Solid State Communications 149 (2009) 1479–1484

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Solid State Communications

journal homepage: www.elsevier.com/locate/ssc

Strong optical field study of a single self-assembled quantum dot

Xiaodong Xu a,1, Bo Sun a, P.R. Berman a, Duncan G. Steel a,∗, Daniel Gammon b, L.J. Sham c

a The H. M. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, MI 48109, USAb Naval Research Laboratory, Washington, DC 20375, USAc Department of Physics, University of California San Diego, La Jolla, CA 92093, USA

a r t i c l e i n f o

Article history:Received 4 February 2009Accepted 27 April 2009 by the Guest EditorsAvailable online 18 May 2009

PACS:78.67.HC42.50.Gy42.50.Hz78.47.Fg

Keywords:A. NanostructureD. Optical propertiesE. Nonlinear optics

a b s t r a c t

We review the investigation of a single quantum dot driven by a strong optical field. By coherent pump-probe spectroscopy, we demonstrate the Autler–Townes splitting and Mollow absorption spectrum ina single neutral quantum dot. Furthermore, we also show the typical Mollow absorption spectrum bydriving a singly charged quantum dot in a strong optical coupling regime. Our results show all the typicalfeatures of an isolated atomic system driven by a strong optical field, such as the AC stark effect, Rabi sidebands and optical gain effect, which indicate that both neutral and charged quantum dots maintain thediscrete energy level states even at high optical field strengths.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Semiconductor Quantum dots (QDs) are customizable solidstate analogs of atoms [1]. Due to their unique atomic like phys-ical properties, QDs have been proposed in numerous quantummechanical based applications, one of which is optically drivenquantum information processing and computation [1,2]. Signifi-cant research progress has been made in this direction, such asinitial qubit state preparation [3–8], coherent optical manipula-tion of a single qubit [9–15], and quantum logic gates based on QDqubits [13,16,17].One topic that has drawn a lot attention recently is the inves-

tigation of QD systems driven by strong optical fields [18–25]. Thecoupling strength between the light field and QD transition is rep-resented by the Rabi frequency ΩR =

µ·Exh , where µ is the tran-

sition dipole moment and Ex is the optical field strength. The QDsystem is driven in the strong coupling regime if ΩR is compa-rable or larger than the transition linewidth. In atomic systems,

∗ Corresponding author.E-mail address: dst@umich.edu (D.G. Steel).

1 Present address: Center for Nanoscale Systems, Cornell University, Ithaca, NewYork 14853, USA.

0038-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2009.04.042

it has been shown that the strong coupling can lead to interest-ing spectral features, such as triplets in the emission spectrum,known as Mollow triplets [26], and an optical gain effect in the ab-sorption spectrum [27,28], known asMollow absorption spectrum(MAS) [29].The opportunities for observing these quantum optics effects in

the QD system come from its atomic properties, which suppressesthe many body effects which exist in higher dimensionalsemiconductor systems. In addition, a QD has a much larger dipolemoment compared to atomic transitions. The research progress onthe strong coupling between laser light and QDs has grown rapidlyin the last couple of years. Autler–Townes splitting [30], optical ACstark effect, Mollow triplets, and MAS all have been demonstratedfor a single neutral exciton [18,21], bi-exciton [20,22], and chargedQD systems [19,25].In this paper, we use coherent pump-probe spectroscopy to

study the dynamics of a strongly driven quantum dot. For asingle neutral QD, we utilize the V system formed by the twopolarized neutral exciton states. A strong pump is fixed to beresonant with one exciton transition. If the weak probe scansacross the coupled exciton transition, we observe two Lorentzianlineshapes, known as the Aulter–Townes (AT) doublet. If the weakprobe is on the same transition as the strong pump, we observeMAS, which is composed of a weak center peak and two Rabiside bands with dispersive lineshapes. The negative part of the

1480 X. Xu et al. / Solid State Communications 149 (2009) 1479–1484

a b c

ΩgR

ΩgR

Fig. 1. Bare state and dressed state pictures of a QD driven by a strong optical field.(a) A strong pump is in resonance with transition H and a weak beam probes eithertransition H or transition V . Dressed state picture (b) without (c) and with light-transition coupling. The dots represent the dressed state populations with a reddetuned pump.

lineshape indicates the probe gain effect. We further demonstratethe MAS of a single charged QD with the pump beam both onand detuned from resonance. When the pump is detuned, theprobe absorption spectrum has three spectral features: one weakdispersive lineshape at the pump frequency, one AC stark shiftedRabi side band due to the absorption, and one Rabi side bandshowing a probe gain effect. All the experimental observationscan be explained by the standard optical Bloch equations used foratomic systems. Our results not only demonstrate the coherentnonlinear interaction between laser light and a singleQDoscillator,they also show that both neutral and singly-charged QDs maintaindiscrete energy levels even when driven by a strong optical field.The paper is organized as following. We first give the physical

model and theoretical explanation of the AT splitting and opticalMAS. Then we present the experimental results of a stronglyoptical driven neutral QD. Finally, we will discuss the MAS of asingly charged QD.

2. Theory

When an atomic transition is driven by a strong optical field,shown in Fig. 1(a), the atomic states are dressed by the laserfields, as shown in Fig. 1(b), where N is the number of laserphotons. Taking into account the coupling between the lightand the atomic transition, the system can be represented by thedressed states in Fig. 1(c) [31], where the separation of the dressedstates in each manifold is the coupling strength hΩgR , whereΩgR is the generalized Rabi frequency. In order to solve for theprobe absorption analytically, we follow the method developedin Refs. [29,32] and describe our system with the optical Blochequations dρdt =

[H, ρ

]+ Decay, where ρ is the density matrix

and H is the Hamiltonian.We first consider the case where the weak beam probes the

transition from state 2 to state 1. By solving the density matrixequations to all orders of the pump and to first order of the probe,we get the weak probe absorption [18]

αAT = αoIm

i4[γ13 − i

(δp − δx

)]ρ(0)22 − 2iΩRρ

(0)32

4(γ21 − iδp

) [γ13 − i

(δp − δx

)]+Ω2R

(1)

where α is a constant, γ13 =γ1+γ32 + Γ13, γ21 =

γ12 + Γ21,

γ23 =γ32 + Γ23, Γij is a dephasing rate of transition ij, γi is the

population relaxation rate of state i, δx = ω32−ωx, δp = ω12−ωp,ωx(ωp) is the pump (probe) field frequency, and ωij is the atomicfrequency of transition ij. Under the condition ofΩR >> γi,Γij, theprobe absorption profile separates into twoRabi side bands located

at E+ and E−, where E± = 12

(δx ±

√Ω2R + δ

2x

), as shown by the

dressed state picture in Fig. 1(c). These two side bands are known

as the Autler-Town doublet. If δx is larger thanΩR, the absorptionpeaks are AC stark shifted. If the pump is tuned to be resonant withtransition ω23, i.e. δx = 0, the two peaks are symmetric, separatedin energy by hΩR.The probe absorption spectrum changes dramatically if the

probe scans the same transition as the pump. Again, solving theoptical Bloch equations to all orders of the pump and to first orderin the probe, we get the absorption coefficient of the probe beam asin Box I, where1 = δx − δp is the probe detuning from the pump,A = γ23 − iδx and B = γ23 + i1. When the pump is in resonanceandΩR is larger than γ23, we can get a simplified form for equationin Box I. If δp ' ±ΩR, we find

αMS = ±αoγ3

ΩR

(δp ∓ΩR

)4(δp ∓ΩR

)2+ (γ3 + γ23)

2. (2)

Eq. (2) shows that there are two dispersive line shapes as a functionof the probe frequency centered at ±ΩR with their zero crossingsat δp = ±ΩR. When

∣∣δp∣∣ > ΩR, the probe experiences absorption.Interestingly, the probe absorption is negative given

∣∣δp∣∣ < ΩR,i.e. the probe sees gain. Since the pump field is in resonance, thiscorresponds to a gain effect without a population inversion in anypicture. Thewings of the dispersive line shapes also provide nearlyconstant gain near δp = 0. However, the gain is canceled by aweakabsorptive component exactly at δp = 0. Explicitly, near δp = 0one finds

αMS = −αo

2

(γ23

ΩR

)2γ3

γ23

(δ2p/γ23

δ2p + γ223

).

The above calculation of the MAS is for a resonant pumpfield. If the pump is detuned from the atomic transition, theprobe absorption lineshapes are quite different. When the pumpdetuning |δx| is comparable or larger than transition linewidth,a simple physical picture can be derived in the fully quantizeddressed state picture. Assuming the pump is tuned to the red ofthe atomic transition, the quantized dressed states can be writtenas [31]|α (N)〉 = c |2,N〉 − s |3,N − 1〉|β (N)〉 = s |2,N〉 + c |3,N − 1〉

where c =√12

(1+ δx

ΩgR

), s =

√12

(1− δx

ΩgR

)and the generalized

Rabi frequencyΩgR =√Ω2R + δ

2x .

The population of the dressed states can be solved by ignoringthe coherence between the dressed states. The steady statesolutions are

ρα,α =c4

c4 + s4, ρβ,β =

s4

c4 + s4.

The above solutions show that under the negative detuning of thepump, δx > 0, the dressed state |α (N)〉 is more populated than|β (N)〉. In Fig. 1(c), we use the size of the dots to represent thedressed state population. There are three transitions shown byFig. 1(c). Since |α (N)〉 is more populated than |β (N + 1)〉, thetransition centered at ωx + Ω

gR is an AC stark shifted absorp-

tion peak (the purple dashed line). The state |α (N + 1)〉 is morepopulated than |β (N)〉and hence, the population inversion of thedressed states induces a probe gain centering at ωx −Ω

gR (the red

dashed line). When the probe frequency approaches the pump,there is a weak dispersive lineshape (light blue dashed line). Thisfeature cannot be derived from the previous method, as it relies onthe coherence between the dressed states.

3. Experiments

The experiment is performed on single self-assembled QDsembedded in a Schottky diode structure [33,34]. There is an Al

X. Xu et al. / Solid State Communications 149 (2009) 1479–1484 1481

αMS = αoIm

iγ3A

[iγ3A∗ (A∗ + i∆)+∆

(Ω2R2 − γ23B− i (B+ i∆) δx + δ

2x

)](γ 23 AA∗ + iΩ

2Rγ23∆

) (B2 + δ2x

)+ γ23Ω

4RB+ γ3

(Ω2R (γ23 + B)

(γ23B+ δ2x

)+

iγ 223B2∆+ iδ2x∆

(B2 + γ 223 + δ

2x

) )

Box I.

mask with micron size apertures on top of the sample to providespatial resolution to do single dot studies. The density of dots isabout 1 dot/µm2. Two continuous wave lasers are used in theexperiment. One serves as a strong pump and the other as a weakprobe. In the experiment, we set both lasers to be linearly polarizedwith orthogonal polarizations. We use a linear polarizer in front ofthe detector to filter out the pump beam so that only the probeabsorption is detected and the background noise on the detectorinduced by the strong pump is eliminated.Using the diode structure, we control the QD potential and the

number of electrons or holes in the QD by varying the voltageacross the sample. In the experiment, we set the voltage such thatthe dot is either uncharged or charged with one electron, allowingus to study neutral QDs and single negatively charged QDs. Inorder to further increase the signal to noise ratio, we adopt theDC Stark shift modulated absorption technique [35]. In additionto the DC voltage applied to the sample, we applied a small ACvoltage. The QD resonance is DC Stark shifted and oscillates at theACmodulation frequency. Therefore,we can use a lock-in amplifierto detect the absorption signal at the modulation frequency. Theamount of the DC stark shift can be controlled by the modulationamplitude. We apply a large modulation amplitude, which shiftsthe resonance by an amount larger than the transition linewidth.This way, the obtained spectrum corresponds directly to the probeabsorption.

4. Results and discussion

We first performed the strong optical field study with a singleneutral QD [18]. Without the optical excitation, the QD is in thecrystal ground state. With the optical excitation, we create anelectron and hole pair, known as an exciton. Due to the QD in-plane anisotropy, a neutral QD has two polarized exciton stateswith linear polarizations [36]. As shown in Fig. 1(a), they form aV system. Fig. 2(a) shows a single beam absorption spectrum withpolarized light excitation. It clearly resolves two polarized excitonstates with a fine structure splitting of∼15µeV. The line on top ofthe data is fit to a Lorentzian.For a demonstration of AT splitting, we set a strong pump

with horizontal polarization on resonance with transition H andprobe transition V with aweak vertically polarized light. The probeabsorption spectra as a function of pump intensity are shown inFig. 2(b). We shift the data vertically for clarity. Compared to thesingle Lorentzian lineshape with one beam excitation, the probelineshape splits into AT doublets with symmetric lineshapes whenthe strong pump is turned on. We noticed that there is a smallenergy shift of the probe spectrum relative to the low pumpintensity excitation. This effect is probably due to a small screeningeffect introduced by the photoexcited charge in the diode whichsaturates at a certain pump power level. When we change thepump intensity, we adjust the pump frequency to follow the shiftof the resonance.The lines on top of the data are theoretical fits, assuming that

condition γ3 = γ1. The good fits of the theory to the data indicatethat the QD behaves like an isolated atomic system. The fitsyield hγ23 and hγ3 of (0.73± 0.07) µeV and (1.48± 0.07) µeV,respectively. The decay rate is almost twice the dephasing rate,

indicating the absence of significant pure dephasing of thisQD. Thisis important for quantum logic applications.Since the separation of the doublet equals the pump Rabi

frequency, the splitting increases as we increase the pump fieldstrength. We plot the peak separation as a function of pump fieldstrength in Fig. 2(c). As expected, a straight line fits the data andcrosses zero in the absence of the pump. As the Rabi frequencyis the product of the dipole moment and pump field strength, weinfer a dipole moment of 30 D for this particular QD.We alsomeasure the AT splitting spectrumwith pump detuned

from transition H. The spectra are shown in Fig. 2(d) with afixed pump Rabi frequency hΩR of ∼4.5 µeV. As we discussedin the theory, the detuning of the pump changes the dressedstate populations. For example, dressed state |α (N)〉 is morepopulated than |β (N)〉 with a negative pump detuning, shownin Fig. 1(c). Therefore, the probe absorption strength is strongerfrom |α (N)〉 to state |1〉 than from |β (N)〉, as we observed in thedata for the negatively detuned pump. The positive detuning canbe understood from the same argument. Since the pump detuningis relative small compared to the transition linewidth, the shift ofthe absorption resonance induced by the pump detuning is notpronounced. AT splitting spectra with large pump detunings canbe found in Ref. [22].When the pump and probe beam are tuned to the same

transition, we are able to get complex MAS lineshapes, whichare composed of a weak center peak and two Rabi side bandswith dispersive lineshapes. The pump intensity dependent MASis shown in Fig. 3(a), where a single beam absorption spectrum isplotted at the bottom for comparison.When the QD is driven in thestrongly coupling regime, the separation of the Rabi sides bandsshould be two times the pump Rabi frequency, which dependslinearly on the pump field strength. We extract the separation ofthe Rabi sidebands and plot it in Fig. 3(b) as a function of pumpfield strength. A line fits the data very well, which indicates theQD is indeed optically driven into a strong coupling regime. Thelines on top of the data are theoretical fits to equation in Box I.Again, a good agreement between theory and experimental dataindicates the isolated atomic features of a single QD system evenwhen driven by a strong optical field.A unique feature of the MAS is the probe gain effect, which

is confirmed by the observations of the negative part of thedispersive lineshapes. This gain is from the coherent energyexchange between the pump and probe beams through the QDnonlinearity. Since the strong pump is on resonance, this is a gaineffect without population inversion in either the bare state pictureor dressed state picture. We define the probe gain efficiency as theratio of the amplitude of the negative signal to the single beamprobe absorption strength. Using the pump Rabi hΩR of 3.3 µeVas an example, the probe gain efficiency corresponds to 3.6%.So far, we have discussed the AT splitting and MAS of a single

neutral QD. These results show a single neutral QD maintains itsdiscrete energy states even when driven by a strong optical field.However, the electronic structure of a single negatively chargedQD is more complicated than a neutral QD. As shown by Fig. 4(a),the excited state of a single charged QD is a many particle system,formed by two electrons and one hole. We can consider the trionsystem as an effective two-level structure due to the degeneracyof the two optically allowed transitions. One import question we

1482 X. Xu et al. / Solid State Communications 149 (2009) 1479–1484

Abs

orpt

ion

(a.u

.)

Pump Detuning (µeV)

0.0

-0.25

-0.50

-1.40

+0.25

+0.50

+1.32

Abs

orpt

ion

(a.u

.)

12

24

6

36

48

60

0

PI (w/cm ) 36

HV

0

8

00

4

Rab

i Spl

ittin

g (µ

eV)

4 8

2PI (w/cm )2

a c

b d

0-25 25

Probe detuning (µeV)

Probe detuning (µeV) Probe detuning (µeV)

Abs

orpt

ion(

1x10

-4)

0-6 6 0-8 8

Field Strength (w/cm ) 2

Fig. 2. Demonstration of AT splitting of a neutral exciton. (a) Polarization resolved exciton states with one beam excitation. The zero of the probe laser energy scale is1329.260 meV. (b) AT splitting spectra as a function of pump intensity (PI). (c) Energy separation of the AT doublets as a function of pump field strength. (d) AT spectra withvarious pump detuning. The zero of the laser energy scale for (b) and (d) is 1329.256 meV.

Probe detuning (µeV)

-8 0 8

ba

Fig. 3. Demonstration of the MAS of a neutral exciton. (a) The MAS as a functionof pump intensity (W/cm2). The zero of the probe laser energy scale is 1329.268meV. (b) The energy separation of the Rabi side bands as a function of pump fieldstrength.

need to ask is how a charged QD behaves when driven by a strongoptical field. Recently, the nonlinear Fano interference effect hasbeen observed in charged QD systems, which indicates a chargedQD could couple to some continuum states when illuminated bya strong optical field [37]. A recent report of a single charged QDdriven by a strong optical field also does not show the typicalfeatures of MAS [23]. All these results indicate that the physicalproperties of a charged QD could be more complex than a neutral

QD system when optically driven in the strong coupling regime.In the following, we demonstrate that a single charged QD drivenby a strong optical field shows the typical MAS features, which arein good agreement with the theory for a two-level atomic systemcoupled to a strong optical field. In the trion study below, we usethe same experimental setup as the above neutral QD, except weuse a different QD.We first set the pump detuning δx to be zero and scan the probe

frequency across the trion transition frequency. Fig. 4(b) showsthe probe absorption lineshapes with various pump intensities.Instead of a Lorentzian absorption lineshape in the absence of thepump, as shown at the bottom of the Fig. 4(b), the lineshape ofthe probe beam in the presence of a strong pump beam showsthe typical MAS: a triplet-like absorption pattern appears withone weak central structure and two Rabi side bands. The negativepart of the absorption signal demonstrates the probe gain effect.Using the pump intensity 95 W/cm2 as a example, the probe gainefficiency is 5.3%.As we discussed in the neutral QD study, the Rabi sidebands

have dispersive lineshapes and zero crossing at ±ΩR providedΩR 2γ , where 2γ is the trion transition linewidth. We extractthe splitting of the Rabi side bands from the data and plot it out as afunction of the pump field strength in Fig. 4(c). SinceΩR is linearlyproportional to Ex, as expected, the extracted splitting dependslinearly on the pump field strength and goes to zero in the absenceof the pump. We also infer a trion dipole moment of 22 D from theabove data,which is comparable to the neutral QD. The observationof the Rabi side bands is a signature of the optical generation of asingle dot trion Rabi oscillations. ThemaximumRabi frequencywedemonstrate in the experiment is hΩR = 6.6µeV, which is limitedby the current experimental configuration.We can extract the trion decay and dephasing rates by fitting

the dispersive lineshapes with the Optical Bloch equations. The

X. Xu et al. / Solid State Communications 149 (2009) 1479–1484 1483

ba

c

Fig. 4. Demonstration of the MAS of a single negative charged QD with pump onresonance. (a) The trion energy level diagramwithout magnetic fields. (b) The MASof a trion state as a function of pump field strength. The zero of the laser energyscale is 1318.797meV. (c) The energy separation of the Rabi side bandswith variouspump intensities.

solid lines in Fig. 4(a) are the theoretical fittings by equation inBox I, which agree well with the data. The fittings yields triondecay and dephasing rates of (2.1±0.4)µeV and (1.4±0.2)µeV,respectively. Since the dephasing rate is almost half of the decayrate, there is no appreciable pure dephasing in this charged QD.As we tune the pump laser frequency away from the trion

transition, the dispersion-like lineshapes of the Rabi side bandsevolve into two Lorentzian-like Rabi side bands and a weakdispersive lineshape in the center. Fig. 5(a) displays the probeabsorption spectrum as a function of the pump detuning with afixed pump intensity of 95W/cm2. We can see that the absorptionstrengths, the lineshapes, and the associated spectral positions arevery sensitive to the pump detuning.We explain the typical spectral features of the probe absorp-

tion spectrum by using the spectrum associated with the pumpdetuning of −6.2 µeV as an example, which is the bottom curveof Fig. 5(a). There is a strong absorption feature located atωx+Ω

gR .

This is the AC Stark shifted absorption peak, corresponding to theblue dashed lines we discussed in Fig. 1(c). A unique feature isthe pure negative absorption lineshape located at ωx −Ω

gR , which

demonstrates the probe gain effect. This gain effect is from the pop-ulation inversion in the dressed state picture, as shown the red linein Fig. 1(c). In the lowest order perturbation theory, this reflects athree photonRamangain effect: theQDabsorbs twopumpphotonsat frequencyωx and emits a photon atωx−Ω

gR . The probe gain effi-

ciency also depends on the pump detuning. AssumingΩR 2γ , ithas been shown theoretically that themaximumprobe gain occursat |δx| =

ΩR3 [38]. For the pump detuning of −1.24 µeV, the data

shows the probe gain efficiency is about 9.7%, which ismuch larger

bΩR

a

Fig. 5. The MAS spectrum of the trion state with a detuned pump. (a) The MASspectrum as a function of pump detunings at a pump intensity of 95W/cm2 . (b) Thespectral positions of the Rabi sides as a function of pump detunings, which showsan anti-crossing feature.

than the probe gain efficiencywhen the pump is on resonancewiththe same pump intensity.Both of the spectral positions of the probe gain and the AC

stark shifted absorption peak can be tuned by varying the pumpdetuning. Fig. 5(b) plots the spectral position of the Rabi side bandsas a function of the pump detuning, where the trion transitionenergy is set as the zero energy point. The photon energy whereprobe gain occurs can be tuned by the simultaneous tuning ofthe pump detuning. Fig. 5(b) also clearly demonstrates a uniquefeature of atomic dressed states: anti crossing behavior of theRabi side bands. The dotted curves in the plot are the theoreticalpredictions of the peak positions as a function of the detuning,which matches the data pretty well. The laser light inducedtransition energy shifts at the large pump detuning are a cleardemonstration of the dynamic, or AC Stark effect.

5. Conclusion

In summary, we have demonstrated AT splitting for a neutralQD andMAS for both a neutral and a single negatively charged QD.Our study shows either a neutral QD or a charged QD can behave

1484 X. Xu et al. / Solid State Communications 149 (2009) 1479–1484

like an isolated quantum oscillator even when driven by a strongoptical field. This opens the door for the demonstration of QDmodulators, dressed state lasers, and quantum logic applicationsinvolving strong optical fields.

Acknowledgements

This work is supported by U.S. ARO, AFOSR, ONR, NSA/LPS, andFOCUS-NSF.

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