Coherent Feedback Control of Two-Dimensional Excitons
Supplemental Methods
Christopher Rogers,1, ∗ Dodd Gray, Jr.,1 Nathan Bogdanowicz,1
Takashi Taniguchi,2 Kenji Watanabe,2 and Hideo Mabuchi1, †
1Ginzton Laboratory, Stanford University,
348 Via Pueblo, Stanford, CA 94305
2National Institute for Materials Science,
1-1 Namiki, Tsukuba 305-0044, Japan
(Dated: January 12, 2020)
1
CONTENTS
I. Supplemental Materials 3
A. Excitation Occupation Number 3
B. Discussion of Other Works 3
C. Discrepancies Between Model and Experiment 4
D. Sample Fabrication 4
E. Experimental Setup 5
F. Reflectance Model 7
G. Reflectance Model Fitting 7
H. Direct Mirror Position Fitting 9
I. Extracted Parameters 10
J. Discussion of Inhomogeneous Broadening 11
K. Linewidth Model 13
L. Uncertainty Analysis 14
2
I. SUPPLEMENTAL MATERIALS
A. Excitation Occupation Number
We excite with ∼15 nW of continuous-wave optical power with a bandwidth of 300 nm.
The photon rate at the sample is thus ∼60 GHz. Further considering only optical power
resonant with X0, the photon rate is ∼0.4 GHz. Taking into account the exciton decay rate
of ∼2 meV ≈ 480 GHz, the excitation occupation number during the measurement is very
low, ∼10−3.
B. Discussion of Other Works
During preparation of this manuscript we became aware of preprints presenting similar
work by You Zhou, et al. [58], H.H. Fang, et al. [59] and J. Horng, et al. [60].
In [58], the electromechanical method of actuating a suspended TMD heterostructure
couples strongly to strain and allows only small (∼ 25 nm) changes in z, whereas our
purely mechanical method of actuating the mirror is entirely free of any induced strain or
electric field effects in the TMD, and allows for large z displacements (∼400 nm) over more
than one full fringe. In [59], the fixed and discrete mirror position necessitates averaging
over different sample areas, convolving uncertainties due to sample inhomogeneity with the
linewidth modulation effect, while the low-reflectivity mirror limits their range of control
over the radiative lifetime. In [60], the data is taken over several fringes and for both
emission and reflection, but the samples are of relatively low quality (that is, only partially
radiatively broadened).
3
C. Discrepancies Between Model and Experiment
In Figs. 2a and 2c, there is some slight mismatch between experiment and theory. One
slight difference is that the deepest reflectance feature obtained over z is smaller in the ex-
periment (8%) than the model (13%), which is likely due to a combination of pure dephas-
ing (which is not included in the model), spectrally structured inhomogeneous broadening,
mode-mismatch between the interfering reflected beams, and diffraction effects. For the
same reasons, the dip in reflectance at zc in the experiment is 43%, while in the model it is
35%
D. Sample Fabrication
We fabricate heterostructures using a dry pickup transfer technique [49, 50]. We first
clean 300 nm SiO2 on Si substrates, and fused silica substrates by sonicating in acetone for
2 minutes, then deionized water for 2 minutes and finally isopropanol for 2 minutes. The
substrates are then subjected to oxygen plasma for 5 minutes. Graphite (NGS Naturegraphit
GmbH), hexagonal Boron Nitride (hBN), and MoSe2 (2D Semiconductors or HQ Graphene)
are then exfoliated onto the freshly cleaned substrates using Scotch tape. The substrates
are observed under an optical microscope to identify monolayer MoSe2, few-layer graphene
and 50-120 nm hBN.
Polydimethylsiloxane (PDMS) with thin polycarbonate (PC) stamps are used to create
the heterostructures. To produce the stamp, a 6% PC solution is used to form a thin film
on a glass slide. This thin film is then transferred onto a 1 mm × 1 mm piece of PDMS on a
different glass slide using Scotch tape with a hole punched in the middle. This stamp is then
used to sequentially pick up the mechanically exfoliated flakes by bringing the stamp slowly
4
into contact with a flake on the exfoliation substrate. In our case, we first pick up the ‘top’
hBN, then the monolayer MoSe2, then the ‘bottom’ hBN, and finally the few-layer graphene
flake. Each flake is picked up at a temperature of about 60 ◦C. This stack (including the
PC film) is transferred to a glass substrate by heating the substrate to 140 ◦C and bringing
the stamp into contact. After letting the sample sit for one day, the PC is removed by
dissolution in chloroform.
The mirrors are prepared by taking a small glass substrate (∼ 1 mm x 1 mm in lateral
dimensions) and affixing it to a larger carrier substrate. This is then coated in 120 nm of
gold, with a 3 nm titanium adhesion layer.
E. Experimental Setup
A detailed experimental schematic is shown in Fig. S1. The experiment is conducted in
an optical cryostat (Montana Instruments Nanoscale Workstation) at a nominal temperature
of 4 K and a pressure of 1 · 10−7 Torr. The sample is attached to a fixed mount while the
gold mirror is actuated by a slip-stick piezo mirror mount (Janssen Precision Engineering).
Light from either a lamp (Thorlabs SLS201) or a supercontinuum laser (NKT Photonics
SuperK) is coupled into the custom confocal microscope through a single mode optical
fiber. Two reflective collimators serve to couple the microscope to the single mode fibers for
excitation and reflection. The excitation and reflection paths are separated by a 50/50 non-
polarizing beamsplitter. Two achromatic lenses with focal length f1 = 75mm form the first 4f
system. The first of these lenses is translated along the optical axis using a motorized stage,
which shifts the focus of the beam at the sample along the optical axis. The range of travel
of the beam focus at the sample is approximately ±200 µm. A tip-tilt mirror mechanically
actuated by motorized stages (Newport U100-A and Newport LTA-HS) at the beginning of
5
the second 4f section (comprised of two achromatic lenses of focal length f2 = 150 mm) shifts
the beam in the transverse plane at the sample. The total travel of the beam focus is about
±300 µm. A microscope objective (20×, 0.4 numerical aperture, Olympus MSPLAN) inside
the optical cryostat focuses the light down on the sample and mirror. Light is collected
back through the same optical path, and sent to a grating spectrometer for measurement.
A removable beamsplitter enables imaging of the sample. Note that the lens imaging onto
the camera in the imaging train is also on a translation stage, allowing the imaging plane
to be matched with that of the excitation spot.
The Gaussian beam diameter of the spot focused on the sample is approximately 2w0 =
1.7 µm, where w0 is the Gaussian beam waist. For a wavelength of 750 nm, this corresponds
to a Rayleigh range of zR = 3.1 µm. Since zR is much larger than the full range of mir-
ror displacement (approximately one fringe, or ∼ 400 nm) the spot size does not change
appreciably during the measurement. This implies that the various materials parameters,
which may in principle vary spatially, are constant for the purposes of this measurement.
Note that the Rayleigh range is also less than the distance between the mirror and sample
(at most 1100 nm), so the beam does not significantly diverge as it propagates between the
mirror and the sample.
The grating spectrometer used to measure the reflectance spectra has an 1800 line/mm
reflective diffraction grating on a motorized rotation stage (Newport RGV100). Spectra are
measured using a camera (Princeton Instruments PIXIS 2048). The nominal resolution of
the spectrometer is approximately 1 cm−1.
Each spectrum is normalized to a spectrum taken at a flake-free area on the substrate.
Measurements were automated using the python instrument control package Instrumental
available on GitHub at https://github.com/mabuchilab/Instrumental.
6
F. Reflectance Model
For simplicity we use a model of stack reflectivity based on a Lorentzian susceptibility
for the MoSe2 exciton, taking into account radiative broadening in vacuum γr,0 and non-
radiative broadening γnr [22, 56]:
χexc = − c
ω0d
γr,0
ω − ω0 + iγnr
2
(S1)
where ω0 is the exciton center frequency, ω is the optical frequency, c is the speed of light,
and d is the MoSe2 thickness. The index of refraction of the MoSe2 is then:
nexc =√n2
0 + χexc (S2)
where n0 is the background index in the MoSe2. Reflectance from the full stack Rω0(ω) in-
cluding the mirror is calculated using a transfer-matrix-method to obtain Fresnel coefficients
[57]. Inhomogeneous broadening effects are included with a characteristic width of γib. To
obtain the reflectance R(ω) including inhomogeneous broadening, Rω0(ω) is calculated for
a range of exciton center frequencies and combined by weighting with a Gaussian of width
γib:
R(ω) =1√
2πγib
∫Rω′
0(ω)e−(ω0−ω′
0)2/2γ2ib dω′0 (S3)
This assumes that the inhomogeneously broadened excitons emit incoherently, so that in-
terference effects average out.
G. Reflectance Model Fitting
We simultaneously fit spectral data in Figs. 2a and 2c from several selected characteristic
z-traces, to the full reflectance model in Eq. S3. Note that the z position of each trace was
7
also treated as a parameter. We use only a selection of z-traces rather than the full set
because the computation is faster.
The maximum mean-squared error (MSE) over the selected traces is used as the opti-
mization metric. Mathematically, this is represented as:
MSE = sup
{∫ λ2λ1dλ(R
(i)Exp −R
(i)Model)
2
λ2 − λ1
∣∣∣∣∣i ∈ A}
(S4)
where A is the set of traces i, R(i)Exp is the measured reflection of trace i, and R
(i)Model is the
model evaluated for trace i. This calculated for the range of λ1, λ2 = 751, 754 nm.
This metric was minimized to find the global fitting parameters ω0 = 1647.74 meV,
γr,0 = 1.06 meV, γnr = 0.45 meV and γib = 0.23 meV. The experimental spectra selected
for the fitting procedure and the corresponding modeled reflectance given by the optimized
fitting parameters are shown in Fig. S2.
We note that the four fitting parameters are highly constrained by the experimental
data, and that other values of the parameters do not produce satisfactory agreement between
experiment and model. For simplicity in the discussion below we ignore the subtle difference
between γib and γib,eff . Doing so does not qualitatively alter the conclusions reached. First, ω0
is set by the position of the reflectance dip at both zc and zd,1/2. Second, the quantity γnr+γib
is constrained by the reflectance linewidth at zd, where radiative broadening is negligible.
Similarly, the total linewidth γtot = γr + γnr + γib is constrained by the linewidth at zc.
Lastly, the magnitude of the on-resonant reflection at zc constrains the ratio γr/(γnr + γib).
Because the modulation of γr by the mirror is independent of the fitting parameters, we can
conceptually replace γr in the above discussion by Aγr,0 (where A is constant). The four
independent relations above then fully constrain the fitting parameters.
The static (unfitted) parameters used in the reflectance model are as follows. The index
of the silica substrate is n = 1.45, and the index of the hBN is n = 1.9. The index of
8
refraction of the gold at ω0 is n = 0.1388 + 4.4909i. The thickness of the gold is 120 nm.
The thickness of the top hBN is 87 nm, and the thickness of the bottom hBN is 128 nm.
The background index of the MoSe2 is n = 4.5. The graphene flake is modeled as a bilayer
with an index at ω0 of n = 2.15 + 1.91i.
Note that a linear interpolation is performed on the fitted z values to obtain the position
of traces not used in the fitting procedure.
H. Direct Mirror Position Fitting
Unfortunately, the slip-stick piezo stage used to actuate the mirror does not have a
position encoder. In order to verify that the z-values extracted while performing the global
fit to the model are accurate, we compare to a more direct method of finding the mirror
position. We use spectra taken over a wide spectral range (and not in the spectral region of
the exciton) to extract an independent measure of z.
We denote the z-values extracted from the global fit to the model as z0. The z values
extracted by fitting traces outside of the spectral range of the exciton are denoted zf .
At each z position for the data in Figs. 2a and 2c, we also took spectra over the range
of 770 nm to 900 nm from the same position on the sample. As can be seen in Fig. S4,
there are broad fringes that vary with mirror position, due primarily to the modulation of
absorption in the gold mirror and the few-layer graphene as the mirror position is changed.
Treating z as a free parameter, we fit the reflectance in this region to the same reflectance
model for the full heterostructure shown in Eq. S3. Note that because these spectra are
off-resonant from X0, the exciton susceptibility has a negligible effect. Several examples of
the measured and fitted spectra are shown in Fig. S4.
We compare these extracted values of zf to those obtained from the global fit (z0) in Fig.
9
S5. The values of z agree reasonably well, within approximately ±25 nm. This verifies that
the z fitting procedure described in Sec. I G is accurate.
Using these zf values obtained more directly, we can produce plots analogous to those
in the main text. Overall, the differences between the plots produced using the different
sets of z values is relatively small. We show a heatmap of reflectance in Fig. S6, which
is analogous to Fig. 2a from the main text. Linecuts of reflectance are shown in Fig. S7,
which is analogous to Fig. 2c from the main text. Two linecuts of reflectance are shown in
Fig. S8, which is analogous to Fig. 2d from the main text. Values of extracted parameters
are shown in Fig. S9, which is analogous to Fig. 3a from the main text. Values of linewidth
comparing to a simplified model are shown in Fig. S10, which is analogous to Fig. 3b from
the main text.
Further, we can compare the experimental linecuts to those from the model, which we
present in Fig. S11. We can see that the agreement is poorer than that using the z values
from the main text, when comparing to the analogous Fig. S2.
I. Extracted Parameters
To extract the model parameters in Fig. 3a we first extract the total FWHM linewidth
of the X0 feature from the reflectance model as a function of mirror position z. Assuming a
Voight line shape [54, 55] we can extract the intrinsic Lorentzian linewidth (γr+γnr) because
the intrinsic Gaussian linewidth (γib) is known directly from the model. This also yields the
effective contribution to the linewidth of inhomogeneous broadening (γib,eff) as the difference
between the total linewidth and the intrinsic Lorentzian linewidth. We can then trivially
extract γr from the intrinsic Lorentzian linewidth because γnr is known.
10
J. Discussion of Inhomogeneous Broadening
Throughout the main text, we have briefly mentioned the constant inhomogeneous broad-
ening parameter γib that appears in the model. This parameter is the Gaussian width of
the inhomogeneous broadening used in the model. Again, we stress that this parameter is
constant with respect to z.
However, in the main text we more often reference γib,eff , which is the contribution of
Gaussian broadening to the total FWHM linewidth of the exciton resonance. That is, γib,eff
is the difference between the total linewidth extracted from the model, and the (Lorentzian)
contributions of γr and γnr. Mathematically, this is γib,eff = γtot− γr − γnr. Despite the fact
that the intrinsic inhomogeneous broadening (γib) is constant, the effective inhomogeneous
broadening (γib,eff) varies with z. This might at first seem counterintuitive or even incorrect,
but is nonetheless true and correct. We again stress that the parameter modeling the
magnitude of the inhomogeneous broadening (γib) is constant. It is only the effect of this
parameter on the total linewidth extracted from the model (γib,eff) that changes with z.
In Fig. S3 we show the various contributions to the linewidth as a function of mirror
position, including γib,eff . The top panel is in analogy to Fig. 3a. In the main text, we
omitted γib,eff for simplicity.
When two individual Lorentzians are convolved, the total linewidth is simply the sum of
their individual linewidths. The same is true when two Gaussians are convolved. However,
when a Gaussian and a Lorentzian are convolved, the total linewidth is not simply the
sum of their individual linewidths. The combination of a Gaussian and a Lorentzian is the
well-known Voight lineshape [54, 55]. When a Lorentzian of width fL is convolved with a
11
Gaussian of width fG, an approximation of the resulting linewidth is given by [55]:
fT ≈ C1fL +√C2f 2
L + f 2G (S5)
where C1 = 0.5346 and C2 = 0.2166. In this case, the linewidths do not directly sum, but
rather are combined in a nonlinear manner. This is illustrated in the bottom panel of Fig.
S3.
This nonlinear addition of linewidths is the underlying reason that γib,eff changes with
z, even though γib is constant. Here, the inhomogeneous broadening γib corresponds to
fG. The Lorentzian contribution to the broadening fL corresponds to the sum of radiative
and nonradiative broadening, γr + γnr. The total linewidth extracted from the model (γtot)
corresponds to fT . The contribution of the inhomogeneous broadening to the total linewidth
(γib,eff) then corresponds to fV − fL.
We again stress that the inhomogeneous broadening in the model is constant as a function
of z, as it should be. However, as γr (and correspondingly fL) varies with mirror position, the
nonlinear manner in which the Lorentzian and Gaussian portions of the lineshape combine
causes γib,eff to vary with mirror position as well.
Lastly, we would like to make a comment regarding the value of η0 = 0.45 chosen for
Fig. 3b since it relates to the nonlinear contribution of the inhomogeneous broadening to
the total linewidth. Note first that the peak ratio γr/(γnr + γib,eff) occurs when γr = 2γr,0.
In this case, γr/(γnr + γib,eff) = 2η0/(1 − η0). However, the peak ratio γr/(γnr + γib,eff) ∼ 3
extracted from the reflectance model does not match that of γr/(γnr + γib,eff) = 1.64 we
expect for η0 = 0.45 from this simplified model. This discrepancy is primarily due to γib,eff
varying with z so as to partially counteract the change in γr, as seen in the top panel of
Fig. S3. Because the intrinsically Lorentzian exciton feature is convolved with a Gaussian
inhomogeneous broadening of width γib to form a Voight profile, the effective inhomogeneous
12
broadening γib,eff is larger when the total linewidth γtot is small [54, 55].
K. Linewidth Model
From [32], for an ideal dipole near and parallel to an ideal mirror:
τxτ0
=
[1− 3 sinx
2x− 3 cosx
2x2+
3 sinx
2x
]−1
(S6)
where τx is the lifetime at normalized distance x = 4πzλ0
from the mirror, τ0 is the lifetime
in vacuum, λ0 is the wavelength in vacuum and z is the optical path length between the
mirror and the dipole. When the dipole has a coherent quantum efficiency η0 in vacuum,
the modified lifetime τ ′x is:
τ ′xτ0
=1
1 + η0
(τ0τx− 1) (S7)
It then follows from Eq. S6 that the radiative decay rate γr for a perfect dipole is:
γrγr,0
= 1− 3 sinx
2x− 3 cosx
2x2+
3 sinx
2x3(S8)
where γr,0 is the radiative decay rate in vacuum. For the more general case with sub-unity
coherent quantum efficiency η0 = γr,0γtot,0
, with γtot,0 being the total linewidth in vacuum, it
follows from Eq. S7 that:
γtot(x)
γtot,0
= 1 + η0
(γrγr,0− 1
)(S9)
Using Eq. S8 we find that:
γtot(x)
γtot,0
= 1 + η0
[3 sinx
2x3− 3 cosx
2x2− 3 sinx
2x
](S10)
For a 2D dipole the case is different. Assuming that the dipole has perfect transverse
coherence and thus emits only into forward and backward plane wave modes, the modifi-
cation of γr is determined by the interference between −eik0dopl and 1. Here k = 2πλ0
is the
13
wavenumber of the light in vacuum and dopl is the total optical path length traversed by
the backwards-emitted wave until it comes back to the 2D dipole. The negative sign comes
from the phase flip on reflection. In this case, the radiative decay rate is proportional to:
γr ∝∣∣E0 − E0e
ik0dopl∣∣2 = 2 |E0|2 [1− cos(k0dopl)] (S11)
where E0 is the electric field magnitude emitted in each direction from the dipole. In the
case with no mirror, the radiative decay rate is proportional to the total intensity of emitted
radiation:
γr,0 ∝ 2 |E0|2 (S12)
The factor of two is due to emission in both the forward and backward directions. Thus, we
can write down the modification of radiative decay rate:
γrγr,0
= 1− cos(k0dopl) (S13)
Identifying that dopl = 2z:
γrγr,0
= 1− cos(2k0z) = 1− cos(x) (S14)
Using Eq. S9, which holds for a 2D dipole as well, we find that:
γtot(x)
γtot,0
= 1− η0 cos(x) (S15)
L. Uncertainty Analysis
In this section we address the motivation and methods for the uncertainties listed in
the main text. First, we address the uncertainty of the minimum and maximum linewidth
extracted from the experimental data (0.9± 0.1 meV and 2.3± 0.1 meV). In this case, the
uncertainty comes primarily from noise in the spectra and the fact that the spectrometer
14
has finite resolution. We estimate that the uncertainty in finding each half-max point is
approximately one spectrometer pixel. Doubling this (once for each half-max point), we
find an uncertainty of approximately 0.1 meV.
To quantify the uncertainty in the parameters and values extracted from the model/fitting
procedure, we vary the model parameters and examine the effect on the MSE. Specifically,
we vary each parameter (ω0, γr,0, γnr, γib) individually (both increasing and decreasing), and
find the value for which the MSE is doubled. This is used as an uncertainty bound on the
value of the parameter.
Plots of the experimental and modeled reflectance from selected traces for the case where
the MSE is doubled are shown for increased (decreased) ω0 in Fig. S12 (Fig. S13). The
model parameters for increased (decreased) ω0 are denoted by Mω0↑ (Mω0↓). These plots
are in analogy to Fig. S2, which is for the original model (denoted M0). We find that the
bounds are ω0 = 1647.74 +0.1−0.09 meV.
Plots of the experimental and modeled reflectance from selected traces for the case where
the MSE is doubled are shown for increased (decreased) γr,0 in Fig. S14 (Fig. S15). The
model parameters for increased (decreased) γr,0 are denoted by Mγr,0↑ (Mγr,0↓). We find that
the bounds are γr,0 = 1.06+0.22−0.28 meV.
Plots of the experimental and modeled reflectance from selected traces for the case where
the MSE is doubled are shown for increased (decreased) γnr in Fig. S16 (Fig. S17). The
model parameters for increased (decreased) γnr are denoted by Mγnr↑ (Mγnr↓). We find that
the bounds are γnr = 0.45+0.06−0.11 meV.
Plots of the experimental and modeled reflectance from selected traces for the case where
the MSE is doubled are shown for increased (decreased) γib in Fig. S18 (Fig. S19). The
model parameters for increased (decreased) γib are denoted by Mγib↑ (Mγib↓). We find that
15
the bounds are γib = 0.23+0.19−0.23 meV.
We also produced maps of the difference between the model and experiment for these
same cases as above. For reference, Fig. S20 shows this type of plot for the original model
parameters M0. Maps of the difference between the model and experiment are shown for
increased (decreased) ω0 in Fig. S21 (Fig. S22). Maps of the difference between the model
and experiment are shown for increased (decreased) γr,0 in Fig. S23 (Fig. S24). Maps of
the difference between the model and experiment are shown for increased (decreased) γnr in
Fig. S25 (Fig. S26). Maps of the difference between the model and experiment are shown
for increased (decreased) γib in Fig. S27 (Fig. S28).
We use these extreme cases MX where the individual fitting parameters are varied to find
uncertainty bounds on the values (linewidth, position, etc.) extracted from the model, which
are shown in Fig. 3a-c. First, theses quantities are also extracted from each of the models
MX . For each quantity of interest, the maximum (minimum) of these values over all of the
different models MX is used as the upper (lower) error bound. These are the uncertainties
plotted in Fig. 3a-c, and also in Fig. S3. For the main text, the most important of these
values is the maximum radiative broadening which occurs at zc. This maximum radiative
broadening varies between approximately 1.4 and 2.1 meV. Hence, we use 1.8± 0.4 meV in
the main text.
16
(a)
FIG. S1. Experimental Setup. Light is coupled from a lamp or a laser through a single mode fiber
into a custom confocal microscope, which focuses light on the sample and collects the reflection.
The collected light is measured using a grating spectrometer.
17
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model M0
MSEMAX = 0.0032
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
Model M0
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S2. Characteristic spectra used to fit the model parameters. The experimental data
is shown in solid lines, and the model in dashed lines. The mirror position z (also fitted as a
parameter) corresponding to each spectrum is shown. The z-label of the trace with the maximum
MSE is shown in bold and red text. The model parameters and the MSE are also labeled in the
figure.
18
0
1
2
3L
inew
idthγ
(meV
)
γtot
γr
γib,eff
γnr
800 900 1000Mirror Position z (nm)
0
1
2
3
Lin
ewid
thγ
(meV
)
γtot ↔ fV
γr + γnr ↔ fL
γib,eff ↔ fV − fL2√
2 ln (2) γib ↔ fG
0.0
0.5
1.0
1.5
Lin
ewid
thγ
(nm
)
0.0
0.5
1.0
1.5
Lin
ewid
thγ
(nm
)
zm,1 zc zm,2
(a)
FIG. S3. Extracted and Modeled Linewidths Including the contribution of Inhomo-
geneous Broadening. In the top panel, the FWHM linewidth γtot both from the model and
extracted from the experimental data. Note that we cannot extract linewidth data over the full
range of the experimental data, since near zd the X0 resonance is almost completely extinguished.
Also shown are γr, γnr and γib,eff from the model. The shading represents the uncertainties in
the model, see Section I L for further explanation of how this uncertainty is calculated. This is
analogous to Fig. 3 from the main text. In the bottom panel, are shown the total linewidth,
the Gaussian contribution to the linewidth, the Lorentzian contribution to the linewidth, and the
intrinsic Gaussian linewidth. The corresponding parameters are indicated in the legend.
19
(a)
FIG. S4. Selected spectra used to extract the mirror position z. Data from the experi-
ment is shown with solid lines, and the model is shown with dashed lines. The mirror position z
corresponding to each spectrum is labeled.
20
800
900
1000
1100
z(n
m)
zf
z0
0 50 100index
−20
0
20
z 0−z f
(nm
)
(a)
FIG. S5. Comparison of z values obtained from independent fitting methods, as a
function of trace index. The top panel shows the z values zf obtained by fitting spectra outside
of the spectral region of the exciton, and the z values z0 obtained from the global fit in the spectral
region of the exciton. The bottom panel shows the difference between the values obtained from
these two methods. The circles indicate the indices of the traces used in the global fit.
21
752
754Experiment
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
)
Model1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S6. Experimental and Modeled Reflectance heatmap for alternative fitting of z.
Measured and modeled reflectance spectra near the X0 resonance as z is varied over a full fringe.
This is using z values zf obtained by fitting spectra outside of the spectral region of the exciton.
Measurements at 4 K. The plot is analogous to Fig. 2a.
22
zm,2
zd,2Experiment
zc
zm,2zm,1
zc
zd,1
zm,1
750 752 754Wavelength (nm)
0.0
0.5
1.0
Refl
ecta
nce
Model
16451650Energy (meV)
16451650Energy (meV)
800850900950100010501100Mirror Position z (nm)
(a)
FIG. S7. Experimental and Modeled Reflectance linecuts for alternative fitting of z.
Selected line cuts of the measured and modeled reflectance in the spectral region of X0. The black
arrows indicate increasing z. This is using z values zf obtained by fitting spectra outside of the
spectral region of the exciton. Measurements at 4 K. The plot is analogous to Fig. 2c.
23
0.6
0.8
1.0
Refl
ecta
nce
z = 1093 nmγt = 0.91 meV
z = 929 nmγt = 2.30 meV
750.0 752.5 755.0Wavelength (nm)
0.0
0.5
1.0
Nor
mal
ized
Refl
ecta
nce
16451650Energy (meV)
(a)
FIG. S8. Experimental and Modeled Reflectance linecuts for alternative fitting of z.
Measured reflectance, both absolute and normalized, at two z positions highlighting the modulation
of total linewidth. This is using z values zf obtained by fitting spectra outside of the spectral region
of the exciton. Measurements at 4 K. The plot is analogous to Fig. 2d.
24
0
1
2
3
Lin
ewid
thγ
(meV
)
γtot
γr
γnr
752.2
752.4
752.6
752.8X
0W
avel
engt
hλX
0(n
m)
800 900 1000Mirror Position z (nm)
0.0
0.2
0.4
0.6
0.8
1.0
Min
imu
mR
eflec
tan
ce
Experiment
Model
0.0
0.5
1.0
1.5
Lin
ewid
thγ
(nm
)
1647
1648 ωX
0(m
eV)
zm,1 zc zm,2
(a)
FIG. S9. Extracted and Modeled Linewidths for alternative fitting of z. The top panel
shows the FWHM linewidth γtot both from the model and extracted from the experimental data.
Also shown are γr and γnr from the model. The middle panel shows the center frequency ωX0 for
both model and experiment. The bottom panel shows the minimum reflectance for both model
and experiment. This is using z values zf obtained by fitting spectra outside of the spectral region
of the exciton. The plot is analogous to Figs. 3a-c.
25
0.0
0.5
1.0
1.5
2.0
Rel
ativ
eL
inew
idthγ
tot/γ
tot,
0
η0 = 1
500 1000Mirror Position z (nm)
0.0
0.5
1.0
1.5
2.0
Rel
ativ
eL
inew
idthγ
tot/γ
tot,
0
η0 = 0.45
Experiment
2D Dipole
Point Dipole
(a)
FIG. S10. Comparison to Simplified Linewidth Model for alternative fitting of z. Sim-
plified models of the total linewidth modulation for both a point and 2D dipole, assuming a perfect
mirror with zero skin depth. The top panel shows the ideal case with coherent quantum efficiency
in vacuum η0 = 1, and the bottom panel shows the case with η0 = 0.45 alongside the experimental
data. This is using z values zf obtained by fitting spectra outside of the spectral region of the
exciton. The plot is analogous to Fig. 3d.
26
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1116 nm
z = 1093 nm
z = 1076 nm
z = 1068 nm
z = 1042 nm
z = 1029 nm
752 754
z = 1002 nm
z = 990 nm
z = 988 nm
z = 990 nm
z = 959 nm
z = 938 nm
752 754
z = 915 nm
z = 882 nm
z = 850 nm
z = 822 nm
z = 797 nm
z = 792 nm
752 754
z = 777 nm
z = 774 nm
z = 769 nm
z = 758 nm
Model M0
MSEMAX = 0.0161
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
Model M0
Raw Exp.
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S11. Characteristic linecut spectra for alternative fitting of z. The experimental
data is shown in solid lines, and the model in dashed lines. The mirror position z corresponding
to each spectrum is shown. This is using z values zf obtained by fitting spectra outside of the
spectral region of the exciton. The z-label of the trace with the maximum MSE is shown in bold
and red text. The model parameters and the MSE are also labeled in the figure.
27
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mω0↑MSEMAX = 0.0064
ω0 = 1647.84 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
Model Mω0↑
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S12. Characteristic spectra plotted for model parameters Mω0↑ from the uncer-
tainty analysis of increasing ω0. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
28
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mω0↓MSEMAX = 0.0064
ω0 = 1647.65 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
Model Mω0↓
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S13. Characteristic spectra plotted for model parameters Mω0↓ from the uncer-
tainty analysis of decreasing ω0. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
29
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγr,0↑MSEMAX = 0.0064
ω0 = 1647.74 meVγr,0 = 1.28 meVγib = 0.23 meVγnr = 0.45 meV
Model Mγr,0↑
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S14. Characteristic spectra plotted for model parameters Mγr,0↑ from the uncer-
tainty analysis of increasing γr,0. in dashed lines. The experimental data is shown in solid
lines, and the model in dashed lines. The mirror position z corresponding to each spectrum is
shown. The z-label of the trace with the maximum MSE is shown in bold and red text. The model
parameters and the MSE are also labeled in the figure.
30
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγr,0↓MSEMAX = 0.0064
ω0 = 1647.74 meVγr,0 = 0.78 meVγib = 0.23 meVγnr = 0.45 meV
Model Mγr,0↓
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S15. Characteristic spectra plotted for model parameters Mγr,0↓ from the uncer-
tainty analysis of decreasing γr,0. in dashed lines. The experimental data is shown in solid
lines, and the model in dashed lines. The mirror position z corresponding to each spectrum is
shown. The z-label of the trace with the maximum MSE is shown in bold and red text. The model
parameters and the MSE are also labeled in the figure.
31
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγnr↑MSEMAX = 0.0064
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.51 meV
Model Mγnr↑
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S16. Characteristic spectra plotted for model parameters Mγnr↑ from the uncer-
tainty analysis of increasing γnr. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
32
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγnr↓MSEMAX = 0.0064
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.34 meV
Model Mγnr↓
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S17. Characteristic spectra plotted for model parameters Mγnr↓ from the uncer-
tainty analysis of decreasing γnr. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
33
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγib↑MSEMAX = 0.0064
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.42 meVγnr = 0.45 meV
Model Mγib↑
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S18. Characteristic spectra plotted for model parameters Mγib↑ from the uncer-
tainty analysis of increasing γib. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
34
750 752 754
Wavelength (nm)
0
1
Refl
ecta
nce
z = 1111 nm
z = 1090 nm
z = 1075 nm
z = 1067 nm
z = 1035 nm
z = 1018 nm
752 754
z = 992 nm
z = 978 nm
z = 968 nm
z = 965 nm
z = 944 nm
z = 925 nm
752 754
z = 910 nm
z = 886 nm
z = 854 nm
z = 846 nm
z = 822 nm
z = 799 nm
752 754
z = 784 nm
z = 769 nm
z = 759 nm
z = 733 nm
Model Mγib↓MSEMAX = 0.0059
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.00 meVγnr = 0.45 meV
Model Mγib↓
Experiment
16451650Energy (meV)
16451650 16451650 16451650
(a)
FIG. S19. Characteristic spectra plotted for model parameters Mγib↓ from the uncer-
tainty analysis of decreasing γib. The experimental data is shown in solid lines, and the model
in dashed lines. The mirror position z corresponding to each spectrum is shown. The z-label of
the trace with the maximum MSE is shown in bold and red text. The model parameters and the
MSE are also labeled in the figure.
35
752
754Experiment/Model M0 Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Experiment
Model M0
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
)
Model M0
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S20. Maps of the difference between model and experiment for model parameters
M0. The top panel shows the difference between experiment and M0. The middle panel shows
the experimental data, and the bottom panel shows the model M0. The model parameters are also
labeled in the figure.
36
752
754Experiment/Model Mω0↑ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mω0↑ Difference
Model Mω0↑
ω0 = 1647.84 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mω0↑
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S21. Maps of the difference between model and experiment for model parameters
Mω0↑. The top panel shows the difference between experiment and Mω0↑. The middle panel shows
the difference between models Mω0↑ and M0. The bottom panel shows model Mω0↑. The model
parameters are also labeled in the figure.
37
752
754Experiment/Model Mω0↓ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mω0↓ Difference
Model Mω0↓
ω0 = 1647.65 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mω0↓
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S22. Maps of the difference between model and experiment for model parameters
Mω0↓. The top panel shows the difference between experiment and Mω0↓. The middle panel shows
the difference between models Mω0↓ and M0. The bottom panel shows model Mω0↓. The model
parameters are also labeled in the figure.
38
752
754Experiment/Model Mγr,0↑ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγr,0↑ Difference
Model Mγr,0↑
ω0 = 1647.74 meVγr,0 = 1.28 meVγib = 0.23 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγr,0↑
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S23. Maps of the difference between model and experiment for model parameters
Mγr,0↑. The top panel shows the difference between experiment and Mγr,0↑. The middle panel
shows the difference between models Mγr,0↑ and M0. The bottom panel shows model Mγr,0↑. The
model parameters are also labeled in the figure.
39
752
754Experiment/Model Mγr,0↓ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγr,0↓ Difference
Model Mγr,0↓
ω0 = 1647.74 meVγr,0 = 0.78 meVγib = 0.23 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγr,0↓
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S24. Maps of the difference between model and experiment for model parameters
Mγr,0↓. The top panel shows the difference between experiment and Mγr,0↓. The middle panel
shows the difference between models Mγr,0↓ and M0. The bottom panel shows model Mγr,0↓. The
model parameters are also labeled in the figure.
40
752
754Experiment/Model Mγnr↑ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγnr↑ Difference
Model Mγnr↑
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.51 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγnr↑
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S25. Maps of the difference between model and experiment for model parameters
Mγnr↑. The top panel shows the difference between experiment and Mγnr↑. The middle panel
shows the difference between models Mγnr↑ and M0. The bottom panel shows model Mγnr↑. The
model parameters are also labeled in the figure.
41
752
754Experiment/Model Mγnr↓ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγnr↓ Difference
Model Mγnr↓
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.23 meVγnr = 0.34 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγnr↓
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S26. Maps of the difference between model and experiment for model parameters
Mγnr↓. The top panel shows the difference between experiment and Mγnr↓. The middle panel
shows the difference between models Mγnr↓ and M0. The bottom panel shows model Mγnr↓. The
model parameters are also labeled in the figure.
42
752
754Experiment/Model Mγib↑ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγib↑ Difference
Model Mγib↑
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.42 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγib↑
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S27. Maps of the difference between model and experiment for model parameters
Mγib↑. The top panel shows the difference between experiment and Mγib↑. The middle panel shows
the difference between models Mγib↑ and M0. The bottom panel shows model Mγib↑. The model
parameters are also labeled in the figure.
43
752
754Experiment/Model Mγib↓ Difference
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Diff
eren
ceR
1−R
2
752
754Model M0/Model Mγib↓ Difference
Model Mγib↓
ω0 = 1647.74 meVγr,0 = 1.06 meVγib = 0.00 meVγnr = 0.45 meV
800 900 1000 1100Mirror Distance z (nm)
750
752
754
Wav
elen
gth
(nm
) Model Mγib↓
0.0
0.2
0.4
0.6
0.8
1.0
Refl
ecta
nce
1645
1650
En
ergy
(meV
)
zd,1 zm,1 zc zm,2 zd,2
(a)
FIG. S28. Maps of the difference between model and experiment for model parameters
Mγib↓. The top panel shows the difference between experiment and Mγib↓. The middle panel shows
the difference between models Mγib↓ and M0. The bottom panel shows model Mγib↓. The model
parameters are also labeled in the figure.
44