aainstitutetext: Departament de Física Quàntica i Astrofísica and Institut de
Ciències del Cosmos (ICC), Universitat de Barcelona, Martí i Franquès 1,
ES-08028, Barcelona, Spain.bbinstitutetext: Institució Catalana de Recerca i
Estudis Avançats (ICREA), Passeig Lluís Companys 23, ES-08010, Barcelona,
Spain.ccinstitutetext: CPHT, CNRS, École Polytechnique, IP Paris, F-91128
Palaiseau, France

# Mega-Hertz Gravitational Waves from Neutron Star Mergers

Jorge Casalderrey-Solana b,a David Mateos a,c Mikel Sanchez-Garitaonandia

###### Abstract

Neutron star mergers provide a unique laboratory for the study of strong-field
gravity coupled to Quantum Chromodynamics in extreme conditions. The
frequencies and amplitudes of the resulting gravitational waves encode
invaluable information about the merger. Simulations to date have shown that
these frequencies lie in the kilo-Hertz range. They have also shown that, if
Quantum Chromodynamics possesses a first-order phase transition at high baryon
density, then this is likely to be accessed during the merger dynamics. Here
we show that this would result in the nucleation of superheated and/or
supercompressed bubbles whose subsequent dynamics would produce gravitational
waves in the Mega-Hertz range. We estimate the amplitude of this signal and
show that it may fall within the expected sensitivity of future
superconducting radio-frequency cavity detectors for mergers at distances up
to tens of Mega-parsecs.

##  1 Introduction

The revolutionary discovery of gravitational waves (GW) has opened two
unprecedented windows into the Universe: one into the strong-field regime of
gravity, and another one into the dynamics of quantum matter. These two
aspects are often intertwined. This is beautifully illustrated by neutron star
(NS) mergers, in which quarks and gluons described by Quantum Chromodynamics
(QCD) interact in the strong gravitational field sourced by the quarks and
gluons themselves.

Most of the merger dynamics can be accurately described by numerical
simulations of magneto-hydrodynamics coupled to general relativity (for a
review, see e.g. Baiotti:2016qnr ). One robust conclusion of these simulations
is that the characteristic lifetime of the system is of order of several,
possibly 10, milliseconds (ms). These simulations take as an input the
properties of QCD, in particular its Equation of State (EoS). At zero baryon
density the EoS can be reliably determined via the lattice formulation of QCD.
In contrast, the famous sign problem deForcrand:2009zkb  renders this tool
inapplicable at the baryon densities of interest for NS physics. In
particular, it is unknown whether QCD possesses first-order phase transitions
(FOPT) in this region. Nevertheless, a variety of arguments suggest that at
least two FOPT may be present Stephanov:2004xs ; Alford:2007xm ;
Fukushima:2010bq ; Guenther:2022wcr , as sketched in Fig. 1. One is the
transition from hadronic matter to deconfined, quark matter. The other is the
transition from a non-superconducting to a color-superconducting phase. We
emphasize that, while these transitions are well motivated, at the theoretical
level their existence cannot be rigorously established at present. NS mergers
have the potential to establish this experimentally, possibly in combination
with heavy ion collision experiments. To realise this potential, a theoretical
understanding of the imprints of a phase transition on the GW spectrum is
indispensable.

Figure 1: Two possible phase transitions in QCD, indicated by the solid red
curves. $T$ and $\mu$ are the temperature and the baryon chemical potential,
respectively. The dotted black curve on the left panel shows a possible
evolution of a region of a NS merger as this region is heated and/or
compressed. The points dubbed $A$, $A^{\prime}$ and $C$ correspond to the
states shown in Fig. 3.

Numerical simulations of NS mergers based on EoS with a hadronic-quark matter
phase transition include Most:2018eaw ; Most:2019onn ; Ecker:2019xrw ;
Prakash:2021wpz ; Weih:2019xvw ; Tootle:2022pvd . These studies have shown
that the dynamics of the merger results in the formation of regions in which
the matter is sufficiently heated and/or compressed so that the
thermodynamically preferred phase is the quark-matter phase. We will
generically refer to these regions as “Hot or Compressed Spots”, or HoCS for
short — see Fig. 2. The characteristic size of these HoCS is Tootle:2022pvd

$L\simeq 5\,\mbox{km}\,.$ (1)

To our knowledge, no simulation based on an EoS that includes a transition to
a color-superconducting phase has been performed. Nevertheless, existing
simulations have shown that the merger leads to the formation of cold quark-
matter regions in which the baryon density can be ten times larger than the
nuclear saturation density. This makes it conceivable that color-
superconducting matter may be formed in NS mergers.

Numerical simulations have shown that the _presence_ of HoCS affects the
overall merger dynamics, mainly because of a softening of the EoS, and leads
to detectable modifications of the GW spectrum in the usual kilo-Hertz (kHz)
frequency range. One may think of this as a “macroscopic” consequence of the
phase transition. However, to the best of our knowledge, the _phase transition
dynamics_ in HoCS has not been previously considered. The purpose of this
paper is to show that this dynamics leads to a characteristic signal in the
Mega-Hertz (MHz) range. One may think of this as a “microscopic” consequence
of the phase transition. We will see that this conclusion is extremely robust
since it only depends on generic properties of a FOPT. Therefore we do not
need to commit ourselves to a particular type of FOPT. The only assumptions we
need to make are that some FOPT is present in QCD and that this is accessed by
the merger dynamics.

Figure 2: Formation of HoCS in the merger of two NS. The picture corresponds
to a snapshot at a time before bubble nucleation begins. This means that the
green region is in a state in the lower stable branch in Fig. 3, whereas the
HoCS are in a state in the lower metastable branch between points $A$ and $B$.
Nucleation and the subsequent dynamics are shown in Fig. 4.

The key idea is that the millisecond characteristic time scale for the
evolution of the merger is much longer than the characteristic nuclear time
scale, 1 fm $\simeq$ $10^{-23}$ s. This huge separation of scales means that,
from the viewpoint of QCD processes, the merger evolution is adiabatic to an
extremely good approximation. In turn, this implies that the HoCS are
initially born as carefully prepared metastable regions of superheated and/or
supercompressed matter. Once the superheating or supercompression are large
enough, namely once the HoCS is sufficiently deep into the metastable branch,
bubbles of the stable phase begin to nucleate. The point where this happens is
labeled “$B$” in Fig. 3. These bubbles then expand to a macroscopic size and
collide, leaving behind long-lived sound waves propagating on top of the
stable phase, as illustrated in Fig. 4. As we will explain below, this
dynamics gives rise to a characteristic GW spectrum whose peak frequency is in
the MHz range.

As usual, we work with units such that $\hbar=c=1$.

##  2 Frequency

We begin by noting that the dynamics of a FOPT in a NS merger is very similar
to that of a cosmological FOPT (for a review, see e.g. Hindmarsh:2020hop ),
except for the fact that in the cosmological case the metastable phase is
supercooled. In this case there is a compelling physical picture that emerges
from the large body of work developed over several decades. Here we will take
advantage of this picture to give an order-of-magnitude estimate of the GW
signal in a NS merger. A more refined analysis will be presented elsewhere
else .

In the cosmological case the separation of scales is provided by the fact that
the expansion rate of the Universe, $H^{-1}$, with $H$ the Hubble rate, is
much longer than the microscopic time scale given by the inverse of the local
temperature, $T^{-1}$. As the Universe expands and cools down it eventually
enters the metastable phase. At some point bubbles of the stable phase begin
to nucleate. These bubbles then grow and collide and eventually leave behind a
superposition of long-lived sound waves propagating on the stable phase. At
sufficiently late times turbulence may also develop. Although each of these
processes contributes to the GW spectrum, for many models of the transition it
has been shown that the dominant contribution comes from collisions of sound
waves with one another Hindmarsh:2013xza . As a first approximation, we will
assume that the same may be true in a NS merger. We will now determine the
peak frequency of the GWs, and in the next section we will estimate their
characteristic strain.

Figure 3: Energy density as a function of the position along the dashed black
curve in Fig. 1. Both $T$ and $\mu$ increase from left to right. The dotted
black vertical line indicates the location of the phase transition, determined
by the condition that the states $A$ and $A^{\prime}$ have the same free
energy density. The solid green curves indicate stable states. States on the
dashed blue curves are metastable. The doted red curve corresponds to unstable
states. As a region is heated or compressed it enters the lower metastable
branch. At the point $B$ the condition (12) is satisfied. At this point
bubbles of the preferred state $C$ on the upper stable branch are quickly
nucleated.

Let $R$ be the mean bubble centre separation at the end of the phase
transition, as illustrated in Fig. 4. This scale is inherited by the sound
waves that are left behind after the bubbles collide, and it is further
imprinted on the GWs produced by the subsequent fluid dynamics. The peak GW
frequency, $\lambda$, is therefore Hindmarsh:2020hop ; Hindmarsh:2015qta

$\lambda\simeq R\,.$ (2)

Nucleated bubbles grow with a constant bubble wall velocity $v_{w}$. A
reasonable ballpark value for this quantity is

$v_{w}\simeq 0.1\,.$ (3)

We will see below that, if $\beta^{-1}$ denotes the duration of the phase
transition, then the mean bubble centre separation is given by

$R=\left(8\pi\right)^{1/3}\frac{v_{w}}{\beta}\,.$ (4)

Consequently, in order to determine the GW wavelength, we must estimate the
duration of the phase transition. For this purpose we need to recall how the
phase transition takes place dynamically.

Suppose that some region of the NS follows the trajectory indicated by the
dashed black curve in Fig. 1(left). The energy density of the available phases
of the system as $T$ and $\mu$ vary along this curve is shown in Fig. 3. The
multivaluedness of this plot is a hallmark of a FOPT. The difference in the
energy density between points $A^{\prime}$ and $A$ corresponds to the latent
heat. A similar multivaluedness and a similar discontinuity across the phase
transition are found in the entropy density and the baryon density, which also
increase as the phase transition is crossed along the trajectory in Fig.
1(left). The free energy density is multivalued but continuous across the
phase transition. The location of the phase transition, indicated by the
dotted black vertical line, is determined by the condition that the free
energy densities of points $A$ and $A^{\prime}$ are equal to one another. The
different colors in Fig. 3 indicate stable, metastable and unstable states.
Note that only the stable states are shown in the phase diagram of Fig.
1(left), whose hadronic and quark-matter phases correspond to the lower and
upper stable branches of Fig. 3, respectively. Points $A$ and $A^{\prime}$ are
approached as the phase transition curve is approached along the hadronic and
the quark-matter phases, respectively. Although the quantitative details of
Fig. 3 depend on the specific trajectory in Fig. 1(left), its qualitative
features only depend on the fact that this trajectory crosses a line of FOPTs.

Figure 4: Phase transition dynamics inside a HoCS. First, microscopic bubbles
of the stable phase are nucleated inside the metastable phase (left). The
bubbles then grow to a macroscopic size $R$ and collide (center), eventually
leaving behind long-lived sound waves of characteristic size $R$ propagating
on top of the stable phase (right). Once some region of the NS enters the
metastable region, as in Fig. 2, it takes a time of order $\tau\simeq 1$ ms
for bubbles to begin to nucleate, as depicted in the left panel of this
figure. It then takes a time $\beta^{-1}\simeq 6\,\mu$s to go from the left
panel to the right panel. Finally, the lifetime of the sound waves in the
right panel is again of order $\tau\simeq 1$ ms.

The reason why metastable (and unstable) states are not represented in Fig. 1
is that, by definition, the phase diagram only captures the _static_
properties of the system, namely the globally thermodynamically stable states.
Yet, metastable states play a crucial role in the way the FOPT takes place
_dynamically_. Indeed, as the region in the NS is heated and/or compressed
along the trajectory of Fig. 1(left), the corresponding state moves along the
lower stable branch in Fig. 3 until it eventually enters the lower metastable
branch. Once this happens, bubbles of the stable phase can begin to nucleate.
On general grounds, the probability per unit volume and per unit time for this
to happen takes the form

$\frac{dP}{dt\,d^{3}x}=\Lambda^{4}\,e^{-S[\Lambda]}\,.$ (5)

In this equation, $\Lambda$ is a linear combination of the temperature and the
chemical potential that depends on the specific point on the metastable
branch. In a NS merger the natural value for this scale is (see e.g.
Annala:2019puf )

$\Lambda^{4}\simeq 1\,\mbox{GeV}/\mbox{fm}^{3}\,.$ (6)

$S$ is the action of the critical bubble at the same point in the metastable
phase. The critical bubble is the bubble for which the inward-pointing force
due to the surface tension is exactly balanced by the outward-pointing force
due to the pressure difference between the inside and the outside of the
bubble. The action of this bubble controls the bubble nucleation probability
in the semiclassical approximation, hence the exponential dependence in (5).
The value of $S$ is infinite at the point $A$ where the metastable phase meets
the stable phase, and it decreases until it reaches zero at the turning point
where the metastable phase meets the unstable phase. At an arbitrary point on
the metastable branch the value of $S$ depends on the corresponding values of
the temperature and the chemical potential.

Let $t_{c}$ be the time when the system enters the metastable phase. At a
later time $t>t_{c}$, the fraction of the HoCS volume still in the metastable
phase is Guth:1981uk ; Enqvist:1991xw

$q(t)=e^{-I(t)}\,,$ (7)

with

$I(t)=\int_{t_{c}}^{t}\,dt^{\prime}\,\frac{4}{3}\pi
v_{w}^{3}(t-t^{\prime})^{3}\,\Lambda^{4}e^{-S(t^{\prime})}\,.$ (8)

Eqn. (7) is easily interpreted. The integrand is the volume of a bubble
nucleated at a time $t_{c}<t^{\prime}<t$ multiplied by the bubble nucleation
probability at that time. At early times $I(t)\ll 1$ and

$q(t)\simeq 1-I(t)\,.$ (9)

This is the expected expression if bubble overlaps are ignored. The
exponentiation in (7) accounts for these overlaps.

Since the integrand in (8) is dominated by times $t^{\prime}$ near $t$, we can
Taylor-expand the action to linear order in $t-t^{\prime}$ to obtain

$q(t)\simeq\exp\left[-\frac{4}{3}\pi
v_{w}^{3}\,\Lambda^{4}e^{-S(t)}\,6\beta^{-4}\right]\,,$ (10)

where

$\beta=-\frac{dS}{dt}\,.$ (11)

The phase transition takes place at the time $t_{f}$ where the exponent in
(10) becomes unity, namely when

$8\pi v_{w}^{3}\,\Lambda^{4}e^{-S(t_{f})}\,\beta^{-4}=1\,.$ (12)

At this time a fraction $1/e\simeq 37\%$ of the volume remains in the
metastable phase. Expanding (10) around $t_{f}$ we see that $q(t)$ takes the
very simple form

$q(t)\simeq\exp\left[-e^{\beta(t-t_{f})}\right]\,.$ (13)

This equation justifies the interpretation of $\beta^{-1}$ (evaluated at
$t=t_{f}$) as the duration time of the transition. This time can be determined
from the condition (12). To do so, we first use the chain rule to rewrite (11)
as

$\beta\tau=-\Lambda\frac{dS}{d\Lambda}\,,$ (14)

where

$\frac{1}{\tau}=\frac{1}{\Lambda}\frac{d\Lambda}{dt}\,.$ (15)

The time $\tau$ is the characteristic evolution time of the system which,
based on NS merger simulations, is of order

$\tau\simeq 1\,\mbox{ms}\,.$ (16)

Generically, we expect the dimensionless derivative of $S$ on the right-hand
side of (14) to be of the same order as $S$ itself. Therefore

$\beta\tau\simeq S\,,$ (17)

where $S$ is understood to be evaluated at $t=t_{f}$. Substituting in (12) we
then have

$S^{4}\,e^{S}\simeq 8\,v_{w}^{3}\tau^{4}\Lambda^{4}\simeq 10^{81}\,,$ (18)

where we have used (3), (6) and (16). The number on the right-hand side comes
from the huge hierarchy between the microscopic scale, $\Lambda$, and the
macroscopic scale, $\tau$. This hierarchy is responsible for the adiabaticity
of the evolution until bubbles begin to nucleate. Solving for the action we
find

$S\simeq 166\,,$ (19)

and using (16) and (17) we determine the duration of the phase transition to
be

$\beta^{-1}\simeq\frac{1}{166}\,\mbox{ms}\simeq 6\,\mu\mbox{s}\,.$ (20)

We can now determine the mean bubble center separation. The number of bubbles
per unit volume at a time $t>t_{c}$ is the integral until that time of the
probability of nucleating bubbles in the metastable phase, times the available
volume of metastable phase. Thus at a time $t>t_{c}$ this is given by

$n_{bubble}(t)=\int_{t_{c}}^{t}dt^{\prime}\,q(t^{\prime})\,\frac{dP}{dt^{\prime}\,d^{3}x}\,.$
(21)

Using the Taylor expansions around $t_{f}$ introduced above, this becomes

$\displaystyle n_{bubble}(t)$ $\displaystyle=$
$\displaystyle\Lambda^{4}e^{-S(t_{f})}\int_{t_{c}}^{t}dt^{\prime}\,\exp\left[-e^{\beta(t-t_{f})}\right]\,e^{\beta(t-t_{f})}$
(22) $\displaystyle=$
$\displaystyle-\Lambda^{4}e^{-S(t_{f})}\beta^{-1}\int_{t_{c}}^{t}dt^{\prime}\,\frac{d}{dt^{\prime}}\exp\left[-e^{\beta(t-t_{f})}\right]$
$\displaystyle=$
$\displaystyle\Lambda^{4}e^{-S(t_{f})}\beta^{-1}\Big{[}1-q(t)\Big{]}\,.$

At late times $q(t\to\infty)\to 0$ and, using (12), we arrive at the final
density of bubbles

$n_{bubble}=\frac{\beta^{3}}{8\pi v_{w}^{3}}\,.$ (23)

Since the mean bubble center separation is defined as $R=(n_{bubble})^{-1/3}$,
this leads to (4). Using (2), (3) and (20) we finally arrive at the value

$\lambda\simeq R\simeq 530\,\mbox{m}$ (24)

for the GW wavelength, corresponding to a frequency

$f\simeq 0.6\,\mbox{MHz}\,.$ (25)

To recap, once some region of the NS enters the metastable region, as in Fig.
2, it takes a time of order $\tau\simeq 1$ ms for bubbles to begin to
nucleate, as depicted in the left panel of Fig. 4. The typical number of
nucleated bubbles in one of these regions is of order

$N_{bubble}=n_{bubble}\times L^{3}=\frac{L^{3}}{R^{3}}\simeq 840\,.$ (26)

Once bubble nucleation starts, it takes a time of order $\beta^{-1}\simeq
6\,\mu$s for the phase transition to complete, namely to go from the left
panel to the right panel of Fig. 4. Finally, the lifetime of the sound waves
left behind after the transition, as illustrated in the right panel of Fig. 4,
is again of order $\tau\simeq 1$ ms. The dominant contribution to GW emission
comes from these sound waves, and these GWs have frequency $f\simeq
0.6\,\mbox{MHz}$.

##  3 Characteristic strain

As explained above, we follow the physical picture developed in the
cosmological case and assume that the dominant GW production mechanism is due
to the long-lived sound waves left behind after all the bubbles have collided.
In the cosmological case, the GW energy density produced by sound waves can be
estimated as

$\rho_{\textrm{\tiny GW}}\simeq(8\pi
G)\,v_{f}^{4}\,(\mathcal{E}+\mathcal{P})^{2}\,\Delta
t\,R\,\bar{\Omega}_{\textrm{\tiny GW}}\,.$ (27)

A detailed derivation of this formula can be found in e.g. Hindmarsh:2015qta .
We will not reproduce the derivation here, but we will provide intuition for
the origin of each term.

$G$ is Newton’s constant, related to the reduced Planck mass through

$(8\pi G)^{-1/2}=M_{p}\simeq 2.4\times 10^{18}\,\mbox{GeV}\,.$ (28)

The factor of $G$ in (27) encodes the fact that there are no GWs if gravity is
turned off.

$\mathcal{E}$, $\mathcal{P}$ and $v_{f}^{2}$ are the energy density, the
pressure and the root mean velocity of the fluid in the HoCS left behind by
the bubble collisions, as illustrated in Fig. 4(right). We take $\mathcal{E}$
and $\mathcal{P}$ to be of order (see e.g. Annala:2019puf )

$\mathcal{E}\simeq\mathcal{P}\simeq 1\,\mbox{GeV}/\mbox{fm}^{3}\,.$ (29)

GWs are metric perturbations, $h_{ij}(t,\bm{x})$, sourced by the transverse-
traceless part of the energy-momentum tensor, $T^{TT}_{ij}(t,\bm{x})$. For
sound waves on a fluid we have

$T^{TT}_{ij}\sim(\mathcal{E}+\mathcal{P})v_{i}v_{j}\,.$ (30)

The energy density in GWs scales as

$\rho_{\textrm{\tiny
GW}}\propto\dot{h}_{ij}(t,\bm{x})\dot{h}^{ij}(t,\bm{x})\,.$ (31)

This leads to the $v_{f}^{4}\,(\mathcal{E}+\mathcal{P})^{2}$ factor in (27).

$\Delta t$ is the effective time duration of the source for the purpose of
determining $\rho_{\textrm{\tiny GW}}$. In the cosmological case $\Delta
t\simeq H^{-1}$. In the NS case one may naively think that this is replaced by
the characteristic time scale of the merger, namely by a few milliseconds.
However, the effective time duration is shorter. The reason is that, once a GW
produced inside the HoCS leaves the HoCS, it no longer contributes to the
growth of $\rho_{\textrm{\tiny GW}}$ inside the HoCS. The effective time scale
to determine the GW energy density inside the HoCS is therefore the light-
crossing time of the HoCS

$\Delta t\simeq L\simeq 1.7\times 10^{-2}\,\mbox{ms}\,,$ (32)

where we have used (1).

$R$ is a characteristic length scale in the fluid flow which must be present
in (27) on dimensional grounds once the rest of the dimensionful factors have
been identified. The presence of this scale is also confirmed by explicit
simulations that show that this scale coincides with the mean separation
between the bubbles whose collisions produce the fluid flow in the first
place.

Finally, $\bar{\Omega}_{\textrm{\tiny GW}}$ is a dimensionless factor
quantifying the efficiency with which shear stress in the fluid is converted
to GWs. Although its precise value depends on the details of the flow,
simulations show that its ballpark value, $\bar{\Omega}_{\textrm{\tiny
GW}}\sim 10^{-2}$, is rather insensitive to these details.

The GW energy density is related to the characteristic strain $h_{c}$ through
(see e.g. Moore:2014lga )

$\rho_{\textrm{\tiny GW}}=\frac{2\pi^{2}}{8\pi
G}\,\frac{h_{c}^{2}}{\lambda^{2}}\,.$ (33)

Equating (27) and (33) we find

$h_{c}^{2}\simeq\frac{10^{6}}{2\pi^{2}}\,v_{f}^{4}\,\left(\frac{\mathcal{E}+\mathcal{P}}{M_{p}^{4}}\right)\,\Big{(}(\mathcal{E}+\mathcal{P})\times
1\mbox{ms}\times 1\mbox{m}^{3}\Big{)}\,\bar{\Omega}_{\textrm{\tiny
GW}}\,\left(\frac{\Delta
t}{1\,\mbox{ms}}\right)\left(\frac{\lambda}{100\,\mbox{m}}\right)^{3}\,,$ (34)

where we have used (2). The second and third factors are

$\frac{\mathcal{E}+\mathcal{P}}{M_{p}^{4}}\simeq 4.8\times
10^{-76}\,,\qquad(\mathcal{E}+\mathcal{P})\times 1\mbox{ms}\times
1\mbox{m}^{3}\simeq 3\times 10^{66}\,.$ (35)

Putting it all together we get

$h_{c}\simeq 8.5\times 10^{-4}\,v_{f}^{2}\,\left(\frac{\Delta
t}{1\,\mbox{ms}}\right)^{1/2}\left(\frac{\lambda}{100\,\mbox{m}}\right)^{3/2}\,.$
(36)

The amplitude of the GW decreases as the inverse distance $d$ to the source,
so the observed characteristic strain would be

$h_{c}^{\textrm{\tiny obs}}=h_{c}\,\frac{L}{d}\simeq 1.4\times
10^{-24}\,v_{f}^{2}\,\left(\frac{100\,\mbox{Mpc}}{d}\right)\left(\frac{\Delta
t}{1\,\mbox{ms}}\right)^{1/2}\left(\frac{\lambda}{100\,\mbox{m}}\right)^{3/2}\,.$
(37)

Using the values (24) and (32) we finally arrive at

$h_{c}^{\textrm{\tiny obs}}\simeq 2.1\times
10^{-24}\,v_{f}^{2}\,\left(\frac{100\,\mbox{Mpc}}{d}\right)\,.$ (38)

The fluid velocity depends on details of the phase transition such as the
strength of the transition, the bubble wall velocity, etc., but a
representative range is Cutting:2019zws

$0.01\lesssim v_{f}\lesssim 0.3\,,$ (39)

with higher values favoured for strong phase transitions, as might be expected
for QCD. It is conceivable that future superconducting radio-frequency GW
detectors may reach a sensitivity of order $h_{c}\simeq 10^{-24}$ private ;
agno . If we require that the observed characteristic strain be at least as
large as this value, then the velocity range (39) translates into a range of
maximum distances from which the signal could be observed given by

$20\,\mbox{kpc}\lesssim d\lesssim 20\,\mbox{Mpc}\,.$ (40)

For comparison, for the first detection GW170817 LIGOScientific:2017vwq  the
distance was $d\simeq 40$ Mpc.

##  4 Discussion

Numerical simulations of NS mergers based on a QCD EoS with a FOPT show that
the phase transition is accessed during the merger dynamics. We have argued
that, because the characteristic time scale of the merger evolution is so much
longer than the nuclear time scale, from the QCD viewpoint the evolution is
adiabatic to an extremely good approximation. This results in the formation of
metastable regions of superheated and/or supercompressed matter in which the
phase transition takes place via the formation and subsequent expansion and
collision of bubbles.

This physics is analogous to that of a cosmological phase transition, except
for the fact that in the cosmological case the metastable region is
supercooled. Based on this analogy we have provided a rough estimate of the
features of the resulting GW spectrum. We have found that the duration of the
signal is of order of milliseconds and the peak frequency is in the MHz range.
Based on the analogy with the cosmological case, we expect the precise GW
power spectrum to take the form a broken power law centered around this
frequency.

The characteristic strain that would be observed coming from a single
superheated and/or supercompressed spot is given by (38). Simulations show
that several spots are typically formed, and that they can transition back to
the original phase. This transition would proceed via the nucleation and
collision of supercooled bubbles, leading to additional emission of GWs in a
similar frequency range. We expect these effects to enhance the signal that we
have estimated.

Our results are interesting for several reasons. First, this is the only known
_astrophysical_ source of GWs in the MHz range. Second, as indicated by eqn.
(40), it is conceivable that signals coming from NS mergers as far as tens of
Mpc might be detectable by future superconducting radio-frequency GW detectors
private ; agno . Given all the uncertainties in our rough estimate, we regard
this as an exciting outcome. Third, the detection of the MHz signal would be
aided (i) by the fact that it would happen at the same time as the usual kHz
signal associated to the merger, and (ii) by the fact that both would be
preceded by the long inspiral signal, which for GW170817 lasted for about 100
s. Fourth, the signal is stochastic in nature, since it originates from the
superposition of the dynamics of many sound waves, but it will come from a
very specific direction in the sky determined by the location of the NS
binary. All these considerations suggest that, with the targeted sensitivity
of future detectors in the MHz range, GWs may provide us with a direct signal
of a FOPT in QCD.

## Acknowledgements

We are grateful to Jordi Miralda for collaboration at the initial stages of
this paper, and to Krishna Rajagopal for comments on the draft. We thank Yago
Bea, Roberto Emparan, Alexander Haber, Luciano Rezzolla, Andreas Schmitt,
Alexandre Serantes, and specially Andreas Ringwald and Carlos Tamarit, for
useful discussions. We are extremely grateful to Diego Blas for extensive
explanations. We are supported by the “Unit of Excellence MdM 2020-2023” award
to the Institute of Cosmos Sciences (CEX2019-000918-M) and by grants
SGR-2017-754, PID2019-105614GB-C21 and PID2019-105614GB-C22. MSG is also
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement No758759).

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A note on Apery’s constant is transcendental

Shekhar Suman
M.Sc., I.I.T. Delhi, India

Email<EMAIL_ADDRESS>
Abstract: Beuker’s [2] considers the following integral

$\int_{0}^{1}\int_{0}^{1}\frac{-\log xy}{1-xy}P_{n}(x)P_{n}(y)\ dxdy$

If $d_{n}=\text{LCM}(1,2,...,n)$, then

$0<\frac{|A_{n}+B_{n}\zeta(3)|}{d_{n}^{3}}<2(\sqrt{2}-1)^{4n}\zeta(3)$

for some $A_{n},B_{n}\in\mathbb{Z}$. We establish that if Apery’s constant is
algebraic then the above inequality fails to be true. This proves that
$\zeta(3)$ is transcendental.
Keywords and Phrases: Riemann zeta function, analytic continuation, Apery’s
constant.
2020 Mathematics Subject Classification: 11M06
1\. Introduction and Definitions The Riemann zeta function, $\zeta(s)$ is
defined as the analytic continuation of the Dirichlet series

$\zeta(s)=\sum_{n=1}^{{\infty}}\frac{1}{n^{s}}$

which converges in the half plane $\Re(s)>1$. The Riemann zeta function is a
meromorphic function on the whole complex plane, which is holomorphic
everywhere except for a simple pole at s = 1 with residue 1. The Riemann
Hypothesis states that all the non trivial zeros of the Riemann zeta function
lie on the critical line $\Re(s)=\frac{1}{2}$. The Riemann xi function is
defined as

$\xi(s)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$

$\xi(s)$ is an entire function whose zeros are the non trivial zeros of
$\zeta(s)$ [1]. Further $\xi(s)$ satisfies the functional equation [1]

$\xi(s)=\xi(1-s)$

It is known that the Apery’s constant, $\zeta(3)$ is irrational [2].

2\. Main Theorems The goal of this article is to prove the following result.

Theorem : Apery’s constant, $\zeta(3)$ is transcendental.

Proof: We prove the claim by the method of contradiction. Let us assume that
$\zeta(3)$ is algebraic. Then it satisfies

$c_{0}+c_{1}\zeta(3)+c_{2}\zeta^{2}(3)+...+c_{m}\zeta^{m}(3)=0$ (1)

where $c_{i}\in\mathbb{Z}$, $c_{0}>0$ and $m\in\mathbb{Z}^{+}$.
It is well known that [2]

$2\zeta(3)=\int_{0}^{1}\int_{0}^{1}\frac{-\log xy}{1-xy}\ dxdy$ (2)

Beuker’s [2] considers the following integral

$I_{n}:=\int_{0}^{1}\int_{0}^{1}\frac{-\log xy}{1-xy}P_{n}(x)P_{n}(y)\ dxdy$
(3)

By Beuker’s [2] proof, we have if $d_{n}=\text{LCM}(1,2,...,n)$, then

$0<\frac{|A_{n}+B_{n}\zeta(3)|}{d_{n}^{3}}<2(\sqrt{2}-1)^{4n}\zeta(3)$ (4)

for some $A_{n},B_{n}\in\mathbb{Z}$. Equation (2) can be rewritten as

$0<\frac{|A_{n}+B_{n}\zeta(3)|}{2(\sqrt{2}-1)^{4n}d_{n}^{3}}<\zeta(3)$ (5)

So we get on squaring equation (3)

$0<\frac{|A_{n}+B_{n}\zeta(3)|^{2}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{2}}<\zeta^{2}(3)$
(6)

Similarly we get

$0<\frac{|A_{n}+B_{n}\zeta(3)|^{3}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{3}}<\zeta^{3}(3)$
(7)

and so on

$0<\frac{|A_{n}+B_{n}\zeta(3)|^{m+1}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{m+1}}<\zeta^{m+1}(3)$
($m$+5)

Now multiplying (3) by $c_{0}$, (4) by $c_{1}$, (5) by $c_{2}$,…, ($m$+3) by
$c_{m}$, we get

$\displaystyle 0$
$\displaystyle<c_{0}\frac{|A_{n}+B_{n}\zeta(3)|}{2(\sqrt{2}-1)^{4n}d_{n}^{3}}+c_{1}\frac{|A_{n}+B_{n}\zeta(3)|^{2}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{2}}+c_{2}\frac{|A_{n}+B_{n}\zeta(3)|^{3}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{3}}$
$\displaystyle+...+c_{m}\frac{|A_{n}+B_{n}\zeta(3)|^{m+1}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{m+1}}<c_{0}\zeta(3)+c_{1}\zeta^{2}(3)+c_{2}\zeta^{3}(3)+...+c_{m}\zeta^{m+1}(3)$
($m$+6) $\displaystyle 0$
$\displaystyle<c_{0}\frac{|A_{n}+B_{n}\zeta(3)|}{2(\sqrt{2}-1)^{4n}d_{n}^{3}}+c_{1}\frac{|A_{n}+B_{n}\zeta(3)|^{2}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{2}}+c_{2}\frac{|A_{n}+B_{n}\zeta(3)|^{3}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{3}}$
$\displaystyle+...+c_{m}\frac{|A_{n}+B_{n}\zeta(3)|^{m+1}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{m+1}}<\zeta(3)(c_{0}+c_{1}\zeta(3)+c_{2}\zeta^{2}(3)+...+c_{m}\zeta^{m}(3))$
($m$+7)

Now using equation (1) we get

$\displaystyle 0$
$\displaystyle<c_{0}\frac{|A_{n}+B_{n}\zeta(3)|}{2(\sqrt{2}-1)^{4n}d_{n}^{3}}+c_{1}\frac{|A_{n}+B_{n}\zeta(3)|^{2}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{2}}+c_{2}\frac{|A_{n}+B_{n}\zeta(3)|^{3}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{3}}$
$\displaystyle+...+c_{m}\frac{|A_{n}+B_{n}\zeta(3)|^{m+1}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{m+1}}<0$
($m$+8)

So we get

$0<\sum_{k=1}^{m+1}c_{k-1}\frac{|A_{n}+B_{n}\zeta(3)|^{k}}{(2(\sqrt{2}-1)^{4n}d_{n}^{3})^{k}}<0$
($m$+9)

which is a contradiction. So our assumption that $\zeta(3)$ is algebraic is
not possible. Hence $\zeta(3)$ is transcendental. This settles the proof of
the Theorem.

References

[1] Edwards, H.M., Riemann’s Zeta Function , Dover Publications (2001).

[2] Beukers, F., A note on the irrationality of $\zeta(2)$ and $\zeta(3)$,
Bulletin of the London Mathematical Society, Vol. 11, issue 3, 268-272 (1979).

[3] Apéry, R. (1979). Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque.
61: 11–13.

Data Availability Statement (DAS)

[1] Books available in the library or online library - for the reference
[1](Click here for [1])

[2] Data openly available in a public repository that does not issue DOIs -
for the reference, [2] (Click here for [2]) and [3] (Click here for [3])


# Fast measurement of group index variation with ultimate precision using
Hong-Ou-Mandel interferometry

Sandeep Singh<EMAIL_ADDRESS>Photonic Sciences Laboratory, Physical
Research Laboratory, Navarangpura, Ahmedabad, 380009 Gujarat, India  Indian
Institute of Technology-Gandhinagar, Ahmedabad 382424, Gujarat, India  Vimlesh
Kumar  Photonic Sciences Laboratory, Physical Research Laboratory,
Navarangpura, Ahmedabad, 380009 Gujarat, India  G. K. Samanta  Photonic
Sciences Laboratory, Physical Research Laboratory, Navarangpura, Ahmedabad,
380009 Gujarat, India

###### Abstract

Hong-Ou-Mandel (HOM) interferometry has emerged as a valuable tool for quantum
sensing applications, particularly in measuring physical parameters that
influence the relative optical delay between pair photons. Unlike classical
techniques, HOM-based quantum sensors offer higher resolution due to their
intrinsic dispersion cancellation property. Despite this advantage, achieving
precise measurements of optical delay crucial for practical applications often
involves time-consuming integration and post-processing with traditional
statistical methods. To address this challenge, our recent work focused on
optimizing optical delay measurements in a time-efficient manner. By carefully
selecting the length of a 1 mm periodically-poled KTP (PPKTP) crystal for pair
photon generation, we achieved a remarkable group index measurement precision
of $\sim 6.75\times 10^{-6}$ per centimeter of sample length, surpassing the
previous maximum precision by over 400$\%$. These current measurements
maintain fast detection and high photon counts, which are essential for
practical quantum sensing applications. The HOM-based method, while limiting
the measurement range, can be extended by compensating for photon delay using
an optical delay stage. As a proof-of-principle, we measured the group index
variation of PPKTP over a temperature range up to 200∘C with a precision in
the range of one part per million ($\sim$10-6). This advancement not only
contributes to quantum sensing but also holds promising implications for high-
precision and long-range measurements in quantum optical coherence tomography.

††preprint: AIP/123-QED

##  I Introduction

Quantum optical coherence tomography (QOCT) represents a non-invasive imaging
modality primarily employed to generate three-dimensional cross-sectional
visualizations of transparent or reflective materials, specifically focusing
on the microstructural surface morphology of biological tissuesNasr  _et al._
(2003); Povazay  _et al._ (2002). In QOCT, unlike conventional optical
coherence tomography with classical light, the utilization of entangled photon
pairs allows probing of the sample surface with an axial resolution enhanced
by a factor of 2 even for the same spectral bandwidth source in both
scenarios, showcasing the quantum advantage in precision imaging. On the other
hand, the immunity against the dispersion effects owing to the frequency-
entangled pair photons enables QOCT to access higher resolution of surface
morphology at sub-micron scalesNasr  _et al._ (2003) through the use of a
broader spectrum of photons compared to classical counterparts.

The conventional QOCT relies on the Hong-Ou-Mandel (HOM) interference, where
two indistinguishable photons coalesce at a balanced beam splitter, with one
photon traversing a reference delay line and the other probing the sample
surface, encoding its features in the relative temporal delay between the pair
photons Hong, Ou, and Mandel (1987). Since the change in temporal delay
results in the variation in coincidence counts and the temporal delay results
from the surface morphology of the specimen, one can map the surface
morphology by recording the coincidence counts in the transverse plane.
However, the axial resolution is constrained by the coherence length of
indistinguishable photons and, thus, the width of the HOM interference dip.
For example, in the case of a stratified specimen, the probe photon acquires
data pertaining to the optical thickness across the distinct layers. As a
result, the precise determination of the thickness profile of individual
layers demands a meticulous understanding of their corresponding group
indices. Even in quantum optical coherence microscopy (QOCM), achieving an
accurate three-dimensional reconstruction of the specimen necessitates a
precise determination of the group index variations within the transverse
plane of the sample. Unfortunately, these group index variations, in general,
are exceedingly small, and conventional methodologies for group index
measurement lack the necessary precision to measure those subtle changes.
Nevertheless, while HOM interference has been applied for measuring group
index through the assessment of optical delay Lyons  _et al._ (2018)
introduced by the sample and has enabled high-precision measurements in
quantum microscopy Ndagano  _et al._ (2022), QOCT Abouraddy  _et al._ (2002),
and vibration sensing Singh  _et al._ (2023) without adequately addressing the
crucial requirement to improve the precision of HOM interferometry. In most
cases, high precision is achieved by exploring the stringent prerequisites,
including the long interaction length of the specimen and subsequent post-
processing of large data accumulated iteratively Reisner  _et al._ (2022). As
a result, such approaches suffer from the common drawback of long measurement
time and limited kinds of samples. For example, the measurement of group index
variation of the nonlinear crystals due to the temperature-dependent
dispersion properties is of paramount interest for designing various
experiments, including the optimization of phase-matching for high nonlinear
conversion efficiency and optimum phase-compensation of the entangled state
generated by the nonlinear crystal. However, the precision measurement using
the existing HOM interferometer-based group index measurement techniques
demands long interaction lengths.

Unfortunately, in many cases, nonlinear crystals exhibit complicated growth
techniques Zhou  _et al._ (2023), inherent fragility, and high cost Dhanaraj
_et al._ (2010), posing challenges in procuring samples with extended
interaction lengths. Therefore, it is imperative to enhance the resolution of
the HOM interferometer. In our recent study, we demonstrated that the
resolution of the HOM interferometer can be improved by increasing the
spectral width of pair photons, achieved through the appropriate selection of
crystal length for generating these pair photons Singh  _et al._ (2023).

In this study, we present the utilization of pair photons with a broad
spectral width to measure the group index variation of a nonlinear crystal,
achieving a resolution of approximately $2.25\times 10^{-6}$. By employing a 1
mm long periodically-poled potassium titanyl phosphate (PPKTP) crystal that
generates down-converted pair photons with a spectral bandwidth of 162 $\pm$
1.12 nm, we developed a HOM interferometer with a full width at half maximum
(FWHM) dip width of 4.05 $\pm$ 0.03 $\mu$m. We successfully measured the
temperature-dependent group index variation of another PPKTP crystal with a
resolution of approximately $2.25\times 10^{-6}$ over a measurement range of
approximately $7.3\times 10^{-5}$. While there is potential for further
resolution enhancement by increasing the spectral bandwidth of pair photons
with a thin nonlinear crystal, we deliberately restricted the crystal length
to maintain a relatively higher photon pair generation rate, even in a short
interaction length, ensuring a faster measurement process Singh  _et al._
(2023).

The group index ($n_{g}$) of a dispersive material is related to its
refractive index ($n$) as follows Miriampally (2016),

$\begin{split}n_{g}&=\frac{c}{v_{g}}\\\
&=n-\lambda\frac{dn}{d\lambda}|_{\lambda=\lambda_{\circ}}\\\ \end{split}$ (1)

where, c, and $v_{g}$ are the velocity of light in vacuum and the group
velocity of light in the material, respectively. The $\lambda_{\circ}$ is the
central wavelength of the photon wave packet traveling through the dispersive
material. It is evident from Equation 1 that in the normal dispersion regime,
the group index of a material, $n_{g}$, is always higher than its refractive
index, $n$. The optical delay, $\Delta x$, associated with the group index of
the material for an effective interaction length of $L$ can be expressed as,

$n_{g}=\frac{\Delta x}{L}\\\ $ (2)

Equation 2 makes it clear that the precision of measuring the group index,
$n_{g}$, of a specimen is contingent upon the precision in measuring the
optical delay introduced by it and its physical length, $L$. Hence, accessing
the ultimate precision, which is the central goal of this report, in optical
delay measurement becomes imperative to accurately measure the group delay
introduced by the material under test for quantum imaging and microscopy.

##  II Experimental setup

The schematic of Fig. 1(a) shows the concept of quantum sensing using HOM
interferometry, while the detailed experimental scheme is the same as reported
in Ref. Singh  _et al._ (2023). A single-frequency, linearly polarized
(vertical polarization), continuous-wave (CW), fiber-coupled diode laser
providing 20 mW of output power at 405.4 nm is used as the pump laser in the
experiment.

Figure 1: a) Schematic of the experimental setup. NC: PPKTP crystal of grating
period = 3.425 $\upmu$m, S: Specimen, M1-2: dielectric mirrors, BS: 50:50 beam
splitter, D1-2: Single photon counting modules. b) HOM interference profiles
for pair photons of different spectral bandwidths. (Inset) Magnified image of
the HOM dips.

The pump laser is focused at the center of the nonlinear crystal for the
optimum rate of pair-photon generation. Two periodically poled KTiOPO4 (PPKTP)
crystals of the same aperture (1 $\times$ 2 mm2) but of different interaction
lengths, $L$ = 1, and 20 are used as the nonlinear crystals (NC) for the
experiment. Both the nonlinear crystals have a single grating period of
$\Lambda$ = 3.425 $\upmu$m designed for the degenerate, type-0 ($e$→$e+e$)
phase-matched parametric down-conversion (PDC) of 405 nm to 810 nm. To set out
quasi-phase-matching, the PPKTP crystals are kept in an oven having dynamic
temperature control over room temperature to 200∘C with temperature stability
of $\pm$0.1∘C. After being separated from the pump, the non-collinear, down-
converted pair photons are collimated and guided in two different paths to
interfere with the balanced (50:50) beam splitter (BS). The relative optical
delay between the pair photons is adjusted using the optical delay line in the
path of one of the photons. On the other hand, the sample, S, under study is
placed in the path of another photon. The photons at the output ports of the
BS are collected using single-mode fiber connected to the single-photon
counting modules, D1 and D2 (AQRH-14-FC, Excelitas). Further, the time-to-
digital converter, TDC (ID-800), is used to measure the single and coincidence
counts. All optical components used in the setup are selected for optimal
performance at both the pump and SPDC wavelengths. All the data is recorded at
a coincidence window of 1.6 ns and a typical pump power of 0.25 mW.

##  III Results and Discussions

We first studied the performance of the HOM interferometer for
indistinguishable photons of four different spectral widths of the pair
photons. It is evident from Ref. Singh  _et al._ (2023) that the pair photons
generated from PPKTP crystal of length, L = 1 mm, and 20 mm, have spectral
widths of 162 $\pm$ 1.12 nm, and 39 $\pm$ 0.2 nm, respectively. To further
reduce the spectral width, we have filtered the pair-photons using
interference filters with transmission bandwidths of 3.2 nm and 10 nm centered
at 810 nm. Using the pair-photons with four different spectral bandwidths, 162
nm, 39 nm, 10 nm, and 3.2 nm, all centered at 810 nm, we measured the HOM
interference by recording the coincidence counts with the relative optical
delay. The results are shown in Fig. 1(b). As expected from the previous
report Singh  _et al._ (2023), the FWHM width of the HOM dips is found to be
4.05 $\pm$ 0.01, 16.9 $\pm$ 0.03, 42.4 $\pm$ 0.06 and 144.5 $\pm$ 0.08
$\upmu$m obtained using pair photons having spectral bandwidth of 162 $\pm$
1.12 nm (crystal length 1 mm), 39 $\pm$ 0.2 nm (crystal length 20 mm), 10
$\pm$ 0.01 (10 nm interference filter) and 3.2 $\pm$ 0.01 nm (3.2 nm
interference filter), respectively. Interestingly, we observe an inverted
Gaussian profile approximates the HOM curve (purple line and dots) for the
pair of photons with the spectral bandwidth of 3.2 nm, substantiating the
inherent Gaussian temporal profile of the pair photons. In contrast, for
broadband pair-photons having a spectral width of 162 nm, we also observe
oscillatory fluctuations (black line and dots) with diminishing amplitudes in
the coincidence counts as one moves away from the minima of the dip,
indicative of the beating phenomenon arising from the interference of
probability amplitudes within the broadband two-photon wave packet Chen  _et
al._ (2019). To gain further insights, we have enlarged the minimum
coincidence section of the HOM curves, as depicted in the inset of Fig. 1(b).
Notably, within the optical delay range of $\pm$2 $\mu$m, the minimum
coincidence counts for pair-photons with spectral widths of 3.2 nm, 10 nm, and
39 nm coincide. This observation highlights the potential for significant
errors in determining the optical delay corresponding to the minimum
coincidence counts, leading to inaccuracies in the measurement of physical
parameters (e.g., the refractive index of a sample) utilizing the HOM dip
shift. Conversely, the rapid variation in coincidence counts with optical
delay, particularly evident in the HOM curve (black dots) of pair-photons with
broad spectra, enhances accuracy in identifying and measuring the optical
delay corresponding to the minimum coincidence. Therefore, for high-precision
measurements in quantum sensing applications, one must look for a narrow HOM
curve.

Figure 2: Variation of coincidence counts with optical delay of HOM
interferometer for pair photons of the spectral width of 162 nm in a) absence
and b) presence of the sample of physical length, $L$ = 5 mm. c) Group index
of five samples measured using HOM interferometer for pair photons of
different spectral bandwidths. d) Variation of group index of KTP crystal of
constant length and temperature as a function of spectral bandwidth of the
pair photons.

Furthermore, to emphasize the necessity of a narrow HOM curve for high-
resolution HOM-based quantum sensors, we employed HOM interferometry to
measure the optical delay between pair photons induced by a dispersive
material with varying group indices. Utilizing the measured group index and
the corresponding dispersion relations, one can deduce the refractive index of
the material Miriampally (2016). For group index measurement, we have used
five different samples (S), including the commonly used periodically poled
nonlinear crystals, KTP, Mgo-doped stoichiometric grown lithium tantalate
(SLT), and MgO-doped congruent grown lithium niobate (CLN) of lengths, 5.07,
5.18, and 10.03 mm, respectively and two glass slabs, named as Gls1 (Schott
glass) and Gls2 (BK7) of lengths, 3.10, and 1.15 mm, respectively. As
expected, the insertion of the sample (S) in one of the arms of the HOM
interferometer shifts the HOM curve. We again retrieved the HOM curve using
the delay stage employed in the other arm of the HOM interferometer.
Subsequently, using the overall shift ($\Delta x$) of the interference minima
due to the presence of the samples for all four spectral widths of the pair-
photons, we calculated the group index using Eq. 2. The results are shown in
Fig. 2. As evident from Fig. 2(a), the pair photons generated by a PPKTP
crystal of length 1 mm have a symmetric HOM curve, in the absence of any
sample, of FWHM width (obtained using Gaussian fit to the experimental data)
of 4.06 $\pm$ 0.02 $\upmu$m. The minimum coincidence point of the HOM curve is
achieved for the linear stage reading of 21.554 mm. However, the insertion of
the sample (PPKTP crystal of length, L = 5 mm) shifts the HOM curve, as shown
in Fig. 2(b). The HOM curve has an asymmetry, but the FWHM width of 4.11 $\pm$
0.03 $\upmu$m is almost same as the FWHM width of the HOM curve in the absence
of the sample. Such observation confirms the dispersion cancellation property
of the HOM interferometer Okano  _et al._ (2015) despite the broad spectral
width (162 nm) of the photons. The minimum coincidence point of the HOM curve
in the presence of a 5 mm long PPKTP sample is obtained at a linear stage
reading of 26.164 mm. The overall change in the optical delay can be written
as $\Delta x$ = ($n_{g}$-1)L = 4.61 mm. Using this optical delay, we can
measure the group index of the sample. We repeated the same measurement
procedure for different samples of varying lengths and spectral widths of the
pair photons and measured their group indices. The results are shown in Fig.
2(c). It is observed that the group indices of all the five samples measured
(dots) using pair-photons of different spectral bandwidths match with their
respective group indices calculated using the respective Sellmeier equation
(line). Such observation establishes the capability and reliability of the HOM
interferometer as a potential refractometer to measure the group index of any
dispersive materials. In this measurement, all the samples (S) were kept at
room temperature.

To scrutinize the accuracy of the measured group indices under the influence
of various parameters contributing to inaccuracies in the measurement of
($\Delta x$), especially while using different spectral widths of pair-
photons, we have amplified the scale of the recorded group index data for the
PPKTP crystal (as an example). The results are shown in Fig. 2(d). As evident
from Fig. 2(d), the group index (dots) of the PPKTP crystal measured using
different spectral widths of the pair-photons lies close to the theoretical
value (blue line) calculated using the dispersion relations Kato and Takaoka
(2002) of the PPKTP crystal. However, upon careful observation, it becomes
evident that the measurement precision increases from $\pm 8.3\times 10^{-4}$
to $\pm 1.0\times 10^{-4}$, representing an 8-fold enhancement when utilizing
HOM interference with pair-photons of broad spectral width (162 nm) compared
to 3.2 nm. The observed enhanced precision in group index measurement can be
attributed to the improved accuracy in assessing the shifts in the HOM curve
with a narrow dip width, a result of using pair photons with a broad spectral
bandwidth. Interestingly, the dispersion of the sample in the presence of
different spectral widths of pair photons showed almost no change in
resolution, affirming the dispersion cancellation characteristics of the HOM
interferometer, consistent with the observation of two-photon interferometry-
based QOCT using ultra-broadband pair-photons Okano  _et al._ (2015). From
these results, it is evident that employing the shift of the HOM curve as a
pointer enables the measurement of the group index of any dispersive medium
with an accuracy in the range of $\pm 1\times 10^{-4}$ when using pair-photons
with a broad spectral bandwidth of 162 $\pm$ 1.12 nm. Consistent performance
is evident in group index measurements across the remaining materials. For a
better understanding, we have tabulated all the group index measurement
parameters in Table 1. The length of the samples is their measured value in
the lab using vernier calipers with a precision of 10 $\upmu$m. The measured
lengths precisely conform to the specified tolerance range provided by the
sample supplier. The spectral width of the pair-photons used in this
measurement is 162 $\pm$ 1.12 nm. As evident from Table 1, the measurement
accuracy of the group index of all the dispersive mediums with different
lengths is expectedly in the range of $\pm 1\times 10^{-4}$, similar to the
order of precision achieved by conventional OCT Singh (2002); Yablon (2013).
However, such low measurement precision may not be suitable for the
measurement of low group indices, for example, the temperature-dependent group
index (thermo-optic dispersion) of different nonlinear crystals particularly
relevant for quasi-phase-matched nonlinear parametric processes. Commonly, the
quasi-phase-matched crystals available in the market have a weak temperature-
dependent refractive index variation in the range of $10^{-4}$ to $10^{-5}$
∘C-1. Therefore, we need the precision at least one order of magnitude higher,
i.e., $\sim 10^{-6}$. On the other hand, we have demonstrated the measurement
of static optical delay of $\sim$ 60 nm using HOM interferometry with pair
photons of broad spectral width Singh  _et al._ (2023). As the optical delay
is defined as the product refractive index, $n$, and physical length, $L$, of
a medium, i.e., $n\times L$, one can easily measure the change in the
refractive index smaller than $\sim$10-6 by suitably selecting the physical
length of the medium.

Table 1: Group index ($n_{g}$) of different materials at room temperature. Sample | Length | Theory | Measured | Accuracy | Sellmeier
---|---|---|---|---|---
| (mm) | $n_{g}$ | $n_{g}$ |  | equation
PPKTP crystal | 5.07 | 1.90949 | 1.90944 | 1.02 $\times$ $10^{-4}$ | Kato and Takaoka (2002)
PPSLT crystal | 5.18 | 2.21997 | 2.21994 | 8.97 $\times$ $10^{-5}$ | Manjooran  _et al._ (2012)
MCLN crystal | 10.03 | 2.27129 | 2.27132 | 8.04 $\times$ $10^{-5}$ | Manjooran  _et al._ (2012)
Schott Glass (Gls1) | 1.15 | 1.52927 | 1.52924 | 1.05 $\times$ $10^{-4}$ | Polyanskiy (2023)
Glass slab (Gls2) | 3.10 | 1.50778 | 1.50773 | 8.97 $\times$ $10^{-5}$ | Gan (1995)

To validate the feasibility of measuring the temperature-dependent group index
(thermo-optic dispersion) of periodically poled crystals, we selected a PPKTP
crystal. This choice is advantageous for two reasons: firstly, experimental
observations can be corroborated using standard data from available Sellmeier
equations Kato and Takaoka (2002), and secondly, the PPKTP crystal exhibits
approximately 15 times less susceptibility to temperature fluctuations
compared to other popular periodically poled crystals such as lithium niobate
and lithium tantalate Manjooran  _et al._ (2012); Lin  _et al._ (2017). This
makes PPKTP an optimal selection for demonstrating the efficacy of this
configuration in discerning subtle variations in group indices.

However, before conducting the experimental observations, we analytically
calculated the variations in the group index of the PPKTP crystal as a
function of crystal temperature. Let us assume that $x(T_{\circ})$ is the
optical delay between the pair photons due to the PPKTP crystal of length $L$
and group index $n_{g}$ at the initial temperature $T_{\circ}$. The
$x(T_{\circ})$ can be written as:

$x(T_{\circ})=n_{g}(T_{\circ})\times L(T_{\circ})$ (3)

Now, the change in crystal temperature by $\Delta T$ leads to the change in
its group index and effective interaction length due to thermo-optic
dispersion and thermal expansion, respectively. Therefore, the temperature-
dependent optical delay, $x(T_{\circ}+\Delta T)$, can be written as,

$x(T_{\circ}+\Delta T)=(n_{g}(T_{\circ})+\Delta n_{g}(\Delta
T))\times(L(T_{\circ})+\Delta L(\Delta T))$ (4)

where, $\Delta n_{g}(\Delta T)$ and $\Delta L(\Delta T)$ denote the change in
group index and the effective interaction length of the crystal, respectively,
due to the change in crystal temperature of $\Delta T$. The net change in the
relative optical delay corresponding to the change in crystal temperature can
be written as,

$\Delta x(\Delta T)=x(T_{\circ}+\Delta T)-x(T_{\circ})$ (5)

Using Eq. 3 and Eq. 4 in Eq. 5 and neglecting the term $\Delta n_{g}(\Delta
T)\Delta L(\Delta T)$ due to its small magnitude ($\sim 10^{-10}$) as compared
to the measurement accuracy, we can represent the temperature-dependent group
index change as,

$\Delta n_{g}(\Delta T)\simeq\frac{\Delta x(\Delta T)-n_{g}\Delta L(\Delta
T)}{L}$ (6)

The second term in the numerator of Eq. 6 accounts for the change in relative
optical delay between the signal and idler of the pair photons due to the
thermal expansion of the PPKTP crystal placed in one of the arms of the HOM
interferometer. Since the refractive index and crystal lengths vary
simultaneously with the temperature, we can measure one of the parameters at a
time for a given value of the other parameter. Therefore, we used the
available thermal expansion data of PPKTP crystal from the literature using
Ref. Emanueli and Arie (2003) and measured the temperature-dependent group
index variation of the PPKTP crystal. As evident from Eq. 6, for a given
crystal length, L, by measuring the change in optical delay, $\Delta x(\Delta
T)$, one can easily find the temperature-dependent group index of the crystal.
In doing so, we have reconstructed the HOM curve in the presence of the sample
(PPKTP crystal of measured length, L = 30.12 mm) at a fixed temperature of
(26∘C). Given the intrinsic dispersion cancellation property as observed in
Fig. 2 and Ref. Okano  _et al._ (2015), we obtained the HOM curve of FWHM
width same as the HOM curve presented in Fig. 2(b)), despite having such a
long sample.

Figure 3: a) Variation of optical delay as a function of crystal temperature
measured using the delay dependent coincidence counts variation of HOM curve
(see Fig. 2). The straight line (red) is the linear fit to the experimental
results. b) Temporal stability of the coincidence counts for a period of 10
minutes at each increment (1∘C) in the sample crystal temperature.

Furthermore, we adjusted the delay line so that the initial point
corresponding to the sample crystal temperature, $T_{\circ}=26^{\circ}$C,
results in an optical delay of +1 $\mu$m (see Fig. 2(b)). This selection is
purposefully made to access most of the linear range of the HOM curve, thereby
enhancing the measurement range. Subsequently, we changed the temperature of
the sample (PPKTP crystal) in minimum steps of 0.1∘C, as allowed by the oven.
Since these small increments are close to the stability range of the oven, for
each temperature rise of 0.1∘C, we allowed the oven and crystal sufficient
time (approximately 2 minutes) to settle before recording the data. We
recorded the coincidence counts data for a total of 200 iterations to enhance
the accuracy of the measurement. Access to such a large dataset for each
measurement was possible due to the high brightness of the source. In fact, it
was possible to acquire substantial coincidence counts even in low integration
time (50 ms), thus drastically reducing the overall time of measurement. Using
these coincidence counts and comparing them with the HOM curve (see Fig.
2(b)), we estimate the optical delay as a function of the crystal temperature.
The results are shown in Fig. 3.

As evident from Fig. 3(a), the optical delay due to the change in the group
index of the sample (PPKTP crystal) increases linearly with the crystal
temperature in the range of 26 - 29∘C with a slope of 1.158 $\upmu$m∘C-1 of
the crystal temperature. Further increase in crystal temperature pushes the
optical delay out of the linearity range and provides arbitrary results.
However, by adjusting the initial point, we can measure the group index of the
nonlinear crystal over a range of 3∘C. To confirm the reliability of the
experiment, we recorded the variation of coincidence counts over 10 minutes
for each step change (0.1∘C) of the crystal temperature across 27.1 - 28.0∘C.
The results are shown in Fig. 3(b). It is evident from Fig. 3(b) that a 0.1∘C
change in crystal temperature results in a substantial change in coincidence
counts, exceeding 100 counts per 100 ms of integration time. This considerable
change, well above the dark count and accidental counts, establishes the
feasibility of resolving group index induced optical delays for a temperature
change smaller than 0.1∘C. We could not access the lower step of the
temperature change due to the unavailability of a suitable oven in our lab. As
observed previously Singh  _et al._ (2023), the temporal variation of the
coincidence counts can be attributed to various external perturbations,
including the laser intensity fluctuation, air current, and local temperature
instability in the laboratory. However, it is intriguing to observe that the
change in coincidence counts resulting from various external perturbations is
considerably smaller than the change in coincidence counts due to the
alteration in the group index of the sample (PPKTP crystal).

Figure 4: Variation of group index of the PPKTP crystal (S) with temperature
measured using a) optical delay-dependent coincidence variation in the linear
region of the HOM curve, and b) optical delay mirror, compensating the
coincidence change mediated by the temperature-dependent group index
variation. (Inset) shows the magnified image of the temperature-dependent
group index variation. Blue and red dots are theoretical and experimental
results, respectively.

Using the optical delay corresponding to the change in sample crystal (L =
30.12 mm) temperature from Fig. 3(a), the group index value from Table 1 and
the thermal expansion of the sample crystal, $\Delta L(\Delta T)$ from
Emanueli and Arie (2003) in Eq. 6, we calculated the temperature-dependent
group index, $\Delta n_{g}$($\Delta$ T), of the PPKTP crystal (S). The results
are shown in Fig. 4. As evident from Fig. 4(a), the temperature-dependent
group index (red dots) measured using the HOM curve of FWHM width around 4.11
$\upmu$m has a temperature range of 3∘C at any initial temperature and the
corresponding measurement range for group index variation of $\sim$7.3
$\times$ 10-5. The measurement of group index for the minimum change of
crystal temperature, $0.1^{\circ}$C, sets the resolution of the HOM-based
quantum sensor to be $\sim 2.25\times 10^{-6}$ for sample length of 30.12 mm.
Using the dispersion relations of the PPKTP crystal from Ref. Kato and Takaoka
(2002); Emanueli and Arie (2003), we calculated the temperature-dependent
group index variation (blue dots) to be in close agreement with the
experimental results (red dots). As compared to previous report on high
precision group index measurements Reisner  _et al._ (2022) of a sample
(optical fiber) length of 50 cm having precision of 3 $\times 10^{-5}$ cm-1
(absolute value of 6 $\times 10^{-7}$ for 50 cm sample) our measurement
precision, 6.75 $\times 10^{-6}$ cm-1, show an unprecedented enhancement by
$\sim$ 444%. Such enhancement was possible due to the use of a narrow HOM-
curve resulting from the control in the length of the nonlinear crystal
producing broadband pair photons Singh  _et al._ (2023).

To evaluate group index variations across an extended temperature range, one
can utilize a HOM curve with a higher full width at half-maximum (FWHM) width.
However, as initially explained, this approach typically leads to poor
precision in group index measurement. To circumvent range-dependent resolution
issues and enhance the measurement range while maintaining high resolution
($\sim 2.25\times 10^{-6}$), we employed the technique reported in Ref. Singh
_et al._ (2023). In this method, we used a HOM curve with an FWHM width of
4.11 $\upmu$m and initially adjusted the optical delay corresponding to the
initial crystal temperature at the middle of the linearity region of the HOM
curve. Knowing the coincidence counts for this initial point, we adjusted the
optical delay stage and compensated for the change in the coincidence count
arising from the temperature-dependent group index change. Since the entire
HOM curve is not utilized for this experiment, we can access an unlimited
range with the finest accuracy determined by the narrow HOM curve. In the
current experiment, we measured the group index variation of the PPKTP crystal
from room temperature to the allowed maximum temperature of 200∘C with a
precision of $\sim 2.25\times 10^{-6}$. The experimental results are shown in
Fig. 4(b). As evident from Fig. 4(b), the group index of the PPKTP crystal
changes by $\sim 350\times 10^{-5}$ for the increase of temperature to 200∘C.
The magnified image, as shown by the inset of Fig. 4(b), shows the resolution
of the measurement to be in the range of $\sim 2.25\times 10^{-6}$. The close
agreement between the calculated values (blue dots) and experimental results
(red dots) signifies the reliability of this delay compensation technique for
conducting group index measurements over an extended range with notable
precision of $\sim 2.25\times 10^{-6}$.

##  IV Conclusion

In conclusion, we have successfully demonstrated a HOM interferometer-based
quantum sensor for measuring group index variations of nonlinear crystals with
high precision. Using the HOM curve of FWHM width of 4.11 $\pm$ 0.01 $\upmu$m
resulting from the pair photons generated by a PPKTP crystal of length 1 mm,
we have measured the temperature-dependent group index of PPKTP crystal over a
temperature range of 3∘C at any arbitrary initial temperature with a
resolution of 6.75 $\times 10^{-6}$ per centimeter length of the sample. This
resolution is $>$ 400$\%$ better than the previous group index measurement
Reisner  _et al._ (2022). Such enhancement was possible due to the use of
optimum length of the nonlinear crystal to control the spectral bandwidth of
the pair photon, thus, the width of the HOM curve, while maintaining a high
generation rate of pair photons to ensure fast measurement. Using compensation
of group index mediated optical delay by the linear optical delay stage, we
observed the possibility of measuring group index over a measurement range of
175∘C with high precision of $\sim 2.25\times 10^{-6}$. The current
demonstration opens the possibility of HOM-based quantum sensors for quantum
optical coherence tomography with high resolution and measurement range.

## ACKNOWLEDGMENTS

All authors acknowledge the support of the Dept. of Space, Govt. of India. G.
K. S. acknowledges the support of Department of Science and Technology, Govt.
of India through Technology Development Program (Project DST/TDT/TDP-03/2022).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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