SReferences

††thanks: These authors contributed equally to this work.††thanks: These
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# Collective super- and subradiant dynamics between distant optical quantum
emitters

Alexey Tiranov† Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Vasiliki
Angelopoulou  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Cornelis
Jacobus van Diepen  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Björn
Schrinski  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Oliver
August Dall’Alba Sandberg  Center for Hybrid Quantum Networks (Hy-Q), The
Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Ying Wang  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Leonardo
Midolo  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Sven Scholz
Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum,
Universitätsstraße 150, D-44801 Bochum, Germany  Andreas Dirk Wieck  Lehrstuhl
für Angewandte Festkörperphysik, Ruhr-Universität Bochum, Universitätsstraße
150, D-44801 Bochum, Germany  Arne Ludwig  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44801
Bochum, Germany  Anders Søndberg Sørensen  Center for Hybrid Quantum Networks
(Hy-Q), The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen
Ø, Denmark  Peter Lodahl  Center for Hybrid Quantum Networks (Hy-Q), The Niels
Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark

###### Abstract

Photon emission is the hallmark of light-matter interaction and the foundation
of photonic quantum 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, 2, 3, 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 emitters (Figure 1a). Observation of collective emission
requires a highly coherent light-matter interface. In particular, long-lived
subradiant features may be elusive in the presence of experimental
imperfections such as dephasing [11].

Collective multi-emitter effects have been studied in the optical [3, 4, 12,
13, 14, 15, 8] and microwave domains [16, 17, 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 transmitted over extended distances [6, 22], making
photonics the backbone technology for the quantum internet [23]. Previous
reports in the optical domain include projective preparation of single-
excitation superradiant states as revealed in two-photon correlation
measurements [24, 25, 26, 27] that does not require emitter-emitter coupling.
Spectroscopic evidence for coherent coupling was reported with closely spaced
dye molecules in a bulk medium [28, 8] and for vacancy centers in a cavity
[15]. Dynamics with modified collective emission would constitute direct
experimental proof of coherent coupling and open new avenues towards
applications. Here we report the observation of coherent dynamics of the
collective excitation of quantum dot (QD) emitters coupled via a photonic
crystal waveguide (PCW) (Figure 1a) by observing both enhanced (superradiant)
and suppressed (subradiant) dynamics. In contrast to cooperative [29] and
amplified [30] spontaneous emission involving multiple excitations, here we
concentrate on a single excitation distributed between a pair of QDs. In the
former case, the resulting emission peak intensity scales as $N^{2}$, while in
the latter, the enhancement is $\propto N$, where $N$ is 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 excitation conditions.

Figure 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 dynamics of the
coupled QD system exhibit super- and subradiance originating from either
constructive (bright line) or destructive (dark line) interference of the
field emitted into the PCW and scattered by each of the QDs. The inset shows a
fluorescence image of the QDs in the PCW where $x_{12}=1.25(3)$ µm,
$x_{23}=0.96(3)$ µm. (b) Energy level diagram of two QDs with a mutual
detuning of $\Delta_{mn}$. Off resonance (left image), the two QDs decay
independently of each other, while on resonance (right image), super- and
subradiant dynamics occur with rates $\Gamma_{\text{sup}}$ and
$\Gamma_{\text{sub}}$, 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 delay corresponding to
the time of excitation of the coupled system. The solid lines are the model
fits to the experimental data. Off resonance, $\Delta_{12}/2\pi=6$ GHz,
$(\Delta_{23}/2\pi=5$ GHz) the decay curve corresponds to the cases of
individual QDs. When two QDs are brought into resonance, two decay components
are observed in the data, a fast $\Gamma_{\text{sup}}$ and a slow
$\Gamma_{\text{sub}}$ corresponding to super- and subradiant dynamics.

Hamiltonian for waveguide-mediated interaction.

The coupled QDs are described by the effective Hamiltonian
$\mathcal{H}_{\text{eff}}$ (frame rotating at the excitation field frequency
and $\hbar=1$) [31, 32]

$\mathcal{H}_{\text{eff}}=\sum_{m,n=1}^{2}\left(J_{mn}-i\frac{\Gamma_{mn}}{2}\right)\sigma_{n}^{+}\sigma_{m}^{-}+\sum_{n=1}^{2}\left(\Delta_{n}-i\frac{\gamma_{n}^{s}}{2}\right)\sigma_{n}^{+}\sigma^{-}_{n},$
(1)

where
$J_{mn}=\frac{1}{2}\sqrt{\beta_{m}\beta_{n}\Gamma_{m}\Gamma_{n}}\sin\phi_{mn}$
and $\Gamma_{mn}=\sqrt{\beta_{m}\beta_{n}\Gamma_{m}\Gamma_{n}}\cos\phi_{mn}$
are the dispersive and dissipative coupling rates connecting QDm and QDn.
$\Gamma_{m}=\gamma^{wg}_{m}+\gamma^{s}_{m}$ contains the decay rate into (out
of) the waveguide $\gamma^{wg}_{m}$ $(\gamma_{m}^{s})$, corresponding to
$\beta$-factors $\beta_{m}=\gamma^{wg}_{m}/\Gamma_{m}$. $\Delta_{m}$ is the
detuning of QDm with respect to the excitation field frequency, such that
$\Delta_{mn}=\Delta_{m}-\Delta_{n}$ is the detuning between the two QDs,
$\phi_{mn}=k\absolutevalue{x_{mn}}$ is the phase lag due to the emitter
separation $x_{mn}$ with $k$ being the effective wavenumber of the PCW mode.
$\phi_{mn}$ determines the character of the coupling between dispersive
($\Gamma_{mn}=0$), which modifies the energy levels, to dissipative
($J_{mn}=0$), 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 $\phi_{mn}$ determines the
cavity resonance condition (Figure 1a). Notably, the system is inherent in
quantum character since the QDs only scatter a single photon at a time.
$\sigma_{m}^{+},\sigma_{n}^{-}$ are the raising and lowering operators for the
optical transition of QDm, QD${}_{n}.$

By selectively pumping one of the QDs, e.g QDn, one excitation is launched to
populate the state $\ket{g_{m}e_{n}}$. On resonance $(\Delta_{mn}=0)$, the
subsequent dynamics is best described in terms of the entangled eigenstates
$\ket{S}=(\ket{e_{m}g_{n}}+\ket{g_{m}e_{n}})/\sqrt{2}$ and
$\ket{s}=(\ket{e_{m}g_{n}}-\ket{g_{m}e_{n}})/\sqrt{2}$ with the associated
decay rates, determined by the phase lag $\phi_{mn}$. The case when
$\phi_{mn}=N\pi$ ($N$ integer) corresponds to a dissipative coupling between
QDs leading to a superradiant $\ket{S}$ and a subradiant $\ket{s}$ state with
modified decay rates (Figure 1b). $\phi_{mn}=(N+1/2)\pi$ results in dispersive
coupling where the collective states are shifted by $J_{mn}$ 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.

Figure 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 $\ket{g_{m}e_{n}}$ is 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 $\Delta_{mn}$. (b) Exemplary state evolution
trajectory for $\Delta_{23}=-5.5\Gamma$, 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 $\ket{g_{m}g_{n}}$ (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
$\ket{g_{m}e_{n}}$. 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 $\Delta_{23}$ 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
$t=\pi/f_{\text{osc}}$, see main text for the definitions.

Super- and subradiance with coherent evolution. 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 Figure 1a). Two
examples of recorded emission signals are shown on Figure 1c,d, where we alter
the detuning to compare the off- and on-resonant cases; see also Supplemental
Notes for further experimental details [33]. On resonance, we observe strongly
modified decay dynamics due to coherent coupling, and both super- and
subradiant features are directly visible. For QD1-QD2, we find radiative
linewidths of $\Gamma_{\text{sup}}/2\pi=1.36(8)$ GHz and
$\Gamma_{\text{sub}}/2\pi=0.280(2)$ 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 visible at the rising edge of the detected pulse. These values
should be referred to the single emitter linewidths of
$\Gamma_{\text{2}}/2\pi=0.79(2)$ GHz and $\Gamma_{\text{1}}/2\pi=0.85(1)$ GHz,
respectively, 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 direct figure-of-merit of the collective coupling
quality and is highly sensitive to experimental imperfections and decoherence;
for a detailed account of the underlying physical parameters, see Supplemental
Notes Table 2 [33]. The long-range nature of the dipole-dipole coupling is
explicitly demonstrated. Using resonant excitation through the PCW, we image
the spatial separation of the QDs and find $x_{12}=1.25(3)$ µm and
$x_{23}=0.96(3)$ µm (see inset of Figure 1a), which should be compared to the
wavelength of $\lambda=270$ nm inside GaAs or the PCW lattice constant of
$a=240$ nm.

Figure 3:  (color online) Controlled preparation of the collective state. (a)
Simultaneous driving of both QDs from QD2-QD3 pair with control of the phase
$\theta$ allows to initialize the system in a collective state. In the case of
$\theta\approx-\pi/2$ the initial state in the single-excitation subspace is
close to $\ket{ge}+i\ket{eg}$. The detuning between the QDs results in the
evolution of the collective state towards $\ket{S}$ when $\Delta_{23}<0$ and
towards $\ket{s}$ for $\Delta_{23}>0$. An example of the state trajectory for
$\Delta_{23}<0$ is depicted on the Bloch sphere. (b) The population of the
super- $\rho_{\text{sup}}$ and subradiant $\rho_{\text{sub}}$ components
during the excitation as a function of the detuning $\Delta_{23}$ for
$\theta\approx-\pi/2$. 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 $\theta\approx-\pi/2$. (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.

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 $\ket{eg}=\left(\ket{S}+\ket{s}\right)/\sqrt{2}$, i.e.,
an equal superposition of the super- and subradiant states. The state
subsequently evolves in time into a collective state (see the Bloch sphere
graphical representation in Figure 2a,b). The color of the Bloch sphere
surface represents the decay time of the respective state, and the collective
state vector precesses due to coherent evolution. Assuming
$\Gamma_{m}=\Gamma_{n}$, the correlated dynamics is described by the
difference between the two eigenvalues of the coupled system
$f_{\text{osc}}=\mathrm{}\sqrt{\Delta_{mn}^{2}+\left(2J_{mn}-i\Gamma_{mn}\right)^{2}}$.
E.g., in the case of pure dissipative coupling ($J_{mn}=0$), two different
regimes are identified (Figure 2c,d). In the underdamped regime,
$\left|\Delta_{mn}\right|>\Gamma_{mn}$, coherent evolution prevails over
dissipation resulting in the observation of an increase in emission intensity.
The dashed lines in Figure 2d-f track the observed intensity maximum for
different values of $\Delta_{mn}$ defined by $t=\pi/f_{\text{osc}}$. In the
overdamped regime, $\left|\Delta_{mn}\right|<\Gamma_{mn}$, the dissipative
coupling damps the excitation to the ground state faster than the coherent
oscillations on the Bloch sphere (leading to the “gap” between two dashed
lines on Figure 2d-f). $\Delta_{mn}=\Gamma_{mn}$ 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 (Figure 2e,f) that are well reproduced by theory (Figure
2d). As for the resonant case, following a fast decay of the superradiant
component, the coupled system evolves toward the subradiant state. In the
detuned case, coherent evolution is observed, leading to modified dynamics
since super- and subradiant components are interchanged. The oscillation
between super- and subradiant components, as defined by the detuning, leads in
an increase of the intensity, corresponding to the superradiant component
being maximal. This is clearly visible as a bright region, as it is tracked by
the dashed line on Figure 2d-f. By comparing the emission from two different
directions (outcoupling port 1 and 2), we observe a similar behavior (Figure
2e,f), which is consistent with a predominantly dissipative coupling
$\phi_{mn}\simeq N\pi$. 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 predominantly
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 waveguide (see
Supplemental Notes [33]). This condition results in the selection of QDs close
to the waveguide center and, therefore, a phase lag between QDs primarily
determined 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 $k/a\simeq\pi$ [9], whereby the phase lag
between neighboring PCW unit cells is $\simeq\pi$, see Supplemental Notes for
further details [33].

The theoretical model (Figure 2d) reproduces the experimental data and fully
captures the complex coherent quantum dynamics observed experimentally (Figure
2e,f). From the analysis, we extract for pair QD2-QD3: $\Gamma_{23}/2\pi=0.61$
GHz and $J_{23}/2\pi=0.03$ GHz, respectively. 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 negligible dispersive
coupling the superradiant $\Gamma_{sup}$ (subradiant $\Gamma_{sub}$) decay
rate can be approximated by
$\Gamma_{sup}\approx\Gamma+\Gamma_{ij}-\sigma_{sd}^{2}/2\beta\Gamma$
($\Gamma_{sub}\approx\Gamma-\Gamma_{ij}+\sigma_{sd}^{2}/2\beta\Gamma$), where
$\sigma_{sd}$ is the spectral diffusion width. The predicted value of
$\Gamma_{sup}/2\pi=1.25$ GHz ($\Gamma_{sub}/2\pi=0.27$ GHz) for the QD2-QD3
pair is in agreement with the experimentally measured $\Gamma_{sup}/2\pi=1.33$
GHz ($\Gamma_{sub}/2\pi=0.22$ GHz). For this QD pair, we obtain
$\beta_{2}=0.88$ and $\beta_{3}=0.83$ 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 detuning is
vanishingly small compared to the optical frequency. This distinction may be
advantageous in applications 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 $\ket{eg}$ or
$\ket{ge}$, coherent oscillations are still observed. This is enabled by the
fast decay of the superradiant component after initialization, leading to the
population of the slower decaying subradiant state that coherently evolves on
the Bloch sphere (Figure 2b).

Control of collective excitations. The deterministic preparation of collective
states is essential in order to pave the way for their applications in
quantum-information processing. To this end, we coherently excite both QDs in
order to control the initial collective state on the Bloch sphere (Figure 3a).
In the magnetic Zeeman field, the coupled QD transitions are orthogonally
(circularly) polarized [34], yet they are efficiently coupled by the optical
mode of the PCW. The phase between the two driving fields $\Omega_{2}$ and
$\Omega_{3}e^{i\theta}$ is adjusted via the polarization of the excitation
laser. Using a single laser, we implement $\theta\approx-\pi/2$, which
prepares an initial state with the single excitation close to
$\ket{eg}+i\ket{ge}$ for $\Delta_{23}=0$. With detuning, the state either
evolves towards the super- (for $\Delta_{23}<0$) or the subradiant (for
$\Delta_{23}>0$) state. This results in a striking difference in the radiation
at a short time, followed by out-of-phase coherent oscillations between the
two components (Figure 3b,c.

The experimental demonstration of this behavior (Figure 3e) is accurately
described by the theory (Figure 3d). We observe a pronounced asymmetry around
zero detuning, where for positive detunings, the emission dynamics is
effectively delayed. This stems from the selective population of the
subradiant state resulting in a lower emission at early times. Subsequently,
the emission intensity increases as the coherent evolution increases the
population of the superradiant state. The reverse behavior is found for
$\Delta_{23}<0$. This is a result of the out-of-phase oscillation of the
population of the super- and subradiant components (Figure 3f,g). As opposed
to the case where a single QD was excited (Figure 2), we observe here multiple
coherent oscillations. This is due to the state $\ket{eg}+i\ket{ge}$, starting
to coherently evolve on the Bloch sphere directly after excitation. This is in
contrast to the case where $\ket{eg}$ is prepared, and the coherent evolution
sets in once the state has partially decayed to $\ket{s}$. We track the
maximum emission intensity (dashed line) for $\Delta_{23}>0$ (and minimum for
$\Delta_{23}<0$) in the plots of Figure 3e. It is consequently found that the
collective light emission intensity can be controlled via the QD detuning. By
simultaneous driving, we populate the doubly excited state $\ket{ee}$
component, which reaches 0.07 for $\approx\pi/3$ excitation pulses that are
used. We note that the doubly excited state $\ket{ee}$, however, does not
contribute to the measured asymmetry between positive and negative detunings.

Concluding remarks. Our observation of super- and subradiant emission dynamics
using pairs of QDs embedded in PCWs and separated by a distance much larger
than the wavelength is facilitated by the PCW offering 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 realizing 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
opportunities when implementing a coherent spin inside the QD [34, 36]. For
instance, advanced photonic cluster states may be generated deterministically
[37], providing a universal resource for measurement-based photonic quantum
computing. To this end, waveguide-mediated dissipative coupling can be
exploited to realize spin-spin entanglement between distant quantum emitters
[38]. Another direction will exploit photon scattering from coupled QDs to
realize efficient Bell-state measurements [39] or photon-photon quantum gates
[40]. Taking a broader perspective, the ability to deterministically couple
multiple 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 quantum 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 support from Danmarks
Grundforskningsfond (DNRF 139, Hy-Q Center for Hybrid Quantum Networks). B.S.
acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), Grant No. 449674892. O.S. acknowledges funding
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No. 801199. S.S., A.L. and A.D.W.
acknowledge 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 developed
the theoretical model, A.T., V.A and C.J.v.D. analyzed 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 company Sparrow Quantum that
commercializes single-photon sources.

Data and materials availability: The data presented in the main text and
Supplementary Materials for this publication are openly available from [42].

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Supplementary materials for
Collective super- and subradiant dynamics between distant optical quantum
emitters

###### Contents

  1. I Theoretical model
  2. II Sample and optical setup
  3. III Optical spectroscopy
  4. IV QD imaging
  5. V ‘Off-resonant’ excitation
  6. VI Resonant excitation

##  I Theoretical model

The theoretical description is based on the master equation for the system,
which reads $(\hbar=1)$

$\displaystyle\dot{\rho}=\mathcal{L}_{\mathrm{tot}}[\rho]=-i\left[H,\rho\right]+\mathcal{L}_{\text{coup}}[\rho]+\mathcal{L}_{\text{decay}}[\rho]+\mathcal{L}_{\text{deph}}[\rho].$
(S1)

Here the Hamiltonian
$H=\sum_{m}\Delta_{m}\sigma^{z}_{m}/2+\Omega_{m}(e^{i\theta_{m}}\sigma_{m}^{+}+e^{-i\theta_{m}}\sigma_{m}^{-})/2$
accounts for the detuning and potential driving of the involved emitters with
relative phases $\theta_{m}-\theta_{n}$. In our treatment, we include three
different Liouvillians. The first one $\mathcal{L}_{\mathrm{coup}}[\rho]$
describes the waveguide mediated coupling. This term is the main focus of this
work and is discussed in detail below. Additionally, we include a Liouvillian
corresponding to the decay of the emitters to modes other than the waveguide
(side modes)

$\displaystyle\mathcal{L}_{\text{decay}}[\rho]=\sum_{m}\gamma^{s}_{m}\sigma_{m}^{-}\rho\sigma_{m}^{+}$
(S2)

with a decay rate $\gamma^{s}_{m}$ for the $m$th emitter. Finally we also
include dephasing described by

$\displaystyle\mathcal{L}_{\text{deph}}[\rho]$
$\displaystyle=\frac{\gamma_{d}}{2}\sum_{m}\left[\sigma^{z}_{m}\rho\sigma_{m}^{z}-\rho\right],$
(S3)

where the dephasing rate $\gamma_{d}$ is assumed similar for both emitters for
simplicity. This is the standard description of pure dephasing acting
individually on each emitter.

To describe the waveguide mediated coupling, we use the well-known rotating-
wave approximated Jaynes-Cummings coupling Hamiltonian in one dimension
$H^{\prime}=\sum_{m}\int\mathrm{d}k(g_{k,m}\sigma_{m}^{+}a_{k}+g^{*}_{k,m}a_{k}^{\dagger}\sigma_{m}^{-})$
with light mode creation operators $a^{\dagger}$ and Pauli operators
$\sigma_{m}^{+}=|e\rangle_{m}\langle g|_{m}$ for different QD sites labeled
with $m=1,2,3$. The coupling of the light modes to the dipoles is given by
$g_{k,m}=-\mathbf{E}_{k}(\textbf{r}_{m})\cdot\mathbf{d}_{m}$, i.e., the
electrical field mode solution in the waveguide at coordinate $\mathbf{r}_{m}$
times the dipole vector of the $m$th quantum dot (QD).

The experiment is conducted in mirror-symmetric photonic-crystal waveguides
(PCWs) that have primarily linear local polarization so that effects of
directional chiral light-matter coupling can be ignored [43], i.e., coupling
coefficients for left and right propagation are of equal magnitude:
$|g_{k,m}|\approx|g_{-k,m}|$. The absence of chirality assumed here is
experimentally confirmed by comparing amplitudes of resonant transmission dips
(Figure S7) while sweeping the external magnetic field in two opposite
directions. For this, we look at the amplitude ratios between two dipoles for
each QD as a function of external magnetic field (Figure S8). We repeated the
same measurement with the direction of the magnetic field reversed (Figure
S8). The results show ‘mirror’ symmetry around zero magnetic field up to 4 T,
confirming the assumption of non-chiral interaction.

According to Bloch’s theorem, we can write the electric field in the periodic
waveguide as $\mathbf{E}_{k}(\mathbf{r})=\mathbf{U}_{k}(\mathbf{r})e^{ikz}$
with a periodic function $\mathbf{U}_{k}(\mathbf{r})$. As we argue in Sec. V
below, the procedure used to select QDs favors QDs located at similar
positions within the unit cell so that $\mathbf{U}_{k}(\mathbf{r}_{m})$ is
similar for all QDs. In combination with the linear polarization, this means
that apart from an overall phase, which can be absorbed by changing the origin
of the $z$-axis, we can choose the mode function to be real
$\mathbf{U}_{k}(\mathbf{r}_{m})\in\mathbb{R}^{3}$ for all QDs. Since the
waveguide obeys time-reversal symmetry, the mode functions must fulfill
$\mathbf{E}_{k}=\mathbf{E}_{-k}^{*}$. This means that the periodic function
$\mathbf{U}_{k}(\mathbf{r}_{m})$ is also real for backward propagation since
$\mathbf{E}_{k}=\mathbf{E}_{-k}^{*}$ implies $\mathbf{U}_{-k}=\mathbf{U}_{k}$.
Furthermore, apart from the propagation phases ($e^{ikz}$), the fact that
$\mathbf{U}_{k}(\mathbf{r}_{m})$ is real means that any phase of the coupling
constant can only arise from the dipole moment of the transition. Since this
dipole moment is the same for the coupling to all modes, any phase in the
dipole moment can be absorbed by a redefinition of $\sigma_{m}^{-}$. We thus
choose real coupling constants in the following, and the phases that appear
will be due to propagation. It is noted that any net phases due to the QDs not
having the same position within the unit cell can be incorporated into the
definition of the phase $\phi_{mn}$ below, but the assumption of real coupling
constants simplifies the description.

Integrating out the photonic degrees of freedom in the Markov approximation
leads to the Liouvillian for the waveguide mediated coupling [44, 45, 32]

$\displaystyle\mathcal{L}_{\mathrm{coup}}[\rho]=i\sum_{mn}J_{mn}[\sigma_{n}^{+}\sigma_{m}^{-},\rho]-\sum_{mn}\Gamma_{mn}\left[\sigma_{m}^{-}\rho\sigma_{n}^{+}-\frac{1}{2}\\{\sigma_{n}^{+}\sigma_{m}^{-},\rho\\}\right]$
(S4)

with the couplings

$\displaystyle\Gamma_{mn}=\frac{4\pi|g_{k_{0},m}||g_{k_{0},n}|}{v_{g}}\,\mathrm{Re}\\{e^{i\phi_{mn}}\\}\quad\mathrm{and}\quad
J_{mn}=\frac{2\pi|g_{k_{0},m}||g_{k_{0},n}|}{v_{g}}\,\mathrm{Im}\\{e^{i\phi_{mn}}\\},$
(S5)

which corresponds to the expressions in the main text with the identification
$\Gamma_{m}\beta_{m}=4\pi|g_{k_{0},m}|^{2}/v_{g}$, where $v_{g}$ is the group
velocity in the waveguide. Here, the coupling depends on quantities evaluated
at the resonant wavenumber $k_{0}$. To ease the notation, we omit the
subscript and simply denote it $k$ in the main text and below. The nature of
the interaction between the emitters depends crucially on the propagation
phase $\phi_{mn}=k|x_{mn}|$ with $|x_{mn}|$ being the distance between
emitters $m$ and $n$. This phase describes the interference of excitations
switching back and forth between the emitters and determines whether the
interaction is dispersive $|\Gamma_{mn}|\ll|J_{mn}|$ for
$\phi_{mn}\approx(N+1/2)\pi$ or dissipative $|J_{mn}|\ll|\Gamma_{mn}|$ for
$\phi_{mn}\approx N\pi$ with $N$ being an integer.

In addition to the master equation for the QDs, we also need to relate the
density matrix to the outgoing fields. These can be determined by the well-
known input-output formalism [46]. In case of excitation from the side, i.e.,
not through the waveguide, the left- and right-going fields are given by
(assuming the three QDs are labeled in ascending order from left to right)

$\displaystyle E_{L}=\sum_{n}e^{i\phi_{1n}}E_{n}\quad\mathrm{and}\quad
E_{R}=\sum_{n}e^{i\phi_{n3}}E_{n},$ (S6)

where $E_{n}=-i\frac{\sqrt{\beta_{n}\Gamma_{n}}\sigma_{n}^{-}}{\sqrt{2}}$. The
intensities are given by the squared modulus of the fields
$I_{L/R}=|E_{L/R}|^{2}$.

The theoretical results depicted throughout the main text are obtained via
numerical integration of Eq. (S1). To account for spectral diffusion, we
average all results over a normal distribution of detunings $\Delta_{m}$ with
standard deviation $\sigma_{sd}$. Alternatively, for a simplified analytical
treatment, we can ignore dephasing and only consider the dynamics in the
single excitation subspace. In this case, the master equation (S1) reduces to

$\displaystyle\dot{\rho}=-i\left[\mathcal{H}_{\text{eff}}\rho-\rho\mathcal{H}_{\text{eff}}^{{\dagger}}\right]$
(S7)

with the effective Hamiltonian given in Eq. (1) of the main text. This
evolution corresponds to the so-called no-jump evolution of the quantum Monte-
Carlo wavefunction approach. The omitted jump terms, representing the state
after a decay, prepare the system in the joint ground state, which doesn’t
emit any light. We can therefore understand the full system dynamics and the
emitted light from the no-jump evolution. On the other hand, when the system
starts in the doubly excited state, it can be reexcited or undergoes
dephasing, the jump evolution is important, and we resort to the full master
equation (S1) for a complete description of the dynamics.

To understand the properties of the emitted light, we now turn to general
symmetry properties that arise for pure dissipative coupling, i.e.
$\phi_{mn}=N\pi$; the experimental scenario observed here. We write the
intensities as

$\displaystyle I_{L}=\sum_{mn}E^{*}_{m}E_{n}e^{-i\phi_{mn}},$ (S8)
$\displaystyle I_{R}=\sum_{mn}E^{*}_{m}E_{n}e^{i\phi_{mn}}.$ (S9)

For purely dissipative coupling we thus have $I_{L}=I_{R}$ since for
$\phi_{mn}=N\pi$ the exponential terms
$e^{i\phi_{mn}}=e^{-i\phi_{mn}}=(-1)^{N}$, making the signal equal at both
detectors.

The symmetry ($\Delta_{mn}\rightarrow-\Delta_{mn}$) in detuning
$\Delta_{mn}=\Delta_{m}-\Delta_{n}$ between the different QDs can be explained
by observing that in case of purely dissipative coupling, the only imaginary
contribution to the master equation (S1), apart from potential phases in the
driving, arises from terms $\propto\Delta_{m}\sigma_{m}^{z}$. Thus, changing
$\Delta_{m}\to-\Delta_{m}$ leads to complex conjugation of the Liouvillian
$\mathcal{L}_{\mathrm{tot}}\to\mathcal{L}_{\mathrm{tot}}^{*}$, and we have
$e^{\mathcal{L}_{\mathrm{tot}}^{*}t}[\rho^{*}]=(e^{\mathcal{L}_{\mathrm{tot}}t}[\rho])^{*}$,
i.e., the master equation is solved by the complex conjugated density matrix
$\rho^{*}$. Since $\rho(t)\to\rho^{*}(t)$ does not affect measurement results
if Eq. (S6) is real ($\phi_{mn}=N\pi$), it follows that for initially real
$\rho=\rho^{*}$, e.g., for a QD quickly excited to the excited state, the time
evolution will produce the same intensity for opposite detunings. On the other
hand, driving two QDs simultaneously can induce a complex phase between the
QDs that breaks the symmetry, as is experimentally observed and discussed in
the main text.

Figure S1: Exemplary calculated time resolved intensities for left (a,c) and
right (b,d) ports according to the analytical results (Eq. (I)), in
qualitative agreement with the experimental data from the main text. The
normalization to the sum of each time trace was used for (b,d). The following
parameters are assumed: $\phi_{12}=0$, $\beta_{1}=\beta_{2}=0.99$,
$\Gamma_{1}=\Gamma_{2}=\Gamma$, $\gamma_{d}=0$.  Figure S2:  Theoretical
population of the different density matrix entries (which are not directly
accessible in the experiment): Due to the decay of the superradiant state
$|S\rangle$, the excitation $|eg\rangle\langle eg|$ gets redistributed to the
other QD $|ge\rangle\langle ge|$ in the shape of the subradiant state
$|s\rangle$, which exhibits an extremely long lifetime if close to $\Delta=0$.
For larger detunings, we observe oscillations between $|S\rangle$ and
$|s\rangle$ that quickly die out due to the fast emission of the superradiant
state. Here, the same parameters as for Figure S1 were used.  Figure S3:
Increasing the relative phase $\phi_{12}=\pi/4$ acquired between the two QDs
leads to asymmetries within and between the left (a,c) and right (b,d) output
ports. The absence of asymmetry between positive and negative detunings in the
experimental data is a clearly visible indicator for $\phi_{12}\simeq 0$. The
normalization to the sum of each time trace was used for (b,d). The following
parameters are assumed: $\phi_{12}=\pi/4$, $\beta_{1}=\beta_{2}=0.99$,
$\Gamma_{1}=\Gamma_{2}=\Gamma$, $\gamma_{d}=0$.

To give analytical expressions for the output fields of two QDs close to
resonance, we assume the initial $\pi$-flip (exciting the left QD) to be
quasi-instantaneous compared to the lifetimes $1/\Gamma_{1},1/\Gamma_{2}$ and
neglect dephasing ($\gamma_{d}=0$). For this simplified scenario, we can give
concise output fields arriving at the right detector at time $t$
$E^{R}(t)=E^{R}_{\mathrm{sup}}(t)+E^{R}_{\mathrm{sub}}(t)$ with the (original)
super- and subradiant contributions

$\displaystyle E^{R}_{\mathrm{sup}}(t)=$
$\displaystyle\frac{-i\sqrt{\Gamma_{1}\beta_{1}}(\Gamma_{2}\beta_{2}-i\Delta+(S+\Gamma_{1}-\Gamma_{2})/2)}{\sqrt{2}S}\exp[-(\Gamma_{1}+\Gamma_{2}+S)\frac{t}{4}]$
$\displaystyle E^{R}_{\mathrm{sub}}(t)=$
$\displaystyle\frac{i\sqrt{\Gamma_{1}\beta_{1}}(\Gamma_{2}\beta_{2}-i\Delta+(-S+\Gamma_{1}-\Gamma_{2})/2)}{\sqrt{2}S}\exp[-(\Gamma_{1}+\Gamma_{2}-S)\frac{t}{4}],$
(S10)

where
$S=\sqrt{4e^{2i\phi}\Gamma_{1}\Gamma_{2}\beta_{1}\beta_{2}+(2i\Delta-\Gamma_{1}+\Gamma_{2})^{2}}$.
We observe that for the ideal setup ($\Gamma_{1}=\Gamma_{2}=\Gamma$,
$\beta_{1}=\beta_{2}=1$, $\Delta=0$, $\phi=2N\pi$) we get
$E^{R}_{\mathrm{sub}}(t)=0$. This corresponds to the subradiant part of the
state $(|e_{1},g_{2}\rangle-|g_{1},e_{2}\rangle)/\sqrt{2}$ being reflected
between the two QDs and never leaving the system, while the superradiant part
$(|e_{1},g_{2}\rangle+|g_{1},e_{2}\rangle)/\sqrt{2}$ decays with double the
decay rate $\Gamma^{R}_{\mathrm{sup}}=2\Gamma$. For $\phi=(2N+1)\pi$, we
achieve the identical scenario, just the super- and subradiant states and
fields change place. The output fields in the left port can be separated into
sub- and superradiant contributions as well:

$\displaystyle E^{L}_{\mathrm{sup}}(t)=$
$\displaystyle\frac{-i\sqrt{\Gamma_{1}\beta_{1}}(e^{2i\phi}\Gamma_{2}\beta_{2}-i\Delta+(S+\Gamma_{1}-\Gamma_{2})/2)}{\sqrt{2}S}\exp[-(\Gamma_{1}+\Gamma_{2}+S)\frac{t}{4}]$
$\displaystyle E^{L}_{\mathrm{sub}}(t)=$
$\displaystyle\frac{i\sqrt{\Gamma_{1}\beta_{1}}(e^{2i\phi}\Gamma_{2}\beta_{2}-i\Delta+(-S+\Gamma_{1}-\Gamma_{2})/2)}{\sqrt{2}S}\exp[-(\Gamma_{1}+\Gamma_{2}-S)\frac{t}{4}],$
(S11)

where the prefactors indicate asymmetry in intensity between the right and
left output fields for $\phi\neq 2N\pi$.

The decay rate of the subradiant state becomes finite as soon as only one of
the parameters $\Delta,\phi$ deviates from the ideal values above. In the case
of $\beta<1$ for the symmetric waveguide, the subradiant state also decays,
but to non-guided ”side modes” that do not contribute to the output field so
that $E^{R}_{\mathrm{sub}}(t)$ is still zero. To the lowest order in those
parameters, the decay rate scales quadratically (assuming for simplicity
$\Gamma_{1}=\Gamma_{2}=\Gamma$ and $\beta_{1}=\beta_{2}=\beta$),

$\displaystyle\Gamma^{R}_{\mathrm{sub}}=\Gamma(1-\beta)+\Gamma\beta\phi^{2}/2+(2-\phi^{2})\Delta^{2}/(4\Gamma\beta)+\mathcal{O}(\Delta^{3}/\Gamma^{3})+\mathcal{O}(\phi^{3}),$
(S12)

while the rate $\Gamma^{R}_{\mathrm{sup}}$ is lowered by the same correction
in $\phi$ and $\Delta$. This manifests in a broad corridor of almost only
superradiant decay around $\Delta=0$; see the exemplary theoretical study
depicted in Figure S1 qualitatively agreeing with the actual experimental
results shown in the main text. The oscillations in intensity for increasing
$\Delta$ stem from the oscillations between the sub- and superradiant states:
in a simplified picture, a phase is acquired $|e_{1},g_{2}\rangle+e^{i\Delta
t}|g_{1},e_{2}\rangle$, letting both states evolve around the Bloch sphere
(Figure 3) and turn into one another. This is depicted in Figure S2 for the
same theoretically ideal scenario showing an oscillation between the super-
and subradiant states $|S\rangle,|s\rangle$. A finite driving time, much
shorter than the emitter lifetimes $\Gamma_{i}$, is assumed during this
calculation. The almost perfect symmetry of the decay tails also indicates a
relative phase $\phi\simeq 0$ since this oscillation is corrected in lowest
order as $(\Delta+\Gamma\beta\phi)/2$, increasing or decreasing the
oscillation frequency depending on the signs of $\Delta$ and $\phi$, see e.g.,
Figure S3.

##  II Sample and optical setup

The sample similar to the one described in [47] is cooled to 4 K in a closed-
cycle cryostat with optical access along the $z$-direction, see Figure S4. A
vector magnet allows us to apply an external magnetic field up to 5 T in the
$z$ direction and 3 T in the $x$-direction. A confocal microscope with 0.8
numerical aperture was used to image the sample, excite the QD from free
space, and couple light into the waveguide. Polarization control is used to
reject the laser scattering while collecting the $x$ or $y$-polarized emission
of collection port 1 and port 2, respectively. Shallow etched gratings [48]
couple to linearly polarized near-gaussian modes and have above 50 nm (FWHM)
bandwidth providing a near-equal broadband collection and excitation of the
QDs coupled to the waveguide. Details about the sample growth and fabrication
can be found in [47].

Acousto-optic modulators (AOMs) are used to stabilize the optical power of two
lasers. The continuous-wave laser used for the excitation of QD optical
transitions is continuously tunable between 900-980 nm. In addition, a pulsed
5 ps laser is utilized for the excited state lifetime measurements and p-shell
excitation. The lasers are combined using non-polarizing beamsplitters (BS) to
address the photonic chip inside the cryostat at 4 K temperature. The
polarization for each laser is controlled using polarizing beamsplitters (PBS)
and a pair of half- ($\lambda$/2) and quarter-waveplates ($\lambda$/4). The
emission from the outcoupling grating of the chip is collected using a BS and
is sent through a spectral filtering setup. The filtering is done using a
reflective diffraction grating with the resulting bandwidth of 25 GHz FWHM,
while the resonant excitation is filtered with a transmission diffraction
grating and an etalon filter with a bandwidth of approximately 3 GHz.
Avalanche single-photon detectors (APD) or superconducting single photon
detectors (SNSPD) are used to detect the filtered signal depending on the
experiment.

In our experiment, the InAs QDs are embedded into a PCW made from a suspended
GaAs membrane and connected to grating couplers at each of its ends to couple
light in and out of the PCW (Figure S4). The intrinsic layer containing self-
assembled InAs QDs is grown inside the membrane comprising a p-i-n diode. The
charge stabilization allows achieving near lifetime-limited emission
linewidths, while the integration into the PCW leads to near-unity $\beta$
factor strongly enhancing the photon-emitter interaction [49]. This makes it
possible to coherently couple emitters separated by a distance much longer
than the wavelength.

Figure S4:  (color online) (a) Energy level diagram of two QDs detuned by
$\Delta_{ij}$ and controlled by an external magnetic field $B_{z}$. (b)
Scanning electron microscope image of the equivalent two-sided PCW structure.
The QD is addressed by lasers propagating from free space, and the emitted
photons are coupled into the waveguide mode, scattered out from right (port 1)
or left (port 2) grating couplers, and coupled into a single-mode fiber. The
inset contains the fluorescence image of the QDs coupled to the PCW (cf.
Section IV below).

##  III Optical spectroscopy

Figure S5 shows the transmission spectra of the PCW and resonant transmission
dips of QDs under study with respect to the PCW band edge. These spectra were
taken at zero magnetic field and at 1.24 V bias voltage.

Figure S5:  (color online) (a) Transmission spectrum of the PCW. (b)
Transmission spectrum of the PCW at a bias voltage of 1.24 V showing
transmission dips of multiple QDs coupled to the optical mode of the PCW. The
spectral region containing the QDs under study is indicated between dashed
lines centered around 940.55 nm. The band edge of the PCW is estimated to be
around 943 nm (dash-dotted line).  Figure S6:  (color online) Fitting of the
resonant transmission spectra for QD1 (a), QD2 (b,c) and QD3 (d,e) for the
high frequency (HF) and low frequency (LF) dipoles for each QD at zero
magnetic field $B=0$. For QD1 only the LF dipole is visible at $B=0$.

In Figure S7, the laser frequency and external magnetic field are scanned. At
each laser frequency, laser background correction is performed by normalizing
to the counts measured at the nonresonant bias voltage of 1 V.

All QDs at zero magnetic field show fine structure splitting (FSS) to be 7.5
GHz (QD1), 6.5 GHz (QD2), and 4.1 GHz (QD3), respectively. The measured FSS
corresponds well to the average value for QDs in bulk InAs. QD1 shows strong
asymmetry for the two $x$ and $y$ dipoles, where only the high frequency (HF)
dipole is well coupled to the PCW at zero magnetic field, while QD2 and QD3
show a similar coupling also for the low frequency (LF) dipole. These
observations are due to the spatial variations of the polarization of the PCW
mode giving different projections of the QD transition dipole moments [9].

The situation changes once the external magnetic field is applied in the
z-direction. Two dipoles are split by the Zeeman interaction, and the relative
ratio in amplitude between the two dipoles is equalized. This is due to the
fact that for an out-of-plane magnetic field, the exciton transition dipoles
become circularly polarized, and therefore the projection on the local linear
PCW polarization implies that both dipoles couple equally. At the high field,
the Zeeman effect is strongly nonlinear because of the diamagnetic shift.

Table S1:  Summary of the extracted QD parameters at zero magnetic field: fine structure splitting (FSS), the exciton gyromagnetic ratio (g-factor) $g_{z}$, radiative recombination rate $\Gamma$, spectral diffusion broadening $\sigma_{sd}$. | QD1 | QD2 | QD3
---|---|---|---
FSS, GHz | 7.5 | 6.5 | 4.1
$g_{z}$ | 1.86 | 1.91 | 1.67
$\Gamma/2\pi$, GHz | 2.3 (LF) | 0.87 (HF), 0.69 (LF) | 0.92 (HF), 0.72 (LF)
$\beta$ | 0.94 (LF) | 0.88 (HF), 0.83 (LF) | 0.84 (HF), 0.83 (LF)
$\sigma_{sd}/2\pi$, GHz | 0.59 | 0.22 | 0.6

The tuning is done by the out-of-plane (Faraday geometry) magnetic field
(Figure S7). The external field gives rise to a pair of Zeeman-split energy
levels for the excited state $\ket{\uparrow\Downarrow}$ and
$\ket{\downarrow\Uparrow}$ connected to the “empty” ground state
$\ket{g}=\ket{0}$, representing two optical dipoles of the QD. The excited
state splitting is used to tune the optical transition in resonance with other
QDs. The external magnetic field leads to circular polarizations for both
dipoles $\sigma_{-}$ and $\sigma_{+}$ for
$\ket{0}\longleftrightarrow\ket{\uparrow\Downarrow}$ and
$\ket{0}\longleftrightarrow\ket{\downarrow\Uparrow}$, respectively. Using the
magnetic field, we are able to tune QD2 and QD3 optical transitions in
resonance around $B_{z}=1.05$ T, QD1 and QD2 close to $B_{z}=2.15$ T and QD1
and QD3 at $B_{z}=3.4$ T.

The absence of chirality is probed experimentally by comparing amplitudes of
resonant transmission dips (Figure S8) for the external magnetic field sweeps
in two opposite directions. The ‘mirror’ symmetry around zero magnetic field
confirms the assumption of non-chiral interaction.

Figure S7:  (color online) The transmission spectrum of the PCW as a function
of the out-of-plane external magnetic field $B$ (Faraday configuration). The
white dashed lines are the theoretical prediction for the Zeeman effect. The
diamagnetic shift $\sigma B^{2}$ is estimated to be $\sigma=2.05$ GHz T-2.
Figure S8:  (color online) Minimum of the transmission spectrum as a function
of the magnetic field $B$ for high (HF) and low frequency (LF) dipole for each
QD. The symmetry around the zero magnetic field illustrates the non-chiral
coupling of each QD to the PCW mode.

The fitting of the resonant transmission spectra at zero magnetic field using
the theoretical model allows extracting the $\beta$-factor to be above
$\beta>0.8$ for all measured dipoles (Table S1).

In Figure S9, the laser frequency and the bias voltage are scanned to probe
the frequency-voltage map of the QD. At each laser frequency, laser background
correction is performed to normalize the counts measured at a non-resonant
bias voltage of 1 V. All QDs show stable excitons between 1.22 V and 1.3 V.
The parallel tuning of the two dipoles for each QD confirms the origin of the
measured optical lines. If not stated explicitly otherwise, the voltage used
for all the measurements presented below was 1.24 V.

Figure S9:  (color online) The frequency-voltage scan of the resonance transmission for (a) QD1, (b) QD2 and QD3 at zero magnetic field with 318.67 THz (a) and 318.75 THz (b) laser frequency offset.  Table S2: Parameters of the photon-mediated coupling for different QD pairs: radiative recombination rates $\\{\Gamma_{i},\Gamma_{j}\\}$, spectral diffusion linewidth $\sigma_{sd}$, phonon dephasing rate $\gamma_{d}$, phase lag $\phi_{ij}$, dissipative coupling rate $\Gamma_{ij}$ and dispersive coupling rate $J_{ij}$. | QD2-QD3 | QD1-QD2 | QD1-QD3
---|---|---|---
$\\{\Gamma_{i},\Gamma_{j}\\}/2\pi$, GHz | 0.79, 0.73 | 0.85, 0.8 | 0.9, 0.65
$\sigma_{sd}/2\pi$, GHz | 0.38 | 0.18 | 0.33
$\gamma_{d}/2\pi$, GHz | 0.03 | 0.03 | 0.03
$\phi_{ij}$, rad | 0.05 | 0.08 | 0.05
$\Gamma_{ij}/2\pi$, GHz | 0.61 | 0.66 | 0.61
$J_{ij}/2\pi$, GHz | 0.03 | 0.05 | 0.03

##  IV QD imaging

To locate the spatial positions of the QDs within the PCW (Figure S4) we
performed imaging of the sample using a CCD camera. For this, six images were
taken, one while exciting each QD on and off resonance, respectively. Another
image was acquired while imaging the sample surface using broadband light
around 940 nm. To excite each QD, the resonant laser light was sent through
the PCW (port 1). The CCD camera (DCC CMOS camera (1280 x 1024)) detected the
light scattered by the QD outside the PCW mode for an acquisition time of $30$
s. An image of the background (measured at 1 V) was then subtracted, see
Figure S10 (first three images from left). A gaussian fit to the peak was used
to identify the position of each QD in pixels with respect to the camera
resolution. The image of the structure, taken with white light on the sample,
was used to calibrate the dimensions and extract relative distances.

Figure S10: Imaging of QD1, QD2, and QD3 in the PCW. The first three (from
left) images were taken during resonant transmission measurements and were
fitted accordingly to extract the maximum of the counts, revealing each QD
position. The fourth image is done using broadband light centered around 940
nm illuminating the chip. The image is superimposed with the first three
images. The frequency and bias voltage used for each QD are noted. The axes
units are in pixels.

The position (in pixels) for each QD was found from the Gaussian fit. To
calibrate the distance, we used the size of the PCW from the white light image
(in pixels), and its size in $\mu$m from the mask used to fabricate the chip.
We got the ratio between the two units: 181(3) nm/px, leading to the
corresponding separations of: 1.25(3) $\mu$m for QD1-QD2 (2nd crossing around
2.15 T), 0.96(3) $\mu$m for QD2-QD3 (1st crossing around 1.05 T) and 2.21(4)
$\mu$m for QD1-QD3 (3rd crossing around 3.4 T).

##  V ‘Off-resonant’ excitation

To characterize the lifetime of the QDs, ‘off-resonant’ excitation through
higher-order QD shells (p-shell, d-shell …) was used. For this, a 5 ps 80 MHz
pulsed laser was used. The initial characterization was performed by measuring
the excitation spectra of the QD using a continuous laser. An example of the
excitation spectra for QD2 is shown on Figure S11. The absence of other QD
spectral lines in the recorded spectra guaranteed single QD excitation with
this method. The excitation frequency was then chosen such that it was far
enough from the resonant one, so that it is easier to filter it out of the
collection path, but close enough, so that the relaxation time to the s-shell
of the conduction band, would not influence the lifetime measurement itself.
Specifically, an excitation around 325.72 THz was used for the lifetime
measurement (920.4 nm corresponding to one of the peaks in the figure). The
power dependence of all three QDs shows the saturation behavior (Figure S12).
An excitation power close to saturation was used during the lifetime
measurements.

Following the pulsed laser excitation, the collected intensity was filtered
using an optical grating setup. The intensity was collected from different
ports of the PCW (Figure S4). The lifetime measurement was performed as a
function of the external magnetic field around each QD crossing. Figure S13,
Figure S14 and Figure S15 show the results of the lifetimes measurements
together with the theoretical model, which uses the extracted parameters from
Table S2. The time trace for each magnetic field was normalized to the sum of
the total counts in the trace.

The detector response function measured independently was used to fit all the
datasets. For QD2-QD3 crossing (Figure S13) the SNSPD with a 200 ps FWHM
temporal jitter was used. While for QD1-QD2 and QD1-QD3 crossings (Figure S14,
Figure S15), it was a fast APD with 40 ps FWHM temporal jitter.

To find one consistent set of parameters, we use the theoretical model
described above. Modeling of the data was done by using Eqs. (I), (I) to
calculate the intensity in each of the ports. Spectral diffusion was taken
into account by integrating the overall intensity over the spectral linewidth
for detunings $\Delta$ between each pair of QDs. Each of the time traces was
normalized to the sum of the trace. For the fitting, five time traces were
used at different detunings around zero detuning. The results of the fit are
shown in Table S2. $\sqrt{\beta_{2}\beta_{3}}=0.8$ is used during the fitting.
This is a conservative value leading to a lower bound on the coherent coupling
constants that are extracted from the fit.

The extracted coupling constants $\Gamma_{ij}$ and $J_{ij}$ are very similar
for all three QD pairs leading to a similar accumulated phase lag $\phi_{ij}$
corresponding to predominantly dissipative coupling. This can be explained by
the fact that all three QDs were selected using the resonant transmission
measurement by giving strong extinction in resonant transmission measurements
(Figure S6). With this approach, we pre-select only the QDs strongly coupled
to the mode of the PCW (i.e., with a high $\beta$-factor). Since the
$\beta$-factor has a strong spatial dependence, this condition results in
having QDs at well-defined positions inside the unit cell of the PCW [50].

Close to the band edge of the PCW, the wavevector is given by $ka=\pi$, where
$a$ is the lattice constant of the photonic crystal. The phase lag between two
QDs separated by the integer number of unit cells $x_{ij}=Na$ is then
$\phi_{ij}=N\pi$, corresponding to dissipative coupling. As explained in
Section I, the effects of directional chiral coupling can be ignored. The
effects from local phases due to QDs positioned off-axis in the PCW can be
incorporated into the definition of the phase $\phi$ while keeping the same
definition for the coupling constants Eq. (S5).

We notice that the separations between QD pairs are very close to an integer
number of unit cells of the photonic crystal with period $a=240$ nm:
$x_{12}=5.2(1)a$, $x_{23}=4.0(1)a$ and $x_{13}=9.2(2)a$, respectively. This
leads to similar contributions from dissipative $\Gamma_{ij}$ and dispersive
$J_{ij}$ coupling for different QD pairs.

Figure S11:  Example of the excitation spectra of QD2 at zero magnetic field
emitting at 940.09 nm (dashed line). The 2D map on the left side shows higher
energy level states of QD2 while the frequency of the continuous excitation
laser is increasing. The excitation around 920.4 nm was used for lifetime
measurements. The plot on the right shows photoluminescence excitation spectra
recorded at a fixed wavelength of 940.09 nm (dashed line in the left plot).
Figure S12: Power saturation measurements for the three QDs, when driven by a
pulsed off-resonant excitation beam.

Figure S13:  (color online) ’Off-resonant’ excitation of QD2 for different
detunings from QD1 and at a magnetic field of 2.1 T. (a-b) Experimental decay
dynamics of QD2 as a function of the detuning between QD1 and QD2
$\Delta_{12}$ for photon collected from port1 (a) and port2 (b). (c-d)
Theoretical calculation is represented by the developed theory model. (e-h)
Decay dynamics for different frequency detunings between QD1 and QD2
$\Delta_{12}$. For the off-resonant condition, the decay is fitted using two
decay processes, where the 2nd decay rate is equal to 0.07(1) GHz, associated
with a spin-flip process. The on-resonance decay consists of two decay
processes corresponding to the super- and subradiant decays. The decay with
detuning close to the linewidth (green dots) contains additional modulation on
top of the decay. Parameters from Table S2 were used for the theory curves.
Figure S14:  (color online) Same as Fig. S13, but for excitation of QD3 and
varying the detuning to QD2 and at a magnetic field strength of 1.0 T.  Figure
S15:  (color online) Same as Fig. S13, but for excitation of QD1 and varying
the detuning to QD3 and at a magnetic field strength of 3.5 T.

##  VI Resonant excitation

The dynamics shown in Figure 2(e),(f), and Figure 3(e) in the main text are
induced by resonant (s-shell) excitation. In this case, the pulsed laser
frequency was tuned into resonance with the emission frequency of the driven
QDs, i.e., about $318.76$ THz. The laser was suppressed in the collection path
with both a transmission grating and a temperature-controlled etalon filter,
where the latter has a narrow linewidth of $3$ GHz. The excitation pulse from
the laser was stabilized with an AOM and frequency filtered. The power for the
laser pulse was calibrated based on Rabi oscillations with the QDs detuned
from each other (Figure S16).

Results of the excitation of QD3 when tuned close to resonance with QD2 are
shown in Figure S17. The recorded behavior is equivalent to the measurement
results with ’off-resonant’ excitation.

The difference becomes clear when exciting both QDs from the pair, see Figure
S18. For these measurements, the same setup is used while the polarization of
the driving laser field is adjusted to pump both QDs. The phase difference
between two excitation fields is $\theta\approx\pi/2$, which is set by the
local polarization projection of the driving field to two dipoles of the
crossing. The two QD transition dipole moments are orthogonally (circularly)
polarized in the Faraday magnetic field configuration, see Figure S5a.,
corresponding to the transitions
$\ket{0}\longleftrightarrow\ket{\uparrow\Downarrow}$ and
$\ket{0}\longleftrightarrow\ket{\downarrow\Uparrow}$, respectively. The
polarization control of the driving field allows to effectively control the
phase $\theta$ between two driving fields.

The fitting of the phase $\theta$ and driving field areas $\Omega_{1}$ and
$\Omega_{2}$ was done using the theoretical model described above. The
parameters extracted from the previous fit Table S2 for the QD2-QD3 pair were
used, while fitting only $\theta$, $\Omega_{1}$ and $\Omega_{2}$ parameters.
The results are given in Table S3.

Table S3: Parameters of the photon-mediated coupling for QD2-QD3 pair while driving both QDs: radiative recombination rates $\\{\Gamma_{2},\Gamma_{3}\\}$, spectral diffusion linewidth $\sigma_{sd}$, phonon dephasing rate $\gamma_{d}$, phase lag $\phi_{23}$, dissipative coupling rate $\Gamma_{23}$ and dispersive coupling rate $J_{23}$, $\Omega_{2}$, $\Omega_{3}$ excitation pulse areas, $\theta$ phase between two driving fields. $\\{\Gamma_{2},\Gamma_{3}\\}/2\pi$, GHz | 0.79, 0.73
---|---
$\sigma_{\text{sd}}/2\pi$, GHz | 0.38
$\gamma_{d}/2\pi$, GHz | 0.03
$\phi_{23}$, rad | 0.05
$\Gamma_{23}/2\pi$, GHz | 0.61
$J_{23}/2\pi$, GHz | 0.03
$\Omega_{2}$, $\Omega_{3}$ | 0.87(6), 1.33(5)
$\theta$ | $-0.48(2)\pi$
Figure S16: Rabi oscillations. Left: Exciting only one dipole (QD3) of the
QD2-QD3 pair. Right: Exciting both dipoles after rotating the polarization of
the excitation laser. The dipole of QD2 is resonant with the filtering etalon
when applying a voltage of around 1.277 V. Maxima appear for two different
values of the bias voltage, indicating that both dipoles are populated for
different powers. The measurement is done at the magnetic field of 0.8 T,
using a 5 ps laser pulse centered around 940.5 nm.  Figure S17:  (color
online) Resonant excitation of QD3 around the resonance with QD2 at 1.05 T.
(a-b) The experimental excitation dynamics of QD3 as a function of the
detuning between QD2 and QD3 $\Delta_{23}$ for photon collected from port1 (a)
and port2 (b). (c-d) Theoretical calculation represented by the developed
theory model. (e-h) Decay dynamics for different frequency detunings between
QD2 and QD3 $\Delta_{23}$. The on-resonance decay consists of two decay
processes corresponding to the super- and subradiant decays. For the detuned
case, an additional coherent modulation is superposed on top of the decay.
Parameters from Table S2 were used for theory curves.  Figure S18:  (color
online) Resonant excitation of the QD2-QD3 pair around their resonance at 1.05
T. (a) Experimental excitation dynamics of the QD2-QD3 pair as a function of
the detuning between QD2 and QD3 $\Delta_{23}$ for photons collected from the
port1. (b)Theoretical calculation represented by the developed theory model.
(c-d) Decay dynamics for different frequency detunings between QD2 and QD3
$\Delta_{23}$. Parameters from Table S3 were used for theory curves.