# Photonic molecule based on coupled ring quantum cascade lasers

Sara Kacmoli<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, Princeton University, Princeton, NJ 08544 USA  Deborah L. Sivco
Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08544 USA  Current address: Trumpf Photonics, Inc., 2601 US
Highway 130, Cranbury, NJ 08512 USA  Claire F. Gmachl  Department of
Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544
USA

###### Abstract

Photonic molecules – particular systems composed of coupled optical resonators
– emulate the behavior of complex physical systems exhibiting discrete energy
levels. In this work, we present a novel photonic molecule composed of two
strongly coupled, mid-infrared ring quantum cascade lasers. We explore both
experimentally and numerically the key features of the photonic molecule such
as the energy level splitting of bonding and antibonding supermodes. Due to
the large size of the resonators, the energy splitting results in bands
containing tens of modes. Each of these modes is furthermore doubly degenerate
with respect to the direction of propagation, namely clockwise and
counterclockwise. We explore several methods to carefully break these
symmetries of the system in a controlled manner by introducing spatial and
temporal asymmetries in the pumping scheme of the ring lasers. By employing
these techniques, we achieve a high degree of precision in the dynamic control
of the photonic molecule. Owing to their inherent suitability for on-chip
integration, this new class of devices may enable applications as varied as
novel mid-infrared sensors or a rich playground for studying non-Hermitian
photonics and quantum optics with quantum cascade lasers.

††preprint: APS/123-QED

##  I Introduction

Discrete energy level schemes are present in many physical systems. Photonic
analogues of such systems can be constructed by coupling resonators together
to form so-called photonic molecules. This nomenclature reflects the fact that
these artificial structures mimic bonded atoms, resulting in hybridized energy
levels that resemble those observed in molecules. Wavelength-scale photonic
molecules have already enabled the investigation and discovery of a number of
complex phenomena in quantum optics [1, 2], non-Hermitian photonics [3, 4] or
condensed matter physics. [5] They have been used to demonstrate new optical
computing paradigms [6] and sensing capabilities. [7, 8, 9] In all of the
examples above, photonic molecules are based on microcavities, where the
energy differences between the individual modes of each resonator are much
larger than the splitting between the hybridized modes. Consequently, these
systems usually contain only two discrete energy levels.

In our work, we demonstrate a photonic molecule based on two large scale and
strongly coupled ring quantum cascade (QC) lasers. Due to the size of the
system, the free spectral range (FSR) of our resonators is small compared to
the splitting between the energy levels. As a result, our photonic molecule
exhibits a new regime of coupling, where two degenerate combs give rise to two
discrete energy bands instead of only two individual levels. In condensed
matter, this is akin to spin splitting of d-level orbitals which occurs, for
example, in transition metal dichalcogenides. [10, 11, 12] Each of the energy
levels in our combs is also doubly-degenerate due to the rotational symmetry
of the ring laser (namely, the clockwise (CW) and counterclockwise (CCW)
propagation directions.) Morever, photonic molecules based on mid-infrared QC
lasers have not yet been demonstrated. In the terahertz QC platform two
studies have explored coupled microdisks, where the resonator size is
comparable to the wavelength, in order to study non-Hermitian photonics,
demonstrating the reversal of pump dependence at an exceptional point. [13, 4]
However, coupling for terahertz QC lasers is significantly easier than in the
mid-infrared due to the long wavelengths and the often-used metal-metal
waveguides of the former. Mid-infrared QC lasers are well suited to sensing
and spectroscopy applications due to the ro-vibrational resonances that many
molecules have in this wavelength range. They are also unique among
semiconductor lasers with a fast gain recovery time of
$\scriptstyle\mathtt{\sim}$ ps scale. Partially owing to this fast gain
recovery, QC lasers are also sources of electrically-pumped frequency combs.
[14] Realizing photonic molecules with mid-infrared QC lasers thus introduces
a new class of lasers which would enable the development of novel and enhanced
sensors. It also presents an opportunity to leverage the unique attributes of
QC lasers in studying non-Hermitian photonics and quantum optics.

In our recent work we have demonstrated efficient evanescent, waveguide-
mediated, in-plane outcoupling from a mid-infrared ring QC laser. Due to their
active nature, these couplers have interesting properties that enable control
over the emission of the ring laser. [15] This development holds significance
especially given the recent demonstrations of ring QC lasers as sources of
frequency combs and solitons. [16, 17, 18] However, in-plane coupling is not
limited to simply coupling light out from ring lasers, but it can also be
utilized to couple multiple identical lasers together, which is the core of
the here discussed work.

We make use of these active couplers to realize our photonic molecule system
operating at a wavelength $\lambda~{}{\raise
0.73193pt\hbox{$\scriptstyle\mathtt{\sim}$}}$
$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with efficient active outcoupling
waveguides. We experimentally and theoretically demonstrate key
characteristics of a photonic molecule such as splitting of the energy bands.
We show that each discrete band corresponds to a bonding and antibonding
supermode. These supermodes are the product of the lifting of the degeneracy
of states caused by specific symmetries in the photonic molecule. We explore
the conditions under which these states are degenerate as well as controlled
degeneracy and symmetry breaking mechanisms. The photonic molecule exhibits
great stability against perturbations of the symmetry of the rotational modes
and the supermodes of the system.

##  II Device design

The photonic molecule system consists of two nominally identical racetrack
lasers evanescently coupled by an air gap, each of which is in turn
evanescently coupled to an active waveguide for the purpose of light
extraction (Fig. 1a). The design of the coupler sections is described in
detail in our previous work [15]; it features adiabatic tapers which enhance
the coupling strength resulting in reasonable coupler lengths for air gaps of
$\scriptstyle\mathtt{\sim}$$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ that are
easily achievable by standard photolithography techniques. Scanning electron
microscope (SEM) images of the air gap, the adiabatic taper and smooth
sidewalls are presented in Fig. 1b. The length of each racetrack is
$3.92\text{\,}\mathrm{mm}$ and the coupler section between the two racetracks
is $190\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Each racetrack can theoretically support independent counter-propagating
modes. Due to the symmetry of the system, these modes are equally favored to
achieve lasing. When the vertical sidewalls have minimal roughness, thus
inducing minimal backscattering, the racetrack is unidirecional – either
lasing in the clockwise (CW) or counter-clockwise (CCW) direction. The two
lasers exchange optical energy through the evanescent coupler between the CW
(CCW) propagating mode of one ring and the CCW (CW) mode of the other.
Therefore, thanks to unidirectionality, the rings strongly tend to oscillate
in opposite orientations (e.g. as depicted in Fig. 1a), resulting in two
equally probable combinations: CW/CCW and CCW/CW. When measuring the light
output at one of the ports in Fig. 1a, only one of these combinations is in
fact observed at a time, as we will show below.

The presence of a coupling mechanism itself can be interpreted as a source of
potential loss and backscattering which raises a concern that adding these
elements will perturb the internal ring dynamics responsible for frequency
comb and dissipative Kerr soliton generation in recent work. [18] To assess
the degree of perturbation, we compare the spectra at similar pumping levels
from three devices: a standalone racetrack laser, a racetrack with one
waveguide coupler and a photonic molecule system all shown in Fig. 1d. We find
that they exhibit the same spectral envelope well captured by a characteristic
$sech^{2}$ fit indicative of the presence of solitons.

To experimentally assess the coupling efficiency we measure the output of each
ring from the same waveguide arm in a device where the coupling between the
rings and between each ring and the waveguides is nominally identical due to
the same length and gap of the coupling region. In this measurement we keep
all components of the system lightly pumped to overcome absorption except for
the ring of interest, which is pumped above threshold. The results of this
experiment are presented in Fig. 1c. We note that the spectra of the two rings
are remarkably similar. The estimated power coupling between the rings is
${\raise 0.73193pt\hbox{$\scriptstyle\mathtt{\sim}$}}8\%$.

Figure 1: (a) Diagram of the photonic molecule circuit, illustrating a
representative steady-state supermode. Gold pads represent electrodes with top
and ground contacts for each element (two pairs per outcoupling waveguide, one
pair per racetrack). (b) SEM images detailing 1. a cross-section of the
evanescent coupler between racetracks (false color marks the active region);
2. a top view of the taper geometry of the couplers, and 3. low sidewall
roughness. (c) Measurements of lasing spectra at top-right port as a function
of pumping current. Note that both racetracks exhibit nearly identical spectra
over the entire current range. In the right plot, the right-hand racetrack is
unpumped and serves as a passive waveguide coupling the optical output of the
left-hand racetrack to the output port. (d) Measured spectra from three
different devices: a standalone racetrack, a racetrack coupled to a waveguide,
and a photonic molecule system as shown in (a). Note that the comb structure
remains intact in all three configurations. The center wavelengths and free
spectral range (FSR) values are different because these are devices of
different sizes fabricated on different wafers.

##  III Photonic molecule ring QC lasers

Having established important basic features of our ring lasers and couplers,
we turn to the photonic molecule implementation of this system. When two
identical resonators with resonance frequency $f_{0}$ couple with strength
$J$, it is well known that the resulting system exhibits two hybridized modes
or supermodes similarly to two atomic orbitals in a molecule. These two
supermodes have two distinct frequencies (or energy levels), $f_{0}\pm J$. The
low energy mode is often called bonding, whereas the high energy mode is
called antibonding. Mathematically, this coupling can be modeled as a
perturbation to the Hamiltonian operator of the system ($\mathcal{H}$), which
causes a change in its eigenvalues (or, in this case, the frequency of the
supermodes).

In an active system, gain and loss terms are also present in the Hamiltonian
causing the eigenvalues to have positive or negative imaginary parts. Taking
these effects into consideration, we use the following Hamiltonian matrix to
describe the photonic molecule system, inspired by previous studies [19]:

$i\frac{\mathop{\mathrm{\mathstrut
d}}\nolimits\\!missing}{\mathop{\mathrm{\mathstrut
d}}\nolimits\\!t}\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}=\mathcal{H}\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}=2\pi\begin{pmatrix}f_{1}+i(\gamma_{1}-\kappa_{1})&J\\\
J&f_{2}+i(\gamma_{2}-\kappa_{2})\end{pmatrix}\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}.$ (1)

In Eq. 1, $E_{1}$ and $E_{2}$ are complex numbers representing the amplitude
of fields in each cavity, $f_{1}$ and $f_{2}$ are the center resonance
frequencies of the two resonators, $\gamma_{1},\kappa_{1}$ and
$\gamma_{2},\kappa_{2}$ represent the gain and loss respectively for each
cavity. We have allowed for the possibility of gain and loss imbalance as well
as non-identical central frequency which is natural, albeit small, in our
system given fabrication variation in the geometry of the resonators. The
Hamiltonian admits two eigenvalues ($f_{\pm}$) and corresponding eigenvectors
($\lambda_{\pm}$) – or “eigenfrequencies” and “supermodes”, respectively – as
shown in Eqs. 2 and 3:

$\displaystyle f_{\pm}=$ $\displaystyle f_{0}+i\eta_{0}\pm\sqrt{\Delta
f^{2}+2i\Delta f\Delta\eta+J^{2}-\Delta\eta^{2}}$ $\displaystyle\approx
f_{0}+i\eta_{0}\pm\sqrt{J^{2}-\Delta\eta^{2}}\textrm{, if }\Delta f\ll J$ (2)

$\lambda_{\pm}\equiv\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}_{\pm}=\begin{pmatrix}(\Delta
f+i\Delta\eta+f_{\pm}-f_{0}-i\eta_{0})/J\\\ 1\end{pmatrix}$ (3)

where we have defined the following quantities for clarity:

$\displaystyle
f_{0}=\frac{f_{1}+f_{2}}{2},\quad\eta_{0}=\frac{(\gamma_{1}-\kappa_{1})+(\gamma_{2}-\kappa_{2})}{2},$
$\displaystyle\Delta
f=\frac{f_{1}-f_{2}}{2},\quad\Delta\eta=\frac{(\gamma_{1}-\kappa_{1})-(\gamma_{2}-\kappa_{2})}{2}.$

In Eq. 2, $\Delta\eta$ represents the imbalance in the gain-loss contrast
between the two rings, and $\Delta f$ the natural center resonance frequency
difference between resonators. In the case of our photonic molecule, $\Delta
f\ll J$.

Experimentally, the ring lasers are pumped electrically, thus the
symmetry/asymmetry of the pumping determines the modal gain terms
($\gamma_{1}$ and $\gamma_{2}$ modified by a confinement factor). Therefore,
we expect $\Delta\eta$ to be linearly dependent on the current injection
imbalance applied between the rings. The two supermodes can be observed by
measuring the optical spectrum emanating from the photonic molecule at any of
its outcoupling ports as a function of pump current imbalance. According to
Eq. 2, we would expect a split between the frequencies of the supermode when
$\Delta\eta<J$, and the collapse to a central frequency otherwise. As shown in
Fig. 2a, we find a gradual splitting between the observed frequencies of the
photonic molecule system. When the gain loss imbalance between the two
racetracks ($\Delta\eta$) is zero, we expect the largest splitting between the
two modes corresponding to be approximately $J$. This situation occurs for a
current imbalance of
${\raise
0.73193pt\hbox{$\scriptstyle\mathtt{\sim}$}}-$79\text{\,}\mathrm{mA}$$, a
small value relative to an average current of $1.45\text{\,}\mathrm{A}$, which
we attribute to a slight loss term ($\kappa$) asymmetry between two ring
lasers.

Based on Eq. 2, we fit the real and imaginary parts of the two calculated
eigenfrequencies (Fig. 2). The real part of the eigenfrequency shown in Fig.
2a explains the frequency split present in the photonic molecule spectrum.
However, we note that for $\Delta I>-$79\text{\,}\mathrm{mA}$$, the lowest
frequency supermode dominates. This can be explained by the imaginary part of
the eigenfrequency, which corresponds to the gain asymmetry between the two
supermodes. When the imaginary parts of the eigenfrequencies shown in Fig. 2b
are equal, the photonic molecule will evolve to either supermode with equal
probability. However, when one eigenfrequency has larger imaginary part than
the other, we expect that the photonic molecule is more likely to converge to
that supermode instead. In this device, since $\Delta f\neq 0$, the imaginary
parts of the eigenfrequency are not equal (Eq. 2 and Fig. 2b). When
$\Delta\eta$ is sufficiently large, the two curves diverge and one supermode
clearly dominates over the other. This divergence can be understood by
examining the values of the eigenvectors (visualized symbolically in Fig. 2b,
where the color represents the sign of the eigenvector and the saturation
represents the amplitude). The amplitude and phase of the eigenvectors for the
entire range of $\Delta\eta$ are given in Figs. S1 and S2. In the lower-
frequency supermode, the fields in both rings are in-phase; whereas the
higher-frequency supermode, the fields have opposite phase. This is analogous
to symmetric and anti-symmetric molecular states that exist in atomic
molecular bonds, where the symmetric state, called ‘bonding’, has lower
energy. Although many systems that exhibit these two states show a preference
for the lower energy one, both states are observed in our system. We later
also explore mechanisms that allow us to select one of them.

Figure 2: (a) Spectral evolution of the photonic molecule as a function of
current imbalance between the two racetracks (blue). Black dots represent the
centers of the spectral envelopes. Cyan and magenta lines represent a fit of
the real part of the two eigenfrequencies calculated based on Eq. 2. (b) Cyan
and magenta lines represent the corresponding imaginary part of the two
eigenfrequencies. For several different values of current imbalance we depict
the eigenvectors symbolically where the color represents the sign of the
eigenvector (blue, positive and red, negative) and the saturation represents
its amplitude. The computed complex eigenvectors are given below each
supermode. Additional simulation details including fitting parameters are
given in the Supplementary Material.

To gain some insight into the physical origin of this energy split, we
consider the intensity distribution of the two supermodes. Bonding supermodes
have an increased intensity of the electric field in the air gap compared to
antibonding supermodes which have a node there (see Fig. S3). Thus, the
effective refractive indices of these two modes are different, leading to the
frequency shift between the two supermodes. We have calculated these two modes
(shown in Fig. S3), and their effective refractive indices are different by
0.45$\%$ which corresponds to a $5.54\text{\,}\mathrm{c}\mathrm{m}^{-1}$ shift
in frequency. This is, in fact, a close match for our observed split of
$\scriptstyle\mathtt{\sim}$ $8\text{\,}\mathrm{c}\mathrm{m}^{-1}$. When
$\Delta\eta$ exceeds a bonding threshold ${\raise
0.73193pt\hbox{$\scriptstyle\mathtt{\sim}$}}J$, i.e. $\Delta\eta>J$, the
photonic molecule dissociates, and the eigenmodes converge to individual ring
modes.

##  IV Rotational Symmetry Breaking

The two supermodes discussed in the previous section fundamentally originate
from a degree of freedom in the phase between the fields in each ring. The
bonding and antibonding supermodes physically manifest as fields that
constructively or destructively interfere at the coupling region,
respectively. Furthermore, the photonic molecule exhibits a degeneracy related
to the propagation orientation of the fields (namely CW and CCW) in each ring.
Due to the rotational symmetry of the ring, both orientations have the same
$\gamma$ and $\kappa$ values. Since we have established from experiment that
each ring is unidirectional, there are theoretically four possible mutually
exclusive propagation orientations: CW/CW, CCW/CCW, CW/CCW and CCW/CW.
Accounting for this extra degree of freedom, the two fields $E_{1}$ and
$E_{2}$ from the previous section can be further distinguished in four field
quantities, $E_{1}^{CW}$, $E_{1}^{CCW}$, $E_{2}^{CW}$, $E_{2}^{CCW}$. We can
observe these four outputs experimentally by placing detectors at the end of
each waveguide arm. When accounting for the mirror symmetry in the coupled-
ring geometry, these four orientations can be classified into two equivalent
modes: CW/CW and CCW/CCW (namely ‘competing’); and CW/CCW and CCW/CW (namely
‘reinforcing’). Since the evanescent directional coupler transfers energy from
the CW mode of one ring into the CCW mode of the other, and vice versa, we
expect that when the photonic molecule oscillates in a CW/CCW (or CCW/CW)
configuration, the fields ‘reinforce’ each other. Otherwise, in the
‘competing’ configuration (CW/CW or CCW/CCW), the coupler transfers energy
from one ring into a counterpropagating mode of the other, which is then
damped under the much stronger laser mode.

Figure 3: Pearson’s correlation coefficient of light pulses measured in the
reinforcing configuration (left panel) and competing configuration (right
panel) for a 2D sweep of currents applied to the two racetracks. Below the
lasing threshold of each racetrack, the photodetector does not capture a
signal, hence the correlation values are random. The schematics of each
measurement configuration depicting the detector placement in red are shown
below each panel.

To demonstrate this effect, we place one detector in the CW port of one ring
and the CCW port of the other. When both rings are pumped above their lasing
threshold at the same time, whenever an output pulse in one CW port is
detected, another pulse is detected in the opposite CCW port. This is shown as
a positive correlation in the left panel of Fig. 3. Conversely, when the
detectors are placed in a CW/CW configuration, these two outputs are
anticorrelated, presented in the right panel of Fig. 3. As this figure shows,
this behavior occurs regardless of the pump levels applied to the two
racetracks, indicating that this behavior is inherent to the photonic
molecule. One exception to this rule is when the pump currents are nearly
identical. In this case we observe that the correlation disappears. This is
the same regime where we find the probabilistic switching between the two
supermodes discussed in the previous section. Generally, we find that the
split in the frequency, i.e. operating in the supermode regime, is accompanied
by rapid fluctuations of the photodetector signal, leading to the distinct
change in correlation along the diagonal. This is due to the rotational mode
degeneracy: when the two rings are pumped equally, there is a probabilisitic
choice between the CW/CCW and CCW/CW configurations of the supermodes.

Because of the pulsed nature of our pumping scheme and the symmetry described
above, each racetrack chooses one direction of emission randomly for every
pulse. Therefore, a typical 2D sweep of the racetrack currents
$I_{\textrm{RT1}}$, $I_{\textrm{RT2}}$ yields the results shown in Fig. 4a1
and Fig. 4a2. The ‘pixelation’ of the output power in these figures indicates
this random selection. However, we can break this rotational symmetry by
asymmetrically pumping the outcoupling waveguide arms as we have shown in
detail in our previous work. [15] This allows us to deterministically select a
particular direction of emission. For example, in Fig. 4a3 and Fig. 4a4, we
have used this mechanism to select the CW mode of racetrack 1 as demonstrated
by the lack of signal in the CCW port (panel 3) and full signal in the CW port
(panel 4).

With this mechanism, we can perturb the natural symmetry of our photonic
molecule and observe the steady state intensity distribution that it selects.
We configure the racetracks in a competing arrangement, such that both
racetracks prefer to emit in CW. We show 20 consecutive photodetector traces
measured on the CW port of racetrack 1 in Fig. 4b along with the pumping
levels of the two racetracks. We note that when the current in racetrack 1 is
higher than racetrack 2 (panel 1), the output pulses are always CW as we
discussed above. As the current levels become more similar, we begin to see
the signal fluctuate between the two ports (panels 2 and 3). Finally, when the
current in racetrack 2 is higher than in racetrack 1, the output switches
fully to CCW (panel 4). We conclude that the photonic molecule only oscillates
in the ‘reinforcing’ configuration, even in the presence of external
stimulation. In this experiment, the photonic molecule never converges to a
‘competing’ configuration.

Figure 4: (a) Panels 1 and 2 show the photodetector power measured at the CCW
and CW ports of racetrack 1, respectively, when no mode selection mechanism is
applied. Panels 3 and 4 show the photodetector power measured at the CCW and
CW ports of racetrack 1, respectively, when a small current imbalance of
$\scriptstyle\mathtt{\sim}$$75\text{\,}\mathrm{mA}$ is applied to the
waveguide arms thus favoring the CW mode. (b) Current pulses of the two
racetracks (black solid and black dashed) along with 20 consecutive
photodetector traces measured at the CW port of racetrack 1 (magenta). In
panels 1 through 4 we gradually increase the applied pump power to racetrack 2
while keeping racetrack 1 at a constant pumping level. As the pumping level of
racetrack 2 exceeds that of racetrack 1, the output switches from fully CW to
fully CCW.

##  V Stability of Photonic Molecule Supermodes

In all previous experiments, both lasers were electrically pumped via
synchronous pulses of widths varying from 50-100 ns with a repetition rate of
$60\text{\,}\mathrm{kHz}$. With every pulse, the rings transition from a zero-
field state and converge to one of two possible supermodes ($\lambda_{+}$ or
$\lambda_{-}$), as well as one of two allowed rotational configurations
(CW/CCW or CCW/CW). Here, we probe the temporal dynamics of our system thanks
to the pulsed pumping scheme. We find that once the photonic molecule
converges to a state, that state is robust to perturbations. To demonstrate
this phenomenon, we set up a similar experiment as shown in the rotational
symmetry breaking case (refer to Fig. 4b): one of the lasers (in this case
racetrack 1) is set up to operate CW. Unlike the previous experiment; however,
this laser’s pump pulse leads the second racetrack’s pump pulse by
$7.5\text{\,}\mathrm{ns}$. During this period, the field in the racetrack 1
cavity starts oscillating only in the CW orientation as shown by the
photodetector traces in Fig. 5a1. Differently from what we described in the
previous section (Fig. 4b), even as the pump amplitude of racetrack 2 (also
primed for CW operation) becomes greater, racetrack 1 still chooses only the
CW mode as shown by the photodetector traces in Fig. 5a4, forcing racetrack 2
to operate CCW. The takeaway from this experiment is that the propagation
orientation is determined at the onset of the current pulse and is stable
against perturbations.

We have also found that a small delay induced in this way will remove any
instabilities in the upper-right quadrant of the correlation plots which we
previously observed in Fig. 3 due to the stable rotational mode selection. One
such correlation plot where the two racetracks have a small delay between them
is shown in (Fig. 5b). We note that along the diagonal, pulses show full
anticorrelation due to the competing configuration setup. In addition to
lifting this degeneracy between the rotational modes, we postulate that the
delay also lifts the equiprobable fluctuation between the two supermodes. This
is supported by measuring the spectra at one section of the diagonal (depicted
with a yellow arrow in Fig. 5b). We perform the same measurement as shown in
Fig. 2a, where the two ring lasers are pumped asymmetrically, but now with the
addition of a small delay between the racetrack pulses. The result of this
experiment is presented in Fig. 5c. We find a lack of the probabilistic region
where the system constantly switches between the two supermodes. Instead, the
system converges from an individual ring mode to one of the two possible
supermodes, revealing a one-sided spectrum (compare to the two-sided spectrum
in Fig. 2a). From this we can conclude that the photonic molecule can be
primed in a particular initial condition to select a rotational state or a
supermode in two ways: spatially, by judicious pumping imbalances of different
elements as well as temporally, by introducing small delays between the two
lasers.

Figure 5: (a) Current pulses of the two racetracks (black solid and black
dashed) along with 20 consecutive photodetector traces measured at the CW port
of racetrack 1 (magenta). As the pumping level of racetrack 2 exceeds that of
racetrack 1 (shown in the transition from panel 1 to 4), the output remains CW
due to the presence of a small delay between the pulses where racetrack 1 is
ahead of racetrack 2 by $7.5\text{\,}\mathrm{ns}$. A magnified view of the
leading edge of the pulses to highlight the delay is shown on the left panel.
(b) Pearson’s correlation coefficient for a competing configuration in the
case of a $7.5\text{\,}\mathrm{ns}$ delay. Note how the instability in the
correlation around the diagonal, which we observe in Fig. 3a, disappears for
this plot. The yellow arrow corresponds to the figure part (c). (c) Spectral
evolution of the photonic molecule when a small delay is introduced between
the two racetrack lasers. This measurment is taken for the region marked with
a yellow arrow in (b) where we would expect to find correlation instability
and a frequency split. Instead, due to the careful current pulse timing, the
system stably selects one of the supermodes.

##  VI Conclusion

In summary, we have demonstrated a mid-infrared photonic molecule based on
coupled ring quantum cascade lasers. Evanescently-coupled active waveguides
are used for in-plane light detection, extracting
$\scriptstyle\mathtt{\sim}$$5\text{\,}\mathrm{mW}$ of output power from these
pulsed, room-temperature devices, without significantly perturbing the inner
dynamics of the ring. We have measured the same spectral envelope and observed
unidirectional lasing in several configurations of ring lasers –– standalone,
coupled, and photonic molecule. The photonic molecule shows spectral mode
splitting in bonding and antibonding supermodes in good agreement with
theoretical predictions. Due to the large size of our resonators, combs of
frequencies are supported and instead of two individual levels, our photonic
molecule exhibits a splitting of two energy bands each containing
$\scriptstyle\mathtt{\sim}$10 modes. This is a novel regime of coupling for
photonic molecules, where the FSR of the cavity is much smaller than the
coupling strength. Additionally, each mode of the photonic molecule is also
doubly degenerate due to the rotational symmetry in the propagation direction
– namely CW and CCW. Due to a combination of the control afforded by waveguide
pumping and assymetric puming of the rings, we can spatially control the
rotational symmetry breaking. Through a series of experiments where we control
the selection of this rotational mode of each ring, we have shown that the
photonic molecule always oscillates in a ‘reinforcing’ configuration. By
introducing a small temporal delay, we find that both the rotational mode and
the supermode selection in the photonic molecule is determined right at the
onset of pumping and is robust against perturbations. The high degree of
dynamic control we have demonstrated in this work coupled with the unique
properties of QC lasers renders these devices a testbed for exploration of
non-Hermitian photonics, quantum optics and coherent instabilities in coupled
laser systems. Our analysis of the photonic molecule provides insight on a new
class of devices in the mid-infrared regime. Given their inherent
compatibility with photonic integration, these devices pave the way towards
compact, chip-scale mid-infrared sensors with enhanced sensitivity.

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