Collective super- and subradiant dynamics between distant optical quantum emitters Alexey Tiranovy1Vasiliki Angelopoulou1Cornelis Jacobus van Diepen1Bj orn Schrinski1Oliver August DallAlba Sandberg1Ying Wang1Leonardo Midolo1Sven Scholz2

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Collective super- and subradiant dynamics between distant optical quantum emitters

Alexey Tiranov†,1, ∗Vasiliki Angelopoulou,1, ∗Cornelis Jacobus van Diepen,1, ∗Bj¨orn

Schrinski,1Oliver August Dall’Alba Sandberg,1Ying Wang,1Leonardo Midolo,1Sven Scholz,2

Andreas Dirk Wieck,2Arne Ludwig,2Anders Søndberg Sørensen,1and Peter Lodahl1, †

1Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr

Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark

2Lehrstuhl f¨ur Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum, Universit¨atsstraße 150, D-44801 Bochum, Germany

(Dated: January 31, 2023)

Photon emission is the hallmark of light-matter interaction and the foundation of photonic quan-

tum science, enabling advanced sources for quantum communication and computing. While single-

emitter radiation can be tailored by the photonic environment, the introduction of multiple emitters

extends this picture. A fundamental challenge, however, is that the radiative dipole-dipole coupling

rapidly decays with spatial separation, typically within a fraction of the optical wavelength. We

realize distant dipole-dipole radiative coupling with pairs of solid-state optical quantum emitters

embedded in a nanophotonic waveguide. We dynamically probe the collective response and identify

both super- and subradiant emission as well as means to control the dynamics by proper excitation

techniques. Our work constitutes a foundational step towards multi-emitter applications for scalable

quantum information processing.

The radiative coupling of multiple optical emitters

has been a long-standing challenge in quantum optics

and atomic physics [1–4]. It oﬀers a route to realizing

quantum gates between emitters [5] thereby constituting

a fundamental building block for quantum-information

processing [6]. Waveguide quantum electrodynamics

(QED) [7] has evolved as a research discipline ideally

suited for overcoming the inherently weak dipole-dipole

coupling. This is because the radiative coupling here is

extended signiﬁcantly beyond the sub-wavelength limit

encountered in unstructured media [8]. The dipole-dipole

interaction can be understood as the absorption and re-

emission of virtual photons, and the waveguide extends

the spatial range to be limited only by the weak leakage

of the waveguide mode due to structural imperfections

[9]. It leads to the formation of collective emitter states

featuring super- or subradiant decay rates [1,2,10], as

controlled by the optical phase lag between the two emit-

ters (FIG. 1a). Observation of collective emission re-

quires a highly coherent light-matter interface. In par-

ticular, long-lived subradiant features may be elusive in

the presence of experimental imperfections such as de-

phasing [11].

Collective multi-emitter eﬀects have been studied in

the optical [3,4,8,12–15] and microwave domains [16–

18]. In microwave QED, multi-qubit interactions have

been realized [19,20] and subradiant collective states

coherently controlled [21]. Realizing such functionalities

in the optical domain is essential: optical photons can be

highly integrated, rapidly processed on-chip, and trans-

mitted over extended distances [6,22], making photonics

the backbone technology for the quantum internet [23].

Previous reports in the optical domain include projective

∗These authors contributed equally to this work.

†Email to: alexey.tiranov@nbi.ku.dk; lodahl@nbi.ku.dk

preparation of single-excitation superradiant states as re-

vealed in two-photon correlation measurements [24–27]

that does not require emitter-emitter coupling. Spectro-

scopic evidence for coherent coupling was reported with

closely spaced dye molecules in a bulk medium [8,28]

and for vacancy centers in a cavity [15]. Dynamics with

modiﬁed collective emission would constitute direct ex-

perimental proof of coherent coupling and open new av-

enues towards applications. Here we report the observa-

tion of coherent dynamics of the collective excitation of

quantum dot (QD) emitters coupled via a photonic crys-

tal waveguide (PCW) (FIG. 1a) by observing both en-

hanced (superradiant) and suppressed (subradiant) dy-

namics. In contrast to cooperative [29] and ampliﬁed

[30] spontaneous emission involving multiple excitations,

here we concentrate on a single excitation distributed be-

tween a pair of QDs. In the former case, the resulting

emission peak intensity scales as N2, while in the lat-

ter, the enhancement is ∝N, where Nis the number

of emitters involved. In the experiment, the QDs are

brought into mutual resonance by Zeeman-tuning with a

magnetic ﬁeld. The coherent oscillations of the collective

state are directly observed and found to be controllable

by spectrally tuning the QDs and by varying the excita-

tion conditions.

Hamiltonian for waveguide-mediated interac-

tion.

The coupled QDs are described by the eﬀective Hamil-

tonian Heﬀ (frame rotating at the excitation ﬁeld fre-

quency and ~= 1) [31,32]

Heﬀ =

m,n=1 Jmn −iΓmn

2σ+

nσ−

n=1 ∆n−iγs

2σ+

nσ−

(1)

where Jmn =1

2√βmβnΓmΓnsin φmn and Γmn =

√βmβnΓmΓncos φmn are the dispersive and dissipative

coupling rates connecting QDmand QDn. Γm=γwg

arXiv:2210.02439v2 [quant-ph] 29 Jan 2023

QD1 QD2 QD3

12 23

1 μm

FIG. 1. (color online) Observation of the super-

and subradiant emission. (a) Illustration of the photon-

mediated coupling between two QDs in a PCW where one

QD is optically pumped. Subsequently, the emission dynam-

ics of the coupled QD system exhibit super- and subradiance

originating from either constructive (bright line) or destruc-

tive (dark line) interference of the ﬁeld emitted into the PCW

and scattered by each of the QDs. The inset shows a ﬂuores-

cence image of the QDs in the PCW where x12 = 1.25(3) µm,

x23 = 0.96(3) µm. (b) Energy level diagram of two QDs with

a mutual detuning of ∆mn. Oﬀ resonance (left image), the

two QDs decay independently of each other, while on reso-

nance (right image), super- and subradiant dynamics occur

with rates Γsup and Γsub, respectively. (c) and (d) Measured

time-resolved emission dynamics of pair QD1-QD2 and QD2-

QD3 both on and oﬀ resonance and after exciting QD2 and

QD3 through higher energy excited states. The measured

count rates are normalized to the maximum at zero time de-

lay corresponding to the time of excitation of the coupled

system. The solid lines are the model ﬁts to the experimental

data. Oﬀ resonance, ∆12/2π= 6 GHz, (∆23/2π= 5 GHz)

the decay curve corresponds to the cases of individual QDs.

When two QDs are brought into resonance, two decay com-

ponents are observed in the data, a fast Γsup and a slow Γsub

corresponding to super- and subradiant dynamics.

γs

mcontains the decay rate into (out of) the waveguide

γwg

m(γs

m), corresponding to β-factors βm=γwg

m/Γm. ∆m

is the detuning of QDmwith respect to the excitation

ﬁeld frequency, such that ∆mn = ∆m−∆nis the de-

tuning between the two QDs, φmn =k|xmn|is the phase

lag due to the emitter separation xmn with kbeing the

eﬀective wavenumber of the PCW mode. φmn deter-

mines the character of the coupling between dispersive

(Γmn = 0), which modiﬁes the energy levels, to dissipa-

tive (Jmn = 0), which aﬀects the decay dynamics. The

system bears a resemblance to the case of a nanocavity

where each QD acts as an end mirror by scattering single

photons into the mode of the waveguide, and φmn deter-

mines the cavity resonance condition (FIG. 1a). Notably,

the system is inherent in quantum character since the

QDs only scatter a single photon at a time. σ+

m,σ−

nare

the raising and lowering operators for the optical transi-

tion of QDm, QDn.

By selectively pumping one of the QDs, e.g QDn, one

excitation is launched to populate the state |gmeni. On

resonance (∆mn = 0), the subsequent dynamics is best

described in terms of the entangled eigenstates |Si=

with the associated decay rates, determined by the phase

lag φmn. The case when φmn =Nπ (Ninteger) corre-

sponds to a dissipative coupling between QDs leading to a

superradiant |Siand a subradiant |sistate with modiﬁed

decay rates (FIG. 1b). φmn = (N+ 1/2)πresults in dis-

persive coupling where the collective states are shifted by

Jmn while leaving their decay rates unchanged compared

to the uncoupled emitters. In the present experiment, we

are primarily studying the regime of dissipative radiative

coupling, leading to modiﬁed emission dynamics.

Super- and subradiance with coherent evolu-

tion. We optically excite a single QD and record the

collective emission dynamics from either collection port

1 or 2, studying three diﬀerent QDs (QD1-3, see inset

of FIG. 1a). Two examples of recorded emission signals

are shown on FIG. 1c,d, where we alter the detuning to

compare the oﬀ- and on-resonant cases; see also Supple-

mental Notes for further experimental details [33]. On

resonance, we observe strongly modiﬁed decay dynam-

ics due to coherent coupling, and both super- and sub-

radiant features are directly visible. For QD1-QD2, we

ﬁnd radiative linewidths of Γsup/2π= 1.36(8) GHz and

Γsub/2π= 0.280(2) GHz, by modeling the data with a

bi-exponential decay. The modeling of the data at short

time delays is limited by the ﬁnite instrument response

function of the single-photon detectors, particularly vis-

ible at the rising edge of the detected pulse. These val-

ues should be referred to the single emitter linewidths of

Γ2/2π= 0.79(2) GHz and Γ1/2π= 0.85(1) GHz, respec-

tively, as observed far oﬀ-resonance where the coupling

is negligible. We derive a super/subradiant enhancement

factor of 1.36/0.28 = 4.8. The enhancement factor is a di-

rect ﬁgure-of-merit of the collective coupling quality and

is highly sensitive to experimental imperfections and de-

coherence; for a detailed account of the underlying physi-

cal parameters, see Supplemental Notes Table 2 [33]. The

FIG. 2. (color online) Coherent dynamics of coupled QDs. (a) The evolution of a single collective excitation can be

represented on a Bloch sphere, with the color indicating the decay rate of the respective collective state. After exciting QDn, the

state |gmeniis populated. Two processes occur: exponential decay by spontaneous emission (black solid arrow) and coherent

evolution between super- and subradiant states (red arrow) as determined by the detuning between the two QDs ∆mn. (b)

Exemplary state evolution trajectory for ∆23 =−5.5Γ, where the length of the Bloch vector shows the population in the

single-excitation subspace. The color bar tracks the evolution time from preparing the coupled system in the initial state until

decaying to the ground state |gmgni(origin of Bloch sphere). (c) Calculated population of the collective states as a function of

time for two values of detuning and comparing super- (yellow curves) and subradiant (grey curves) contributions for an initial

state |gmeni. The corresponding emission intensity is plotted in (d). The calculation is done with the experimentally extracted

parameters of the QD2-QD3 pair (see Supplemental Notes). (e) Full experimental dataset for a continuous scan of detuning

∆23 and comparing measurements from outcoupling port 1 (e) and port 2 (f). The dashed lines trace the maximum population

after the ﬁrst oscillation and are given by t=π/fosc, see main text for the deﬁnitions.

long-range nature of the dipole-dipole coupling is explic-

itly demonstrated. Using resonant excitation through the

PCW, we image the spatial separation of the QDs and

ﬁnd x12 = 1.25(3) µm and x23 = 0.96(3) µm (see inset of

FIG. 1a), which should be compared to the wavelength

of λ= 270 nm inside GaAs or the PCW lattice constant

of a= 240 nm.

Next, we show the coherent evolution of collective

states by precisely controlling the detuning between two

emitters. We start by resonantly exciting a single QD

to prepare |egi= (|Si+|si)/√2, i.e., an equal super-

position of the super- and subradiant states. The state

subsequently evolves in time into a collective state (see

the Bloch sphere graphical representation in FIG. 2a,b).

The color of the Bloch sphere surface represents the de-

cay time of the respective state, and the collective state

vector precesses due to coherent evolution. Assuming

Γm= Γn, the correlated dynamics is described by the

diﬀerence between the two eigenvalues of the coupled

system fosc =q∆2

mn + (2Jmn −iΓmn)2. E.g., in the

case of pure dissipative coupling (Jmn = 0), two diﬀerent

regimes are identiﬁed (FIG. 2c,d). In the underdamped

regime, |∆mn|>Γmn, coherent evolution prevails over

dissipation resulting in the observation of an increase in

emission intensity. The dashed lines in FIG. 2d-f track

the observed intensity maximum for diﬀerent values of

∆mn deﬁned by t=π/fosc. In the overdamped regime,

|∆mn|<Γmn, the dissipative coupling damps the excita-

tion to the ground state faster than the coherent oscilla-

tions on the Bloch sphere (leading to the “gap” between

two dashed lines on FIG. 2d-f). ∆mn = Γmn is the case

of critical damping where the dissipation and coherent

oscillation rates are balanced. These examples quantify

how precise emitter tuning provides an experimentally

accessible “control knob” of coupled collective quantum

states.

The rich coherent dynamics of the collectively-coupled

system is evident from the experimental data (FIG. 2e,f)

that are well reproduced by theory (FIG. 2d). As for the

resonant case, following a fast decay of the superradiant

component, the coupled system evolves toward the sub-

radiant state. In the detuned case, coherent evolution

is observed, leading to modiﬁed dynamics since super-

and subradiant components are interchanged. The os-

cillation between super- and subradiant components, as

deﬁned by the detuning, leads in an increase of the inten-

sity, corresponding to the superradiant component being

maximal. This is clearly visible as a bright region, as it is

tracked by the dashed line on FIG. 2d-f. By comparing

FIG. 3. (color online) Controlled preparation of the collective state. (a) Simultaneous driving of both QDs from

QD2-QD3 pair with control of the phase θallows to initialize the system in a collective state. In the case of θ≈ −π/2 the

initial state in the single-excitation subspace is close to |gei+i|egi. The detuning between the QDs results in the evolution of

the collective state towards |Siwhen ∆23 <0 and towards |sifor ∆23 >0. An example of the state trajectory for ∆23 <0 is

depicted on the Bloch sphere. (b) The population of the super- ρsup and subradiant ρsub components during the excitation as

a function of the detuning ∆23 for θ≈ −π/2. The sign of the detuning determines whether the super- or subradiant states are

preferentially populated. (c) The two populations oscillate out of phase as a function of time after excitation. (d) Calculated

and (e) measured intensity decay dynamics as a function of detuning and collected from the outcoupling port 1. The asymmetric

dependence on detuning is well reproduced by the theory using parameters extracted from the experiment with θ≈ −π/2.

(f) Examples of emission intensity decay curves for two values of detuning explicitly displaying the coherent oscillations being

out-of-phase of each other as determined by the sign of the detuning. (g) Normalised counts as a function of detuning and for

two representative times quantifying the asymmetry.

the emission from two diﬀerent directions (outcoupling

port 1 and 2), we observe a similar behavior (FIG. 2e,f),

which is consistent with a predominantly dissipative cou-

pling φmn 'Nπ. The experiment was repeated on

in total three pairs of QDs, where the additional data

and detailed modeling can be found in the Supplemental

Notes [33]. For all three data sets, we observe predomi-

nantly dissipative coupling, which likely results from the

fact that QD candidates, featuring eﬃcient coupling to

the PCW, were pre-selected in the experiment based on

resonant transmission measurements through the waveg-

uide (see Supplemental Notes [33]). This condition re-

sults in the selection of QDs close to the waveguide center

and, therefore, a phase lag between QDs primarily deter-

mined by the periodicity of the photonic crystal lattice.

Since the PCW coupling is large and QDs are spectrally

close to the band edge of the waveguide mode, this leads

to k/a 'π[9], whereby the phase lag between neighbor-

ing PCW unit cells is 'π, see Supplemental Notes for

further details [33].

The theoretical model (FIG. 2d) reproduces the ex-

perimental data and fully captures the complex coherent

quantum dynamics observed experimentally (FIG. 2e,f).

From the analysis, we extract for pair QD2-QD3:

Γ23/2π= 0.61 GHz and J23/2π= 0.03 GHz, respec-

tively. The dissipative coupling rate is comparable to

the intrinsic linewidth of the respective QDs, while the

dispersive part is almost vanishing. In the case of neg-

ligible dispersive coupling the superradiant Γsup (subra-

diant Γsub) decay rate can be approximated by Γsup ≈

Γ+Γij −σ2

sd/2βΓ (Γsub ≈Γ−Γij +σ2

sd/2βΓ), where

σsd is the spectral diﬀusion width. The predicted value

of Γsup/2π= 1.25 GHz (Γsub/2π= 0.27 GHz) for the

QD2-QD3 pair is in agreement with the experimentally

measured Γsup/2π= 1.33 GHz (Γsub/2π= 0.22 GHz).

For this QD pair, we obtain β2= 0.88 and β3= 0.83

from the resonant transmission data, see Supplementary

Materials [33] for further details.

As opposed to the case of superconducting qubits [16],

changing the detuning between the QDs has a negligible

eﬀect on the phase lag, whereby the coupling remains

dissipative. This is a consequence of the fact that the de-

tuning is vanishingly small compared to the optical fre-

quency. This distinction may be advantageous in appli-

cations of radiative collective coupling, since it allows the

detuning to be exploited as a control parameter without

changing the coupling, which is explicitly demonstrated

in the present work. Interestingly, even when the emitter-

emitter system is initialized in |egior |gei, coherent os-

cillations are still observed. This is enabled by the fast

decay of the superradiant component after initialization,

leading to the population of the slower decaying subra-

diant state that coherently evolves on the Bloch sphere

(FIG. 2b).

Control of collective excitations. The determin-

istic preparation of collective states is essential in or-

der to pave the way for their applications in quantum-

information processing. To this end, we coherently ex-

cite both QDs in order to control the initial collective

state on the Bloch sphere (FIG. 3a). In the magnetic

Zeeman ﬁeld, the coupled QD transitions are orthogo-

nally (circularly) polarized [34], yet they are eﬃciently

coupled by the optical mode of the PCW. The phase be-

tween the two driving ﬁelds Ω2and Ω3eiθ is adjusted via

the polarization of the excitation laser. Using a single

laser, we implement θ≈ −π/2, which prepares an initial

state with the single excitation close to |egi+i|geifor

∆23 = 0. With detuning, the state either evolves towards

the super- (for ∆23 <0) or the subradiant (for ∆23 >0)

state. This results in a striking diﬀerence in the radia-

tion at a short time, followed by out-of-phase coherent

oscillations between the two components (FIG. 3b,c.

The experimental demonstration of this behavior

(FIG. 3e) is accurately described by the theory (FIG. 3d).

We observe a pronounced asymmetry around zero detun-

ing, where for positive detunings, the emission dynamics

is eﬀectively delayed. This stems from the selective pop-

ulation of the subradiant state resulting in a lower emis-

sion at early times. Subsequently, the emission intensity

increases as the coherent evolution increases the popula-

tion of the superradiant state. The reverse behavior is

found for ∆23 <0. This is a result of the out-of-phase

oscillation of the population of the super- and subradiant

components (FIG. 3f,g). As opposed to the case where a

single QD was excited (FIG. 2), we observe here multiple

coherent oscillations. This is due to the state |egi+i|gei,

starting to coherently evolve on the Bloch sphere directly

after excitation. This is in contrast to the case where |egi

is prepared, and the coherent evolution sets in once the

state has partially decayed to |si. We track the maxi-

mum emission intensity (dashed line) for ∆23 >0 (and

minimum for ∆23 <0) in the plots of FIG. 3e. It is conse-

quently found that the collective light emission intensity

can be controlled via the QD detuning. By simultaneous

driving, we populate the doubly excited state |eeicompo-

nent, which reaches 0.07 for ≈π/3 excitation pulses that

are used. We note that the doubly excited state |eei,

however, does not contribute to the measured asymme-

try between positive and negative detunings.

Concluding remarks. Our observation of super- and

subradiant emission dynamics using pairs of QDs embed-

ded in PCWs and separated by a distance much larger

than the wavelength is facilitated by the PCW oﬀer-

ing broadband spectral operation and long-range photon-

mediated coherent interaction between QDs. Our work

can constitute a foundational step towards multi-emitter

applications of technological importance, e.g., for real-

izing quantum transduction between microwave qubits

and the optical domain [35] or for quantum memories

with exponential improvement in photon storage ﬁdelity

[31]. The ability to manipulate the super- and subradiant

state dynamics by controlling the detuning and pumping

conditions will lead to a whole new range of opportuni-

ties when implementing a coherent spin inside the QD

[34,36]. For instance, advanced photonic cluster states

may be generated deterministically [37], providing a uni-

versal resource for measurement-based photonic quantum

computing. To this end, waveguide-mediated dissipa-

tive coupling can be exploited to realize spin-spin en-

tanglement between distant quantum emitters [38]. An-

other direction will exploit photon scattering from cou-

pled QDs to realize eﬃcient Bell-state measurements [39]

or photon-photon quantum gates [40]. Taking a broader

perspective, the ability to deterministically couple mul-

tiple quantum emitters opens a new arena of studying

non-equilibrium quantum many-body physics of strongly

correlated light and matter, which could be used in quan-

tum simulations of strongly correlated condensed matter

systems [41].

Acknowledgments: The authors thank Xiao-Liu

Chu for fruitful discussions at the beginning of the

project.

Funding: We gratefully acknowledge ﬁnancial sup-

port from Danmarks Grundforskningsfond (DNRF 139,

Hy-Q Center for Hybrid Quantum Networks). B.S. ac-

knowledges ﬁnancial support from Deutsche Forschungs-

gemeinschaft (DFG, German Research Foundation),

Grant No. 449674892. O.S. acknowledges funding from

the European Union’s Horizon 2020 research and innova-

tion programme under the Marie Sk lodowska-Curie grant

agreement No. 801199. S.S., A.L. and A.D.W. acknowl-

edge ﬁnancial support of the German-French University

DFH/UFA within the CDFA-05-06 as well as BMBF

QR.X project 16 KISQ 009.

Author contributions: A.T., V.A., and C.J.v.D.

carried out the measurements. Y.W. and L.M. designed

and fabricated the sample. O.S., B.S. and A.S devel-

oped the theoretical model, A.T., V.A and C.J.v.D. an-

alyzed the data and prepared the ﬁgures. S.S., A.D.W.,

and A.L. carried out the growth and design of the wafer.

A.T., V.A., C.J.v.D., and P.L. wrote the manuscript with

input from all the authors. P.L. supervised the project.

Competing interests: P.L. is founder of the com-

pany Sparrow Quantum that commercializes single-

photon sources.

Data and materials availability: The data pre-

sented in the main text and Supplementary Materials

for this publication are openly available from [42].

[1] R. H. Dicke, Physical Review 93, 99 (1954).

[2] M. Gross and S. Haroche, Physics Reports 93, 301

(1982).

[3] R. G. DeVoe and R. G. Brewer, Physical Review Letters

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Collectivesuper-andsubradiantdynamicsbetweendistantopticalquantumemittersAlexeyTiranovy,1,VasilikiAngelopoulou,1,CornelisJacobusvanDiepen,1,BjornSchrinski,1OliverAugustDall'AlbaSandberg,1YingWang,1LeonardoMidolo,1SvenScholz,2AndreasDirkWieck,2ArneLudwig,2AndersSndbergSrensen,1andPeterLodahl1,y...

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Collective super- and subradiant dynamics between distant optical quantum emitters Alexey Tiranovy1Vasiliki Angelopoulou1Cornelis Jacobus van Diepen1Bj orn Schrinski1Oliver August DallAlba Sandberg1Ying Wang1Leonardo Midolo1Sven Scholz2

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