Coherent excitation of a strongly coupled quantum dot - cavity system

Abstract

We have studied the coherent excitation of a strongly coupled QD/photonic crystal cavity system. Time-resolved reflectivity measurements show the vacuum Rabi oscillation of the dot in the cavity. Next, we considered the resonant driving of a cavity-detuned dot, which efficiently populates the cavity mode. This cavity-controlled read-out channel allows high-resolution single quantum dot spectroscopy. Autocorrelation measurements on the cavity mode show antibunching and suggest the use of the resonantly driven QD/cavity system as an on-demand source of single photons with potentially near-unity indistinguishability.

Full text

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Coherent excitation of a strongly coupled quantum dot -
cavity system
Dirk Englund1,2, Arka Majumdar1, Andrei Faraon1,3, Mitsuru Toishi4, Nick Stoltz5, Pierre Petroff5
& Jelena Vuˇ ckovi´ c1
1. Department of Electrical Engineering, Stanford University, Stanford CA 94305
2. Department of Physics, Lyman Laboratory, Harvard University, Harvard MA 02138
3. Department or Applied Physics, Stanford University, Stanford CA 94305
4. Sony Corporation, Shinagawa-ku, Tokyo, Japan, 141-0001
5. Department of Electrical and Computer Engineering, University of California, Santa Barbara,
CA 93106
Photonic nanocavities coupled to semiconductor quantum dots are becoming well developed
systems for studying cavity quantum electrodynamics and constructing the basic architec-
ture for quantum information science. One of the key challenges is to coherently control the
state of the quantum dot/cavity system for quantum memory and gates that exploit the non-
linearity of such a system1–4. Recently, coherent control of quantum dots has been studied
in bulk semiconductor5–7. Here we investigate the coherent excitation of a strongly coupled
InAs quantum dot - photonic crystal cavity system. When the quantum dot and cavity are
on resonance, we observe time-domain Rabi oscillation in the transmission of a laser pulse.
This coherent excitation promises to enable an all-optical method to observe and manipulate
the state a single quantum dot in a cavity. When the detuned dot is resonantly excited, we
show that the resonantly driven quantum dot efficiently emits through the cavity mode, an
effect that is explained in part by an incoherent dephasing mechanism similar to recent the-
oretical models8–10. When the detuned quantum dot is resonantly excited, the cavity signal
represents a spectrally separated read-out channel for high resolution single quantum dot
spectroscopy. In this case, we observe antibunching of the cavity mode. Such a single photon
source could allow photon indistinguishability that approaches unity10as could lift the lim-
1
arXiv:0902.2428v2 [quant-ph] 19 Feb 2009

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itation due to dephasing and timing jitter11,12. The single photon emission is controlled by
the cavity resonance, which relaxes the demands for spectrally matching quantum dots for
two-photon interference and may therefore be of use in linear optics quantum computation13
and quantum communication14,15.
The optical system consists of a photonic crystal (PC) cavity fabricated in a 160-nm thick
GaAs membrane by a combination of electron beam lithography and dry/wet etching steps, as
discussed in Ref.17. The membrane contains a central layer of self-assembled InGaAs quantum
dots (QDs) with an estimated density of 50/µm2. The completed photonic crystal is shown in the
scanning electron micrograph in Fig.1(c).
Weemployagrating-integratedcavity(GIC)designforimprovedinputandoutputcoupling16.
The cavity design is based on a linear three-hole defect cavity18. The perturbed cavity design in-
creases the directionality of the unperturbed cavity emission and improves the in- and out-coupling
efficiency16. The integrated grating has a second-order periodicity as indicated in Fig.1(b).
We first characterize the QD/cavity system by its photoluminescence. The sample is main-
tained between 10K and 50K in a liquid-He continuous flow cryostat. A continuous-wave (cw)
laser beam at 860nm excites electron-hole pairs which can relax through a phonon-mediated pro-
cess into radiative levels of the QD. This above-band driving case corresponds to the pump fre-
quency ωptuned above the single-exciton frequency that is labeled as X in Fig.1(d). Fig.1(e) plots
the photoluminescence spectrum. As we sweep the temperature, we observe the anticrossing be-
tween the QD emission and the cavity emission that is characteristic of the strong cavity-emitter
coupling. From the QD/cavity spectrum, we estimate the system parameters summarized in the
table in Fig.1(d).
Next, we resonantly excite the QD by coupling a tunable cw-laser beam into the cavity. The
laser has a linewidth below 300kHz. Using the cross-polarized setup illustrated in Fig.1(a), we ob-
serve the transmission of the incident vertically polarized laser via the cavity into the horizontally
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Figure 1: (a) Cross-polarized confocal microscope setup. (b) Eyfield in the L3-cavity structure. The structure is
perturbed at the sites indicated by dashed circles, as described in Ref.16. (c) Scanning electron micrograph of fabri-
cated structure. (d) Energy levels of the coupled QD/cavity system showing the two polarization states of the single
exciton (X,Y) and the bi-exciton state (XX). The table lists the system parameters derived from the measurements,
where g,κ,γ,γ∗are the vacuum Rabi frequency, cavity field decay rate, dipole decay rate, and dipole dephasing
rate, respectively. (e) The photoluminescence (PL) shows the QD/cavity anticrossing as the QD is temperature-tuned
through the cavity. (f) Vacuum Rabi splitting observed in the reflectivity from the strongly coupled QD/cavity system.
For comparison, we show the reflectivity of an empty cavity. (g) PL lifetime ∼ 17 ps when the QD is tuned into
the cavity and excitation wavelength λp= 878nm. The emission that is expected theoretically, based on the system
parameters in (d) and a 10-ps relaxation time into the single exciton state, is shown in the solid line. The bottom panel
plots the corresponding expected excited state population |ce(t)|2.
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polarized component. We clearly observe the vacuum Rabi splitting19–21, which demonstrates that
we are coherently probing the QD/cavity system. The reflectivity signal nearly vanishes when the
laser field is resonant with the QD single exciton (X) frequency, showing that the QD has a very
high probability of being in the optically bright state. We obtain good agreement with theory (solid
line fit in Fig.1(f)). In this fit, we used the same parameters as derived from the photoluminescence
(PL) data in Fig.1(e). When the quantum dot is on resonance with the cavity and is pumped at
878nm, the PL decays with a characteristic time of 17ps. The decay time matches a theoretical
model based on the system parameters, shown in the solid line in Fig.1(g). The model considers
a quantum dot which is pumped into the single-exciton excited state (including timing jitter) and
then decays into free space and the cavity mode (see Methods). The state of the QD is described
by a superposition of ground and excited states, |ψ(t)? = cg(t)|g? + ce(t)|e?. The expected ex-
cited state population |ce(t)|2that is predicted by the fitting model is plotted in the bottom panel
of Fig.1(g). Previous measurements of the decay time gave values exceeding 60ps in the strong
coupling regime22, which is longer than expected for the strong coupling regime where the decay
time should be on the order of the cavity ring-down time of ∼ 5ps for a Q ∼ 104. We attribute the
short decay time in this case to the observation that nearly all emission collected from the cavity
originates from the QD. For a very similar system, we previously showed that the cavity mode is
strongly antibunched to (g(2)(0) ∼ 0.05)23. The perturbed cavity lifts the QD signal far above the
background and thus eliminates the collection of the long lived emission lines that would degrade
antibunching.
In the PL and reflectivity measurements described above, we characterized the QD/cavity in
the frequency regime. We will now describe a method to observe the system dynamics by direct
time domain measurements of the vacuum Rabi frequency. Instead of the cw probe laser, we
now use a spectrally filtered Ti:Sapphire laser with 40ps pulses at 80 MHz repetition rate. Since
the reflected beam intensity is weak when detuned from the cavity (as evident from Fig.1(f)),
we performed this measurement when the QD was resonant with the cavity, although a range of
detunings and vacuum Rabi frequencies are in general possible. In Fig.2, we plot the time-resolved
reflected pump intensity, which is acquired with a streak camera (see Methods). The period of 39
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Figure 2: Time-resolved reflectivity measurement shows Rabi oscillation frequency g/2π = 29GHz. We show three
excitation powers in (a,b,c). The observations are fit with a full master equation model using the parameters given
above; in all three plots, the photon flux into the cavity is scaled equally from the measured incident power (0.1nW,
0.23 nW, 1 nW before the lens).
ps closely matches the expected Rabi period T = 2π/g = 40ps. At higher pump power, the Rabi
oscillation becomes less visible (Fig.2(c)). In all observed cases, we found good agreement with
theory (plotted in solid lines), which is obtained using a quantum Monte Carlo simulation (see
Methods). The fits assume the values of g,γ, and κ obtained from spectral measurements, i.e., PL
and reflectivity. The time-resolved reflectivity presented here offers a direct tool for coherently
manipulating the state of the QD/cavity system.
We will now consider the resonant driving of the quantum dot when it is detuned from the
cavity and will demonstrate the use of the cavity mode as a convenient read-out and pumping
channel for resonant single quantum dot spectroscopy and single photon generation. We first lower
the temperature to 10K, which blue-detunes the dot by δ = λd− λc= −1.17nm from the cavity
resonance. The laser excitation is polarized at 45◦to the cavity mode, where its alignment can be
optimized for the reflectivity signal shown in Fig.1(f). Then we scan the laser across the QD and
cavity resonances, as shown in Fig.3. The excitation laser power is ∼ 12nW before the objective
lens. Precisely when the laser becomes resonant with the QD, we observe a strong emission into
the cavity mode. Thus the cavity represents a strong read-out channel for resonant quantum dot
spectroscopy: the resonantly driven QD emits into the cavity mode, which is far detuned and easy
to separate spectrally. Alternatively, if the cavity mode is pumped, the QD single-excition line
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radiates. Figs.3(b,c) plot spectra when the (QD,cavity) are pumped.
Fig.3(d) plots the integrated cavity emission as a function of the laser pump wavelength
λp. The QD absorption linewidth is measured to be lower than 0.006 nm (2 GHz) – about five
times narrower than the ∼ 0.03 nm resolution of our 0.75 m spectrometer. The excitation laser
showed slight mode-hopping; if improved, the resolution should be considerably below 2 GHz.
This cavity-enhanced spectroscopy technique adds an important tool to the repertoire for resonant
single quantum dot spectroscopy24–27. Resonance fluorescence from a QD in a cavity was pre-
viously reported in a planar optical cavity28; however, the excitation geometry used in Ref.28 is
difficult to realize in cavity designs for high Purcell regime or strong coupling, such as photonic
crystals or microdisks. The cavity-enhanced spectroscopy shown here should be applicable for
solid state cavity QED systems with many cavity designs, so long as the QD has a large enough
pure dephasing rate to drive the cavity. Fig. 3(e) plots the integrated QD intensity as the laser is
scanned over the cavity resonance. The QD dot then emits with a linewidth that appears limited by
the our spectrometer resolution of (0.03 nm).
The mechanism that allows the quantum dot to drive the far off-resonant cavity is not yet
completely clear. It has previously been reported that quantum dots that were pumped through
higher excited states or the QD wetting layer can drive the cavity even when it is far detuned22,29,30.
Several recent theoretical models attribute the off-resonant driving of the cavity mode to a pure
dephasing mechanism of the quantum dot8–10. We describe our experimental data with a quantum
master equation model that considers the dephasing as an additional Liouvillian Ldwhich depends
on a dephasing rate γ∗(see Methods). The dephasing term allows driving of the cavity (QD)
through the QD (cavity), as is shown in the fit in Figs.3(c,d), where we used γ∗= 0.1g. This value
of γ∗was measured independently (see Fig.4(f)) and agrees with values cited in the literature for
resonant excitation studies31.
To explore the effect of pumping the off-resonant dot, we measured the cavity emission at
various detunings of the QD exciton. At each cryostat temperature given in Fig.3(h), we tune the
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cw excitation laser to the QD, keeping the power constant at 50 ± 0.5 nW before the lens. The
integrated cavity intensity is plotted with detuning δ = ωcav− ωqd. In our model, we assume a
temperature-dependent dephasing32rate γ = γ0+ α0T, with α0= 0.5µ eV K−1and γ0= κ/100.
The theory does not fully explain the observation, suggesting that pure dephasing is only a part of
the off-resonant driving mechanism between the QD and cavity. Phonon-mediated and two-photon
absorption processes probably also play a role, but are not captured in our model.
We now explore the pump power dependence of coherent excitation. Fig.3(f) shows the
spectrometer intensity at Pin= 200nW. One striking difference is that the features are far more
blurred; we believe this results in part because of increased spectral diffusion at high intensity33,34.
A careful scan across the QD, where the tail of the excitation laser is subtracted, gives the cavity
emission spectrum shown in Fig.3(g). The scan reveals a second peak when the pump is tuned to
λp= 919.85nm. In Fig.3(h), we plot the cavity emission as a function of excitation power when
λpis tuned to the lines one by one. The single-exciton line shows a linear pump dependence, as
expected. The second line shows a quadratic pump dependence, which suggests a bi-exciton state
that is resonantly pumped by two-photon absorption. The power of this line was too low to confirm
this identification by a cross-correlation measurement with the single exciton emission.
We next turn to measurements of the second-order correlation function g(2)(τ) to study the
quantum nature of the QD/cavity system. We estimate g(2)(τ) by a measurement of the autocor-
relation using a Hanbury-Brown-Twiss (HBT) setup. With the half wave plate in Fig.1(a) aligned
to maximize the cavity emission, we obtain a high isolation of the pump, as shown in Fig.4(a).
Under this polarization setting, we observed only the single-excition absorption line through the
cavity emission. We spectrally filter the cavity emission using a 0.2 nm grating filter before the
emission is sent to the HBT setup. The autocorrelation histogram of the cavity emission when the
QD is resonantly excited is shown in Fig.4(b). The antibunching depth is limited to g(2)(0) ∼ 0.76
because the ∼ 300ps detector resolution is longer than the excited state lifetime τ0≈ 118ps, which
is independently measured (see below) and is slower than the 17ps shown in Fig.1(g) because the
QD is detuned from the cavity.
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Figure 3: Coherent QD spectroscopy through cavity mode. (a) Intensity on spectrometer when the pump wavelength
λpis scanned across the detuned QD/cavity system (the main laser excitation is accompanied by a red-detuned side
mode that is 200 times lower in intensity; temperature is 10K). When λpis resonant with the dot (cavity), the cavity
(QD) intensity rises. (b) Spectrum when QD is pumped (the middle peak corresponds to the laser side mode) (c)
spectrum when cavity is pumped. In (b) and (c), we model the driving mechanism by a pure dephasing process with
γ∗= 0.1g. The model agrees well with the emission through the cavity (QD) when the QD (cavity) is pumped. (d)
The integrated cavity emission as a function of the pump wavelength λpshows the single exciton absorption resolved
to 3GHz. (e) The integrated QD emission shows a non-lorentzian dependence on the pump frequency into the cavity.
(f) At high pump power, a second line becomes visible at λp= 919.87nm. (The lines are slightly shifted since the
temperature was, for technical reasons, raised to 18 K). (e) Integrated cavity emission as λpis scanned. Inset: The
power dependence of the two lines suggests exciton (X) and bi-exciton (XX) states. (g) Dependence of integrated
8
cavity emission on QD detuning when the QD is resonantly pumped.

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The detector resolution is less problematic under pulsed excitation. To limit the pump over-
lap with the cavity emission, we use 40-ps probe pulses at 80 MHz repetition rate. The center
excitation wavelength is resonant with the QD at 919.5 nm. Fig.4(c) shows the autocorrelation
histogram of the cavity emission, which indicates g(2)(0) = 0.19(1). We believe the main con-
tribution to counts near τ = 0 is due to the tail of the pulsed excitation laser. Because both the
excitation and the emission into the cavity are faster than the detector resolution, it is not possible
to distinguish them temporally. We believe the τ = 0 peak should be significantly lower with
better spectral filtering or higher QD-cavity detuning.
The autocorrelation measurements demonstrate the use of the resonantly driven QD as an
on-demand single photon source. Such a single photon source has several advantages over pre-
viously reported QD-based single photon sources. The coherent driving mechanism eliminates
timing jitter which results from the random relaxation time of electron-hole pairs under above-
resonant excitation. Furthermore, the emission is stabilized by the cavity frequency. The photon
indistinguishability can approach unity, albeit at the cost of efficiency10. In quantum dots that
are incoherently pumped through a higher excited state, the combination of dephasing and timing
jitter12appears to limit the mean wavefunction overlap to about ∼ 90% for the types of QDs em-
ployed here35. Since the emission occurs through the cavity, it is easier to match the emission from
distant QDs for applications such as remote QD entanglement via interference of single photons
emitted14,15.
We performed time resolved measurements of the emission of the resonantly excited QD into
the cavity mode. Fig.4(d) shows the 40-ps pump pulse and cavity emission, measured simultane-
ously on a streak camera. The QD-driven cavity emission lifetime is τ ∼ 118ps when the dot is
detuned by δ = −1.2nm. We can use this lifetime measurement to infer the dephasing rate. We
fit the decay time by a Monte Carlo simulation of the master equation, where the QD begins in the
ground state and is driven by the 40-ps pump pulse (see Methods). All parameters except for γ∗
are fixed as before. The fit gives γ∗= 0.10(1)g.
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Figure 4: (a) Resonant QD excitation at 10K. For the autocorrelation measurements, the cavity emission is filtered
to a width of 0.2nm. (b) Autocorrelation histogram of cavity emission when the QD is pumped resonantly at a power
of 12nW in cw mode. (c) Pulsed autocorrelation measurement. (d) Time-resolved resonant pumping of the quantum
dot and emission into the cavity mode. A theoretical fit to the cavity emission yields an estimate of the pure dephasing
rate.
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We have studied coherent excitation of a strongly coupled QD/photonic crystal cavity sys-
tem, with three main results. First, time-resolved reflectivity measurements on the QD/cavity
show the vacuum Rabi oscillation of the dot in the cavity and enable a direct means for observ-
ing and manipulating the QD. Second, we considered the resonant driving of a cavity-detuned dot
which efficiently populates the cavity mode, adding insight to previous non-resonant studies of
this phenomenon22,29,30. This cavity-controlled read-out channel allows high-resolution, resonant
single quantum dot spectroscopy. Third, we demonstrated an on-demand single photon source
relying on a resonantly driven quantum dot. This source promises unity indistinguishability10.
Since the emission frequency is set by the cavity resonance, which is easier to control than the
inhomogenously distributed QD frequency, this source is furthermore appealing for creating en-
tanglement by photon interference14,15. In the future, much larger coupling efficiencies will be
required. The all-optical techniques discussed here are compatible with integrated photonic crystal
structures, where cavities coupled to single quantum dots may be connected through networks of
waveguides36–39and other chip-integrated elements.
Acknowledgements
We thank Hideo Mabuchi for helpful discussions. Financial support was provided by
the Office of Naval Research (PECASE and ONR Young Investigator awards), National Science Foundation,
Army Research Office, and DARPA Young Faculty Award. A.M. was supported by the SGF (Texas Instru-
ments Fellow). Work was performed in part at the Stanford Nanofabrication Facility of NNIN supported by
the National Science Foundation.
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to Dirk Englund (email:
englund@stanford.edu).
Methodology
HBT In autocorrelation measurements, light is filtered to 0.2nm, directed through a 50:50 beam
splitter, and coupled through multi mode fibers to single photon counter modules. Coincidences
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are recorded on a time interval analyzer.
Time-domain dynamics of QD/cavity system Fig.2: We reflect 40 ps pulses from the cavity.
The temperature of the cryostat is adjusted to tune the single exciton transition into the cavity.
The spectral and spatial alignment are optimized with the scanning cw laser in the cross-polarized
arrangement shown in Fig.1(a). The reflected beam is now recorded on a Hamatsu C5680 streak
camera.
A quantum Monte Carlo simulation based on Ref.40 is used to model the time-dependent
reflectivity. The values for g,κ,γ,∆λ are taken from the spectral characterization. A Gaussian
classical field E(t) = E0exp(−t2/2σ2
is initialized into the ground state |g?. With all other system parameters fixed, the amplitude E0
is adjusted for the experiment with lowest power of Pin = 0.1nW. This calibration fixes E0for
t) drives the cavity mode with a FWHM of 40ps. The QD
experiments with higher power. Each simulation yields the time-dependent cavity photon number
?a(t†(t)a(t)?.
From the reflectivity in Fig.1(f), we estimate the QD has a probability of being in an opti-
cally dark state of pD≈ 0.2, meaning that a background signal corresponding to an empty cavity
reflectivity must be added, giving ?a†(t)a(t)??= pD?a†(t)a(t)?g=0+ (1 − pD)?a†(t)a(t)?, where
we set g = 0 for the first term and pD= 0.2. The final fit is obtained by convolving ?a†(t)a(t)??
with the streak camera response of 3ps.
Fig.1(g): To model the PL when the QD is tuned to the cavity and excited incoherently
throughahigherenergylevel, weusetheMonteCarlosimulationdescribedabove, butinitializethe
QD in the excited state |e?. The time jitter is modeled by finding a weighted average of relaxation
times, with a 1/e time of 10ps.
Fig.4(d): In the Monte Carlo simulation, the QD is detuned by δ and driven resonantly. The
QD is initialized into the ground state and the single-excition transition is driven by the classical
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field. The Hamiltonian now has a driving term E(t)(σ++ σ−), where σ+,−are the raising and
lowering operators of the quantum dot.
Supplemental Material
Dephasing model The Master equation describing a QD (lowering operator σ = |g??e|) coupled
to a cavity mode (with annihilation operator a) is given by
dρ
dt= −i[H,ρ] +κ
2(2aρa†− a†aρ − ρa†a) +γ
2(2σρσ†− σ†σρ − ρσ†σ) +γd
2(σzρσz− ρ) (1)
where γ, κ and γdaccounts for QD population decay, cavity population decay and QD pure de-
phasing; σz = [σ†,σ]. H is the hamiltonian of the system without considering the losses and is
given by
H =∆
2a†a −∆
2(σz) + ig(σa†− aσ†)
(2)
∆ is QD-cavity detuning and the reference energy is the mean of QD and cavity energy. Using eq.
1 one can write10
?da
?
dt
= (i∆
?
= (−i∆
2−γ
2−κ
2)?a? + g?σ?
(3)
?dσ
dt2− γd)?σ? + g?σza?
(4)
(5)
Thesetwocoupledequationsaresolved. Initialconditiona(0) = 1;σ(0) = 0anda(0) = 0;σ(0) =
1 correspond to pumping the cavity and the dot respectively. Let us define A =?−i∆
2−κ
2
?and
B =?i∆
When the dot is pumped, the solution in time domain is given by
2−γ
2− γd
?and λ−= A + B −?(A − B)2− 4g2, λ+= A + B +?(A − B)2− 4g2.
a(t) =
(eλ+t− eλ−t)g
?(A − B)2− 4g2
(6)
and
σ(t) =(A − B)(eλ−t− eλ+t) +?(A − B)2− 4g2(eλ−t+ eλ+t)
2?(A − B)2− 4g2
13
(7)

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So the spectrum emitted by the cavity (Scav) and by the QD (SQD) are given by (by virtue of
Quantum Regression theorem):
Scav(ω) =
g2
(A − B)2− 4g2
A − B +?(A − B)2− 4g2
ω − λ−
????
1
ω − λ−
−
1
ω − λ+
−A − B −?(A − B)2− 4g2
ω − λ+
????
2
(8)
SQD(ω) =
1
2((A − B)2− 4g2)
?????
?????
2
(9)
Similarly, if the cavity is pumped, then depending on the other initial condition one can solve and
get:
a(t) =(B − A)(eλ−t− eλ+t) +?(A − B)2− 4g2(eλ−t+ eλ+t)
(eλ−t− eλ+t)g
?(A − B)2− 4g2
So the spectrum becomes:
?????
SQD(ω) =
(A − B)2− 4g2
In our experimental setup as the collection efficiency from cavity is much higher compared to QD,
2?(A − B)2− 4g2
(10)
σ(t) =
(11)
Scav(ω) =
1
2((A − B)2− 4g2)
B − A +?(A − B)2− 4g2
ω − λ−
−B − A −?(A − B)2− 4g2
ω − λ+
????
?????
2
(12)
g2
????
1
ω − λ−
−
1
ω − λ+
2
(13)
hence we can assume that in experiment what we are observing is Scav(ω).
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