Document [original]

ARTICLE

Mutual coupling and synchronization of optically

coupled quantum-dot micropillar lasers at ultra-low

light levels

Sören Kreinberg1, Xavier Porte1, David Schicke2, Benjamin Lingnau2, Christian Schneider3, Sven Höﬂing3,4,

Ido Kanter5, Kathy Lüdge2& Stephan Reitzenstein1

Synchronization of coupled oscillators at the transition between classical physics and

quantum physics has become an emerging research topic at the crossroads of nonlinear

dynamics and nanophotonics. We study this unexplored ﬁeld by using quantum dot micro-

lasers as optical oscillators. Operating in the regime of cavity quantum electrodynamics

(cQED) with an intracavity photon number on the order of 10 and output powers in the 100

nW range, these devices have high β-factors associated with enhanced spontaneous emis-

sion noise. We identify synchronization of mutually coupled microlasers via frequency locking

associated with a sub-gigahertz locking range. A theoretical analysis of the coupling behavior

reveals striking differences from optical synchronization in the classical domain with negli-

gible spontaneous emission noise. Beyond that, additional self-feedback leads to zero-lag

synchronization of coupled microlasers at ultra-low light levels. Our work has high potential

to pave the way for future experiments in the quantum regime of synchronization.

https://doi.org/10.1038/s41467-019-09559-2 OPEN

1Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany. 2Institut für Theoretische Physik, Technische

Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany. 3Technische Physik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany.

4SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, UK. 5Gonda Brain Research Center and Department of Physics,

Bar-Ilan University, Ramat-Gan 52900, Israel. Correspondence and requests for materials should be addressed to X.P. (email: javier.porte@tu-berlin.de)

or to S.R. (email: [email protected])

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 1

1234567890():,;

Synchronization is an ubiquitous phenomenon in mutually

coupled systems1which—under appropriate conditions—

leads to a spontaneous self-organization of the coupled

elements2. A multitude of different physical, biological, or che-

mical systems can exhibit synchronization, making it a funda-

mental interdisciplinary property of interacting nonlinear

systems1,3,4. The complexity of this phenomenon is well depicted

by the variety of existing synchronization scenarios. One pro-

minent example is chaos synchronization, where the individual

coupled elements all follow the same chaotic trajectory5. In this

context, semiconductor lasers are attractive table-top devices to

study fundamental aspects of nonlinear dynamics and synchro-

nization6–15 with proposed applications in random number

generation and secure key exchange16,17.

Recently, the prospect of exploring synchronization in coupled

nanoscale oscillators has received increasing attention. Enabled

by important technological advances, it has become feasible to

investigate nonlinear dynamics and synchronization at ultra-low

energies in systems previously only explored from a quantum

mechanical perspective. For instance, mutual synchronization of

the Kuramoto type has been demonstrated in optomechanical

structures18 and in nanomechanical oscillators19,20. Most inter-

esting is the quantum limit of nonlinear interaction and syn-

chronization, which has been addressed experimentally for

instance in 2D Josephson junction arrays21. It has also triggered

numerous theoretical studies which predict novel phenomena

such as partial locking and synchronization blockade22,23, even

elucidating interesting connections between entanglement and

synchronization24–26.

Situated at the crossroads between nonlinear dynamics,

nanophotonics and quantum optics, cavity-enhanced microlasers

are interesting devices to drive research on synchronized oscil-

lators toward the quantum regime. They offer a rich spectrum of

exciting physics with potential applications as coherent light

sources in system-on-chip quantum technologies27. Due to their

low-mode volume on the order of the cubic wavelength, micro-

lasers usually operate in the regime where cavity quantum elec-

trodynamics (cQED) effects such as enhanced spontaneous

coupling in terms of high (β) factors become important. Up until

now, microlaser studies have focused almost exclusively on the

properties of individual devices, not considering coupling inter-

actions with external passive or active elements. However, this

situation is changing and recent works report on interesting

effects like spontaneous symmetry-breaking due to local coupling

between cavity modes in nanophotonic structures28,29 and on

tailoring of the mode-switching dynamics and photon statistics in

feedback coupled microlasers30,31. At the same time it has

become interesting to theoretically describe the dynamics and

stability of microlasers when mutually coupled with delay32–34.

Beyond that, if it comes to scaling effects and the pursuit for

better understanding of coupling and synchronization in complex

small-scale systems in the presence of enhanced noise, our work

may also foster progress in related disciplines like socio-

economics, biology, ecology, and hydrodynamics35–37.

Here, we present the experimental implementation of optically

coupled microlasers with incoherent optical coupling delay at far-

below µW output power levels. We apply bimodal semiconductor

quantum dot (QD) micropillar lasers driven with intracavity

photon numbers on the order of ten to study mutual coupling at

ultralow light levels. This detailed investigation on the dynamics

of coupled micro- or nanolasers is of interdisciplinary and

immediate importance for scientists working on the dynamics of

nonlinear oscillators and for those interested in microscopic or

nanophotonic lasers. In the studied devices the energy degeneracy

of the fundamental cavity mode is lifted by slight structural

asymmetries resulting in two orthogonal linearly polarized

fundamental mode components38–40. The related bimodal beha-

vior is of essential importance for the dynamic properties of

micropillar lasers and leads to a plethora of exciting physical

phenomena such as gain competition41, unconventional normal-

mode coupling42, and mode switching43 as an instance of

Bose–Einstein condensation of photons44. Using bimodal QD-

microlasers we experimentally demonstrate mutual locking and

synchronization via a detailed analysis of their spectral properties

and photon statistics in face-to-face conﬁguration and in presence

of an additional passive delay. Accurate numerical modeling

supports our ﬁndings and allows us to reveal the underlying time-

resolved character of the synchronized dynamics.

Results

Emission characteristics of bimodal micropillars. The QD

microlasers, which we used for coupling experiments as sketched

in Fig. 1, were realized by means of molecular beam epitaxy of

planar microcavity structures and subsequent nanoprocessing of

electrically driven micropillar cavities as we detail in Supple-

mentary Note 1. These devices are delicate objects at the forefront

of science and technology and their optical properties, for

instance in terms of the emission energy, vary within certain

bounds from microlaser to microlaser. While this variation is not

critical for fundamental studies of individual microlasers as done

in many previous works, it becomes a major issue in case

of coupling scenarios between different microlasers with an

effective injection locking range on the order of a few gigahertz45.

In our present work the situation is even more problematic since

we do not only require spectral matching of the microlasers, they

also need to show very similar emission properties with respect

to their output power and linewidth to enable symmetric mutual

coupling and synchronization experiments. For this purpose we

fabricated large linear arrays each consisting of 120 QD-

microlasers from which we selected pairs of suitable candidates

as described in Supplementary Note 1 in more detail.

Injection current emission characteristics of a particular

suitable pair of microlasers with almost identical emission

properties are depicted in Fig. 2. The experiments were performed

using the spectroscopic setup described in the Methods section

and in Supplementary Note 1 in more detail. The strong mode

(SM) of each of the two lasers, which are used in mutual coupling

experiments, shows a characteristic s-shape in log–log presenta-

tion of the input–output curves and the slight nonlinearity

μpillar 1 μpillar 2

bμpillar 1

μpillar 2

Fig. 1 Illustration of the studied experimental conﬁgurations. aFace-to-face

mutual coupling and bmutual coupling via passive relay of two micropillar

(μpillar) lasers. The setup is arranged to couple the two perpendicularly

polarized emission modes of each micropillar (i.e. strong mode (SM) and

weak mode (WM)) to their respective counterparts

ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2

2NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications

indicates a high β-factor. The weak mode (WM) of each laser

loses the intermodal gain competition43, saturates at intermediate

injection current and decreases in intensity at highest pump

conditions. The description of the experimental data by our

theoretical model (see Methods) yields experimentally not

accessible parameters (summarized in Table 1) such as the

spontaneous emission factor of β≈4×10

−3for both microlasers

and the injection current dependent intracavity photon number.

As can be seen in Fig. 2a, b this photon number is as low as 1–20

in the working range (shaded areas) of our coupling experiments.

The SM and WM emission frequencies plotted in the middle

panels of Fig. 2shift to higher values with increasing pump due to

the plasma effect until a red shift sets in at high injection currents

because of sample heating. The frequency splitting between SM

and WM is 26 GHz for pillar 1 and 21 GHz for pillar 2 and stays

constant over the investigated pump current range (see Methods

for high resolution spectra and more information). As presented

in the lower panels of Fig. 2the SM linewidths decrease by more

than two orders of magnitude eventually narrowing down to less

than 100 MHz at the highest injection currents of about 30 μA. In

contrast, the WM’s above-threshold linewidths increase, which

indicates a noncomplete transition to laser action for these

modes. Noteworthy, previous injection-locking experiments on

QD-micropillar lasers revealed a locking range of approximately

1 GHz45. Thus, to resolve possible locking effects between two

mutually coupled micropillar lasers, emission linewidths smaller

100

15 20 25 30

0.1

100

15 20 25 30

Output power (nW)

Exp.

Sim.

P1 SM

Frequency

–333 THz (GHz)

Spectral

FWHM (GHz)

Current (μA) Current (μA)

0.1

Cavity photon number

Exp.

Sim.

Fig. 2 Input–output characteristics of two QD-micropillar lasers used in mutual coupling experiments. Experimental (dots) and numerical (lines)

injection current dependence of output power, emission frequency and linewidth for the strong modes (SM) and weak modes (WM) of pillar P1 (panels

a,c, and e) and pillar P2 (panels b,d, and f). Dashed areas indicate the operation regime for the mutual coupling experiments which require emission

linewidths smaller than one gigahertz (yellow area) and can be achieved for injection currents exceeding 20 μA. The calculated intracavity photon number

presented in panels aand branges from about 1 and 20 in the relevant current regime and was determined numerically by our theory

Table 1 Parameters used for the simulations if not stated

otherwise

Fitted parameter Value

Optical cavity losses,

strong (weak) mode

(κ

) 39 (38.5) ns−1

Optical gain coefﬁcient,

strong (weak) mode

s(g0

w) 5.35 (5.21) m2

V2s

Self gain compression,

strong (weak) mode

(ε

)10ð12Þ´1010 m2

Cross gain compression,

strong (weak) mode

(ε

)16ð17:8Þ´1010 m2

Spontaneous emission factor βP1(P2) 3.5 (4) × 10−3

Parasitic current JP1ðP2Þ

p2.3 (7.3) μA

Injection efﬁciency ηP1(P2) 0.0596 (0.0674)

Linewidth enhancement factor αP1(P2) 1.7 (1.0)

Reservoir carrier lifetime τ

1ns

Given parameter Value

Effective scattering rate Sin 7×10

−15 m2ps−1

Effective lasing mode area A15 µm2

Lasing mode volume V5µm3

Number of (in)active QDs ZQD

inact 312 (938)

Background refractive index n

3.34

QD lifetime, μ-laser 1 (2) τP1ðP2Þ

sp 155 (185) ps

Photon energy ℏω1.38 eV

Coupling delay time τ3.85 ns

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2 ARTICLE

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 3

than the expected locking range of approximately 1 GHz are

required (yellow areas in Fig. 2) which deﬁnes the working range

(shaded areas in Fig. 2) of the microlasers for the coupling

experiments discussed in the following.

Frequency locking of mutually coupled microlasers.Weﬁrst

study the spectral properties of our coupled optical oscillators in

face-to-face conﬁguration, unveiling a coherence behavior and

locking properties particular to high-βmicrolasers. Therefore, we

mutually couple the selected pair of micropillar lasers and vary

the relative detuning between the two microlasers as shown in

Fig. 3a–d. The emission frequency of pillar 1 (P1) is kept constant

(at constant temperature of 32 K). Meanwhile the frequency of

pillar 2 (P2) is precisely scanned across the emission frequency of

pillar 1 by sweeping its temperature in the range T

∈[32, 36 K].

While the temperature is swept, emission spectra of pillar 1 are

recorded by using a Fabry–Perot scanning interferometer. A

matrix is formed from the spectra, such that each column of the

matrix corresponds to one spectrum. Emission spectra of pillar 2

are recorded in the same way in a second run. The matrices are

then plotted as 2D heat maps. The detuning ranges of ±3 GHz

displayed in Fig. 3a–d correspond to a temperature scan ranging

from 34.9 to 33.9 K. When tuning the two lasers close to reso-

nance, i.e., for detunings ≲0:5 GHz, clear mutual frequency

locking can be identiﬁed as a change in slope of the relative

frequency vs. detuning characteristics: within the locking range,

the emission of both lasers is shifted toward a common frequency,

returning to their free-running values outside of the locking

range. A comparison between panels a–d of Fig. 3illustrates that

the locking range depends on the mutual coupling strength

(varied by adjusting the variable attenuator in the coupling path),

which has a transmittance Tof 90% (38%) in a–d.

Deeper insight into the locking behavior requires a more

detailed study of the locking range as a function of the coupling

strengths. In agreement with previous reports on externally

controlled micropillar lasers45 and coupled semiconductor

lasers33,46, the obtained locking-range width is proportional to

–3

–2

–1

Relative frequency f (GHz)

Pillar 1

IP1 = 27.2 μA

T = 90%

Pillar 2

IP2 = 27.4 μA

T = 90%

Pillar 2

IP2 = 27.4 μA

T = 38%

Pillar 1

IP1 = 27.2 μA

T = 38% 0.0

0.1

0.3

0.4

0.5

0.6

0.8

0.9

1.0

Norm. int.

–3 –2 –1 0 1 2 3

–3

–2

–1

0.0

0.5

1.0

1.5

2.0

Locking range (GHz)

Square root attenuator transmittance T½

Experiment

Simulation

Fit through (0,0)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

IP2 = 24.4 μA

eCoupling strength K

IP1 = 27.1 μA

IP2 = 27.4 μA

IP1 = 27.2 μA

Nominal detuning s (GHz)

–3 –2 –1 0 1 2 3

Fig. 3 Mutually coupled strong modes of the two micropillar lasers for different coupling strengths. aDetuning scans of the strong modes with high (panels

aand b) and low (panels cand d) coupling strengths K.Tis the transmittance of the attenuator in the coupling path. eDependence of the locking range on

the square root of the attenuator transmission (lower axis) and the coupling strength Kused in numerical simulations (upper axis). The horizontal axes are

scaled such that experimental and simulated data both lie on a common linear function

ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2

4NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications

the injected electric ﬁeld strength, i.e., to the square root of light

intensity controlled by the attenuator transmittance (T). Figure 3e

depicts such dependence for both experimental (symbols) and

simulations (solid line) data. In order to plot both datasets

together, we match the linear dependence of the locking range on

K(numerics) with the experimental data to ﬁnd the proportion-

ality factor between Kand the square root attenuator transmit-

tance. Even though measuring the free space optical losses (beam

splitters, polarization optics, lenses and cryostat windows) is in

principle feasible, this matching is necessary because it is not

possible to quantify the coupling efﬁciency into the pillar and to

the laser ﬁeld. The maximum experimental amplitude coupling

strength (at T=1) is thus estimated as K≈2.5%. In the

simulations (solid lines), the coupling strength Kis studied over

a larger range.

Identiﬁcation and analysis of mutual coupling. The presence of

locking between the microlasers emission unequivocally indicates

coupling. However, it is the slope mdescribed by the microlasers’

emissions inside the locking region (see Fig. 3a–d), which

determines the direction of the coupling. Figure 4a depicts this

slope as the ratio of frequency change of the locked signal Δfand

the nominal detuning Δνbetween the laser modes. We use this

slope as the indicator for having achieved not only unidirectional

but mutual coupling. Consider for instance the limiting case of a

unidirectional injection experiment: here the emission of the

injecting master laser by deﬁnition must not be inﬂuenced by the

slave laser subjected to injection. Strictly speaking this condition

can only be fulﬁlled by placing optical isolators in the coupling

path. However, provided that the output powers of the two

mutually coupled lasers are at least strongly imbalanced, there

will be one “master-like”laser and one “slave-like”laser, even

without an optical isolator. While the former is almost unaffected

by the mutual coupling, the latter is strongly inﬂuenced by the

injected light. In this situation, when tuning the master-like laser,

the slave-like laser will perfectly follow the injected signal in the

locking region, which results in a locking slope of m=1. On the

contrary, if only the slave-like laser is tuned, the locking slope will

have a value of 0, because its emission frequency is locked to the

master-like laser. If the output power imbalance between master

and slave is reduced, the locking slope will start turning away

from these extreme values and eventually reach m=0.5 for

evenly balanced coupling (cf. horizontal dotted line in Fig. 4b).

Based on these considerations, for phase-locked (or frequency-

locked) lasers under mutual coupling conditions, the locking

slopes of both oscillators, m

and m

, should be equal, as both

lasers are locked to each other and emit light on a frequency in

between the two free-running laser lines. Surprisingly, the two

microlasers exhibit different locking slopes, both in experiment

and simulations. In Fig. 4b, the inverse slopes m1

P1 and m1

P2 can

be seen to differ especially for low-output powers of pillar P2,

which resembles a master–slave setup for which m

=0is

expected. This means that inside the locking range, the average

emission frequency of the two microlasers is deviating propor-

tionally to the nominal detuning. Importantly, these deviations

are not expected to occur for classical coupled oscillators and are

attributed to the effect of partial locking in high-β

microlasers45,47. The fact, that the locking slopes get more

similar when the output power of pillar P2 is increased, can be

explained by two converging factors: (a) the stronger injection

into pillar P1 and (b) the decreasing relative contribution of

quantum noise to the output power of pillar P2.

The experimental and numerical observations in Fig. 4b are

further analyzed reducing our laser model to a system of coupled

phase oscillators as detailed in the Methods section. Based on Eq.

(10) we conclude that for ﬁxed output power of pillar P1 this

description yields that m

depends on the output power P

out,2

pillar P2 via

ðPout;2ÞA/m1

P2 1;ð1Þ

with a scaling factor A¼1

2. Thus, theory predicts that the

common frequency within the locking range gets pulled closer

toward the free-running frequency of P2, i.e., m1

P2 1!0, with

increasing power of pillar P2. The experimental scaling coefﬁcient

exp

is obtained by ﬁtting the equation

BðPout;2ÞAexp ¼m1

P2 1;ð2Þ

to the experimental data. In contrast to the analytic expectations,

the experimental data and numerical simulations suggest an

exponent of A≈−2 instead of the expected A¼

2(see

respective dashed and gray continuous lines in Fig. 4b. The

different A coefﬁcients of −1/2 and −2 are explained by the fact

that the solution of the simpliﬁed coupled phase equations in Eq.

(2) describes the behaviour of individual Fabry–Perot ﬁne

structure modes (c.f. Supplementary Note 2) that are not resolved

in experiment. However, in the experiment and in our numerical

QD laser model (Eqs. (4)–(6)), all coupled-cavity modes

–3 –2 –1 0 1 2 3

–3

–2

–1

Nominal detuning 

(GHz)

Relative frequency f (GHz)



Δf

m =

50 200 500100

P1: 237 nW

–1

– 1

P2 optical output power (nW)



Δf

Fig. 4 Locking slopes of the two mutually coupled micropillar lasers. Panel aillustrates how the slope mis calculated. bExperimental (symbols) and

numerically simulated (lines) locking slopes in dependence of the optical output power of pillar P2. The horizontal dotted line depicts the classically

expected slope m=0.5 and the oblique dashed and continuous gray lines respectively correspond to the slopes of A=−2 and A=−0.5 (Please see the

main text for the deﬁnition of A)

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2 ARTICLE

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 5

contribute and, after ﬁtting a Lorentzian proﬁle to the envelopes,

both results agree very well and lead the slopes of −2 in Fig. 4.

We refer to Supplementary Note 3 for more information on the

numerical simulations of the scaling coefﬁcient. In future

experiments, it will be interesting to study the locking slopes

with higher spectral resolution (of better than 10 MHz) to access

the regime predicted by the solution of the coupled phase

equations.

Identiﬁcation of synchronization by correlation studies. In the

ﬁeld of cavity-enhanced nano- and microlasers a detailed study of

the photon statistics of emission is of particular interest. Mea-

suring the power-dependent photon autocorrelation function g(2)

(τ) allows for instance to unambiguously prove lasing emission in

high-βlasers operating close to the limit of thresholdless opera-

tion48, for the identiﬁcation of superradiant emission49, or for

ruling out chaotic mode switching45. In addition, it is also highly

beneﬁcial for the identiﬁcation of chaotic dynamics in feedback

coupled microlasers operating at ultra low-emission powers30.

Determining the photon auto- and cross-correlation function is

also highly interesting in the present case of mutually coupled

microlasers to obtain profound insight into the underlying

emission dynamics and possible synchronization of intensity

ﬂuctuations. Due to the intrinsic mode competition in micropillar

lasers, we expect mode-switching events during which the SM is

dark and the WM is bright. The frequency of these mode-

switching events and the related intensity variance are enhanced

in the present case of high-βlasing43.

In the respective experiment the output intensities of pillar P1

and pillar P2 are cross correlated via single-photon counting

module (SPCM) 1 and SPCM 2 as presented in the Methods

section and more detailed in Supplementary Figure 1, respec-

tively. Polarization optics are used to ﬂexible detect photons from

any polarization mode of pillars P1 and P2. We focus in our study

on the case where the WMs are resonantly coupled and feature

pronounced intensity ﬂuctuations, as the SMs show only

marginal signatures of photon bunching and no signiﬁcant

cross-correlation peaks when resonantly coupled. As reference we

present auto-correlation functions of noncoupled pillars P1 and

P2 in Supplementary Figure 7 and brieﬂy discuss the observations

in Supplementary Note 5. Noteworthy, pronounced photon

bunching in auto- and cross-correlation functions is a typical

behavior in delay-coupled micropillar lasers30. We denote the

second-order photon correlation function of the WMs as gð2Þ

wiwj,

giving the auto-correlation for pillar iwhen i=j, and the cross-

correlation for i≠j. An example of a WM–WM cross-correlation

measurement is shown in Fig. 5a for pump currents of I

=27.7

μA and I

=24.5 μA. Clear peaks can be observed at t

−t

≈4

ns, corresponding to the coupling delay of 3.85 ns between the

microlasers. The double-peak structure indicates leader-laggard

intensity synchronization of the two micropillars, i.e., if a

ﬂuctuation occurs in pillar P1, there is a chance that it will be

repeated in pillar P2 and vice versa. The numerical time series

depicted in Fig. 5b conﬁrm this interpretation of the experimental

data in terms of leader-laggard dynamics32, showing a strong

similarity between the time-series when either of the time-series is

shifted in time by the coupling delay τ. The laser coupling can be

observed to irregularly induce short mode-switching events in

both lasers (e.g., near t=153 ns for pillar P1 in Fig. 5b). The

relatively low-peak values of the cross-correlation gð2Þ

w1w2ðτÞin

comparison to the free-running auto-correlation (gð2Þ

w1w1ð0Þ¼1:5

for pillar 1 and gð2Þ

w2w2ð0Þ¼1:6 for pillar 2) proves imperfect

synchronization between the lasers, and suggests that only a small

ratio (≈13%) of switching events are repeated in the respective

other laser.

The intensity cross-correlation depends on the dynamical

susceptibility of the lasers to a perturbing signal, and thus on their

ability to reproduce and synchronize to the signal of the other

laser. We, therefore, investigate the dependence of the cross-

correlation on the mutual laser detuning ν

of the WMs.

Figure 6a, b shows the measured cross-correlation of the WMs of

the two lasers and the FPI spectra of the WM, respectively. Since

the SM is much more intense than the WM, it is still visible in the

FPI spectra on a logarithmic scale even after attenuation by the

polarizing beam splitter (PBS). Interestingly, as we discuss in

Supplementary Note 6, mutual optical coupling does not

signiﬁcantly inﬂuence the overall intensity in detuning dependent

locking experiments. The mutual locking of the WMs around a

WM detuning of 0 leads to a strong enhancement of the WM

signals, while suppressing the SM intensity. Near the locking

range of the SMs, at a WM detuning of ν

≈−5 GHz, the reverse

effect is observed together with a strong suppression of the WMs.

This can be understood by the reduction of effective optical losses

of the WM by 2.5%, thus reducing the required inversion of the

QDs to maintain lasing and reducing the available gain for the

SMs. This is a well-known effect from two-mode lasers in other

148 152 156

Pillar P1 intensity

152 156 160

Pillar P2 intensity

Time (ns)

–40 –20 0 20 40

0.98

1.00

1.02

1.04

1.06

1.08

g(2)

w1w2

(t2–t1)

Time delay (t2–t1) / ns

Fig. 5 Intensity cross-correlations of two coupled micropillar lasers. a

Intensity cross-correlation gð2Þ

w1w2ðt2t1Þof the weak modes of pillars P1 and

P2. The weak modes were tuned to resonance (ν

=0) in face-to-face

conﬁguration (see conﬁguration a in Fig. 1) at injection currents I

=27.7 μA

and I

=24.5 μA. The two main peaks at 4.3 ns suggest leader-laggard

synchronization of the intensity ﬂuctuations between the lasers. One

roundtrip (7.7 ns) further, at 12 ns, weaker revival peaks are barely

observable. b, c Simulated intensity dynamics, showing the leader-laggard

behavior of the two coupled micropillars P1 (panel b, upper trace: strong

mode, lower trace weak mode) and P2 (panel c, upper trace: strong mode,

lower trace weak mode) at injection currents of I

=27.1 μAandI

=24.2 μA.

The time axis for pillar P2 has been shifted with respect to pillar 1 by 3.85

ns, i.e., the optical distance between the two micropillars. This illustrates

the delayed correlation of the intensity ﬂuctuations

ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2

6NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications

setups50–52. In Fig. 6c, d, the corresponding simulated cross-

correlation and optical spectra are depicted, matching the

experimental data very well. In order to reproduce the conditions

from Fig. 6b, simulations of the attenuated strong-mode spectra

were superimposed onto the simulated WM spectra in Fig. 6d.

Within the locking range of the WMs, intensity ﬂuctuations are

generally suppressed, thus leading to smaller delay peaks in the

cross-correlation. At either edge of the locking range (ν

≈±1.5

GHz), the signature of the dynamic unlocking of both lasers

becomes evident, leading to stronger peaks in the gð2Þ

w1w2cross-

correlation. Depending on the detuning, the cross-correlation

peak ±τcan be enhanced, i.e., the role of the leader in the leader-

laggard synchronization of the microlasers is mainly taken on by

the laser that is positively frequency-detuned with respect to the

other laser. This asymmetry in the frequency detuning is due to

the amplitude-phase coupling, i.e., nonzero α12. In Supplemen-

tary Note 4, we discuss the impact of the α-factor in more detail

and compare experimental locking results with simulations

considering a constant α-factor. An enhancement of the weak-

mode correlations can be observed also within the locking range

of the SMs, as the WMs are suppressed and driven further toward

thermal (bunched) emission. For scenarios where strong correla-

tion between the coupled laser emission is required, a detuning

near the locking boundaries of the WMs or within the SM locking

range should be preferred.

Zero-lag synchronization of microlasers with self-feedback.

Previous work showed the possibility of zero-lag synchronization

of chaotic intensity ﬂuctuations in small networks of mutually

coupled semiconductor lasers, in particular if the lasers are also

subject to feedback14. We explore this important regime of cou-

pled nonlinear oscillators. Noteworthy, this setting (see Fig. 1b) in

the single-photon regime could eventually be linked to entan-

glement of mutually coupled quantum systems14.

Here, we explore the possibility of zero-lag synchronization by

introducing a mirror relay in the center of the beam path between

the two oscillators. The length of the feedback beam path is

chosen to introduce additional passive optical feedback to each

cavity-enhanced microlaser. The feedback delay is equal to the

coupling delay time between the pillars. A semipermeable mirror

is thus placed at half distance in the coupling path, such that it

introduces feedback of the required time delay. As seen in the

previous discussion of Fig. 6, a strong cross-correlation between

the coupled lasers can be expected in regions of dynamical

instabilities. We therefore choose two other micropillar lasers, P1′

and P2′(see Supplementary Notes 6 and 7 for more details), from

the same arrays and couple them with a semipermeable mirror in

the aforementioned setup. These pillars show a crossing of their

SM and WM intensity in their current dependence at pump

currents far above threshold and exhibit more frequent mode-

switching events between their respective SM and WMs43. The

SM competition at this operating point results in a striking

increase in the autocorrelation gð2Þ

wiwið0Þand an enhanced

sensitivity with respect to optical feedback31, which should

enhance the correlation signatures when coupling the two

microlasers. In order to quantify the cross-correlation gð2Þ

w1′w2′ðτÞ,

we calculate the linear intensity cross-correlation coefﬁcient for

the two coupled pillars

ρðτÞ¼ gð2Þ

w1′w2′ðτÞ1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

gð2Þ

w1′w2′ð0Þ1



gð2Þ

w1′w2′ð0Þ1



r;ð3Þ

and expect a value of 1 (−1) for fully linearly correlated (anti

correlated) dynamics and a value of 0 for uncorrelated dynamics.

The resulting time-dependent correlation coefﬁcient is displayed

in Fig. 7. With the passive optical feedback from the micropillars

to the semipermeable mirror, additional correlation peaks at a

time delay of zero appear (blue line) if compared to Fig. 5a, along

with revival peaks after integer multiples of the coupling delay.

While the cross-correlation measurement shows zero-lag correla-

tion coefﬁcients of up to 34%, a strong peak of up to 50% at the

coupling delay time (red and green lines) can be seen,

corresponding to simultaneously occurring leader-laggard type

synchronization. The coexistence of both zero-lag and leader-

–4 0 4

–8

–4

–4 0 4

–8

–4

Experiment

0.96

1.00

1.04

1.08

1.12

(2)

2()

(2)

2()

–4 0 4

Experiment

Strong modes

suppressed

Strong modes

locked

Weak modes

locked

Weak modes

suppressed

101

102

103

Simulation

Time delay  (ns)

0.94

0.99

1.04

1.09

1.14

1.19

1.24

–4 0 4

–2

Simulation

Nominal detuning w (GHz)

Relative frequency f (GHz)

5 × 103

5 × 102

5 × 101

5 × 100

5 × 10–1

Fig. 6 Correlation and locking maps of two coupled micropillar lasers.

Measured aand simulated cweak-mode intensity cross-correlation

gð2Þ

w1w2ðτ¼τ2τ1Þ(color-coded) in face-to-face conﬁguration as function

of the time delay τfor different detunings ν

=28.0 μA and I

25.6 μA). b,dCorresponding log-intensity Fabry–Perot interferometer (FPI)

spectra (color-coded) of the laser output in dependence of the detuning

=28.6 μA, I

=27.0 μA in the experiments, I

=27.1 μA, I

24.4 μA in the simulations

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2 ARTICLE

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 7

laggard synchronization peaks in the cross-correlation of the

high-βmicrolasers can be interpreted as a coexistence or

stochastic transition between the two types of dynamics. In that

direction, strong noise is known to perturb coupled lasers away

from the synchronization manifold53, leading to intermittent

desynchronization events known as bubbling.

Discussion

Synchronization of coupled systems is at the heart of nonlinear

dynamics and can lead to a plethora of dynamical patterns ran-

ging from leader-laggard behavior to zero-lag synchronization. It

plays a vital role in our brain activity and can be applied for

secure data communication. We set out to push the ﬁeld of

optical synchronization toward the few-photon regime by

studying the joint dynamics of mutually coupled microlasers with

cavity-enhanced functionality and sub-µW output powers. In our

microlasers, spontaneous emission couples orders of magnitude

more efﬁciently to the lasing mode compared to semiconductor

lasers. Thus, their dynamics is crucially inﬂuenced by enhanced

spontaneous emission noise which is negligible in the classical

counterparts but plays an important role in our studies. Indeed,

we merged the topical areas of nanophotonics and nonlinear

physics by mutually coupling quantum-dot microlasers with a

small intracavity photon number (<20), similar to numbers

observed in micropillar lasers very recently by means of photon-

number resolving detectors54, to explore the noise-governed

regime of synchronization for the ﬁrst time.

When our microlasers are not too far detuned, clear mutual

locking of their emission frequencies is observed. Due to the high-

spontaneous emission noise in the cavity-enhanced micropillar

lasers, the locking remains imperfect, manifesting itself in a

deviation of the locking slopes of both lasers. This behavior is in

striking contrast to macroscopic coupled laser setups, where the

unlocking transition is abrupt. The time-resolved intensity cross-

correlation measurements show a partial synchronization of the

intensity patterns, reaching correlation coefﬁcients of up to 50%.

When coupled with a passive relay, signatures of both zero-lag

synchronization as well as leader-laggard type can be observed.

Our experimental results are described in excellent agreement

by numerical simulations based on semiclassical rate equations.

The simulations support the interpretation of noise-driven

dynamics in our coupled system of cavity-enhanced optical

oscillators and reveals a ﬁne structure of the optical spectra of the

locked microlasers comprising several compound laser modes,

forming a frequency comb with a broad Gaussian envelope. We

interpret this mode structure as a stochastic switching between

different compound laser modes, which are individually locked

between the coupled microlasers. A detailed experimental ver-

iﬁcation of this predicted behavior is subject to future work.

Noteworthy, the investigated QD microlasers exhibit charge

carrier lifetimes in the order of 0.1 ns due to the enhancement of

the spontaneous recombination rate and the carrier-density

dependent scattering processes. Consequently, the carrier lifetime

differs from the photon lifetime of 0.01 ns only by a factor of ten,

which, compared to macroscopic quantum well lasers, is very low.

The microlasers thus exhibit behaviour similar to class-A lasers,

encompassing strongly damped relaxation oscillations and also a

higher stability to external feedback55,56. Most of the existing

literature on coupled lasers is devoted to the quantum-well laser

case and thus new coupled dynamics might occur in our case. A

theoretical bifurcation analysis for delay coupled lasers in57

already suggest that a smaller time-scale ratio leads to much wider

locking ranges and new phase locked solutions, however,

experimental results in this regime are still missing. Thus, besides

the inﬂuence of stochastic effects on the dynamics, the strongly

damped internal dynamics also plays a crucial role for the

observed locking dynamics.

In summary, our experiments prove mutual coupling and zero-

lag synchronization in the few-photons regime of interacting

optical oscillators. We have revealed that in this regime with on

the order of ten intracavity photons and high β-factors quantum

noise starts to become signiﬁcant and classical synchronization

features get smeared out. We conﬁrm these effects by highly

sensitive single-photon cross-correlations. Interestingly, the issue

of realizing synchronization in noise quantum systems has

recently been explored theoretically and it was shown that it can

be overcome by application of squeezing-driven oscillators58.As

such the present experimental and theoretical results have high

potential to open up new perspectives to explore synchronization

at the crossroad between classical and quantum physics. Espe-

cially interesting in this sense would be investigations on mutual

coupling of nanoscale oscillators, where the fascinating bound-

aries between classical synchronization and quantum entangle-

ment phenomena24,25,59 can be experimentally explored.

Methods

Sample technology and experimental setup. The microlasers under study are

5μm diameter electrically contacted micropillars based on AlGaAs heterostructures

consisting of a single layer of In

0.3

0.7

As QDs with a density of 5 × 109cm−2

enclosed by two high-quality AlAs/GaAs distributed Bragg reﬂectors (DBR) (see

Supplementary Note 1 for more technological details). This conﬁguration ensures a

small mode volume and pronounced light–matter interaction that result in cQED-

enhanced coupling of spontaneous emission into the lasing mode60.

Using advanced nanofabrication technology and an optimized sample design

we realized dense arrays of 120 QD-micropillars each. For the coupling

experiments sample pieces each containing one of these arrays were placed into

two independent He-ﬂow cryostats separated by 700 mm and operated in a

temperature range between 31 and 36 K. The lasers in the two selected arrays stem

from neighboring parts of the same semiconductor wafer to ensure similar

emission characteristics. All micropillars in the array share one common gold

contact bar and are thus driven in parallel. Therefore, we chose voltage-driven

operation (instead of the commonly preferable current-driven operation), in order

to decouple the operating point of each micropillar from random electrical changes

in other micropillars. The electrical current through each micropillar under

investigation was estimated to be 1/120 of the current through the corresponding

–20 –15 –10 –5 0 5 10 15 20

–0.1

0.0

0.1

0.2

0.3

0.4

0.5

P1′ leading

Zero lag

P2′ leading

Cross correlation coefficient 

Time delay (t

2′

–t

1′

) / ns

Fig. 7 Cross-correlation coefﬁcient of synchronized micropillar lasers. The

plot shows the delay-dependent cross-correlation coefﬁcient ρ(τ=t

2′

−t

1′

)

of mutually coupled micropillar lasers with an additional mirror relay. In the

delay range [−10 ns, 10 ns] the sum (bright yellow curve) of ﬁve peaks of

the form Aexp (−|τ−τ

center

|/τ

corr

) are ﬁtted to the data (black). The zero-

lag peak is depicted in blue, the leader-laggard peaks where pillar 1′or pillar

2′is leading are depicted in red and green, respectively

ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2

8NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications

120 micropillar array, but the exact current through a speciﬁc micropillar is not

known

Figure 8a presents the experimental setup which is used to study the mutual

coupling of micropillar lasers via symmetric paths. Emission of each microlaser is

ﬁrst collimated by an aspheric lens with signiﬁcantly reduced transmission losses if

compared to usually used long working-distance microscope objectives and is then

directed by beam-splitters with 90% reﬂectivity to the other microlaser of the

selected pair. We would like to note that the use of an aspheric lens is crucial to

achieve a high enough optical power level for the mutual coupling experiments

between the microlasers. The transmitted light (10%) is directed via a PBS toward

the two detection paths. Using polarization optics, it is possible to independently

select the micropillar modes (strong and weak) being coupled and also those being

detected. An optional variable attenuator (VarAtt) in the coupling path enables

control of the coupling strength. In an alternative version of the experiment, which

aims at the demonstration of zero-lag synchronization, the variable attenuator is

substituted by a passive relay (pellicle mirror with 50% transmittance) placed in the

center of the beam path between both pillars. The required submicrometer

mechanical stability of the coupling beam path between the pillars with microscale

upper facets is ensured by a customized video control loop in which the

microscopic image of each sample was constantly monitored by a computer such

that (unavoidable) temperature-induced sample shifts were automatically

compensated by tracking the motorized linear stages of the corresponding cryostat.

A sample spectrum of the emission modes of the selected pair of micropillar

lasers is shown in Fig. 8b. The diagram illustrates the deﬁnition of the nominal

detuning νs¼νP2

sνP1

sand νw¼νP2

wνP1

wof the SM and WMs, respectively.

Due to a different frequency splitting between strong and WMs in the two lasers,

the SM and WM can be precisely and independently tuned in and out of resonance

individually by centikelvin temperature changes. We would like to note that the

SM–WM splitting of each individual pillar depends mainly on structural

asymmetries and was found to be independent of temperature and injection

current. This can also be seen in Fig. 2(panels cand d), showing the pump current

dependence of the individual mode frequencies. In laser P1 (P2), a splitting of 26

GHz (21 GHz) is found. Consequently, the nominal detunings of WMs and SMs of

the selected micropillar lasers differ by a constant value of 5 GHz, ν

=ν

+5 GHz.

By changing the injection current and thus, the output intensities of the two

microlasers the coupling conﬁguration can be continuously tuned from a master/

slave scenario (where the output power of one laser is much larger and drives the

weak laser) to a mutual coupling scenario (where the output powers of both lasers

are similar)57.

Theoretical model. The model used in this paper is based on semiclassical sto-

chastic rate equations61 taking into account the electron scattering mechanisms

into the QDs as derived in our previous works43,62,63. The description of micro-

and nanolasers with semiclassical equations was recently shown to be valid down to

a surprisingly low number of emitters on the order of ten64. Our chosen theoretical

framework should therefore be suited to accurately describe the dynamical prop-

erties of the micropillar lasers considered here. In our model we account for the

two orthogonal linearly polarized micropillar modes by two separate complex

electric ﬁeld equations, denoted as WM and SM, corresponding to their respective

output power above threshold as discussed in the previous section. As the

microlaser output is predominantly linearly polarized and dominated by strong

spontaneous emission, we couple both laser modes to a single-charge carrier type

and describe the mode interaction by phenomenological gain compression terms.

We thus neglect spin-ﬂip dynamics required to model the behaviour in lower-β

VCSEL devices65,66. For each of the two coupled lasers, we model the electrical

ﬁelds E

of the two modes j∈w, s, the occupation probability of the active and

inactive QDs ρ

(in)act

, and the reservoir carrier density n

. Here we denote as active

the portion of QDs within the inhomogeneous distribution that couple to the lasing

mode via stimulated emission.

dtEjðtÞ¼

hωZQD

ϵ0ϵbgVgjð2ρactðtÞ1Þκj

ð1þiαÞEjðtÞ

þ∂

∂tEjjjsp þ∂

∂tEjjcoup

;ð4Þ

dtρactðtÞ¼ P

j2fs;wg

gj½2ρactðtÞ1EjðtÞ



2ρactðtÞ

τsp

SinnrðtÞ½1ρactðtÞ

;ð5Þ

dtnrðtÞ¼ η

e0AðJJpÞSinnrðtÞ2ZQD

A½1ρðtÞ

nrðtÞ

τr2ZQD

inactρinact

Aτsp

:ð6Þ

The laser is pumped by injecting an electric current Jinto the reservoir n

from

where electrons may either recombine without contributing to the lasing mode or

scatter into QDs with the rate Sin ×n

(t). We account for experimental details in

the pumping process by assuming a laser dependent injection efﬁciency η(see ηP1

(P2) in Table 1), and a parasitic current J

, determined from ﬁts to the experimental

input–output curves, see also Fig. 2and Table 1. The occupation of inactive dots is

calculated from the steady-state value of Eq. (5) without stimulated emission,

taking into account only spontaneous recombination within these dots:

ρinactðtÞ¼ðτspSinnrÞð1þτspSinnrÞ1

:ð7Þ

The electric ﬁelds of WM and SM both interact with the active QDs by

stimulated emission. Since the frequencies of the two modes differ by only a few

tens of μeV, we consider only one carrier population that is interacting with both

optical modes, which leads to gain competition, modeled as

gj¼g0

j1þε0nbgc0X

i2fs;wg

εjijEiðtÞj2

1

:ð8Þ

The gain g

s,w

of strong and WMs respectively depends on the individual

intensity of both modes and the compression factors ε

with i,j∈{w, s}. A mode

with high intensity reduces (compresses) the gain for both modes.

Spontaneous emission into the lasing modes is modeled via a Gaussian white

noise source ξðtÞ2C, where 〈ξ(t)〉=0 and 〈ξ(t)ξ(t′)〉=δ(t−t′), such that

∂

∂tEjjsp ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

hω

ε0εbg

2ZQD

ρ2

act

τsp

sξðtÞ:ð9Þ

We simulate the two coupled micropillar lasers each with its own set of

differential Eqs. (4)–(6), with the two lasers indicated by an index P1, P2,

respectively. In the rotating frame of the free-running emission frequency of P2, the

Normalized intensity

70 80 90 100

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Frequency - 333 THz (GHz)

900.1 900.0

bWavelength (nm)

Through monochromator to SPCM / HBT / FPI

Through monochr. PBS

LinPol

VarAtt

50:50

Cryostat 2

Cryostat 1

Pillar 2

Pillar 1

90:10

225 mm

λ/2 λ/2

λ/2

LinPol

350 mm

700 mm

P1

sP2

P1

w P2

w

s

Fig. 8 Experimental setup and high-resolution emission spectra.

aExperimental setup showing the coupling beam path (solid red) and the

detection beam paths (pale red). Each micropillar laser sample is placed in a

cryostat at temperatures of T

=32 K and T

∈[32 K, 36 K]. Linear polarizers

(LinPol) in combination with half-wave plates (λ/2) are used for mode

selection, a variable attenuation (VarAtt) is used to control the mutual

coupling strength, and a 50/50 polarizing beam splitter (PBS) directs the

pillar emission to the monochromators with attached detectors, which

include single-photon counting modules (SPCM) that are used to measure

high-resolution spectra by a Fabry–Perot interferometer (FPI) and intensity

auto-correlations in a Hanbury Brown and Twiss (HBT) conﬁguration or

cross-correlations. In addition, 90/10 (90:10) beam splitters are used for

white light illumination and monitoring of the sample surface. bFabry–Perot

interferometer (FPI) spectra of the noncoupled micropillar lasers P1 and P2.

The strong mode and weak mode (respectively normalized to 1 and to 0.7)

are depicted in the same color-coding as used in Fig. 2

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2 ARTICLE

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 9

mutual coupling of the two lasers is expressed by

∂

∂tEP1

jjcoup ¼KκP1

jEP2

jðtτÞþ2πiνjEP1

∂

∂tEP2

jjcoup ¼KκP2

jEP1

jðtτÞ;

where Kis the coupling strength and τthe time delay after which the light from

one laser arrives at the other. The term ν

accounts for the relative frequency

detuning between the two SMs, with an additional 5 GH detuning between the

WMs due to the mode splitting mentioned above: ν

=ν

+5 GHz.

Using the above model, we can accurately reproduce the measured

input–output characteristics and current-dependent linewidths (see lines in Fig. 8),

and allows for an accurate extraction of model parameters from the measured data.

The slight differences in the laser characteristics between the two microlasers lead

also to slightly different input parameters for the modeled devices. The parameters

used in the simulations are listed in Table 1.

Description of locking slopes by coupled phase oscillators description.To

theoretically analyze our experimental and numerical observations in Fig. 4b), we

reduce our laser model to a system of coupled phase oscillators4.Wedosoby

neglecting the amplitude dynamics of the electric ﬁelds within the microlasers and

setting the linewidth enhancement factor α=0. Dropping the Henry factor αis

required to obtain clear analytical solutions. The resulting phase equations read

φ1ðtÞ¼ε2!1sinðφ2ðtτÞφ1ðtÞÞ

φ2ðtÞ¼ε1!2sinðφ1ðtτÞφ2ðtÞÞ þ 2πν

In order to quantify the locking dynamics, we deﬁne the locking slope m

m¼df

dν

;

where 2πf¼_

φ1¼_

φ2is the common phase velocity of the mutually locked

oscillators. A locking slope of m=0orm=1 denotes the limit cases where the

locked oscillation frequency of both oscillators is given by the free-running

frequency of oscillator 1 or 2, respectively.

Within this approach, the locking slope mdepends on the quotient of the

coupling strengths

εn!m¼KκPm

EPn

EPm



;

and can be calculated approximately to

m1

P2 1ε1!2

ε2!1

þ2τε1!2:ð10Þ

The second term on the right hand side dominates the locking slope for all cases

considered in this work.

Data availability

The data supporting the ﬁndings presented in this study are available from the

corresponding author upon request.

Received: 28 August 2018 Accepted: 19 March 2019

References

1. Rosenblum, M., Pikovsky, A. & Kurths, J. Synchronization, A Universal

Concept In Nonlinear Sciences. (Cambrige University Press, Cambridge, 2003).

2. Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The

Kuramoto model: a simple paradigm for synchronization phenomena. Rev.

Mod. Phys. 77, 137–185 (2005).

3. Winfree, A. T The Geometry of Biological Time. (Springer: New York, 1980). .

4. Kuramoto, Y Chemical Oscillations, Waves, and Turbulence, vol. 19 of Springer

Series in Synergetics. (Springer: Berlin, Heidelberg, 1984.

5. Winful, H. G. & Rahman, L. Synchronized chaos and spatiotemporal chaos in

arrays of coupled lasers. Phys. Rev. Lett. 65, 1575 (1990).

6. Heil, T., Fischer, I., Elsässer, W., Mulet, J. & Mirasso, C. Chaos

synchronization and spontaneous symmetry-breaking in symmetrically delay-

coupled semiconductor lasers. Phys. Rev. Lett. 86, 795–798 (2001).

7. Javaloyes, J., Mandel, P. & Pieroux, D. Dynamical properties of lasers coupled

face to face. Phys. Rev. E 67, 036201 (2003).

8. Erzgräber, H., Krauskopf, B. & Lenstra, D. Compound laser modes of mutually

delay-coupled lasers. SIAM J. Appl. Dyn. Syst. 5,30–65 (2006).

9. Liu, Y.-Y., Xia, G.-Q., Deng, T., He, Y. & Wu, Z.-M. Experimental

investigation on the nonlinear dynamic characteristics of mutually delay-

coupled semiconductor lasers system. Optoelectron. Adv. Mater. 13, 613

(2011).

10. Mirasso, C. R., Vicente, R., Colet, P., Mulet, J. & Pérez, T. Synchronization

properties of chaotic semiconductor lasers and applications to encryption. C.

R. Phys. 5, 613–622 (2004).

11. Fischer, I. et al. Zero-lag long-range synchronization via dynamical relaying.

Phys. Rev. Lett. 97, 123902 (2006).

12. Ozaki, M. et al. Leader-laggard relationship of chaos synchronization in

mutually coupled vertical-cavity surface-emitting lasers with time delay. Phys.

Rev. E 79, 026210 (2009).

13. Tiana-Alsina, J. et al. Zero-lag synchronization and bubbling in delay-coupled

lasers. Phys. Rev. E 85, 026209 (2012).

14. Aviad, Y., Reidler, I., Zigzag, M., Rosenbluh, M. & Kanter, I. Synchronization

in small networks of time-delay coupled chaotic diode lasers. Opt. Express 20,

4352–4359 (2012).

15. Soriano, M. C., García-Ojalvo, J., Mirasso, C. R. & Fischer, I. Complex

photonics: dynamics and applications of delay-coupled semiconductors lasers.

Rev. Mod. Phys. 85, 421–470 (2013).

16. Kanter, I. et al. Synchronization of random bit generators based on coupled

chaotic lasers and application to cryptography. Opt. Express 18, 18292–18302

(2010).

17. Porte, X., Soriano, M. C., Brunner, D. & Fischer, I. Bidirectional private key

exchange using delay-coupled semiconductor lasers. Opt. Lett. 41, 2871–2874

(2016).

18. Heinrich, G., Ludwig, M., Qian, J., Kubala, B. & Marquardt, F. Collective

dynamics in optomechanical arrays. Phys. Rev. Lett. 107, 043603 (2011).

19. Zhang, M. et al. Synchronization of micromechanical oscillators using light.

Phys. Rev. Lett. 109,1–5 (2012).

20. Matheny, M. H. et al. Phase synchronization of two anharmonic

nanomechanical oscillators. Phys. Rev. Lett. 112,1–5 (2014).

21. Vinokur, V. M. et al. Superinsulator and quantum synchronization. Nature

452, 613–615 (2008).

22. Walter, S., Nunnenkamp, A. & Bruder, C. Quantum synchronization of two

Van der Pol oscillators. Ann. Phys. 527, 131–138 (2015).

23. Lörch, N., Nigg, S. E., Nunnenkamp, A., Tiwari, R. P. & Bruder, C. Quantum

synchronization blockade: energy quantization hinders synchronization of

identical oscillators. Phys. Rev. Lett. 118, 243602 (2017).

24. Mari, A., Farace, A., Didier, N., Giovannetti, V. & Fazio, R. Measures of

quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111,

103605 (2013).

25. Galve, F., Luca Giorgi, G. & Zambrini, R. Quantum Correlations and

Synchronization Measures. In Fanchini, F., Soares Pinto, D. & Adesso, G.

(eds.) Lectures on General Quantum Correlations and their Applications,

393–420 (Springer, Cham, 2017).

26. Witthaut, D., Wimberger, S., Burioni, R. & Timme, M. Classical

synchronization indicates persistent entanglement in isolated quantum

systems. Nat. Commun. 8, 14829 (2017).

27. Munnelly, P. et al. A pulsed nonclassical light source driven by an integrated

electrically triggered quantum dot microlaser. IEEE J. Sel. Top. Quantum

Electron.21, 681–689 (2015).

28. Hamel, P. et al. Spontaneous mirror-symmetry breaking in coupled photonic-

crystal nanolasers. Nat. Photonics 9, 311–315 (2015).

29. Marconi, M., Javaloyes, J., Raineri, F., Levenson, J. A. & Yacomotti, A. M.

Asymmetric mode scattering in strongly coupled photonic crystal nanolasers.

Opt. Lett. 41, 5628 (2016).

30. Albert, F. et al. Observing chaos for quantum-dot microlasers with external

feedback. Nat. Commun. 2, 366 (2011).

31. Holzinger, S. et al. Tailoring the mode-switching dynamics in quantum-dot

micropillar lasers via time-delayed optical feedback. Opt. Express 26,

22457–22470 (2018).

32. Mulet, J., Mirasso, C. R., Heil, T. & Fischer, I. Synchronization scenario of two

distant mutually coupled semiconductor lasers. J. Opt. B 6,97–105 (2004).

33. Wünsche, H.-J. et al. Synchronization of delay-coupled oscillators: a study of

semiconductor lasers. Phys. Rev. Lett. 94, 163901 (2005).

34. Han, H. & Shore, K. A. Analysis of high-frequency oscillations in mutually-

coupled nano-lasers. Opt. Express 26, 10013 (2018).

35. Vicsek, T. A question of scale. Nature 411, 421 EP (2001).

36. Lai, Y. M., Newby, J. & Bressloff, P. C. Effects of demographic noise on the

synchronization of a metapopulation in a ﬂuctuating environment. Phys. Rev.

Lett. 107, 118102 (2011).

37. Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461,53

(2009).

38. Gayral, B., Gérard, J. M., Legrand, B., Costard, E. & Thierry-Mieg, V. Optical

study of GaAs/AlAs pillar microcavities with elliptical cross section. Appl.

Phys. Lett. 72, 1421–1423 (1998).

39. Whittaker, D. M. et al. High Q modes in elliptical microcavity pillars. Appl.

Phys. Lett. 90, 161105 (2007).

40. Reitzenstein, S. et al. AIAs/GaAs micropillar cavities with quality factors

exceeding 150.000. Appl. Phys. Lett. 90,1–4 (2007).

ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2

10 NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications

41. Leymann, H. A. M. et al. Intensity ﬂuctuations in bimodal micropillar lasers

enhanced by quantum-dot gain competition. Phys. Rev. A 87, 053819 (2013).

42. Khanbekyan, M. et al. Unconventional collective normal-mode coupling in

quantum-dot-based bimodal microlasers. Phys. Rev. A 91, 043840 (2015).

43. Redlich, C. et al. Mode-switching induced super-thermal bunching in

quantum-dot microlasers. New J. Phys. 18, 063011 (2016).

44. Leymann, H. A. M. et al. Pump-power-driven mode switching in a

microcavity device and its relation to bose-einstein condensation. Phys. Rev. X

7, 021045 (2017).

45. Schlottmann, E. et al. Injection locking of quantum-dot microlasers operating

in the few-photon regime. Phys. Rev. Appl.6, 64030 (2016).

46. Vicente, R., Tang, S., Mulet, J., Mirasso, C. R. & Liu, J.-M. Synchronization

properties of two self-oscillating semiconductor lasers subject to delayed

optoelectronic mutual coupling. Phys. Rev. E 73, 047201 (2006).

47. Cresser, J. D., Hammonds, D., Louisell, W. H., Meystre, P. & Risken, H.

Quantum noise in ring-laser gyros. II. Numerical results.Phys. Rev. A 25,

2226–2234 (1982).

48. Ota, Y., Kakuda, M., Watanabe, K., Iwamoto, S. & Arakawa, Y. Thresholdless

quantum dot nanolaser. Opt. Express 25, 19981–19994 (2017).

49. Jahnke, F. et al. Giant photon bunching, superradiant pulse emission and

excitation trapping in quantum-dot nanolasers. Nat. Commun. 7, 11540 EP

(2016).

50. Osborne, S., Heinricht, P., Brandonisio, N., Amann, A. & O’Brien, S.

Wavelength switching dynamics of two-colour semiconductor lasers with

optical injection and feedback. Semicond. Sci. Technol. 27, 094001 (2012).

51. Virte, M., Panajotov, K. & Sciamanna, M. Mode competition induced by

optical feedback in two-color quantum dot lasers. IEEE J. Quantum Electron.

49, 578–585 (2013).

52. Meinecke, S., Lingnau, B., Röhm, A. & Lüdge, K. Stability in optically injected

two-state quantum-dot. Lasers Ann. Phys. 529, 1600279 (2017).

53. Flunkert, V., D’Huys, O., Danckaert, J., Fischer, I. & Schöll, E. Bubbling in

delay-coupled lasers. Phys. Rev. E 79, 065201 (R) (2009).

54. Schlottmann, E. et al. Exploring the photon-number distribution of bimodal

microlasers with a transition edge sensor. Phys. Rev. Appl. 9, 400 (2018).

55. Huyet, G. et al. Quantum dot semiconductor lasers with optical feedback.

Phys. Stat. Solidi A 201, 345–352 (2004).

56. Globisch, B., Otto, C., Schöll, E. & Lüdge, K. Inﬂuence of carrier lifetimes on

the dynamical behavior of quantum-dot lasers subject to optical feedback.

Phys. Rev. E 86, 046201 (2012).

57. Bonatto, C., Kelleher, B., Huyet, G. & Hegarty, S. P. Transition from

unidirectional to delayed bidirectional coupling in optically coupled

semiconductor lasers. Phys. Rev. E 85, 026205 (2012).

58. Sonar, S. et al. Squeezing enhances quantum synchronization. Phys. Rev. Lett.

120, 163601 (2018).

59. Roulet, A. & Bruder, C. Quantum synchronization and entanglement

generation. Phys. Rev. Lett. 121, 063601 (2018).

60. Reitzenstein, S. et al. Low threshold electrically pumped quantum dot-

micropillar lasers. Appl. Phys. Lett. 93, 061104 (2008).

61. Haken, H Light and Matter Ic. (Springer: Berlin, Heidelberg, 1970). .

62. Lüdge, K. & Schöll, E. Quantum-dot lasers—desynchronized nonlinear

dynamics of electrons and holes. IEEE J. Quantum Electron. 45, 1396–1403

(2009).

63. Lingnau, B. Nonlinear and Nonequilibrium Dynamics of Quantum-Dot

Optoelectronic Devices. (Springer International Publishing, Switzerland, 2015).

Springer Theses.

64. Mørk, J. & Lippi, G. L. Rate equation description of quantum noise in

nanolasers with few emitters. Appl. Phys. Lett. 112, 141103 (2018).

65. van der Sande, G. et al. The effects of stress, temperature, and spin ﬂips on

polarization switching in vertical-cavity surface-emitting lasers. IEEE J.

Quantum Electron. 42, 898–906 (2006).

66. Virte, M., Panajotov, K., Thienpont, H. & Sciamanna, M. Deterministic

polarization chaos from a laser diode. Nat. Photonics 7,60–65 (2012).

Acknowledgements

The research leading to these results has received funding from the European Research

Council (ERC) under the European Union’s Seventh Framework (ERC Grant Agreement

No. 615613). B.L. and K.L. acknowledge support from DFG (Deutsche For-

schungsgemeinschaft) within CRC787. D.S. acknowledges support from DFG within

GRK1558.

Author contributions

S.R. initiated the research and conceived the experiments together with S.K. and I.K. S.K.

and X.P. performed the experiments under supervision by S.R. D.S., B.L., and K.L.

developed the theoretical models and performed the numerical modeling. C.S. and S.H.

realized the samples. All the authors discussed the results. S.K., X.P., and S.R. wrote the

manuscript with contributions from all other authors.

Additional information

Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467-

019-09559-2.

Competing interests: The authors declare no competing interests.

Reprints and permission information is available online at http://npg.nature.com/

reprintsandpermissions/

Journal peer review information:Nature Communications thanks K. Alan Shore and the

other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in

published maps and institutional afﬁliations.

Open Access This article is licensed under a Creative Commons

Attribution 4.0 International License, which permits use, sharing,

adaptation, distribution and reproduction in any medium or format, as long as you give

appropriate credit to the original author(s) and the source, provide a link to the Creative

Commons license, and indicate if changes were made. The images or other third party

material in this article are included in the article’s Creative Commons license, unless

indicated otherwise in a credit line to the material. If material is not included in the

article’s Creative Commons license and your intended use is not permitted by statutory

regulation or exceeds the permitted use, you will need to obtain permission directly from

the copyright holder. To view a copy of this license, visit http://creativecommons.org/

licenses/by/4.0/.

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09559-2 ARTICLE

NATURE COMMUNICATIONS | (2019) 10:1539 | https://doi.org/10.1038/s41467-019-09559-2 | www.nature.com/naturecommunications 11