# Testing The Existence of Single Photons

Quynh M. Nguyen  Asad Khan  University of Minnesota, Minneapolis, MN, USA,
55455

(May 12, 2014)

###### Abstract

We demonstrated the existence of single photon by counting correlation of
laser output of a birefringent $\beta$-Barium Borate (BBO) crystal. The best
result of anti-correlation parameter violates the classical prediction for
wavelike behavior of light by 80 standard deviations.

††preprint: AIP/123-QED

##  I Introduction

Classically, light is an electromagnetic wave. Quantum mechanically, light has
both wave-like and particle-like properties, and the particle-like quanta are
called photons. Experiments such as the photo-electric effect can make this
particle-like nature of the field apparent but that is still not enoughStanley
to prove that light is made of photonsThorn . The reason is, we never measure
light directly but always measure the associated photocurrent instead. It is
possible for the granularity appearance to be caused by the discrete nature of
the electron in the detectors. Thus, observing granularity in measurements of
the field is necessary but not a sufficient proof for the existence of photon.

In 1986, Grangier, Roger, and AspectGrangier  performed an experiment in which
they examined correlations between photodetections at the transmission and
reflection outputs of a 50/50 beamsplitter. If a single quantum of light is
incident on the beamsplitter (BS), it should be detected at the transmission
output or at the reflection output, but not both: there should be no
coincident detections between the two outputs. In fact, Grangier et al.
measured fewer coincidences than predicted by a classical wave theory. Their
result violates a classical inequality by 13 standard deviations,
demonstrating that the field incident on the beamsplitter was well described
by a single-photon state. The key difficulty in such a measurement is to
create a field that truly has a single-photon incident on the BS. A dim source
of light containing on average a single photon is not sufficient.

Since photons are quantum mechanical objects, an experiment which requires a
quantum mechanical explanation would imply that there is more to the field
than just classical wave. If classical theories of electromagnetism cannot
explain the result, we take it to mean that single photons exist.

Our experiment is an updated version of the one performed originally by
Grangier et al., using a higher count rate and more advanced equipment. The
increase count rate allows a violation of classical inequality by 80 standard
deviations with 10 minutes of counting time. The experiment is well described
by the quantum mechanics of a field in a single photon state incident upon a
BS.

Figure 1: Coincidence measurement. The incident beam is split into transmitted
(detected at B) and reflected (detected at B′) at a 50/50 beamsplitter (BS).
Detection at B and B′ are examined to se whether or not they occur
simultaneously

##  II Theory

###  II.1 Early measurements

We are interested in examining correlations between the photocounts on two
detectors situated at the output ports of a BS (Fig. 1). The first experiment
to examine these correlations was carried out by Hanbury Brown and Twiss
Hanbury  who found a positive correlation between the outputs of the two
detectors. However in their experiment, Hanbury et al. were not counting
individual photons, but were instead measuring correlations between continuous
currents from photomultiplier tubes (PMTs) Hanbury . This positive correlation
only indicated that when the current from one PMT increased, the current on
the second tended to increase simultaneously.

In contrast, when Brannen and Ferguson performed a similar experiment, they
observed no positive correlation, and then claimed “that if such a positive
correlation did exist, it would call for a major revision of some fundamental
concepts in quantum mechanics.”Brannen  However, Hanbury Brown and
TwissHanbury2  noted that the experimental parameters used by Brannen and
Ferguson precluded the observation of positive correlations. They showed that
positive correlations are not only allowed by quantum mechanics, but also are
a natural consequence of the tendency for photons and other bosons to “bunch”
together.

The first experiment to observe positive correlations using coincidence
detection of individual photocounts (not just photocurrents) from PMTs was
performed by Twiss, Little, and Hanbury Brown Hanbury3 . They observed
positive correlations of a few percent. This amount of correlation was
expected, given their experimental parameters.

###  II.2 Classical field

By classical field, we mean an electromagnetic wave that is completely
described by Maxwell’s equations. Referring to Fig. 1, the intensity of the
field incident on the polarizing beamsplitter (PBS) is I(t), the transmitted
and reflected fields from the PBS go to detectors B and B′, and their
intensities are $I_{B}(t)$ and $I_{B^{\prime}}(t)$. The correlation between
$I_{B}(t)$ and $I_{B^{\prime}}(t)$ are given by anti-correlation
parameter111as known as degree of second-order coherence $\alpha^{2d}(\tau)$,
which is a function of the time delay $\tau$ between the intensity
measurements:

$\alpha^{2d}(\tau)=\frac{\langle
I_{B}(t+\tau)I_{B^{\prime}}(t)\rangle}{\langle I_{B}(t+\tau)\rangle\langle
I_{B^{\prime}}(t)\rangle}$ (1)

The superscript 2d indicates a two-detectors set up. If the light source is
stationary, then the brackets are interpreted as ensemble averages rather than
time averages. We are interested in simultaneous ($\tau$ = 0) intensity
correlations. If the beamsplitter has an intensity transmission of Tr, and
reflection coefficient of Re, then the transmitted, reflected, and incident
intensities are related by

$I_{B}(t)=Tr\times I(t)\text{ and }I_{B^{\prime}}=Re\times I(t)$ (2)

Inserting these expressions into eq. (1) gives

$\alpha^{2d}(0)=\frac{\langle[I(t)]^{2}\rangle}{\langle I(t)\rangle^{2}}$ (3)

which is independent of the splitting ratio of the beamsplitter. The variance
of the intensity, which can be written as $\Delta I^{2}=\langle
I^{2}\rangle-\langle I\rangle^{2}$, must be a positive numberLoudon . Thus, it
must be true that

$\alpha^{2d}(0)\geq 1\text{ (classical fields)}$ (4)

This result has been derived using classical wave theory. In eq. (4), equality
with 1 is achieved if the input intensity is perfectly stable with no
fluctuations. The closet experimental realization of a stable, classical wave
is laser light. Light from a stable laser yields $\alpha=1$, no matter how dim
it is.

###  II.3 Semi-classical theory of photodetection

In the previous section, anti-correlation is based on intensities of the
fields leaving the beamsplitter. However, in practice, one does not measure
intensities directly but rather the photocurrent from detectors. Therefore, it
is neccesary to model the photodetection process. In semi-classical theory of
photoelectric detection, the field is treated classically and the
photodetector is treated quantum mechanicallyMandel . Accordingly, the
conversion of continuous electromagnetic waves into discrete photoelectrons is
a random process. The probability of obtaining a single photocount from a
single detector (for example, detector B), within a short time window $\Delta
t$, is proportional to the average intensity of the incident field:

$P_{B}=\eta_{B}\langle I_{B}(t)\rangle\Delta t$ (5)

where $\eta_{B}$ is a constant that characterizes the the detection efficiency
of detector B. The joint probability of getting a photocount at detector B’
(within a short time window $\Delta t$), and then after a time $\tau$
obtaining a photocount at detector B (also within the time window $\Delta t$)
is given by

$P_{BB^{\prime}}=\eta_{B}\eta_{B^{\prime}}\langle
I_{B}(t+\tau)I_{B^{\prime}}(t)\rangle\Delta t$ (6)

Combining eqs. (1), (4), (5), and (6) gives

$\alpha^{2d}(0)=\frac{P_{BB^{\prime}}(0)}{P_{B}P_{B}^{\prime}}\geq 1\text{
(classical fields) }$ (7)

In summary, it is possible to measure anti-correlation parameter between the
fields leaving a BS by measuring the probability of joint and individual
photocounts at detectors B and B′. And the anti-correlation parameter must
satisfy $\alpha^{2d}(0)\geq 1$. The probability and $\alpha^{2d}(0)$ are
modeled after our experiment set up.

##  III Experimental technique

###  III.1 The experimental probabilities

Experimentally, probabilities must be expressed in terms of measured count
rates. For examples, the probability of a detection at B in a short time
interval $\Delta t$ (characteristic of the hardware in use) is simply given by
the average rate of detections multiplied by $\Delta t$. The average rate of
detections at B is just the number of detection $N_{B}$ divided by the total
counting time T. The probabilities for B’ detections and BB’ coincidences are
given similarly:

$P_{B}=\frac{N_{B}}{T}\Delta t\text{,
}P_{B^{\prime}}=\frac{N_{B^{\prime}}}{T}\Delta t\text{,
}P_{BB^{\prime}}=\frac{N_{BB^{\prime}}}{T}\Delta t$ (8)

Here $N_{BB^{\prime}}$ is the number of coincidence counts between the two
detectors. Substituting eq. (8) into eq. (7) gives

$\alpha^{2d}(t)=\frac{N_{BB^{\prime}}}{N_{B}N_{B^{\prime}}}\times\frac{T}{\Delta
t}\text{ (2-detectors)}$ (9)

Thus, with classical field statistics, it is possible to measure anti-
correlation between the fields leaving PBS by measuring the probability of
joint and individual photoncounts at detector B and B′. The time window
$\Delta t$ and the total measuring time T are adjustable. Alpha must satisfy
the inequality $\alpha\geq 1$. When the photocounts at B and B′ are completely
uncorrelated $\alpha=1$, which occurs when the input field to the PBS is
perfectly stable. When the input field fluctuates, then $\alpha>1$ indicating
positive correlations.

###  III.2 Experimental apparatus

In order to measure $\alpha=0$, it is necessary to have a single photon
incident on the PBS shown in Fig. 1. We exploited spontaneous parametric
downconversion process to produce single-photon states. The experiment use a
$\beta$-barium borate (BBO) crystal cut at $29^{o}$ so that the twin photons
emerge with approximately equal energy ($\lambda$ = 810nm) at $3^{o}$ from the
initial pump direction Pearson . Anti-correlation parameter was measured for
the two down-converted beams using detector A and B (Fig. 2). These two beams
need to be higly correlated in order to use photodetections at A as a
condition for single photon incident on the PBS.

Figure 2: The set up for anti-correlation measurement of parametric
spontaneous downconverted lights. Major components include 405nm pump laser
(blue), Downconversion Crystal (BBO), single photon counting modules (SPCMs),
fiber-coupling lens A, B and half wave plate (HWP). Optical fibers direct the
light from A,B to their corresponding SPCMs. Downconverted lights make a
$3^{o}$ angle with respect to the main beam and are shown in yellow. The
distance from the BBO to detector A is 100cm. The correlation between photons
dectected at detector A and B must be established to ensure single photon
incidence on the PBS.

In the final set up (shown in Fig. 3), a single photon incident on the PBS is
achieved conditionally upon the detection of a photon at detector A. Thus,
three-detectors measurements are necessary to see $\alpha=0$. Using the same
source without conditioning, detectors B and B′ would detects non-single
photon fields and give $\alpha\geq 1$. In a real experiment, $\alpha$ is
expected to be non-zero because there will always be some accidental three-
fold coincidence counts $N_{ABB^{\prime}}$. Next, $\alpha$ needs to be modeled
after a three-detectors set up.

Figure 3: Three-detectors experimental apparatus. A polarized beamsplitter
(PBS) and detector B′ were added to the two-detectors apparatus.

###  III.3 Three-detectors anti-correlation

If the measurement of $\alpha^{2d}$ is conditioned on a measurement of
photocount on a third detector A, in place of eq. (7), we have

$\alpha^{3d}(0)=\frac{P_{ABB^{\prime}}(0)}{P_{AB}(0)P_{AB^{\prime}}(0)}\text{
(3-detectors)}$ (10)

Where $P_{ABB^{\prime}}$ is the probability of threefold coincidence
detection, and $P_{AB}(0)$ and $P_{AB^{\prime}}(0)$ are the probabilities of
coincidence detection between detector A and detectors B and $B^{\prime}$,
respectively. The superscript 3d indicates three-detector anti-correlation
parameter. Since the events of interest are conditioned on events at A, the
number of detections at A, $N_{A}$, serves as the total number of trials,
which can be used to normalized probabilities :

$P_{AB}(0)=\frac{N_{AB}}{N_{A}}\text{,
}P_{AB^{\prime}}(0)=\frac{N_{AB^{\prime}}}{N_{A}}\text{,
}P_{ABB^{\prime}}(0)=\frac{N_{ABB^{\prime}}}{N_{A}}$ (11)

Using these probabilities, $\alpha^{3d}$ becomes

$\alpha^{3d}(0)=\frac{N_{ABB^{\prime}}}{N_{AB}N_{AB^{\prime}}}N_{A}\text{
(3-detectors)}$ (12)

For classical fields, $\alpha^{3d}\geq 1$. If a single quantum of light is
incident on the PBS, it should be detected at the transmission output or at
the reflection output, but not both: one photon should not be detected twice.
In other words, $N_{ABB^{\prime}}=0$ in eq. (12). Quantum mechanics predicts
$\alpha=0$ for 3 detectors.

$\Delta t$ | $R_{A}$ | $R_{B}$ | $R_{AB}$ | $R_{AB}^{acc}$ | $\alpha^{2d}$
---|---|---|---|---|---
(ns) | (Hz) | (Hz) | (Hz) | (Hz) |
10 | 14,800 | 16,700 | 223 | 2.48 | $90.2\pm 4$
20 | 14,600 | 16,700 | 231 | 4.77 | $48.5\pm 0.6$
40 | 14,200 | 15,700 | 238 | 8.99 | $26.5\pm 0.4$
60 | 14,100 | 15,700 | 244 | 13.4 | $18.3\pm 0.9$
10 | 46,100 | 51,300 | 719 | 23.7 | $17.9\pm 0.2$
20 | 45,600 | 50,700 | 756 | 46.2 | $16.4\pm 0.2$
40 | 44,300 | 49,200 | 803 | 87.1 | $9.2\pm 0.1$
60 | 43,800 | 48,600 | 843 | 128 | $6.6\pm 0.1$
Table 1: Correlation results for the two arms of our down-converted light sources at two different attenuation levels. We report averages of 20 runs, each 30 s long. The values of of the anti-correlation parameter indicate that the two arms are correlated. $\Delta t$ | $R_{A}$ | $R_{AB}$ | $R_{AB^{\prime}}$ | $R_{ABB^{\prime}}$ | $R_{ABB^{\prime}}^{acc}$ | $\alpha^{3d}$ | violation
---|---|---|---|---|---|---|---
(ns) | (Hz) | (Hz) | (Hz) | (Hz) | (Hz) |  | $(\sigma)$
10 | 14,800 | 128 | 95 | 0.016 | 0.018 | $0.019\pm 0.012$ | 80
20 | 14,600 | 132 | 99 | 0.031 | 0.037 | $0.035\pm 0.015$ | 62
40 | 14,200 | 136 | 102 | 0.062 | 0.075 | $0.064\pm 0.018$ | 53
60 | 14,100 | 140 | 104 | 0.10 | 0.11 | $0.097\pm 0.023$ | 40
10 | 46,100 | 412 | 306 | 0.19 | 0.18 | $0.71\pm 0.02$ | 47
20 | 45,600 | 431 | 325 | 0.34 | 0.38 | $0.112\pm 0.04$ | 22
40 | 44,300 | 458 | 345 | 0.76 | 0.78 | $0.213\pm 0.04$ | 18
60 | 43,800 | 477 | 366 | 1.1 | 1.2 | $0.268\pm 0.33$ | 22
Table 2: Correlation results for a three-detectors measurement using a down-
converted light source (all rates measure in Hz) at two attenuation levels. We
report averages of 20 runs, each 30 s long. Last column shows violation of the
classical inequality $\alpha^{3d}\geq 1$ in unit of $\sigma$

###  III.4 Accidental coincidence

In principle, our detectors can measure single photons. However, all detectors
and their associated electronics have physical limitations such as efficiency
and resolution that can affect the experimental result. Single-photon
detectors convert an incoming photon to an electronic pulse. The pulse widths
$\tau_{p}$ used were in order of 10 ns. For the counting circuit to register a
coincidence, the pulses from two detectors must overlap at least partially. If
the pulse widths are the same, the time window is $\Delta t=2\tau_{p}$. For
two random and independent sources, all coincidence counts are purely
accidental. If the average count rates for detector A and B are $R_{A}$ and
$R_{B}$, the expected rate of accidental coincidences is given byEckart

$R^{acc}(2d)=\Delta tR_{A}R_{B}\text{ (2 detectors)}$ (13)

With a three-detectors set up, one possibility of accidental coincidence is
for three uncorrelated photons to end up at detector A, B, B′ within a small
enough time such that the pulses overlap. However, for small pulse widths,
such threefold accidentals are extremely rare because it depends on the square
of $\tau_{p}:R_{acc}=3\tau_{p}^{2}R_{A}R_{B}R_{B}^{\prime}$.Eckart . Much more
likely is a twofold random accidental between a real twofold coincidence and a
random third single. With background negligible (by using filters), the
threefold accidental rate can be estimated by

$R^{acc}(3d)\approx\Delta tR_{A}R_{B}\text{ (3 detectors)}$ (14)

Using eq. (13) and eq. (14), the rate of accidental coincidences were
estimated and included in calculating $\alpha$.

##  IV Result

A correlation measurement using two detectors A and B corresponding to both
arms of down-converted light was performed. Because the beams are correlated,
it was expected that number of coincidence to be greater than those of random
source, that is, $\alpha^{2d}>1$. Typical results from 20 30s-long runs are
summarized in Table 1. Data in Table 1 clearly shows that $\alpha^{2d}$ is
larger than one, indicating the down-converted light is correlated and
appropriate for producing single-photon state.

Typical results for the three-detector correlation experiment using the
verified single photon source are shown in Table 2. Values of $\alpha^{3d}$
were found to be less than one, which provides clear evidence for the quantum
nature of light. $\alpha^{3d}<1$ means light consists of quanta that can be
transmitted or reflected at a beam splitter, but not both.

##  V Analysis

###  V.1 Down-conversion correlation

In theory, down-converted lights incident on detector A and B are perfectly
correlated, and $\alpha^{2d}$ should be much larger than the result shown in
Table 1. Consider an ideal experiment with perfect alignment and detectors and
no background noise. In this case, every twin pair produced will be measured
and all pairs are in perfect coincidence. If the rate of twins incident on the
detector is R(t), then $R_{A}=R_{B}=R_{AB}=R(t)$. Using eq. (LABEL:2d_prob)
and the fact that the rate of event is just the total number of count divided
by the total integration time T: $R_{i}=\frac{N_{i}}{T}$, $\alpha^{2d}$ for
the ideal experiment is given by

$\alpha^{2d}=\frac{R_{AB}}{R_{A}R_{B}\Delta t}=\frac{1}{\Delta tR(t)}$ (15)

For a rate of 14,800 Hz and $\Delta t=$ 10ns, perfect equipment would give
$\alpha^{2d}=6757$. The efficiency of this kind of system is about 4% Pearson
, leaving $\alpha^{2d}$ at most 270, in the same order of magnitude with our
measurement.

Interestingly, $\alpha^{2d}$ can be increased by decreasing the rate of down-
converted lights, as can be seen in Table 1. The reason is that the accidental
coincidence rate depends quadratically on R(t) in eq. (13), while the measured
coincidence rates depends linearly on R(t) in eq. (15).

###  V.2 Anti-correlation parameter with three detectors

If a truly single-photon state were incident on the PBS, quantum mechanics
would predict that $\alpha^{3d}=0$. A consequence of defining a “coincidence”
within a finite time window $\Delta t$ is an expected nonzero anti-correlation
parameter. This is because the possibility that uncorrelated photon from
different downconversion events may hit the B and B′ within the time window.
These are accidental coincidences, which increase with the time window. Data
in Table 1 ad Table 2 show that as the count rate and time window increase, so
do the rate of accidental coincidence $R_{AB}^{acc}$ and
$R_{ABB^{\prime}}^{acc}$.

The dependence on time window of accidental coincidence also explains why the
conindicence count $R_{ABB^{\prime}}$ decreases as the time window $\Delta t$
decreases. A perfectly correlated twin coincidence should not depend on
$\Delta t$, but in this case, a portion of the experimental coincidences is
accidental.

To improve correlation measurement, accidental coincidence must be reduced.
One method is to reduce the coincidence window. This is supported by the date
in Table 2, but it is limited by hardware. A second method is to reduce the
power of the pump laser, which effectively lowers all count rates in eq. (14).

###  V.3 Other considerations

Our best value of $\alpha^{2d}$ for the downconverted beams is about three
times smaller than theoretical prediction (90 compare to 270). The efficiency
of our system is lower than that of Pearson et al.,Pearson  who used similar
equipment. Better precision of measuring $\alpha$ can be made by improving
alignment technique. Especially, we found difficulty in defining the beam path
(a necessary step for positioning the PBS) of a non-visible laser using
irises.

Table 1 also shows that $R_{A}$ and $R_{B}$ are not equal on average. The
problem pertained after exchanging the fiber-coupling lens. Thus, the SPCM
corresponds to detector B must have a higher efficiency than that of detector
A. Coincidence rate can be increased if detector A is improved.

Given the beam path’s length (100cm from the BBO to fiber-coupling lens A),
the time delay effect due to uncertainty in path’s length (2mm) is $\approx
0.01$ns, which is negligible compared to our smallest time window (10ns).

##  VI conclusion

We have performed an experiment whose results cannot be explained using
classical theory of electromagnetic wave. The results are consistent with a
quantum mechanical description of light in which a field in a single-photon
state is incident on a beamsplitter. Therefore, we take the result as a
verification of the existence of photons.

##  VII acknowledgement

We are grateful to Kurt Wick for his support and guidance. We also would like
to thank professor G. Pawloski and professor C. Pryke for their reviews of our
Technical Design Report.

## References

  * (1) M. Beck, _Quantum Mechanics: Theory and Experiment_ Oxford University Press, New York, first ed., 2012.
  * (2) E. Brannen and H. I. S. Ferguson, _”The question of correlation between photons in coherent light rays,_ Nature (London) 178, 481–482 (1956).
  * (3) C. Eckart et al., _”Accidental coincidences in counter circuits,”_ Phys. Rev. 53, 752-756 (1938).
  * (4) P. Grangier et al., _”Experimental evidence for a photon anti-correlation effect on a beamsplitter: A new light on single photon inteferences,”_ Nature (London) 177, 27–29 (1956).
  * (5) R. Hanbury Brown and R. Q. Twiss, _”Correlation between photons in two coherent beams of light,”_ Am. J. Phys. 72, 1210-1219 (2004).
  * (6) R. Hanbury Brown and R. Q. Twiss, _”Interferometry of the intensity fluctuations in light I. Basic theory: the correlation between photons in coherent beams of radiation,”_ Proc. R. Soc. London, Ser.  A 242, 300–324 (1957).
  * (7) R. Hanbury Brown and R. Q. Twiss, _”Correlation between photons in coherent beams of light, detected by a coincidence counting technique,”_ Nature (London) 180, 324–326 (1957).
  * (8) R. Loudon, _The Quantum Theory of Light,_ Oxford, 3rd ed., 2000.
  * (9) 23L. Mandel and E. Wolf, _Optical Coherence and Quantum Optics,_ Cambridge University Press, Cambridge, 1995.
  * (10) B. J. Pearson and D. P.Jackson, _”A hands-on introduction to single photons and quantum mechanics for undergraduate,”_ Am. J. Phys. 78, 471-484 (2010).
  * (11) R. Q. Stanley., _”Question 45: What if anything! does the photoelectric effect teach us?,”_. Am. J. Phys. 64, 839 (1996).
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$\displaystyle\bm{\mathbf{d}}_{3}$
$\displaystyle=\frac{a}{2}\begin{pmatrix}\sqrt{3}\\\
-1\end{pmatrix},\quad\bm{\mathbf{d}}_{2}=\frac{a}{2}\begin{pmatrix}-\sqrt{3}\\\
-1\end{pmatrix},\quad\bm{\mathbf{d}}_{1}=-(\bm{\mathbf{d}}_{2}+\bm{\mathbf{d}}_{3})=\bm{\mathbf{a}}\begin{pmatrix}0\\\
1\end{pmatrix},$ (S108) $\displaystyle\bm{\mathbf{a}}_{3}$
$\displaystyle=\bm{\mathbf{d}}_{3}-\bm{\mathbf{d}}_{2},\quad\bm{\mathbf{a}}_{2}=\bm{\mathbf{d}}_{1}-\bm{\mathbf{d}}_{3},\quad\bm{\mathbf{a}}_{1}=\bm{\mathbf{d}}_{1}-\bm{\mathbf{d}}_{2},$
(S109)

where $a$ is the lattice constant. The reciprocal lattice (quasimomentum
space) is spanned by the reciprocal lattice vectors $\bm{\mathbf{b}}_{i}$

$\displaystyle\bm{\mathbf{b}}_{1}$
$\displaystyle=\frac{2\pi}{3a}\begin{pmatrix}\sqrt{3}\\\
1\end{pmatrix},\quad\bm{\mathbf{b}}_{2}=\frac{2\pi}{3a}\begin{pmatrix}-\sqrt{3}\\\
1\end{pmatrix},\quad\bm{\mathbf{b}}_{3}=\bm{\mathbf{b}}_{1}+\bm{\mathbf{b}}_{3},$
(S110)

such that $\bm{\mathbf{a}}_{i}\cdot\bm{\mathbf{b}}_{i}=2\pi\delta_{ij}$ for
$i,j\in\\{1,2\\}$ [see Fig. S1].

Figure S1: (a) Illustration of the Bogoliubov-Haldane model (S122) on a
hexagonal lattice with two sublattice sites $A$ ($B$) with potential energy
offset $\Delta$ ($-\Delta$), real (complex-valued) tunneling matrix element
$J_{1}$ ($J_{2}e^{i\theta}$) for the nearest (next-) nearest neighbors, and
on-site interaction strength $U$. The translation vectors connecting the
(next-) nearest neighbors ($\bm{\mathbf{d}}_{i}$) $\bm{\mathbf{a}}_{i}$ (S109)
are shown. (b) Illustraction of reciprocal lattice of the hexagonal lattice
structure of (a) including the reciprocal lattice vectors
$\bm{\mathbf{b}}_{i}$ (S110). In the 1st Brillouin zone, we depict a high-
symmetry path along the high-symmetry points $\Gamma$, $K$, $M$, and
$K^{\prime}$.

Having discussed the non-interacting part, we turn to the Bogoliubov theory,
which accounts for additional quantum (and thermal) fluctuations on top of a
mean-field solution for a weakly interacting bosonic system [123, 124, 134].
For this, we first obtain a mean-field solution via a minimization of an
associated Gross-Pitaevskii (GP) energy functional. The stationary solutions
gained by minimizing the GP energy functional will then yield the mean-field
ground state, which is assumed to be macroscopically occupied. Then, we will
describe quantum fluctuations on top of the condensate, by considering non-
condensed deviations from this mean-field solution. This procedure can be
formally understood by replacing the operators within the Hamiltonian
$\hat{H}$ (S104) via $\hat{a}_{\ell s}\to\sqrt{n_{0}}\zeta_{\ell
s}+\hat{a}_{\ell s}$, where the first term denotes the complex-valued mean-
field solution, $\zeta_{\ell s}\in\mathbb{C}$, and the second term the
additional quantum fluctuations where $n_{0}$ denotes condensate density of
the system. Expanding the Bose-Hubbard Hamiltonian (S104) in orders of the
condensate density $n_{0}$ then yields [20]

$\displaystyle\hat{H}=n_{0}E_{\text{GP}}+n_{0}^{1/2}\hat{H}^{(L)}+\hat{H}^{(B)}+\mathcal{O}\left(n_{0}^{-1/2}\right).$
(S111)

The first term denotes the GP energy functional $E_{\text{GP}}$ which is a
function of the mean-field solutions $\zeta_{\ell s}$. The next two terms
$\hat{H}^{(L)}$ and $\hat{H}^{(B)}$ are linear and quadratic in the bosonic
operators $\hat{a}^{(\dagger)}_{\ell s}$, where the latter is known as the
Bogoliubov Hamiltonian, which is sought after in the following.

First, the energy functional $E_{\text{GP}}$ is found by replacing the
operators $\hat{a}^{(\dagger)}_{\ell s}$ in $\hat{H}$ by complex-valued
functions $\sqrt{n}\zeta_{\ell s}^{(*)}$, where it is assumed that almost all
bosons are condensed, i.e., $n_{0}\approx n=N/V$, with $V=2N_{\text{uc}}$ and
$N$ denoting the total number of bosons in the system and $N_{\text{uc}}$ the
number of unit cells in the lattice [20]. For the parameters of the bosonic
Haldane model considered here, the condensate is formed at the single zero
quasimomentum mode $\bm{\mathbf{q}}=\bm{0}$ in the lower band such that a
suitable ansatz for the ground state is given by a spatially homogeneous
function $\zeta_{\ell s}=\chi_{s},\ s\in\\{A,B\\}$. This leads to the
following energy functional in quasimomentum space

$\displaystyle E_{GP}=\begin{pmatrix}\chi_{A}^{*}\
\chi^{*}_{B}\end{pmatrix}(\mathcal{H}(\bm{\mathbf{q}}=\bm{0})-\mu_{\text{eff}}\mathbbm{1}_{2})\begin{pmatrix}\chi_{A}\\\
\chi_{B}\end{pmatrix}+\frac{U}{2}n\left(\left\lvert\chi_{A}\right\rvert^{4}+\left\lvert\chi_{B}\right\rvert^{4}\right),$
(S112)

where we have introduced $\mu_{\text{eff}}$ as a Lagrange multiplier for
satisfying the conservation of the total number of bosons in the system, i.e.,
$\sum_{\ell s}\left\lvert\zeta_{\ell s}\right\rvert^{2}=N$. We minimize
$E_{GP}$ by taking the variational derivative of $E_{GP}$ with respect to
$\chi_{s}^{*}$ which leads to the so-called GP equations,

$\displaystyle\big{(}\mathcal{H}(\bm{\mathbf{q}}=\bm{0})-\mu_{\text{eff}}\mathbbm{1}_{2}\big{)}\begin{pmatrix}\chi_{A}\\\
\chi_{B}\end{pmatrix}+nU\begin{pmatrix}\left\lvert\chi_{A}\right\rvert^{2}\chi_{A}\\\
\left\lvert\chi_{B}\right\rvert^{2}\chi_{B}\end{pmatrix}=0.$ (S113)

Since the non-interacting condensate is formed at the
${\lvert}\bm{\mathbf{q}}=\bm{0},-{\rangle}$ mode we parametrize $\chi_{s}$ via
the Bloch sphere representation ${\lvert}\bm{\mathbf{q}}-{\rangle}$ (S106)
[21]

$\displaystyle\begin{pmatrix}\chi_{A}\\\
\chi_{B}\end{pmatrix}=\begin{pmatrix}\sin(\theta/2)\\\
-\cos(\theta/2)e^{i\varphi}\end{pmatrix}.$ (S114)

Multiplying (S113) by $\begin{pmatrix}\chi_{A}^{*}\,\chi_{B}^{*}\end{pmatrix}$
and $\begin{pmatrix}-\chi_{B}^{*}\,\chi_{A}^{*}\end{pmatrix}$ from the left
yields three equations [21, 135]

$\displaystyle(h_{0}(\bm{0})-\mu_{\text{eff}})-h(\bm{0})[\cos(\theta_{0})\cos(\theta)+\sin(\theta_{0})\sin(\theta)\cos(\varphi-\varphi_{0})]+2Un(\left\lvert\chi_{A}\right\rvert^{4}+\left\lvert\chi_{B}\right\rvert^{4})=0,$
(S115) $\displaystyle
h(\bm{0})[\cos(\theta_{0})\sin(\theta)\cos(\varphi)-\cos(\varphi_{0})\sin(\theta_{0})\cos(\theta)]-Un\sin(\theta)\cos(\theta)\cos(\varphi)=0,$
(S116)
$\displaystyle(h_{0}(\bm{0})-\mu_{\text{eff}})\sin(\theta)\sin(\varphi)+h(\bm{0})\sin(\theta_{0})\sin(\varphi_{0})+Un\sin(\theta)\sin(\varphi)=0,$
(S117)

where
$\bm{\mathbf{h}}(\bm{0})=h(\bm{0})\begin{pmatrix}\cos(\varphi_{0})\sin(\theta_{0}),\sin(\varphi_{0})\sin(\theta_{0}),\cos(\theta_{0})\end{pmatrix}^{T}$
and $h_{0}(\bm{0})$ stem from (S105). For a given interaction energy $U$ and
density $n$, we numerically minimize the Eqs. S115 to S117 and gain the
stationary solutions $(\theta,\varphi,\mu_{\text{eff}})$ specifying the mean-
field ground state in (S114).

Since the BEC occurs in the lowest band at $\bm{\mathbf{q}}=\bm{0}$, we
further specify the unitary rotation (provided by the eigenstates of the non-
interacting system (S106)) for the obtained solution [21]

$\displaystyle\begin{pmatrix}\hat{a}_{\bm{0},A}\\\
\hat{a}_{\bm{0},B}\end{pmatrix}=U(\theta,\varphi)\begin{pmatrix}\hat{a}_{\bm{0},+}\\\
\hat{a}_{\bm{0},-}\end{pmatrix}=\begin{pmatrix}f_{A,+}&f_{A,-}\\\
f_{B,+}&f_{B,-}\end{pmatrix}\begin{pmatrix}\hat{a}_{\bm{0},+}\\\
\hat{a}_{\bm{0},-}\end{pmatrix},$ (S118) $\displaystyle\text{with}\
U(\theta,\varphi)=\begin{pmatrix}\cos(\theta/2)&\sin(\theta/2)\\\
\sin(\theta/2)e^{i\varphi}&-\cos(\theta/2)e^{i\varphi}\end{pmatrix},$ (S119)

where $f_{s,\pm}(\theta,\varphi)$ with $s\in\\{A,B\\}$ denote the stationary
solutions of the GP functional (S112).

The next step is to go beyond this mean-field description and derive the
corresponding Bogoliubov Hamiltonian $\hat{H}^{(B)}$ (S111). The starting
point is the quasimomentum space representation of the Bose-Hubbard
Hamiltonian (S104)

$\displaystyle\hat{H}=\sum_{\bm{\mathbf{q}}}\sum_{s,s^{\prime}}\hat{a}^{\dagger}_{\bm{\mathbf{q}},s}\mathcal{H}_{s,s^{\prime}}(\bm{\mathbf{q}})\hat{a}_{\bm{\mathbf{q}},s^{\prime}}+\frac{U}{V}\sum_{\bm{\mathbf{q}},\bm{\mathbf{q}}^{\prime},\bm{\mathbf{q}}^{\prime\prime}}\sum_{s}\hat{a}^{\dagger}_{\bm{\mathbf{q}}+\bm{\mathbf{q}}^{\prime\prime},s}\hat{a}^{\dagger}_{\bm{\mathbf{q}}^{\prime}-\bm{\mathbf{q}}^{\prime\prime},s}\hat{a}_{\bm{\mathbf{q}},s}\hat{a}_{\bm{\mathbf{q}}^{\prime},s},$
(S120)

where
$\hat{a}_{\bm{\mathbf{q}},s}=\sum_{\ell}{\langle}\bm{\mathbf{q}}s{\,|\,}\ell
s{\rangle}\hat{a}_{\ell s}$ denote the bosonic operators transformed into
quasimomentum space. Utilizing another rotation into the eigenbasis (S106) the
bosonic operators in quasimomentum space are decomposed into a condensed and a
non-condensed part via [21]

$\displaystyle\sum_{\bm{\mathbf{q}}}\hat{a}_{\bm{\mathbf{q}},s}$
$\displaystyle=\hat{a}_{\bm{0},s}+\sum_{\bm{\mathbf{q}}\neq\bm{0}}\hat{a}_{\bm{\mathbf{q}},s}=f_{s,-}\hat{a}_{\bm{0},-}+f_{s,+}\hat{a}_{\bm{0},+}+\sum_{\bm{\mathbf{q}}\neq\bm{0}}\hat{a}_{\bm{\mathbf{q}},s},$
(S121)

where the condensation mode is assumed to be located at
$\bm{\mathbf{q}}=\bm{0}$ in the lowest band and $f_{s,\pm}\equiv
f_{s,\pm}(\varphi,\theta)$ are functions depending on the stationary solutions
of the energy functional $E_{GP}$ (S112).

Next, the decomposition (S121) is inserted into $\hat{H}$ (S120) and the
Hamiltonian is separated into parts of $\bm{\mathbf{q}}=\bm{0}$ and
$\bm{\mathbf{q}}\neq\bm{0}$ while only terms up to second order in
$\hat{a}^{\dagger}_{\bm{\mathbf{q}}\neq\bm{0},s}$ are retained, leading to an
approximate Hamiltonian. For the actual Bogoliubov approximation, the
condensation mode is then replaced via $\hat{a}_{\bm{0},-}\to\sqrt{N_{0}}$,
where $N_{0}$ denotes the number of particles in the condensate mode.
Performing all these steps will eventually lead to the following Bogoliubov
Hamiltonian in quasimomentum space

$\displaystyle\begin{split}\hat{H}^{(B)}=&\frac{1}{2}\sum_{\bm{\mathbf{q}}\neq\bm{0}}\begin{pmatrix}\hat{a}^{\dagger}_{\bm{\mathbf{q}},A}\,\hat{a}^{\dagger}_{\bm{\mathbf{q}},B}\,\hat{a}_{-\bm{\mathbf{q}},A},\hat{a}_{-\bm{\mathbf{q}},B}\end{pmatrix}\begin{pmatrix}\mathcal{H}(\bm{\mathbf{q}})+\mathcal{H}_{1}&2\mathcal{H}_{2}\\\
2\mathcal{H}_{2}^{*}&(\mathcal{H}(-\bm{\mathbf{q}})+\mathcal{H}_{1})^{*}\end{pmatrix}\begin{pmatrix}\hat{a}_{\bm{\mathbf{q}},A}\\\
\hat{a}_{\bm{\mathbf{q}},B}\\\ \hat{a}^{\dagger}_{-\bm{\mathbf{q}},A}\\\
\hat{a}^{\dagger}_{-\bm{\mathbf{q}},B}\end{pmatrix}\\\
+&\frac{1}{2}\begin{pmatrix}\hat{a}^{\dagger}_{\bm{0},+}\,\hat{a}_{\bm{0},+}\end{pmatrix}\begin{pmatrix}h_{3}&2h_{4}\\\
2h_{4}^{*}&h_{3}^{*}\end{pmatrix}\begin{pmatrix}\hat{a}_{\bm{0},+}\\\
\hat{a}^{\dagger}_{\bm{0},+}\end{pmatrix}+\text{const.}.\end{split}$ (S122)

We can also compactly write this Bogoliubov Hamiltonian (S122) in terms of the
Nambu spinor representation

$\displaystyle\hat{H}^{(B)}=\frac{1}{2}\sum_{\bm{\mathbf{q}}\neq\bm{0}}\Psi_{\bm{\mathbf{q}}}^{\dagger}\mathcal{M}_{\bm{\mathbf{q}}}\Psi_{\bm{\mathbf{q}}}+\frac{1}{2}\Psi^{\dagger}_{+}\mathcal{M}_{+}\Psi_{+}+\text{const.},$
(S123)

where
$\Psi^{\dagger}_{\bm{\mathbf{q}}}\equiv\begin{pmatrix}\hat{a}^{\dagger}_{\bm{\mathbf{q}},A}\,\hat{a}^{\dagger}_{\bm{\mathbf{q}},B}\,\hat{a}_{-\bm{\mathbf{q}},A},\hat{a}_{-\bm{\mathbf{q}},B}\end{pmatrix}$
and
$\Psi^{\dagger}_{+}\equiv\begin{pmatrix}\hat{a}^{\dagger}_{\bm{0},+}\,\hat{a}_{\bm{0},+}\end{pmatrix}$.
Here, $\mathcal{M}_{+}$ is the $2\times 2$ matrix and
$\mathcal{M}_{\bm{\mathbf{q}}}$ the quasimomentum dependent $4\times 4$ matrix
from (S122), which are explicitly given by [21, 135]

$\displaystyle\begin{split}\mathcal{M}_{\bm{\mathbf{q}}}&=\begin{pmatrix}\mathcal{H}(\bm{\mathbf{q}})+\mathcal{H}_{1}&2\mathcal{H}_{2}\\\
2\mathcal{H}_{2}^{*}&(\mathcal{H}(-\bm{\mathbf{q}})+\mathcal{H}_{1})^{*}\end{pmatrix},\quad\mathcal{M}_{+}=\begin{pmatrix}h_{3}&2h_{4}\\\
2h_{4}^{*}&h_{3}^{*}\end{pmatrix},\\\\[8.5359pt] \text{with}\
\mathcal{H}_{1}&=4Un\left\lvert
F_{-}\right\rvert^{2}-\mu_{\text{eff}}\mathbbm{1}_{2},\quad\mathcal{H}_{2}=Un(F_{-})^{2},\quad\end{split}$
(S124)

where $F_{-}=\text{diag}\begin{pmatrix}f_{A,-},f_{B,-}\end{pmatrix}$ and where
the entries of $\mathcal{H}(\bm{\mathbf{q}})$ are given by (S105) and (S107).
The entries of $\mathcal{M}_{+}$ are given by

$\displaystyle h_{3}=4Un\sum_{s}\left(\left\lvert
f_{s,-}\right\rvert^{2}\left\lvert
f_{s,+}\right\rvert^{2}\right)+\sum_{s,s^{\prime}}f_{s,+}^{*}\mathcal{H}_{s,s^{\prime}}(0)f_{s^{\prime},+}-\mu_{\text{eff}},\quad
h_{4}=Un\sum_{s}(f_{s,-})^{2}\,(f_{s,+}^{*})^{2},$ (S125)

where $\mu_{\text{eff}}$ follows the stationary solutions of Eqs. S115 to
S117.

The Bogoliubov Hamiltonian (S122) can now be diagonalized by applying the
paraunitary (Bogoliubov transformation) $W_{\bm{\mathbf{q}}}\,(W_{+})$ [21]

$\displaystyle\begin{pmatrix}\hat{a}_{\bm{\mathbf{q}},A}\\\
\hat{a}_{\bm{\mathbf{q}},B}\\\ \hat{a}^{\dagger}_{-\bm{\mathbf{q}},A}\\\
\hat{a}^{\dagger}_{-\bm{\mathbf{q}},B}\end{pmatrix}=W_{\bm{\mathbf{q}}}\begin{pmatrix}\hat{b}_{\bm{\mathbf{q}},+}\\\
\hat{b}_{\bm{\mathbf{q}},-}\\\ \hat{b}^{\dagger}_{-\bm{\mathbf{q}},+}\\\
\hat{b}^{\dagger}_{-\bm{\mathbf{q}},-}\end{pmatrix}\ \Leftrightarrow\
\Psi_{\bm{\mathbf{q}}}=W_{\bm{\mathbf{q}}}\beta_{\bm{\mathbf{q}}},\quad\begin{pmatrix}\hat{a}_{\bm{0},+}\\\
\hat{a}^{\dagger}_{\bm{0},+}\end{pmatrix}=W_{+}\begin{pmatrix}\hat{b}_{\bm{0},+}\\\
\hat{b}^{\dagger}_{\bm{0},+}\end{pmatrix}\ \Leftrightarrow\
\Psi_{+}=W_{+}\beta_{+}$ (S126)

with
$\beta_{\bm{\mathbf{q}}}\equiv\begin{pmatrix}\hat{b}_{\bm{\mathbf{q}},+}\,\hat{b}_{\bm{\mathbf{q}},-}\,\hat{b}^{\dagger}_{-\bm{\mathbf{q}},+}\,\hat{b}^{\dagger}_{-\bm{\mathbf{q}},-}\end{pmatrix}^{T}$
and
$\beta_{+}\equiv\begin{pmatrix}\hat{b}_{\bm{0},+}\,\hat{b}^{\dagger}_{\bm{0},+}\end{pmatrix}^{T}$
consisting of the new bosonic quasi-particle annihilation (creation) operators
$\hat{b}^{(\dagger)}_{\bm{\mathbf{q}},\pm}$ at quasimomentum $\bm{\mathbf{q}}$
in the upper ($+$) or lower Bogoliubov energy band ($-$). The $4\times 4$
($2\times 2$) Bogoliubov transformation matrix $W_{\bm{\mathbf{q}}}$ ($W_{+}$)
can be parametrized (S6) as

$\displaystyle W_{+}$
$\displaystyle=\begin{pmatrix}u_{\bm{0}}&v_{\bm{0}}^{*}\\\
v_{\bm{0}}&u_{\bm{0}}^{*}\end{pmatrix},\quad
W_{\bm{\mathbf{q}}}=\begin{pmatrix}U_{\bm{\mathbf{q}}}&V_{-\bm{\mathbf{q}}}^{*}\\\
V_{\bm{\mathbf{q}}}&U_{-\bm{\mathbf{q}}}^{*}\end{pmatrix}$ (S127)
$\displaystyle\text{with}\ U_{\bm{\mathbf{q}}}$
$\displaystyle=\begin{pmatrix}u_{A,+}^{\bm{\mathbf{q}}}&u_{A,-}^{\bm{\mathbf{q}}}\\\
u_{B,+}^{\bm{\mathbf{q}}}&u_{B,-}^{\bm{\mathbf{q}}}\end{pmatrix},\quad
V_{\bm{\mathbf{q}}}=\begin{pmatrix}v_{A,+}^{\bm{\mathbf{q}}}&v_{A,-}^{\bm{\mathbf{q}}}\\\
v_{B,+}^{\bm{\mathbf{q}}}&v_{B,-}^{\bm{\mathbf{q}}}\end{pmatrix}.$ (S128)

Since these quasi-particle operators still have to obey the bosonic
commutation relations, the Bogoliubov transformation matrix is paraunitary
[cf. Eq. S20], i.e., $W^{\dagger}\tau W=W\tau W^{\dagger}=\tau,\ \text{with}\
\tau=\sigma_{z}\otimes\mathbbm{1}_{4}$
($\tau=\sigma_{z}\otimes\mathbbm{1}_{2}$) for $W=W_{\bm{\mathbf{q}}}$
($W=W_{+}$). These paraunitary matrices are specifically constructed [2, 7]
such that

$\displaystyle
W^{\dagger}_{\bm{\mathbf{q}}}\mathcal{M}_{\bm{\mathbf{q}}}W_{\bm{\mathbf{q}}}=\text{diag}\begin{pmatrix}E_{+}(\bm{\mathbf{q}}),E_{-}(\bm{\mathbf{q}}),E_{+}(-\bm{\mathbf{q}}),E_{-}(-\bm{\mathbf{q}})\end{pmatrix},\quad
W^{\dagger}_{+}\mathcal{M}_{+}W_{+}=\text{diag}\begin{pmatrix}E_{+}(\bm{0}),E_{+}(\bm{0})\end{pmatrix},$
(S129)

holds, by which the Bogoliubov Hamiltonian is eventually diagonalized [21] to

$\displaystyle\hat{H}^{(B)}=\sum_{\bm{\mathbf{q}}}E_{+}(\bm{\mathbf{q}})\hat{b}^{\dagger}_{\bm{\mathbf{q}},+}\hat{b}_{\bm{\mathbf{q}},+}+\sum_{\bm{\mathbf{q}}\neq
0}E_{-}(\bm{\mathbf{q}})\hat{b}^{\dagger}_{\bm{\mathbf{q}},-}\hat{b}_{\bm{\mathbf{q}},-}+\text{const.}\,.$
(S130)

Here, $E_{\pm}(\bm{\mathbf{q}})$ is the Bogoliubov energy spectrum of the
upper ($+$) and lower ($-$) band.

The Bogoliubov energy spectrum can be obtained by (numerically) solving the
energy eigenvalue problem (S29) of the (non-hermitian) dynamical matrix
$D(\bm{\mathbf{q}})=\tau_{z}\mathcal{M}_{\bm{\mathbf{q}}}$, i.e.,
$D(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})=\tilde{\Omega}_{j}\bm{w}^{n}(\bm{\mathbf{q}})$.
However, the eigenmodes obtained from this diagonalization procedure do _not_
lead to the correct paraunitary conditions for the Bogoliubov modes (S20).

As described in Colpa1978,Shindou2013,Refs. Ref. [2, 7], the correct
paraunitary Bogoliubov matrices can be obtained via an algorithm, which
includes a Cholesky decomposition of the coefficient matrices
($\mathcal{M}_{\bm{\mathbf{q}}},\ \mathcal{M}_{+}$). For completeness, we
briefly review this algorithm in the next section.

###  S10A Algorithm for constructing paraunitary Bogoliubov transformation
matrix

Here, mainly following Colpa1978,Shindou2013,Refs. Ref. [2, 7], we briefly
discuss how to (numerically) obtain a paraunitary Bogoliubov transformation
matrix such as $W_{\bm{\mathbf{q}}}$ and $W_{+}$ (S127) when given a bosonic
BdG Hamiltonian.

We discuss the algorithm for a translationally invariant bosonic BdG
Hamiltonian such that we have quasimomentum dependent $2N\times 2N$
(hermitian) coefficient matrix $\mathcal{M}_{\bm{\mathbf{q}}}$ (S1) of the
form

$\displaystyle\mathcal{M}_{\bm{\mathbf{q}}}=\begin{pmatrix}K_{\bm{\mathbf{q}}}&G_{\bm{\mathbf{q}}}\\\
G^{*}_{-\bm{\mathbf{q}}}&K^{*}_{-\bm{\mathbf{q}}}\end{pmatrix},$ (S131)

with $K_{\bm{\mathbf{q}}}$ and $G_{\bm{\mathbf{q}}}$ being quasimomentum
dependent $N\times N$ matrices with $N$ denoting the number internal degrees
of freedom (e.g., the number sublattice sites within a unit cell). For each
quasimomentum $\bm{\mathbf{q}}$ the bosonic system can then be diagonalized by
a paraunitary matrix $W_{\bm{\mathbf{q}}}$ (S18) such that

$\displaystyle
W^{\dagger}_{\bm{\mathbf{q}}}\mathcal{M}_{\bm{\mathbf{q}}}W_{\bm{\mathbf{q}}}=\begin{pmatrix}\bm{\omega}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}_{-\bm{\mathbf{q}}}\end{pmatrix},\ \text{with}\
\bm{\omega}_{\bm{\mathbf{q}}}=\text{diag}\begin{pmatrix}\omega_{1,\bm{\mathbf{q}}},\ldots,\omega_{N,\bm{\mathbf{q}}}\end{pmatrix},$
(S132)

where $W_{\bm{\mathbf{q}}}$ again fulfill the paraunitary conditions (S20),

$\displaystyle
W^{\dagger}_{\bm{\mathbf{q}}}\tau_{z}W_{\bm{\mathbf{q}}}=W_{\bm{\mathbf{q}}}\tau_{z}W^{\dagger}_{\bm{\mathbf{q}}}=\tau_{z}\
\text{with}\ \tau_{z}=\sigma_{z}\otimes\mathbbm{1}_{N}.$ (S133)

Now, given that $\mathcal{M}_{\bm{\mathbf{q}}}$ is a positive-definite matrix
(all eigenvalues of $\mathcal{M}_{\bm{\mathbf{q}}}$ are strictly positive),
the paraunitary matrix $W_{\bm{\mathbf{q}}}$ can then be constructed via the
following steps [2].

First, we apply a Cholesky decomposition to the coefficient matrix
$\mathcal{M}_{\bm{\mathbf{q}}}$ (which is possible since
$\mathcal{M}_{\bm{\mathbf{q}}}$ is positive-definite) (S131), which decomposes
$\mathcal{M}_{\bm{\mathbf{q}}}$ into a product between an upper triangular
matrix $K_{\bm{\mathbf{q}}}$ and its hermitian conjugate
$K_{\bm{\mathbf{q}}}^{\dagger}$, i.e.,
$\mathcal{M}_{\bm{\mathbf{q}}}=K_{\bm{\mathbf{q}}}^{\dagger}K_{\bm{\mathbf{q}}}$.
We note that the positive-definiteness of $\mathcal{M}_{\bm{\mathbf{q}}}$ also
ensures the existence of the inverse $K_{\bm{\mathbf{q}}}^{-1}$.

Second, we introduce a hermitian matrix $T_{\bm{\mathbf{q}}}\equiv
K_{\bm{\mathbf{q}}}\tau_{z}K_{\bm{\mathbf{q}}}^{\dagger}$ from which we obtain
the unitary matrix $U_{\bm{\mathbf{q}}}$ via the eigendecomposition of
$T_{\bm{\mathbf{q}}}$,

$\displaystyle
U^{\dagger}_{\bm{\mathbf{q}}}T_{\bm{\mathbf{q}}}U_{\bm{\mathbf{q}}}=\begin{pmatrix}\bm{\omega}_{\bm{\mathbf{q}}}&0\\\
0&-\bm{\omega}_{-\bm{\mathbf{q}}}\end{pmatrix}.$ (S134)

These two diagonal matrices $\bm{\omega}_{\bm{\mathbf{q}}}$ and
$\bm{\omega}_{-\bm{\mathbf{q}}}$ already provide the Bogoliubov energy
spectrum of Eq. S132 [7]. One can further show that both
$\bm{\omega}_{\bm{\mathbf{q}}}$ and $\bm{\omega}_{-\bm{\mathbf{q}}}$ are real
and strictly positive [2].

Third, the paraunitary matrix $W_{\bm{\mathbf{q}}}$ is then constructed via

$\displaystyle
W_{\bm{\mathbf{q}}}=K_{\bm{\mathbf{q}}}^{-1}U_{\bm{\mathbf{q}}}\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix},$ (S135)

which will diagonalize the coefficient matrix $\mathcal{M}_{\bm{\mathbf{q}}}$
(S131) and importantly fulfill the paraunitary conditions (S133). This
concludes the construction of the paraunitary matrix $W_{\bm{\mathbf{q}}}$ for
each quasimomentum $\bm{\mathbf{q}}$.

One can then show the diagonalization of Eq. S132 by the direct application of
(S135)

$\displaystyle
W^{\dagger}_{\bm{\mathbf{q}}}\mathcal{M}_{\bm{\mathbf{q}}}W_{\bm{\mathbf{q}}}=W^{\dagger}_{\bm{\mathbf{q}}}K_{\bm{\mathbf{q}}}^{\dagger}K_{\bm{\mathbf{q}}}W_{\bm{\mathbf{q}}}=\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}U_{\bm{\mathbf{q}}}^{\dagger}(K_{\bm{\mathbf{q}}}^{\dagger})^{-1}K_{\bm{\mathbf{q}}}^{\dagger}K_{\bm{\mathbf{q}}}K_{\bm{\mathbf{q}}}^{-1}U_{\bm{\mathbf{q}}}\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}=\begin{pmatrix}\bm{\omega}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}_{-\bm{\mathbf{q}}}\end{pmatrix}.$ (S136)

Moreover, we can see that the construction of $W_{\bm{\mathbf{q}}}$ (S135)
also fulfills the paraunitary conditions (S133),

$\displaystyle W^{\dagger}_{\bm{\mathbf{q}}}\tau_{z}W_{\bm{\mathbf{q}}}$
$\displaystyle=\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}U_{\bm{\mathbf{q}}}^{\dagger}(K_{\bm{\mathbf{q}}}^{\dagger})^{-1}\tau_{z}K_{\bm{\mathbf{q}}}^{-1}U_{\bm{\mathbf{q}}}\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}=\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}U_{\bm{\mathbf{q}}}^{\dagger}T_{\bm{\mathbf{q}}}^{-1}U_{\bm{\mathbf{q}}}\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}$ (S137)
$\displaystyle=\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}\begin{pmatrix}\bm{\omega}^{-1}_{\bm{\mathbf{q}}}&0\\\
0&-\bm{\omega}^{-1}_{-\bm{\mathbf{q}}}\end{pmatrix}\begin{pmatrix}\bm{\omega}^{1/2}_{\bm{\mathbf{q}}}&0\\\
0&\bm{\omega}^{1/2}_{-\bm{\mathbf{q}}}\end{pmatrix}=\begin{pmatrix}\mathbbm{1}_{N}&0\\\
0&-\mathbbm{1}_{N}\end{pmatrix}=\tau_{z},$ (S138)

where in the first step we have just inserted the definition of
$W_{\bm{\mathbf{q}}}$ (S135). In the second equality, we used that
$T_{\bm{\mathbf{q}}}^{-1}=(K_{\bm{\mathbf{q}}}^{\dagger})^{-1}\tau_{z}K_{\bm{\mathbf{q}}}^{-1}$,
which itself directly follows from
$T_{\bm{\mathbf{q}}}=K_{\bm{\mathbf{q}}}\tau_{z}K_{\bm{\mathbf{q}}}^{\dagger}$.
In the second equality, following from (S134) we used that
$U_{\bm{\mathbf{q}}}^{\dagger}T_{\bm{\mathbf{q}}}^{-1}U_{\bm{\mathbf{q}}}=\begin{pmatrix}\bm{\omega}^{-1}_{\bm{\mathbf{q}}}&0\\\
0&-\bm{\omega}^{-1}_{-\bm{\mathbf{q}}}\end{pmatrix}$.

###  S10B SQGT from integrated Bogoliubov excitation rates

Figure S2: Integrated and ideal symplectic quantum metric component
$g_{yy}(\bm{\mathbf{q}})$ and $g_{xy}(\bm{\mathbf{q}})$ for the upper particle
band of the Bogoliubov-Haldane model (S122). (a) Integrated (left) and ideal
(right) symplectic quantum metric component $g_{yy}(\bm{\mathbf{q}})$ for
$Un/J_{1}=3$ shown in 2D quasimomentum space. (b) Same as in (a) but for the
off-diagonal component $g_{xy}(\bm{\mathbf{q}})$. (c) Ideal (solid line) and
integrated (crosses) symplectic quantum metric component
$g_{yy}(\bm{\mathbf{q}})$ along a high-symmetry path in quasimomentum space
for two interactions strengths $Un/J_{1}=0$ (non-interacting Haldane model)
and $Un/J_{1}=3$. (d) Same as in (c) but for the off-diagonal component
$g_{xy}(\bm{\mathbf{q}})$. All the simulations have been performed for driving
strengths $A/J_{1}=0.01$ up to a final time of $J_{1}t/\hbar=10$ for a
frequency range of $\hbar\omega/J_{1}\in[0.1(0.2),5.5]$, for $Un/J_{1}=3\,(0)$
with a spacing of $\delta\omega=0.05$ over which the excitation rates have
been integrated. The red-colored hexagon in (a) and (b) indicates the first
Brillouin zone. Other parameters of the underlying Haldane model (S107) are
the same as in Fig. 3 in the main text, i.e., $J_{2}=0.1J_{1}$,
$\theta=0.5\pi$, $\Delta/(3\sqrt{3}J_{2})=0.5$.

Here, we briefly discuss how we numerically implement our proposed method to
extract the SQGT from the integrated Bogoliubov excitation rate applied to the
Bogoliubov-Haldane model (S122). We start with an initially populated
Bogoliubov particle mode $\bm{w}^{n}(\bm{\mathbf{q}})$ which is obtained from
the paraunitary Bogoliubov transformation matrix (S126). The time evolution of
the Bogoliubov particle mode $\bm{w}^{n}(\bm{\mathbf{q}},t)$ is performed via
(S48) in the translationally invariant co-moving frame. Specifically, when the
system is shaken along the $x$-direction the time-dependent dynamical matrix
is given by Eq. S64. The total excitation rate into other Bogoliubov in
response to this perturbation is then computed via

$\displaystyle\Gamma^{n}_{\bm{\mathbf{q}}}(\omega)=\frac{1}{t}\sum_{m}s_{m}\left\lvert(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}\tau_{z}\bm{w}^{n}(\bm{\mathbf{q}},t)\right\rvert^{2},$
(S139)

where $t$ is the final integration time and $\omega$ the frequency of the
perturbation (S64). Integrating (S139) over a range of probing frequencies
$\omega$, $\Gamma^{n}_{\text{int},\bm{\mathbf{q}}}=\int
d\omega\,\Gamma^{n}_{\bm{\mathbf{q}}}(\omega)$, then allows us to obtain the
symplectic quantum metric component $g_{xx}(\bm{\mathbf{q}})$ via (S67). This
similarly applies to the $g_{yy}(\bm{\mathbf{q}})$ component of the SQGT when
the system is shaken along the $y$-direction where the dynamical matrix in co-
moving frame becomes
$D_{y}^{\prime}(t)=D(\bm{\mathbf{q}})+(2A/\omega)\sin(\omega
t)\partial_{q_{y}}D(\bm{\mathbf{q}})+\mathcal{O}((A/\omega)^{2})$. For the
off-diagonal components the lattice is shaken along both directions (S68), the
time-dependent dynamical matrix is given by Eq. S70. The total excitation
rates $\Gamma^{n,\pm}_{\bm{\mathbf{q}}}(\omega)$ are obtained in the same way
as in (S139). Taking the differential integrated rate,
$\Delta\Gamma^{n}_{\text{int}}=\Gamma^{n,+}_{\text{int}}-\Gamma^{n,-}_{\text{int}}=\int
d\omega\,(\Gamma^{n,+}_{\bm{\mathbf{q}}}(\omega)-\Gamma^{n,-}_{\bm{\mathbf{q}}}(\omega))$
over a range of probing frequencies $\omega$ eventually allows to obtain the
off-diagonal component of the quantum metric $g_{xy}(\bm{\mathbf{q}})$ (for
$\varphi=0$) and the Berry curvature $B_{xy}(\bm{\mathbf{q}})$ (for
$\varphi=\pi/2$) via (S72).

The results of this simulation for $g_{xx}(\bm{\mathbf{q}})$ and
$B_{xy}(\bm{\mathbf{q}})$ are shown in Fig. 3 in the main text. For
completeness, here we show the results for the remaining components of the
SQGT, $g_{yy}(\bm{\mathbf{q}})$ and $g_{xy}(\bm{\mathbf{q}})$ in Fig. S2.

###  S10C Signatures for the distribution of the original particles

Our measurement protocol for extracting the SQGT requires measuring the
excitation fraction in specific Bogoliubov (particle) modes. Focusing on the
Bogoliubov-Haldane model (S122), we discuss the signatures for the
quasimomentum distribution of the original particles $c_{\bm{\mathbf{q}},s}$
given the presence of the single or multiple quasiparticles excitations
$\hat{b}^{\dagger}_{\bm{\mathbf{q}},\pm}$ on top of the Bogoliubov vacuum,
which can then be measured e.g., via time-of-flight techniques in optical
lattice platforms [95, 91, 96, 102].

We start by noting that the ground state ${\lvert}\psi_{0}{\rangle}$ of the
Bogoliubov-Hamiltonian (S130) is given by the vacuum state of the Bogoliubov
quasiparticles fulfilling the conditions,

$\displaystyle\hat{b}_{\bm{\mathbf{q}},+}{\lvert}\psi_{0}{\rangle}=0,\
\forall\bm{\mathbf{q}},\quad\text{and}\quad\hat{b}_{\bm{\mathbf{q}},-}{\lvert}\psi_{0}{\rangle}=0,\
\forall\bm{\mathbf{q}}\neq 0,$ (S140)

where in the second equality the condensation mode has to be explicitly
excluded, as the Bogoliubov theory is only concerned with excitations outside
the condensate mode at $\bm{\mathbf{q}}=\bm{0}$ in the lower band. Even though
the Bogoliubov ground state contains no quasiparticles, there is still a
finite amount of original particles outside the condensation mode which are
(virtually) excited by interparticle interactions. Since these excitations are
only driven by quantum fluctuations, this result is called quantum depletion.

We now consider the more general situation where a finite number of $l$
quasiparticles are excited on top of the Bogoliubov vacuum at quasimomentum
$\bm{\mathbf{q}}$ in band $n\in\\{\pm\\}$, i.e,
$(b_{\bm{\mathbf{q}},n}^{\dagger})^{l}/\sqrt{l!}\,{\lvert}\psi_{0}{\rangle}\equiv{\lvert}\phi_{0}{\rangle}$.
For $l=0$ this state is again the Bogoliubov ground state (S140). Now, for the
state ${\lvert}\phi_{0}{\rangle}$ with $l$ quasiparticle excitations, we
compute the original particle contributions outside the condensation mode
${\langle}\phi_{0}{\,|}\hat{a}^{\dagger}_{\bm{\mathbf{q}},s}\hat{a}_{\bm{\mathbf{q}},s^{\prime}}{|\,}\phi_{0}{\rangle}$
where $s,s^{\prime}\in\\{A,B\\}$ is again the sublattice index (S122). To this
end, we explicitly use the Bogoliubov transformations for
$\bm{\mathbf{q}}\neq\bm{0}$ (S126)-(S128) and arrive at

$\displaystyle\langle\hat{a}^{\dagger}_{\bm{\mathbf{q}},s}\hat{a}_{\bm{\mathbf{q}},s}\rangle$
$\displaystyle=l\,\left\lvert
u_{s,m}^{\bm{\mathbf{q}}}\right\rvert^{2}\delta_{m,n}+\left\lvert
v_{s,+}^{-\bm{\mathbf{q}}}\right\rvert^{2}+\left\lvert
v_{s,-}^{-\bm{\mathbf{q}}}\right\rvert^{2},\quad\langle\hat{a}^{\dagger}_{-\bm{\mathbf{q}},s}\hat{a}_{-\bm{\mathbf{q}},s}\rangle=l\,\left\lvert
v_{s,m}^{\bm{\mathbf{q}}}\right\rvert^{2}\delta_{m,n}+\left\lvert
v_{s,+}^{\bm{\mathbf{q}}}\right\rvert^{2}+\left\lvert
v_{s,-}^{\bm{\mathbf{q}}}\right\rvert^{2},$
$\displaystyle\langle\hat{a}^{\dagger}_{\bm{\mathbf{q}},A}\hat{a}_{\bm{\mathbf{q}},B}\rangle$
$\displaystyle=l\,\left[(u_{A,m}^{\bm{\mathbf{q}}})^{*}u_{B,m}^{\bm{\mathbf{q}}}\right]\delta_{m,n}+(v_{A,+}^{-\bm{\mathbf{q}}})(v_{B,+}^{-\bm{\mathbf{q}}})^{*}+(v_{A,-}^{-\bm{\mathbf{q}}})(v_{B,-}^{-\bm{\mathbf{q}}})^{*}=\langle\hat{a}^{\dagger}_{\bm{\mathbf{q}},B}\hat{a}_{\bm{\mathbf{q}},A}\rangle^{*},$
$\displaystyle\langle\hat{a}^{\dagger}_{-\bm{\mathbf{q}},A}\hat{a}_{-\bm{\mathbf{q}},B}\rangle$
$\displaystyle=l\,\left[v_{A,m}^{\bm{\mathbf{q}}}(v_{B,m}^{\bm{\mathbf{q}}})^{*}\right]\delta_{m,n}+(v_{A,+}^{\bm{\mathbf{q}}})(v_{B,+}^{\bm{\mathbf{q}}})^{*}+(v_{A,-}^{\bm{\mathbf{q}}})(v_{B,-}^{\bm{\mathbf{q}}})^{*}=\langle\hat{a}^{\dagger}_{-\bm{\mathbf{q}},B}\hat{a}_{-\bm{\mathbf{q}},A}\rangle^{*}.$
(S141)

For $l=0$ initial quasiparticle excitation present in band $n$ in (S141), we
recover the quantum depletion result. As an example, let us consider $l$
initial quasiparticle excitations in the lower Bogoliubov energy band at
$\bm{\mathbf{q}}$, i.e.,
${\lvert}\phi_{0}{\rangle}=(b_{\bm{\mathbf{q}},-}^{\dagger})^{l}/\sqrt{l!}\,{\lvert}\psi_{0}{\rangle}$.
Then, for the sublattice site $s=A$, we have
$l\times{\lvert}u_{A,-}^{\bm{\mathbf{q}}}{\rvert}^{2}+{\lvert}v_{A,+}^{-\bm{\mathbf{q}}}{\rvert}^{2}+{\lvert}v_{A,-}^{-\bm{\mathbf{q}}}{\rvert}^{2}$
original particles moving with quasimomentum $+\bm{\mathbf{q}}$ and
$(l+1)\times{\lvert}v_{A,-}^{\bm{\mathbf{q}}}{\rvert}^{2}+{\lvert}v_{A,+}^{\bm{\mathbf{q}}}{\rvert}^{2}$
original particles moving with quasimomentum $-\bm{\mathbf{q}}$. These results
(S141) directly relate the presence of initial Bogoliubov quasiparticle
excitations with the quasimomentum distribution of the real bosonic particles.

In optical lattice platforms, the momentum distribution can be observed via
time-of-flight (TOF) imaging techniques. The momentum distribution
$N(\bm{\mathbf{k}})$ of the system after TOF is given by [102, 136]

$\displaystyle
N(\bm{\mathbf{k}})=\left\lvert\tilde{w}(\bm{\mathbf{k}})\right\rvert^{2}\frac{1}{N_{\text{uc}}}\sum_{\ell\ell^{\prime}}\sum_{ss^{\prime}}e^{-i\bm{\mathbf{k}}\cdot(\bm{\mathbf{r}}_{\ell
s}-\bm{\mathbf{r}}_{\ell^{\prime}s^{\prime}})}\langle\hat{a}^{\dagger}_{\ell
s}\hat{a}_{\ell^{\prime}s^{\prime}}\rangle,$ (S142)

where $\left\lvert\tilde{w}(\bm{\mathbf{k}})\right\rvert^{2}$ is the Wannier
function envelope with the Fourier-transformed Wannier function
$\tilde{w}(\bm{\mathbf{k}})$, $N_{\text{uc}}$ the number of unit cells, and
$\hat{a}_{\ell s}=\sum_{\bm{\mathbf{q}}}{\langle}\ell
s{\,|\,}\bm{\mathbf{q}}s{\rangle}\hat{a}_{\bm{\mathbf{q}},s}=(N_{\text{uc}})^{-1/2}\sum_{\bm{\mathbf{q}}}e^{-i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}_{\ell
s}}\hat{a}_{\bm{\mathbf{q}},s}$ the real bosonic operators in the original
real space basis with the unit cell index $\ell$ and sublattice index $s$
(S104). Then, for the initial state with multiple Bogoliubov quasiparticle
excitations in band $n$ (S141) the TOF momentum distribution
$N(\bm{\mathbf{k}})$ (S142) becomes

$\displaystyle N(\bm{\mathbf{k}})=$
$\displaystyle\sum_{s,s^{\prime}}\langle\hat{a}^{\dagger}_{\bm{\mathbf{k}}s}\hat{a}_{\bm{\mathbf{k}}s^{\prime}}\rangle=\langle\hat{a}^{\dagger}_{\bm{\mathbf{k}}A}\hat{a}_{\bm{\mathbf{k}}A}\rangle+\langle\hat{a}^{\dagger}_{\bm{\mathbf{k}}B}\hat{a}_{\bm{\mathbf{k}}B}\rangle+\left(\langle\hat{a}^{\dagger}_{\bm{\mathbf{k}}A}\hat{a}_{\bm{\mathbf{k}}B}\rangle+\text{c.c.}\right)$
$\displaystyle=$ $\displaystyle\sum_{s\in\\{A,B\\}}\left\\{l\,\left\lvert
u_{s,m}^{\bm{\mathbf{k}}}\right\rvert^{2}\delta_{m,n}+\left\lvert
v_{s,+}^{-\bm{\mathbf{k}}}\right\rvert^{2}+\left\lvert
v_{s,-}^{-\bm{\mathbf{k}}}\right\rvert^{2}\right\\}$ $\displaystyle+$
$\displaystyle\left[\left(l\,\left[(u_{A,m}^{\bm{\mathbf{k}}})^{*}u_{B,m}^{\bm{\mathbf{k}}}\right]\delta_{m,n}+(v_{A,+}^{-\bm{\mathbf{k}}})(v_{B,+}^{-\bm{\mathbf{k}}})^{*}+(v_{A,-}^{-\bm{\mathbf{k}}})(v_{B,-}^{-\bm{\mathbf{k}}})^{*}\right)+\text{c.c.}\right],$
(S143)

where we neglected the overall Wannier envelope function
$\left\lvert\tilde{w}(\bm{\mathbf{k}})\right\rvert^{2}$ for now. Thus,
equipped with the quasimomentum distribution results obtained from Bogoliubov
theory (S141) and the TOF images (S143), the distribution of Bogoliubov
quasiparticle excitations can be inferred to extract the SQGT.

We believe that the following experimental protocol can be used to measure the
Bogoliubov excitation fraction after periodic modulation the systems’
parameters to extract the SQGT as described in the main text. First, one
should measure the TOF momentum distribution without performing the periodic
modulation and without exciting any quasiparticles to obtain a reference
measurement. Then, as a second step, we imagine exciting Bogoliubov
quasiparticles at a desired quasimomentum $\bm{\mathbf{q}}$ in band $n$ on top
of the Bogoliubov vacuum with the protocol described in the main text or the
one of Stamper-Kurn1999,Vogels2002,Refs. Ref. [91, 95], whose momentum
distribution should then be measured via TOF. The difference of these first
two measurements then provides a measure of the excited Bogoliubov
quasiparticle distribution of step two. As a third step, after exciting
initial Bogoliubov quasiparticles, one should perfom the periodic parameter
modulation protocol to induce scattering of Bogoliubov quasiparticles into the
other band. Then, the momentum distribution of the system should be measured
again via TOF. Eventually, the difference of this final momentum distribution
and the one after step two gives us the induced Bogoliubov quasiparticle
excitation fraction which can be used to extract the SQGT as described in the
main text.


# WS2 Band Gap Renormalization Induced by Tomonaga Luttinger Liquid Formation
in Mirror Twin Boundaries

Antonio Rossi  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, United States of America  Advanced Light Source, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, United States of America
John C. Thomas  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, United States of America  Johannes T. Küchle  Molecular
Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United
States of America  Physics Department E20, Technical University of Munich,
James-Franck-Str. 1, 85748 Garching, Germany  Elyse Barré  Molecular Foundry,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States of
America  Zhuohang Yu  Department of Materials Science and Engineering, The
Pennsylvania State University, University Park, PA 16082 United States of
America  Center for Two-Dimensional and Layered Materials, The Pennsylvania
State University, University Park, PA, 16802 United States of America  Da Zhou
Department of Materials Science and Engineering, The Pennsylvania State
University, University Park, PA 16082 United States of America  Center for
Two-Dimensional and Layered Materials, The Pennsylvania State University,
University Park, PA, 16802 United States of America  Shalini Kumari
Department of Materials Science and Engineering, The Pennsylvania State
University, University Park, PA 16082 United States of America  Center for
Two-Dimensional and Layered Materials, The Pennsylvania State University,
University Park, PA, 16802 United States of America  Hsin-Zon Tsai  Department
of Physics, University of California at Berkeley, Berkeley, CA, United States
of America  Ed Wong  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, United States of America  Chris Jozwiak  Advanced Light
Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United
States of America  Aaron Bostwick  Advanced Light Source, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, United States of America  Joshua A.
Robinson  Department of Materials Science and Engineering, The Pennsylvania
State University, University Park, PA 16082 United States of America  Center
for Two-Dimensional and Layered Materials, The Pennsylvania State University,
University Park, PA, 16802 United States of America  Department of Physics and
Department of Chemistry, The Pennsylvania State University, University Park,
PA, 16802 United States of America  Mauricio Terrones  Department of Materials
Science and Engineering, The Pennsylvania State University, University Park,
PA 16082 United States of America  Center for Two-Dimensional and Layered
Materials, The Pennsylvania State University, University Park, PA, 16802
United States of America  Department of Physics and Department of Chemistry,
The Pennsylvania State University, University Park, PA, 16802 United States of
America  Archana Raja  Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, United States of America  Adam Schwartzberg
Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
United States of America  D. Frank Ogletree  Molecular Foundry, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, United States of America
Jeffrey B. Neaton  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, CA, United States of America  Department of Physics,
University of California at Berkeley, Berkeley, CA, United States of America
Michael F. Crommie  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, CA, United States of America  Department of Physics,
University of California at Berkeley, Berkeley, CA, United States of America
Kavli Energy Nano Sciences Institute, University of California Berkeley,
Berkeley, CA, United States of America  Francesco Allegretti  Physics
Department E20, Technical University of Munich, James-Franck-Str. 1, 85748
Garching, Germany  Willi Auwärter  Physics Department E20, Technical
University of Munich, James-Franck-Str. 1, 85748 Garching, Germany  Eli
Rotenberg  Alexander Weber-Bargioni  Molecular Foundry, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, United States of America

## Abstract

Tomonaga-Luttinger liquid (TLL) behavior in one-dimensional systems has been
predicted and shown to occur at semiconductor-to-metal transitions within two-
dimensional materials. Reports of mirror twin boundaries (MTBs) hosting a
Fermi liquid or a TLL have suggested a dependence on the underlying substrate,
however, unveiling the physical details of electronic contributions from the
substrate require cross-correlative investigation. Here, we study TLL
formation in MTBs within defectively engineered WS2 atop graphene, where band
structure and the atomic environment is visualized with nano angle-resolved
photoelectron spectroscopy, scanning tunneling microscopy and scanning
tunneling spectroscopy, and non-contact atomic force microscopy. Correlations
between the local density of states and electronic band dispersion elucidated
the electron transfer from graphene into a TLL hosted by MTB defects. We find
that MTB defects can be substantially charged at a local level, which drives a
band gap shift by $\sim$0.5 eV.

## Main

One-dimensional (1D) systems in condensed matter physics provide unique
insight into a variety of quasi particle excitations, including charge density
waves (CDWs) that arise due to Peierls instabilities,1, 2 lossless transport
through electronic wires in topological edge states,3 quantum spin liquids,4
as well as more exotic phenomena such as Marjorana modes in nanowires5 and the
emergence of a Tomonaga-Luttinger liquid (TLL). The latter has been realized
in both nanotubes and transition metal dichalcogenides (TMDs).6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17 These quasi particle excitations not only host new
condensed matter physics phenomena but hold the promise to become major
pillars of quantum electronics and quantum information applications.18 We show
the capability to controllably create 1D confined systems and to directly
observe TLL formation at its native length scale, which is key to
understanding the governing principles behind such a strongly-correlated
system.

The hallmarks of a TLL are the independent dispersion of charge and spin,
fractional charge transport, and power law suppression of the density of
states (DOS) near the Fermi energy (EF).19, 14, 15, 16 The formation of a TLL
was first observed in 1D carbon nanotubes,8, 9 where the conductance of
bundled single-walled carbon nanotubes showed power law scaling with respect
to bias voltage and temperature. This has since been extended to a number of
1D systems such as semiconductor single-channel wires, nanowires, organic
conductors, and fractional quantum Hall edge channels.19, 13, 4, 8, 20, 21,
22, 23, 24, 15, 25 Recently, this phenomenon has also been shown to exist
within 1D defects in two-dimensional (2D) TMDs at a plane of lattice points
where the crystal structures on either side of the interface are mirrored,
which defines a mirror twin boundary (MTB) within monolayer TMDs.14, 15, 16
MTBs have been predicted to exhibit metallic properties and to form out of
sub-stoichiometric metal (M = Mo, W) or from depleted chalcogen (X = S, Se,
Te) in MX2 materials.26, 27, 22, 28 TLL formation at MTB defects within TMDs
adds to the the fascinating and vast array of material properties that 2D
monolayer TMDs hold such as single photon emission, tunable band gaps, and
strong spin-orbit coupling, to name a few.29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39 Zhu et al. have applied scanning tunneling microscopy and scanning
tunneling spectroscopy (STM/STS) to directly map out the local density of
states (LDOS) associated with TLLs,16 where measurements show a gap opening
near EF, a length dependence on the highest occupied and lowest unoccupied
state band gap, and spin-charge dispersion observable in Fourier transform
(FT) STS maps.14, 15

The role of the substrate for the formation of TLLs in MTBs has yet to be
fully ascertained, and, in addition, a comprehensive understanding of the
macroscopic band structure with the local electronic structure in TLL
formation requires explanation. In order to address these questions, we
engineer 1D defects into 2D WS2 grown on graphene with tunable control over
both length and density via a post-synthesis, _in-situ_ approach. Measurement
techniques cross-correlating STM/STS together with spatially- and nano angle-
resolved photo emission spectroscopy (nARPES) help to give a broad range of
information both on the electronic band structure of the system and the nature
of the defects.40 Non-contact atomic force microscopy (ncAFM) further enables
structure identification of MTBs that host a TLL, where a metallic tip is
functionalized with a CO molecule.28, 41

We report new insights into the role of the graphene substrate in TLL
formation within 1D TMD MTB heterostructures. Two 1D defect symmetries are
unamibiguously identified, where their controlled introduction enables TLL
formation. Exclusive doping (charge transfer) from graphene into an MTB gives
on the order of $\sim$4-5 electrons per MTB, which brings the EF of graphene
near the Dirac point reducing its screening power and, therefore, increasing
the electron-electron interaction in the MTBs. We also observe an atomically
localized band gap renormalization over the MTB/TLL systems, with indications
of the conduction band taking part in TLL formation.

In order to study 1D MTB defects that may host a TLL, we perform STM/STS
studies on both unmodified WS2, grown epitaxially via chemical vapor
deposition (CVD) on a graphene/SiC substrate, and the same sample after Ar+
sputtering and annealing _in-situ_ to induce defectivity.42, 43, 44
Comparative results of low-energy sputtering in contrast to a low-temperature
anneal are shown (Fig. 1 (a-b)). As-grown samples are not significantly
modified by annealing up to 250 ∘C, while chalcogen vacancies (VS) start to
form at 600 ∘C.37 Low-energy Ar+ sputtering at chalcogen creation sample
temperatures ($SA_{step}$) greatly increases the density of VS and MTB defects
(see Supplementary Notes 1 and 2 and Supplementary Figs. 1 and 2 for both
density calculations and SRIM simulations) compared to only annealing. Post
$SA_{step}$ annealing at 600 ∘C ($SAA_{step}$) substantially reduces point
defect density, where MTBs with increased length (L${}_{SA_{step}}$ = 3.37
$\pm$ 2.87 nm, L${}_{SAA_{step}}$ = 8.75 $\pm$ 4.01 nm) are formed (see
Supplementary Fig. 1).

We confirm local structure via ncAFM performed with a functionalized CO tip in
Fig. 1 (c and d). Under the conditions of chalcogen depletion and excess
metal, the asymmetric mirror glide boundary (AMGB) and the symmetric mirror
twin boundary (SMTB), amongst other possible formations, are predicted to be
favorable in TMDs,27, 22, 45 where we measure the formation of both AMGB and
SMTB defects under the conditions described (see Supplementary Fig. 3) that
corresponds to predicted structures. As ncAFM frequency shifts are not
affected by the electronic structure changes near EF,28 the atomically-
resolved images obtained of SMTB in WS2 are described by a WS2 edge sharing
chalcogen point sites, and, in addition, the acquired AMGB is characterized by
an edge shifted by half of a lattice spacing (a = 0.315 nm) with W bonded to
four sulfur atoms across the MTB (see Supplementary Fig. 4).46 It may be
important to note that while MTBs have been more extensively studied in MoS2
and MoSe2, high formation energy in WS2 has hindered atomic-scale
investigations of MTBs. We are able to take advantage of sulfur reduction
techniques coupled with tandem atomic-scale measurements. Upon further
investigation with STS, the two types of defects introduced during both
$SA_{step}$ and $SAA_{step}$ are verified to be MTBs and VS, as detailed in
Fig. 1 (e). Here, the band gap of surrounding WS2 is measured to be on the
order of 2.5 eV and VS shows deep unoccupied defect states split by spin-orbit
coupled W _d_ states, which is consistent with previously measured values.37
Conversely, MTBs show a semiconducting band gap between the highest occupied
state (HOS) and lowest unoccupied state (LUS) on the order of 100 meV.

In order to better investigate the physics of the MTB bandgap formed in WS2,
we make use of both point STS and differential conductance mapping, which are
powerful tools for screening defects.28, 37, 47 Electronic band-edge state
differential conductance maps, acquired by dI/dV imaging (Fig. 2 (a-f)), at
the LUS ($\psi_{-}$) and the HOS ($\psi_{+}$) are spatially out-of-phase for
the SMTB case, and are a spatially in-phase for an AMGB. Additionally, the
number of orbital nodes changes as a function of bias for both the AMGB and
the SMTB. This 1D particle-in-a-box behavior, which has been demonstrated for
both types of defects in other TMDs, such as MoSe2 and MoS2, shows an
increasing number of nodes moving further past the LUS (with decreasing
period) and a decreasing number of nodes moving below the HOS (with increased
period) for the SMTB or the reverse behavior for the AMGB,14, 16 which is
consistent with our measurements. We find that the nodal periodicity at the
HOS and LUS for a measured SMTB structure (Fig. 2 (a-b)) is near 2 nm and 1.5
nm, respectively, which is far above the lattice constant of 0.315 nm.
Additionally, the measured periodicity for the inspected AMGB structure is 0.5
nm (Fig. 2 (d-e)) at both the HOS and the LUS, which is not within an integer
relationship with the lattice parameter of WS2. Supplemental Fig. 5 and 6
further show conductance maps as a function of energy both below HOS and above
the LUS for both structures, where the presence of a neighboring VS does not
show significant hybridization with an SMTB across measured energy regimes. We
also highlight nodal increase as a function of bias in Fig. 2 (c), where the
number of nodes moves from 6 at the LUS up to 10 nodes at 0.4 eV before
nearing the CBM of WS2 for the SMTB. This is also shown for an AMGB in Fig. 2
(f), where 8 nodes are measured at the HOS and 9 nodes are measured at -0.3
eV. A dense (1$\times$128$\times$500) linescan of point spectroscopy is
captured in Supplementary Fig. 7, where nodes increase by an integer number as
the bias is swept from below the HOS to above the LUS for an SMTB, which is
depicted both spatially and in Fourier space. Here, the HOS has 4 nodes and
the LUS has 5, increasing up to 10 nodes at 0.31 eV. As low-energy excitations
are not Fermi liquid quasiparticles in TLL theory, the spin- and charge-
density waves exhibit different dispersions and velocities, denoted as
$\upsilon_{s}$ and $\upsilon_{c}$, respectively. Their ratio can be
experimentally measured in the FT-STS measurement as
$K_{c}=\upsilon_{s}/\upsilon_{c}$, where $K_{c}$ is the Luttinger parameter.
From our FT-STS, we extract a $K_{c}$ of 0.5, which is consistent with
previously acquired values on other TMD systems.14, 16 Band gap ($E_{gap}$) as
a function of length is shown in Supplementary Fig. 8, where $E_{gap}$ is
dependent upon length ($L$), scaling linearly with $L^{-1}$. This behaviour
stands in contrast to Peierls instability, where the $E_{gap}$ is constant in
the CDW case and does not exhibit a length dependence.14 Finally, we also look
at the DOS measured in topographic STM images. TLL nodal oscillations are
indistinguishable above CBM of WS2, however, a measure of signal intensity as
a function of distance is feasible. Here, electron density is expected to
decay according to $K_{c}$ as $\rho\sim x^{-K_{c}}$,15 where the DOS ($\rho$)
measured under constant-current as a function of distance (x), in nanometers,
is shown in Supplementary Fig. 9. The fitted exponential function yields a
value of 0.5 that is well-matched to the values obtained in the FT-STS and in
agreement with previously measured MTBs that host a TLL.14, 15, 16 Overall, we
identify the TLL properties of these defects by observing a Luttinger
parameter of 0.5, measuring an $E_{gap}$ opening near EF, identifying an
$E_{gap}$ dependence on MTB length, visualizing 1D particle in a box behavior,
and extracting evidence of spin-charge separation.

We next perform nARPES to directly visualize the crystal band dispersion. The
sub-$\mu$m probe in nARPES offers a spatial resolution capable of capturing
the local inhomogeneity in the sample. Fig. 3 showcases the as-measured band
structure of the unmodified crystal versus the sample exposed to Ar+
bombardment. The two spectra are collected from the same sample, where a small
region is found to be unaffected by the $SA_{step}$, due to sample holder
shadowing during preparation. The spectrum obtained from the non-defective
structure is displayed in Fig. 3 (a). It is collected along the $\Gamma$-K
direction. Graphene and WS2 keep epitaxial registry, therefore WS2 also has
the same crystal orientation.48 A sketch of the two Brillouin zones (BZ)
(graphene in black, WS2 in green) is highlighted in the inset of Fig. 3 (a).
Graphene exhibits sharp bands with the EF $\sim$400 meV above the Dirac point.
This is consistent with graphene prepared from thermal decomposition of SiC,
where the carbon-rich buffer layer between graphene and SiC substrate creates
an electric dipole at the interface affecting the chemical potential of
graphene.49 Such a native gating also affects the WS2 band, whose top of the
valence band appears to be $\sim$1.48 eV below the EF.48 The local maximum at
$\Gamma$ appears below the maximum at K, confirming the monolayer nature of
the TMD.50 The $SA_{step}$ deeply affects the bands of WS2. Fig. 3 (b) shows
the spectrum obtained from the region with high defect density due to the
$SA_{step}$. A substantial band gap renormalization is observed, where both
the magnitude and chemical potential position are affected. The WS2 occupied
electron bands are shifted upwards by $\sim$500 meV. The position of the top
of the valence band is further confirmed by the spectrum collected with linear
vertical polarization (see Supplementary Fig. 10). The graphene bands are also
shifted up, although not by the same magnitude, with a subsequent change in
the doping level. This is clear analyzing the Dirac bands collected near the K
point of the graphene BZ along the direction highlighted in red within the
inset of panel (a) (Fig. 3 (c and d)). The momentum distribution curvers
(MDCs), collected at the EF (Fig. 3 (e and f)) can be fit with two Lorentzian
functions. The position of their peaks defines a distance that approximates
the diameter of the circle fitting the Fermi surface of graphene. Using
Luttinger theorem,51, 52 it is possible to extract the doping level being
n${}_{p}=1.2\times 10^{13}$ $\textrm{cm}^{-2}$ and n${}_{d}=2.5\times 10^{12}$
$\textrm{cm}^{-2}$ for the as-grown and defective crystal, respectively (Fig.
3 (c-f)). The unmodified doping level agrees well with the value found in
literature.51 From these values and locally-measured defect densities
(Supplementary Fig. 1 and Supplementary Note 2), we are able to determine that
each MTB hosts 3-8 electrons (0.021 $\pm$ 0.009 $\frac{MTB}{nm^{2}}$). Our
Monte Carlo calculations further suggest that our sample treatment does not
have a direct impact on graphene (Supplementary Fig. 2). As a matter of fact,
the presence of defects in graphene would open a gap at the Dirac point caused
by an alteration of the system symmetry rather than shifting its chemical
potential.53 The shift of the Dirac point in graphene is the result of charge
transfer from graphene to the newly formed MTBs in WS2. The chemical potential
difference with respect to the as-grown crystal does not match the magnitude
observed for WS2. Our findings suggest that graphene plays a significant role
in the formation of a TLL, both donating charge to the newly formed defects
and providing a weaker electronic screening due to the lower carrier density
near neutrality point. This, overall, increases the electron-electron
interaction strength in the TMD.

In Fig. 3 (g), we analyze the energy distribution curves (EDC) crossing the
top of the valence band (vertical dashed line in Fig. 3 (a-b)) by overlapping
the EDCs with the STS curves collected from the unmodified crystal, a single
sulfur vacancy, and an MTB formed on the TMD. The band shift is caused mostly
by a band gap renormalization related to the presence of the MTB as opposed to
isolated sulfur vacancies. Indeed, the onset of the valence band displayed by
the STS obtained from the sulfur vacancy (dashed green line) does not match
the onset of the valence band extracted by the EDC taken from Fig. 3 (b).
Conversely, the STS curve obtained from the 1D MTB (dark red line) displays
the onset of the valence band matching the EDC from nARPES data taken after
the $SA_{step}$ (light red line). The blue lines show the EDC and STS
comparison from the pristine structure, where in this case the alignment
between the two curves is well-matched at the expected onset of the valence
band. This behavior is additionally measured with spatially-localized STS
acquisition in Fig. 4, where a shift in the VBM is seen approaching an MTB
with an STM tip. The measured onsets of the VBM (-1.16 $\pm$ 0.04 eV) and the
CBM (0.74 $\pm$ 0.09 eV) was measured locally with STS, which corresponds with
the appearance of the HOS and LUS. The relative shift of the VBM is +530 meV
(CBM is shifted by -80 meV) and is further depicted in Fig. 4 (a), which
matches well to the band structure acquired by nARPES. Both nARPES
measurements and STS highlight the importance in choice of substrate to
determine the electronic properties of 1D defects in TMDs. Where other
findings have shown a substrate dependence,15 cross-correlated measurements
made in this report are able to directly visualize the EF position related to
the presence of MTBs over graphene. W 4$f$ and S 2$p$ core levels in
Supplementary Fig. 11 show both core levels are rigidly shifted to lower
binding energy in a similar fashion as the bands in the valence region. This
suggests that the band gap renormalization is overall electrostatically
driven. W and S peak fitting upon MTB formation displays a weaker component
that is further shifted towards a lower binding energy of an additional few
hundreds of meV due to the local chemical environment.

This work presents a new, controllable way to create to create MTBs in WS2
epitaxially grown on graphene is presented. We show how MTBs host a TLL, in
which spectroscopic and topological signature is shown via STM/STS and ncAFM.
A band gap opening of roughly 100 meV is observed near the EF confirming the
correlated nature of the electronic state inside the 1D defect. We
demonstrated the formation of two kinds of MTBs, AMGB and SMTB, which display
similar spectroscopic features but a distinct spatial difference in
conductance image mapping. By means of nARPES, we were able to correlate
scanning probe spectroscopic MTB features to the band structure of the
crystal. We observed how defective states behave as an acceptor of graphene
electrons and that they cause a massive band gap renormalization of WS2. The
same effect is reflected on core levels where we observe a similar chemical
shift in binding energies.

Encoding information with two different quantum degrees of freedom across 1D
structures can have immediate relevance for quantum information processing,
where application of such materials has yet to be manifested in functional
devices. Additionally, electron transport across semiconductor to metal
transitions may be beneficial in ultrafast electronic systems. Here, we
provide a step-by-step approach to produce 1D metallic structures within a 2D
semiconducting material, which holds relevance in atomic-scale tailorable
systems and electronic modification at the nanoscale. We further anticipate
engineered TMD materials to be relevant in spin-polarized measurements, charge
state effects, and spin transport.

##  1 Methods

###  1.1 Scanning probe microscopy (SPM) measurements

All measurements were performed with a Createc GmbH scanning probe microscope
operating under ultrahigh vacuum (pressure < 2x$10^{-10}$ mbar) at liquid
helium temperatures (T < 6 K). Either etched tungsten or platinum iridium tips
were used during acquisition. Tip apexes were further shaped by indentations
into a gold substrate. STM images are taken in constant-current mode with a
bias applied to the sample. STS measurements were recorded using a lock-in
amplifier with a resonance frequency of 683 Hz and a modulation amplitude of 5
mV.

In ncAFM measurements, a qPlus quartz-crystal cantilever was used (resonance
frequency, f0 ≈ 30 kHz; spring constant, k ≈ 1800 N/m; quality factor, Q >
18,000; and oscillation amplitude, A ≈ 1 Å).36 The metallic tip was
functionalized with a CO molecule for enhanced resolution.41

###  1.2 Angle-resolved photoemission spectroscopy (ARPES) measurements

ARPES experiments were performed in ultra high vacuum at T= 6K at beamline
7.0.2 (MAESTRO) at the Advanced Light Source. The beam-spot size was ≈ 1 μm.
The photon energy for the valence band structure is $h\nu$ = 150 eV and $h\nu$
= 350 eV for core levels. The XPS curve-fitting analysis was performed using a
convolution of Doniach-Sunjic and Gaussian line shapes superimposed on a
background built of a constant, a linear component, and a step-function. For
each S 2$p$ spin-orbit doublet, a spin-orbit splitting of 1.2 eV and a
branching ratio I(2$p$3/2) : I(2$p$1/2) = 2 : 1 (defined in terms of peak
areas) were used. The W 4$f$ spin-orbit doublets were fit using individual
components with a spin-orbit splitting of 2.2 eV, to take into account non-
linearities.

###  1.3 Sample preparation

Monolayer islands of WS2 were grown on graphene/SiC substrates with an ambient
pressure CVD approach. A monolayer graphene (MLG)/SiC substrate with 10 mg of
WO3 powder on top was placed at the center of a quartz tube, and 400 mg of
sulfur powder was placed upstream. The furnace was heated to 900 ∘C and the
sulfur powder was heated to 250 ∘C using a heating belt during synthesis. A
carrier gas for process throughput was used (Ar gas at 100 sccm) and the
growth time was 60 min. The CVD grown WS2/MLG/SiC was further annealed _in
vacuo_ at 400 ∘C for 2 hours.

Monolayer WS2 was sputtered with an argon ion gun (SPECS, IQE 11/35) that
operated at 0.1 keV energy with 60∘ off-normal incidence at a pressure of
5$\times$10-6 mbar and held at 600 ∘C. A rough measure of current
(0.6$\times$10-6 A) enabled the argon ion flux to be estimated at
(1.5$\times$1013 $\frac{ions}{cm^{2}s}$), where the sample was irradiated for
up to 30 seconds.

Samples were transferred from the STM to the nARPES chamber using an Ar
suitcase that prevented air exposure to enable cross-correlative studies
without risking sample degradation.

### Data availability

All data needed to evaluate the conclusions exhibited are present in the paper
and/or the supplemental information.

### Code availability

Software used for analysis are either presented in the supplemental
information or can be provided upon reasonable request from the authors.

### Acknowledgements

The authors thank Nino Hatter and Sebastian Baum from CreaTec Fischer & Co.
GmbH for helpful discussions. This work was supported as part of the Center
for Novel Pathways to Quantum Coherence in Materials, an Energy Frontier
Research Center funded by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences. Work was performed at the Molecular Foundry and at the
Advanced Light Source, which was supported by the Office of Science, Office of
Basic Energy Sciences, of the U.S. Department of Energy under contract no. DE-
AC02-05CH11231. S.K and J.A.R. acknowledge support from the National Science
Foundation Division of Materials Research (NSF-DMR) under awards 2002651 and
2011839. J.T.K and F.A. acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG) through the TUM International Graduate School of
Science and Engineering (IGSSE), GSC 81.

### Author Contributions

A.R., J.C.T., and A.W.-B. conceived and carried out the experiments. A.R.,
J.C.T., and J.T.K carried out nARPES/XPS measurements with the assistance of
C.J., A.B., and E.R. E.B. and J.C.T. contributed to SRIM/TRIM simulations.
A.R., J.C.T., and J.T.K. carried out ncAFM measurements with the assistance of
H.-Z.T. and M.F.C. A.R., J.C.T., and J.T.K. performed all STM/STS experiments
with additional contributions from A.R., E.W., A.S., D.F.O., F.A., W.A., and
A.W.-B. A.R. and J.C.T. performed all nARPES related data analysis with
support from C.J., A.B., and E.R. A.R. and J.C.T. performed all ncAFM, STM,
and STS related data analysis with support from A.R., E.W., A.S., D.F.O.,
J.B.N., M.F.C., F.A., W.A., and A.W.-B. Z.Y., T.Z., S.K., J.A.R. and M.T.
synthesized the samples. All authors discussed the results and contributed
towards the manuscript.

### Competing interests

The authors declare that they have no competing interests.

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Fig. 1: WS2 Defect Introduction. (a) Scanning tunneling micrograph depicting
pristine WS2 before Ar+ bombardment ($I_{tunnel}$ = 30 pA, $V_{sample}$ = 1.2
V). Scale bar, 10 nm. (b) After the pristine sample is heated and exposed to
an Ar+ sputter ($I_{tunnel}$ = 30 pA, $V_{sample}$ = 1.2 V), both VS and MTBs
are present. Scale bar, 2 nm. (c) Lattice structure is measured by ncAFM
($V_{sample}$ = 0.0 V), with a CO-functionalized tip, which showcases the SMTB
that is placed next to a structural schematic of (d) of the MTB. Scale bar,
0.25 nm. (e) Point spectroscopy, in the form of the LDOS, for unmodified WS2,
VS, and an SMTB. Fig. 2: LDOS Mapping. Conductance maps ($V_{modulation}$ = 5
mV) performed across an SMTB show dual-line orbital behavior at (a) 0.132 eV
(LUS) and (b) -0.025 eV (HOS) that is spatially out of phase. (c) Accumulated
conductance maps across a single-line of the SMTB ($V_{modulation}$ = 5 mV)
are further shown as a function of bias, where the number of nodes increase as
bias voltage is increased. Scale bars, 1 nm. Conductance maps (dI/dV) of the
(d) 0.047 eV (LUS) and (e) -0.047 eV (HOS) that are spatially in phase within
a single-line AMGB. (f) Compiled conductance maps ($V_{modulation}$ = 5 mV)
across the single-line AMGB are further shown, where the number of nodes
decreases as the voltage is increased. Fig. 3: nARPES Band Structure
Comparison. (a) Unmodified and (b) defective band structure of WS2 on
graphene. The inset displays the WS2 (green) and the graphene (black) BZ. The
spectra are collected along the $\Gamma$-K orientation. nARPES spectra of as-
grown (c) and defective (d) crystals that are collected along the red line of
the inset in panel (a), with respective MDCs at the EF shown in (e) and (f).
(g) EDCs overlapped with STS from unmodified and defective WS2. Dark blue and
dark red are the STS signals collected on pristine WS2 and on an MTB,
respectively. Dashed green is the STS spectrum collected on an isolated sulfur
vacancy. The light blue and light red lines are the EDCs collected at the $K$
point of the BZ (dashed vertical lines in panels (a) and (b) respectively).
Fig. 4: Tomonoga Luttinger Liquid Spatial Dependence. (a) LDOS spectra
collected from pristine WS2 to an MTB ($V_{modulation}$ = 5 mV, $I_{set}$ =
150 pA) and then recorded in reverse. The as-measured VBM, CBM, HOS, and LUS
are highlighted with relative positions to EF. (b) The process of charge
transfer from graphene into an MTB that is able to host highly correlated
electron states is depicted, where the EF of graphene is shifted and locally
modified due to the presence of an MTB.