#  Negative Radiation Pressure Scheme for Simultaneous Suppression of Arduous
Back-Action Evasion and Shot Noise in Gravitational Wave Detectors

Souvik Agasti<EMAIL_ADDRESS>Abhishek Shukla  Milos Nesladek
Physics Department, Hasselt University

###### Abstract

Aiming at application for gravitational wave (GW) detection, we propose a
novel scheme how to obtain quantum back action evading measurements performed
on an opto-mechanical cavity, by introducing a negative radiation pressure
coupling between the cavity field and the end mirror. The scheme consists of
introducing a double cavity with end mirrors interlocked by a pivot and moving
in opposite directions. The measurement is performed by sending a two-mode
squeezed vacuum to both cavities and detecting the output through the
heterodyne detection. Compared to the previously proposed hybrid negative mass
spin-optomechanical system in [1], we see that our scheme is capable to
suppress back action noise by nearly two orders of magnitude more in the lower
frequency region. Overall, the setup has been able to squeeze the output noise
below the standard quantum limit, with more efficiency. In addition, the
scheme has also proven to be beneficial for reducing thermal noise by a
significant amount. We confirm our result by a numerical analysis and compared
it with the previous proposal [1].

††preprint: APS/123-QED

##  I Introduction

The second-generation gravitational wave (GW) detectors (such as advanced LIGO
[2], advanced Virgo [3] and advanced GEO [4]) and others aim at increasing the
GW detection sensitivity and significantly reducing the detection noise. The
noise is composed of mainly electronic, seismic, thermal, and quantum noise
components. For example, thermal noise can be reduced using a cryogenic setup
or low thermal noise coatings [5]. At the current state of the art, the
quantum noise is still the major concern of limitation [6]. The arms of a GW
detector are typically considered to be optomechanical systems. The main
sources of quantum noise in optomechanical systems are the shot noise, which
is dependent on the number of incident photons injected into the system
$\delta N\propto\sqrt{N}$, and the quantum back-action (QBA) noise which is
generated due to the optomechanical interaction of the cavity field with the
end mirror. Further progress in the quantum noise suppression in GW
interferometers has been achieved by using homodyne detection. In this case,
the essence of quantum noise and its spectral dependence is mathematically
encapsulated in the quantum noise spectral density, described in terms of the
expectation value of quadrature field operators as
$S_{a}(\omega)=\frac{1}{2K_{a}(\omega)}\langle\frac{1}{2}\\{X_{\phi_{L}}^{out},X_{\phi_{L}}^{out}\\}\rangle=\frac{h^{2}_{SQL}}{2}\left[\frac{(K_{a}(\omega)-\cot\phi_{L})^{2}+1}{K_{a}(\omega)}\right]$,
where
$X_{\phi_{L}}^{out}=\frac{1}{\sqrt{2}}(a^{out}e^{-i\phi_{L}}+{a^{\dagger}}^{out}e^{i\phi_{L}})$
is the output quadrature expressed in terms of output annihilation (creation)
field operators: $a^{out}({a^{\dagger}}^{out})$. $K_{a}(\omega)$ is the
optomechanical coupling strength and $\phi_{L}$ is the angle of output
quadrature and $h_{SQL}$ is the standard quantum limit (SQL) of a free mass
for GW strain [6]. This scheme is currently being installed in LIGO-Virgo.
However, this approach is limited by the shot noise. Also, the response
quadrature is extremely narrow and highly dependent on the operational
frequency [6]. The shot noise can be reduced by brutally increasing the power
of the pump laser, but, that increases the QBA noise, as two noises are
correlated. The QBA noise becomes significant when the measurement time goes
long, i.e. it dominates at lower frequencies (below $\approx$ 100 Hz), as QBA
noise spectral density is proportional to the susceptibility of the massive
suspended mirror ($\chi_{M}=-1/\omega^{2}$, where $\omega$ is the reference
frequency in homodyne detection). In addition to above two dominant quantum
noises the low-frequency band is also affected by seismic, electronic, and
thermal noises.

A further improvement in increasing the sensitivity is done by using squeezed
light as a probe. Using this idea, another scheme has been in a proposal for
noise reduction, by injecting squeezed vacuum as an input
($S_{a}(\omega)=\frac{h^{2}_{SQL}}{2}\left[e^{-2r}K_{a}(\omega)+\frac{e^{2r}}{K_{a}(\omega)}\right]$,
where $r$ is the squeezing parameter) [7, 6]. Even though the input quadrature
is frequency-dependent, the scheme has been able to reduce either the shot or
QBA noise, but at the cost of an equivalent increment of each other.
Therefore, the major problem still remains the reduction of both noises,
simultaneously. Working in this direction, recently a highly interesting
hybrid quantum scheme has been proposed, introducing a concept of the negative
mass, which involves an auxiliary system that consists of an atomic spin
ensemble [1, 8]. Here, a two-mode squeezed vacuum (TMSV) is sent to the spin-
optomechanical hybrid system as an input, and the output response of the
hybrid system is measured through heterodyne detection. This combined
measurement suppresses both the shot noise and QBA noise by a factor $\cosh
2r$ (r is the squeezing amplitude), by back-action evading (BAE) measurements.
Even though that this scheme has triggered high interests in the design of GW
detectors, it faces severe implementation issues. The major issue is to design
such a negative mass atomic spin system working at low frequencies and
bandwidth so that both the susceptibilities of the spin system and mechanical
mirror can match perfectly each other. The problem becomes more lethal,
especially, at lower frequencies as the susceptibilities are more likely to
deviate from each other [1, 8].

To overcome these problems, here, we propose a new idea of introducing two
cavities with mutually interlocked mirrors moving in opposite directions. We
show, that by this design, which we call a negative radiation pressure
coupling, one can very effectively suppress both the shot and QBA noise,
simultaneously. This approach is based on geometrical considerations and it
omits the auxiliary spin system [1]. Similarly to the spin-cavity hybrid
system, the proposed scheme thus suppresses both noises simultaneously but
with much higher efficiency, which we discuss below. The scheme is
significantly simpler to implement practically, by modifying the currently
well-developed mirror hanging and suspension systems, such as in [9]. The
proposed principle is depicted in Fig. 1(i). According to this scheme two
photon flux are sent to two cavities which bounces back from same end mirror.
The end mirror is constructed by such a way that it is double faced so each
part of the split beam arrives to the opposite face of the mirror. By this way
the back action noise is compensated and we called this principle the negative
radiation pressure coupling. However as this geometry is practically difficult
to realize GW signal, instead, we work out further this principle by involving
two cavities shown in Fig. 1(ii). In this case, a second cavity is introduced
by bringing another interferometer alongside the primary (main) one and
attaching and suspending the end mirrors of a both cavities on a joint pivot,
so that both mirrors can move in opposite directions with respect to each
other under the influence of QBA and in the same direction ( due to the pivot
movement) under the influence of GW. Our scheme proves to be significantly
more efficient at low frequencies, especially when it comes to suppressing the
QBA noise. Besides, our model is also capable of suppressing the thermal
noise, which turns out to be an extra benefit of our scheme. The system works
by injecting TMSV generated throughout parametric down conversions (PDC) as
explained in detail below (Fig. 1(iii)).
In some other approaches, local readout based upon a dual laser beam has been
suggested in the context of the increment of the sensitivity of the GW
detector [10]. However, though this concept could remove the QBA noise from
the front mirror, the secondary laser beam does not resonate in the arm
cavities, and so does not handle the noise produced by the end mirror. In
another approach, a dual beam optical spring has been proposed to increase the
sensitivity of the GW detector, through modifying mechanical susceptibility by
tuning the power, detuning frequency and bandwidth which effectively acts as a
negative inertia [11]. However, this approach did not lead to suppressing QBA
or the thermal noise but rather amplifying the signal. Another proposition,
that has also been experimentally tested is based on injecting squeezed
vacuum, frequency-dependent output squeezing [12] but the suppression is the
same as for the spin-optomechanical hybrid scheme [1] and effective in a
comparatively narrower range of frequency. However, our work highly
efficiently suppresses both quantum noises and in addition reduces thermal
noise simultaneously.

In this letter, we start with a brief description of our double cavity
mechanical system, followed by the method of transferring the input TMSV into
the cavities and engineering the light-matter interaction to obtain squeezed
output below SQL. The investigation ends up with a numerical analysis to see
the scheme potential in the search for future implementations on advanced GW
interferometer installations.

Figure 1:  (i) Schematic diagram of the principle of the negative radiation
pressure realized by sharing a common two-faced mechanical mirror. However,
for practical realization, we replace this configuration with a scheme (ii)
composed of two mirrors supported by a joint pivot. (ii) Implementation of the
optomechanical double cavity, on a realistic GW interferometer; where the end
mirrors are attached and suspended on a pivot so that they can move together
in opposite directions. (iii) Block diagram of the double cavity BAE
measurement setup, fed by input TMSV generated through the PDC process, and
the outputs are detected through spatially separated heterodyne detection.
(iv) The top panel image shows the movement of the end mirrors on a pivot P
for thermal noise and quantum back action noise and makes the end mirrors move
in opposite directions cancelling this noise. However, at the same time, the
pivot P itself moves with respect to the front mirrors under the influence of
GW tidal force with both end mirrors in the same direction in the bottom
(lower panel) .  Figure 2:  The spectral densities of quantum noise for
$r=1.7,L_{a}=4000m,m=40kg,\kappa_{a}/2\pi=500Hz,\Theta_{a}/(2\pi)^{3}=100^{3}s^{-3}$.
In both cases, 2.5% of input losses and 2.5% of output losses are assumed. In
addition, for both optical cavities and spin system 0.01% of intracavity
roundtrip losses are assumed.

##  II Model

As our dual cavity optomechanical system shares a common pivot (see Fig.
1(i)), the increment of the length of one cavity created by the movement of
the mirror decreases the length of the other. This brings opposite signs to
the radiation pressure couplings between the mechanical oscillator and two
optical cavities. However, under the influence of GW tidal force, both the
cavities stretch or squeeze together simultaneously. Therefore, when it
applies on realistic interferometer in Fig. 1(ii), under the tidal GW force,
the pivot moves synchronously inducing the same length changes in both
cavities (Fig. 1(iv)).

The Hamiltonian of the entire system can be then described as:

$\displaystyle\hat{H}$
$\displaystyle=\hbar\omega_{a}a^{\dagger}a+\hbar\omega_{b}b^{\dagger}b+\frac{p^{2}}{2m}+\frac{m\omega^{2}_{m}x^{2}}{2}-\hbar\frac{\omega_{a}}{L_{a}}xa^{\dagger}a$
$\displaystyle+$
$\displaystyle\hbar\frac{\omega_{b}}{L_{b}}xb^{\dagger}b+i\hbar\left(a^{\dagger}E_{a}e^{-i\omega_{L_{a}}t}+b^{\dagger}E_{b}e^{-i\omega_{L_{b}}t}-h.c.\right)$
(1)

where $L_{a}$ and $L_{B}$ are the lengths, $\omega_{a}$ and $\omega_{b}$ are
the frequencies, and $a(a^{\dagger})$ and $b(b^{\dagger})$ are the
annihilation(creation) operators of the cavity modes A and B, respectively.
$x$ and $p$ are position and momentum operators of the mechanical mode. The
driving field with strength $(E_{a,b})$ and with frequencies
$\omega_{L_{a},L_{b}}$ is applied on optomechanical cavities, which in turn
determines the normalized optical power
$\Theta_{a,b}=\frac{8\omega_{L_{a,b}}I_{a,b}}{mcL_{a,b}}$ in the linear
coupling regime. $m,\omega_{m}$ are the mass and oscillation frequency of the
mechanical resonator, respectively. The mechanical resonator and both the
optical cavities interact continuously with their corresponding bosonic
thermal bath which is of Markovian nature, and can be approximated as having a
flat and infinite spectral density [13]. Our scheme advances a previous very
interesting concept that was using an active feedback control to lock the
motion of a mirror [14, 15]. In this case, another photon beam impinging on
the main interferometer mirror is fed through a small control mirror providing
the feedback. The external feedback force is optimized to reduce
optomechanical back action, which is controlled by the output response of
cavity B. However, this concept is limited by the shot noise. Unlike the
feedback control, our model, therefore, remains fundamentally different, as we
used two cavities to generate the negative radiation pressure and TMSV as
input to both cavities so that both the shot and QBA noise can be suppressed
at the output. Our scheme has thus an important advantage of suppressing both
the shot and QBA noise, simultaneously and to a degree significantly higher
than another concept of negative mass [1]. In addition, our concept also
reduces the thermal noise on mechanical mirrors at the output, which is a
significant problem in the realistic construction of the GW interferometers.
For example, approaches towards the reduction of thermal noise are being
developed using low thermal noise coatings, but these are still rather far
from practical realization. Here, with a rather simple optical concept, we can
suppress these noises, in addition to back action noise and the shot noise.

Considering the interferometer tuned on resonance and without taking optical
losses into account (see Appendix A for details), each of the outputs of the
interferometer are measured in their phase quadrature $(\phi_{L}=\pi/2)$
individually by setting up a homodyne detector, presented by the output
quadratures as:

$\displaystyle Y_{a}^{out}$
$\displaystyle=e^{2i\beta_{a}}Y_{a}^{in}+K_{ab}X_{b}^{in}-K_{a}X_{a}^{in}$
(2a)
$\displaystyle+i\sqrt{2K_{a}}\frac{F^{S}_{a}+F^{th}}{F^{sql}}\,\,\text{for
A},$ $\displaystyle Y_{b}^{out}$
$\displaystyle=e^{2i\beta_{b}}Y_{b}^{in}+K_{ab}X_{a}^{in}-K_{b}X_{b}^{in}$
(2b)
$\displaystyle+i\sqrt{2K_{b}}\frac{F^{S}_{b}-F^{th}}{F^{sql}},\,\,\text{for
B}$

where
$K_{a,b}(\omega)=-e^{(2i\beta_{a,b})}\frac{2\kappa_{a,b}\Theta_{a,b}\chi_{M}}{\kappa_{a,b}^{2}+\omega^{2}}$
and $K_{ab}=\sqrt{K_{a}K_{b}}$, $\kappa_{a,b}$ is the rate of dissipation of
the corresponding cavity and $\beta_{a,b}=\arctan(\omega/\kappa_{a,b})$ .
$F^{th}$ is the thermal force acting on the mirror and $F^{S}_{a,b}$ is the
signal force on each cavity which is invoked by the gravitational wave. In the
above equation, the GW force acts via the pivot movement in the same direction
in both cavities, but the thermal as well as radiation pressure forces act in
opposite directions. $F^{sql}=\sqrt{\hbar m/\chi_{M}}$ is the standard quantum
limit of the force which in turn fixes the minimum limit of the spectral
density of the sum of the shot and QBA noises $(S^{sql}_{F}=\hbar
m\omega^{2})$

We probe two cavities of the optomechanical system with entangled TMSV as an
input signal which is generated by non-linear optical crystal ($\chi$) through
the PDC and measure the outputs parallel through heterodyne detection (Fig.
1(iii)). Considering $r$ as the arbitrary amplitude of the squeezing parameter
and fixing the phase to $\pi/2$ in this PDC process, so that the quadratures
$X^{in}_{a}$ and $Y^{in}_{a}$ remain uncorrelated and their spectral densities
become $e^{2r}/2$ and $e^{-2r}/2$. The regenerated entangled modes satisfy

$\displaystyle X^{in}_{a,b}$ $\displaystyle=\cosh rX^{\chi}_{a,b}+\sinh
rX^{\chi}_{b,a}$ $\displaystyle Y^{in}_{a,b}$ $\displaystyle=\cosh
rY^{\chi}_{a,b}-\sinh rY^{\chi}_{b,a},$ (3)

where $X^{\chi}_{a,b},Y^{\chi}_{a,b}$ are the amplitude and phase quadratures
of two independent vacuum modes, probed into the nonlinear crystal.

To suppress both the shot and QBA noises, we express the optomechanical
coupling strengths to be $K_{b}=WK_{a}$, where $W$ is a weight factor.
Hereafter, we parameterize the weight factor and minimize the quantum noise
spectra with respect to it, which imposes a condition $K_{b}=(\tanh
2r)^{2}K_{a}$, and therefore, $K_{ab}=(\tanh 2r)K_{a}$. Therefore, the
condition mathematically provides the maximum suppression of the quantum
noise. This condition can be easily achieved by tuning the strength of the
pump lasers. This leads to obtaining the combined output measurement, weighted
by $\tanh 2r$ on the cavity $B$, as

$\displaystyle Y_{a}^{out}$ $\displaystyle+Y_{b}^{out}\tanh
2r=e^{2i\beta_{a}}Y_{a}^{in}+e^{2i\beta_{b}}Y_{b}^{in}\tanh 2r$
$\displaystyle-\frac{K_{a}}{\cosh^{2}2r}(X_{a}^{in}-\tanh 2rX_{b}^{in})$
$\displaystyle+\frac{i\sqrt{2K_{a}}}{\cosh^{2}2r}\frac{F^{th}}{F^{sql}}+i\sqrt{2K_{a}}\frac{F^{S}_{a}+\tanh^{2}2rF^{S}_{b}}{F^{sql}},$
(4)

which corresponds to the suppression of both shot and QBA noises by the
factors $\cosh 2r$ and $\cosh^{5}2r$, respectively
($S(\omega)=\frac{1}{2K_{a}(\omega)}\langle\frac{1}{2}\\{Y_{a}^{out}+Y_{b}^{out}\tanh
2r,Y_{a}^{out}+Y_{b}^{out}\tanh 2r\\}\rangle$). In addition, the first element
of the third line of the Eq. (II) shows that the scheme is also been able to
suppress the thermal noise by $\cosh^{4}2r$. Besides, as discussed before, the
structure of the double cavity GW interferrometer in Fig. 1(ii) ensures both
the end mirrors, including the suspension pivot, move same direction at a time
under the influence of GW tidal force (Fig. 1(iv)), which leads to satisfying
$F^{S}_{a}=F^{S}_{b}$, for an equal length of the cavities A and B. This
causes an enhancement in the balanced detection of GW signal with a factor
$(1+\tanh^{2}2r)^{2}$. Note that optical losses are not taken into account so
far in this theoretical model, which can easily be introduced as it is shown
in [1]. The modifications of input/output formalism in the presence of losses,
and the associated balancing conditions to obtain BAE measurements, are given
in Appendix A. While evaluating the efficiency of this scheme through
numerical estimation in the next section, we included several possible kinds
of optical losses corresponding to a realistic scenario.

In order to implement the scheme in realistic cavity optomechanical systems,
one can make the cavities as close to identical to each other, by engineering
their finesse, cavity resonance frequencies, and the single photon coupling to
the mechanical motion. As both cavities are stabilized by driving pump of the
same frequencies ($\omega_{L_{a}}=\omega_{L_{b}}$), the deviation of the
cavity parameters from their identical behavior can be adjusted by properly
tuning the intensities of the pump lasers so that the balancing condition
$K_{b}=(\tanh 2r)^{2}K_{a}$ remains uncompromised.

##  III Numerical Estimation

Using numerical simulations, we determined both the shot and QBA noise
together and compare them to the well-balanced hybrid spin-optomechanical
system, proposed in [1, 8], to visualize the efficiency of the scheme. In this
simulation, as explained above, we consider the cavity B to be identical to
the cavity A. Keeping in mind the possibility of a practical implementation,
we have chosen modeling parameters corresponding to those of the LIGO
interferometer and a realistic GW detector [1]. As anticipated, Fig 2 shows
that the shot and the QBA noise together are limited by the SQL for a simple
cavity optomechanical system. However, in the case of the balanced hybrid
spin-optomechanical system (for a situation when the susceptibility of atomic
spin and mechanical mirror are perfectly matched), for input squeezing $r=1.7$
($\approx 15$ db) the fluctuation goes below SQL and the sensitivity gain is
around 7.5 db across the entire frequency range of interest. As seen in Fig.
2, our double cavity optomechanical system scheme, reducing both noises, can
suppress the fluctuation below SQL for a much larger range of frequencies. As
anticipated, the reduction of the shot noise remains the same with the spin-
optomechanical hybrid system ($\approx 7.5db$), however the QBA noise
reduction increases for our scheme to 56 db (9 db for the hybrid spin-
optomechanical system). As the QBA noise dominates at a lower frequency, the
scheme is seen to be much more efficient, especially at lower frequencies (
$<$ 100 Hz).

##  IV Conclusion

We present a novel scheme suppressing at the same time both the shot and the
optomechanical back action noise, reaching beyond the standard quantum limit.
Compared to the schemes proposed before, our scheme appears to be much more
efficient for, firstly, not demanding any auxiliary spin system, therefore
there is no need to design a negative mass spin system with a lower Larmor
frequency and bandwidth. Secondly, it has the ability to suppress the QBA
noise (56 db) more than the spin-optomechanical hybrid scheme (9db), along
with the same rate of suppression of shot noise. Thirdly, it suppresses the
thermal noise simultaneously with good efficiency. We believe that our scheme
can bring remarkable advantages for designing the second generation GW
detectors, operating especially at lower frequencies.

###### Acknowledgements.

SA would like to thank Philippe Djorwe for his fruitful suggestions. The work
has been supported by the E-Test (EMR) project and European Union, Nuclear
singlet state in diamond for overcoming the standard quantum limit in
gravitational wave detectors (SingletSQL) GA no 101065991.

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##  Appendix A Inclution of input, output and internal losses

Taking into account optical losses, the Eq. (2a) can be modified as

$\displaystyle Y_{a,b}^{out}=$
$\displaystyle\sqrt{\eta_{a,b}^{out}}\big{[}R_{a,b}Y_{a,b}^{inp}+\sqrt{\eta_{a}^{int}\eta_{b}^{int}}K_{ab}X_{b,a}^{inp}-\eta_{a,b}^{int}K_{a,b}X_{a,b}^{inp}+T_{a,b}{v_{Y}}_{a,b}^{int}+\sqrt{\eta_{a,b}^{int}(1-\eta_{b,a}^{int})}K_{ab}{v_{X}}_{b,a}^{int}$
$\displaystyle-\sqrt{\eta_{a,b}^{int}(1-\eta_{a,b}^{int})}K_{a,b}{v_{X}}_{a,b}^{int}+i\sqrt{2\eta_{a,b}^{int}K_{a,b}}\frac{F^{S}_{a,b}\pm
F^{th}}{F^{sql}}\big{]}+\sqrt{1-\eta_{a,b}^{out}}{v_{Y}}_{a,b}^{out}$ (5)

where $R_{a,b}=\frac{2\eta_{a,b}^{int}\kappa_{a,b}}{\kappa_{a,b}-i\omega}-1$
and
$T_{a,b}=\frac{2\kappa_{a,b}\sqrt{\eta_{a,b}^{int}(1-\eta_{a,b}^{int})}}{\kappa_{a,b}-i\omega}$
and

$\displaystyle
X_{a,b}^{inp}=\sqrt{\eta_{a,b}^{inp}}X_{a,b}^{in}+\sqrt{(1-\eta_{a,b}^{inp})}{v_{X}}_{a,b}^{in}$
(6a) $\displaystyle
Y_{a,b}^{inp}=\sqrt{\eta_{a,b}^{inp}}Y_{a,b}^{in}+\sqrt{(1-\eta_{a,b}^{inp})}{v_{Y}}_{a,b}^{in}$
(6b)

are the effective incident light quadratures for the interferometer cavities,
${v_{X}}_{a,b}^{in}$ and ${v_{Y}}_{a,b}^{in}$, ${v_{X}}_{a,b}^{out}$ and
${v_{Y}}_{a,b}^{out}$, and ${v_{X}}_{a,b}^{int}$ and ${v_{Y}}_{a,b}^{int}$ are
the vacuum amplitude and phase noise operators associated with the input,
output, and internal optical losses of the interferometer cavities,
respectively, and $\eta_{a,b}^{inp},\eta_{a,b}^{out}$ and $\eta_{a,b}^{int}$
are their corresponding quantum efficiencies. Therefore, the cross-correlation
spectral densities of the input signals are

$\displaystyle
2S[X_{a}^{inp},X_{a}^{inp}]=2S[Y_{a}^{inp},Y_{a}^{inp}]=1+2\eta_{a}^{inp}\sinh^{2}r$
(7a) $\displaystyle
2S[X_{b}^{inp},X_{b}^{inp}]=2S[Y_{b}^{inp},Y_{b}^{inp}]=1+2\eta_{b}^{inp}\sinh^{2}r$
(7b)

$\displaystyle
2S[X_{a}^{inp},X_{b}^{inp}]=\sqrt{\eta_{a}^{inp}\eta_{b}^{inp}}\sinh 2r$ (7c)
$\displaystyle
2S[Y_{a}^{inp},Y_{b}^{inp}]=-\sqrt{\eta_{a}^{inp}\eta_{b}^{inp}}\sinh 2r$ (7d)

$\displaystyle
S[X_{a}^{inp},Y_{a}^{inp}]=S[X_{a}^{inp},Y_{b}^{inp}]=S[Y_{a}^{inp},X_{b}^{inp}]$
$\displaystyle=S[X_{b}^{inp},Y_{b}^{inp}]=0$ (7e)

Considering the input signal is a highly squeezed vacuum, i.e. very high value
of $r$ and therefore $e^{r}$ dominates in the Eq. (II), the impact of the
vacuum noise operators is negligible with respect to the signal. In order to
suppress QBA noise, it requires a modified weight factor
$\sqrt{\frac{\eta_{a}^{out}\eta_{a}^{inp}}{\eta_{b}^{out}\eta_{b}^{inp}}}\tanh
2r$ on the measurements of the output of cavity B, without putting any weight
on cavity A. It also requires a new balancing condition

$K_{b}=\frac{\eta_{a}^{int}\eta_{a}^{inp}}{\eta_{b}^{int}\eta_{b}^{inp}}\tanh^{2}2rK_{a},\hskip
5.69054ptK_{ab}=\sqrt{\frac{\eta_{a}^{int}\eta_{a}^{inp}}{\eta_{b}^{int}\eta_{b}^{inp}}}\tanh
2rK_{a}$ (8)

which suppresses the QBA noise by the factor
$\frac{\cosh^{5}2r}{\eta_{a}^{inp}}$

##  Appendix B Thermal force on mirror, pivot and coating

The thermal noise on the end mirror majorly has three different sources: the
Brownian motion of the individual mirrors and the coatings on the mirrors and
the thermal oscillations of the pivot. The thermal forces on individual
mirrors which have already been taken into account in Eq. (II), are seen to be
suppressed by the factor $\cosh^{2}2r$. Here, we discuss the possibilities of
two other sources of noise. The thermal force acting on the suspension pivot
also contributes to the noise, which can be taken care of by considering the
signal force on each cavity modified by
$F^{S}_{a}=F^{S}_{b}=(F^{GW}+F^{th}_{pivot})$ where $F^{GW}$ and
$F^{th}_{pivot}$ are the forces by GW signal and thermal force on pivot,
respectively. This modifies the Eq. (II), as

$\displaystyle Y_{a}^{out}$ $\displaystyle+Y_{b}^{out}\tanh
2r=e^{2i\beta_{a}}Y_{a}^{in}+e^{2i\beta_{b}}Y_{b}^{in}\tanh 2r$
$\displaystyle-\frac{K_{a}}{\cosh^{2}2r}(X_{a}^{in}-\tanh
2rX_{b}^{in})+\frac{i\sqrt{2K_{a}}}{\cosh^{2}2r}\frac{F^{th}}{F^{sql}}$
$\displaystyle+i\sqrt{2K_{a}}\frac{(1+\tanh^{2}2r)(F^{GW}+F^{th}_{pivot})}{F^{sql}}$
(9)

which shows, even though the thermal forces acting on the individual mirror
are being suppressed by the factor $\cosh^{2}2r$, the thermal force on the
pivot has been increased by $1+\tanh^{2}2r$. The thermal force on the pivot is
dependent on temperature and frequency. Therefore, one way to decrease thermal
noise is by increasing the stiffness of its suspension $(k\propto\omega^{2}$
and $n_{th}\propto e^{-\beta\omega})$. However, the higher operational
frequency can affect the susceptibility $(\chi_{M})$. In particular, the
natural oscillation frequency and the damping of the suspension pivot has an
influence on the susceptibility as
$\chi_{M}=-\left(-\omega_{m}^{2}+\omega(\omega+i\eta/m)\right)^{-1},$ where
$\omega_{m},m$ and $\eta$ are respectively the natural frequency, mass and the
frictional coefficient of the mirror of the GW detector. In practical
scenarios, detectors are designed where the mechanical mirror is hung through
a multistage pendulum to obtain seismic isolation. For example, Virgo mirrors
are suspended via a long chain of pendulum, hosted in towers 10 meters tall,
therefore the natural frequency of the mirror is much lesser than the
frequency of interest $(\omega_{m}<<\omega)$. For example, in the case of the
E-TEST project [9], 0.23 Hz will be the frequency for the longitudinal and
transversal modes. Heavy masses of the mirrors (40 kg for LIGO and 100 kg
planned for Einstein telescope) will lead to suppressing the frictional losses
( by a high-quality factor). In order to make the detector efficient at low
frequencies ($\omega<100$ Hz) the influence of susceptibility on the the noise
should be minimal ($<<\omega$). Anticipating that the thermal noise
($n_{th}\propto e^{-\beta\omega}$) will be dealt with by designing the natural
frequency of the suspension and therefore the susceptibility, we calculate and
plot the quantum noise with and without considering motions of the suspension
pivot in Fig. 3 . From this plot we can deduce that the changes of quantum
noise, considering the mechanical parameters of the mirrors and suspension
appear to be minimal and can be easily be neglected. Further on, the thermal
noise can be additionally reduced by controlling temperature, e.g. by a
cryogenic cooler.

Figure 3:  The spectral densities of quantum noise considering without and
with the motion of pivot. The natural oscillation frequency and the frictional
coefficient are considered as $\omega_{m}=0.25,\eta/m=0.05$ Hz. The rest of
all other parameters are considered the same with Fig. 2


# Chiral symmetry breaking through spontaneous dimerization in kagomé metals

Riccardo Ciola  Institute for Theoretical Physics and Center for Extreme
Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC
Utrecht, The Netherlands  Kitinan Pongsangangan  Institute for Theoretical
Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht
University, Princetonplein 5, 3584 CC Utrecht, The Netherlands  Ronny Thomale
Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074
Würzburg, Germany  Lars Fritz  Institute for Theoretical Physics and Center
for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein
5, 3584 CC Utrecht, The Netherlands

###### Abstract

Due to an uprise in the variety of candidate compounds, kagome metals have
recently gained significant attention. Among other features, kagome metals
host Dirac cones as a key band structure feature away from half filling, and
potentially yield an exceptionally large fine structure, beyond values found
in other 2D Dirac materials such as graphene. We investigate the possibility
of chiral symmetry breaking in kagome metals. Based on a heuristic lattice
model, we determine the critical coupling strength and the ordering pattern by
means of a Schwinger-Dyson mean-field analysis. As the leading instability we
identify a dimerization pattern which spontaneously opens an excitation gap at
the Dirac point and breaks the chiral symmetry.

##  I Introduction

Recent years have seen a plethora of condensed matter research activities in
the field of Dirac- and Weyl-type systems, where effective Lorentz covariance
is preserved due to the linear dispersion relation in the crystal. One of the
first systems that gained massive attention is graphene [1], but soon
thereafter many more followed. Dirac systems exhibit a linear bandcrossing at
isolated points in the Brillouin zone, which is locally protected by chiral
and time reversal symmetry. As a result, Dirac systems for two spatial
dimensions and higher show a semi-metallic density of states [1, 2]. A
discussion that predates graphene, and has its counterpart in high energy
physics [3], centers around the possibility to spontaneously break the chiral
symmetry in a two-dimensional layer of graphite, first formulated by
Khveshchenko [4, 5]. In contemporary condensed matter terms, this question
translates into whether graphene at zero temperature is an (excitonic)
insulator or not. The driving force behind the spontaneous gap formation in
this scenario is Coulomb interaction whose strength is controlled by the
dimensionless fine structure constant

$\displaystyle\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}v_{F}}\;.$ (1)

Here, $e$ is the electric charge of the electron, $\epsilon_{0}$ the
dielectric constant of vacuum, $\epsilon_{r}$ the relative permittivity of the
medium, and $v_{F}$ the Fermi velocity of the electronic system. In typical
condensed matter systems, $\alpha$ is a quantity that is below unity.

Figure 1: Schematic phase diagram of two dimensional Dirac theories with long-
range Coulomb interactions. The $x$-axis denotes the number of Dirac fermion
flavors whereas the $y$-axis shows the value of the dimensionless coupling
constant $\alpha$ (Eq. (1)). For $N$ smaller than $N_{c}$, there exists a
critical coupling strength $\alpha_{c}(N)$ where a quantum phase transition
between a semimetal and an insulator takes place.

The problem of $N$ flavors of two-dimensional massless Dirac fermions with an
effective speed $v_{F}$ coupled via long-range interactions has been
investigated by a plethora of methods, ranging from renormalization group [6]
to full lattice simulations using quantum Monte Carlo (for Dirac fermions this
is possible without the sign problem due to the particle-hole symmetry of the
system) [7]. There is a curious dichotomy in this field: perturbation theory
based methods, including large-N approaches, suggest the irrelevance of
Coulomb interaction irrespective of the value of the fine structure constant;
on the other hand, strong coupling methods suggest the existence of a critical
$\alpha_{c}$ beyond which there is spontaneous gap formation due to
spontaneous chiral symmetry breaking, at least if $N$ is not too large. The
finding of the strong coupling methods can be summarized in the schematic
phase diagram shown in Fig. 1. The $x$-axis is labeled with $N$, which is the
number of Dirac fermion flavors. In the case of graphene, this corresponds to
$N=4$, which includes valley ($K$ and $K^{\prime}$) and spin ($\uparrow$ and
$\downarrow$). The $y$-axis denotes the strength of the Coulomb interaction,
$\alpha$. There is a line where a quantum phase transition takes place. The
critical coupling $\alpha_{c}$ associated with it goes to $\infty$ at $N_{c}$.
This implies that there is a quantum phase transition from a semimetallic
phase to an insulator at a critical coupling $\alpha_{c}$ as long as
$N<N_{c}$. Depending on the calculational method, there is a large variation
of $\alpha_{c}$ for a given $N$ [8, 9, 10, 11].

In our work, we turn to kagomé metals as an intriguing instance of 2D Dirac
materials, and their potential to show the interaction-driven phenomenon of
chiral symmetry breaking. Kagomé metals have been investigated in a number of
contexts. Descending from the established domain of Mott phases in kagomé
layers as a host for quantum paramagnetism [12], kagomé metals such as FeSn
and Fe2Sn3 were found to exhibit itinerant magnetism and, in particular,
anomalous Hall responses [13, 14]. A host material for kagomé metals with a
symmetry unbroken Fermi liquid parent state was most recently found in AV3Sb5
(A=K,Cs,Rb) featuring Vanadium kagomé nets [15, 16]. At half filling, the
generic kagomé band structure is close to (possibly multiple) van Hove
singularities, and hence prone to exhibiting Fermi surface instabilities [17,
18, 19]. While the degree of experimental exploration is still at an early
stage, unconventional chiral charge order and superconductivity have already
been theoretically ascribed to AV3Sb5 [20, 21, 22, 23, 24], which might appear
possibly along with an onset of nematic order [16]. Reaching the Dirac cone
filling in a kagomé metal, however, in general is a challenging task.
Depending on the sign of the hybridization which dictates the flat kagome band
to locate above (below) the dispersive bands, a filling of 1/3 (2/3) has to be
reached. While as of yet one is still short of an experimental realization,
Sc-substituted Herbertsmithite has been identified as a promising candidate
material [25]. In the context of establishing the notion of turbulent electron
hydrodynamics, Sc-Herbertsmithite has subsequently been the pivotal starting
point to investigate the fine structure constant inherent to kagomé Dirac
materials, and was shown that it potentially has a very large fine structure
constant, up to three times larger than graphene [26]. This gives credence to
the assertion that if one were to look for strong correlation phenomena of
Dirac cones, kagomé Dirac materials appear as an eminently suited domain.

By employing a Schwinger-Dyson mean-field approach on the kagome lattice, we
show that if the chemical potential of a kagomé metal is tuned to its Dirac
point, there is an instability towards the spontaneous formation of a
dimerization pattern (related patterns have previously been discussed in Refs.
[27, 28]). This pattern breaks the sublattice symmetry and hence parity. As
such, it opens a gap in the single-particle excitation spectrum on the mean-
field level. We further find that the critical coupling for the dimerization
formation is $\alpha_{\rm{cr}}\approx 1.22$, which from theoretical
investigations for Sc-Herbertsmithite could be in realistic reach for tailored
kagomé candidate materials.

Our paper is organized as follows. In Section II, we introduce our kagome
model consisting of a generic tight-binding model supplemented by Coulomb
interactions. We then proceed to the mean-field treatment of the model, and
derive both the ordering pattern as well as the critical coupling in Section
III. In Section IV we conclude that kagome metals promise to be among the most
preferable systems to study spontaneous chiral symmetry breaking in Dirac
materails.

##  II Model

The starting point is the tight-binding model on the kagomé lattice. It has
the generic form

$\displaystyle
H=t\sum_{<i,j>,\sigma}\left(c_{i\sigma}^{\dagger}c^{\phantom{\dagger}}_{j\sigma}+c_{j\sigma}^{\dagger}c^{\phantom{\dagger}}_{i\sigma}\right)\;.$
(2)

The kagomé lattice has a three-site unit cell which we account for by
introducing the three component vector $\psi_{i}=(A_{i},B_{i},C_{i})^{T}$
which contains the wave-functions of one unit-cell. We furthermore introduce
the three lattice vectors $\vec{a}_{1}=a(2,0)^{T}$,
$\vec{a}_{2}=a(1,\sqrt{3})^{T}$, and $\vec{a}_{3}=a(-1,\sqrt{3})^{T}$ as well
as the nearest neighbor vectors $\vec{\delta}_{AB}=a(1,0)^{T}$,
$\vec{\delta}_{AC}=a/2(1,\sqrt{3})^{T}$, and
$\vec{\delta}_{BC}=a/2(-1,\sqrt{3})^{T}$.

Figure 2: (a) Kagomé lattice with a three site unit cell. The sites within the
unit cell are labelled $A$, $B$, and $C$. The vectors $\vec{a}_{1}$,
$\vec{a}_{2}$, and $\vec{a}_{3}$ are the elementary lattice vectors defined in
the main text. (b) The hexagonal first Brillouin zone of the kagomé lattice.

Assuming periodic boundary conditions we can solve the Hamiltonian (2) via
Fourier transform. The resulting Hamiltonian assumes the form

$\displaystyle
H=\sum_{\sigma}\sum_{\rm{1stB.Z.}}\psi^{\dagger}_{\sigma,\vec{k}}\mathcal{H}(\vec{k})\psi^{\phantom{\dagger}}_{\sigma,\vec{k}}$
(3)

with the Bloch Hamiltonian given by

$\displaystyle\mathcal{H}(\vec{k})=2t\left(\begin{array}[]{ccc}0&\cos\left(\vec{k}\cdot\vec{\delta}_{AB}\right)&\cos\left(\vec{k}\cdot\vec{\delta}_{AC}\right)\\\
\cos\left(\vec{k}\cdot\vec{\delta}_{AB}\right)&0&\cos\left(\vec{k}\cdot\vec{\delta}_{BC}\right)\\\
\cos\left(\vec{k}\cdot\vec{\delta}_{AC}\right)&\cos\left(\vec{k}\cdot\vec{\delta}_{BC}\right)&0\end{array}\right)\;.$
(7)

The spectrum of this system is given by

$\displaystyle E_{1}(\vec{k})$ $\displaystyle=$ $\displaystyle
t\left(1+\sqrt{3+2\sum_{i}\cos\left(2\vec{k}\cdot\vec{\delta}_{i}\right)}\right)$
(9) $\displaystyle E_{2}(\vec{k})$ $\displaystyle=$ $\displaystyle
t\left(1-\sqrt{3+2\sum_{i}\cos\left(2\vec{k}\cdot\vec{\delta}_{i}\right)}\right)$
(10) $\displaystyle E_{3}(\vec{k})$ $\displaystyle=$ $\displaystyle-2t$ (11)

with $\vec{\delta}_{i}=\vec{\delta}_{AB},\vec{\delta}_{AC},\vec{\delta}_{BC}$.
The corresponding band structure is shown in Fig. 3.

Figure 3: Band structure of a kagomé nearest-neighbor tight-binding model. The
band touching points at $2/3$-filling and their vicinities resemble the Dirac
cones of graphene.

This Hamiltonian harbors a flat band at energy $-2t$ as well as a Dirac type
band-structure reminiscent of graphene. The Dirac points are located at
momenta $\vec{K}=(2\pi/3,0)^{T}$ and $\vec{K}^{\prime}=(-2\pi/3,0)^{T}$. In
the following we show how this Dirac structure emerges from the lattice
Hamiltonian.

###  II.1 Low Energy Expansion

In graphene, it is sufficient to expand the Bloch Hamiltonian around one of
the two independent Dirac point in order to obtain the linear Dirac cone
structure that determines the massless electron dispersion. For the kagomé
lattice, the tight-binding model, Eq. (7), does not show the same simple
structure and one has to explicitly decouple the flat band. If we expand the
Bloch Hamiltonian up to the linear order in $\vec{K}+\vec{k}$, we get

$\displaystyle\mathcal{H}(\vec{k})\simeq-t\left(\begin{array}[]{ccc}0&1+\sqrt{3}ak_{x}&-1+\frac{\sqrt{3}}{2}ak_{x}+\frac{3}{2}ak_{y}\\\
1+\sqrt{3}ak_{x}&0&-1+\frac{\sqrt{3}}{2}ak_{x}-\frac{3}{2}ak_{y}\\\
-1+\frac{\sqrt{3}}{2}ak_{x}+\frac{3}{2}ak_{y}&-1+\frac{\sqrt{3}}{2}ak_{x}-\frac{3}{2}ak_{y}&0\end{array}\right)\,.$
(15)

We take a set of orthogonal eigenvectors of the linear Hamiltonian
$\mathcal{H}(\vec{k}=0)$ corresponding to the different bands to decouple the
flat band and isolate the effective Dirac point theory. The proper
transformation matrix is given by

$\displaystyle U=\begin{bmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 1\\\
1\end{pmatrix}\ ,\ \frac{1}{\sqrt{6}}\begin{pmatrix}2\\\ -1\\\ 1\end{pmatrix}\
,\ \frac{1}{\sqrt{3}}\begin{pmatrix}-1\\\ -1\\\ 1\end{pmatrix}\end{bmatrix}\
.$ (16)

which transforms $U\ h(\vec{k})\ U^{-1}=\mathcal{H}(\vec{k})$ with

$\displaystyle
h(\vec{k})=t\left(\begin{array}[]{ccc}1-\frac{\sqrt{3}}{2}ak_{x}+\frac{3}{2}ak_{y}&-\frac{3}{2}ak_{x}-\frac{\sqrt{3}}{2}ak_{y}&\frac{3}{2\sqrt{2}}ak_{x}+\frac{\sqrt{3}}{2\sqrt{2}}ak_{y}\\\
-\frac{3}{2}ak_{x}-\frac{\sqrt{3}}{2}ak_{y}&1+\frac{\sqrt{3}}{2}ak_{x}-\frac{3}{2}ak_{y}&\frac{\sqrt{3}}{2\sqrt{2}}ak_{x}+\frac{3}{2\sqrt{2}}ak_{y}\\\
\frac{3}{2\sqrt{2}}ak_{x}+\frac{\sqrt{3}}{2\sqrt{2}}ak_{y}&\frac{\sqrt{3}}{2\sqrt{2}}ak_{x}+\frac{3}{2\sqrt{2}}ak_{y}&-2\end{array}\right)\;.$
(20)

In order to better identify the Dirac theory, we make the following coordinate
transformation

$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(k_{x}-\sqrt{3}k_{y}\right)\;,$ $\displaystyle
k_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{3}k_{x}+k_{y}\right)\;.$ (21)

which has a Jacobian of unity. We furthermore introduce $v_{F}=\sqrt{3}ta$ as
the effective Fermi velocity. This allows to rewrite

$\displaystyle
h(\vec{k})=t\left(\begin{array}[]{ccc}t-v_{F}k_{1}&-v_{F}k_{2}&\frac{v_{F}}{\sqrt{2}}k_{2}\\\
-v_{F}k_{2}&t+v_{F}k_{1}&\frac{v_{F}}{\sqrt{2}}k_{1}\\\
\frac{v_{F}}{\sqrt{2}}k_{2}&\frac{v_{F}}{\sqrt{2}}k_{1}&-2t\end{array}\right)\;.$
(25)

The $2\times 2$ matrix in the upper-left corner resembles the structure of the
Dirac theory of graphene which we are going to show now. The other terms
couple the effective Dirac theory to the flat band of energy $-2t$. To
decouple the flat band, we integrate it out and derive an effective low energy
theory. We do this perturbatively order by order, since the energy of the flat
band is far removed from the Dirac point. For reasons of conciseness, we only
sketch the procedure here: The leading order corrections are of the order
$k_{1}^{2}/(3t)$, $k_{2}^{2}/(3t)$, and $k_{1}k_{2}/(3t)$ and can consequently
safely be ignored in the linear limit. To conclude, to leading order we can
ignore the rest and focus on the $2\times 2$ submatrix, the upper left block:

$\displaystyle\mathcal{H}^{\rm{Dirac}}(\vec{k})$ $\displaystyle=$
$\displaystyle t\
\mathbb{1}+v_{F}\left(k_{1}\sigma_{z}+k_{2}\sigma_{x}\right)\ .$ (26)

The effective dispersion of this Hamiltonian is given by the energy
eigenvalues $E_{\pm}=t\pm v_{F}\sqrt{k_{1}^{2}+k_{2}^{2}}=t\pm
v_{F}\sqrt{k_{x}^{2}+k_{y}^{2}}$, where the last step is easy to verify.

The linearized Hamiltonian (26), valid around $\vec{K}$, provides an excellent
starting point for the discussion of potential mass terms. Excluding the
possibility of hybridizing spin or valleys (for instance through disorder or
short range interactions) for now, we concentrate on the two-dimensional
matrix. The electron momenta are locked to the Pauli matrices $\sigma_{x}$ and
$\sigma_{z}$, which leaves $\sigma_{y}$ as the ’free’ Pauli matrix. A term
that cannot be eliminated by a momentum shift consequently is given by
$\Delta\sigma_{y}$. This leads to the dispersion
$E_{\pm}=t\pm\sqrt{v_{F}^{2}\vec{k}^{2}+\Delta^{2}}$ and consequently a gap of
size $2|\Delta|$ at the Dirac points.

###  II.2 Energy gap through dimerization

In a next step, we transform the mass term back to the full lattice problem
which allows for a straightforward physical interpretation. Using the unitary
matrix $U$, we transform the mass term back to the $3\times 3$ lattice problem
and end up with

$U\ \begin{pmatrix}\Delta\sigma_{y}&\begin{matrix}0\\\ 0\end{matrix}\\\
\begin{matrix}0&0\end{matrix}&0\end{pmatrix}\
U^{-1}=\frac{i\Delta}{\sqrt{3}}\begin{pmatrix}0&1&1\\\ -1&0&-1\\\
-1&1&0\end{pmatrix}\ .$ (27)

Figure 4: Real space representation of the dimerization pattern. Thick line
corresponds to an enhanced hopping element whereas broken line corresponds to
a reduced one, compare Eq. (32).

This term corresponds to an inequivalence in the hopping between the
sublattices inside a unit cell. It can therefore be identified with a
dimerization pattern. In view of our later lattice analysis it is useful to
compare it with a lattice version of a dimerization pattern [28, 27] which is
shown in Fig. 4. After Fourier transformation, the dimerization matrix reads

$\displaystyle\delta\mathcal{H}(\vec{k})=$ $\displaystyle\frac{2\Delta
i}{3}\left(\begin{array}[]{ccc}0&\sin(\vec{k}\cdot\vec{\delta}_{AB})&\sin(\vec{k}\cdot\vec{\delta}_{AC})\\\
-\sin(\vec{k}\cdot\vec{\delta}_{AB})&0&\sin(\vec{k}\cdot\vec{\delta}_{BC})\\\
-\sin(\vec{k}\cdot\vec{\delta}_{AC})&-\sin(\vec{k}\cdot\vec{\delta}_{BC})&0\end{array}\right)\;.$
(31) (32)

At the Dirac point $\vec{K}$, we reproduce Eq. (27) whereas at
$\vec{K}^{\prime}$ we obtain a relative sign (this is required by time-
reversal symmetry, otherwise the system would have a finite Chern number and
consequently be a Chern insulator). The total Bloch Hamiltonian reads
$\mathcal{H}_{\rm{tot}}(\vec{k})=\mathcal{H}(\vec{k})+\delta\mathcal{H}(\vec{k})$
and has the spectrum

$\displaystyle E_{1,2}(\vec{k})$ $\displaystyle=$ $\displaystyle
t\left(1\pm\sqrt{3\left(1+\frac{2}{3}\sum_{j}\cos(2\vec{k}\cdot\vec{\delta}_{j})\right)+\frac{2\Delta^{2}}{3t^{2}}\left(1-\frac{1}{3}\sum_{j}\cos(2\vec{k}\cdot\vec{\delta}_{j})\right)}\right)\;,$
$\displaystyle E_{3}(\vec{k})$ $\displaystyle=$ $\displaystyle-2t\;.$ (33)

We observe that this dimerization process does not influence the flat band at
all.

Figure 5: Band structure of a kagomé nearest-neighbor tight-binding model with
a dimerization term ($t=1$ and $\Delta=0.15$). The band touching points are
removed in the presence of the dimerization term.

To conclude: we found that if we introduce a mass function proportional to the
dimerization matrix, Eq. (32), we will be able to open an energy gap at the
Dirac point.

###  II.3 Effective interaction

In non-relativistic electronic system ($v_{F}\ll c$ with $c$ being the speed
of light), Coulomb interaction assumes the form

$\displaystyle\frac{e^{2}}{8\pi\epsilon_{0}\epsilon_{r}}\int
d^{d}rd^{d}r^{\prime}\rho(\vec{r})\frac{1}{|\vec{r}-\vec{r}^{\prime}|}\rho(\vec{r}^{\prime})$
$\displaystyle=\frac{\alpha v_{F}}{2}\int
d^{d}rd^{d}r^{\prime}\rho(\vec{r})\frac{1}{|\vec{r}-\vec{r}^{\prime}|}\rho(\vec{r}^{\prime})\;,$
(34)

where $\rho(\vec{r})$ is the electron density at site $\vec{r}$. The measure
for the strength of the electronic interaction is the dimensionless fine-
structure constant $\alpha$ which was introduced in Eq. (1). For typical Dirac
systems it is a quantity $\mathcal{O}(1)$, since $v_{F}/c\approx
10^{-2}-10^{-3}$ and $\epsilon_{r}\approx 2-10$. Typically, in finite density
electronic systems, Coulomb interaction is dynamically screened through
particle-hole excitations. Technically, this can described by the random phase
approximation (RPA). RPA resums the whole diagrammatic series shown in Fig. 6
(a) and can formally be justified in the limit of a large number of fermion
flavors. The double wiggly (single) line denotes the effective RPA potential
(unscreened potential), $V^{\rm{RPA}}\left(\vec{k},\omega\right)$, whereas the
straight directed line is the fermion propagator. The effective RPA potential
assumes the form

$\displaystyle
V^{\rm{RPA}}\left(\vec{k},\omega\right)=\frac{1}{V^{-1}_{C}(\vec{k})-\Pi\left(\vec{k},\omega\right)}\;,$
(35)

where $\Pi\left(\vec{k},\omega\right)$ is the polarization function and the
bare Coulomb interaction in two dimensions is given by

$\displaystyle V_{C}(\vec{k})=\frac{e^{2}}{2\epsilon}\frac{1}{|\vec{k}|}\;.$
(36)

Figure 6: (a) Diagrammatic representation of the RPA potential equation. The
double wiggly line is the renormalized Coulomb potential, whereas the single
wiggly line is the bare Coulomb potential. The simple line is the fermion
propagator and the closed bubble the polarization function. (b) The self-
consistent Schwinger Dyson equation. The double line is the full fermion
propagator.

The polarization function at zero temperature can be expressed in terms of the
fermion Green function according to

$\displaystyle\Pi(\vec{k},\omega)=-2\int\frac{d\nu}{2\pi}\int\frac{d^{2}q}{(2\pi)^{2}}\;{\rm{tr}}\;G_{0}(\vec{k}+\vec{q},\omega+\nu)G_{0}(\vec{q},\nu)\;.$

In the above expression, the factor of two is related to the spin degree of
freedom. We use the Green function of the fermions in Eq. (II.3) whose inverse
is given by

$\displaystyle
G^{-1}_{0}(\vec{k},\omega_{n})=-i\omega_{n}+\mathcal{H}(\vec{k})\;,$ (38)

where $\mathcal{H}(\vec{k})$ is the Bloch Hamiltonian of the kagomé system in
absence of dimerization and $\omega_{n}=(2n+1)\pi T$ is the fermionic
Matsubara frequency with $n$ being an integer.

In pure two-dimensional Dirac systems at zero temperature, it is well known
that the polarization function in terms of Matsubara frequencies follows

$\displaystyle\Pi_{\rm{2D-Dirac}}(\vec{k},\omega)\propto\frac{\vec{k}^{2}}{\sqrt{v_{F}^{2}\vec{k}^{2}+\omega^{2}}}\;,$
(39)

which shows the absence of screening. We could not find a closed analytical
expression for Eq. (II.3) for the full three band system, so we had to resort
to a numerical solution instead. In order to simplify the representation and
also in view of the later treatment, we use the on-shell condition, meaning
$\omega=v_{F}k$, which is valid for low values of $\omega$. The result of the
numerical integration is shown in Fig. 7. In the respective plot, the chemical
potential is tuned to the Dirac point. Importantly, it shows the
characteristic $\propto|\vec{k}|/v_{F}$ at low values of $|\vec{k}|$ behavior
also observed in the two-dimensional Dirac theory, see Eq. (39). We checked
that the slope agrees with the inverse value of the effective Fermi velocity
relevant for the lattice system.

Figure 7: Polarization function for the Kagome lattice within the on-shell
condition.

##  III The Schwinger-Dyson equation

###  III.1 Meanfield analysis

The Schwinger-Dyson mean-field technique is a self-consistent technique which
allows to estimate the critical coupling of a (quantum) phase transition. On a
formal level, it can be derived from a Hubbard-Stratonovich decoupling of the
Coulomb interaction in the respective instability channel, in this case the
dimerization. It leads to a self-consistency equation of the type

$\delta\mathcal{H}(\textbf{k})=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{d\nu}{2\pi}\
V^{\rm{RPA}}(\vec{k}+\vec{p})G(\vec{p},\nu)\;,$ (40)

which is diagrammatically shown in Fig. 6 (b). In this expression,

$\displaystyle
G^{-1}(\vec{k},\omega)=-i\omega+\mathcal{H}(\vec{k})+\delta\mathcal{H}(\vec{k})\;,$
(41)

is the Green function of the system taking into account the presence of the
dimerization function. In Eq. (40), we have employed the on-shell condition
for the interaction potential $V^{\rm{RPA}}$, i.e., it is only a function of
$\vec{k}+\vec{p}$. The poles in the $\nu$-integral reside at

$\displaystyle\nu_{1,2}(\vec{p},\Delta)=-iE_{1,2}(\vec{p})\;,$ (42)

see Eq. (33). Interestingly, the flat band vanishes exactly from the gap
equation, leaving only the Dirac cones. This is to be expected since according
to Eq. (32) a finite dimerization leaves the flat band inert. We can now shift
the frequency according to the chemical potential $t$, namely
$\nu^{\prime}\rightarrow\nu+it$. In this way the poles become symmetric around
the real axis, i.e., $\nu^{\prime}_{\pm}=\nu_{\pm}-it$, and we can integrate
the Schwinger-Dyson Eq. (40) using the residue theorem. The self-consistent
equation for the gap becomes

$\displaystyle\Delta(\vec{k})=\int_{BZ}\frac{d\vec{p}}{(2\pi)^{2}}\ \
\frac{V^{\rm{RPA}}(\vec{k}+\vec{p})\Delta(\vec{p})}{2i\nu^{\prime}_{+}\left(\vec{p},\Delta(\vec{p})\right)}\
\frac{\sin(\vec{p}\cdot\vec{\delta}_{j})}{\sin(\vec{k}\cdot\vec{\delta}_{j})}\
.$ (43)

It is important to note that compared to Eq. (32) we allow for a mean-field
parameter $\Delta$ that is a function of $\vec{k}$, i.e., $\Delta(\vec{k})$.
This expression is independent of the specific direction j. Without loss of
generality we choose $\vec{\delta}_{AB}$ leading to a gap function
$\Delta(\vec{p})$ that is symmetric in all the Dirac points, i.e., symmetric
under a rotation of the Brillouin zone of $2\pi/3$ radians, in agreement with
the symmetry of the underlying lattice. One comment about Eq. (43) is in order
here: we are mostly interested in the gap function $\Delta$ for
$\vec{k}\approx\vec{K}$, meaning at the Dirac point. Since
$V^{\rm{RPA}}(\vec{k}+\vec{p})$ gives the maximal contribution at
$\vec{p}=-\vec{k}$, this requires $\vec{p}\approx-\vec{K}$. This implies that
the self-consistency equation couples opposite Dirac points. In that
situation,
$\sin(\vec{p}\cdot\vec{\delta}_{j})/\sin(\vec{k}\cdot\vec{\delta}_{j})\simeq-1$.
This can be used to show that it is justified to carry out the Schwinger-Dyson
mean-field treatment in the effective Dirac low-energy setting, as one should
expect.

It is worth noting here that we calculated the RPA interaction,
$V^{\rm{RPA}}$, using the bare propagator, $G_{0}$. While this results in a
slightly inaccurate estimate of the gap term in the ordered phase, it does not
influence the location of the critical point, which is our focus.

###  III.2 Spontaneous dimerization

The gap equation, Eq. (43), can now be solved numerically for different values
of the coupling parameter $\alpha$. On a technical level, we use Gaussian
quadrature for the integration and find the solution through an iteration of
the gap equation, which stops when convergence is reached. A representative
example of the gap solution profile is shown in Fig. 8 for a value of
$\alpha=2.0$.

Figure 8: Numerical solution of the gap equation with coupling parameter
$\alpha=2.0$.

We have determined the phase diagram of this semimetal-insulator transition as
a function of $\alpha$. It shows a transition from a gapless semi-metallic
phase to a gapped insulating phase. The value of the gap as a function of
$\alpha$ is shown in Fig. 9 which is the main result of this paper.

Figure 9: Mass gap at the Dirac point as a function of the coupling parameter
$\alpha$. We find that there is a spontaneous gap formation for values of
$\alpha>1.224$

The critical coupling parameter is given by $\alpha_{c}\approx 1.224$: for
$\alpha>\alpha_{c}$ there is spontaneous mass generation, whereas for
$\alpha<\alpha_{c}$ the systems remains semi-metallic. It should be noted
here, that this critical value is in good agreement with the value predicted
from a simple two-component Dirac theory.

##  IV Conclusion

Kagomé metals offer an exciting new platform to observe many-body physics,
mostly due to the existence of van Hove singularities. In this work we have
focused on the Dirac metal structure also present in kagomé systems: at
$2/3$-filling, a kagomé metal such as Sc-Herbertsmithite, is expected to show
physics characteristic of a Dirac metal. We have found that in that situation
one can potentially observe the phenomenon of chiral symmetry breaking of
Dirac fermions.This dynamic instability can be driven by long-range Coulomb
interactions and on the microscopic level it corresponds to the formation of a
dimerization pattern that opens a gap at the Dirac points. This instability
occurs for values of $\alpha$ greater than $\alpha_{c}\approx 1.224$. It is
the analogue of the excitonic insulator which was hypothesized to exist in
suspended graphene. This has not been observed in experiments to date and a
possible explanation is that realistic values of $\alpha$ are either too small
($\mathcal{O}(1)$) or the gap is too small to be detected. Given that e.g.
recent estimates for the fine structure constant in Sc-Herbertsmithite suggest
$\alpha\approx 2.9$ leads us to conclude that kagomé Dirac semi-metals
constitute a promising system to look for spontaneous chiral symmetry breaking
of Dirac fermions.

## Acknowledgements

The authors thank Johanna Erdmenger and Henk Stoof for discussions. This work
is part of the D-ITP consortium, a program of the Netherlands Organisation for
Scientific Research (NWO) that is funded by the Dutch Ministry of Education,
Culture and Science (OCW). R.T. is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID
258499086-SFB 1170 and the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter – ct.qmat Project-ID 390858490-EXC
2147.

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# MeetDot: Videoconferencing with Live Translation Captions

Arkady Arkhangorodsky, Christopher Chu, Scot Fang, Yiqi Huang,
Denglin Jiang, Ajay Nagesh, Boliang Zhang, Kevin Knight
DiDi Labs
4640 Admiralty Way
Marina del Rey, CA 90292
{arkadyarkhangorodsky,chrischu,scotfang,denglinjiang,
<EMAIL_ADDRESS>
###### Abstract

We present MeetDot, a videoconferencing system with live translation captions
overlaid on screen. The system aims to facilitate conversation between people
who speak different languages, thereby reducing communication barriers between
multilingual participants. Currently, our system supports speech and captions
in 4 languages and combines automatic speech recognition (ASR) and machine
translation (MT) in a cascade. We use the re-translation strategy to translate
the streamed speech, resulting in caption flicker. Additionally, our system
has very strict latency requirements to have acceptable call quality. We
implement several features to enhance user experience and reduce their
cognitive load, such as smooth scrolling captions and reducing caption
flicker. The modular architecture allows us to integrate different ASR and MT
services in our backend. Our system provides an integrated evaluation suite to
optimize key intrinsic evaluation metrics such as accuracy, latency and
erasure. Finally, we present an innovative cross-lingual word-guessing game as
an extrinsic evaluation metric to measure end-to-end system performance. We
plan to make our system open-source for research purposes.111The system will
be available at https://github.com/didi/meetdot

##  1 Introduction

As collaborations across countries is the norm in the modern workplace,
videoconferencing is an indispensable part of our working lives. The recent
widespread adoption of remote work has necessitated effective online
communication tools, especially among people who speak different languages. In
this work, we present MeetDot, a videoconferencing solution with live
translation captions. Participants can see an overlaid translation of other
participants’ speech in their preferred language. Currently, we support speech
and captions in English, Chinese, Spanish and Portuguese.

The system is built by cascading automatic speech recognition (ASR) and
machine translation (MT) components. We process the incoming speech signal in
a streaming mode, transcribe it in the speaker’s language to be used as input
to an MT system to decode in the listener’s language during the call. We have
very tight latency requirements to be able to provide good quality captions in
a live video call. Our framework has several features to enable a better user
experience and reduce the cognitive load on the participants such as smooth
pixel-wise scrolling of the captions, fading text that is likely to change and
biased decoding of machine translation output Arivazhagan et al. (2020) to
reduce the flicker of the captions. In addition, we present preliminary work
towards identifying named-entity mentions in speech by interpolating a
pronunciation dictionary with the ASR language model.

We present the key metrics to measure the quality of the captions such as
accuracy, latency and stability (flicker in captions). Our system provides an
integrated evaluation suite to enable fast development through hill climbing
on these metrics. In addition to these intrinsic metrics, we present an
interesting online cross-lingual word guessing game, where one of the
participants is given a word that they describe and the other participants
have to guess the word by reading the captions in their respective languages.

Figure 1: MeetDot system architecture. The ASR and MT modules are designed to
be plug-and-play with different services in the backend (e.g. in-house DiDi MT
system/Google MT API). Frontend consists of different “views” - e.g. live
captions view - captures audio from another browser window/zoom/system to show
translated captions.

We present the following as the key contributions of our work:

  * •

A video conference system with live translation of multilingual speech into
captions overlaid on participants’ videos. The system has several features to
enhance user experience.

  * •

A comprehensive evaluation suite closely integrated with the system and a set
of metrics to reduce latency, caption flicker and accuracy.

  * •

A cross-lingual word-guessing game that can be used as an extrinsic metric to
evaluate end-to-end system performance.

We are in the process of releasing the system as open-source software for the
purpose of furthering research in the area.

##  2 Related Work

The area of simultaneous translation has attracted a lot of attention is
recent years. The recent editions of the Workshop on Automatic Simultaneous
Translation (Wu et al., 2021, 2020) and the tutorial in EMNLP 2020 Huang et
al. (2020) provide us an overview of the state-of-the-art and challenges in
live translation. Recent technical advances include newer architectures such
as prefix-to-prefix Ma et al. (2019) and adaptive policy methods such as
imitation learning Zheng et al. (2019) and monotonic attention Raffel et al.
(2017). The solutions in this space are differentiated into speech-to-speech
and speech-to-text, in the former there is a speech synthesis component. Our
work falls in the latter bucket, since we only display captions to the user.
Since we have access to separate audio-input channels (one per participant),
we do not need speaker diarization Park et al. (2021).

Re-translation is a common strategy applied, wherein we translate from scratch
every new extended source sentence, transcribed through the ASR module as the
participant speaks. The ability to modify previously displayed captions when
new ASR output is available can lead to flicker in displayed captions.
Following previous work Arivazhagan et al. (2020), we measure this using the
erasure metric and reduce it using biased beam search during MT decoding. To
better capture our use case of short exchanges in a meeting scenario compared
to long speech, we introduce additional metrics initial lag, incremental
caption lag, mean word burstiness and max word burstiness.

There have been recent working systems in this space, such as Ma et al.
(2019); Cho et al. (2013); Wang et al. (2016) that provide live captions for
single-speaker lectures and does not focus on multi-party meetings. Very
recently, there are news reports of systems that offer live translations of
multi-party meetings ZDNet (2021), but their technical details are unclear.

##  3 System Description

Figure 2: MeetDot room creation. Landing page (top, left panel) Any user can
set up a MeetDot room and share its URL with potential participants (bottom,
left panel). Admin users can select parameters that control captioning, speech
recognition, and translation (right panel, §3).

The overall system architecture is shown in Figure 1. It mainly consists of
two components (1) Frontend: runs on the user’s computer locally (2) Backend:
runs on the server and consists of ASR and MT modules that interact with the
Flask server. The modular architecture allows us to swap the ASR and MT
components from different service providers (such as Google ASR or in-house
Kaldi-based ASR services). These are explained below.

MeetDot User Interface:  We implemented a simple web-based user interface that
allows users to have meetings with automatically translated captions overlaid.
Our interface consists of:
\- A home page (Figure 2, top-left panel) for creating a meeting room with
default settings, or a game (§ 4).
\- A meeting creation page, where the room settings can be configured in
different ways for experimentation. (Figure 2, right panel)
\- A meeting landing page, where a user specifies their speaking language and
caption language (usually the same) before joining the meeting. (Figure 2,
bottom-left panel)
\- A meeting page, where the video call takes place (Figure 3)

Figure 3: MeetDot videoconference interface. Translated captions are
incrementally updated (word-by-word, phrase-by-phrase) on top of participant
videos. Translations also appear in the transcript panel (on right, not
shown), updated utterance-by-utterance. Choosing a caption language (4th
button from left at the bottom, in green) displays all captions in that
particular language. This depicts the view of the English caption user.

The meeting page features real-time translated captions overlaid on each
speaker’s video, displayed to each user in their selected language (Figure 3).
The user’s spoken language and caption language can also be changed on-the-
fly. The meetings support full video-conferencing functionality, including
microphone and camera toggles, screen sharing, and gallery/focused views.
Finally, there is a panel that shows the automatically transcribed and
translated conversation history in the user’s preferred language, which can
also be saved.

MeetDot also supports “Live Translation” mode that can translate audio from
any audio feed, e.g. a microphone or the user’s own computer. We provide
instructions for the latter, so users can see captions for audio in either
another browser tab or an ongoing Zoom call or audio from their system.

The frontend is implemented in Vue,222https://vuejs.org/ and the backend is
implemented using Flask. Video-conferencing functionality is built with
WebRTC,333https://webrtc.org/ using Daily.co444https://www.daily.co/ as the
signaling server to initiate real-time audio and video connections. The
frontend sends audio bytes to the Flask backend through a WebSocket
connection, which routes them to the speech recognition and translation
services and returns translated captions to be displayed.

Speech Recognition and Machine Translation:  Our backend services of ASR and
MT are joined in a cascaded manner. Each participant’s speech is fed in a
streaming fashion to the ASR module of the appropriate language selected by
the user. Additionally, we show the ASR output as captions to the speaker as
feedback/confirmation to them. Each of the transcribed text returned by ASR is
fed to the MT system to translate it into the caption language selected by the
reader, from scratch. This strategy is termed as re-translation. Since the ASR
stream continually returns a revised or an extended string, the input to MT is
noisy and will lead to the captions overwritten frequently (termed as flicker)
leading to a high cognitive load on the reading. We employ several techniques
to have better user experience while they are reading the captions (elaborated
more below).

At present, we support English, Chinese, Spanish and Portuguese languages for
both speech and captions. The modular architecture of our system allows us to
plug-and-play different ASR and MT service components in the backend. We
develop two different ASR systems based on the Kaldi framework and WeNet Yao
et al. (2021). We can also swap either of these to use Google ASR API instead.
For MT, we have the option of using our in-house DiDi MT system Chen et al.
(2020) as well as call Google MT API. For Kaldi ASR, we adapt pre-trained
Kaldi models to videoconferencing domain by interpolating the pre-trained
language model with our in-domain language models. For WeNet, we use use the
Unified Conformer model and language model interpolation is planned for future
work.555We use a pre-trained checkpoint for English ASR and trained a Chinese
ASR model from scratch using the multi-cn and TAL datasets. Following
Arivazhagan et al. (2020), we modify the decoding procedure of MT in OpenNMT’s
ctranslate toolkit666https://github.com/OpenNMT/CTranslate2 to mitigate the
issue of flicker mentioned above.

Direction | Systems | Final | Translation | Normalized | Initial | Incremental | Mean word | Max word
---|---|---|---|---|---|---|---|---
|  | bleu | lag (s) | erasure | lag (s) | caption lag (s) | burstiness | burstiness
En-to-Zh | Google ASR, Google MT | 17.81 | 5.84 | 2.69 | 3.82 | 0.71 | 5.26 | 11.83
” | Kaldi ASR, DiDi MT | 19.59 | 2.72 | 0.33 | 2.59 | 0.43 | 4.34 | 9.20
” | WeNet ASR, DiDi MT | 23.76 | 2.39 | 0.21 | 2.73 | 0.47 | 4.76 | 9.62
Zh-to-En | Google ASR, Google MT | 9.99 | 2.33 | 0.47 | 4.31 | 2.12 | 5.20 | 9.46
” | Kaldi ASR, DiDi MT | 7.88 | 3.33 | 0.73 | 2.70 | 0.42 | 2.42 | 6.13
” | WeNet ASR, DiDi MT | 9.76 | 2.27 | 0.37 | 2.52 | 0.32 | 2.42 | 5.65
Table 1: Baseline results on our daily work conversation dataset. Note that
the Google API does not have biased-decoding available to reduce flicker.
Metrics are explained in §4. Right 4 metrics are introduced by our work.

We include several additional features to enhance user experience. We use
NVIDIA NeMo toolkit777https://github.com/NVIDIA/NeMo to punctuate the captions
and predict if the word should be capitalized or not, which makes the captions
more readable. We have an initial named entity recognition module, to
recognize mentions of participants’ names in speech when using the Kaldi ASR
system. This is performed by interpolating the ASR’s language model with a
language model trained on a synthesised corpus that contains participants’
names. The name pronunciation dictionary required by Kaldi ASR is generated
automatically by rules. Profanity is detected using a standard list of
keywords and starred in both ASR and translation output. We have an
experimental module to detect the speaker’s language automatically (instead of
being manually set by the user). The advantage of such a feature is to allow
code-switching between multiple languages which is a common behavior among
multilingual speakers Solorio and Liu (2008); Sivasankaran et al. (2018).

Figure 4: Cross-lingual word guessing game for extrinsic evaluation. Roles:
one player is given a word to describe in their language, one or more players
look at the captions displayed to guess the correct word. Screenshot shown is
that of the describer of the word. Word is “eyelash” (left panel) - the
participants communicate hints and guesses through MeetDot translation
captions, and the system itself spots correct guesses.

Captioning Strategies:  Here we describe how we deploy ASR and MT capabilities
to create translation captions that are incrementally updated in real time.
Since we display translation captions on top of a speaker’s video feed, we
have limited screen real-estate—for example, 3 lines of 60 characters each.

Each time we receive an incremental ASR update (hypothesis extension or
revision), we translate the updated string from scratch Arivazhagan et al.
(2020). If the translation exceeds the available real-estate, we display the
longest suffix that fits. ASR also signals utterance boundaries; we only send
the current, growing utterance through MT, caching the MT results on previous
utterances.

The basic system exhibits large amounts of flicker Niehues et al. (2016,
2018), which we mitigate with these methods:

Translate-k. We only send every $k$th ASR output to MT. This results in
captions that are more stable, but which update more slowly.

Translate-t. Improving on translate-k, we send an ASR output to MT if at least
$t$ seconds have elapsed since the last MT call.

Mask-k. We suppress the last $k$ words from the MT output, providing time for
the translation to settle down Cho and Esipova (2016); Gu et al. (2017); Ma et
al. (2019); Arivazhagan et al. (2020). We often use Mask-4 in practice.

Biased MT decoding. We encourage word-by-word MT decoding to match the string
output by the previous MT call Arivazhagan et al. (2020), avoiding translation
variations that, while acceptable, unfortunately introduce flicker.

Preserve linebreaks. When possible, we prevent words from jumping back & forth
across linebreaks.

Smooth scrolling. We reduce perception of flicker by scrolling lines smoothly,
pixel-wise.

##  4 Evaluation

Dataset: In order to evaluate our system in the most-appropriate deployment
scenario, we construct an evaluation dataset based on meetings within our
team. We have a total of 5 meetings, 3 meetings involving 7 participants and
in English and 2 meetings involving 2 participants in Chinese. The content of
the meetings are daily work conversation and contains a total of 135 mins (94
mins English and 41 mins Chinese). We manually transcribe and translate these
meetings using a simple and uniform set of guidelines. The resulting dataset
consists of 494 English utterances ($\sim 11k$ words, translated into Chinese)
and 183 Chinese utterances ($\sim 9.8k$ characters, translated into English).
We use this dataset to measure our intrinsic evaluation metrics.

Intrinsic metrics:  We adopt the following metrics from previous work on
streaming translation Papineni et al. (2002); Niehues et al. (2016);
Arivazhagan et al. (2020):

\- Final Bleu. We measure the Bleu MT accuracy of final, target-language
utterances against a reference set of human translations. Anti-flicker devices
that “lock in” partial translations will generally decrease final Bleu.
\- Translation lag. We measure (roughly) the average difference, in seconds,
between when a word was spoken and when its translation was finalized on the
screen.
\- Normalized erasure. We quantify flicker as $m$/$n$, the number of words $m$
that get erased during the production of an $n$-word utterance.

To support our videoconferencing application, we introduce other intrinsic
metrics:

\- Initial lag. In videoconferencing, we find it valuable to generate
translation captions immediately after a speaker begins, even if the initial
translation involves some flicker. Here we measure the time between initial
speaking time and the first displayed word. We improve initial lag by adopting
a  Mask-0 policy at the start of an utterance, transitioning to our usual
Mask-4 policy later.
\- Incremental caption lag. The average time between caption updates.
\- Mean (& Max) word burstiness. The mean number of words or characters for
Chinese (maximum, respectively) that are added/subtracted in a single caption
update (for a given utterance’s translation–averaged over all utterances,
respectively).

In Table 1, we present baseline results of the various intrinsic metrics for
the English to Chinese and Chinese to English systems. We present results for
3 different module combinations, namely, (i) Google API888Accessed on
2021-09-09. for ASR999https://cloud.google.com/speech-to-text and
MT101010https://cloud.google.com/translate (ii) Kaldi ASR and DiDi MT (iii)
WeNet ASR and DiDi MT. From the results table, we would like to highlight that
the biased decoding modification for machine translation has a positive impact
on several metrics such as translation lag, normalized erasure and word
burstiness. Biased decoding is absent in the Google MT API, hence has higher
numbers in these metrics. This results in increasing flicker leading to a
poorer user experience. Our final bleu score when using WeNet ASR and DiDi MT
is several points better than Google ASR and Google MT in the English to
Chinese direction and has comparable performance in Chinese to English
direction. ASR system’s performance is an important factor in a cascaded
system. Kaldi ASR has a word error rate (WER) rate of 43.88 for English
(character error rate (CER) of 49.75 for Chinese, respectively) compared to
WeNet ASR’s WER of 34.74 (CER of 38.03 for Chinese, respectively). This has a
direct impact on final bleu scores as seen from the results.

Cross-lingual word guessing game:  Extrinsic metrics for speech translation
are not as popular as intrinsic ones Arivazhagan et al. (2020); Niehues et al.
(2016). However, given the broad range of techniques for displaying
translation captions, we would like to measure things that are closer to the
user experience.

Here, we introduce a cross-lingual, cooperative word game for A/B testing
different captioning algorithms. Players who speak different languages use
MeetDot’s translation captions to communicate with each other. If the players
obtain higher scores under one condition, we conclude that their communication
is, in some way, made more effective and efficient.

The game is a variation on
Taboo,111111https://www.hasbro.com/common/instruct/Taboo(2000).PDF in which
one player receives a secret word (such as “racoon” or “scary”) and must
describe it without mentioning the word or variant of it. The other player
tries to guess the word. The players are awarded a point for every word
guessed in a 4-minute period. The first player is allowed to skip upto three
words.

In our variation, the first player may receive a Chinese word and describe it
in Chinese, while the second player sees English captions and makes guesses in
English. When translation is quick, accurate, and readable, players score
higher.

We design our bilingual wordlists to contain words with limited ambiguity.
This way, we are able to build a reliable, automatic scorer that rewards
players and advances them to the next word.

We also implement a competitive, multi-player version of the game, where
players are assigned points for making correct guesses faster than others, and
for giving clues that lead to fast guesses.

##  5 Conclusion and Future Work

We describe MeetDot, a videoconferencing system with live translation
captions, along with its components: UI, ASR, MT, and captioning. We implement
an evaluation suite that allows us to accurately compute metrics from the
sequence of captions that users would see. We also describe a cross-lingual
word game for A/B testing different captioning algorithms and conditions.

Our future work includes improved ASR/MT, extrinsic testing, and an open
source release. Our overall goal is to provide a platform for developing
translation captions that are accurate and “right behind you”.

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