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PHYSICAL REVIEW B 96, 041124(R) (2017)

Coherent coupling of individual quantum dots measured with phase-referenced

two-dimensional spectroscopy: Photon echo versus double quantum coherence

Valentin Delmonte,1,2Judith F. Specht,3Tomasz Jakubczyk,1,2Sven Höfling,4,5Martin Kamp,4Christian Schneider,4

Wolfgang Langbein,6Gilles Nogues,1,2Marten Richter,3and Jacek Kasprzak1,2,*

1Université Grenoble Alpes, F-38000 Grenoble, France

2CNRS, Institut Néel, “Nanophysique et Semiconducteurs” Group, F-38000 Grenoble, France

3Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin,

Hardenbergstrasse 36, D-10623 Berlin, Germany

4Technische Physik, University of Würzburg, Würzburg 97074, Germany

5SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom

6School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom

(Received 9 March 2017; published 26 July 2017)

We employ two-dimensional (2D) coherent, nonlinear spectroscopy to investigate couplings within individual

InAs quantum dots (QDs) and QD molecules. Swapping pulse ordering in a two-beam sequence permits one

to distinguish between rephasing and nonrephasing four-wave mixing (FWM) configurations. We emphasize

the nonrephasing case, allowing one to monitor two-photon coherence dynamics. Respective Fourier transform

yields a double quantum 2D FWM map, which is corroborated with its single quantum counterpart, originating

from the rephasing sequence. We introduce referencing of the FWM phase with the one carried by the driving

pulses, overcoming the necessity of its active stabilization, as required in 2D spectroscopy. Combining single and

double quantum 2D FWM provides a pertinent tool in detecting and ascertaining coherent coupling mechanisms

between individual quantum systems, as exemplified experimentally.

DOI: 10.1103/PhysRevB.96.041124

Nuclear magnetic resonance (NMR) spectroscopy con-

ceived phase-locked, multipulse techniques, yielding multidi-

mensional spectra by Fourier transforming temporal sequences

into respective frequency coordinates [1,2]. The possibility

to spread the response of biological or chemical molecules

of high structural complexity, especially proteins, across

many axes enabled one to assess their spatial form and to

understand interatomic interactions and couplings. The idea

to selectively address and evolve subsets of transitions from

congested spectra via a multipulse toolbox, and then projecting

the results onto two-dimensional (2D) or three-dimensional

diagrams, is a far-reaching legacy of NMR. At a juncture

of coherent spectroscopy and condensed matter physics, 2D

spectroscopy provided insight into the dynamics and couplings

of many-body optical excitations in solids, in particular, of

excitons in semiconductor quantum wells [3–5] and novel 2D

layered materials [6], as well as in ensembles of quantum

dots [7,8] (QDs) or nanocrystals [9]. A principal tool in

these investigations is k-resolved four-wave mixing (FWM)

spectroscopy and its extensions probing multiwave mixing

processes [4].

FWM spectroscopy has been exploited over the years to

study Coulomb interactions and related ultrafast coherent

dynamics of excitons in semiconductors [10–14]. In these

ensemble experiments, a strong inhomogeneous broadening

usually is an obstacle to implement coherent control protocols.

*[email protected]

Published by the American Physical Society under the terms of the

Creative Commons Attribution 4.0 International license. Further

distribution of this work must maintain attribution to the author(s)

and the published article’s title, journal citation, and DOI.

This issue is largely overcome when restricting the study

to individual excitons. Moreover, in an exciting context of

optical information processing in solids, establishing con-

trolled channels of coupling within a set of few-level systems

represents a truly challenging goal. For these reasons, it is

necessary to access the coherence of individual excitons, and

then to ascertain the mechanism of their coherent interactions,

involving both Coulomb (local) and radiation (long-range)

mediated phenomena. The present Rapid Communication

represents a step forward in this field.

FWM microscopy of single QD excitons [15] was previ-

ously accomplished by phase-sensitive optical heterodyning

combined with interferometric detection, efficiently subtract-

ing resonant background and permitting colinear geometry

of the excitation pulses. Recently, the detection sensitivity

of intrinsically weak single QD FWM has been enhanced

substantially by using photonic nanostructures, improving

the QD coupling with external laser beams [16–18]. Here,

we perform FWM spectroscopy of individual InAs QDs

embedded in a low-Qsemiconductor microcavity [19]. We

point out two advancements with respect to our recent reports

[16,20]: First, we demonstrate 2D FWM constructed from

two-photon coherences—known as double quantum 2D FWM

[21–24]—driven on individual transitions, specifically QD

exciton-biexciton systems (GXB)[25]. Second, we introduce

referencing of the FWM phase, offering a convenient alter-

native for its active stabilization, which is widely believed

to be required in 2D spectroscopy. Using the one-quantum

and two-quantum spectroscopy, we have measured single QDs

and a QD molecule. A comparison of the spectra signatures to

theory allowed us to identify the nature of the internal coupling

mechanism in the QD molecule system. Our work shows that

the combined single and double quantum 2D spectroscopy is

a powerful tool to reveal and understand coherent coupling

2469-9950/2017/96(4)/041124(6) 041124-1 Published by the American Physical Society

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VALENTIN DELMONTE et al. PHYSICAL REVIEW B 96, 041124(R) (2017)

1356 1358 1360 1362

(b)

(a)

delay τ

(ps)

photon energy (meV)

GXGX

X BX B

X B

-10

-5

X B

nonrephasing

FWM amplitude

FWM

scan

rephasing

pathway

time

FIG. 1. Rephasing and nonrephasing pathways in two-beam

four-wave mixing of individual quantum dots. (a) Two possible

pulse sequences in the so-called positive (negative) delays τ12,

corresponding to the rephasing (nonrephasing) FWM pathways. The

nonrephasing pathway involves a two-photon coherence between

the ground state (G) and a two-particle state, here, a quantum dot

biexciton (B). (b) Measured FWM amplitude as a function of τ12

on a few InAs QDs embedded in a low-Qmicrocavity. Impinging

E1,E2intensities of (150,600) nW correspond to pulse areas of

around (0.4π,0.8π), significantly beyond the χ(3) limit, generating a

pronounced exciton-biexciton beating.

and excitation transfer mechanisms—an interdisciplinary is-

sue spanning from biology and photochemistry, to quantum

engineering. The results are especially pertinent for the latter

area, as we open different avenues of research in the quantum

control of optically active nanoscopic two-level and few-level

systems in solids.

To acquire the FWM spectra [16], we use a pair of

100 fs laser pulses, E1and E2, with a variable delay τ12,

positive for E1leading. They are frequency shifted by 1=

80 MHz and 2=80.77 MHz, respectively, using acousto-

optic deflectors. A FWM heterodyne beat with a reference

field ERis retrieved at 22−1=81.54 MHz frequency,

carrying the lowest-order response E

1E2E2(where denotes

complex conjugate) and also higher orders with the same

phase evolution. The signal is spectrally dispersed using a

spectrometer, detected with a CCD camera and retrieved

in amplitude and phase by applying spectral interferometry.

ERarrives a few picoseconds prior to E2, unless specified

otherwise.

As shown in Fig. 1(a), in two-beam FWM, the first pulse

E1induces coherence, which evolves during τ12,tobethen

converted into FWM by the second pulse E2.Thelowest

electronic excitations of a neutral QD can be cast into three

categories of states: a ground state (G), single excitons (X), and

two-exciton states, known as biexcitons (B). GX transitions

are addressed by one-photon coherence driven by E1, which

is converted to FWM of GX and XB by a density grating

E

1E2on Gand X[20,26]. Inverting the temporal ordering of

the two light pulses, a GB transition can be inspected by a

two-photon coherence induced by E2, transformed into FWM

of both transitions at the arrival of E1[20,26]. The simple

three-level system of Fig. 1(a) illustrates the case of a neutral

QD driven along one of its polarization axes. For a single

two-level system, such as a QD trion, FWM can be only

0 100 200 300 400

FWM

amplitude

τ12

0.5ps, 100ps, 200ps

FWM amplitude

(ps)

1358 1360

energy (meV)

12=1ns

τR2= -0.85ns

FIG. 2. Photon-echo formation on a single QD exciton measured

upon a FWM rephasing pathway. The delay τR2, between the reference

ERand E2, is scanned for different values of τ12, as indicated.

Formation of the photon echo is observed: A Gaussian form of the

FWM transient is fully recovered for τ12 >¯h/σ. Temporal width of

the echo yields the inhomogeneous broadening σ. Inset: By adjusting

τR2, one shifts the temporal detection window towards the echo, such

that the FWM signal can be retrieved via spectral interference even

for delays exceeding the temporal resolution of the setup, defined

by the spectrometer. This is here exemplified for τ12 =1ns and

τR2 =−0.85 ns.

created for τ12 >0 from one-photon coherence induced by

E1, since the trion system cannot be doubly excited within

the employed spectral bandwidth. In fact, two transitions

in Fig. 1(b) show strictly no signal for τ12 <0 and are

attributed to trion transitions. Therein, we also recognize pairs

of exciton biexcitons, labeled as GX1-X1B1,GX2-X2B2, and

GX3-X3B3, occurring in three distinct QDs. FWM exhibits

a pronounced beating as a function of τ12 >0, with a period

corresponding to Bbinding energy, which is induced beyond

the χ(3) regime by high-order contributions propagating at the

FWM frequency [20,27]. Instead, for τ12 <0, FWM is equally

created on GX and XB transitions, with no beating.

A time-resolved FWM transient created upon the two pulse

configurations displays different characteristics. For τ12 >0,

there is a phase conjugation between E1and FWM. Owing

to the rephasing, FWM of an inhomogeneously broadened

system has a Gaussian form, with a maximum at t=τ12

and temporal width inversely proportional to the probed

spectral inhomogeneous broadening σ. Importantly, the time-

integrated amplitude of such a photon echo is not sensitive on

σ, instead, the homogeneous broadening is probed through the

τ12 dependence. At a level of individual transitions, σis accu-

mulated due to a residual spectral wandering in time-averaged

measurements [17,18,26,28]. For σin the μeV range, which

is a case even for high-quality QD systems, the echo width

becomes comparable to or larger than the temporal sensitivity,

given by the spectrometer resolution (here about 120 ps). To

demonstrate the formation of such a broad echo [18], we

scan the delay τR2, between ERand E2, for three different

τ12, as shown in Fig. 2. The echo develops fully only for

τ12 =200 ps, and from its width [full width at half maximum

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0 -50 -100 -150 -200

(d)

(b)

TTP (X1B1) = (63 3)ps

FWM amplitude

delay τ12 (ps)

FWM amplitude

TTP (GX1) = (75 3)ps

(a) (c)

0 -50 -100 -150 -200

noise level

X1B1

time (ps)

GX1

0-10-20

time (ps)

delay τ12 (ps)

FIG. 3. Coherent dynamics of an exciton-biexciton system mea-

sured at the nonrephasing FWM configuration, τ12 <0. (a), (b)

Time-resolved FWM transient measured at GX1and X1B1for

negative delays. Due to the nonrephasing configuration, the FWM

decay becomes more pronounced when increasing delay. (c), (d)

Two-photon coherence dynamics, induced between Gand B(also

known as a biexciton coherence), measured at the nonrephasing FWM

configuration. The two-photon dephasing time is retrieved from the

exponential decay of GX1and X1B1transitions.

(FWHM)] tσ=¯h/σ =(214 ±33) ps we retrieve spectral

inhomogeneous broadening 8 ln (2)σ=8ln(2)¯h/tσ=(17 ±

3) μeV (FWHM). By advancing τR2 sufficiently close to the

maximum of the echo, the temporal detection window is

brought toward its maximum, permitting one to retrieve FWM

spectral interference for delays significantly exceeding the

temporal sensitivity of the spectrometer, as shown in the inset

for τ12 =1 ns and τR2 =−0.85 ns.

For τ12 <0 there is no strict phase conjugation between

two-photon coherence and FWM, and therefore the photon

echo is absent. In Figs. 3(a) and 3(b) we show (t,τ12)-

resolved maps of the FWM amplitude measured on the

GX1and X1B1transitions, respectively. As τ12 is increased

towards more negative values, FWM decay becomes more

pronounced, owing to a nonrephasing character of the signal.

The two-photon coherence dynamics of GX1and X1B1,

i.e., respective time-integrated FWM vs τ12, are presented in

Figs. 3(a) and 3(b). From the exponential decay of GX1and

X1B1we retrieve two-photon (biexciton) dephasing [29,30]

TTP(GX1,X1B1)=(75 ±3,65 ±3) ps. Similar values of TTP

are obtained by analyzing two other GX-XB pairs. Note that,

in homogenously broadened GXB systems, TTP should be the

same when inferring it either from the GX or XB transition.

A slightly faster biexciton dephasing evaluated from the X1B1

decay is attributed to its stronger inhomogeneous broadening

via spectral wandering with respect to GX1: The latter is

due to energy fluctuations of the exciton level only, whereas

the former is sensitive on wandering of both the exciton and

biexciton levels [31,32]. These spectral fluctuations do not

have to be correlated, and thus yield a shorter TTP when reading

it out from X1B1.

1356 1358 1360 1362

2716

2718

2720

2722

(b)

nonrephasing frequency

(meV)

FWM frequency, ω

(meV)

(a)

1356

1358

1360

1362

1000

rephasing frequency

(meV)

FIG. 4. Two-dimensional FWM spectroscopy of exciton com-

plexes in a few InAs QDs probed along the (a) rephasing and

(b) nonrephasing pathways. Four exciton-biexciton systems in dif-

ferent QDs are indicated dashed-dotted, dotted, dashed, and solid

lines, respectively.

To illustrate couplings in the probed system of a few QDs,

we Fourier transform FWM(ω3,τ12) sequences with respect to

the delay τ12. The experimental setup is encapsulated, provid-

ing a passive stabilization of the phase during the acquisition.

However, the phase relationship between FWM measured for

subsequent τ12 is inevitably lost and can only be achieved via

active stabilization [33,34], which is not implemented here.

Knowledge of the FWM phase for subsequent delays τ12 is

a precondition to execute the Fourier transform yielding 2D

FWM. In our previous works [20,35], we have circumvented

this issue by imposing a phase relationship onto the data

by choosing a separated transition in the spectral domain,

acting as a local oscillator, and setting its phase to zero

for all delays. We then applied this phase factor globally to

the full spectrum, adjusting all other frequencies versus τ12,

accordingly. Such a transformation remains justified, as long as

the guiding transition to correct for, in particular, exhibiting no

coherent coupling, is available in the spectrum. This generally

is not the case. To overcome this experimental limitation,

we have conceived a post-treatment protocol permitting

one to reference the FWM phase, using auxiliary spectral

interferences of ERwith the driving pulses (see Sec. II A

of the Supplemental Material [36] for a description of the

phase-referencing protocol).

In Fig. 4we present 2D FWM obtained from the set of

QDs highlighted in Fig. 1.Forτ12 >0, FWM generated by

all resonances driven by E1forms a diagonal in the resulting

2D spectrum. This includes single trions and neutral excitons,

but also biexcitons—the latter can directly be driven by E1

beyond the χ(3) limit [20], as applied here (an example of the

2D FWM of GXB systems in the χ(3) limit is provided in

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(meV)

1359

1360

(meV)

(d)

1356 1358 1360

2716

2718

FWM frequency, ω

(meV)

(a)

1359

1360

(c)

(meV) ω

(meV)

(b)

2716

2718

FIG. 5. Quantum dot molecule, consisting of two electrostatically

coupled InAs QDs, observed in single and double quantum 2D FWM.

Measured (a) rephasing and (c) nonrephasing 2D FWM spectra

revealing coherent couplings between two QDs. Corresponding

simulations (see Sec. I C of the Supplemental Material [36]for

details regarding the model parameters) are shown in (b) and (d).

The signatures belonging to this QD molecule are marked by

dashed lines. An additional exciton-biexciton pair in (a) and (c)

at (1360.3,1356.8) meV occurs in other QDs not involved in the

molecule formation, thus not included in the calculated spectra.

Fig. S6 of the Supplemental Material [36]). The biexcitons are

off-diagonally shifted by their respective binding energies of

a few meV, and form squarelike features in 2D FWM under

strong excitation, i.e., close to the (π/2,π)areaof(E1,E2)

pulses. 2D FWM resulting from τ12 <0 is shown in Fig. 4(b).

FWM originates from a corresponding two-photon resonance

driven by E2. Here, the two-photon energy corresponds to the

sum of GX and XB transition energies. In such nonrephasing

2D FWM, we retrieve the response of GXB systems, whereas

exciton complexes without doubly excited states within the

excitation bandwidth, such as singly charged QDs, do not

contribute.

Figures 5(a) and 5(c) show the measured rephasing and

nonrephasing 2D spectra recorded at another position at the

sample. In the following, we focus on the two QDs that show up

as transitions GX1and GX2on the diagonal of the rephasing

spectrum with resonance energies E1=1359.7 meV and

E2=1358.95 meV—via hyperspectral imaging these are

found to be within a 0.5μm vicinity [35](seeFig.S7ofthe

Supplemental Material [36] for experimental results regarding

the FWM imaging). The peak pattern highlighted by the

dashed lines differs from the signatures observed in Fig. 4

in two major respects: First, the spin-orbit coupling of the two

circularly polarized excitons within each QD leads to linearly

polarized exciton eigenstates, where each QD is described by

a four-level system [20]. This causes a splitting of each exciton

resonance on the diagonal of the rephasing spectrum into

clusters of four peaks, observed in Figs. 5(a) and 5(c). Second,

besides the X1B1and X2B2peaks that are redshifted along the

FWM axis by the intradot biexciton binding energies 1=

−3.3 meV and 2=−3.6 meV, respectively, we observe

two off-diagonal cross peaks labeled X2X1and X1X2at the

spectral positions (ω3=E2;ω1=E1) (upper cross peak) and

(ω3=E1;ω1=E2) (lower cross peak). The appearance of

these cross peaks clearly indicates a coherent interdot coupling

between the two QDs: The electrostatic Coulomb coupling

leads to an energy renormalization of the interdot biexciton

B12 consisting of one exciton in each QD. The biexciton shift

lifts the symmetry of the lower GX1(GX2) and higher X1B12

(X2B12) transitions, such that the quantum pathways involving

these transitions no longer destructively interfere and cross

peaks show up [35]. (See Sec. I of the Supplemental Material

[36] for theoretical details. A level scheme of the considered

QD molecule, including all coupling-induced energy shifts, is

shown in Fig. S1.) The electrostatic interaction 12 between

two excitons located in two different QDs is small compared

to the intradot biexciton binding energies 1and 2. In fact,

the spectrally resolved FWM amplitude [see Fig. S7 of the

Supplemental Material [36] for experimental results regarding

the spectrally resolved FWM amplitude retrieved from the

2D FWM spectrum shown in Fig. 5(a)] reveals that it is

only of the order of 12 =90 μeV and it shifts the interdot

biexciton towards higher energies, showing up as blueshifted

[37] high-energy shoulders of the exciton resonance peaks.

This interpretation is supported by calculations [38]ofthe

rephasing and nonrephasing 2D signals depicted in Figs. 5(b)

and 5(d) (see Sec. I of the Supplemental Material [36]for

details regarding calculations of the FWM response).

In the nonrephasing two-quantum spectrum, the coupling

of the two QDs manifests itself in a peak pair labeled X1B12

and X2B12 at the interaction-shifted two-exciton transition

GB12 (energy ω2=E1+E2+12 =2718.74 meV) with

FWM frequencies ω3=E1=1359.7 meV and ω3=E2=

1358.95 meV, respectively. Theoretical calculations (see Sec.

I of the Supplemental Material [36] for details regarding

calculations of the FWM response) also suggest that exciton

transfer processes between the two QDs such as a dipole-

induced (Förster) interaction and Dexter-type coupling via a

wave-function overlap are negligible [39]: First, these coupling

types are expected to be in the μeV range [35] and therefore

difficult to detect considering our spectrometer resolution of

25 μeV. Second, they would lead to additional peaks for an

intradot biexciton in one QD after the first pulse has created a

single exciton in the other QD. These peaks are not observed

in the spectra, indicating that exciton transfer elements are

negligible (see Sec. I C and Fig. S3 of the Supplemental

Material [36] for theoretical simulations of the FWM spectra

involving different origins of the coherent coupling).

An interesting feature about the observed QD molecule

is that, in contrast to the other isolated exciton-biexciton

systems, the two coupled QDs show a pronounced fine-

structure splitting (FSS) of the order of 60 and 140 μeV,

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respectively. This is around five times higher than the FSS

typically present in these QDs [20]. Moreover, the FSS of the

other isolated exciton-biexciton systems in our sample (see

also Fig. 4) is not visible since the direction of the linear

excitation/reference polarization was chosen to be parallel to

the anisotropy axis. The observation of such a pronounced

FSS only for the resonances associated with the QD molecule

therefore suggests that the spatial proximity of the two coupled

QDs altered the local symmetry of the confinement, changing

the magnitude of the FSS and the polarization of the excitonic

transitions.

In summary, we have implemented phase-referenced dou-

ble quantum 2D FWM spectroscopy of individual quantum

systems. By merging it with a single quantum counterpart,

we have ascertained coherent couplings between excitons,

the structure of (bi)exciton states, and coupling energies in

single InAs QDs and in a quantum dot molecule. The optical

selection rules of the latter were investigated theoretically.

This methodology is appealing to infer electronic couplings

and charge transfer in deterministically defined QD molecules

[40,41] and propagative coherence in photonic molecules

[42]. By merging it with a recently developed multiwave

mixing toolbox [16], it could be also used to visualize and

control polaritonic couplings in solid state cavity-quantum

electrodynamics [43].

Note added. Recently, we became aware of a report

of 2D FWM in rephasing and nonrephasing configurations

of individual transitions in interface fluctuation QDs in

Ref. [44].

We gratefully acknowledge the financial support by the

European Research Council (ERC) Starting Grant PICSEN

(Grant No. 306387). J.F.S. and M.R. acknowledge financial

support by Deutsche Forschungsgemeinschaft through SFB

787 B1 and GRK 1558 A4. Financial support by the state of

Bavaria is acknowledged.

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