Viability of rotation sensing using phonon interferometry in Bose-Einstein
condensates

Charles W. Woffinden1,2,*,$\dagger$, Andrew J. Groszek1,2,3,$\dagger$,
Guillaume Gauthier1,2, Bradley J. Mommers1,2, Michael W. J. Bromley1,4, Simon
A. Haine5, Halina Rubinsztein-Dunlop1,2, Matthew J. Davis1,2,3, Tyler W.
Neely1,2, Mark Baker1,2,6

1 School of Mathematics and Physics, University of Queensland, St. Lucia, QLD
4072, Australia

2 Australian Research Council Centre of Excellence for Engineered Quantum
Systems, School of Mathematics and Physics, University of Queensland, St.
Lucia, QLD 4072, Australia

3 Australian Research Council Centre in Future Low-Energy Electronics
Technologies, School of Mathematics and Physics, University of Queensland, St.
Lucia, QLD 4072, Australia

4 School of Sciences, University of Southern Queensland, Toowoomba, QLD 4350,
Australia

5 Department of Quantum Science and Technology, Research School of Physics and
Engineering, The Australian National University, Canberra, ACT 2601, Australia

6 Quantum Technologies Group, Sensors and Effectors Division, Defence Science
& Technology Group

*<EMAIL_ADDRESS>
$\dagger$ These authors have contributed equally to this work and share first
authorship.

## Abstract

We demonstrate the use of a ring-shaped Bose-Einstein condensate as a rotation
sensor by measuring the interference between two counter-propagating phonon
modes imprinted azimuthally around the ring. We observe rapid decay of the
excitations, quantified by quality factors of at most
$\mathbf{\textit{Q}\boldsymbol{\approx}27}$. We numerically model our
experiment using the c-field methodology, allowing us to estimate the
parameters that maximise the performance of our sensor. We explore the damping
mechanisms underlying the observed phonon decay, and identify two distinct
Landau scattering processes that each dominate at different driving amplitudes
and temperatures. Our simulations reveal that Q is limited by strong damping
of phonons even in the zero temperature limit. We perform an experimental
proof-of-principle rotation measurement using persistent currents imprinted
around the ring. We demonstrate a rotation sensitivity of up to
$\boldsymbol{\Delta\Omega\approx}\mathbf{0.3\,rad\,s^{-1}}$ from a single
image, with a theoretically achievable value of
$\boldsymbol{\Delta\Omega\approx}\mathbf{0.04\,rad\,s^{-1}}$ in the atomic
shot-noise limit. This is a significant improvement over the shot-noise-
limited $\boldsymbol{\Delta\Omega\approx}\mathbf{1\,rad\,s^{-1}}$ sensitivity
obtained by Marti et al.[1] for a similar setup.

##  1 Introduction

High precision rotation sensing is critical for applications such as inertial
navigation [2], geodesy [3], and tests of fundamental physical theories such
as general relativity [4]. One mature technology used for rotation sensing is
the hemispherical resonator gyroscope (HRG) [5, 2], which relies on standing
mechanical waves generated around a hemispherical body akin to those produced
by tapping the rim of a wine glass. When the hemispherical body is rotated,
the standing waves precess around the circumference due to the Coriolis force.
This precession is proportional to the rotation rate of the hemisphere and can
thus be used to measure the rotation rate. HRGs are used for rotation sensing
in applications where high precision and reliability are required, such as on
board the James Webb Space Telecope[6]. HRG excitations are robust to damping,
quantified by high quality factors of $Q\sim 10^{7}$. However, their signal is
susceptible to drift if used over long periods, and they only provide a
measure of relative—rather than absolute—rotation.

In recent decades, cold atoms have emerged as an attractive alternative to
traditional optical techniques for performing high precision interferometry.
The associated atomic wavelengths and velocities make it possible to realise
extremely high precision interferometric devices such as gravimeters,
gradiometers, and gyroscopes [7]. For rotation sensing, ‘free-space’ atom
interferometry configurations have proved successful, where the atoms are in
free-fall for the duration of the interrogation [8, 9, 10, 11, 12]. However,
such setups require a large apparatus and are therefore not suitable for many
practical sensing applications. More recently, guided or trapped atom
interferometers have been demonstrated [13, 1, 14, 15, 16, 17, 18]. In these
systems, an ultracold gas or superfluid Bose-Einstein condensate (BEC) is
confined to a geometry, such as a ring trap, using optical or magnetic
potentials [19, 20]. These systems are compact, have an improved lifetime, and
produce strong signals due to their high atomic densities. However,
measurements are susceptible to systematic phase shifts arising from atom–atom
interactions and trap imperfections.

Recently, it was proposed that a ring-shaped BEC could be used for rotation
sensing [1, 14] using similar operating principles to an HRG. In this scheme,
standing waves are formed using the collective excitations (phonons) of the
condensate. Because these phonon modes are the lowest energy excitations
available [21], they are anticipated to be resistant to losses into other
excitation branches, as well as being minimally affected by interactions and
trap inhomogeneity. An added advantage is that superfluids are irrotational,
and can therefore serve as an absolute frame of reference for rotation sensing
with no need for calibration [22]. While Ref. [1] successfully implemented
this interferometry scheme, their sensitivity was too low to perform a
practical rotation measurement. Furthermore, the phonons were found to decay
at a rate much faster than that predicted by theory, severely limiting the
potential sensitivity of the system.

Here, we further explore the properties of the phonon-based rotation scheme
proposed in Ref. [1] to determine whether such sensors may be of practical
use. Primarily, we aim to improve the sensitivity of this device over the
implementation of Ref. [1] and perform a measurement of an applied rotation.
We experimentally create a quasi-two-dimensional (2D) ring-shaped BEC using a
digital micromirror device (DMD) and perform a sinusoidal phase imprint to
excite azimuthal phonon modes. We characterise the performance of our
interferometer as both the wavelength and the amplitude of the imprinted mode
are varied, observing quality factors up to a maximum of $Q\approx 27$. We
numerically model our setup using classical field simulations at finite
temperature [23], obtaining $Q$ factors that are in good agreement with the
experiment. Using the numerics, we perform a detailed exploration of the
physical processes responsible for the rapid phonon damping observed in this
system. Our analysis reveals a rich variety of decay behaviour resulting from
the interplay between two distinct forms of Landau damping [24, 25, 26]. By
simulating our system in the limit of zero temperature and weak perturbations
where these damping mechanisms are minimised, we predict that the quality of
our setup could be improved to $Q\sim 150$, but even in this idealised
scenario, the performance is many orders of magnitude below modern HRGs for an
experimentally realistic fixed area. Both experimentally and numerically, we
apply the phonon interferometry technique to directly measure the rotation
rate of persistent currents in the ring. Our measurements imply that our
experiment could achieve a rotation sensitivity of up to
$\Delta\Omega\approx$0.3\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$ from
a single measurement.

##  2 Phonon interferometry for rotation sensing

A phonon interferometer functions in much the same way as a hemispherical
resonator gyroscope, whereby the beat frequency between counter-propagating
excitations can be used to provide a measurement of the external rotation rate
[2, 5]. The superfluid nature of the medium additionally provides a stationary
inertial frame of reference (FoR) for the phonons, and in this frame the
counter-propagating modes travel at equal and opposite angular velocities. The
azimuthal density of the atoms as a function of time $t$ and angle $\theta$
around the ring can therefore be written:

$n_{\theta}(\theta,t)=\bar{n}_{\theta}+\delta
n_{\theta}(\theta,t)=\bar{n}_{\theta}+\delta n_{\theta}^{+}(\theta,t)+\delta
n_{\theta}^{-}(\theta,t),$ (1)

where $\bar{n}_{\theta}$ is the ground state density of the ring, $\delta
n_{\theta}$ is the total density perturbation, and $\delta n_{\theta}^{\pm}$
are the perturbations due to the anticlockwise (clockwise) propagating modes
in the medium. For the case of azimuthal standing waves, the density
perturbations have the form [14]

$\delta
n_{\theta}^{\pm}(\theta,t)=\mathcal{A}\sin{\left(m\theta\pm\omega_{m}t\right)},$
(2)

with $\mathcal{A}$ the perturbation amplitude, $m$ the mode number
(corresponding to the number of wavelengths around the ring), and $\omega_{m}$
the corresponding angular frequency. The total density perturbation in the
frame of the superfluid is therefore

$\delta n_{\theta}(\theta,t)=2\mathcal{A}\sin{(m\theta)}\cos{(\omega_{m}t)}.$
(3)

If an observer is in the laboratory FoR rotating at a rate $\Omega$ relative
to the superfluid, the azimuthal co-ordinate from their perspective shifts,
$\theta\to\theta-\Omega t$, and hence they will see the standing wave precess
at a frequency $m\Omega$:

$\delta
n_{\theta}^{\mathrm{(lab)}}(\theta,t)=2\mathcal{A}\sin{\left[m\left(\theta-\Omega
t\right)\right]}\cos{(\omega_{m}t)}.$ (4)

By measuring the azimuthal density of the condensate, the observer can
therefore deduce the rotation rate $\Omega$.

##  3 Experiment

###  3.1 BEC preparation

The BEC experimental setup and preparation process have been described in
detail in Ref. [19]. In brief, a BEC of 87Rb atoms in the
$|F=1,m_{F}=-1\rangle$ state, formed using a hybrid optical and magnetic
trapping technique [27], is tightly trapped in the vertical ($z$) direction
using a far red-detuned $1064\text{\,}\mathrm{nm}$ sheet beam potential with a
trapping frequency $\omega_{z}=2\pi\times$133\text{\,}\mathrm{Hz}$$. A far
blue-detuned $532\text{\,}\mathrm{nm}$ laser is used to create the horizontal
trapping potential and is shaped by direct projection from a DMD. By switching
the DMD mirrors to an on or off position, arbitrary and dynamic patterning of
the BEC at sub-micron resolution can be achieved [19].

We use a ring-shaped potential in the horizontal plane (see Sec. 3.2), with
typical atom numbers of $N_{\mathrm{atoms}}\approx 2\times 10^{6}$. We expect
the system to be in the Thomas–Fermi regime, with a parabolic density profile
in the vertical direction of width
$\approx$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Using the Thomas–Fermi
approximation, we estimate the chemical potential to be
$\mu\approx$55\text{\,}\mathrm{nK}$\times k_{\mathrm{B}}$. Although the system
should be fully 3D in a thermodynamic sense (with a true BEC transition), the
tight vertical and radial confinements restrict the relevant dynamics to be
along the azimuthal direction. Throughout this work, the condensate fraction
of the gas is $n_{0}\approx 0.8$, which we find to be the highest achievable
in this setup. We estimate this value by letting the cloud evolve in time-of-
flight and calculating the fraction of atoms contained within respective fits
to the condensed and thermal components, as measured from absorption imaging.

###  3.2 Experimental sequence

The experimental sequence is summarised in Fig. 1. We initially load the atoms
into an annulus with inner and outer radii of
$r_{\mathrm{in}}\approx$35\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$r_{\mathrm{out}}\approx$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
respectively, and a potential depth of $\approx$330\text{\,}\mathrm{nK}$\times
k_{\mathrm{B}}$. The inner radius is then expanded to
$r_{\mathrm{in}}\approx$45\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, increasing
the atomic density and the chemical potential. This lessens the impact of trap
imperfections. We find that $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is the
minimum ring thickness we can achieve that avoids segmentation of the
condensate due to light leak through from the DMD. After this step, the atoms
are left to equilibrate in the final ring potential for
$100\text{\,}\mathrm{ms}$.

Figure 1: Timing diagram for experimental sequence of ring formation, phonon
imprinting, hold time, and imaging. The depth of the trapping potential is
indicated with the red line, where
$t_{\mathrm{ramp}}=$500\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ (time not to
scale—see text). Insets show the applied DMD potential for different parts of
the sequence, with white areas corresponding to an attractive potential. In
the example shown, the imprinted mode number $m^{*}=5$.

We then imprint an azimuthally varying phase pattern around the ring. This is
done by applying a sinusoidal potential $V_{\mathrm{imp}}(\theta)$ for a time
$t_{\mathrm{imp}}$, causing the atoms to acquire a phase factor
$\exp[iV_{\mathrm{imp}}(\theta)t_{\mathrm{imp}}/\hbar]$. Our imprinting
potential takes the form $V_{\mathrm{imp}}(\theta)=V_{0}\sin(m^{*}\theta)$,
where $m^{*}$ is the mode number being excited111Throughout this work, we use
$m$ for the general azimuthal mode index and denote the imprinted mode
$m^{*}$.. This pattern excites two counter-propagating waves, which, due to
the superfluid’s irrotational nature, must have equal amplitudes. The
amplitude $V_{0}$ is set to $\approx 90\%$ of the trap depth—this value
ensures a fast imprint (relative to the response time of the atoms) while also
minimising the loss of atoms from the BEC. We apply this imprinting potential
via the DMD by adding it to the existing trapping potential.

Immediately prior to the phase imprint, the trapping potential height is
doubled to $\approx$660\text{\,}\mathrm{nK}$\times k_{\mathrm{B}}$ over a time
$t_{\mathrm{ramp}}=$500\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, in order to
prevent atoms spilling out of the trap when the imprint is applied. The
imprinting potential is switched on for a variable time
$$60\text{\,}\mathrm{\SIUnitSymbolMicro s}$\leq
t_{\mathrm{imp}}\leq$300\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, with longer
imprints generating higher amplitude phase patterns (for significantly longer
imprint times, a density perturbation forms, which is the regime explored in
Ref. [1]). After this time, the imprinting potential is switched off, and the
trap is lowered to its original height over time $t_{\mathrm{ramp}}$, so that
stray-light induced losses are minimised during the subsequent evolution. The
BEC is then allowed to evolve for a variable hold time $t_{\mathrm{hold}}$ (up
to $400\text{\,}\mathrm{ms}$), after which time the atomic density is measured
in situ using Faraday imaging.

##  4 Numerical modelling

We numerically model the Bose gas using three-dimensional c-field simulations
[23], allowing us to approximate the effects of the thermal cloud on the
dynamics. In this approach, a complex-valued classical field
$\psi(\textbf{r},t)$ is used to describe the condensate and a chosen set of
macroscopically occupied low-energy excitations. Excitations above the energy
cutoff are excluded from the field and are not modelled dynamically. Although
c-field models cannot quantitatively capture all aspects of the Bose gas
simultaneously [28], they have nonetheless been shown to match experiments
when used appropriately [29, 30, 31]. Here we use the c-field simulations to
provide insight into the mechanisms of the damping behaviour of the system at
non-zero temperature, and find that the results are in semi-quantitative
agreement with experiments when the condensate fractions are matched.

To sample thermal states, we use the stochastic projected Gross–Pitaevskii
Equation [32, 33, 34], in which the above-cutoff excitations are treated as a
thermal reservoir with temperature $T$ and chemical potential $\mu$, coupled
to the field via the dimensionless collision rate $\gamma$. The time evolution
is given by:

$d\psi=\mathcal{P}\biggl{\\{}-\frac{i}{\hbar}L_{\rm GP}\psi
dt+\frac{\gamma}{\hbar}(\mu-L_{\rm GP})\psi dt+dW\biggr{\\}},$ (5)

where the operator $L_{\rm GP}=-(\hbar^{2}/2m_{\mathrm{a}})\nabla^{2}+V_{\rm
trap}+g|\psi|^{2}$, and the complex local white noise $dW$ has correlations
$\langle dW^{*}(\textbf{r},t)dW(\textbf{r}^{\prime},t)\rangle=(2\gamma k_{\rm
B}T/\hbar)\delta(\textbf{r}-\textbf{r}^{\prime})dt$. The interaction parameter
$g=4\pi\hbar^{2}a_{\rm s}/m_{\mathrm{a}}$, with the 87Rb scattering length
$a_{\rm s}\approx$5.3\text{\,}\mathrm{nm}$$ and atomic mass
$m_{\mathrm{a}}\approx$1.44\text{\times}{10}^{-25}\text{\,}\mathrm{kg}$$. The
projection operator $\mathcal{P}$ ensures that the field is confined to the
classical field region, defined here as the set of three-dimensional plane
waves with single-particle energies satisfying
$\hbar^{2}|\textbf{k}|^{2}/2m_{\mathrm{a}}<E_{\rm cut}$, with $E_{\rm cut}$
the energy cutoff.

We use a trapping potential $V_{\rm trap}(x,y,z)=V_{xy}(x,y)+V_{z}(z)$, where
$V_{z}(z)=m_{\mathrm{a}}\omega_{z}^{2}z^{2}/2$ is a tight harmonic trapping
potential, and the planar potential $V_{xy}$ is chosen to be an annulus of the
form:

$V_{xy}(x,y)=\tilde{V}_{0}\left\\{1+\frac{1}{2}\tanh\left[\frac{2}{\sigma}(r-r_{\rm
out})\right]-\frac{1}{2}\tanh\left[\frac{2}{\sigma}(r-r_{\rm
in})\right]\right\\},$ (6)

with $r=\sqrt{x^{2}+y^{2}}$ the radial co-ordinate. This potential has maximum
height $\tilde{V}_{0}$, inner (outer) radius of $r_{\mathrm{in}}$
($r_{\mathrm{out}}$), and barrier width set by $\sigma$.

To generate the initial condition for our simulations, we first obtain the
ground state of the potential by evolving Eq. (5) with $\gamma=1$ and $T=0$.
We then set the temperature $T\geq 0$ and continue to evolve in time until
thermal equilibrium is reached. Finally, we imprint a phase profile:

$\psi\to\psi\exp\left[i\Phi_{\mathrm{imp}}\cos(m^{*}\theta)\right],$ (7)

where $\Phi_{\mathrm{imp}}$ is the imprinted amplitude of the phase profile
and $m^{*}$ is the imprinted mode number. We then evolve this initial state in
time using the projected Gross–Pitaevskii equation [23], obtained by setting
$\gamma=0$ in Eq. (5). In this model, the classical field is decoupled from
the thermal reservoir, and hence energy and particle number are both conserved
under time evolution. The decay of the imprinted phonon therefore results from
the internal redistribution of population among the available modes, rather
than loss of energy to the reservoir.

In Eq. (5), we use a chemical potential $\mu=$58\text{\,}\mathrm{nK}$\times
k_{\mathrm{B}}$ and (unless otherwise stated) a temperature
$T\approx$170\text{\,}\mathrm{nK}$$. These values are chosen such that the
number of atoms approximately matches the experiment, and the decay times of
the imprinted phonons are on the order of those measured experimentally. With
these parameters, we find that the condensate fraction is $n_{0}\approx
0.81$222Determined from the largest eigenvalue of the one-body density matrix,
in accordance with the Penrose–Onsager criterion [35, 23]., although we note
that this parameter should be treated as a self-consistent measure of
coherence within the c-field model, and may be different from the
experimentally measured value.

The planar potential is chosen to have barrier height
$\tilde{V}_{0}=5\mu=$290\text{\,}\mathrm{nK}$\times k_{\mathrm{B}}$, wall
width $\sigma\approx$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and inner
(outer) radius of $r_{\rm in}=$45\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
($r_{\rm out}=$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). We use a numerical
grid of $256\times 256\times 24$ points, corresponding to a simulation domain
of $$132\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$132\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$16\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. We numerically evolve
Eq. (5) using XMDS2 [36]. A second-order semi-implicit differential equation
solver is used when $\gamma>0$, and a fourth-order Runge–Kutta scheme is used
when $\gamma=0$. The energy cutoff is set to $E_{\mathrm{cut}}=\hbar^{2}k_{\rm
cut}^{2}/2m_{\mathrm{a}}\approx$62\text{\,}\mathrm{nK}$\times k_{\mathrm{B}}$
(corresponding to wavenumber cutoff $k_{\rm
cut}\approx$4.7\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$). The imprinted
mode number is varied between $3\leq m^{*}\leq 13$, and the amplitude is
varied in the range of $0.1\pi\leq\Phi_{\mathrm{imp}}\leq 8\pi$. We note that
although we have used a 3D model for these simulations, we find that 2D
simulations with the same atom number and condensate fraction provide
quantitatively similar predictions for the decay of the imprinted excitation.

##  5 Analysis

###  5.1 Measurement protocol

For both experimental and numerical data, we analyse the evolution of the
azimuthal excitations by measuring the atomic density. Following the method
presented in Ref. [1], we perform a Fourier transform of the density to obtain
the complex Fourier amplitude $A_{m}$ of the $m^{\mathrm{th}}$ azimuthal mode:

$A_{m}(t)=\frac{\iint
n_{\mathrm{2D}}(\textbf{r},t)\mathrm{e}^{-im\theta}\mathrm{d}\textbf{r}}{\iint
n_{\mathrm{2D}}(\textbf{r},t)\mathrm{d}\textbf{r}}=A^{(R)}_{m}(t)+iA^{(I)}_{m}(t),$
(8)

where $n_{\mathrm{2D}}(\textbf{r},t)$ is the two-dimensional ($z$-integrated)
atomic density profile at time $t$, and $A^{(\nu)}_{m}$ are the real ($\nu=R$)
and imaginary ($\nu=I$) components of the amplitude. Throughout this work,
$A_{m}(t)$ is normalised by the atom number, such that $0\leq|A_{m}(t)|\leq
1$.

Treating the radial density profile as time-independent, the two-dimensional
density can be expressed as a product
$n_{\mathrm{2D}}(\textbf{r},t)=n_{r}(r)n_{\theta}(\theta,t)$. Substituting
this into Eq. (8) and using $n_{\theta}$ given by Eqs. (1) and (4), the
amplitude of the excited phonon mode $m=m^{*}$ as measured from the lab frame
will have the form

$A_{m^{*}}(t)=i\alpha(t)\sin(\omega_{m^{*}}t+\phi_{0})\mathrm{e}^{im^{*}\Omega
t}.$ (9)

For generality, we have included a phase offset $\phi_{0}$ and a time-varying
envelope $\alpha(t)$ to account for any damping. As in previous works [1, 31,
37], we find that the envelope $\alpha(t)$ is well described by exponential
decay in most cases. We will therefore assume

$\alpha(t)=\alpha_{0}\mathrm{e}^{-t/\tau},$ (10)

throughout most of this work. Here, $\tau$ is the decay constant and
$0\leq\alpha_{0}\leq 1$ is the imprint amplitude, controlled by either
$t_{\mathrm{imp}}$ (in the experiment) or $\Phi_{\mathrm{imp}}$ (in the
numerics). In the case $|\Omega|>0$ considered in Sec. 7, we extract the
rotation rate by measuring the angle $\theta(t)\equiv\arg\\{A_{m^{*}}(t)\\}$,
which grows linearly in time as $\theta(t)\propto m^{*}\Omega t$.

###  5.2 Quality factor of the phonon interferometer

The quality or $Q$ factor of a resonator is typically defined as [38]

$Q=\frac{\omega_{m}\tau}{2},$ (11)

where $\omega_{m}$ is the angular frequency of the excitation, and $\tau$ is
its lifetime. The frequency of the $m^{\mathrm{th}}$ mode can be approximated
as $\omega_{m}\approx m\omega_{0}$, where $\omega_{0}\approx c_{\mathrm{s}}/R$
is the fundamental frequency of the ring, determined by the speed of sound
$c_{\mathrm{s}}=\sqrt{\mu/m_{\mathrm{a}}}$ and the ring radius $R$.

Figure 2: Typical experimental sequence for an $m^{*}=7$ excitation with
imprint time $t_{\mathrm{imp}}=$400\text{\,}\mathrm{\SIUnitSymbolMicro s}$$
(imprint amplitude $\alpha_{0}\approx 0.11$). (a)–(d) Density images for hold
times of 0, 6, 9 and 15 ms respectively. The colour scale denotes the squared
density in arbitrary units (a.u.). (e) Decaying oscillations of the imaginary
component of the Fourier amplitude of the $m^{*}=7$ azimuthal mode.
Experimental data (open circles) have been fitted with a decaying sine wave
(black line); the grey shaded region shows the 95% confidence interval for the
fit. Data points corresponding to frames (a)–(d) are labelled accordingly.

In principle, the quality may be improved by either increasing $\omega_{m}$ or
$\tau$. However, in this system it has previously been demonstrated that
$\tau\propto m^{-1}$ [1], and thus the product $\omega_{m}\tau$ should remain
constant. The $Q$ factor can therefore only be improved through other means,
such as by varying the ring properties.

##  6 Characterising the phonon interferometer

###  6.1 Quality factor measurements

We first explore the properties of the system with no applied rotation,
$\Omega=0$. This allows us to benchmark the performance of the interferometer,
which will determine its maximum achievable sensitivity to rotation (see Sec.
7.5). We have performed experiments using the method described in Sec. 3.2 for
a range of imprint amplitudes $0.0135\lesssim\alpha_{0}\lesssim 0.19$ and mode
numbers $m^{*}\in\\{3,5,7,13\\}$. An example series of density images for an
$m^{*}=7$ phase imprint is shown in Fig. 2(a–d), depicting the evolution of
the oscillation over one period. The time evolution of the Fourier amplitude
$A_{m^{*}}^{(I)}(t)$ obtained from Eq. (8) is shown in Fig. 2(e) (open
circles), demonstrating the decaying oscillatory behaviour. From a fit to Eqs.
(9)–(10) (solid black line), we obtain measurements of the oscillation
frequency $\omega_{m^{*}}$ and the decay time constant $\tau$, giving a
quality factor $Q\approx 5$ for this configuration [Eq. (11)].

Figure 3: $Q$ factor as a function of imprint amplitude $\alpha_{0}$ for four
different mode numbers $m^{*}$ (log–log scale). (a) Experimental data; (b)–(e)
experimental (open coloured points) and c-field simulation (filled black
points) data for $m^{*}=\\{3,5,7,13\\}$, respectively. In all panels, the
power-law fit to the experimental data, $Q\propto\alpha_{0}^{-1}$, is included
for comparison. Standard errors in $Q$ are presented as vertical bars.

The $Q$ factors measured in this way for each run of the experiment are
displayed in Fig. 3(a). In agreement with Ref. [1], we find $Q$ to be largely
independent of the imprinted mode number $m^{*}$, meaning that performance
cannot be improved through the choice of $m^{*}$ (see Sec. 5.2). Additionally,
we find that the data approximately follow an inverse relationship with
initial imprint amplitude, $Q\propto\alpha_{0}^{-1}$, indicating that we are
in a nonlinear damping regime where $\tau$ is $\alpha_{0}$-dependent. In the
limit $\alpha_{0}\to 0$, we expect the system to evolve approximately
linearly, corresponding to a constant $Q(\alpha_{0})$. The $Q$ data do show a
flattening at low $\alpha_{0}$ consistent with this expectation, although the
spread of the data is too large to reveal a clear picture. From these results,
we conclude that our interferometer should perform best in the limit of low
$\alpha_{0}$. However, even the best performing configuration only achieves
$Q\approx 27$—orders of magnitude lower than other modern rotation sensing
technologies [5]. Moreover, using such weak imprints reduces the signal-to-
noise ratio, until the oscillation becomes indistinguishable from noise at
initial amplitudes $\alpha_{0}\lesssim 0.013$.

To gain further insight into these results, we simulate our system using the
c-field method outlined in Sec. 4 using a similar range of parameters
$\alpha_{0}$ and $m^{*}$. We fit Eqs. (9)–(10) to the measured amplitude
$A_{m^{*}}^{(I)}(t)$ of the imprinted mode over the first
$\sim$100\text{\,}\mathrm{ms}$$ of evolution. The resulting $Q$ measurements
are shown as black points in Fig. 3(b)–(e), corresponding to
$m^{*}=\\{3,5,7,13\\}$, respectively. For comparison, the equivalent
experimental data from (a) are also shown in each frame (b)–(e), along with
the same power-law $Q\sim\alpha_{0}^{-1}$. We find broad agreement between our
experimental and numerical results, to within the precision expected from the
c-field methodology. For large $\alpha_{0}$ the numerical data follow the
power-law quite convincingly. For small $\alpha_{0}$, the numerical results
show a clear plateauing behaviour, particularly for $m^{*}=7$ and $m^{*}=13$,
providing convincing evidence of linear decay behaviour in this regime.
Importantly, the simulations do not predict significantly higher $Q$ factors
for this experimental configuration, and hence the low $Q$ values observed
cannot be attributed to imperfections in the experiment. Rather, the excited
phonons are being damped by physical processes that are present in both
simulation and experiment. In the next section, we attempt to understand these
decay processes, as well as the observed transition between linear and
nonlinear behaviour as $\alpha_{0}$ is varied.

###  6.2 Damping of phonons

####  6.2.1 Scattering processes

The imprinted excitation decays via mode mixing processes arising from the
nonlinear interactions in the Bose gas. The dominant scattering processes
involve interactions between three azimuthal quasiparticle excitations and the
condensate mode333Interactions between four quasiparticles are also possible,
but are less frequent.. In such a process, the imprinted mode $m^{*}$ will
interact with two other quasiparticle modes (denoted by integers $p$ and $q$),
as well as the condensate (denoted $0$), leading to four potential events
[24]: $m^{*}+p\to q+0$ (Landau damping [26, 25]), $m^{*}+0\to p+q$ (Beliaev
damping [39, 40]), $q+0\to m^{*}+p$ (Landau growth) and $p+q\to m^{*}+0$
(Beliaev growth). In the absence of quantum fluctuations (which are neglected
in the c-field simulations), these scattering events can only occur if both
incoming modes and at least one of the outgoing modes have nonzero population.
Additionally, the collisions must conserve both energy and angular momentum.
Since we are interested in long wavelength excitations, the relevant part of
the energy spectrum scales linearly with mode number, $E(m)\sim|m|$. In the
case of Landau damping, the two conservation conditions are therefore
$|m^{*}|+|p|=|q|$ (energy) and $m^{*}+p=q$ (angular momentum). Energy
conservation prohibits interactions between counter-rotating modes, and hence
we restrict our treatment to positive (anticlockwise) modes without loss of
generality.

In the experiment, we expect Landau damping to dominate over all other
processes because of the large initial population in the $m^{*}$ mode, as well
as the large number of decay channels available for such collisions. (Beliaev
damping, on the other hand, is highly restricted due to the low values of
$m^{*}$, in combination with the conservation laws stated above.) We do,
however, identify two unique Landau-type damping processes in the dynamics.
Generally, the incoming mode $p$ will be thermally populated (i.e. $p\neq
m^{*}$), leading to population transfer into another thermal mode $q=m^{*}+p$.
However, in the special case $p=m^{*}$, a phenomenon known as frequency
doubling occurs, where the outgoing mode $q=2m^{*}$. We find that these two
processes lead to significantly different damping behaviour of the imprinted
mode, as explored in the next section.

Figure 4: Exemplar of two decay rates in the c-field model for an imprint with
$m^{*}=5$ and $\alpha_{0}\approx 0.18$, and an initial condensate fraction
$n_{0}\approx 0.81$. (a) Raw Fourier amplitude oscillation (black circles),
and (b) the decay envelope function $\alpha(t)$ (black circles), obtained by
dividing the data by $\sin(\omega_{m^{*}}t+\phi_{0})$, with fitted frequency
$\omega_{m^{*}}=2\pi\times 28$ Hz and phase offset
$\phi_{0}=$3.3\text{\,}\mathrm{rad}$$ (blue shaded region) and
$\phi=$3.1\text{\,}\mathrm{rad}$$ (red shaded region). Note the semi-log scale
in (b). In both (a) and (b), the envelope function is well described by Eq.
(10), with time constants $\tau_{1}\approx$ $50\text{\,}\mathrm{ms}$ for
$t<$90\text{\,}\mathrm{ms}$$ (blue shaded region), and
$\tau_{2}\approx$143\text{\,}\mathrm{ms}$$ for $t>$90\text{\,}\mathrm{ms}$$
(red shaded region). The fits are shown with a cyan solid line and a red
dashed line, respectively. (c) Amplitudes of the imprinted mode $m^{*}=5$ and
higher harmonics (solid lines), vertically offset for clarity. Dashed lines
show fifth-order polynomial fits capturing the envelope $\alpha(t)$ of the
amplitude data. The index $\nu\in\\{R,I\\}$, with real component plotted for
even modes, and imaginary component plotted for odd modes. (d) Comparison of
fitted amplitude envelope polynomials (solid lines), as well as the thermal
mode population $\Lambda_{\mathrm{th}}(t)$ (see text). The red circles
correspond to time-averaged values of $\Lambda_{\mathrm{th}}$, while the
shaded region denotes the standard deviation. The vertical dashed line denotes
the $90\text{\,}\mathrm{ms}$ timescale where the decay rate crosses over, as
measured from (a) and (b).

####  6.2.2 Explanation of observed phonon damping

The two Landau-type decay processes described above become visibly distinct in
the simulations for large imprints ($\alpha_{0}\gtrsim 0.05$), manifesting as
two separate exponential decay constants $\tau_{1}$ and $\tau_{2}$ in the
decay of the envelope function $\alpha(t)$. This behaviour is demonstrated in
Fig. 4 for an example c-field simulation with $m^{*}=5$ and $\alpha_{0}\approx
0.18$. In (a), the raw amplitude of the imprinted mode is plotted as a
function of time (black circles), alongside two fits to Eqs. (9)–(10)—one for
$t<$90\text{\,}\mathrm{ms}$$ (cyan solid line), and one for
$t>$90\text{\,}\mathrm{ms}$$ (red dashed line). In (b), we plot the time-
varying envelope $\alpha(t)$, which we have isolated by dividing by the
sinusoidal component of the fit, $\sin(\omega_{m^{*}}t+\phi_{0})$, with
fitting parameters $\omega_{m^{*}}$ and $\phi_{0}$444We have implemented a
moving-median binning procedure after dividing by the sinusoid in order to
discount singularities that occur where it passes through zero.. The
exponential parts of the fit are also plotted for comparison, clearly
demonstrating that two decay constants are required to describe the data. We
have been unable to definitively observe this phenomenon in the experiment due
to limitations in the signal-to-noise ratio, although the strongest imprint
($m^{*}=13$, $\alpha_{0}\approx 0.18$) shows behaviour consistent with two
exponential decay rates.

Concurrent with this change in decay rate, we notice a change in the behaviour
of the other azimuthal modes in the simulation. Immediately after $t=0$, the
higher harmonics of the $m^{*}=5$ imprint ($m=10$, $15$, …) begin to grow in
population, as seen in Fig. 4(c). To quantify this effect, we have fitted
quintic polynomial curves to approximate the envelopes $\alpha(t)$ of each of
these modes. The fits are plotted in Fig. 4(d) [also shown as dashed lines in
(c)]. The higher harmonics are seen to rapidly grow in amplitude at early
times, before peaking at $\approx$20\text{\,}\mathrm{ms}$$ with a maximum
population that decreases for increasing $m$. This growth of amplitude can be
explained as a form of high harmonic generation (HHG) [41, 42] arising from a
cascade of Landau-type processes $im^{*}+jm^{*}\rightarrow(i+j)m^{*}+0$.
Initially, the frequency doubling process described above ($i=j=1$) converts
population into the $2m^{*}$ mode. This then stimulates ($i=1$, $j=2$)
collisions into the $3m^{*}$ mode, followed by growth of the $4m^{*}$ mode via
($i=1$, $j=3$) and ($i=2$, $j=2$), and so on. In this way, the population
rapidly transfers from the imprinted mode to the higher harmonics. Other modes
that are not integer multiples of $m^{*}$ are initially thermally populated,
and their amplitudes remain at $|A_{m}(t)|\lesssim 0.01$ throughout the
simulation.

After peaking at $\approx$20\text{\,}\mathrm{ms}$$, the populations of the
higher harmonics begin to decay again, facilitated by scattering processes
involving the thermally populated modes. Eventually, they decrease to the
level of the thermal modes, indicating that the HHG process is no longer
dominant. Importantly, around the observed crossover time of
$\approx$90\text{\,}\mathrm{ms}$$, the population in the imprinted mode
becomes almost equal to the population across all the thermal modes,
$\Lambda_{\mathrm{th}}(t)\equiv\sum_{m\neq jm^{*}}|A_{m}(t)|$, as seen in Fig.
4(d)555Note that the thermal mode amplitudes oscillate in time, meaning that
this measure will on average underestimate the true thermal population.. Once
this occurs, the thermal Landau damping process should begin to dominate for
mode $m^{*}$, because collisions with thermal quasiparticles at least as
probable as collisions with imprinted quasiparticles.

Figure 5: $Q$ factor as a function of the initial imprint amplitude
$\alpha_{0}$ for different imprinted mode numbers $m^{*}$ from the c-field
simulations with $n_{0}\approx 0.81$. For $\alpha_{0}\gtrsim 0.05$, two decay
times $\tau_{1}$ and $\tau_{2}$ emerge, giving rise to two distinct $Q$
values, depending on the temporal window used for the fit (‘early’ or ‘late’,
respectively). We attribute the source of the $Q\sim\alpha_{0}^{-1}$ power-law
to be the HHG damping process, while the approximately
$\alpha_{0}$-independent behaviour reflects the Landau scattering from thermal
modes. The ‘early’ data are the same as those plotted in Fig. 3(b–e).

With this understanding, we revisit the numerical results presented in Fig.
3(b–e). By performing additional fits of Eqs. (9)–(10) to $A_{m^{*}}^{(I)}(t)$
using a later temporal window, we find that a second decay time $\tau_{2}$
systematically emerges at large $\alpha_{0}$. Figure 5 shows the $Q$ factor as
measured for this later fitting window (dark circles), with the data from the
earlier fitting window (light squares) also shown for comparison. Two distinct
values of $Q$ are obtained for $\alpha_{0}\gtrsim 0.05$. While the early time
data follow $Q\sim\alpha_{0}^{-1}$ in this region (as identified in Sec. 6.1),
the late time data show little dependence on $\alpha_{0}$. This suggests that
while the early HHG damping process gives rise to nonlinear decay, once the
late time thermal Landau damping process takes over, the dynamics enter an
approximately linear regime. The weak downward trend of $Q(\alpha_{0})$ from
the late fits is also consistent with an increase in heating resulting from
the redistribution of increasingly large imprinted quasiparticle populations
into the thermal modes.

####  6.2.3 Temperature dependence of damping processes

Due to their different origins, the HHG and thermal Landau damping processes
should give rise to distinct behaviour as both the imprint amplitude and the
temperature of the system are varied. For large $\alpha_{0}$ where HHG
initially dominates, lowering the temperature should not significantly affect
the decay rate, since the frequency doubling process is unaffected by the
thermal populations. By contrast, small $\alpha_{0}$ imprints should be
strongly temperature dependent, since thermal Landau damping is the dominant
process. In the limit of both zero temperature and small $\alpha_{0}$, damping
should be minimised, because only very weak HHG should be possible.

To test this hypothesis, we have performed additional simulations of the
$m^{*}=5$ imprint at zero temperature (i.e. condensate fraction $n_{0}=1$).
Figure 6 compares the envelope functions $\alpha(t)$ for $n_{0}=1$ (blue
squares) and $n_{0}\approx 0.81$ (orange circles) for two imprint amplitudes
$\alpha_{0}$. The envelopes have been obtained in the same way as in Fig.
4(b). For weak imprints [panel (a)], we find that the decay rate is much
higher at finite temperature than zero temperature. Consistent with our
prediction, the envelope function for $n_{0}=1$ is approximately flat at early
times, showing a gradual approach to exponential decay as the population
transfers to higher harmonics of $m^{*}$. At the highest amplitudes [panel
(b)], we see that the early-time decay rate is similar at the two
temperatures, demonstrating the temperature-independence of HHG. The late-time
decay rate, on the other hand, is significantly higher at finite temperature,
consistent with the prediction that thermal Landau processes are more
important here.

Figure 6: Envelope functions $\alpha(t)$ from the c-field simulations for an
$m^{*}=5$ imprint at condensate fractions $n_{0}=1$ (blue squares) and
$n_{0}\approx 0.81$ (orange circles), obtained by dividing each amplitude
curve $A_{m^{*}}^{(I)}(t)$ by the sinusoidal component of the fit to Eqs.
(9)–(10). (a) Low initial amplitude imprint, $\alpha_{0}\approx 0.02$; (b)
high initial amplitude imprint, $\alpha_{0}\approx 0.25$. Solid lines show
exponential fits to Eq. (10). In (b), the early-time fits are performed in the
window $t\leq$75\text{\,}\mathrm{ms}$$; all other fits correspond to windows
$t\geq$150\text{\,}\mathrm{ms}$$. Figure 7: $Q$ factor as a function of
condensate fraction $n_{0}$ in the c-field model, for an $m^{*}=5$,
$\alpha_{0}\approx 0.02$ imprint. ‘Early’ and ‘late’ fits correspond to
temporal windows $t\leq$100\text{\,}\mathrm{ms}$$ and
$$150\text{\,}\mathrm{ms}$\leq t\leq$400\text{\,}\mathrm{ms}$$, respectively.
Standard errors are shown.

####  6.2.4 Change in quality factor with temperature

The data in Fig. 6(a) indicate that it should be possible to improve the $Q$
factor of our system by reducing the temperature. To explore this possibility
further, we have repeated our simulations of the $m^{*}=5$, $\alpha_{0}\approx
0.02$ imprint for a range of condensate fractions $n_{0}$. As seen in Fig.
6(a), the envelope $\alpha(t)$ exhibits a slow approach to exponential decay
at the lowest temperatures. We therefore measure the decay constant $\tau$
using two temporal windows: $t\leq$100\text{\,}\mathrm{ms}$$ and
$$150\text{\,}\mathrm{ms}$\leq t\leq$400\text{\,}\mathrm{ms}$$. The resulting
$Q$ factors are presented in Fig. 7. Evidently, the early time $Q$ factors are
significantly larger at the highest condensate fractions, $n_{0}\gtrsim 0.95$.
In this limit, $Q$ factors as large as $\sim 150$ are attainable, provided
that the measurement only probes the first $\sim$100\text{\,}\mathrm{ms}$$ of
evolution. If longer interrogation times are required, the maximum possible
$Q$ drops to $\sim 50$, which is only a minor improvement over our best
experimentally measured values.

##  7 Rotation measurement with a persistent current

###  7.1 Persistent currents

We now turn to testing the performance of our superfluid ring as a rotation
sensor in the case rotation is applied, i.e. $|\Omega|>0$. Due to the
challenges involved with rotating the laboratory FoR, we have instead opted to
induce rotation in the ring by generating persistent currents. This provides a
proof-of-principle demonstration of rotation sensing using phonon
interferometry [14]. However, we note that in a realistic rotation sensing
application, persistent currents are undesirable. Such currents will be
generated if the atoms are rotating faster than the lowest rotation rate the
ring can support, $\Omega_{0}$, when the BEC is formed, introducing ambiguity
into the rotation measurement.

A superfluid confined to a ring geometry is capable of supporting persistent
currents, which arise due to the quantisation of circulation around the ring
in units of $h/m_{\mathrm{a}}$, where $h$ is Planck’s constant. The rotation
rate $\Omega$ is therefore also quantised such that:

$\Omega=l\Omega_{0}\approx\frac{l\hbar}{m_{\mathrm{a}}R^{2}},$ (12)

where $l$ is number of circulation quanta and $R$ the radius of the ring. With
a ring radius of $$45\text{\,}\mathrm{\SIUnitSymbolMicro m}$\lesssim
R\lesssim$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the presence of a single
quantum of circulation (i.e. $l=1$) indicates a rotation rate that varies
within the range
$$0.24\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$\lesssim\Omega_{0}\lesssim$0.36\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$
as a function of radius.

###  7.2 Experimental stirring protocol

We generate persistent currents experimentally by stirring with a potential
barrier [14], as depicted in Fig. 8. First, we load the atoms into a ring-
shaped trap with inner and outer radii
$r_{\mathrm{in}}\approx$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$r_{\mathrm{out}}\approx$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, including
an additional barrier to be used for azimuthal stirring [panel (a)]. We then
expand the inner radius to
$r_{\mathrm{in}}\approx$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in order to
increase the atomic density [panel (b)]. A radial barrier of width
$\approx$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ is then ramped on,
splitting the system into two concentric rings of
$\approx$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness [panel (c)]. The
barrier is then swept around the outer ring for $<1$ cycle, moving with an
angular acceleration of $1.53\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-2}$
until a desired angular velocity is achieved in the outer ring. At this point,
we remove the azimuthal barrier and allow the system to relax for
$500\text{\,}\mathrm{ms}$ so that any excitations generated by the stirring
can dissipate [panel (d)].

To use the system as a rotation sensor, we then imprint phonons into the outer
rotating ring in the same way as described in Sec. 3.2, ignoring the inner
ring [panel (e)]. We can then proceed to measure the rotation rate of the
imprinted phonon mode by measuring the rate of change of the angle
$\theta(t)=\arg\\{A_{m^{*}}(t)\\}$ (see Sec. 5). Alternatively, we can perform
a ‘control’ measurement of the system’s rotation rate by removing the trapping
potential and letting the two rings interfere before taking an image [panel
(f)]. The resulting interference pattern is expected to contain $l$ fringes,
enabling a direct measurement of the circulation [43, 44]. By comparing these
two measurements, we can confirm that the phonon interferometry scheme matches
an established method of measuring rotation.

Figure 8:  Absorption images of the squared density during the stirring
sequence. (a) A thick ring is created, with a barrier for azimuthal stirring
(top right); (b) the inner radius is expanded to increase the atomic density;
(c) a radial barrier is introduced and the azimuthal barrier is rotated to
generate a winding in the outer ring; (d) the azimuthal barrier is removed;
(e) either a sinusoidal phase is imprinted in the outer ring (in this case,
$m^{*}=13$), or; (f) the inner and outer ring are allowed to interfere for a
direct measurement of the winding number (in this case, $l=1$). The squared
density is measured in the same arbitrary units (a.u.) across all frames. The
color scale in (f) applies to frames (a–d,f), while the range of the color
scale for frame (e) has been increased for clarity.

###  7.3 Experimental rotation measurement

We perform rotation measurements for a range of circulation quanta around the
ring, up to $l=9$ (corresponding to
$\Omega\approx$2.5\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$). For
these measurements, we use phonon modes $m^{*}=7$ and $m^{*}=13$ to maximise
the signal-to-noise ratio in the measurement of the phase accumulation rate
$\mathrm{d}\theta/\mathrm{d}t=m^{*}\Omega$. Figure 9 displays an example
measurement for an $m^{*}=13$, $l=9$ imprint, with both the imaginary
component of the Fourier amplitude $A_{m^{*}}^{(I)}(t)$ and the angle
$\theta(t)$ shown. The dashed orange line shows a linear fit to $\theta(t)$,
from which the rotation rate $\Omega$ can be measured. Points where the
imprinted mode amplitude is small ($|A_{m^{*}}^{(I)}(t)|\leq 0.01$, vertical
shaded regions) are excluded from the fit, as these correspond to times when
the density profile is approximately uniform around the ring, resulting in an
undefined angle $\theta$.

Figure 9: Evolution of the Fourier amplitude for an $m^{*}=13$ imprint, with a
persistent current of
$\approx$2.5\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$ around the ring.
Blue squares show the imaginary component of $A_{m^{*}}(t)$ (left axis), while
orange circles show the phase $\theta(t)/m^{*}$ (right axis). The orange
dashed line shows a linear fit to the phase, with the shaded region denoting
one standard error. Vertical grey shading corresponds to regions with small
imaginary component ($|A_{m^{*}}^{(I)}(t)|\leq 0.01$) where phase data are
excluded from the fit. Figure 10: Measured rotation rates
$\Omega_{\mathrm{phonon}}$ from phonon interferometry with $m^{*}=7$ and
$m^{*}=13$ imprints, as a function of applied rotation. (a) Experimentally
measured rotation rates as a function of the rotation as measured from the
fringe count $\Omega_{\mathrm{fringe}}$; (b) rotation rates measured from the
c-field simulations as a function of the applied rotation
$\Omega_{\mathrm{applied}}$. In both frames, the ideal result is shown as a
black line. Note that the winding number $l$ is nondeterministic in the
experiment, whereas the simulations provide precise control over $l$. The
imprint amplitude $0.04\lesssim\alpha_{0}\lesssim 0.08$ for the experimental
data, while $\alpha_{0}\approx 0.1$ was chosen for the simulations. The error
bars indicate the standard error.

Figure 10(a) displays the rotation rate $\Omega_{\mathrm{phonon}}$ as measured
from phonon interferometry, compared to $\Omega_{\mathrm{fringe}}$ obtained
from fringe counting (see Sec. 7.2). All points on the graph are located close
to the ideal scenario $\Omega_{\mathrm{phonon}}=\Omega_{\mathrm{fringe}}$
(solid line), demonstrating a successful implementation of phonon
interferometry for rotation sensing. However, our stirring sequence does not
provide deterministic control of the winding number, introducing variability
in the applied rotation rate. This may explain the large spread in $\theta(t)$
seen in Fig. 9, which is in turn reflected in the vertical error bars in Fig.
10(a). The horizontal error bars are dominated by the variation in rotation
frequency over the finite width of the ring due to the irrotational nature of
the flow (see Sec. 7.1). We note that both of these effects are artefacts of
the use of persistent currents for rotation. In a realistic rotation sensing
scenario, the BEC would rotate as a rigid body relative to the laboratory FoR,
and these sources of uncertainty would not be present. We discuss the
fundamental rotation sensitivity of this system in Sec. 7.5.

###  7.4 Numerical rotation measurement

We simulate the rotation measurements using the c-field model for comparison
with the experimental results. We use the same simulation procedure outlined
in Sec. 4, with an additional step of phase imprinting $l$ circulation quanta,
$\psi\to\psi\mathrm{e}^{il\theta}$, onto the ground state before evolving Eq.
(5) to equilibrium at finite temperature. We perform simulations with
$l=\\{1,2,3,6,9\\}$ and imprinted mode numbers $m^{*}=\\{7,13\\}$.

Following the same analysis procedure as the experiment, we extract the
rotation rate $\Omega_{\mathrm{phonon}}$ by measuring the rate of change of
$\theta(t)$. As for the experimental data, we remove points where
$|A_{m^{*}}(t)|\leq 0.01$ in order to avoid singularities, and we restrict the
fits to $t\lesssim$100\text{\,}\mathrm{ms}$$. We then bootstrap our fit by
repeatedly sampling $10$ randomly selected points out of the remaining $\sim
30$ points in the time sequence, performing a linear fit to each subsample.
The rotation rate and its uncertainty are then obtained from the mean and
standard deviation of the resulting distribution of gradients. These
measurements are plotted as a function of the applied rotation
$\Omega_{\mathrm{applied}}=l\Omega_{0}$ in Fig. 10(b). The data follow the
expected behaviour, although the singularities do still affect the quality of
the fitting, leading to some slight disagreement with the expected trend.
Evidently, the simulation results are more precise than those from the
experiment, with smaller vertical error bars in particular due to the absence
of uncertainty in the applied winding number. However, the horizontal error
bars remain approximately the same size as the experimental data, due to the
aforementioned variation in rotation frequency over the width of the ring.
This result demonstrates the viability of the sensor for practical rotation
measurement and confirms our understanding of the experiment.

###  7.5 Sensitivity of rotation sensor

The performance of the interferometer as a rotation sensor can be quantified
by its sensitivity $\Delta\Omega$, defined as the smallest rotation frequency
it is capable of resolving. Unlike in the previous sections, here we focus on
a scenario where just a single run of the experiment is performed, and a
sample is obtained at an optimal time $t$. The uncertainty in a measurement of
$\Omega$ made from this sample is $\Delta\Omega(t)=\Delta
A_{m^{*}}\left|\mathrm{d}A_{m^{*}}/\mathrm{d}\Omega\right|^{-1}$, which from
Eqs. (9)–(10) gives

$\Delta\Omega(t)=\Delta
A_{m^{*}}\frac{\exp(t/\tau)}{m^{*}\alpha_{0}t|\sin(\omega_{m^{*}}t+\phi_{0})|}.$
(13)

If we make a measurement at time $t=\tau$, the ratio $\exp(t/\tau)/t$ will be
minimised, and $\Delta\Omega(\tau)\propto
1/\tau|\sin(\omega_{m^{*}}\tau+\phi_{0})|$. The global minimum of the single-
shot sensitivity will therefore be achieved if $t=\tau$ also corresponds to a
maximum of the sinusoid (i.e. a maximum of the density perturbation), giving:

$\Delta\Omega_{\mathrm{min}}=\Delta
A_{m^{*}}\frac{\mathrm{e}}{m^{*}\alpha_{0}\tau}.$ (14)

This optimal scenario could be engineered via a change in the frequency
$\omega_{m^{*}}$, for example by adjusting the atomic density.

Expressing this ideal sensitivity in terms of the $Q$ factor (11),

$\Delta\Omega_{\mathrm{min}}\approx\Delta
A_{m^{*}}\frac{\mathrm{e}\omega_{0}}{2\alpha_{0}Q},$ (15)

it becomes evident that the imprinted mode number $m^{*}$ does not affect the
sensitivity, and that $\Delta\Omega_{\mathrm{min}}$ will improve with larger
$Q$. However, in Sec. 6.1 we found that $Q$ and $\alpha_{0}$ are not
independent quantities in general, and in particular that
$Q\propto\alpha_{0}^{-1}$ for strong imprints. In this regime, the sensitivity
becomes completely independent of the properties of the imprinted mode,
leaving only its dependence on the system properties via $\omega_{0}$.

If the density profile measurement is shot-noise limited, we expect
fluctuations in the Fourier amplitude of order $\Delta
A_{m^{*}}=(2N_{\mathrm{atoms}})^{-1/2}$ [1]. In this limit, Eq. (15) predicts
an optimal sensitivity of
$\Delta\Omega_{\mathrm{ASN}}\approx$0.04\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$
for our experiment. For comparison, we directly estimate the fluctuations in
$A_{m^{*}}(t)$ by experimentally measuring the spread $\Delta
A_{m^{*}}^{(R)}(t)$ for zero imprinted rotation, in which case $\langle
A_{m^{*}}^{(R)}(t)\rangle=0$ is expected due to the orientation of the phase
imprint. By doing this for all non-rotating datasets presented in Fig. 3(a),
we estimate that $\Delta A_{m^{*}}$ is approximately eight times larger than
than the shot-noise limited value, leading to an achievable sensitivity of
$\Delta\Omega_{\mathrm{min}}\approx$0.3\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$
from a single measurement at $t=\tau$.

##  8 Conclusions

We have established the sensitivity of a rotation sensing schemed based on the
interference of counter-propagating phonon modes imprinted in the phase of a
ring-shaped Bose–Einstein condensate. Such a device is fundamentally limited
by the lifetime of the imprinted excitations, and the system achieves quality
factors up to a maximum of $Q\approx 27$. This is an improvement over
previously reported values in similar experimental setups [1, 37], but still
many orders of magnitude poorer than commercially available sensors.

By performing c-field simulations of the experiment, we have confirmed that
the low quality factors obtained do not arise from experimental deficiencies.
Rather, the decay of the imprinted excitations is caused by physical damping
processes facilitated by the nonlinearity of the BEC. Intriguingly, the
numerics have revealed two distinct decay processes at work. Scattering
between imprinted and thermal quasiparticles provides a form of damping that
depends only on temperature, while collisions between imprinted quasiparticles
give rise to damping that depends on the imprinting amplitude but not on
temperature. In the limit of zero temperature and weak imprints, where both of
these damping mechanisms should be minimised, our simulations suggest that $Q$
factors up to $\sim 150$ may be attainable in this system.

We have successfully performed a rotation measurement using this phonon
interferometry scheme by creating a persistent current in the ring BEC,
thereby demonstrating a resolution below the fundamental frequency of the
ring,
$\Omega_{0}\approx$0.29\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$, when
fitting to a sequence of measurements. We predict that the optimal rotation
sensitivity attainable from a single run of our experiment is
$\Delta\Omega_{\mathrm{min}}\approx$0.3\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$.
While this is an improvement over previous work [1], it is significantly above
the atomic shot-noise limit of
$\Delta\Omega_{\mathrm{ASN}}\approx$0.04\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$
of our apparatus.

The rotation sensing precision could therefore be improved by reducing the
noise floor, for example via the reduction of optical noise sources. Improving
the signal-to-noise ratio of the measurement would also allow for weaker
imprints to be used, increasing the sensitivity [see Eq. (15)]. Further gains
could be made in the sensitivity by implementing non-destructive imaging,
which would allow for multiple sequential measurements without increasing the
experimental run time [45]. Finally, it might be possible to alter the trap
geometry to limit the available scattering channels, making the imprinted
excitations resistant to decay.

Significant improvements in sensitivity would be required to produce a sensor
of comparable precision to those already commercially available. However, the
superfluid’s absolute FoR does prevent measurement drift, potentially
providing a very significant potential advantage for operation over extended
periods. These devices may therefore prove useful if used in conjunction with
conventional rotation sensors as a means of calibration to reduce uncertainty
in classical systems over prolonged operation [46, 47, 48]. This could be
achieved by repeatedly performing single-shot rotation rate measurements using
the phonon system. The difference between these measurements and those from
the classical system from individual measurements would accumulate to provide
a long term drift value that could thus be subtracted from the classical
measurement [46, 47, 48].

##### Funding information

Funding for this work was provided by a Next Generation Technologies Fund –
Quantum Technologies grant (QT95) by the Commonwealth Defence Science and
Technology Group, Australian Recearch Council Centre of Excellence for
Engineered Quantum Systems (EQUS, CE170100009), and ARC Discovery Projects
grant DP160102085. This research was also partially supported by the
Australian Research Council Centre of Excellence in Future Low-Energy
Electronics Technologies (project number CE170100039). T.W.N. and S.A.H.
acknowledge the support of Australian Research Council Future Fellowships
FT190100306 and FT210100809, respectively.

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# IMPROVEMENT OF AUDIOVISUAL QUALITY ESTIMATION USING A NONLINEAR
AUTOREGRESSIVE EXOGENOUS NEURAL NETWORK AND BITSTREAM PARAMETERS

###### Abstract

With the increasing demand for audiovisual services, telecom service providers
and application developers are compelled to ensure that their services provide
the best possible user experience. Particularly, services such as
videoconferencing are very sensitive to network conditions. Therefore, their
performance should be monitored in real time in order to adjust parameters to
any network perturbation. In this paper, we developed a parametric model for
estimating the perceived audiovisual quality in videoconference services. Our
model is developed with the nonlinear autoregressive exogenous (NARX)
recurrent neural network and estimates the perceived quality in terms of mean
opinion score (MOS). We validate our model using the publicly available INRS
bitstream audiovisual quality dataset. This dataset contains bitstream
parameters such as loss per frame, bit rate and video duration. We compare the
proposed model against state-of-the-art methods based on machine learning and
show our model to outperform these methods in terms of mean square error
(MSE=0.150) and Pearson correlation coefficient (R=0.931).

Index Terms—  nonlinear autoregressive exogenous (NARX), recurrent neural
network (RNN), audiovisual quality estimation, no reference, videoconferencing

##  1 Introduction

A recent study conducted by Cisco predicts that IP video traffic will account
for 82% of all Internet traffic by 2022. The sum of all forms of IP videos,
which include video streaming, video files exchanged through file sharing,
video-streamed gaming, and videoconferencing, will continue to grow in the
range of 80 to 90% of total IP traffic [1].
With this increasing demand for video services, audiovisual quality assessment
becomes an important task to succeed in such competitive markets. The standard
way of measuring perceived audiovisual quality is to conduct subjective tests
where the end user is asked to assess the audio/video quality as a whole or in
various parts on a continuous or discrete scale. However, assessing quality by
conducting subjective tests is resource consuming and is not feasible in real-
time communications.
When studying the characteristics of a real-time multimedia communication, it
is necessary to have a clear understanding of the impact of the available
quality of service (QoS) parameters on perceived audiovisual quality [2]. The
QoS handles purely the technical aspects of a service [3] while the perceived
quality incorporates human-related factors.
In this paper, we propose a nonlinear autoregressive exogenous (NARX)-based
model, which uses QoS parameters to estimate the perceived audiovisual quality
from for real-time audiovisual services such as videoconference. Specifically,
we consider the persistence aspect of videoconferences and add the temporal
dimension to the perceived audiovisual quality. We suppose that the feedback
from recent events is important to estimate the current perceived audiovisual
quality. In addition, we show that this feedback information increases the
accuracy of the quality estimation model. This could encourage service
providers to adopt deep learning algorithms to optimize and control their
services.
To the best of our knowledge, the NARX algorithm has not yet been applied to
estimate the perceived audiovisual quality. This is primarily due to the
quality perception simplification that results from ignoring its temporal
aspect. Another reason is the fact that most of existing video databases have
not considered the sequential aspect of videos. In this paper, we
experimentally show that the NARX model can outperform state-of-the-art
machine learning algorithms for audiovisual quality estimation.

##  2 Related work

A large body of work was conducted on video quality assessment based on QoS
parameters, however, most of them were performed for video streaming services.
In [4], the authors present a broad review of perceived quality modeling and
methodologies. In the specific case of audiovisual services, standardized
models have achieved good prediction performance. More recently, various
machine learning methods, for instance based on deep learning, were proposed
to estimate the perceived audiovisual quality. In this section, we briefly
present some of these standardized methods developed for streaming services
and video telephony applications. Afterwards, we summarize the recent works
using machine learning to estimate the perceived audiovisual quality.
The ITU-T G.1070 [5] is a model recommended for video telephony service
planning. This model is derived from the E-model [6] which was standardized
for telephony over IP. The model takes as inputs some video parameters such as
codec type, packet loss rate, bit rate, key frame interval, frame rate, video
format and display size. Its main goal is to estimate audiovisual quality,
while taking into consideration human perception so that service planners can
avoid under-engineering. The use of this model is limited to quality planning,
and applications such as monitoring are not covered by the recommendation.
The ITU-T P.1201 [7] model is a non-intrusive model recommended for estimating
the audiovisual quality of streaming services. This model provides separate
estimation for audio, video, and audiovisual quality, where the output or
quality estimation is rated on the five-point mean opinion score (MOS) scale.
It considers parameters such as compression, packet loss, re-buffering, frame
rate, video resolution and bit rate affect audiovisual quality. For mobile TV
(lower-resolution application) and IPTV (higher-resolution application), the
Root Mean Square Error (RMSE) and Pearson correlation values for audiovisual
modeling were evaluated as 0.470 and 0.852, 0.435 and 0.911, respectively.
However, by design, these methods have limited application areas and cover
limited coding technologies such as IPTV. Therefore, with the rise of the deep
learning algorithms, many recent studies have focused on extending and
improving these models.
Goudarzi et al. [8] used a regression model to estimate audiovisual quality
using two parameters corresponding to packet loss rate and frame rate. Their
work also investigated how network and application parameters influence the
overall audiovisual quality. The proposed model was evaluated under 60
different test conditions, where videos were degraded with packet loss and
using different bit rates, and achieved a Pearson correlation coefficient of
0.93 and RMSE of 0.237 for the audiovisual database used in experiments.
Recently, Demirbilek et al. [9] extended the work of Goudarzi et al. for
videoconferencing services using machine learning algorithms such as decision
trees, random forest and multi-layer perceptron (MLP). In [10], the same
authors updated the database of Gourdazi et al. [8] with additional QoS
parameters than packet loss and frame rate, and estimated the perceived
audiovisual quality of different machine learning algorithms on this extended
database. Futhermore, these two authors extend their audiovisual database with
information from the stream (bitstream).
While autoregressive nonlinear networks like the NARX have led to impressive
results in various applications [11, 12], their potential for videoconference
audiovisual quality assessment based on quality of service (QoS) parameters
(no-reference model) has so far not been investigated.

##  3 PROPOSED NARX-BASED APPROACH

###  3.1 NARX model

The nonlinear autoregressive with exogenous inputs (NARX) is a class of
recurrent neural network capable of efficiently modeling time series data with
feedback connections [13].

Fig. 1: NARX network with delayed inputs ($d_{u}$) and delayed outputs
($d_{y}$); $z^{1}$ is unit time delay.

The NARX model has become popular in recent years because of its performance
and fast convergence [11, 12]. Some researchers even listing this model among
the most efficient neural networks [14]. The main difference between the NARX
and conventional recurrent neural networks (RNNs) is that it uses feedback
connections from the outputs to the inputs while RNNs have recurrent
connections in the hidden layers [15]. Siegelmann et al. [14] have shown that
gradient descent learning is more effective in NARX networks than in standard
recurrent neural network architectures. When using the information from the
exogenous inputs, the NARX model provides more flexibility compared with the
feedforward network. This makes the model more effective than other neural
networks and motivates its choice to solve our audiovisual quality estimation
problem.
Studies have shown the robust performance of NARX for approximating almost
every nonlinear function [15]. The output $y(n)$ (Fig. 1) of the NARX model is
the result of a nonlinear operation on multiple past outputs and an external
variable:

$\begin{split}y(n)\,=f\,(\,y(n-1),\,y(n-2),\ldots.,\,y(n-d_{y});\\\
\,\,u(n),\,u(n-1),\,\ldots.,\,u(n-d_{u}))\end{split}$ (1)

From this equation, we see that there are two types of inputs to a NARX: past
outputs $y(n-1)$, $y(n-2)$, … , $y(n-d_{y})$ that are fed back as future
inputs (illustration of the dynamic model), and previous (and current)
external variables $u(n)$, $u(n-1$), … , $u(n-d_{u})$, where $d_{y}$ is the
number of lags in the outputs and $d_{u}$ is the number of lags in the
external variables (inputs). The autoregression function $f$ is approximated
by a feed-forward multi-layer perceptron (MLP) neural network.
One should note that other neural networks can be derived from the NARX
architecture. For example, without lag ($d_{y}=0$ and $d_{u}=0$), the model
becomes a simple feed-forward multi-layer perceptron (MLP) network. Moreover,
when there is no feedback loop of delayed outputs, the model is equivalent to
a time delay neural network (TDNN) [16].

###  3.2 Proposed NARX model to estimate the perceived audiovisual quality

In this section, we propose a NARX-based model for estimating the perceived
audiovisual quality using QoS parameters. As shown in Fig.1, the basic NARX
model consists of inputs corresponding to QoS-related parameters, a hidden
layer and an output layer. In this work, we consider QoS parameters from the
INRS bitstream audiovisual dataset such as aFrame count, pFrame count, video
bit rate as inputs (see more details in Table 2). The output of the NARX
network is the mean opinion score (MOS) ranging from 1 to 5. To reduce the
model’s complexity, we use a single hidden layer with $N$ nodes. As
recommended for the design of MLP networks [15], we set the number of neurons
in the hidden layer equal to the total numbers of inputs parameters.
The hyperbolic tangent is used for the transfer function in the hidden layer,
while a linear transfer function is employed for the output layer. For
training, we selected the Bayesian regularization [17, 18] training algorithm
which reduces overfitting by penalizing the $L2$ norm of network parameters.
During training, the inputs parameters and their corresponding MOS (from the
INRS Bitstream dataset) are used to learn the model’s parameters. The inputs
or QoS parameters such as packet loss are collected periodically in the
videoconferencing gateway through RTCP (Real-Time Transport Control Protocol)
reports. For example, the RFC 3550 [19] recommends the collection of these
metrics every 5 seconds. Thus, the inputs data are sequential and could be
considered as time-series data.
For the NARX model, in the presence of labeled data (supervised learning), the
training method is called open loop (OL) configuration. During the testing,
the estimated MOS value is obtained from the trained model by feeding it the
inputs parameters.

##  4 SIMULATION RESULTS AND ANALYSIS

In this section, we evaluate the performance of our proposed NARX-based model
for perceived quality estimation and compare it against state-of-the-art
machine learning methods for videoconferencing [9].

###  4.1 INRS Bitstream Audiovisual Dataset

Recently, INRS designed an audiovisual quality dataset namely “INRS
Audiovisual Quality Dataset” [9, 10]. This dataset was proposed to update the
PLYM dataset [8] which considered the importance of compression and network
distortion in perceived quality estimation. Studies have shown that, for the
videoconference service, the change in the transmission packet loss can have a
dramatic impact on perceived quality. In addition, the video frame rate and
quantization parameter may also affect quality, although more moderately.
Moreover, the authors of “INRS Audiovisual Quality Dataset” released two
versions of this dataset called parametric and bitstream. While the parametric
version is limited to 4 parameters namely packet loss rate, video
quantization, noise reduction filters and video frame rate (see Table 1), the
bitstream version collects information on group of frames for each video and
contains 140 parameters retrieved in terms of $I$ and $P$ frames. These 140
parameters reported in the bitstream dataset are also referred as “Bitstream
parameters”.

Table 1:  QoS parameters used to build INRS Audiovisual Dataset [10] Parameters | Values
---|---
Frame per second (FPS) | 10, 15, 20, 25
Quantization parameter (QP) | 23, 27, 31, 35
Noise reduction filters (NRF) | 0.999
Audiovisual packet loss rate (PLR) (%) | 0, 0.1, 0.5, 1, 5

Note that the two versions of “INRS Audiovisual Quality Dataset” are publicly
available and for our study, we selected the bitstream version as the
parameters in terms of $I$ and $P$ frames are relevant for the quality
estimation. Furthermore, the bitstream version of the “INRS Audiovisual
Quality Dataset” could be considered as the improvement of the parametric
version because it contains both parametric values and information of headers
and the payload of the video bitstream for each video. Thus, the bitstream
version provides a maximum of information about the audio and video.
The video used to create the datasets were encoded with H.264 and a total of
160 degraded videos were obtained by applying various network conditions and
application parameters to the reference or original video (see the values in
Table 1). There is one $I$ frame every 10 seconds in the reference video and
the total video has three $I$ frames. The subjective method used to obtain the
MOS for each video is ACR (Absolute Rate Scaling) and the observers were
allowed to provide their subjective scores after listening and watching to the
first 10 seconds of the video [9, 20].
More detailed information about the INRS audiovisual quality dataset regarding
the selected video sequence, test methodology and comparison to the other
publicly available datasets is given in [9, 10]

As the bitstream dataset contains 140 parameters, we use random forest
parameters features selection algorithm to obtain the top important
parameters. From this operation, we got 9 important parameters as shown and
defined in Table 2.

Table 2:  Nine (9) important parameters selected from INRS Bitstream
Audiovisual Dataset [9] Parameters
---
aFrame Count | Total count of audio frames
pFrame Count Diff |  Total loss rate in the count of pframes.
S2 pFrame Count Diff |  Loss rate in the count of P-frames for the 2nd second.
S3 pFrame Mean Diff |  Loss rate in the average size of the p-frames for the 10th second.
S10 pFrame Mean Diff |  Loss rate in the average size of the p-frames for the 10th second.
Audio NB Frames
Audio Bit Rate
Video Bit Rate
Video Packet Loss Rate

To have a better estimation of performance, we employed a $k$-fold cross-
validation strategy [21]. In this strategy, the available data is split in $k$
even-sized subsets and, for each cross-validation fold, a different subset is
set aside for testing while remaining examples are used to train the model.
Training examples in each fold are further divided following a 70%/30%
training-validation split for learning the network weights (training set) and
selecting optimal training hyper-parameters (validation set). In this work, we
used $k=5$ folds in our experiments.

###  4.2 Results and Discussion

In the following experiments, the performance of our NARX-based model is
compared to that of recent work on audiovisual services using the same
database [10]. This previous works showed that machine learning algorithms
such as random forest (RF) perform better than the MLP algorithm for
estimating the perceived audiovisual quality.
To evaluate the performance of our model under various configurations and
compare it to the state-of-the-art algorithms, we chose two metrics
recommended by the ITU-T P.1401 [22]: Pearson correlation coefficient (R) and
Mean squared error (MSE). For selecting the model applied on test examples, we
stopped the training when the validation error reached its minimum, as shown
in Figure 2.

Fig. 2: Example of training, validation and test errors for a single fold.

Different topologies were tested for our NARX model, using either no feedback
of delayed outputs ($d_{y}=0$) or a feedback of delayed outputs equal to that
of delayed inputs ($d_{y}=d_{u}$). In the latter case, the model estimates the
perceived audiovisual quality (estimated MOS) at time $t$ based on the values
of the QoS parameters (inputs) at time $t-1$, $t-2$, …, $t-d_{u}$ and past
outputs $y(t-1)$, $y(t-2)$, …, $y(t-d_{y})$.

Table 3 reports the performance obtained for three settings of our NARX model:
no feedback of delayed inputs with $d_{u}=3$, feedback of delayed inputs and
outputs with $d_{y}=d_{u}=3$, and feedback of delayed inputs and outputs with
$d_{y}=d_{u}=4$. Note that other settings were tested, however, they did not
improve the performance; hence we do not report them in the tables. As in [9],
we simulated and compared our results with the random forest, linear
regression, bagging and MLP algorithms.

Table 3:  Performance Comparison of Various Models on the INRS Bitstream Audiovisual Dataset [10] Model | MSE |  Pearson Correlation
---|---|---
Multi Layer Perceptron (MLP) | 0.270 | 0.851
Random Forest (RF) | 0.143 | 0.921
Bagging | 0.160 | 0.912
Our NARX ($d_{y}=0,d_{u}=3$) | 0.203 | 0.911
Our NARX ($d_{y}=d_{u}=3$) | 0.144 | 0.932
Our NARX ($d_{y}=d_{u}=4$) | 0.170 | 0.931

We observe that our proposed NARX-based perceived audiovisual quality
estimation model outperforms the three other audiovisual quality models for
all three tested settings of delayed inputs and outputs. Comparing the
settings together, the best performance is achieved for input and output lags
of $d_{y}=d_{u}=3$, with Pearson correlation coefficient of 0.932 and MSE of
0.144 (see Table 3). When using the bitstream dataset, we obtain a clear
understanding of these setting as the database is set with a reference video
that has three $I$ frames. So, the input and output lags of $d_{y}=d_{u}=3$ is
the best setting for the videos in this dataset. Thus, the best performance is
obtained when using the information about the three $I$ frames in the video
during the quality prediction. When, we increase the input and output lags i.e
$d_{y}=d_{u}=4$, there is no increasing in Pearson correlation and also we
obtained the value of MSE increased. The others frames don’t add any
improvement to the prediction.

Fig. 3: Performance in terms of Pearson Correlation (R) and Mean Squared Error
(MSE) for 5-fold

Results also show the usefulness of performing autoregression on both inputs
and outputs, as highlighted by the improvement in MSE and Pearson correlation
compared to the setting with $d_{y}=0$ and $d_{u}=3$. Increase the number of
delayed inputs and outputs also leads to a higher performance, although this
trend does not continue beyond a lag of 4. Figure 3 gives the MSE and Pearson
correlation values obtained for individual folds, while using $d_{y}=d_{u}=3$.
As can be seen, the performance remains stable across folds with a small
variance in both metrics.

##  5 CONCLUSION

In this paper, we presented a no-reference NARX-based perceptual quality
estimation model for audiovisual service. In the proposed model, the
application and network parameters such as the frame rate, quantization
parameter, packet loss rate and noise reduction filters were taken into
consideration. We evaluated and compared our proposed NARX-based perceptual
quality estimation model using the INRS audiovisual dataset. Simulation
results showed that our proposed NARX-based model outperforms the state-of-
the-art models in terms of Pearson correlation coefficient and MSE. The
results indicate that our NARX-based perceived audiovisual quality estimation
is a practical method for the assessment of video perceived quality. Moreover,
with only a single hidden layer, our NARX-based model can well estimate the
perceived quality. Future works will target other QoS parameters having an
impact on perceived quality and use them to improve the results of the
perceived audiovisual quality estimation.

###  5.1 ACKNOWLEDGEMENT

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada (NSERC), and in part by Dr. Richard J. Marceau
Chair on Wireless IP Technology for Developing Countries, particularly through
our industrial partner, Media5 Corporation.

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