Linewidth broadening of a quantum dot coupled to an off-resonant cavity
Arka Majumdar, Andrei Faraon, Erik Kim, Dirk Englund, Hyochul Kim, Pierre Petroff, Jelena Vuckovic
LLinewidth broadening of a quantum dot coupled to an off-resonant cavity
Arka Majumdar, ∗ Andrei Faraon, † Erik D. Kim, Dirk Englund, ‡ and Jelena Vuˇckovi´c E.L.Ginzton Laboratory,Stanford University, Stanford, CA, 94305
Hyochul Kim and Pierre Petroff
Materials Department, University of California,Santa Barbara, CA 93106
We study the coupling between a photonic crystal cavity and an off-resonant quantum dot underresonant excitation of the cavity or the quantum dot. Linewidths of the quantum dot and thecavity as a function of the excitation laser power are measured. We show that the linewidth of thequantum dot, measured by observing the cavity emission, is significantly broadened compared tothe theoretical estimate. This indicates additional incoherent coupling between the quantum dotand the cavity.
Recent demonstrations of cavity quantum electrody-namics (CQED) with a single quantum dot (QD) coupledto a semiconductor micro-cavity show the great potentialof this system for developing robust, scalable quantum in-formation processing devices [1, 2, 3, 4]. However, unlikeultra-cold atoms, QDs constantly interact with their lo-cal environments and this interaction plays a significantrole in CQED experiments with QDs. For example, sev-eral experiments have reported the observation of cavityemission even when the QD is far detuned ( ∼ − ∗ Electronic address: [email protected] † Currently at H.P Laboratories, Palo Alto, CA-94304 ‡ Currently at Columbia University, New York City, NY both the cavity and the laser field. In the absence of adriving laser, the dynamics of a coupled QD-cavity sys-tem is described by the Jaynes-Cummings Hamiltonian H JC = ~ ω c a † a + ~ ω d σ † σ + ~ g ( σ † a + σa † ) (1)Here, ω c and ω d are the cavity and the QD resonancefrequency, respectively, σ is the lowering operator for theQD, a is the annihilation operator for the cavity pho-ton and g is the coherent interaction strength betweenthe QD and the cavity. The eigen-frequencies ω ± of thecoupled system are given by [1] ω ± = ω c + ω d − i κ + γ ± r g + 14 ( δ − i ( κ − γ ) ) (2)where 2 κ and 2 γ are the cavity energy decay rate andthe QD spontaneous emission rate, respectively and δ is the QD-cavity detuning ω d − ω c . When the coherentinteraction strength g is greater than the decay rates κ and γ , the system is in strong coupling regime, and theeigen-states of H JC are polaritons possessing the charac-teristics of both the cavity and the QD. In this regime,when the QD-cavity detuning δ = 0, the linewidth ofthe polaritons is κ + γ . However, when the QD-cavitydetuning δ is much greater than g , the system is in thedispersive CQED regime. In this regime, one polaritondevelops a cavity-like character while the other becomesmore QD-like. The linewidths Γ c and Γ qd of the cavity-like and QD-like polaritons, respectively, are given by(with a pure QD dephasing rate of γ d ) [13]Γ c ≃ κ + 2 (cid:16) gδ (cid:17) γ (3)Γ qd ≃ γ + γ d ) + 2 (cid:16) gδ (cid:17) κ (4)The linewidth Γ qd can be interpreted as a combinationof the QD spontaneous emission rate (2 γ ) and the QDemission rate into the cavity mode 2 ( g/δ ) κ .n the other hand, when a QD is coherently drivenby a laser field in the absence of any optical cavity, thesystem dynamics is described by the Master equation dρdt = − i ~ [ H, ρ ] + 2 γ L [ σ ] + γ d L [ σ † σ ] (5)Here ρ is the density matrix of the QD optical transitionand γ d is the pure dephasing rate. L [ D ] is the Lindbladoperator for an operator D and is given by L [ D ] = DρD † − D † Dρ − ρD † D (6)The Hamiltonian H describing the coherent dynamics ofthe driven QD is given by H = ~ ω d σ † σ + ~ Ω2 ( σe − iω l t + σ † e iω l t ) (7)where, ω d and ω l are the QD resonance and the drivinglaser frequency, respectively, and Ω is the Rabi frequencyof the driving laser field. In solving the Master equation(Eq. 5), it is found that the intensity I of the QD reso-nance fluorescence for ω d = ω l is given by I = Ω γ ( γ + γ d ) Ω γ ( γ + γ d ) ∝ ˜ P P (8)where ˜ P = Ω γ ( γ + γ d ) . The QD linewidth ∆ ω is given by∆ ω = 2( γ + γ d ) s γ ( γ + γ d ) ∝ p P (9)The broadening of the QD linewidth with laser excita-tion power occurs due to increasing stimulated emissioncaused by the laser field and is known as power broad-ening. Such power broadening of the QD linewidth hasbeen reported by several other groups [14, 15].Following the discussion above, the linewidth ∆ ω of aresonantly driven QD that is coupled to an off-resonantcavity has contributions from both the increased emis-sion rate in the cavity mode and the increasing stimu-lated emission due to the driving laser. As the QD isdetuned from the cavity (and hence the laser driving theQD resonantly is also detuned from the cavity), the QDemission into cavity mode and the stimulated emissioninto the driving laser mode are independent and ∆ ω isgiven by ∆ ω = 2 (cid:16) gδ (cid:17) κ + 2( γ + γ d ) p P = ∆ ω c + ∆ ω p P (10)Here, ∆ ω c = 2 ( g/δ ) κ and ∆ ω = 2( γ + γ d ). Similarly,as the cavity is coupled to the QD, the cavity-like polari-ton linewidth contains a contribution from the QD emis-sion, as evident from Eq. 3. However as the cavity loss Probelaser SpectrometerPBSOLHWPPhotonic crystal (a) (b)
FIG. 1: (color online)(a) Scanning electron micrograph ofthe fabricated photonic crystal cavity. (b) The experimen-tal setup. An objective lens (OL) with a numerical apertureof 0 .
75 is used in front of the cryostat to image the chip. Ahalf wave plate (HWP) is used to adjust the excitation polar-ization relative to the cavity axis. A polarizing beam splitter(PBS) is used to perform cross-polarized reflectivity measure-ments. Details of the experimental setup are given in [1]. rate 2 κ is much greater than the QD spontaneous emis-sion rate 2 γ , the modification of the cavity linewidth isnegligible. From now on, we will refer to the cavity-likepolariton as the “cavity” and QD-like polariton as the“QD”.Experiments are performed in a helium-flow cryostat atcryogenic temperatures ( ∼ −
55 K) on self-assembledInAs QDs embedded in a GaAs photonic crystal cavity[1]. The 160nm GaAs membrane used to fabricate thephotonic crystal is grown by molecular beam epitaxy ontop of a GaAs (100) wafer. The GaAs membrane sitson a 918 nm sacrificial layer of Al . Ga . As. Under thesacrificial layer, a 10-period distributed Bragg reflector,consisting of a quarter-wave AlAs/GaAs stack, is used toincrease the collection into the objective lens. The pho-tonic crystal was fabricated using electron beam lithog-raphy, dry plasma etching, and wet etching of the sacrifi-cial layer in hydrofluoric acid (6%). A scanning electronmicrograph of a photonic crystal cavity along with a di-agram of the experimental setup is shown in Fig. 1.We perform two different types of experiments to studythe off-resonant QD-cavity coupling. For the first type,a narrow bandwidth ( ∼
300 kHz) laser is scanned acrossthe QD optical transition while the emission at the cavitywavelength is observed. In the second type, the laser isscanned across the cavity linewidth and the QD emissionis observed. Figs. 2 (a), (b) show the cavity and QDemission spectra for the first and second experiments,respectively. Figs. 2 (c), (d) show the integrated cavityand QD intensities as we scan the laser across the QDand the cavity, respectively. Lorentzian fits to the cavityand the QD intensities as a function of laser wavelengthenable estimation of the QD and the cavity linewidths,respectively.The first type of experiment is performed on three dif-ferent QD-cavity systems for different detunings between
31 932 933 934 935 λ (nm) I n t e n s i t y ( a . u . ) (a) λ (nm) I n t e n s i t y ( a . u . ) (b) Laser wavelength (nm) C av i t y I n t e n s i t y ( a . u . ) (c)
931 931.2 931.40102030
Laser wavelength (nm) Q D I n t e n s i t y ( a . u . ) (d)Cavity Laseron QD QDLaseron Cavity FIG. 2: (color online)(a) Cavity emission when the QD is reso-nantly excited. (b) QD emission when the cavity is resonantlyexcited. Experiments to obtain (a) and (b) are performed at55 K. The cavity wavelength is 931 . .
15 nm and (b) 931 . .
15 nm is very weak under resonant excitation ofthe cavity. Hence, for (b) another QD at 931 . . . the cavity and the QD transition. Details of three sys-tems are given in the Table I. The detuning betweenthe cavity and a particular QD transition is controlledby varying the sample temperature. As the limited tem-perature tuning range limits the range of achievable QD-cavity detunings, multiple QDs must be chosen to coveran extended range of detunings. However, all three sys-tems show similar qualitative behavior.In the first experiment, we observe saturation of thecavity emission with increasing power of the laser usedto excite the QD. We fit the cavity intensity with themodel given by Eq. 8 [Figs. 3 (a),(c), and (e) (solidline)]. In actual experiments, Ω ∝ ηP , where P is themeasured laser excitation power in front of the objectivelens and η is a constant factor signifying the percentageof incident light coupled to the QD. Hence, assuming thatboth the QD spontaneous emission rate 2 γ and the puredephasing rate γ d are independent of the laser excitationpower, ˜ P = αP , where α is a constant factor, indepen-dent of the laser power. α is determined from the fit tothe cavity intensity with the excitation laser power. Inaddition to emission saturation, we see broadening of theQD linewidth with increasing excitation laser power, asmeasured from Lorentzian fits similar to the one shownin Fig. 2 (c). Measurements of the QD linewidth as afunction of the laser power for the three different QDsstudied are plotted in Figs. 3 (b), (d), and (f). Using the TABLE I: Details of the QD-cavity systems employed in thefirst experiment, when the cavity emission is observed by res-onantly exciting the QD. Also shown are the fits for two dif-ferent contributions to the QD linewidth, ∆ ω c and ∆ ω , andthe theoretical estimate for ∆ ω c (see Eq. 10).QD Tempe- QD Cavity ∆ ω c / π ∆ ω / π ∆ ω c / π rature Wave- Wave- (Fit) (Fit) (Theory)length length(K) (nm) (nm) (GHz) (GHz) GHz S .
15 934 . . .
96 1 . S . . . . . S .
15 931 . . . extracted values of ˜ P = αP (as previously explained),the linewidths are fit with the model given by Eq. 10[Figs. 3 (b),(d), and (f) (solid line)]. The fitting param-eters are shown in Table I.We note that for the QD S
1, the value of ∆ ω obtainedfrom the fit is of the same order of magnitude as thelinewidth of a resonantly driven QD without a cavity(∆ ω/ π ∼ . ω for the second ( S
2) and the third ( S
3) QD canbe attributed to high dephasing rate at higher sampletemperature [16] and the vicinity of etched surfaces ofthe photonic crystal.To theoretically estimate ∆ ω c / π (contribution fromthe increased emission into the cavity mode as given byEq. 4) in Table I , we assume g = κ . This is an overes-timated value of g as our system is not strongly coupled(which is confirmed by bringing the QD onto resonancewith the cavity). The overestimated g leads to an overes-timate of ∆ ω c . However, we find that even those theoret-ically overestimated ∆ ω c values are still much lower thanthe experimental data shown in Table I. Just pure QDdephasing cannot explain this finding as dephasing con-tributes only to the term ∆ ω . The increased broadeningindicates a higher coupling strength between the QD andthe cavity exceeding what our theoretical model predicts.One possible explanation of this incoherent coupling isthat the resonantly excited QD couples to the continuumstates provided by the wetting layer or neighboring GaAslayers via tunneling [17] or Auger process [18]. This con-tinuum of states then couples to the off-resonant cavityleading to the observation of cavity emission.We now analyze the linewidth of the process [Fig. 2(d)]responsible for transferring photons from the resonantlyexcited cavity to the QD. We perform the second type ofexperiment (exciting the cavity and collecting emissionfrom the QD) on two QD-cavity systems (Table II). TheQD described in the first row of Table II is the same asthe QD used in the first experiment (second row of TableI). The other two systems shown in Table I could not beemployed in this experiment, as they either showed noemission or very weak emission from QD line under cavityexcitation. Hence, we employed another QD system ( S µ W) C a v i t y I n t en s i t y ( c . c . ) (a) 0 1 2 395100105110115 Laser Power ( µ W) ∆ ω ( G H z ) (b)0 2 4 60100200 Laser Power ( µ W) C a v i t y I n t en s i t y ( c . c . ) (c) 0 2 4 6130140150160 Laser Power ( µ W) ∆ ω ( G H z ) (d)2 4 6 8 1002040 Laser Power ( µ W) C a v i t y I n t en s i t y ( c . c . ) (e) 2 4 6 8 10135140145150 Laser Power ( µ W) ∆ ω ( G H z ) (f) S1 S1S2 S2S3 S3
FIG. 3: (color online) (a),(c),(e): Integrated cavity emissionas a function of the excitation power of the laser resonantlypumping the QD, for the three QD-cavity systems studied.(c.c. stands for CCD count.) The solid lines are fits to thedata using the model given by Eq. 8. (b),(d),(f): Corre-sponding measured linewidths [as in Fig. 2 (c)] as a functionof the laser excitation power. The solid lines are fits to thedata using the model given by Eq. 10. The excitation laserpower is measured in front of the objective lens.TABLE II: Details of the QD-cavity systems employed in thesecond experiment, when the QD emission is observed by res-onantly exciting the cavity. Also shown are the values of the∆ ω c . QD Tempe- QD Cavity ∆ ω c / π rature Resonance Resonance (GHz)(K) (nm) (nm) S . . . S . . . described in Table II.Figs. 4 (a),(c) show the QD intensity as a function ofthe power of the laser resonantly pumping the cavity. Weobserve saturation of the integrated QD emission and thedata fit well with the model given by Eq. 8. In this ex-periment, we also measure the cavity linewidth ∆ ω c , buthere we scan the laser wavelength across the cavity andcollect the integrated emission from the QD. In addition,we also measure the intrinsic cavity linewidth ∆ ω c fromcavity reflectivity measurements at low laser power. Inreflectivity measurements, the laser is scanned across thecavity linewidth and the cavity reflected laser power is observed, as in our previous work [1]. For both cavities,the linewidths ∆ ω c , extracted from the second type of ex-periment (exciting cavity resonantly and imaging emis-sion at QD wavelength) are larger than the linewidth∆ ω c obtained in reflectivity measurements. Figs. 4(b),(d) show the difference between two linewidths, i.e., Laser Power ( µ W) Q D I n t e n s i t y ( c . c . ) (a) Laser Power ( µ W) ∆ ω c − ∆ ω c0 ( G H z ) (b) Laser Power ( µ W) Q D I n t e n s i t y ( c . c . ) (c) Laser Power ( µ W) ∆ ω c − ∆ ω c0 ( G H z ) (d)S2 S2S4 S4 FIG. 4: (color online) (a),(c): Integrated QD emission as afunction of the excitation power of the laser resonantly pump-ing the cavity, for the two QD-cavity systems studied (see Ta-ble II). (c.c. stands for CCD count.) The solid lines are fitsto the data using model given by Eq. 8. (b),(d): The differ-ence between the cavity linewidth ∆ ω c measured by observingthe QD emission and the cavity linewidth ∆ ω c obtained fromthe cavity reflectivity measurements, as a function of the laserpower. The solid line is a linear fit to the difference. The ex-citation laser power is measured in front of the objective lens. (∆ ω c − ∆ ω c ), which increases linearly with laser power.This additional broadening is attributed to the free car-riers generated by the laser excitation.In conclusion, we studied the off-resonant QD-cavitycoupling under resonant excitation of both the QD andthe cavity. We found that pure dephasing along withpower broadening and coherent coupling between thecavity and the QD underestimate the QD linewidth. Thisindicates a higher incoherent coupling strength betweenthe QD and the cavity, possibly resulting from the cou-pling to the continuum of states of the wetting layer orneighboring GaAs [17, 18].The authors acknowledge financial support providedby the National Science Foundation, Army Research Of-fice and Office of Naval Research. A.M. was supportedby the Stanford Graduate Fellowship (Texas Instrumentsfellowship). [1] D. Englund, Andrei Faraon, Ilya Fushman, Nick Stoltz,Pierre Petroff, and Jelena Vuckovic. Controlling cavity reflectivity with a single quantum dot. Nature , 450:857,2007.2] K. Hennessy, A. Badolato, M. Winger, D. Gerace,M. Atature, S. Gulde, S. Falt, E. L. Hu, andA. Imamoglu. Quantum nature of a strongly-coupled sin-gle quantum dot-cavity system.
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