coherent feedback control of two-dimensional excitons

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)

∗ [email protected]

† [email protected]

1

mailto:[email protected]

mailto:[email protected]

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

https://github.com/mabuchilab/Instrumental

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.


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.


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.


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



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


(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


Model M0

Raw Exp.


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


Model Mω0↑

Experiment


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


Model Mω0↓

Experiment


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




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


Model Mγr,0↑

Experiment


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


Model Mγr,0↓

Experiment


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


Model Mγnr↑

Experiment


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




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


Model Mγnr↓

Experiment


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




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


Model Mγib↑

Experiment


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




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


Model Mγib↓

Experiment


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




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



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

)


(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↑



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

)


(a)


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↓



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

)


(a)


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


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↑



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

)


(a)


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↓



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

)


(a)


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


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↑



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

)


(a)


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


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↓



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

)


(a)


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


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↑



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

)


(a)


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


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↓



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

)


(a)


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


44

coherent feedback control of two-dimensional excitons

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