in the ultimatelaser suppression. Instead, the amount of light classi-cally scattered into the detection channel is determinedby the fidelity of the self-homodyne effect. Experimen-tally, we previously found that this effect was capableof interferometrically cancelling >
95% of the light scat-tered through the L3 cavity’s fundamental mode [15]. Inlight of this strong suppression, no background has beensubtracted from the experimental data.Throughout the measurements we use a picosecondpulsed laser with 80 . F. APPENDIX F: DETAILS ON THESIMULATIONS
We used the same simulation techniques as in our Op-tica paper [9] and have reproduced the details from thesupplemental information describing these simulationshere. While describing these simulations, we fully elabo-rate on how the self-homodyne interference works.0The quantum-optical simulations were performed us-ing density matrix master equations with the QuantumOptics Toolbox in Python (QuTiP) [29], where the stan-dard Jaynes-Cummings model was used as a startingpoint. The effects of phonons were incorporated throughthe addition of incoherent decay channels with rates thatwere previously extracted [8]. To simulate the first or-der spectra of our system under excitation with a singlepulse, we compute the one-sided spectrum S ( ω ) = Re (cid:20)Z Z R dtdτ h A † ( t + τ ) A ( t ) i e − iωτ (cid:21) (5)of the free-field mode operator A ( t ). Input-output the-ory can relate the internal cavity mode operator a ( t ) tothe external field operator by the radiative cavity fielddecay rate κ/
2. Hence, for a JC system in the solidstate where the QD radiative decay rate γ plays an in-significant role compared to κ [8], spectral decomposi-tion of the cavity mode operator yields the spectrum ofthe detected light. Therefore, we can compute an un-normalised version of this spectrum with A ( t ) → a ( t ) inequation 5. We can also compute an unnormalised ver-sion of the incoherent spectrum with h A † ( t + τ ) A ( t ) i →h A † ( t + τ ) A ( t ) i − h A † ( t + τ ) ih A ( t ) i in equation 5. Toarrive at the version measured by a spectrometer of fi-nite bandwidth, we convolve S ( ω ) with the spectrometersresponse function. To simulate selfhomodyne suppres-sion (SHS), we replace A ( t ) → a ( t ) + α ( t ) in equation 5.Physically, α ( t ) is a slightly phase- and amplitude-shiftedversion of the incident laser pulse (originating from thecontinuum-mode scattering)[15]. In order to simulate thenormalized measured degree of second-order coherence, g (2) D (0) = g (2) D (0) N with N = R R dt h A † ( t ) A ( t ) i , we calculate g (2) D (0) = RR R dtdτ hT − [ A † ( t ) A † ( t + τ )] T + [ A ( t + τ ) A ( t )] i (cid:0)R R dt h A † ( t ) A ( t ) i (cid:1) (6)under excitation by a single pulse [7, 25]. The operators( T ) ± indicate the time ordering required of a physicalmeasurement [30]. We can likewise replace A ( t ) → a ( t )in equation 6 and also model SHS with the replacement of A ( t ) → a ( t ) + α ( t ) in equation 6. Despite the simplic-ity of equation 5, adding spectral filtering to equation 6is analytically and numerically quite challenging. Thespectral decomposition of this equation requires a fourthorder integral that is often intractable even numerically.Fortunately, the newly discovered sensor formalism [30]allows for efficient calculation of the spectrally filteredversion of the measured degree of second-order coherence.Here, we coherently attach a pair of two-level sensors tothe system Hamiltonian with the addition of the sensorHamiltonian to the Jaynes-Cummings Hamiltonian: H = H JC + X i =1 h ω s ς † i ς i + ǫ (cid:16) aς † i + a † ς i (cid:17)i (7)where ω s is the sensor frequency, ς the sensor annihilationoperator, and ǫ the sensor coherent coupling strength.The sensor coupling is chosen small enough so that itsbackaction on the system is negligible, i.e. ǫ Γ / ≪ γ f ,where γ f is the fastest transition rate in the un-sensedsystem. Additionally, the sensor decay terms of rate Γ areadded to the total Liouvillian. Here, in order to simulateSHS, we replace aς † i + a † ς i → ( a ( t ) + h α ( t ) i ) · ς † i + (cid:0) a † + h α ∗ ( t ) i (cid:1) · ς i (8)in equation 7. To arrive at the physically measured andspectrally filtered second-order coherence functions, thetotal degree of second-order coherence is computed be-tween the two sensors: g (2) D (0) = RR R dtdτ hT − [ ς † ( t ) ς † ( t + τ )] T + [ ς ( t + τ ) ς ( t )] i (cid:16)R R dt h ς † ( t ) ς ( t ) i (cid:17) (9)As the sensors are degenerate in every manner, the or-dering of their operation is arbitrary. In our model, thesensors are used as filters while the detector is assumedto be sufficiently broadband to integrate the correlationsover our entire experimental domain. This approxima-tion is accurate as the detector has a timing resolutionof greater than 200 ps compared with the system decaytime of approximately 50 ps. [1] J. L. O’Brien Optical Quantum Computing , Science ,1567 (2007).[2] J. L. O’Brien, A. Furusawa & J. Vuˇckovi´c
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