# Functional renormalization group study of the quark-meson model with
$\omega$ and $\rho$ vector mesons

Mohammed Osman<EMAIL_ADDRESS>Defu Hou
<EMAIL_ADDRESS>Wentao Wang<EMAIL_ADDRESS>Institute of
Particle Physics (IOPP) and key laboratory of Quark and lepton physics (MOE),
Central China Normal University, Wuhan 430079, China  Hui Zhang
<EMAIL_ADDRESS>Key Laboratory of Atomic and Subatomic Structure
and Quantum Control (MOE), Guangdong Basic Research Center of Excellence for
Structure and Fundamental Interactions of Matter, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Guangdong Provincial
Key Laboratory of Nuclear Science, Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China

###### Abstract

The functional renormalization group (FRG) is a non-perturbative method that
considers quantum and thermal fluctuations. Using the FRG flow equations, the
critical region of the two-flavor quark-meson model in a finite isospin
chemical potential with omega and rho vector mesons interactions is
investigated in this work. We also use the traditional mean-field method to
calculate the phase diagram in the chiral limit for comparison. The results
show that the influences of the omega meson and rho meson on the phase
structure are quite different. The existence of the isospin chemical potential
also causes significant changes in the phase structure.

††preprint: APS/123-QED

##  I Introduction:

The phase diagram of Quantum chromodynamics (QCD) is a widely studied topic in
both experimental and theoretical physics and of great interest [1, 2]Since
its initial conjecture by N. Cabibbo and G. Parisi, much has been learned
about its properties [3]. While experimental studies and lattice Monte-Carlo
simulations made it possible to study the high-temperature region
quantitatively, our understanding of the phase diagram at high baryon density
remains uncertain, in part because the lattice simulations are not directly
applicable due to the infamous fermion-sign problem [4] in addition, nuclear
interactions at finite density are extremely complex.

STAR Collaboration heavy ion collision (HIC) experiments at the Relativistic
Heavy Ion Collider at Brookhaven National Laboratory. (RHIC) [5, 6, 7] and by
NA61/SHINE Collaboration in the Super Proton Synchrotron (SPS) at CERN [8, 9],
in addition to the properties of the quark-gluon plasma (QGP), we are
presently mapping the phase boundary of QCD. Future facilities will consist of
the Nuclotron-based Ion Collider fAcility (NICA) at the Joint Institute for
Nuclear Research [10], The Antiproton and Ion Research Facility (FAIR) at GSI
Helmholtzzentrum für Schwerionenforschung [11] and J-PARC Heavy Ion Project at
Japan Proton Accelerator Research Complex (J-PARC) [12], it will also
contribute to the collective effort to better understand the nuclear
properties and quark matter under extreme temperature conditions, density
during magnetic field presence.

Model calculations, using the Nambu–Jona–Lasinio (NJL) model or the quark-
meson (QM) model, can be improved by going beyond the common mean field
approximation (MF) [13, 14, 15, 16, 17, 18, 19, 20, 21]. One way to go beyond
the MF approximation is using an application of the renormalization group to
continuous field theories. The Functional Renormalization Group (FRG) is a
potent non-perturbative method that allows quantum and thermal fluctuations to
be incorporated into a field theory. The FRG has been utilized extensively to
investigate the QCD phase diagram with chiral effective models beyond the MF,
such as the NJL model [22, 23, 24, 25] and the QM model [26, 27, 28, 29, R021,
30, 31]. See Ref. [32, 33, 34] for detailed evaluations of the FRG method.
The application of the FRG method to the 2-flavor QM model results in the
presence of an unphysical negative entropy density region near the critical
region where the model predicts a first-order chiral phase transition and CEP
[35]. A back-bending behavior of the chiral phase transition line at low
temperatures and a negative entropy density in the symmetric regime was
observed in Ref. [36].

Typically, the FRG is applied to quark models with two flavors, scalar
$(\sigma)$ and pseudoscalar $(\pi)$ fluctuations [37, 38, 39].In addition,
studies have been conducted on the vector $(\rho)$ and axial vector (a1)
fluctuations in the isovector channels [40, 41, 42]. However, to the best of
our knowledge, the $\omega$ and $\rho$-fluctuations were only considered in
the context of Walecka-type nucleon-$\sigma$-$\omega$ and $\rho$ models whose
primary focus is a nuclear matter at low temperature and density [43, 44]. In
the context of the quark model, it is known that the mean field of the
$\omega$ and $\rho$ mesons have a significant effect on the phase boundary and
the location of the critical end point [45, 46, 47], Consequently, it is
natural to investigate the stability of the mean-field picture against the
$\omega$ and $\rho$ fluctuations.

In this work, we will consider the 2-flavor QM model with $\omega$ and $\rho$
vector mesons to explore the combination between these vectors and their
effect on the critical endpoint predicted by the model, and we will take into
account the $(\sigma,\pi,\omega)$ and $\rho$ \- fluctuations and study their
impacts on the phase diagram. In the chiral limit, the phase diagram is
typical of the second order at high temperature and low chemical potential and
the first order at low temperature and high chemical potential. There exists a
tricritical point where the second order line changes into the first order one
[48, 49, 50].

Following is the structure of this paper:

##  II THE QUARK-MESON MODEL WITH $\omega$ AND $\rho$ VECTOR MESONS

The Lagrangian of the two-flavors Quark-Meson model with $\omega$ and $\rho$
vector mesons in Minkowski space is

$\begin{array}[]{l}\begin{aligned}
\mathcal{L}&=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}+\frac{\mu_{I}}{2}\gamma_{0}\tau_{3}+\mu\gamma_{0}\right)\psi\\\
&-\bar{\psi}\left[g_{s}\left(\sigma+i\gamma_{5}\bm{\tau}\cdot\bm{\pi}\right)+\gamma_{\mu}\left(g_{\omega}\omega^{\mu}+g_{\rho}\bm{\tau}\cdot\bm{\rho}^{\mu}\right)\right]\psi\\\
&+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma+\frac{1}{2}\partial_{\mu}\bm{\pi}\partial^{\mu}\bm{\pi}-\frac{1}{4}F_{\mu\nu}^{(\omega)}F^{(\omega)\mu\nu}-\frac{1}{4}F_{\mu\nu}^{(\rho)}F^{(\rho)\mu\nu}\\\
&-U(\sigma,\bm{\pi},\omega,\bm{\rho})\end{aligned}\end{array}$ (1)

The field strength tensors of the vector bosons $\omega_{\mu}$ and
$\bm{\rho_{\mu}}$ are generally given as:
$F^{(\omega)}_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$
and
$F^{(\rho)}_{\mu\nu}=\partial_{\mu}\bm{\rho}_{\nu}-\partial_{\nu}\bm{\rho}_{\mu}-g_{\rho}\bm{\rho}_{\mu}\times\bm{\rho}_{\nu}$
respectively. A field $\psi$ is the light two-flavor quark field $=(u,d)^{T}$.
And coupled to isoscalar-scalar $(\sigma)$ and pion $(\pi)$ fields
transforming as a four-component field $(\sigma,\pi)^{T}$ under the chiral
group. A bold symbol stands for a vector, and
$\bm{\tau}=(\tau_{1},\tau_{2},\tau_{3})$, are the Pauli matrices in isospin
space, and introducing isospin chemical potential $\mu_{I}=\mu_{u}-\mu_{d}$.
Treated as background fields, only $\omega_{0}$ and $\rho^{3}_{0}$ are non-
vanishing and the non-abelian of $F^{(\rho)}_{\mu\nu}$ does not contribute in
practice. A Hubbard-Stratonovich transformation bosonizes these interactions,
introducing effective vector-isoscalar and vector-isovector fields,
$\omega_{\mu}$ and $\rho_{\mu}$, respectively. While the fluctuations of the
$\pi$ and $\sigma$ fields will be included non-perturbatively, $\omega_{\mu}$
and $\rho_{\mu}$ will be treated as mean fields. These vector bosons
conveniently parametrize unresolved short-distance physics. They are not to be
identified with the physical omega and rho mesons. The potential for
$\sigma,\pi,\omega$ and $\rho$ is

$\begin{array}[]{l}\displaystyle
U(\sigma,\bm{\pi},\omega,\bm{\rho})=\frac{\lambda}{4}(\sigma^{2}+\bm{\pi}^{2}-f_{\pi}^{2})^{2}-\frac{m^{2}_{\omega}}{2}\omega_{\mu}\omega^{\mu}-\frac{m^{2}_{\rho}}{2}\bm{\rho}_{\mu}\bm{\rho}^{\mu},\\\\[5.0pt]
\end{array}$ (2)

where $f_{\pi}$ is the pion decay constant. We will use the value $f_{\pi}=93$
MeV, and $m_{\omega}=m_{\rho}\sim 1$ Gev. The parameters in our model are
$g_{s}$, $g_{\omega}$, $g_{\rho}$ and $\lambda$. The values of these
parameters can be different for the MF and FRG calculations when we try to
reproduce the same value for quantities such as the constituent quark mass of
$\sim 300MeV$. As for the value of $g_{\omega}$, $g_{\rho}$ and $m_{\nu}$,
$m_{\rho}$, in our calculation, they always appear in the form of
$g_{\omega}$/$m_{\nu}$, $g_{\rho}$/$m_{\rho}$, so we will not discuss their
values independently.

###  II.1 Mean-field approximation

The chiral symmetry is explicitly broken in the vacuum, and the expectation
values of the meson fields are $\left\langle\sigma\right\rangle$ =$f_{\pi}$
and $\pi=0$. Due to the rotational symmetry, only the zero component of the
vector fields $\omega_{\mu}$ and $\bm{\rho}_{\mu}$ can have an expectation
value [51]. Only considering the time component $\omega_{0}$ and
$\rho^{3}_{0}$ of the vector fields $\omega_{\mu}$ and $\bm{\rho}_{\mu}$, the
mean-field potential reads as

$\begin{array}[]{l}\displaystyle
U_{MF}(\sigma,\omega,\rho)=\frac{\lambda}{4}(\sigma^{2}-f^{2}_{\pi})^{2}-\frac{m^{2}_{\omega}}{2}\omega_{\mu}\omega^{\mu}-\frac{m^{2}_{\rho}}{2}(\rho_{0}^{3})_{\mu}(\rho_{0}^{3})^{\mu},\\\\[5.0pt]
\end{array}$ (3)

The mean-field effective potential is

$\begin{array}[]{l}\displaystyle\Omega_{MF}=\Omega_{\psi\overline{\psi}}+U_{MF}(\sigma,\omega_{0},\rho^{3}_{0}),\\\\[5.0pt]
\end{array}$ (4)

With thermal quark and antiquark contributions $\mu$ :quark chemical potential
; $T:$ temperature ;$\beta=\frac{1}{T}$

$\displaystyle\Omega_{\psi\bar{\psi}}=-$
$\displaystyle\mathrm{v}_{\mathrm{q}}\int\frac{\mathrm{d}^{3}\mathrm{p}}{(2\pi)^{3}}\left\\{\mathrm{E}_{\mathrm{q}}\theta\left(\Lambda_{\mathrm{MF}}^{2}-\mathrm{p}^{2}\right)\right\\}$
(5)
$\displaystyle-\frac{\mathrm{V}_{\mathrm{q}}}{2}\mathrm{~{}T}\int\frac{d^{3}\mathrm{p}}{(2\pi)^{3}}\left\\{\ln\left[1+\mathrm{e}^{-\beta\left(\mathrm{E}_{\mathrm{q}}-\mu_{\mathrm{eff}}^{+}\right)}\right]\right.$
$\displaystyle+\ln\left[1+\mathrm{e}^{-\beta\left(\mathrm{E}_{\mathrm{q}}+\mu_{\text{eff
}}^{+}\right)}\right]$
$\displaystyle\left.+\ln\left[1+\mathrm{e}^{-\beta\left(\mathrm{E}_{\mathrm{q}}-\mu_{\mathrm{eff}}^{-}\right)}\right)+\ln\left[1+\mathrm{e}^{-\beta\left(\mathrm{E}_{\mathrm{q}}+\mu_{\mathrm{eff}}^{-}\right)}\right]\right\\}$

The first term is the fermion vacuum fluctuation. The mean fields of the
vector bosons enter the partition function just by shifting the chemical
potentials; if we eliminated it, the chiral limit transition would always be
of first order [52]. We introduce the effective quark(anti-quark) chemical
potential given as:

$\begin{array}[]{l}\displaystyle\mu^{\pm}_{eff}=(\mu-
g_{\omega}\omega_{0})\pm(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0}),\\\\[5.0pt]
\end{array}$ (6)

The effect of the mean field generates an in-medium quark (anti-quark) mass
(effective mass)

$\begin{array}[]{l}\displaystyle m_{eff}=g_{s}\sigma,\\\\[5.0pt] \end{array}$
(7)

For a given temperature $T$ and chemical potential $\mu$, the mean-field
effective potential the gap equation for $\omega_{0}$ and $\rho^{3}_{0}$ can
be driven by solving quantum equation of motion for $\omega_{0}$ and
$\rho^{3}_{0}$:

  1. 1.

For $\omega_{0}$

$\begin{array}[]{l}\displaystyle\omega_{0}=\frac{1}{2}\frac{g_{\omega}}{m^{2}_{v}}[{\omega_{q}}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\partial}{\partial\mu}(\ln[1+e^{-\beta(E_{q}-\mu^{+}_{eff})}]+\\\\[15.0pt]
\quad\quad\quad\ln[1+e^{-\beta(E_{q}+\mu^{+}_{eff})}]+\\\\[15.0pt]
\quad\quad\quad\ln[1+e^{-\beta(E_{q}-\mu^{-}_{eff})}]+\ln[1+e^{-\beta(E_{q}+\mu^{-}_{eff})}]),\end{array}$
(8)
$\begin{array}[]{l}\displaystyle\partial_{k}\omega_{0,k}=\frac{1}{2}\frac{g_{\omega}}{m^{2}_{\nu}}[n^{+}+n^{-}],\end{array}$
(9)

  2. 2.

For $\rho^{3}_{0}$

$\begin{array}[]{l}\displaystyle\rho^{3}_{0}=\frac{1}{2}\frac{g_{\rho}}{m^{2}_{\rho}}[{\nu_{q}}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\partial}{\partial\mu}(\ln[1+e^{-\beta(E_{q}-\mu^{+}_{eff})}]+\\\\[15.0pt]
\quad\quad\quad\ln[1+e^{-\beta(E_{q}+\mu^{+}_{eff})}])+\\\\[15.0pt]
\quad\quad\quad\frac{\partial}{\partial\mu^{-}}(\ln[1+e^{-\beta(E_{q}-\mu^{-}_{eff})}]+\ln[1+e^{-\beta(E_{q}+\mu^{-}_{eff})}])],\end{array}$
(10)

$\begin{array}[]{l}\displaystyle\partial_{k}\rho^{3}_{0,k}=\frac{1}{2}\frac{g_{\rho}}{m^{2}_{\rho}}[n^{-}-n^{+}],\end{array}$
(11)

and the up-quark and down-quark densities, respectively, are determined by

$\begin{array}[]{l}\displaystyle n^{-}(T,(\mu-
g_{\rho}\rho^{3}_{0}))-(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0}))\\\\[15.0pt]
\quad\quad\quad=-\frac{\partial}{\partial\mu}\Omega_{\psi\overline{\psi}}(T,(\mu-
g_{\omega}\omega_{0})-(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0})),\end{array}$
(12)

and down-quark

$\begin{array}[]{l}\displaystyle n^{+}(T,(\mu-
g_{\omega}\omega_{0}))-(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0}))\\\\[15.0pt]
\quad\quad\quad=-\frac{\partial}{\partial\mu}\Omega_{\psi\overline{\psi}}(T,(\mu-
g_{\omega}\omega_{0})+(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0})),\end{array}$
(13)

the vector couplings $g_{\omega}$ and $g_{\rho}$ and the masses of the
$\omega$, $\rho$ fields respectively, are not independent,
$g_{\omega}$$\omega_{k}$, $g_{\rho}$$\rho_{k}$ are proportional to
$(\frac{g_{\omega}}{m_{\omega}})^{2}$, $(\frac{g_{\rho}}{m_{\rho}})^{2}$
respectively. Only their ratio $(\frac{g_{\omega}}{m_{\omega}})$,
$(\frac{g_{\rho}}{m_{\rho}})$ appear in calculations for both MF and FRG.

In our calculation, we follow the choice of Ref. [53] and set the parameters
$g_{s}$=3.3 and $\lambda$ = 20, with which the constituent quark mass in a
vacuum is $M_{vac}$ = $g_{s}$$f_{\pi}$ $\simeq$ 307 MeV, and the sigma mass is
$m_{\sigma}$ = $\sqrt{2\lambda f_{\pi}^{2}}$ $\simeq$ 588 MeV.

Remember that the strength of $\omega_{0}$ and $\rho^{3}_{0}$ field in MF
calculations is proportional to the quark density $n$. After incorporating the
fluctuations, Nevertheless, the quark number density is equal to the sum of
single particle contribution and contributions from other fluctuations;
therefore, such a proportional relationship no longer holds.
In the MF calculation, the self-consistent equation for the $\omega_{0}$ field
is directly related to the sum of the quark densities while the one for the
$\rho^{3}_{0}$ field is related to the difference in the quark densities. This
means that the $\rho^{3}_{0}$ field is zero for symmetric matter
($\rho^{3}_{0}$ =0), i.e. if considering $\mu_{u}$ = $\mu_{d}$. Similar
results are produced by the FRG calculation.
Indeed, in Ref. [44], It was demonstrated that disregarding recovery is
possible for the MF results. However, the choice of the non-zero ultraviolet
value of the $\rho^{3}_{0}$ vector field, $\rho^{3}_{0,\Lambda}$, resulting in
an explicit isospin breaking interaction and to a non-zero $\rho^{3}_{0}$
field, even for symmetric matter. Indeed, in Ref. [54], non-zero values for
$\omega_{0,\Lambda}$ were regarded as their influence on the phase diagram was
investigated. Nevertheless, there is no cause to regard an ultraviolet
potential with explicit isospin breaking by the $\rho^{3}_{0}$ field.

In order to study the influence of the $\rho^{3}_{0}$ field on the phase
diagram and, in particular, the first order phase transition line at low
temperature.

###  II.2 FRG flow equation

Functional Renormalization Group (FRG), a powerful non-perturbative method
that allows incorporating quantum and thermal fluctuations in a field theory
[55] and has been extensively applied to effective QCD models [R0046, 56, 57].
The effective average action $\Gamma_{k}$ with a scale k obeys the exact
functional flow equation

$\begin{array}[]{l}\displaystyle\partial_{k}\Gamma_{k}=\frac{1}{2}Tr\left[\frac{\partial_{k}R_{k}}{\Gamma^{(2)}_{k}+R_{k}}\right],\end{array}$
(14)

Where $\Gamma^{(2)}_{k}$ is the second functional derivative of the effective
average action with respect to the fields, the trace includes momentum
integration as well as traces of overall inner indices. An infrared regulator
$R_{k}$ was introduced to suppress fluctuations at momenta below the scale k.
The regulator may assume any functional form because it will only interfere
with the arbitrarily chosen path in the theory space between these points. In
this investigation, quarks serve as the dynamical fields in the flow equation,
$\sigma$, and $\pi$, and they affect the effective potential and the size of
$\omega_{0}$ and $\rho_{0}$ fields. Contrary to vector fields’ spatial
components, Because it is not coupled to the time derivative, the $\omega_{0}$
and $\rho_{0}$ fields are not dynamical.

Therefore, the value of $\omega_{0}$ and $\rho_{0}$ are completely fixed by
specifying the values of other fields. At each scale $k$ in the flow equation,
we determine the values of $\omega_{0}$ and $\rho_{0}$ field by solving the
consistency equation for given $\sigma$, and $\pi$, so the resultant
$\omega_{0}$ and $\rho_{0}$ may be written as
($\omega_{0,k}$,$\rho^{3}_{0}$)($\sigma$,$\pi$). These $\omega_{0,k}$ and
$\rho^{3}_{0}$ fields in turn appear in the effective chemical potential for
quarks, affecting the dynamical fluctuations in the flow equations. Throughout
our study we neglect the flow of all wave-function renormalization factors.

The scale-dependent effective potential can be expressed by replacing the
potential U with the scale-dependent one $U_{k}$,

$\begin{array}[]{l}\displaystyle\Gamma_{k}=\int
d^{4}x{\mathcal{L}}|U\rightarrow U_{k},\end{array}$ (15)

with the Euclidean Lagrangian from Eq.(1), Finite temperatures and chemical
potentials are treated within the Matsubara formalism. The time-component is
Wick-rotated, ${t}\rightarrow-i\tau$ and the imaginary time $\tau$ is
compactified on a circle with radius $\beta=\frac{1}{T}$, where T is the
temperature, for which after introduced $\int
d^{4}x\equiv\int^{1/T}_{0}dx_{0}\int_{V}d^{3}x$. Due to the chiral symmetry,
the potential $U$ depends on $\sigma$ and $\pi$ only through the chiral
invariant

$\begin{array}[]{l}\displaystyle\phi^{2}=\sigma^{2}+\bm{\pi}^{2},\end{array}$
(16)

As mentioned, the vector fields $\omega_{0},\rho^{3}_{0}$ appear here only as
mean fields. The complete k-dependence is in the effective potential $U_{k}$.
In analogy to the mean-field potential, the effective potential has a chirally
symmetric piece, $U_{k}^{\phi}$ the explicit chiral symmetry breaking term and
the mass terms of the vector bosons:

$\begin{array}[]{l}\displaystyle
U_{k}=U^{\phi}_{k}+U^{\omega}_{k}+U^{\rho}_{k},\end{array}$ (17)

starting with some ultraviolet $(UV)$ potentials $U_{\Lambda}$ as our initial
conditions, we integrate fluctuations and obtain the scale-dependent $U_{k}$.
The form of $U_{k}^{\phi}$ will be determined without assuming any specific
forms, while for the potential of the $\omega$ and $\rho$ fields, we use the
same form as in Eq (3):

$\begin{array}[]{l}\displaystyle
U^{\omega,\rho}_{k}=-\frac{m^{2}_{\omega}}{2}\omega_{\mu}\omega^{\mu}-\frac{m^{2}_{\rho}}{2}\rho_{\mu}\rho^{\mu}\end{array}$
(18)

to use Wetterich’s equation, a regulator function, that respects the
interpolating limits of the effective average action, has to be chosen. We
employ the so-called optimized or Litim regulator function [58], for bosons
and fermions, respectively given by:

$\begin{array}[]{l}\displaystyle
R^{B}_{k}(p)=(k^{2}-p^{2})\theta(k^{2}-p^{2}),\end{array}$ (19)
$\begin{array}[]{l}\displaystyle
R^{F}_{k}(p)=\left(\begin{array}[]{cc}0&ip_{i}(\gamma^{E}_{i})^{T}\\\
ip_{i}\gamma^{E}_{i}&0\\\
\end{array}\right)(\sqrt{\frac{k^{2}}{p^{2}}-1})\theta(k^{2}-p^{2})\end{array}$
(20)

because of the structure of the regulators, the dependence on three-momenta is
eliminated, and only integral over theta function remains. The flow equation
for the potential $U_{k}^{\phi}$ can be obtained as:

$\begin{array}[]{l}\displaystyle\partial_{k}U^{\phi}_{k}(T,\mu)=\frac{k^{4}}{12\pi^{2}}\Bigg{[}\left\\{\frac{3[1+2n_{B}(E_{\pi})]}{E_{q}}+\frac{[1+2n_{B}(E_{\sigma})]}{E_{\sigma}}\right\\}\\\
\displaystyle\quad\quad\quad\quad\quad-
v_{q}\left\\{\frac{1-n_{F}(E_{q},\mu^{+}_{eff})-n_{F}(E_{q},-\mu^{-}_{eff})}{E_{q}}\right\\}\\\
\displaystyle\quad\quad\quad\quad\quad-
v_{q}\left\\{\frac{1-n_{F}(E_{q},-\mu^{+}_{eff})-n_{F}(E_{q},\mu^{-}_{eff})}{E_{q}}\right\\}\Bigg{]}\end{array}$
(21)

where the effective energies are given by:

$\begin{array}[]{l}\displaystyle E_{\pi}=\sqrt{k^{2}+M^{2}_{\pi}},\\\
\end{array}$ (22) $\begin{array}[]{l}\displaystyle
E_{\sigma}=\sqrt{k^{2}+M^{2}_{\sigma}},\\\ \end{array}$ (23)
$\begin{array}[]{l}\displaystyle E_{q}=\sqrt{k^{2}+M^{2}_{q}},\\\ \end{array}$
(24)

for pion, sigma-meson, and quark, respectively. And the scale-dependent
particle masses are:

$\begin{array}[]{l}\displaystyle M^{2}_{q}=g^{2}\phi^{2},\\\ \end{array}$ (25)

$\begin{array}[]{cc}\displaystyle M^{2}_{\pi}=2U^{\prime}_{k}(\phi^{2}),\\\
\end{array}$ (26)

$\begin{array}[]{cc}\displaystyle
M^{2}_{\sigma}=2U^{\prime}_{k}(\phi^{2})+4\phi^{2}U^{\prime\prime}_{k}(\phi^{2}),\\\
\end{array}$ (27)

and we also defined $U^{\prime}_{k}=\frac{\partial U_{k}}{\partial\phi^{2}}$.

The effective chemical potential,

$\begin{array}[]{l}\displaystyle\mu^{\pm}_{eff}=(\mu-
g_{\omega}\omega_{0,k})\pm(\frac{\mu_{I}}{2}+g_{\rho}\rho^{3}_{0,k}),\\\\[5.0pt]
\end{array}$ (28)

depends also on the field $\omega_{0,k}$ ,$\rho_{0,k}$ , and both depend on
the scale $k$. The extended occupation numbers simplify to the usual Fermi-
Dirac distribution functions for boson and fermion occupation numbers are:

$\begin{array}[]{l}\displaystyle n_{B}(E)=\frac{1}{e^{\beta E}-1},\ \
n_{F}(E,\mu)=\frac{1}{e^{\beta(E-\mu)}+1},\\\ \end{array}$ (29)

The flow equation for the effective action, the $\omega_{0},\rho$ fields must
be calculated self-consistently. Therefore, at each momentum scale $k$ [44,
54] we solve the mean-field equation for $\omega_{0,k}$ and $\rho^{3}_{0,k}$.

$\begin{array}[]{l}\displaystyle\frac{\partial
U_{k}}{\partial\omega_{0,k}}=0,\\\ \displaystyle\frac{\partial
U_{k}}{\partial\rho^{3}_{0,k}}=0.\\\ \end{array}$ (30)

The dependence on $\omega_{0,k},\rho^{3}_{0,k}$ appears in the mass term and
the fermion loop. The flow equation reads:

  1. 1.

For $\rho^{3}_{0,k}$

$\begin{array}[]{l}\displaystyle\partial_{k}\rho^{3}_{0,k}=-\frac{g_{\rho}\
k^{4}}{\pi^{2}\ m^{2}_{\rho}\ E_{q}}\frac{\partial}{\partial{\mu}}\
[n_{F}(E_{q},\mu^{+}_{eff})\ +n_{F}(E_{q},-\mu^{+}_{eff})\\\ \displaystyle+\
\frac{\partial}{\partial\mu}\ [n_{F}(E_{q},\mu^{-}_{eff})\
+n_{F}(E_{q},-\mu^{-}_{eff})],\\\ \end{array}$ (31)

On the right side, the first term of down-quark and the other of up-quark.This
equation constitutes our flow equations for the $\rho$ field functions of
$\phi$.

  2. 2.

For $\omega_{0,k}$

$\displaystyle\partial_{k}\omega_{0,k}=-\frac{g_{\omega}k^{4}}{\pi^{2}m_{\omega}^{2}E_{q}}\frac{\partial}{\partial\mu}\left[n_{F}\left(E_{q},\mu_{eff}^{+}\right)+n_{F}\left(E_{q},-\mu_{eff}^{+}\right)\right.$
(32)
$\displaystyle+\frac{\partial}{\partial\mu}\left[n_{F}\left(E_{q},\mu_{eff}^{-}\right)+n_{F}\left(E_{q},-\mu_{eff}^{-}\right)\right],$

This equation constitutes our flow equations for the $\omega_{0,k}$ field
functions of $\phi$.

Note that the flow equations for $\omega_{0}$ and $\rho^{3}_{0}$ can be solved
for a given $\phi$, independently of the potential $U^{k}_{\phi}$ (which only
tells us where the minimum of $\phi$ is).

To understand the behavior of $\omega_{0}$ and $\rho^{3}_{0}$, for the moment
we ignore the $k$ dependence in $\mu_{eff}$, and carry out the integration
over $k$. Then the resulting expression for $\omega_{0}$ and $\rho^{3}_{0}$
are some factor times the MF expression for the number density. But unlike the
MF case, $\omega_{0}$ and $\rho^{3}_{0}$ are not directly proportional to the
physical number density because the baryon density gets contributions not only
from single particles but also fluctuations (see Eq.( 36)). Moreover, as we
will see in Sec.3 if we include the $k$ dependence in $\mu^{eff}_{k}$, the
$\omega_{0,k}$ and $\rho^{3}_{0,k}$ fields at $k_{IR}$ are not even
proportional to the single-particle contribution. Therefore, the extrapolation
of the MF relation $\omega_{0}\sim{n}$ and $\rho_{0}\sim{n}$ does not work
without exception to understand the FRG results.

Finally, the initial conditions for the flow equations must be set up. The UV
scale $\Lambda$ should be sufficiently large in order to take into account the
relevant fluctuation effects and small enough to render the description in
terms of the model degrees of freedom realistic [59]. In our calculation we
follow the choice of Ref. [38] , $\Lambda$ = 500 MeV. The initial for the
potential is

$\begin{array}[]{l}\displaystyle
U^{\phi}_{\Lambda}=\frac{\lambda}{4}{\phi^{4}},\\\ \end{array}$ (33)

and set the parameters $g_{s}$ = 3.2, $\lambda$ = 8 with the vacuum effective
potential from the FRG computation having the minimum at $\sigma_{vac}\simeq
93$ MeV, which is regarded as $f_{\pi}$. We note that the value of $\lambda$,
which enforces $\phi$ to stay near $f_{\pi}$, is considerably smaller than the
MF case $(\lambda\sim 20)$. If we start with another initial condition with an
additional $\phi^{2}$ term to give the mass, we need to readjust $\lambda$ but
obtain qualitatively similar results; in fact, starting with the condition
Eq.( 33), the scale evolution first generates the $\phi^{2}$ terms, reflecting
the universality.

The initial condition for the $\omega$ and $\rho$ fields has not been examined
in detail, and we simply try.

$\begin{array}[]{cc}\displaystyle\omega_{0,\Lambda}(\phi)=0,\displaystyle\rho^{3}_{0,\Lambda}(\phi)=0,\end{array}$
(34)

Later we will also present the result of another different initial condition,
but it will turn out that such a modification does not change the main story
in this paper.

Assembling all these elements, we calculate the effective potential with the
fluctuations integrated to $k_{IR}$ = 0. The final step is to find $\Phi$ =
$\sigma^{*}$, which minimizes the effective potential. At the minimum, the
effective potential is identified as the thermodynamic potential,

$\begin{array}[]{l}\displaystyle\Omega(\mu,T)=\Gamma_{IR=0}(\mu,T,\sigma^{*}),\par\end{array}$
(35)

In practice, it is numerically expensive to reduce the $IR$ cutoff and we
typically stop the integration around $K_{IR}\simeq 20MeV$.

The baryon number density is then obtained by taking the derivative with
respect to $\mu_{B}$ = $N_{c}\mu$,

$\begin{array}[]{l}\displaystyle
n_{B}(\mu,T)=-\frac{1}{N}\frac{\partial\Gamma_{{k}_{IR=0}}(\mu,T,\sigma^{*})}{\partial\mu},\end{array}$
(36)

The derivative is taken numerically with the interval $\Delta\mu$ = 0.1 MeV.

##  III RESULS

###  III.1 Effective potential

Figure 1:  $T=10$ $MeV$, $\mu=283$, $MeV$ $\mu_{I}=150$ $MeV$ Figure 2:
$T=10$ $MeV$, $\mu=133.2$, $MeV$ $\mu_{I}=550$ $MeV$

By numerically solving Eqs. ( 21, 31, 32), the results that can be obtained
directly are the effective potentials at different energy scales. The
evolution of the energy scale ranges from the set 500 MeV to the infrared
cutoff less than or equal to 10 MeV. The evolution after the infrared cutoff
is considered to remain unchanged within the accuracy. Fig. 1 and Fig. 2 show
the evolution of the effective potential with energy scale using different
isospin chemical potentials, revealing the influence of isospin chemical
potential on the effective potential. The quark chemical potentials are taken
around the critical points. We compared the cases where the isospin chemical
potential is 150 MeV and 550 MeV, respectively.

It can be found that during the process of gradually integrating the
fluctuations, the isospin chemical potential significantly changes the value
of the effective potential. When focusing on the local characteristics of the
effective potential, the influence of fluctuations on the extreme value near
$\phi=0$ is more obvious, but it does not affect the extreme value around
$\phi=f_{\pi}$. On the other hand, as the RG flow evolves with the
renormalization time, the effective potential in Fig. 2 flattens at an earlier
stage. This implies that isospin asymmetry leads to the inclusion of a greater
number of fluctuations in the system at the “early” stage.

Figure 3:

Fig. 3 shows the relationship between the effective potential and the chemical
potential under different isospin chemical potentials at low temperatures.
Isospin chemical potential introduced in the system does significantly lower
the effective potential at the same temperature and at less quark chemical
potential.

###  III.2 Chiral condensates

To investigate the peculiar chiral phase transition in the region of low
temperature and high chemical potential for isospin asymmetric matter, we can
observe the behavior of the order parameter in the low-temperature region
under different parameters. This is a typical region where a first-order phase
transition should occur, implying that with an increase in quark chemical
potential, the order parameter decreases discontinuously from a finite value
to a small one, during which the chiral symmetry of the system is restored.

Fig. 4 illustrates the evolution of chiral condensation with quark chemical
potential under the same vector coupling but different isospin chemical
potentials. As the isospin chemical potential increases, the quark chemical
potential inducing the chiral phase transition gradually decreases. Generally,
at higher temperatures, the initial value of chiral condensation is less than
at low temperature, and higher temperatures are closer to the region of
second-order phase transition. We note that an increase in isospin chemical
potential also leads to a decrease in the initial value of chiral
condensation, corresponding to a reduction in the critical temperature for a
system with isospin asymmetry.

Fig. 5 and Fig. 6 describe the low-temperature behavior of the order parameter
under different coupling strengths of rho mesons with finite isospin chemical
potential. First, when the coupling constant is zero (including omega mesons),
but there exists finite isospin chemical potential (depicted by the black line
in Fig. 5), it is fundamentally different from matter that lacks both vector
coupling and isospin chemical potential (depicted in Fig. 7). The former
undergoes a second-order phase transition followed by a first-order phase
transition, while the latter undergoes a first-order phase transition followed
by a second-order phase transition. In our calculations, we find that this
difference brought about by isospin chemical potential is temperature-
independent. Even when considering vector coupling, the two cases exhibit
distinct characteristics. After accounting for the influence of vector
coupling, a system with finite isospin chemical potential may undergo a first-
order or second-order phase transition followed by a first-order phase
transition, while isotopically symmetric matter will certainly undergo a
first-order phase transition, followed by either a first-order or second-order
phase transition. On the other hand, in Fig. 5 and Fig. 6, we simultaneously
consider the effects of both isospin chemical potential and vector coupling.
We find that with an increase in vector coupling, the transition types at
smaller quark chemical potential positions continuously change from second-
order to first-order and then revert to first-order until they disappear. As
we have observed in the results of the effective potential, these phenomena
are evidently due to the modification of the vacuum near $\phi=0$ induced by
isospin chemical potential.

Figure 4:  Chiral condensates as a function of quark chemical potential under
different isospin chemical potentials, calculated at $T=10$ $MeV,$
$g_{\omega}/m_{\omega}=g_{\rho}/m_{\rho}=0.006$ $\left[MeV\right]^{-1}$. The
different colored lines correspond to different isospin chemical potentials.
Figure 5:  Chiral condensates as a function of quark chemical potential
calculated at $T=10$ $MeV,$ $\mu_{I}=200$ $MeV,$ $g_{\omega}/m_{\omega}=0$
$\left[MeV\right]^{-1}$. The plot shows the variation for different
$g_{\rho}/m_{\rho}$, different colored lines corresponding to different vector
coupling strengths. Figure 6:  Chiral condensates as a function of quark
chemical potential calculated at $T=10$ $MeV,$ $\mu_{I}=200$ $MeV,$
$g_{\omega}/m_{\omega}=0$ $\left[MeV\right]^{-1}$. The plot shows the
variation for different $g_{\rho}/m_{\rho}$, different colored lines
corresponding to different vector coupling strengths. Figure 7:  Chiral
condensates as a function of quark chemical potential, calculated at $T=10$
$MeV,$ $g_{\omega}/m_{\omega}=g_{\rho}/m_{\rho}=0$ $\left[MeV\right]^{-1}$.

###  III.3 Baryon number density

Figure 8:  The change of the baryon number densities with chemical potential
when $T=10$ $MeV$. The black line is corresponding to $\mu_{I}=0$ $MeV$, the
blue line is corresponding to $\mu_{I}=200$ $MeV$, the red line is in
$\mu_{I}=300$ $MeV$, the green line is in $\mu_{I}=400$ $MeV$ respectively.
The baryon number density normalized by the nuclear saturation density
$n_{0}=0.16fm^{-1}$.

Fig. 8 shows the relationship between baryon number density and chemical
potential at the same temperature under different isospin chemical potentials.
At low isospin chemical potential, there is an obvious first-order phase
transition, but when the isospin chemical potential gradually increases, the
phase transition type shows a tendency to approach the second-order phase
transition. In addition, we note that isospin chemical potential enables
chiral phase transition to happen in a smaller quark chemical potential. It is
assumed that the stronger isospin asymmetry causes the instability of the
system, just as the chemical potential plays a role in the interaction.

###  III.4 Phase diagram

In this section, we investigate the influence of different isospin chemical
potentials and couplings on chiral phase transition in the chiral limit. The
FRG equations here we used are the flow equation Eq.( 21) for the rho meson
given equation Eq.( 31) and for the omega meson equation given Eq.( 32) for
different values of temperature and chemical potential. In addition, we also
calculate the results obtained by solving the mean-field effective potential
equation Eq.( 5) using different parameters for qualitative comparison.

Figure 9:  The FRG ${T-\mu}$ phase diagram with different vector couplings.
The solid lines show the first-order phase transition, and the dashed lines
show the second-order phase transition. The stars show the TCPs. The
parameters are set as: $f_{\pi}$ = 93 MeV, $g_{s}$ = 3.2, $\lambda$ = 8, the
ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV, ${\mu_{I}}$ = 200 MeV.  Figure
10:  The mean-field ${T-\mu}$ phase diagram including vacuum fluctuation
($\Lambda_{MF}$=260 MeV) for the two-flavor massless QCD with different vector
couplings. Solid lines show the first-order phase transitions, and dashed
lines show the second-order phase transition. Dashed lines show the critical
points, and the stars show the TCPs. Figure 11:  The FRG ${T-\mu}$ phase
diagram with different $\mu_{I}$. The solid lines represent the first-order
phase transition, and the dashed lines represent the second-order phase
transition. The stars show the TCPs. The parameters are set as: $f_{\pi}$ = 93
MeV, $g_{s}$ = 3.2, $\lambda$ = 8, the ultraviolet cutoff $\Lambda_{FRG}$ =500
MeV, the coupling constants $g_{\omega}$ $m^{-1}_{\omega}$ = $g_{\rho}$
$m^{-1}_{\rho}$ = 0.006 $[MeV]^{-}1$. Figure 12:  The mean-field ${T-\mu}$
phase diagram including vacuum fluctuation ($\Lambda_{MF}$=260 MeV) for the
two-flavor massless QCD with different isospin chemical potentials. Solid
lines show the first-order phase transitions, and dashed lines show the
second-order phase transition. Dashed lines show the critical points, and the
stars show the position of the TCPs.

Fig. 9 and Fig. 10 show that with an increase in the coupling constant.
$\omega$ and $\rho$ take the same coupling strength in the calculation for
easy discussion of the effects of the two vector mesons on the phase
structure. Focusing on a fixed temperature, the restoration of chiral symmetry
in the hadronic phase requires a higher chemical potential. In the low
temperature region of Fig. 8, we find an idiosyncratic second-order boundary,
different from the one shown in [26]. Due to $\mu_{I}$ changes the vacuum
around $\phi=0$, isospin-asymmetric matter exhibits distinct chiral phase
transition behavior at low temperatures compared to isospin-symmetric matter.
The results calculated by mean-field approximation are given with different
vector couplings in Fig. 10. Due to the inclusion of the vacuum term in our
calculation, there is a second-order phase transition [54]. The phase boundary
moves gradually to the high chemical potential region as the vector coupling
increases. At the same time, the tricritical point(TCP)’s temperature
decreases gradually. This is in contrast to the results presented in Fig. 9,
but it aligns with the conclusions drawn by previous work in Ref. [54], the
interaction of vector mesons does not alter the temperature of the chiral
phase transition (TCP).

Fig. 11 and Fig. 12 show the chiral phase diagram obtained by different
isospin chemical potentials. However, different from the mean-field method in
Fig. 12, Fig. 11 shows the “back-bending” feature unique to the functional
renormalization group calculation method in the low-temperature region as well
as Fig. 9. In this region, as the temperature decreases gradually, the
chemical potential required for phase transition gradually decreases, while
the mean field method shows normal characteristics, the lower the temperature,
the higher the chemical potential of phase transition. The “back-bending”
feature of the phase diagram obtained by FRG equations has always been a hot
topic. We expect that the introduction of isospin chemical potential and
$\rho$ meson can modify this behavior, but judging from the results, this
seems to make the back-bending effect more pronounced. It is quite
understandable that, as the isospin chemical potential increases, fluctuations
in the low-temperature region are incorporated into the RG flow evolution.
This further underscores that the bending effect in the phase diagram is a
consequence of excessive fluctuations considered in the low-temperature
region.

Figure 13:  The FRG ${T-\mu}$ phase diagram with different $\omega$ meson’s
couplings. The solid lines show the first-order phase transition, and the
dashed lines show the second-order phase transition. The stars show the TCPs.
The parameters are set as: $f_{\pi}$ = 93 MeV, $g_{s}$ = 3.2, $\lambda$ = 8,
the ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV, ${\mu_{I}}$ = 200 MeV,
$g_{\rho}$ $m^{-1}_{\rho}$=0 $MeV^{-1}$. Figure 14:  The FRG ${T-\mu}$ phase
diagram with different $\rho$ meson’s couplings. The solid lines show the
first-order phase transition, and the dashed lines show the second-order phase
transition. The stars show the TCPs. The parameters are set as: $f_{\pi}$ = 93
MeV, $g_{s}$ = 3.2, $\lambda$ = 8, the ultraviolet cutoff $\Lambda_{FRG}$ =500
MeV, ${\mu_{I}}$ = 200 MeV, $g_{\omega}$ $m^{-1}_{\omega}$=0 $MeV^{-1}$.

Fig. 13 and Fig. 14 depict the impact of coupling constants for omega and rho
mesons on the phase boundaries, respectively. We observe that at fixed
temperature, the chiral phase transition occurs at a higher chemical
potential. Although both two vector mesons contribute to the system’s
increased stability, the influence of omega meson on the phase boundaries is
significantly stronger than that of rho meson at the same coupling strength.
Furthermore, in the extreme cases, the influence of omega coupling and rho
coupling on the phase transition chemical potential differs. With the
enhancement of omega coupling, the quark chemical potential required for the
phase transition at a given temperature increases more rapidly. In contrast,
the impact of rho coupling on the phase boundary becomes increasingly limited
with the increase in coupling strength.Additionally, by combining the results
from Fig. 5, Fig. 6 and Fig. 14, one can investigate the specific behaviors in
the low-temperature region. In the low-temperature region, the pattern
observed in the phase boundary at lower chemical potentials with varying rho
coupling strength is as follows: as the rho coupling strength gradually
increases from 0, the boundary of the first-order phase transition appears at
lower temperatures, and the region of the second-order phase transition
becomes smaller until the second-order phase transition disappears. The
boundary of the first-order phase transition merges with the boundary at
higher chemical potentials.

##  IV SUMMARY

We have calculated the two-flavor quark meson model with omega and rho mesons
through the FRG method and the mean-field method to obtain two-phase diagrams
and studied the influence of vector mesons and isospin chemical potential on
the phase structure of the chiral phase diagram. Therefore, the main
conclusions are divided into two aspects: one is the influence of vector
couplings on phase structure; and the other one is the influence of isospin
chemical potential on phase structure.

For vector couplings, we find that the influences of omega meson and rho meson
on phase boundaries are different. First, under the joint action of all vector
couplings, the results given by the mean-field and FRG methods are
qualitatively consistent in the high-temperature region. With the increase of
coupling strength, the phase boundary moves to the high chemical potential
region, but the results given by the FRG show that the temperature of TCP does
not decrease with the increase of chemical potential. Then we compare the
phase structures when the two vector couplings exist independently: We observe
that the presence of vector couplings causes a shift of the phase boundary
towards higher chemical potentials. However, the influence of the omega meson
is significantly greater than that of the rho meson. For TCPs, both two types
of vector couplings don’t change their temperature of them. Comparing the
influence of two types of vector couplings on the phase boundary, in extreme
cases, the impact of omega coupling and rho coupling on the phase transition
chemical potential is different. As the omega coupling strengthens, the quark
chemical potential required for the phase transition at a given temperature
increases more rapidly. On the other hand, the influence of rho coupling on
the phase boundary becomes increasingly limited with the increase in coupling
strength.

For isospin chemical potential, with the vector coupling strength unchanged,
the increase of isospin chemical potential causes the phase boundary to move
as a whole to the low temperature and low-density area, and the temperature of
TCP also decreases gradually. Different from only changing the vector
coupling, the isospin chemical potential also significantly reduces the
temperature at which the phase transition occurs at the extremely low chemical
potential. By analyzing the effective potential and baryon number density
under different isospin chemical potentials, we believe that increasing
isospin chemical potential increases the influence of fluctuations, they
change the vacuum around $\phi=0$, resulting in corresponding changes in phase
structure. Besides considering the combined effects of both types of vector
couplings, we have investigated the chiral phase structure of dense isospin
asymmetric matter at low temperatures. We have identified another distinct
phase boundary, different from the one reported in the literature. Despite
both being secondary phase boundaries emerging in the low-temperature region
at lower baryon numbers, the transition types and positions in the phase
diagram are different from those previously observed. The reasons for these
differences can be attributed to the contribution of isospin chemical
potential to fluctuations in the effective potential.

Furthermore, the FRG and the mean-field methods show significant differences
in our calculations; that is, the “bending back” effect appeared in the low-
temperature region of the phase diagram without exception in the FRG results.
We try to analyze the causes of this “back bending”. It can be found that
increasing the vector coupling will aggravate the ”bending back” behavior,
which is consistent with Ref. [54]. Increasing the isospin chemical potential
will also make the “back bending” phenomenon more obvious. Ref. [56] pointed
out that this effect of the QM model is likely due to the regulator adopted by
the FRG, which includes too many fluctuations in the evolution process. This
likely means that when the conventional FRG method is used to calculate the
model, any factor that increases the fluctuation effect will lead to increased
“back bending”. Our work further confirms this point.

###### Acknowledgements.

Hui Zhang acknowledges the financial support from the Guangdong Major Project
of Basic and Applied Basic Research (Grant No. 2020B0301030008) and the
National Natural Science Foundation of China (Grant No. 12047523, and
12105107). We are thankful to Haicang Ren for valuable discussions on this
work.

## References

  * Braun-Munzinger and Wambach [2009] P. Braun-Munzinger and J. Wambach, The Phase Diagram of Strongly-Interacting Matter, Rev. Mod. Phys. 81, 1031 (2009), arXiv:0801.4256 [hep-ph] .
  * Friman  _et al._ [2011] B. Friman, C. Hohne, J. Knoll, S. Leupold, J. Randrup, R. Rapp, and P. Senger, eds.,  _The CBM physics book: Compressed baryonic matter in laboratory experiments_, Vol. 814 (2011).
  * Cabibbo and Parisi [1975] N. Cabibbo and G. Parisi, Exponential Hadronic Spectrum and Quark Liberation, Phys. Lett. B 59, 67 (1975).
  * de Forcrand [2009] P. de Forcrand, Simulating QCD at finite density, PoS LAT2009, 010 (2009), arXiv:1005.0539 [hep-lat] .
  * Adamczyk  _et al._ [2014] L. Adamczyk _et al._ (STAR), Beam-Energy Dependence of the Directed Flow of Protons, Antiprotons, and Pions in Au+Au Collisions, Phys. Rev. Lett. 112, 162301 (2014), arXiv:1401.3043 [nucl-ex] .
  * Adamczyk  _et al._ [2018] L. Adamczyk _et al._ (STAR), Collision Energy Dependence of Moments of Net-Kaon Multiplicity Distributions at RHIC, Phys. Lett. B 785, 551 (2018), arXiv:1709.00773 [nucl-ex] .
  * Adamczyk  _et al._ [2017] L. Adamczyk _et al._ (STAR), Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program, Phys. Rev. C 96, 044904 (2017), arXiv:1701.07065 [nucl-ex] .
  * Acharya  _et al._ [2022] A. Acharya _et al._ (NA61/SHINE), $K^{*}(892)^{0}$ meson production in inelastic p+p interactions at $40$ and $80$ GeV/$c$ beam momenta measured by NA61/SHINE at the CERN SPS, Eur. Phys. J. C 82, 322 (2022), arXiv:2112.09506 [nucl-ex] .
  * Grebieszkow [2017] K. Grebieszkow (NA61/SHINE), New results on fluctuations and correlations from the NA61/SHINE experiment at the CERN SPS, PoS EPS-HEP2017, 167 (2017), arXiv:1709.10397 [nucl-ex] .
  * Blaschke  _et al._ [2016] D. Blaschke, J. Aichelin, E. Bratkovskaya, V. Friese, M. Gazdzicki, J. Randrup, O. Rogachevsky, O. Teryaev, and V. Toneev, Topical issue on Exploring Strongly Interacting Matter at High Densities - NICA White Paper, Eur. Phys. J. A 52, 267 (2016).
  * Ablyazimov  _et al._ [2017] T. Ablyazimov _et al._ (CBM), Challenges in QCD matter physics –The scientific programme of the Compressed Baryonic Matter experiment at FAIR, Eur. Phys. J. A 53, 60 (2017), arXiv:1607.01487 [nucl-ex] .
  * Sako  _et al._ [2014] H. Sako _et al._ , Towards the heavy-ion program at J-PARC, Nucl. Phys. A 931, 1158 (2014).
  * Nikolov  _et al._ [1996] E. N. Nikolov, W. Broniowski, C. V. Christov, G. Ripka, and K. Goeke, Meson loops in the Nambu-Jona-Lasinio model, Nucl. Phys. A 608, 411 (1996), arXiv:hep-ph/9602274 .
  * Nemoto  _et al._ [2000] Y. Nemoto, K. Naito, and M. Oka, Effective potential of O(N) linear sigma model at finite temperature, Eur. Phys. J. A 9, 245 (2000), arXiv:hep-ph/9911431 .
  * Oertel  _et al._ [2001] M. Oertel, M. Buballa, and J. Wambach, Meson loop effects in the NJL model at zero and nonzero temperature, Phys. Atom. Nucl. 64, 698 (2001), arXiv:hep-ph/0008131 .
  * Baacke and Michalski [2003] J. Baacke and S. Michalski, O(N) linear sigma model beyond the Hartree approximation at finite temperature, in  _6th Workshop on Quantum Field Theory under the Influence of External Conditions (QFEXT03)_ (2003) pp. 282–287, arXiv:hep-ph/0312031 .
  * Andersen and Brauner [2008] J. O. Andersen and T. Brauner, Linear sigma model at finite density in the 1/N expansion to next-to-leading order, Phys. Rev. D 78, 014030 (2008), arXiv:0804.4604 [hep-ph] .
  * Müller  _et al._ [2010] D. Müller, M. Buballa, and J. Wambach, The Quark Propagator in the NJL Model in a self-consistent 1/Nc Expansion, Phys. Rev. D 81, 094022 (2010), arXiv:1002.4252 [hep-ph] .
  * Yamazaki and Matsui [2013] K. Yamazaki and T. Matsui, Quark-Hadron Phase Transition in the PNJL model for interacting quarks, Nucl. Phys. A 913, 19 (2013), arXiv:1212.6165 [hep-ph] .
  * Zacchi and Schaffner-Bielich [2018] A. Zacchi and J. Schaffner-Bielich, Effects of Renormalizing the chiral SU(2) Quark-Meson-Model, Phys. Rev. D 97, 074011 (2018), arXiv:1712.01629 [hep-ph] .
  * Câmara Pereira and Costa [2020] R. Câmara Pereira and P. Costa, One-meson-loop NJL model: Effect of collective and noncollective excitations on the quark condensate at finite temperature, Phys. Rev. D 101, 054025 (2020), arXiv:2003.08430 [hep-ph] .
  * Fukushima and Pawlowski [2012] K. Fukushima and J. M. Pawlowski, Magnetic catalysis in hot and dense quark matter and quantum fluctuations, Phys. Rev. D 86, 076013 (2012), arXiv:1203.4330 [hep-ph] .
  * Braun [2012] J. Braun, Fermion Interactions and Universal Behavior in Strongly Interacting Theories, J. Phys. G 39, 033001 (2012), arXiv:1108.4449 [hep-ph] .
  * Aoki  _et al._ [2014] K.-I. Aoki, S.-I. Kumamoto, and D. Sato, Weak solution of the non-perturbative renormalization group equation to describe dynamical chiral symmetry breaking, PTEP 2014, 043B05 (2014), arXiv:1403.0174 [hep-th] .
  * Aoki and Yamada [2015] K.-I. Aoki and M. Yamada, The RG flow of Nambu–Jona-Lasinio model at finite temperature and density, Int. J. Mod. Phys. A 30, 1550180 (2015), arXiv:1504.00749 [hep-ph] .
  * Schaefer and Wambach [2005a] B.-J. Schaefer and J. Wambach, The Phase diagram of the quark meson model, Nucl. Phys. A 757, 479 (2005a), arXiv:nucl-th/0403039 .
  * Herbst  _et al._ [2013a] T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phase structure and thermodynamics of QCD, Phys. Rev. D 88, 014007 (2013a), arXiv:1302.1426 [hep-ph] .
  * Fu and Pawlowski [2015] W.-j. Fu and J. M. Pawlowski, Relevance of matter and glue dynamics for baryon number fluctuations, Phys. Rev. D 92, 116006 (2015), arXiv:1508.06504 [hep-ph] .
  * Herbst  _et al._ [2014] T. K. Herbst, M. Mitter, J. M. Pawlowski, B.-J. Schaefer, and R. Stiele, Thermodynamics of QCD at vanishing density, Phys. Lett. B 731, 248 (2014), arXiv:1308.3621 [hep-ph] .
  * Jung  _et al._ [2017a] C. Jung, F. Rennecke, R.-A. Tripolt, L. von Smekal, and J. Wambach, In-Medium Spectral Functions of Vector- and Axial-Vector Mesons from the Functional Renormalization Group, Phys. Rev. D 95, 036020 (2017a), arXiv:1610.08754 [hep-ph] .
  * Andersen  _et al._ [2014] J. O. Andersen, W. R. Naylor, and A. Tranberg, Chiral and deconfinement transitions in a magnetic background using the functional renormalization group with the Polyakov loop, JHEP 04, 187, arXiv:1311.2093 [hep-ph] .
  * Gies [2012] H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852, 287 (2012), arXiv:hep-ph/0611146 .
  * Pawlowski [2007] J. M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322, 2831 (2007), arXiv:hep-th/0512261 .
  * Delamotte [2012] B. Delamotte, An Introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852, 49 (2012), arXiv:cond-mat/0702365 .
  * Câmara Pereira  _et al._ [2020a] R. Câmara Pereira, R. Stiele, and P. Costa, Functional renormalization group study of the critical region of the quark-meson model with vector interactions, Eur. Phys. J. C 80, 712 (2020a), arXiv:2003.12829 [hep-ph] .
  * Otto  _et al._ [2022] K. Otto, C. Busch, and B.-J. Schaefer, Regulator scheme dependence of the chiral phase transition at high densities, Phys. Rev. D 106, 094018 (2022), arXiv:2206.13067 [hep-ph] .
  * Schaefer and Wambach [2008] B.-J. Schaefer and J. Wambach, Renormalization group approach towards the QCD phase diagram, Phys. Part. Nucl. 39, 1025 (2008), arXiv:hep-ph/0611191 .
  * Schaefer and Wambach [2005b] B.-J. Schaefer and J. Wambach, The Phase diagram of the quark meson model, Nucl. Phys. A 757, 479 (2005b), arXiv:nucl-th/0403039 .
  * Herbst  _et al._ [2011] T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, The phase structure of the Polyakov–quark–meson model beyond mean field, Phys. Lett. B 696, 58 (2011), arXiv:1008.0081 [hep-ph] .
  * Rennecke [2015] F. Rennecke, Vacuum structure of vector mesons in QCD, Phys. Rev. D 92, 076012 (2015), arXiv:1504.03585 [hep-ph] .
  * Eser  _et al._ [2015] J. Eser, M. Grahl, and D. H. Rischke, Functional Renormalization Group Study of the Chiral Phase Transition Including Vector and Axial-vector Mesons, Phys. Rev. D 92, 096008 (2015), arXiv:1508.06928 [hep-ph] .
  * Jung  _et al._ [2017b] C. Jung, F. Rennecke, R.-A. Tripolt, L. von Smekal, and J. Wambach, In-Medium Spectral Functions of Vector- and Axial-Vector Mesons from the Functional Renormalization Group, Phys. Rev. D 95, 036020 (2017b), arXiv:1610.08754 [hep-ph] .
  * Drews and Weise [2014] M. Drews and W. Weise, Functional renormalization group approach to neutron matter, Phys. Lett. B 738, 187 (2014), arXiv:1404.0882 [nucl-th] .
  * Drews and Weise [2015] M. Drews and W. Weise, From asymmetric nuclear matter to neutron stars: a functional renormalization group study, Phys. Rev. C 91, 035802 (2015), arXiv:1412.7655 [nucl-th] .
  * Fukushima [2008] K. Fukushima, Phase diagrams in the three-flavor Nambu-Jona-Lasinio model with the Polyakov loop, Phys. Rev. D 77, 114028 (2008), [Erratum: Phys.Rev.D 78, 039902 (2008)], arXiv:0803.3318 [hep-ph] .
  * Lourenco  _et al._ [2012] O. Lourenco, M. Dutra, T. Frederico, A. Delfino, and M. Malheiro, Vector interaction strength in Polyakov-Nambu-Jona-Lasinio models from hadron-quark phase diagrams, Phys. Rev. D 85, 097504 (2012), arXiv:1204.6357 [nucl-th] .
  * Bratovic  _et al._ [2013] N. M. Bratovic, T. Hatsuda, and W. Weise, Role of Vector Interaction and Axial Anomaly in the PNJL Modeling of the QCD Phase Diagram, Phys. Lett. B 719, 131 (2013), arXiv:1204.3788 [hep-ph] .
  * Lu  _et al._ [2015] Y. Lu, Y.-L. Du, Z.-F. Cui, and H.-S. Zong, Critical behaviors near the (tri-)critical end point of QCD within the NJL model, Eur. Phys. J. C 75, 495 (2015), arXiv:1508.00651 [hep-ph] .
  * Adhikari  _et al._ [2017] P. Adhikari, J. O. Andersen, and P. Kneschke, Inhomogeneous chiral condensate in the quark-meson model, Phys. Rev. D 96, 016013 (2017), [Erratum: Phys.Rev.D 98, 099902 (2018)], arXiv:1702.01324 [hep-ph] .
  * Herbst  _et al._ [2013b] T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phase structure and thermodynamics of QCD, Phys. Rev. D 88, 014007 (2013b), arXiv:1302.1426 [hep-ph] .
  * Floerchinger and Wetterich [2012] S. Floerchinger and C. Wetterich, Chemical freeze-out in heavy ion collisions at large baryon densities, Nucl. Phys. A 890-891, 11 (2012), arXiv:1202.1671 [nucl-th] .
  * Skokov  _et al._ [2010] V. Skokov, B. Friman, E. Nakano, K. Redlich, and B. J. Schaefer, Vacuum fluctuations and the thermodynamics of chiral models, Phys. Rev. D 82, 034029 (2010), arXiv:1005.3166 [hep-ph] .
  * Scavenius  _et al._ [2001] O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H. Rischke, Chiral phase transition within effective models with constituent quarks, Phys. Rev. C 64, 045202 (2001), arXiv:nucl-th/0007030 .
  * Zhang  _et al._ [2017] H. Zhang, D. Hou, T. Kojo, and B. Qin, Functional renormalization group study of the quark-meson model with $\omega$ meson, Phys. Rev. D 96, 114029 (2017), arXiv:1709.05654 [hep-ph] .
  * Berges  _et al._ [2002] J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363, 223 (2002), arXiv:hep-ph/0005122 .
  * Tripolt  _et al._ [2018] R.-A. Tripolt, B.-J. Schaefer, L. von Smekal, and J. Wambach, Low-temperature behavior of the quark-meson model, Phys. Rev. D 97, 034022 (2018), arXiv:1709.05991 [hep-ph] .
  * Strodthoff and von Smekal [2014a] N. Strodthoff and L. von Smekal, Polyakov-Quark-Meson-Diquark Model for two-color QCD, Phys. Lett. B 731, 350 (2014a), arXiv:1306.2897 [hep-ph] .
  * Litim [2001] D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001), arXiv:hep-th/0103195 .
  * Drews  _et al._ [2013] M. Drews, T. Hell, B. Klein, and W. Weise, Thermodynamic phases and mesonic fluctuations in a chiral nucleon-meson model, Phys. Rev. D 88, 096011 (2013), arXiv:1308.5596 [hep-ph] .
  * Tripolt  _et al._ [2014] R.-A. Tripolt, N. Strodthoff, L. von Smekal, and J. Wambach, Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group, Phys. Rev. D 89, 034010 (2014), arXiv:1311.0630 [hep-ph] .
  * Câmara Pereira  _et al._ [2020b] R. Câmara Pereira, R. Stiele, and P. Costa, Functional renormalization group study of the critical region of the quark-meson model with vector interactions, Eur. Phys. J. C 80, 712 (2020b), arXiv:2003.12829 [hep-ph] .
  * Strodthoff and von Smekal [2014b] N. Strodthoff and L. von Smekal, Polyakov-Quark-Meson-Diquark Model for two-color QCD, Phys. Lett. B 731, 350 (2014b), arXiv:1306.2897 [hep-ph] .


# Method for in-solution, high-throughput $T_{1}$ relaxometry using
fluorescent nanodiamonds

Erin. S. Grant  School of Physics, The University of Melbourne  Mina. Barzegar
Amiri Olia  Bio21 Molecular Science and Biotechnology Institute  Ella. P.
Walsh  School of Chemistry, The University of Melbourne  Gawain McColl  Florey
Institute of Neuroscience and Mental Health  Liam T. Hall  School of
Chemistry, The University of Melbourne  David A. Simpson  School of Physics,
The University of Melbourne<EMAIL_ADDRESS>
###### Abstract

Fluorescent nanodiamonds (FNDs) have been exploited as sensitive quantum
probes for nanoscale chemical and biological sensing applications, with the
majority of demonstrations to date relying on the detection of single FNDs.
This places significant limits on the measurement time, throughput and
statistical significance of a measured result as there is usually marked
inhomogeneity within FND samples. Here we have developed a measurement
platform that can report the $T_{1}$ spin relaxation time from a large
ensemble of FNDs in solution. We first describe a refined sensing protocol for
this modality and then use it to identify the optimal FND size for the
detection of paramagnetic targets. Our approach is simple to set up, robust
and can be used for rapid material characterisation or a variety of in-situ
quantum sensing applications.

††preprint: APS/123-QED Figure 1: Schematic of the in-solution measurement
apparatus. The set-up consists of a 532 nm laser which is modulated using an
acousto-optical modulator. NV fluorescence is extracted through an optical
fibre and measured on an single photon avalanche photodiode. Top left inset: A
photo of the cuvette, NV fluorescence is visible as a thin red line through
the optical filter at the front of the image. Top right inset: Quantum sensing
is performed using dispersed FNDs containing the nitrogen-vacancy defect which
consists of a substitutional nitrogen atom and an adjacent vacancy.

Fluorescent nanodiamonds (FNDs) containing the nitrogen-vacancy (NV) defect
have been developed over the last decade as a novel nanoscale sensor for a
wide range of targets including pH [1, 1], temperature [2, 3, 4, 5], free
radicals [6, 7, 8, 9], and paramagnetic targets such as manganese, gadolinium
and iron [10, 11, 12, 13, 14]. The majority of characterisation and chemical
sensing studies using FNDs address individual particles as they offer high
sensitivity and spatial resolution [15, 13, 16, 6, 17, 18]. However, these
approaches are often limited in their statistical power due to the
inhomogeneous nature of FND size, shape, brightness and coherence [15].
Therefore, alternate approaches which measure the average behaviour of an
ensemble of FNDs offer robustness over individually sampling a small subset by
reducing measurement bias. To date, chemical sensing using FND ensembles has
been performed with particles deposited onto a 2D substrate. However, this
approach can produce inhomogeneous coverage of the target across the FND
sample, introducing large variabilities that reduce the quantitative utility
of these measurements [19].

Here, we have established an all-optical, high-throughput protocol for
reporting on the quantum properties of FNDs in-solution. This cuvette-based
approach results in a more uniform interaction between FNDs and a target,
facilitating in situ measurement and characterisation using $T_{1}$ spin
relaxometry. While this study focuses on the development of this novel
modality for measuring the $T_{1}$ spin relaxation time of dispersed FNDs,
this apparatus can also be modified to provide quantification of the average
brightness and spectra of samples, as previously demonstrated [20].

This work contains two sections. First a protocol for $T_{1}$ spin relaxometry
is established, which can account for fluctuations in the fluorescence
intensity of the NV centres that are involved in a given measurement. We then
apply this protocol to rapidly explore the best sized FNDs for paramagnetic
sensing of chelated Gd3+ ions in the form of gadobutrol - a chemical contrast
agent used in magnetic resonance imaging (MRI). This proof of principle
demonstration shows the simplicity and high-throughput of the technique as
well as highlighting the ease of performing a robust control $T_{1}$
calibration measurement.

##  I  A Protocol for In-Solution $T_{1}$ Spin Relaxometry

Measurements of dispersed FNDs were performed on a custom built set-up shown
in Figure 1. The apparatus consisted of a cuvette holder (Thorlabs), 532 nm
(GEM, Laser Quantum) excitation laser, acousto-optical modulator (AOM) (AA
Electronics), and a single photon avalanche photodiode (SPAD) (Excelitas
Technologies). Laser light of 160 mW was focused to the centre of the cuvette
using a 150 mm plano convex lens, forming a focused beam with a waist of
$\sim$100 µm in diameter. NV fluorescence was extracted perpendicularly to the
excitation beam through a 731$\pm$137 nm band pass filter. The NV fluorescence
was coupled into an optical fibre (400 µm core diameter) using an achromatic
collimator with an NA of 0.54 (F950FC-A, Thorlabs) followed by detection with
the SPAD. Micro cuvettes (BRAND 759200, Sigma Aldrich) were used to hold a 170
µL suspension of FNDs. The left inset of Figure 1 shows a photo of the cuvette
and holder. NV fluorescence is visible through the filter at the front of the
holder as a red streak.

Figure 2: In-solution $T_{1}$ relaxometry with FNDs. A) Schematic of the
energy levels of the NV defect in diamond. These consist of triplet ground and
excited states and two intermediate singlet states. B) The NV spin population
is determined by taking the ratio of the integrated intensity in the first 2
us of the excitation pulse by the integrated intensity in the final 10 us of
the pulse. The signals are normalised to account for the differences in the
integration window. C) A typical T1 relaxation curve measured using the pulse
sequence in B). D) Images of 140 nm FND suspensions in ultrapure water at
concentrations ranging from 1 to 300 µg/ml. E) and F) The effect of the
concentration of the FNDs on the measured $T_{1}$ spin contrast and $T_{1}$
relaxation time. The FND concentration has a negligible effect for
concentrations below 50 µg/mL.

$T_{1}$ spin relaxation measurements are facilitated by the intrinsic NV
defects in the FNDs. The NV defect consists of a substitutional nitrogen atom
and an adjacent vacancy, see the right inset of Figure 1. It is a spin-1
system, with an electronic structure shown in Figure 2A. The $m_{s}=0$ and
$m_{s}=\pm 1$ ground spin state levels are separated by the zero-field
splitting, $D=2.87$ GHz. When excited from the $m_{s}=0$ ground state, the NV
defect emits 30% more fluorescence than the corresponding transitions from the
$m_{s}=\pm 1$ states which provides a convenient method to optically report
the ground spin state populations [21, 22]. This can be used to perform
$T_{1}$ spin relaxometry which measures the degree of longitudinal relaxation
of the NV spin polarisation, where $T_{1}$ is defined as the time at which the
emission intensity reaches $1/e$ of the maximum value. The characteristic time
may also be expressed as a rate, $\Gamma=1/T_{1}$, which will be used here.
Fluctuating magnetic field sources act to increase the $T_{1}$ spin relaxation
rate. These magnetic field sources can come from electronic spins on or within
the FND [16, 23], or from external targets such as paramagnetic molecules [24]
or magnetic materials [25].

To measure the $T_{1}$ spin relaxation process, a simple protocol was
employed, as shown in Figure 2B. First, the polarising laser was used to
initialise the ensemble of NV defects within the beam path into the bright,
$m_{s}=0$, ground state. Then, after an evolution time, $\tau$, a second laser
pulse was used to read out the average spin-state of the NV ensemble, as well
as repolarising for the next measurement. Each laser pulse in the protocol was
applied for the same length of time, but by varying $\tau$, a $T_{1}$ spin
relaxation curve of the form shown in Figure 2C can be generated.

For an individual NV defect, the decay profile of the measured emission
intensity will be determined by its intrinsic relaxation rate, $\Gamma_{\rm
intrinsic}$, as well as that produced by the presence of $N_{T}$ target spins,
$\Gamma_{\rm target}$. Giving emission intensity:

$I(\tau)=\mathcal{C}\exp[-\Gamma_{\rm
intrinsic}\tau-\sum_{j}^{N_{T}}\Gamma^{j}_{\rm target}\tau]+c\\\ $ (1)

This is complicated further when considering an ensemble of FNDs, containing
$N_{NV}$ NV defects in total:

$I(\tau)=\sum_{i}^{N_{NV}}\mathcal{C}^{i}\exp[-\Gamma^{i}_{\rm
intrinsic}\tau-(\sum_{j}^{N_{T}}\Gamma^{j}_{\rm target}\tau)^{i}]+c\\\ $ (2)

In this situation the emission intensity, $I(\tau)$, will deviate from a
single exponential lineshape and the short time behaviour can be well
approximated by fitting with a stretched exponential:

$\approx\mathcal{C}\exp[-(\Gamma_{\rm intrinsic}\tau)^{p}-(\Gamma_{\rm
target}\tau)^{q}]+c$ (3)

where $\mathcal{C}$ is the spin contrast, $p$ and $q$ are stretch factors that
are influenced by the variance in $\Gamma_{\rm intrinsic}$ and $\Gamma_{\rm
target}$ across the FND ensemble, and $c$ is a constant offset. The
distributions of both $\Gamma_{\rm intrinsic}$ and $\Gamma_{\rm target}$
depend on the geometric arrangement of the noise sources which contribute to
each, i.e target spins in the case of $\Gamma_{\rm target}$ or intrinsic noise
sources for $\Gamma_{\rm intrinsic}$. It is reasonable to assume that
$p\approx q$ because $\Gamma_{\rm intrinsic}$ is dominated by surface spin
noise in FNDs [12, 26, 23], while the influence of the target material will be
skewed towards spins in close proximity to the surface due to the $r^{-6}$
dependence [12]. The fitted decay curve then becomes:

$I(\tau)=\mathcal{C}\exp[-(\Gamma_{\rm measured}\tau)^{p}]+c,$ (4)

where $\Gamma_{\rm
measured}=\Gamma_{\text{intrinsic}}+\Gamma_{\text{target}}$, allowing the
influence of the target to be isolated via:

$\Gamma_{\text{target}}=\Gamma_{\text{measured}}-\Gamma_{\text{intrinsic}}.$
(5)

To control for laser fluctuations, other environmental noise, and potential NV
charge state changes across the measurement time, each data point in the
$T_{1}$ curve is found by taking the ratio of the integrated emission
intensity in a 2 µs window at the beginning of each laser pulse (which informs
on the average spin state population of the ensemble) with the integrated
intensity in a 10 µs window at the end of the pulse, which represents the
steady state population of the repolarised state. The respective pulses are
shown by the dotted boxes in the SPAD trace of Figure 2B. Each integrated
intensity is normalised to account for the difference in width.

Figure 3: A) The intrinsic $T_{1}$ relaxation rate as a function of FND size.
The error associated with the 30 nm sample has been truncated for clarity, it
had $\Gamma_{\rm intrinsic}=40,000\pm 70,000$. B) The target $T_{1}$
relaxation rate, $\Gamma_{\rm target}$, due to Gd3+ for the 50, 100 and 140 nm
samples. Inset: example curves for the control (red) and target (blue)
measurements, using the 140 nm FNDs.

##  II  Proof of Principle Measurements

To establish the feasibility of this new modality, initial $T_{1}$ relaxation
measurements were performed on 140 nm FNDs purchased from Adamas
Nanotechnologies and dispersed in ultrapure 18.2 Mresistant H2O (Milli-Q,
Millipore). The first consideration for this protocol was whether the low
optical power density, approximately three orders of magnitude less than
traditional confocal microscopy, is sufficient for polarising the NV centres
in the beam path. Secondly, it was important to understand whether charge
state changes, which have been shown to cause artefacts to $T_{1}$ spin
relaxation measurements [27], are still important in this low power regime. To
test these considerations we first measured the emission intensity over 100 µs
of continuous laser illumination to determine the polarisation timescale at
this power. As shown in SI1, a steady-state is achieved after approximately
70-80 µs of constant illumination. The optimal read out pulse length,
$t_{RO}$, was then determined by measuring the steady-state intensity of each
data point during a $T_{1}$ spin relaxation curve with different values of
$t_{\rm RO}$. For the optimal readout pulse length of 80 µs, the steady state
population from each measurement point was within 2-3% over the measurement
period. To understand whether this small variation in steady-state emission
intensity was caused by spin-independent processes such as NV charge state
changes, we conducted a separate $T_{1}$ spin relaxation measurement under the
influence of a strong external magnetic field (on the order of 0.1 T) [28].
The introduction of a DC magnetic field induces spin-mixing, removing the
ability to spin polarise the NV ensemble. Therefore, any observed variation in
emission intensity, both in the reference window, as well as the signal
window, should be spin-independent. As shown in SI2 the 2-3% changes to the
steady-state intensity are accounted for by spin-independent processes that
appear to be occurring during the dark evolution times of the measurement,
rather than within a read out pulse. Therefore, dividing the normalised
intensity in the signal window by that of the reference window adequately
functions as a common mode rejection method to account for these small charge
state population changes [29].

After establishing a $T_{1}$ measurement protocol, we studied the effect of
the FND concentration on the measured $T_{1}$ relaxation curve. Our findings
show that as the concentration of the FND solution is increased, the NV spin
contrast and signal to noise (SNR) of the measurement decreases. This is
attributed to photon scattering at concentrations greater than 25 µg/ml as
illustrated in Figure 2E. The scattering effects also mask the $T_{1}$
relaxation effects, as seen in Figure 2F, leading to significant measurement
errors. Therefore, we work with FND concentrations less than 25 µg/ml for the
remainder of this work. We note this concentration limit is likely to be a
lower bound as scattering increases with particle volume [30]. It should also
be noted that efficacy at low FND concentrations is important for chemical
sensing applications where the concentration of target molecules is low. For
example, an over abundance of FNDs for a given target concentration will
reduce the number of NVs in the ensemble that experience an additional
relaxation process, $\Gamma_{\rm target}$. Therfore the measured relaxation
rate, $\Gamma_{\rm measured}$, will be skewed towards intrinsic values,
limiting SNR.

##  III Paramagnetic Sensing in Solution

In-solution measurement of $T_{1}$ relaxatometry has the potential to offer
high-throughput measurements of large numbers of FNDs, making it useful for
both characterisation and quantum sensing studies. Choosing a particle for
quantum sensing purposes is often about balancing dynamic range, sensitivity,
and the limit of detection (LOD). Ideally, a sensor will respond consistently
to a wide range of concentrations with a low LOD, but it must also be able to
distinguish between two samples with very similar concentrations. In the
context of $T_{1}$ spin relaxometry, optimal performance within this metric
will be determined by the intrinsic $T_{1}$ spin relaxation rate of the FNDs,
their optical noise and the depth distribution of the NV defects within each
FND. If all other characteristics are equal, FNDs with a lower intrinsic
$T_{1}$ spin relaxation rate, $\Gamma_{\rm intrinsic}$, offer a greater
dynamic range, as the operating band width will span
$\Gamma_{\text{max}}-\Gamma_{\text{intrinsic}}$, where $\Gamma_{\rm max}$ is
the maximum possible measured relaxation rate, which is determined by the spin
polarisation time (on the order of microseconds). Meanwhile, the measurement
sensitivity depends on the NV spin contrast, photon count rate, and the total
acquisition time of the measurement. For a given evolution time, $\tau$,
assuming shot-noise limited data, the signal-to-noise ratio of a $T_{1}$ spin
relaxation measurement is given by:

$\text{SNR}(\tau)\approx\sqrt{\frac{\mathcal{R}t_{\rm RO}T_{\rm
tot}}{\tau}}\frac{3\mathcal{C}}{4}e^{-\Gamma_{\rm
intrinsic}\tau}(1-e^{-\Gamma_{\rm measured}\tau}),$ (6)

where $\mathcal{R}$ is the photon count rate, $t_{\rm RO}$ is the laser read-
out length, $T_{\rm tot}$ is the total acquisition time, and $\mathcal{C}$ is
the $T_{1}$ spin contrast [31]. The protocol presented here is approximately a
factor of 2 away from the shot noise limit when comparing the measured photon
statistics with the expected shot noise limited value (see SI3). The
additional optical noise may arise from photon scattering and diffusion of the
FNDs themselves as well as residual laser intensity fluctuations not accounted
for by the common mode rejection method. Nevertheless, we are able to
determine the $T_{1}$ spin relaxation rates with measurement errors less than
10% for a measurement time of $\sim$50 minutes. Finally, FNDs with the
shallowest depth distributions will offer the lowest LOD, as the magnetic
interaction between a single NV defect and a target molecule scales as
$1/r^{6}$.

To explore the interplay between FND size, LOD, sensitivity, and dynamic
range, we measured the intrinsic $T_{1}$ spin relaxation rates of four
different FND sizes as well as their sensitivity to Gd3+ ions. FNDs with a
nominal diameter of 30, 50, 100 and 140 nm were sourced from Adamas
Nanotechnologies. Particles were suspended in ultrapure water and sonicated
for 30 seconds to ensure optimal dispersion. Initially, the intrinsic $T_{1}$
relaxation rate of each FND sample was measured using a 170 µL volume.
Following this initial measurement, the sensitivity was determined by adding a
30 µL solution of 26.6 mM of gadobutrol monohydrate (Sigma Aldrich). The Gd3+
ion bound to this ligand has an electron spin of $S=7/2$, providing a
fluctuating magnetic signal that increases the $T_{1}$ spin relaxation rate of
the NV defect as shown by the example $T_{1}$ relaxation curves in the inset
of Figure 3B. To ensure a similar level of photon shot noise between
measurements, data was acquired until $\sim$1.2 million photons were collected
in the reference window of the first data point. The brightness of each FND
sample varied, with the 30 and 50 nm samples producing ¡1 million counts per
second. The 100 and 140 nm samples however, produced an order of magnitude
more NV fluorescence. Therefore, to ensure the SPAD detector operated within
its linear range, a neutral density filter with an optical density of 1.6 was
added to reduce the count rate to ¡2 million counts/sec.

Table 1: Properties of each FND size and resulting SNR. The contrast value is taken from the intrinsic relaxation curve. Size | Brightness | Contrast | SNR
---|---|---|---
(nm) | (mil photons/sec) | (%) |
30 | 0.7 | 2 $\pm$ 8 | N/A
50 | 0.9 | 4 $\pm$ 1 | 16 $\pm$ 9
100 | 11 | 9 $\pm$ 2 | 43 $\pm$ 9
140 | 13 | 10 $\pm$ 2 | 37 $\pm$ 10

Figure 3A shows the intrinsic $T_{1}$ spin relaxation rate, $\Gamma_{\rm
intrinsic}$, of each FND sample in ultrapure water, while Figure 3B
illustrates the effective $T_{1}$ spin relaxation rate, $\Gamma_{\rm target}$,
resulting from the addition of Gd3+ ions to the solution. The error bars
represent one standard deviation of the error on the $T_{1}$ fitted parameter.

There is a relatively strong negative correlation between $\Gamma_{\rm
intrinsic}$ and the size of the particle; as the size of the particle
increases the intrinsic $T_{1}$ spin relaxation rate decreases. This may be
related to the higher surface-area-to-volume ratio of the smaller particles,
which increases the deleterious effects of unwanted surface spin noise. The 30
nm sample was especially affected, with an intrinsic $T_{1}$ spin relaxation
rate significantly higher than any of the other samples and a prohibitively
large fit error ($\Gamma_{\rm intrinsic}=40,000\pm 70,000$). For this reason
gadobutrol was not added to the 30 nm FNDs.

Overall, the 100 nm FNDs outperformed the larger 140 nm particles, suggesting
that the 100 nm sample has the highest dynamic range and greatest sensitivity.
This is confirmed by the values of $\Gamma_{\rm target}$ recorded for each FND
size and the corresponding SNR (calculated at $\tau=T_{1}/2$ for each
particle), shown in Table 1. In contrast, $\Gamma_{\rm target}$ is much larger
for the 50 nm FNDs compared to the 100 or 140 nm samples, suggesting they can
offer a lower LOD. However, these particles also have a much higher
measurement error, likely because the addition of gadobutrol increased
$\Gamma_{\rm measured}$ to a value close to $\Gamma_{\rm max}$. At this level
of perturbation the system cannot be as effectively spin polarised and hence
there is a reduction in spin contrast (see SI6) which reduces the SNR.
Additionally, the intrinsic relaxation rate of these particles is high, which
means they have a much smaller dynamic range compared to the 100 and 140 nm
FNDs. Importantly however, if $\Gamma_{\rm intrinstic}$ can be reduced for the
smaller particles, they have the potential to sense very low concentrations of
target material with a wider dynamic range due to their more favourable NV
depth distribution. Nevertheless, in their current form, we haave identified
the 100 nm FNDs to be the most attractive particles for paramagnetic quantum
sensing.

This simple in-solution set-up has demonstrated the feasibility of measuring
the average $T_{1}$ spin relaxation rate of dispersed FNDs. We note there is
room to optimise the current configuration further. For example, the use of a
SPAD for detecting the NV fluorescence offers highly time-resolved data, which
was useful for validating the $T_{1}$ spin relaxation protocol, however it
puts constraints on the amount of fluorescence that can be detected.
Consequently, the acquisition speed is currently limited to measurements on
the order of half an hour to an hour. Throughput could therefore be improved
by utilising an alternative detector, such as a photodiode. For this to be
feasible the collection efficiency must be further improved. Pre-detector
elements to achieve this include aspheric lenses [32], integrating spheres
[33], or parabolic lenses, which may provide up to 65% collection efficiency
[34], substantially increasing acquisition speed.

In this work we have demonstrated that $T_{1}$ spin relaxometry can be
performed on dispersed FNDs in an all-optical cuvette-based system, reporting
from millions of particles simultaneously. We have shown how this system can
be used for rapid assessment of the suitability of current commercially
available FND materials for paramagnetic quantum sensing applications. From
our measurements 100 nm FNDs currently offer the largest dynamic range and
highest sensitivity for chemical sensing, however smaller particles have the
potential to reach lower limits of detection. This measurement platform will
also be useful for understanding the effects of surface functionalisation on
$T_{1}$ spin relaxation times [35, 36] \- providing a method for assessing the
impacts of a particular termination with high statistical power and
throughput, enabling more streamlined advancement to FND materials. Finally,
the in-solution nature of the measurement allows for in-situ chemical sensing
which would take orders of magnitude longer if pursued using standard confocal
or widefield microscopes. With straight forward improvements to the collection
efficiency, or by implementing a single point $T_{1}$ protocol, this system
can be improved further, with the potential to offer sub second acquisition
speeds. This in situ technique opens the door to new sensing paradigms for
these nanoscale fluorescent quantum sensors providing a path to well
controlled, time-resolved measurements of dynamic chemical processes.

##  IV Supplementary Material

See Supplementary Material for further information on how the all-optical
protocol was optimised.

##  V Acknowledgements

The authors would like to acknowledge Dr P. Reineck for supplying the
nanodiamond material used in this study. E.G acknowledges the Graeme Clark
Institute for Biomedical Engineering at the University of Melbourne for
funding support.

## Data Availability Statement

The data that support the findings of this study are openly available in
[repository name] at http://doi.org/[doi], reference number [reference
number].

## References

  * Fujisaku  _et al._ [2019] T. Fujisaku, R. Tanabe, S. Onoda, R. Kubota, T. F. Segawa, F. T. So, T. Ohshima, I. Hamachi, M. Shirakawa, and R. Igarashi, PH Nanosensor Using Electronic Spins in Diamond, ACS Nano 13, 11726 (2019).
  * Yukawa  _et al._ [2020] H. Yukawa, M. Fujiwara, K. Kobayashi, Y. Kumon, K. Miyaji, Y. Nishimura, K. Oshimi, Y. Umehara, Y. Teki, T. Iwasaki, M. Hatano, H. Hashimoto, and Y. Baba, View Article Online View Journal — View Issue A quantum thermometric sensing and analysis system using fluorescent nanodiamonds for the evaluation of living stem cell functions according to intracellular temperature, Nanoscale Advances 2, 1859 (2020).
  * Nishimura  _et al._ [2021] Y. Nishimura, K. Oshimi, Y. Umehara, Y. Kumon, K. Miyaji, H. Yukawa, Y. Shikano, T. Matsubara, M. Fukiwara, Y. Baba, and Y. Teki, Wide‑field fluorescent nanodiamond spin measurements toward real‑time large‑area intracellular thermometry, Scientific Reports 11, 1 (2021).
  * Simpson  _et al._ [2017] D. A. Simpson, E. Morrisroe, J. M. McCoey, A. H. Lombard, D. C. Mendis, F. Treussart, L. T. Hall, S. Petrou, and L. C. Hollenberg, Non-Neurotoxic Nanodiamond Probes for Intraneuronal Temperature Mapping, ACS Nano 11, 12077 (2017).
  * Tsai  _et al._ [2017] P. C. Tsai, C. P. Epperla, J. S. Huang, O. Y. Chen, C. C. Wu, and H. C. Chang, Measuring Nanoscale Thermostability of Cell Membranes with Single Gold–Diamond Nanohybrids, Angewandte Chemie - International Edition 56, 3025 (2017).
  * Nie  _et al._ [2021] L. Nie, A. C. Nusantara, V. G. Damle, R. Sharmin, E. P. P. Evans, S. R. Hemelaar, K. J. van der Laan, R. Li, F. P. Perona Martinez, T. Vedelaar, M. Chipaux, and R. Schirhagl, Quantum monitoring of cellular metabolic activities in single mitochondria, Science Advances 7, 1 (2021).
  * Barton  _et al._ [2020] J. Barton, M. Gulka, J. Tarabek, Y. Mindarava, Z. Wang, J. Schimer, H. Raabova, J. Bednar, M. B. Plenio, F. Jelezko, M. Nesladek, and P. Cigler, Nanoscale Dynamic Readout of a Chemical Redox Process Using Radicals Coupled with Nitrogen-Vacancy Centers in Nanodiamonds, ACS Nano 14, 12938 (2020).
  * Sharmin  _et al._ [2021] R. Sharmin, T. Hamoh, A. Sigaeva, A. Mzyk, V. G. Damle, A. Morita, T. Vedelaar, and R. Schirhagl, Fluorescent Nanodiamonds for Detecting Free-Radical Generation in Real Time during Shear Stress in Human Umbilical Vein Endothelial Cells, ACS Sensors 6, 4349 (2021).
  * Norouzi  _et al._ [2022] N. Norouzi, A. C. Nusantara, Y. Ong, T. Hamoh, L. Nie, A. Morita, Y. Zhang, A. Mzyk, and R. Schirhagl, Relaxometry for detecting free radical generation during Bacteria’s response to antibiotics, Carbon 199, 444 (2022).
  * Rendler  _et al._ [2017] T. Rendler, J. Neburkova, O. Zemek, J. Kotek, A. Zappe, Z. Chu, P. Cigler, and J. Wrachtrup, Optical imaging of localized chemical events using programmable diamond quantum nanosensors, Nature Communications 8, 10.1038/ncomms14701 (2017).
  * McGuinness  _et al._ [2013] L. P. McGuinness, L. T. Hall, A. Stacey, D. A. Simpson, C. D. Hill, J. H. Cole, K. Ganesan, B. C. Gibson, S. Prawer, P. Mulvaney, F. Jelezko, J. Wrachtrup, R. E. Scholten, and L. C. Hollenberg, Ambient nanoscale sensing with single spins using quantum decoherence, New Journal of Physics 15, 0 (2013), arXiv:1211.5749 .
  * Tetienne  _et al._ [2013a] J. Tetienne, T. Hingant, L. Rondin, A. Cavaill, L. Mayer, G. Dantelle, T. Gacoin, J. Wrachtrup, J. Roch, and V. Jacques, Spin relaxometry of single nitrogen-vacancy defects in diamond nanocrystals for magnetic noise sensing, Physical Review B 87, 235436 (2013a).
  * Ermakova  _et al._ [2013] A. Ermakova, G. Pramanik, J. M. Cai, G. Algara-Siller, U. Kaiser, T. Weil, Y. K. Tzeng, H. C. Chang, L. P. McGuinness, M. B. Plenio, B. Naydenov, F. Jelezko, and F. Jelezko, Detection of a few metallo-protein molecules using color centers in nanodiamonds, Nano Letters 13, 3305 (2013).
  * Akther  _et al._ [2021] A. Akther, E. P. Walsh, P. Reineck, B. C. Gibson, T. Ohshima, H. Abe, G. McColl, N. L. Jenkins, L. T. Hall, D. A. Simpson, A. R. Rezk, and L. Y. Yeo, Acoustomicrofluidic Concentration and Signal Enhancement of Fluorescent Nanodiamond Sensors, Analytical Chemistry 93, 16133 (2021).
  * Reineck  _et al._ [2019a] P. Reineck, L. F. Trindade, J. Havlik, J. Stursa, A. Heffernan, A. Elbourne, A. Orth, M. Capelli, P. Cigler, D. A. Simpson, and B. C. Gibson, Not All Fluorescent Nanodiamonds Are Created Equal: A Comparative Study, Particle and Particle Systems Characterization 36, 1 (2019a).
  * Ryan  _et al._ [2018] R. G. Ryan, A. Stacey, K. M. O’Donnell, T. Ohshima, B. C. Johnson, L. C. Hollenberg, P. Mulvaney, and D. A. Simpson, Impact of Surface Functionalization on the Quantum Coherence of Nitrogen-Vacancy Centers in Nanodiamonds, ACS Applied Materials and Interfaces 10, 13143 (2018), arXiv:1711.09972 .
  * Padamati  _et al._ [2022] S. K. Padamati, T. A. Vedelaar, F. P. Martinez, A. C. Nusantara, and R. Schirhagl, Insight into a Fenton-like Reaction Using Nanodiamond Based Relaxometry, Nanomaterials 12 (2022).
  * Li  _et al._ [2022] R. Li, T. Vedelaar, A. Mzyk, A. Morita, S. K. Padamati, and R. Schirhagl, Following Polymer Degradation with Nanodiamond Magnetometry, ACS Sensors 7, 123 (2022).
  * Gorrini  _et al._ [2019] F. Gorrini, R. Giri, C. E. Avalos, S. Tambalo, S. Mannucci, L. Basso, N. Bazzanella, C. Dorigoni, M. Cazzanelli, P. Marzola, A. Miotello, and A. Bifone, Fast and Sensitive Detection of Paramagnetic Species Using Coupled Charge and Spin Dynamics in Strongly Fluorescent Nanodiamonds, ACS Applied Materials & Interfaces 11, 24412-24422 (2019).
  * Reineck  _et al._ [2019b] P. Reineck, L. F. Trindade, J. Havlik, J. Stursa, A. Heffernan, A. Elbourne, A. Orth, M. Capelli, P. Cigler, D. A. Simpson, and B. C. Gibson, Not All Fluorescent Nanodiamonds Are Created Equal: A Comparative Study, Particle and Particle Systems Characterization 36, 1 (2019b).
  * Hopper  _et al._ [2018] D. A. Hopper, H. J. Shulevitz, and L. C. Bassett, Spin readout techniques of the nitrogen-vacancy center in diamond, Micromachines 9, 1 (2018).
  * Jayakumar  _et al._ [2018] H. Jayakumar, S. Dhomkar, J. Henshaw, and C. A. Meriles, Spin readout via spin-to-charge conversion in bulk diamond nitrogen-vacancy ensembles, Applied Physics Letters 113, 10.1063/1.5040261 (2018).
  * Laraoui  _et al._ [2012] A. Laraoui, J. S. Hodges, and C. A. Meriles, Nitrogen-vacancy-assisted magnetometry of paramagnetic centers in an individual diamond nanocrystal, Nano Letters 12, 3477 (2012).
  * Tetienne  _et al._ [2013b] J. P. Tetienne, T. Hingant, L. Rondin, A. Cavaillès, L. Mayer, G. Dantelle, T. Gacoin, J. Wrachtrup, J. F. Roch, and V. Jacques, Spin relaxometry of single nitrogen-vacancy defects in diamond nanocrystals for magnetic noise sensing, Physical Review B - Condensed Matter and Materials Physics 87, 1 (2013b), arXiv:1304.1197 .
  * McCoey  _et al._ [2020] J. M. McCoey, M. Matsuoka, R. W. de Gille, L. T. Hall, J. A. Shaw, J. P. Tetienne, D. Kisailus, L. C. Hollenberg, and D. A. Simpson, Quantum Magnetic Imaging of Iron Biomineralization in Teeth of the Chiton Acanthopleura hirtosa, Small Methods 4, 1 (2020).
  * Casabianca  _et al._ [2011] L. B. Casabianca, A. I. Shames, A. M. Panich, O. Shenderova, and L. Frydman, Factors affecting DNP NMR in polycrystalline diamond samples, Journal of Physical Chemistry C 115, 19041 (2011).
  * Bluvstein  _et al._ [2019] D. Bluvstein, Z. Zhang, and A. C. Jayich, Identifying and Mitigating Charge Instabilities in Shallow Diamond Nitrogen-Vacancy Centers, Physical Review Letters 122, 76101 (2019), arXiv:1810.02058 .
  * Giri  _et al._ [2019] R. Giri, C. Dorigoni, S. Tambalo, F. Gorrini, and A. Bifone, Selective measurement of charge dynamics in an ensemble of nitrogen-vacancy centers in nanodiamond and bulk diamond, Physical Review B 99, 155426 (2019), arXiv:1812.02702 .
  * Broadway  _et al._ [2016] D. A. Broadway, J. D. A. Wood, L. T. Hall, A. Stacey, M. Markham, D. A. Simpson, J.-P. Tetienne, and L. C. L. Hollenberg, Spin dynamics of diamond nitrogen-vacancy centres at the ground state level anti-crossing and all-optical low frequency magnetic field sensing, Physical Review Applied 6, 1 (2016), arXiv:1607.04006 .
  * Austin  _et al._ [2020] J. Austin, C. Minelli, D. Hamilton, M. Wywijas, and H. J. Jones, Nanoparticle number concentration measurements by multi-angle dynamic light scattering, Journal of Nanoparticle Research 22, 10.1007/s11051-020-04840-8 (2020).
  * Wood  _et al._ [2016] J. D. A. Wood, D. A. Broadway, L. T. Hall, A. Stacey, D. A. Simpson, J.-p. Tetienne, and L. C. L. Hollenberg, Wide-band nanoscale magnetic resonance spectroscopy using quantum relaxation of a single spin in diamond, Physical Review B 94, 1 (2016).
  * Webb  _et al._ [2020] J. L. Webb, L. Troise, W. Hansen, J. Ahcard, O. Brinza, R. Staacke, M. Kieschnick, J. Meijer, J.-F. Perrier, K. Berg-Sorensen, A. Huck, and U. Andersen Lund, Optimization of a Diamond Nitrogen Vacancy Centre Magnetometer for Sensing of Biological Signals, Frontiers in Physics 8, doi: 10.3389/fphy.2020.522536 Optimization (2020), arXiv:arXiv:2004.02279v1 .
  * Porres  _et al._ [2006] L. Porres, A. Holland, L.-O. Palsson, A. P. Monkman, C. Kemp, and A. Beeby, Absolute Measurements of Photoluminescence Quantum Yields of Solutions Using an Integrating Sphere, Journal of Fluorescence 16, 4 (2006).
  * Wolf  _et al._ [2015] T. Wolf, P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, and J. Wrachtrup, Subpicotesla Diamond Magnetometry, Physical Review X 041001, 1 (2015).
  * Wolcott  _et al._ [2014] A. Wolcott, T. Schiros, M. E. Trusheim, E. H. Chen, D. Nordlund, R. E. Diaz, O. Gaathon, D. Englund, and J. S. Owen, Surface structure of aerobically oxidized diamond nanocrystals, Journal of Physical Chemistry C 118, 26695 (2014).
  * Stacey  _et al._ [2019] A. Stacey, N. Dontschuk, J. P. Chou, D. A. Broadway, A. K. Schenk, M. J. Sear, J. P. Tetienne, A. Hoffman, S. Prawer, C. I. Pakes, A. Tadich, N. P. de Leon, A. Gali, and L. C. Hollenberg, Evidence for Primal sp 2 Defects at the Diamond Surface: Candidates for Electron Trapping and Noise Sources, Advanced Materials Interfaces 6, 1 (2019), arXiv:1807.02946 .