# Single-quadrature quantum magnetometry in cavity electromagnonics

M. S. Ebrahimi  Department of Physics, Faculty of Science, University of
Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran  Ali Motazedifard  Department
of Physics, Faculty of Science, University of Isfahan, Hezar Jerib,
81746-73441, Isfahan, Iran  Quantum Optics Group, Department of Physics,
Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan,
Iran  Quantum Communication and Quantum Optics group, Iranian Center for
Quantum Technologies (ICQTs), Tehran, Iran  M. Bagheri Harouni
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, University
of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran  Quantum Optics Group,
Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib,
81746-73441, Isfahan, Iran

###### Abstract

A scheme of an ultrasensitive magnetometer in the cavity quantum
electromagnonics is proposed, where the intracavity microwave mode is coupled
to a magnonic mode via magnetic dipole interaction. It is shown that by
driving both magnonic and microwave modes with external classical fields and
controlling the system parameters, one can reduce the added noise of magnetic
field measurement below the standard quantum limit (SQL). Surprisingly, we
show that beyond the rotating wave approximation (RWA), not only can the added
noise be suppressed but also the output cavity response to the input signal
can be substantially amplified in order to achieve a precise magnetic field
measurement. The estimated theoretical sensitivity of the proposed magnetic
amplifier-sensor is approximately on the order of $10^{-18}\rm{T/\sqrt{\rm
Hz}}$, which is competitive compared to the current state-of-the-art
magnetometers like superconducting quantum interference devices (SQUIDs) and
atomic magnetometers. The advantages of the proposed sensor in comparison with
the other magnetometers is its high sensitivity at room temperature, sensing
in a wide range of frequencies up to MHz, and its capability for signal-
response amplification.

##  I Introduction

The precise measurement of physical quantities like force, magnetic field,
etc., is a challenging problem in quantum sensing and metrology since noise
sources usually destroy the measurements and cause uncertainty in measurement
results. Therefore, applying methods for suppressing or reducing the added
noise of measurement, notably below the standard quantum limit (SQL), is an
important key in quantum sensing and metrology. As an example, one approach is
coherent quantum noise cancellation (CQNC) 1 , in which the antinoise path in
the quantum dynamics of the system can cancel the original noise path via
destructive quantum interference. Very recently, the backaction noise due to
the radiation pressure has been observed in the Advanced Virgo gravitational
wave detector on the macroscopic scale at low frequency virgo2020 .

Recent decades have witnessed theoretical and experimental progress in
studying force sensing in different kinds of systems, such as bare and hybrid
optomechanical systems 1 ; 2 ; 3 ; 4 ; 5 ; Review of force sensing . It has
been shown that one can control the quantum noise for simultaneous noise
suppression and signal amplification 5 . Quantum magnetometry is another
important quantum metrology topic with undeniable practical applications in
different areas such as geology, navigation, archeology, magnetic storage,
medicine 6 ; 7 ; AVS magnetometry , and even the search for fundamental
physics issues like dark matter axions one ; two ; three . Quantum
magnetometry, which has an inevitable role in human life, has been
investigated in different kinds of systems such as cavity magnon–polaritons
one ; phasemodulatedmagnetometry ; 8 , cavity optomechanics 9 ; 10 ; 11 ; four
, atomic magnetometers 12 ; 13 ; Noisy atomic magnetometry in real time , and
superconducting quantum interference devices (SQUIDs) 14 ; 15 ; size of SQUID
. Although subfemto-Tesla magnetic sensing has been achieved in atomic
magnetometers and SQUIDs, there still exist some limitations and challenges
such as low-frequency range of sensing 13  and high working temperature 15 .

In recent years, besides the cavity optomechanical systems, studying light-
matter interaction based on spin wave collective excitation magnon–photon
interaction in ferromagnetic materials like yttrium iron garnet (YIG) with
very high spin density and low magnon dissipation rate has attracted a great
deal of attention 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 ; 23 . Magnons as
elementary excitations of magnetically ordered systems can efficiently
interact with external magnetic fields and their frequency is tunable, which
yields more controllability. These excitations in some ferromagnetic materials
like YIG have long lifetimes and long coherence times 24 ; 25 ; 26 , which
make them suitable for different applications in quantum information and
quantum metrology 16 ; 25 . Very recently, it has been experimentally shown
that magnon-based sensing is possible with a high sensitivity through
dispersively coupling between the magnetostatic mode and a superconducting
qubit magnon sensing . A homogeneous magnonic mode can interact with microwave
photons via magnetic dipole interaction 16 ; 17 ; 18 ; Highcooperativity ;
Nori . It should be noted that YIG has a high Curie temperature as $559$K, and
thus is ferromagnetic at both cryogenic and room temperatures Nori .
Therefore, coupling between a microwave cavity mode and the collective spin
excitations is experimentally possible in different YIG samples at either
cryogenic 16 ; 18 ; Highcooperativity ; Nori  or room temperature 17 ; Nori .
Microwave photon–magnon interaction that is studied in the context of cavity
electromagnonics 28  has been led to some interesting effects including
classical Rabi oscillation, magnetically induced transparency, and the Purcell
effect 17 . Moreover, microwave photon–magnon interaction affects the dynamics
of cavity magnomechanics 29 ; Dynamical Backaction Magnomechanics  where
magnons interact with microwave photons and phonons respectively via magnetic
dipole and magnetostrictive interactions. This interaction in different hybrid
quantum magnonic-based systems such as cavity electromagnonics and cavity
magnomechanics has important effects including microwave-to-optical quantum
transducer 30 , quantum entanglement and correlation 31 ; 32 ; 33 ; 34 ; 35 ;
36 ; last ; electromagnonics-optomechanics ; foroudcrystalentanglement ,
generation of magnon and microwave photon squeezed states 37 ; microwace field
squeezing , phononic laser 38 , quantum thermometry 39 , quantum magnetometry
one ; phasemodulatedmagnetometry ; 8 , magnon blockade magnonblockade1 ;
magnonblockade2 , quantum illumination quantumillumination , magnon-assisted
photon-phonon conversion photon-phononconversion , and storage and retrieval
of quantum states Cavity magnomechanical storage and retrieval of quantum
states .

Motivated by the above-mentioned investigations on force sensing in
optomechanics and capability of magnetic sensing in electromagnonic systems as
well as the current challenges in magnetometers such as low frequency and high
temperature, here, we theoretically propose a feasible experimental scheme of
a sensor for high-precision quantum magnetometry. We consider an
electromagnonic system whose magnons as quantum excitations of spin wave
interact with microwave photons via the magnetic dipole interaction. In the
proposed system, both magnonic and microwave cavity modes are externally
driven by classical fields. We analyze the system both beyond and under the
RWA and show that beyond the RWA by controlling system parameters, the added
noise of magnetic measurement can be suppressed far below the SQL. In
addition, surprisingly, the signal response can be simultaneously amplified,
which results in a precise magnetic field measurement. The estimated
sensitivity of the theoretical proposed magnetometer is on the order of
$10^{-18}\rm{T/\sqrt{\rm Hz}}$, which is competitive or even better than
current state-of-the-art SQUIDs and atomic magnetometers. Interestingly, the
advantages of our proposed magnetometer compared to the SQUIDs and atomic
based magnetometers are that our magnetometer can even operate at room
temperature and over a wide range of frequency with high sensitivity and
precision.

The paper is organized as follows. In Sec. II, the system Hamiltonian is
described, and then in Sec. III, the dynamics of the system is investigated
through the quantum Langevin-Heisenberg equations of motion. The mechanisms of
the magnetic sensing, noise suppression, and signal amplification are
presented in Sec. IV. In Sec. V, the sensitivity, the signal-to-noise ratio
(SNR), and comparison to other magnetometers are discussed. Finally, the
summary, conclusion, and outlooks are mentioned in Sec. VI.

##  II  The system and Hamiltonian

Figure 1 shows a schematic of the proposed magnetometer: A YIG sphere is
placed inside a microwave cavity with a uniform external bias magnetic field,
$B_{b}$, which is applied to the YIG sphere to produce a homogeneous magnonic
mode, the so-called Kittel mode. Therefore, magnetic dipole interaction can
mediate microwave photon–magnon interaction 16 ; 17 ; 27 ; 28 . Assuming that
both the magnonic and photonic modes are driven by external magnetic and
electromagnetic fields, respectively, the Hamiltonian of the system is given
by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\omega_{a}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{m}\hat{m}^{\dagger}\hat{m}+\hbar
g_{0}(\hat{a}+\hat{a}^{\dagger})(\hat{m}+\hat{m}^{\dagger})$ (1)
$\displaystyle+i\hbar\epsilon_{L}(\hat{a}^{\dagger}e^{-i\omega_{L}t}-\hat{a}e^{i\omega_{L}t})+i\hbar\epsilon_{d}(\hat{m}^{\dagger}e^{-i\omega_{d}t}-\hat{m}e^{i\omega_{d}t})$
$\displaystyle+\hat{H}_{B}.\ $

Figure 1: (Color online) Schematic of a magnetic field sensor based on cavity
electromagnonics, in which a YIG sphere is placed where the magnetic field of
the microwave cavity mode is maximum. The YIG sphere is also biased with a
uniform magnetic field $B_{b}$ in a direction which at least has a component
along the direction orthogonal to the cavity axis. The cavity and magnonic
modes are driven by the classical cavity and magnon drive fields,
respectively. $B$ is an unknown weak external magnetic field which is applied
to the YIG sphere along the $x$ direction and should be measured.

The first two terms describe the free energy of photonic and magnonic modes
where $\hat{a}$ $(\hat{a}^{\dagger})$ and $\hat{m}$ $(\hat{m}^{\dagger})$
respectively denote the annihilation (creation) operators of these two modes
and satisfy bosonic algebra $[\hat{o},\hat{o}^{\dagger}]=1$,
$\hat{o}=\hat{a},\hat{m}$. $\omega_{a}$ and $\omega_{m}$ indicate the
resonance frequency of the cavity and magnonic modes, respectively. It is
worth noting that the magnon frequency is determined by the external bias
magnetic field $B_{b}$ and the gyromagnetic ratio
$\gamma/2\pi={28}~{}\mathrm{GHz/T}$, i.e., $\omega_{m}=\gamma B_{b}$, and can
be easily adjusted via $B_{b}$. The third term in the Hamiltonian represents
the microwave photon–magnon coupling obtained through the magnetic dipole
interaction, which is written without resorting to the rotating wave
approximation (RWA) 27 ; 28 . Note that the RWA breaks in the ultrastrong
coupling (USC) and deep strong coupling (DSC) regimes, which respectively can
be achieved when conditions $g_{0}\lesssim\omega_{a(m)}$ and
$g_{0}\gtrsim\omega_{a(m)}$ are satisfied USC . Therefore, this interaction
Hamiltonian can be considered for either USC or DSC regime. Note that the
strength of magnon-photon coupling is given by $g_{0}=\frac{\gamma
B_{0}}{2}\sqrt{2sN}$, where $\gamma$ is the gyromagnetic ratio and
$B_{0}=\sqrt{\frac{\hbar\omega_{a}\mu_{0}}{V_{a}}}$ indicates the amplitude of
the magnetic field inside the microwave cavity in which $\omega_{a}$ is the
microwave mode frequency, $\mu_{0}$ is the vacuum permeability, and $V_{a}$
stands for the mode volume of the microwave cavity resonance. Also, $s$ and
$N$, respectively, stand for the spin angular momentum $s=5/2$ on each unit
cell of the magnetic material and the number of unit cells with spin $s$ in
the YIG sphere 27 . The contribution of the two external drive fields can be
seen in the fourth and fifth terms of the Hamiltonian. The cavity drive field
with frequency $\omega_{L}$ and power $P_{in}$ drives the cavity mode with
rate $\epsilon_{L}=\sqrt{\frac{2P_{in}\kappa_{a}}{\hbar\omega_{L}}}$ (with
$\kappa_{a}$ being the cavity decay rate). The coupling strength of the magnon
drive field (whose frequency and amplitude are respectively given by
$\omega_{d}$ and $B_{d}$) with the magnonic mode in the YIG sphere is
$\epsilon_{d}=\frac{\eta}{2}B_{d}$, where $\eta=\frac{\gamma}{2}\sqrt{5N}$ 31
. $\hat{H}_{B}$ accounts for the coupling of the magnonic mode in the YIG
sphere to an unknown weak external magnetic field $B$, which should be
measured, and is given by

$\hat{H}_{B}=-\gamma\vec{S}\cdot\vec{B}=-\hbar\eta
B(t)(\hat{m}+\hat{m}^{\dagger}),$ (2)

with $\vec{S}$ being the collective spin angular momentum. The above
Hamiltonian is obtained using the Holstein-Primakkoff transformations and
assuming that the time-dependent external magnetic field is along the $x$
direction 31 . The Hamiltonian in the frame rotating at frequencies
$\omega_{L}$ and $\omega_{d}$ is given by

$\displaystyle\hat{H}^{(rot)}$ $\displaystyle=$
$\displaystyle\hbar\Delta_{a}\hat{a}^{\dagger}\hat{a}+\hbar\Delta_{m}\hat{m}^{\dagger}\hat{m}+\hbar
g_{0}(\hat{a}e^{-i\omega_{L}t}+\hat{a}^{\dagger}e^{i\omega_{L}t})$ (3)
$\displaystyle(\hat{m}e^{-i\omega_{d}t}+\hat{m}^{\dagger}e^{i\omega_{d}t})+i\hbar\epsilon_{L}(\hat{a}^{\dagger}-\hat{a})+i\hbar\epsilon_{d}(\hat{m}^{\dagger}-\hat{m})$
$\displaystyle-\hbar\eta
B(t)(\hat{m}e^{-i\omega_{d}t}+\hat{m}^{\dagger}e^{i\omega_{d}t}),\ $

where $\Delta_{a}\equiv\omega_{a}-\omega_{L}$ and
$\Delta_{m}\equiv\omega_{m}-\omega_{d}$ are the cavity and magnonic detunings,
respectively. We assume that the coupling strength $g_{0}$ is modulated
coherently as $g_{0}(t)=g(1+2\cos\Omega t)$ with $g=g_{0}\mathcal{E}$ where
$\mathcal{E}$ and $\Omega$ are the amplitude and the frequency of time
modulation, respectively. Assuming resonance condition,
$\Omega=\omega_{L}+\omega_{d}$, and also considering the fact that the
magnonic and microwave modes are near resonance in the microwave photon–magnon
interaction $(\omega_{a}\simeq\omega_{m})$ 17  and consequently
$\omega_{L}\simeq\omega_{d}$ can be provided, then, $\Omega=2\omega_{L}$. In
this manner, the photon–magnon interaction term [the third term in the
Hamiltonian in Eq. (3)], by considering the time modulation $g_{0}(t)$ and
using the RWA, which is valid when $g\equiv g_{0}\mathcal{E}\ll\Omega$, can be
simplified as

$\hat{H}_{int}=\hbar g(\hat{a}+\hat{a}^{\dagger})(\hat{m}+\hat{m}^{\dagger}).$
(4)

As mentioned, the magnon–photon coupling strength $g_{0}$ depends on the
number of spins and the amplitude of the magnetic field inside the microwave
cavity. Therefore, by modulating the amplitude of magnetic field inside the
microwave cavity, one can modulate the magnon–photon coupling strength. By
controlling the modulation amplitude of magnetic field, we can change
$\mathcal{E}$ and hence $g$ so that $g\ll\ \Omega$ is valid. On the other
hand, the effective magnon–photon coupling strength can be tuned via the
modulation amplitude $\mathcal{E}$. This controllability on coupling strength
is an advantage of the time modulation in this configuration. It should be
noted that recently the time modulation technique has been theoretically and
experimentally applied in quantum optical systems in order to generate the
nonclassical state, squeezing and also improving the sensing 3 ; 4 ; 5 ;
aliDCEsqueezing ; aliDCE1 ; aliDCE2 ; aliDCE3 ; NoriDCE1 ; NoriDCE2 ; NoriDCE3
; pontinmodulation .

Now, the total Hamiltonian of the system is given by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\Delta(\hat{a}^{\dagger}\hat{a}+\hat{m}^{\dagger}\hat{m})+\hbar
g(\hat{a}+\hat{a}^{\dagger})(\hat{m}+\hat{m}^{\dagger})$
$\displaystyle+i\hbar\epsilon_{L}(\hat{a}^{\dagger}-\hat{a})+i\hbar\epsilon_{d}(\hat{m}^{\dagger}-\hat{m})-\hbar\eta
B(t)(\hat{m}e^{-i\omega_{d}t}+\hat{m}^{\dagger}e^{i\omega_{d}t}),$

in which we have assumed $\Delta_{a}\simeq\Delta_{m}\equiv\Delta$ since
$\omega_{a}\simeq\omega_{m}$ and $\omega_{L}\simeq\omega_{d}$ and we have
considered the first notation of Hamiltonian ($\hat{H}$ instead of
$\hat{H}^{(rot)}$).

##  III  DYNAMICS OF THE SYSTEM

Considering Hamiltonian (II), the quantum Langevin equations (QLEs) describing
the system dynamics are given by

$\displaystyle\delta\dot{\hat{m}}$ $\displaystyle=$
$\displaystyle-i\Delta\delta\hat{m}-ig(\delta\hat{a}^{\dagger}+\delta\hat{a})-\frac{\kappa_{m}}{2}\delta\hat{m}+\sqrt{\kappa_{m}}\hat{m}^{\mathrm{in}}$
(6) $\displaystyle+i\eta B(t)e^{i\omega_{d}t},$
$\displaystyle\delta\dot{\hat{a}}$ $\displaystyle=$
$\displaystyle-i\Delta\delta\hat{a}-ig(\delta\hat{m}^{\dagger}+\delta\hat{m})-\frac{\kappa_{a}}{2}\delta\hat{a}+\sqrt{\kappa_{a}}\hat{a}^{\mathrm{in}},$
(7)

in which $\kappa_{a}$ and $\kappa_{m}$ indicate the cavity and magnonic
dissipation rates, respectively. The input quantum noise of the cavity and
magnonic modes are respectively shown by $\hat{a}^{\mathrm{in}}$ and
$\hat{m}^{\mathrm{in}}$ which satisfy the following correlation functions

$\displaystyle\langle\hat{o}^{\mathrm{in}}(t)\hat{o}^{\mathrm{in}\dagger}(t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle(1+\bar{n}_{o})\delta(t-t^{\prime}),$ (8)
$\displaystyle\langle\hat{o}^{\mathrm{in}\dagger}(t)\hat{o}^{\mathrm{in}}(t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\bar{n}_{o}\delta(t-t^{\prime}),$ (9)

with $o=a,m$ and $\bar{n}_{o}=[\rm
exp(\frac{\hbar\omega_{o}}{k_{B}T})-1]^{-1}$ being the mean number of thermal
excitations of the cavity and magnon modes at temperature $T$. Here, we assume
$\bar{n}_{a}\simeq\bar{n}_{m}\equiv\bar{n}$, which is experimentally
realizable for the microwave photon–magnon interaction
$\omega_{a}\simeq\omega_{m}$ 17 . By defining the quadratures
$\delta\hat{X}_{o}=\frac{\hat{o}+\hat{o}^{\dagger}}{\sqrt{2}}$ and
$\delta\hat{P}_{o}=\frac{i(\hat{o}^{\dagger}-\hat{o})}{\sqrt{2}}$
$(\hat{o}=\delta\hat{a},\delta\hat{m})$, Eqs. (6) and (7) can be written as
follows:

$\displaystyle\delta\dot{\hat{X}}_{m}$ $\displaystyle=$
$\displaystyle\Delta\delta\hat{P}_{m}-\frac{\kappa_{m}}{2}\delta\hat{X}_{m}+\sqrt{\kappa_{m}}\hat{X}^{\prime\mathrm{in}}_{m},$
(10) $\displaystyle\delta\dot{\hat{P}}_{m}$ $\displaystyle=$
$\displaystyle-\Delta\delta\hat{X}_{m}-2g\delta\hat{X}_{a}-\frac{\kappa_{m}}{2}\delta\hat{P}_{m}+\sqrt{\kappa_{m}}\hat{P}^{\prime\mathrm{in}}_{m},$
(11) $\displaystyle\delta\dot{\hat{X}}_{a}$ $\displaystyle=$
$\displaystyle\Delta\delta\hat{P}_{a}-\frac{\kappa_{a}}{2}\delta\hat{X}_{a}+\sqrt{\kappa_{a}}\hat{X}^{\mathrm{in}}_{a},$
(12) $\displaystyle\delta\dot{\hat{P}}_{a}$ $\displaystyle=$
$\displaystyle-\Delta\delta\hat{X}_{a}-2g\delta\hat{X}_{m}-\frac{\kappa_{a}}{2}\delta\hat{P}_{a}+\sqrt{\kappa_{a}}\hat{P}^{\mathrm{in}}_{a},$
(13)

where
$\hat{X}^{\mathrm{in}}_{o}=\frac{\hat{o}^{\mathrm{in}}+\hat{o}^{\mathrm{in}\dagger}}{\sqrt{2}}$
and
$\hat{P}^{\mathrm{in}}_{o}=\frac{i(\hat{o}^{\mathrm{in}\dagger}-\hat{o}^{\mathrm{in}})}{\sqrt{2}}$
($o=a,m$). Also, we have defined $\hat{X}^{\prime\mathrm{in}}_{m}(t)$ and
$\hat{P}^{\prime\mathrm{in}}_{m}(t)$ as follows:

$\displaystyle\hat{X}^{\prime\mathrm{in}}_{m}(t)$ $\displaystyle=$
$\displaystyle\hat{X}^{\mathrm{in}}_{m}-\sqrt{\frac{2}{\kappa_{m}}}\eta
B(t)\sin(\omega_{d}t),$ (14) $\displaystyle\hat{P}^{\prime\mathrm{in}}_{m}(t)$
$\displaystyle=$
$\displaystyle\hat{P}^{\mathrm{in}}_{m}+\sqrt{\frac{2}{\kappa_{m}}}\eta
B(t)\cos(\omega_{d}t),$ (15)

which show the effects of the external magnetic field.

In the system under consideration, the external magnetic field that we are
going to measure, $B$, is coupled to the magnonic mode. On the other hand, the
magnon mode is coupled to microwave photons through the magnetic dipole
interaction. Consequently, the signal corresponding to the external magnetic
field, $B$, can be detected by measuring the optical output phase quadrature,
$\hat{P}^{\mathrm{out}}_{a}$. According to the input-output relation, i.e.,
$\hat{a}^{\mathrm{out}}=\sqrt{\kappa_{a}}\delta\hat{a}-\hat{a}^{\mathrm{in}}$,
the output phase quadrature is given by

$\hat{P}^{out}_{a}=\sqrt{\kappa_{a}}\delta\hat{P}_{a}-\hat{P}^{in}_{a}.$ (16)

By solving the QLEs [Eqs. (10) -(13)] in the Fourier space, we can find
$\delta\hat{P}_{a}(\omega)$. Then, according to Eq.(16),
$\hat{P}^{\mathrm{out}}_{a}$ in the Fourier space can be simplified as

$\displaystyle\hat{P}^{\mathrm{out}}_{a}(\omega)$ $\displaystyle=$
$\displaystyle\mathcal{A}(\omega)\hat{X}^{\prime\mathrm{in}}_{m}(\omega)+\mathcal{B}(\omega)\hat{P}^{\prime\mathrm{in}}_{m}(\omega)$
(17)
$\displaystyle+\mathcal{C}(\omega)\hat{X}^{\mathrm{in}}_{a}(\omega)+\mathcal{D}(\omega)\hat{P}^{\mathrm{in}}_{a}(\omega),$

where $\hat{X}^{\prime\mathrm{in}}_{m}(\omega)$ and
$\hat{P}^{\prime\mathrm{in}}_{m}(\omega)$ are respectively the Fourier
transform of $\hat{X}^{\prime\mathrm{in}}_{m}(t)$ and
$\hat{P}^{\prime\mathrm{in}}_{m}(t)$ and are given by

$\displaystyle\hat{X}^{\prime\mathrm{in}}_{m}(\omega)$ $\displaystyle=$
$\displaystyle\hat{X}^{\mathrm{in}}_{m}(\omega)+\sqrt{\frac{2}{\kappa_{m}}}\eta\frac{i}{2}[B(\omega+\omega_{d})-B(\omega-\omega_{d})],$
(18) $\displaystyle\hat{P}^{\prime\mathrm{in}}_{m}(\omega)$ $\displaystyle=$
$\displaystyle\hat{P}^{\mathrm{in}}_{m}(\omega)+\sqrt{\frac{2}{\kappa_{m}}}\eta\frac{1}{2}[B(\omega+\omega_{d})+B(\omega-\omega_{d})].$
(19)

In addition, the coefficients
$\mathcal{A}(\omega),\mathcal{B}(\omega),\mathcal{C}(\omega)$, and
$\mathcal{D}(\omega)$ are defined as

$\displaystyle\mathcal{A}(\omega)$ $\displaystyle\equiv$ $\displaystyle
4g\sqrt{\kappa_{a}\kappa_{m}}(\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}})\chi^{\prime}_{m}(\omega),$
(20) $\displaystyle\mathcal{B}(\omega)$ $\displaystyle\equiv$ $\displaystyle
4g\Delta\chi_{m}(\omega)\chi^{\prime}_{m}(\omega)\sqrt{\kappa_{a}\kappa_{m}}(\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}}),$
(21) $\displaystyle\mathcal{C}(\omega)$ $\displaystyle\equiv$
$\displaystyle-4g^{2}\Delta\chi_{m}(\omega)\chi^{\prime}_{m}(\omega)\chi_{a}(\omega)\kappa_{a}(\frac{\Delta\chi_{a}(\omega)-1}{\Delta})$
(22) $\displaystyle-\kappa_{a}\chi_{a}(\omega),$
$\displaystyle\mathcal{D}(\omega)$ $\displaystyle\equiv$
$\displaystyle-8g^{2}\Delta\chi_{m}(\omega)\chi^{\prime}_{m}(\omega)\chi_{a}(\omega)\kappa_{a}(\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}})$
(23)
$\displaystyle-\frac{2\kappa_{a}(\Delta\chi_{a}(\omega)-1)}{2i\omega+\kappa_{a}}-1,$

where we have defined $\chi_{a}(\omega)$, $\chi_{m}(\omega)$, and
$\chi^{\prime}_{m}(\omega)$ as follows:

$\displaystyle\chi_{a}(\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{\Delta}{\Delta^{2}-\omega^{2}+i\omega\Delta+\frac{\kappa_{a}^{2}}{4}},$
(24) $\displaystyle\chi_{m}(\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{i\omega+\frac{\kappa_{m}}{2}},$ (25)
$\displaystyle\chi^{\prime}_{m}(\omega)$ $\displaystyle\equiv$
$\displaystyle\chi_{m}(\omega)[1+\Delta^{2}\chi^{2}_{m}(\omega)-4\Delta
g^{2}\chi_{a}(\omega)\chi^{2}_{m}(\omega)]^{-1}.$ (26)

##  IV  MAGNETIC FIELD SENSING

Generally, similar to the cavity quantum electrodynamics systems, there could
be different kinds of light-matter interaction in cavity electromagnonic
systems ultra strong coupling regime nori ; 17  such as DSC, USC, strong
coupling (SC), and weak coupling (WC) regimes, which depend on the light-
matter coupling strength, $g_{0}$. The DSC and USC regimes are satisfied when
$g_{0}\gtrsim\omega_{a(m)}$ and $g_{0}\lesssim\omega_{a(m)}$, respectively.
Note that the RWA breaks down in these two regimes. In SC and WC regimes, in
addition to the frequency and coupling, the damping rates play the important
role such that they can be labeled by $\kappa_{a},\kappa_{m}\ \ll\ g_{0}\ll\
\omega_{a(m)}$ and $g_{0}\ll\ \kappa_{a},\kappa_{m},\omega_{a(m)}$,
respectively. It should be noted that in these two later regimes the RWA is
valid. Furthermore, the intermediate coupling (IMC) regime where also the RWA
is valid is satisfied when $(g_{0}\ll\ \omega_{a(m)}$, $\kappa_{a}<\ g_{0}<\
\kappa_{m})$ or $(g_{0}\ll\ \omega_{a(m)}$, $\kappa_{m}<\ g_{0}<\
\kappa_{a})$.

In the following, we first investigate the magnetic field sensing in the
cavity electromagnonics beyond the RWA, which can be considered for the DSC or
USC regime. Then, we study the possibility of magnetic sensing under the RWA,
which is valid in SC, WC, and IMC regimes.

###  IV.1  Beyond the RWA

In this section, we consider the system beyond the RWA and show that the
signal response corresponding to the external magnetic field $B$ can be
detected via the output cavity phase-quadrature $P^{\mathrm{out}}_{a}$.
Therefore, we calculate the output cavity phase spectrum and show how by
selecting suitable values of system parameters, one can simultaneously
suppress the added noise and amplify the response to the input signal.

The spectrum of the optical output phase quadrature that helps to detect the
input magnetic field is as follows:

$\displaystyle S_{P^{\mathrm{out}}_{a}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d\omega^{\prime}e^{i(\omega+\omega^{\prime})t}\langle\delta
P^{\mathrm{out}}_{a}(\omega)\delta P^{\mathrm{out}}_{a}(\omega^{\prime})$ (27)
$\displaystyle+\delta P^{\mathrm{out}}_{a}(\omega^{\prime})\delta
P^{\mathrm{out}}_{a}(\omega)\rangle.$

Let us now set aside the external magnetic field, $B$, for simplification
since the response of the system to the external magnetic field is independent
of the classical input magnetic field and only depends on the quantum
properties of the system. In this manner, the output phase quadrature spectrum
is obtained as

$\displaystyle S_{P^{\mathrm{out}}_{a}}(\omega)$ $\displaystyle=$
$\displaystyle[|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}](\bar{n}_{m}+\frac{1}{2})$
(28)
$\displaystyle+[|\mathcal{C}(\omega)|^{2}+|\mathcal{D}(\omega)|^{2}](\bar{n}_{a}+\frac{1}{2}).$

We can rewrite this spectrum as follows

$S_{P^{\mathrm{out}}_{a}}(\omega)=R_{m}(\omega)[(\bar{n}_{m}+\frac{1}{2})+n_{\mathrm{add}}(\omega)],$
(29)

in which $R_{m}(\omega)$ and $n_{\mathrm{add}}(\omega)$ show respectively the
magnonic response and the added noise of magnetic field measurement beyond the
RWA and are given by

$\displaystyle R_{m}(\omega)$ $\displaystyle=$
$\displaystyle[|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}],$ (30)
$\displaystyle n_{\mathrm{\mathrm{add}}}(\omega)$ $\displaystyle=$
$\displaystyle(\bar{n}_{a}+\frac{1}{2})\frac{[|\mathcal{C}(\omega)|^{2}+|\mathcal{D}(\omega)|^{2}]}{[|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}]}.$
(31)

So, according to Eqs. (20)–(23), the magnonic response and the added noise can
be simplified as

$\displaystyle R_{m}(\omega)$ $\displaystyle=$ $\displaystyle
16g^{2}\kappa_{a}\kappa_{m}|\chi^{\prime}_{m}(\omega)|^{2}\big{|}\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}}\big{|}^{2}\big{[}\Delta^{2}|\chi_{m}(\omega)|^{2}\\!+\\!\\!1\big{]},$
(32) $\displaystyle n_{\mathrm{add}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\Big{(}\big{|}8g^{2}\kappa_{a}\Delta\chi_{m}(\omega)\chi^{\prime}_{m}(\omega)\chi_{a}(\omega)(\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}})+1+\frac{2\kappa_{a}(\Delta\chi_{a}(\omega)-1)}{2i\omega+\kappa_{a}}\big{|}^{2}+\big{|}4g^{2}\kappa_{a}\chi_{m}(\omega)\chi^{\prime}_{m}(\omega)\chi_{a}(\omega)(\Delta\chi_{a}(\omega)-1)+\kappa_{a}\chi_{a}(\omega)\big{|}^{2}\Big{)}(\bar{n}_{a}+\frac{1}{2})}{16g^{2}\kappa_{a}\kappa_{m}|\chi^{\prime}_{m}(\omega)|^{2}\big{|}(\frac{\Delta\chi_{a}(\omega)-1}{2i\omega+\kappa_{a}})\big{|}^{2}(\Delta^{2}|\chi_{m}(\omega)|^{2}+1)}.$

Figure 2: (Color online) (a) The magnonic response to the input signal,
$R_{m}(\omega)$, and (b) the added noise of magnetic field measurement,
$n_{\mathrm{add}}(\omega)$, vs the normalized frequency $\omega/\kappa_{m}$.
All curves in these two panels have been plotted beyond the RWA corresponding
to the cooperativity $C=1000$. The thin blue solid, purple dashed, thick
orange solid, and brown dotted curves, respectively, correspond to detunings
$\Delta=0,0.05\kappa_{m},0.5\kappa_{m},5\kappa_{m}$. Here, we have set zero
thermal noises, $\bar{n}\simeq 0$, corresponding to the zero temperature.

Figure 3: (Color online) (a) The magnonic response to the input signal,
$R_{m}(\omega)$, and (b) the added noise of magnetic field measurement,
$n_{\mathrm{add}}(\omega)$, versus the normalized frequency
$\omega/\kappa_{m}$. All curves have been plotted beyond the RWA for the
optimized zero detuning, $\Delta=0$. The thin blue solid, magenta dashed,
thick red solid, gray dotted, and green solid curves are respectively referred
to $C=1000,100,10,1,0.5$. Here, we have set zero thermal noises,
$\bar{n}\simeq 0$, corresponding to the zero temperature.

First, let us consider the on-resonance case $\omega=0$. By considering Eqs.
(32) and (LABEL:34), one finds the optimized condition $\Delta=0$ in order to
be able simultaneously amplify the signal and suppress the noise. In this
manner, the magnonic response and the added noise are given by

$\displaystyle R_{m}(\omega=0)|_{\Delta=0}$ $\displaystyle=$
$\displaystyle\frac{64g^{2}}{\kappa_{a}\kappa_{m}}=16C,$ (34) $\displaystyle
n_{\mathrm{add}}(\omega=0)|_{\Delta=0}$ $\displaystyle=$
$\displaystyle(\bar{n}_{a}+\frac{1}{2})\frac{\kappa_{a}\kappa_{m}}{64g^{2}}=\frac{(\bar{n}_{a}+\frac{1}{2})}{16C},$
(35)

in which we have defined $C\equiv\frac{4g^{2}}{\kappa_{a}\kappa_{m}}$ as the
electromagnonic cooperativity beyond the RWA. It is evident that in the on-
resonance case, one can linearly amplify the response and suppress the noise,
simultaneously, by increasing the electromagnonic cooperativity $C$. Note that
beyond the RWA, the cooperativity is controlled via the effective coupling
$g=g_{0}\mathcal{E}$ which itself can be controlled via the modulation
amplitude of the external magnetic drive.

In order to obtain more insights about the effect of different values of
detuning, $\Delta$, on signal amplification and noise suppression beyond the
RWA, in Fig. 2 the magnonic response of the cavity output phase spectrum
$R_{m}(\omega)$ [Fig. 2(a)] and the added noise of magnetic field measurement
$n_{\mathrm{add}}(\omega)$ [Fig. 2(b)] have been plotted versus the normalized
frequency $\omega/\kappa_{m}$ for different values of detuning. As is evident,
at zero temperature, $\bar{n}=0$, and for an arbitrary selected
electromagnonic cooperativity, for example $C=1000$, by choosing the frequency
of the external drive fields equal to the cavity field frequency, the magnonic
response can be increased to high value for a wide range of frequency (for
$\omega\lesssim\kappa_{m}$) and simultaneously the added noise of magnetic
field measurement can be suppressed below the SQL. As the results show, by
increasing the detuning the amount of signal amplification and noise reduction
will decrease. For small detunings like $\Delta=0.05\kappa_{m}$, the frequency
range in which we could at the same time amplify the signal and suppress the
added noise is very short. While for larger detunings like
$\Delta=5\kappa_{m}$, both signal amplification and noise reduction are
impossible. Therefore, the optimized case in which we could amplify the signal
and simultaneously suppress the added noise of measurement, especially for a
wide range of frequency is zero detuning $\Delta=0$. As is evident, although
simultaneous signal amplification and noise reduction have the maximum value
in the on-resonance frequency, they are still possible and appreciable far
from the on-resonance case.

In Fig. 3, we have plotted the magnonic response of the cavity output phase
spectrum $R_{m}(\omega)$ [Fig. 3(a)] and the added noise of magnetic field
measurement $n_{add}(\omega)$ [Fig. 3(b)] versus the normalized frequency
$\omega/\kappa_{m}$ for different cooperativities in order to see how
different values of cooperativity affect the signal amplification and noise
reduction. These results that are obtained for the optimized detuning,
$\Delta=0$, and at zero temperature show that by increasing the
electromagnonic cooperativity signal amplification and noise reduction, which
occur simultaneously, will increase. Although the large cooperativity regime
like $C=1000$ has more signal amplification and noise suppression, the weak
cooperativity regime such as $C=0.5$ can also simultaneously lead to signal
amplification and noise suppression in both on-resonance and off-resonance
frequencies. Note that beyond the RWA, using the coherent time modulation of
amplitude of magnetic drive which yields to the modulation of the
magnon–photon coupling, one can control and tune the electromagnonic
cooperativities via the amplitude of modulation.

Finally, to understand the roles of temperature or thermal noise on the added
noise of magnetic field measurement, in Fig. 4 we have plotted the added noise
of magnetic field measurement $n_{\mathrm{add}}(\omega)$ versus the normalized
frequency $\omega/\kappa_{m}$ at zero detuning, $\Delta=0$, and for two
different temperatures, nearly zero and room temperature. As is clearly seen
and expected from Eq. (LABEL:34), lower temperature corresponding to the
smaller thermal noises enables us to more noise suppression. Interestingly,
our sensor has capability to work sensitively even at room temperature in a
wide range of frequency (in on and off resonance) by controlling the
cooperativity to be large enough, like $C=1000$. As is evident, at room
temperature in off-resonance frequencies, the noise suppression is still
possible by controlling the cooperativity.

It worth remembering again that beyond the RWA, our proposed magnetic sensor
can work sensitively even at room temperature in a wide range of frequency
$\omega\lesssim\kappa_{m}$, so that simultaneously the noise and signal can be
suppressed and amplified, respectively, to obtain a desired precise
measurement.

Figure 4: (Color online) The effect of thermal noises, temperature, on the
added noise of magnetic field measurement. All curves have been plotted beyond
the RWA and for optimized zero detuning. The blue solid, magenta dashed, and
gray dotted curves correspond to zero temperature, $\bar{n}=0$, respectively
for $C=1000,100,1$. Moreover, the blue dot-dashed, magenta solid and gray
solid curves are referred to room temperature corresponding to $\bar{n}=166$
respectively for $C=1000,100,1$. For obtaining thermal noises, we have chosen
$\omega_{a(m)}/2\pi=37.5~{}\rm{GHz}$ as an achievable bare frequency in the
coupled system beyond the RWA 17 .

###  IV.2  Under the RWA

Considering the system with coupling strength, $g_{0}$, much smaller than the
bare frequencies of the uncoupled system
$(g_{0}\ll\omega_{a}\simeq\omega_{m})$ , one can use the RWA to investigate
the system dynamics. Following the same method as Sec. IV.1, the magnonic
response and the added noise of measurement under the RWA and in on-resonance
frequency can be obtained as follows (to derive the Hamiltonian and the system
dynamics, see appendix)

$\displaystyle R^{\prime}_{m}(\omega=0)|_{\Delta=0}$ $\displaystyle=$
$\displaystyle\frac{4C^{\prime}}{(1+C^{\prime})^{2}},$ (36) $\displaystyle
n^{\prime}_{\mathrm{add}}(\omega=0)|_{\Delta=0}$ $\displaystyle=$
$\displaystyle(\bar{n}_{a}+\frac{1}{2})\frac{(1-C^{\prime})^{2}}{4C^{\prime}},$
(37)

where $R^{\prime}_{m}$, $n^{\prime}_{\mathrm{add}}$, and
$C^{\prime}\equiv\frac{4g_{0}^{2}}{\kappa_{a}\kappa_{m}}\big{|}^{\mathrm{RWA}}$,
respectively, indicate the magnonic response, the added noise, and the
electromagnonic cooperativity under the RWA. Note that the above equations are
written in the optimized zero detuning. Furthermore, it should be noted that
under the RWA, in which we use no time modulation, controllability on the
coupling strength $g_{0}$ decreases and can be tuned only by controlling the
magnetic field inside the cavity. It means that the controllability on the
system via the cooperativity under the RWA decreases compared to the system
beyond the RWA where the cooperativity can be manipulated easily through
$g=g_{0}\mathcal{E}$ via the modulation amplitude $\mathcal{E}$.

As is evidenced from Eqs. (36) and (37), unlike beyond the RWA, here, under
the RWA the signal response in on-resonance frequency never can be amplified
such that at the best condition it can be reached to the unity for the
optimized cooperativity $\mathcal{C}^{\prime}_{\rm opt}=1$. The added noise
on-resonance can be still suppressed perfectly for this optimized
cooperativity $\mathcal{C}^{\prime}_{\rm opt}=1$. In other words, the proposed
magnetic sensor under the RWA never can amplify the signal response but still
suppresses the noise below the SQL.

Similar to Sec. IV.1, in order to see how different values of detuning affect
the signal amplification and noise suppression beyond the RWA, in Fig. 5 we
have plotted the magnonic response of the cavity output phase spectrum
$R^{\prime}_{m}(\omega)$ [Fig. 5(a)] and the added noise of magnetic field
measurement $n^{\prime}_{add}(\omega)$ [Fig. 5(b)] with respect to the
normalized frequency $\omega/\kappa_{m}$ for different values of detuning. The
obtained results for the zero temperature and optimized cooperativity
$C^{\prime}=1$ show that increasing the detuning, nonzero detuning, leads to
signal response decrement. Although for small detunings, like
$\Delta=0.05\kappa_{m}$, the magnonic response is nearly the same as for zero
detuning, Fig. 5(b) shows the strongest noise suppression occurs for the
optimized detuning $\Delta=0$. Also, to have a precise measurement one should
control the detuning and cooperativity to be their optimized values to achieve
the maximum response, ${R^{\prime}}_{m}^{\rm max}=1$, and lower noise in a
pretty wide range of frequency $(\omega\leq\ 0.1\kappa_{m})$.

Figure 5: (Color online) (a) The magnonic response to the input signal,
$R^{\prime}_{m}(\omega)$, and (b) the added noise of magnetic field
measurement, $n^{\prime}_{\mathrm{add}}(\omega)$ under the RWA, vs the
normalized frequency $\omega/\kappa_{m}$ for the different detunings in
optimized cooperativity $C^{\prime}=1$. The gray dotted, thin cyan solid,
brown dashed, and thick purple solid curves, respectively, referred to
different detunings $\Delta=0,0.05\kappa_{m},0.5\kappa_{m},5\kappa_{m}$. Here,
we have assumed $\bar{n}\simeq 0$, corresponding to the zero temperature.

To follow the influences of cooperativity on signal amplification and noise
reduction, by selecting the optimized detuning, $\Delta=0$, and considering
zero temperature, $\bar{n}=0$, in Fig. 6 we have plotted the magnonic response
of the cavity output phase spectrum $R^{\prime}_{m}(\omega)$ [Fig.6(a)] and
the added noise $n^{\prime}_{add}(\omega)$ [Fig. 6(b)] versus the normalized
frequency $\omega/\kappa_{m}$ for different values of cooperativity. As is
clearly evident and expected from Eqs. (36) and (37), the optimized value of
cooperativity is $C^{\prime}=1$, which simultaneously leads to the maximum
value of magnonic response, $R^{\prime}_{m}(\omega)=1$, and strong noise
suppression over a wide range of frequency $(\omega\leq\ 0.1\kappa_{m})$. For
smaller cooperativities like $C^{\prime}=0.5$, we can achieve the signal
response near to its maximum, but the noise suppression decreases with respect
to the optimized cooperativity. Moreover, for higher cooperativity than its
optimized value, for example, $C^{\prime}=100$, simultaneous signal
amplification and noise suppression is impossible in a moderate range of
frequencies except for a very short range. The situation is worse for very
high cooperativities, for example, $C^{\prime}=1000$.

To see the effect of temperature on added noise suppression under the RWA, in
Fig. 7 we have plotted $n^{\prime}_{\mathrm{add}}(\omega)$ versus the
normalized frequency $\omega/\kappa_{m}$ for zero and room temperature.
Similar to the results obtained beyond the RWA, by choosing the optimized
parameters ($\Delta=0$, and $C^{\prime}=1$) the noise suppression is still
possible even at room temperature and over a pretty wide range of frequency
($\omega\sim 0.1\kappa_{m}$) in on- and off-resonance frequencies.

To conclude, under the RWA, no signal amplification occurs, i.e.,
${R^{\prime}}_{m}^{\rm max}=1$. However, in the optimized system parameters
($\Delta=0$ and $C^{\prime}=1$), the noise suppression below the SQL is still
possible in a pretty wide range $\omega\sim 0.1\kappa_{m}$ at room
temperature. Therefore, one can infer that the system beyond the RWA is the
optimized case for operational the proposed magnetic sensor. To illustrate
this, in the next subsection we compare the signal response and noise
suppression of the both cases (beyond and under the RWA) for the fixed
optimized achievable parameters.

Figure 6: (Color online)(a) The magnonic response to the input signal,
$R^{\prime}_{m}(\omega)$, and (b) the added noise of magnetic field
measurement , $n^{\prime}_{\mathcal{add}}(\omega)$ under the RWA, vs the
normalized frequency $\omega/\kappa_{m}$ for the different cooperativities.
All curves have been plotted for optimized zero detuning, $\Delta=0$. The thin
blue solid, magenta dashed, thick red solid, gray dotted, and green solid
curves are respectively referred to $C^{\prime}=1000,100,10,1,0.5$. Here, we
have assumed $\bar{n}\simeq 0$, corresponding to the zero temperature. Figure
7: (Color online)The effect of thermal noise, temperature, on the added noise
of magnetic field measurement under the RWA. All curves have been plotted for
the zero detuning, $\Delta=0$. The magenta dashed and gray dotted curves are
referred to zero temperature, $\bar{n}=0$, respectively for $C^{\prime}=100$
and $C^{\prime}=1$, while the magenta and gray solid curves are referred to
room temperature corresponding to $\bar{n}=793$ respectively for
$C^{\prime}=100$ and $C^{\prime}=1$. For obtaining thermal noise, we have
considered $\omega_{a(m)}/2\pi=7.875~{}\rm{GHz}$ as an experimental reported
bare frequencies in the coupled system under the RWA 17 .

###  IV.3  Optimized operational regime for the proposed sensor: Comparison
between system beyond the RWA and under the RWA

First, compare Figs. 3 and 6, which show the results respectively beyond the
RWA and under the RWA at zero temperature and optimized zero detuning. Beyond
the RWA, all values of cooperativity can lead to simultaneous signal
amplification and noise suppression in a wide frequency range
($\omega<\kappa_{m}$), while under the RWA, only by selecting $C^{\prime}=1$
in a short frequency range ($\omega<0.1\kappa_{m}$) can one achieve the strong
noise suppression and the maximum value of magnonic response,
${R^{\prime}}_{m}^{\rm max}=1$, which means no signal amplification occurs, in
contrast to the obtained results beyond the RWA. This can be interpreted by
the role of counter-rotating terms (CRTs) of the microwave photon–magnon
interaction Hamiltonian, which are omitted under the RWA. This can be seen as
symmetry breaking in the interaction term. Thus, CRTs are responsible for high
signal amplification of the system beyond the RWA, which yields to precise
measurement. In fact, under the RWA, one magnon’s worth of input external
magnetic field produces at most one photon’s worth of output light. Therefore,
in this case although the quantum noise can be suppressed below the SQL, the
response cannot be amplified and the system provides only transduction of the
external magnetic field 3 ; 5 ; Review of force sensing . In contrast, beyond
the RWA, i.e., in the presence of the nonlinear terms, one magnon’s worth of
input external magnetic field can produce more than one photon’s worth of
output light. Consequently, not only can the system suppress the quantum noise
below the SQL, but it can behave as a quantum linear amplifier with
controllable gain 3 ; 5 ; Review of force sensing .

To better compare, in Fig. 8 we have plotted the ratio of the magnonic
response beyond the RWA to the corresponding one under the RWA,
$\mathcal{R}(\omega)=\frac{R_{m}(\omega)}{R^{\prime}_{m}(\omega)}$ [Fig. 8(a)]
as well as the ratio of their added noise,
$\mathcal{N}(\omega)=\frac{N(\omega)}{N^{\prime}(\omega)}$ [Fig. 8(b)], versus
the normalized frequency, $\omega/\kappa_{m}$, for different ratios of
cooperativities, $C/C^{\prime}=10,1,0.1$. Here, we have considered
$N(\omega)=[(\bar{n}+\frac{1}{2})+n_{\mathrm{add}}(\omega)]$ and
$N^{\prime}(\omega)=[(\bar{n}+\frac{1}{2})+n^{\prime}_{\mathcal{add}}(\omega)]$
as the induced total noise to the magnetic field measurement beyond the RWA
and under the RWA, respectively. For true comparison, we have set parameters
for zero temperature and optimized zero detuning in both cases while we have
changed cooperativities in their optimized values range such that the
cooperativities ratio can be tuned from small to large values ($0.1$ to $10$).
We note that the optimum cooperativity under the RWA is $C^{\prime}=1$, and
then is always fixed and cannot be changed for plotted curves in Fig. 8.
However, as has been shown, beyond the RWA one can set all values for the
cooperativity since for all values simultaneous signal amplification and noise
suppression are possible. Thus, we have considered three different
cooperativities set as ($C^{\prime}=1,C=10$), ($C^{\prime}=1,C=1$), and
($C^{\prime}=1,C=0.1$).

As illustrated in Fig. 8(a), the magnonic response to the external input in
the optimized parameters set for the system beyond the RWA is always larger
than the system under the RWA, notably for higher ratio of cooperativities,
$C/C^{\prime}$. Also, higher cooperativity ratios correspond to the better
operation of the sensor, i.e., more signal amplification for the system beyond
the RWA compared to that under the RWA. Another advantage of the system beyond
the RWA is its higher response in off resonance frequencies. It means that in
far off-resonance, for example, $\omega>0.1\kappa_{m}$ for considered
parameters in Fig. (8), the proposed sensor beyond the RWA can work better
than the sensor under the RWA. On the other hand, one can conclude that the
room temperature operation of the proposed sensor beyond the RWA is much
better than the RWA especially for far off-resonance frequencies. (Although it
is not shown here, the results of comparison at room temperature are the same
as the obtained results at zero temperature.)

Figure 8(b) shows the ratio of added noises for different values of
cooperativity ratio. As is evident, $\mathcal{N}(\omega)=1$ for frequency
range $\omega\leq\ 0.1\kappa_{m}$ which can be considered as near on-resonance
range. It means that for near on-resonance frequencies the noise suppression
in both cases is approximately the same. [Note that near the resonance the
added noise of the sensor under the RWA in the optimized parameters zero
detuning and $C^{\prime}=1$ is exactly zero; see Eq. 37.] But, in off-
resonance range $\omega>0.1\kappa_{m}$, the added noise ratio
$\mathcal{N}(\omega)$ decreases which means that for off-resonance range noise
suppression of the sensor beyond the RWA is stronger than the sensor under the
RWA.

Finally, we conclude that the operation of our proposed sensor beyond the RWA
is more appropriate for on-resonance and especially off-resonance quantum
magnetometry in a wide frequency range. Furthermore, it should be mentioned
that the results for negative detuning are exactly the same as presented
results for positive detuning.

Figure 8: (Color online)(a) The ratio of the magnonic response,
$\mathcal{R}(\omega)=R_{m}(\omega)/R^{\prime}_{m}(\omega)$, and (b) the noise
of magnetic measurement ratio,
$\mathcal{N}(\omega)=N(\omega)/N^{\prime}(\omega)$, versus the normalized
frequency, $\omega/\kappa_{m}$. The dark blue-solid-thick, red-dashed, and
dark green-solid-thin curves respectively correspond to cooperativities ratios
$C/C^{\prime}=0.1,1.10$. Here, we have considered the optimized zero detuning
and also zero-temperature for both cases. All parameters have been chosen so
the system to be in the optimized sensing regime where the added nosies are
suppressed and signal responses are amplified.

##  V  SENSITIVITY, SIGNAL-TO-NOISE RATIO, and experimental discussion

In this section, we calculate the SNR and also the sensitivity of the system.
Beyond the RWA, we can write the output phase quadrature of the cavity field
as

$\displaystyle P^{\mathrm{out}}_{a}(\omega)$ $\displaystyle=$ $\displaystyle
P^{\mathrm{out}}_{a}(\omega)|_{B=0}+i\mathcal{A}(\omega)\tilde{B}_{1}(\omega)\eta\sqrt{\frac{2}{\kappa_{m}}}$
(38)
$\displaystyle+\mathcal{B}(\omega)\tilde{B}_{2}(\omega)\eta\sqrt{\frac{2}{\kappa_{m}}},\
$

where

$\displaystyle P^{\mathrm{out}}_{a}(\omega)|_{B=0}$ $\displaystyle=$
$\displaystyle\mathcal{A}(\omega)X^{\mathrm{in}}_{m}(\omega)+\mathcal{B}(\omega)P^{\mathrm{in}}_{m}(\omega)+\mathcal{C}(\omega)X^{\mathrm{in}}_{a}(\omega)$
(39) $\displaystyle+\mathcal{D}(\omega)P^{\mathrm{in}}_{a}(\omega)$

is the contribution of the output phase quadrature in the absence of the
external magnetic field (that we are going to sense). $\tilde{B}_{1}(\omega)$
and $\tilde{B}_{2}(\omega)$ are the contribution of the external magnetic
field and are defined as follows

$\displaystyle\tilde{B}_{1}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2}[B(\omega+\omega_{d})-B(\omega-\omega_{d})],$ (40)
$\displaystyle\tilde{B}_{2}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2}[B(\omega+\omega_{d})+B(\omega-\omega_{d})].$ (41)

In order to calculate the sensitivity of the device to the external magnetic
field and the SNR, we follow the method used in Ref. 5 . To this end, we
define the magnetic field operators as

$\displaystyle\delta\tilde{B}_{1}(\omega)$ $\displaystyle=$
$\displaystyle\frac{P^{\mathrm{out}}_{a}(\omega)}{\partial
P^{\mathrm{out}}_{a}(\omega)/\partial\tilde{B}_{1}(\omega)}=\delta\hat{N}_{1}+\tilde{B}_{1}(\omega),$
(42) $\displaystyle\delta\tilde{B}_{2}(\omega)$ $\displaystyle=$
$\displaystyle\frac{P^{\mathrm{out}}_{a}(\omega)}{\partial
P^{\mathrm{out}}_{a}(\omega)/\partial\tilde{B}_{2}(\omega)}=\delta\hat{N}_{2}+\tilde{B}_{2}(\omega),$
(43)

in which $\delta\hat{N}_{1}(\omega)$ and $\delta\hat{N}_{2}(\omega)$ are the
noise operators and affect the system through two different channels
$\hat{X}^{\mathrm{in}}_{m}(\omega)$ and $\hat{P}^{\mathrm{in}}_{m}(\omega)$
independently and are defined as follows:

$\displaystyle\delta\hat{N}_{1}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(\omega)\hat{X}^{\mathrm{in}}_{m}+\mathcal{B}(\omega)\hat{P}^{\mathrm{in}}_{m}+\mathcal{C}(\omega)\hat{X}^{\mathrm{in}}_{a}+\mathcal{D}(\omega)\hat{P}^{\mathrm{in}}_{a}}{i\mathcal{A}(\omega)\eta\sqrt{\frac{2}{\kappa_{m}}}},$
(44) $\displaystyle\delta\hat{N}_{2}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(\omega)\hat{X}^{\mathrm{in}}_{m}+\mathcal{B}(\omega)\hat{P}^{\mathrm{in}}_{m}+\mathcal{C}(\omega)\hat{X}^{\mathrm{in}}_{a}+\mathcal{D}(\omega)\hat{P}^{\mathrm{in}}_{a}}{\mathcal{B}(\omega)\eta\sqrt{\frac{2}{\kappa_{m}}}}.$
(45)

Considering the probability amplitudes of the noise injection through
different channels, the total noise operator can be written as follows

$\delta\hat{N}(\omega)=P_{1}\delta\hat{N}_{1}(\omega)+P_{2}\delta\hat{N}_{2}(\omega),$
(46)

where the probability amplitudes of noise injection ($P_{1}$ and $P_{2}$ )
through $\hat{X}^{\mathrm{in}}_{m}$ and $\hat{P}^{\mathrm{in}}_{m}$ are given
by

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(\omega)}{\sqrt{|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}}},$
(47) $\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}(\omega)}{\sqrt{|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}}}.$
(48)

Therefore, the total noise operator is obtained as follows:

$\displaystyle\delta\hat{N}(\omega)$ $\displaystyle=$
$\displaystyle\frac{(1-i)}{\eta}\sqrt{\frac{\kappa_{m}}{2}}$
$\displaystyle(\frac{\mathcal{A}(\omega)\hat{X}^{\mathrm{in}}_{m}+\mathcal{B}(\omega)\hat{P}^{\mathrm{in}}_{m}+\mathcal{C}(\omega)\hat{X}^{\mathrm{in}}_{a}+\mathcal{D}(\omega)\hat{P}^{\mathrm{in}}_{a}}{\sqrt{|\mathcal{A}(\omega)|^{2}+|\mathcal{B}(\omega)|^{2}}}),$

which yields to noise power spectrum as

$\displaystyle
S_{N}(\omega)=\frac{1}{2}\langle\delta\hat{N}(\omega)\delta\hat{N}(\omega)^{\dagger}\rangle,$
(50)

which can be simplified as

$S_{N}(\omega)=\frac{\kappa_{m}}{\eta^{2}}[(\bar{n}_{m}+\frac{1}{2})+n_{\mathrm{add}}(\omega)].$
(51)

SNR, the ratio of the signal to the variance of the noise, can be written as

$r(\omega)=\frac{|\tilde{B}(\omega)|}{\sqrt{S_{N}(\omega)}}=\frac{|\tilde{B}(\omega)|}{\frac{\sqrt{\kappa_{m}}}{\eta}[(\bar{n}_{m}+\frac{1}{2})+n_{\mathrm{add}}(\omega)]^{\frac{1}{2}}}.$
(52)

The sensitivity or the minimum detectable input of the device is the minimum
magnitude of the input signal required to produce an output with
$r(\omega)=1$. Therefore, sensitivity of the system beyond the RWA,
$\mathcal{S}(\omega)=\sqrt{S_{N}(\omega)}$, is simplified as

$\mathcal{S}(\omega)^{\rm
bRWA}=\frac{\sqrt{\kappa_{m}}}{\eta}[(\bar{n}_{m}+\frac{1}{2})+n_{\mathrm{add}}(\omega)]^{\frac{1}{2}}.$
(53)

The feasible experimental parameters beyond the RWA 17  are
$\omega_{m}/2\pi\simeq\omega_{a}/2\pi=37.5~{}\rm GHz$, $g_{0}/2\pi=2.5~{}\rm
GHz$, $\kappa_{m}/2\pi=15~{}\rm MHz$, and $\kappa_{a}/2\pi=33~{}\rm MHz$. This
coupled microwave photon–magnon system that yields to a ratio of
$g_{0}/\omega_{a(m)}=0.067$ has been realized by placing a YIG sphere (which
spin and orbital angular momentum of its unit cell are respectively
$s=\frac{5}{2}$ and $l=0$ 26 ) with diameter of $2.5~{}\rm mm$ inside a
microwave cavity with dimension $(7.0\times 5.0\times 3.2)~{}\rm mm^{3}$ and
with a bias magnetic field as $B_{b}=1.34~{}\rm{T}$, which is exerted to the
YIG sphere. Therefore, the obtained electromagnonic cooperativity is
$C=\frac{4g^{2}}{\kappa_{a}\kappa_{m}}\simeq 5\mathcal{E}^{2}\times 10^{4}$ in
which $\mathcal{E}$ can be controlled so that the condition
$g=g_{0}\mathcal{E}\ll\ \Omega(\Omega=2\omega_{L})$ to be satisfied. It is
worth pointing out that in practice, for controlling the $\mathcal{E}$ and
therefore the magnon–photon coupling strength one can use a Josephson
parametric amplifier (JPA). By changing the flux bias through the SQUID loop
in the JPA, the inductance of the SQUID is modulated. Accordingly, the modes
in the resonator can be deformed. By placing a YIG sphere where the magnetic
field is maximum and controlling the modulation amplitude of magnetic field,
one can change $\mathcal{E}$ and hence $g$ so that $g\ll\ \Omega$ being valid.
In this manner, control of the cooperativity beyond the RWA is possible via
controlling the time modulation amplitude $\mathcal{E}$ which leads to the
different cooperativities considered in the figures in Sec. IV.1. The
gyromagnetic ratio is $\gamma/2\pi=28~{}\frac{\rm GHz}{\rm{T}}$ 17 , and the
spin density of YIG is $\rho_{s}=4.22\times 10^{27}~{}\rm m^{-3}$ which yield
the total number of spins $N=3.5\times 10^{19}$ 17 . Now, by considering these
feasible experimental parameters in the optimized detuning, $\Delta=0$, the
sensitivity of the theoretical proposed sensor near to on-resonance frequency
beyond the RWA corresponding to USC or DSC regime is obtained as
$\mathcal{S}_{0}^{\rm bRWA}=5.90\times 10^{-18}\frac{\rm{T}}{\sqrt{\rm Hz}}$
and $\mathcal{S}_{\rm room}^{\rm bRWA}=1.08\times
10^{-16}\frac{\rm{T}}{\sqrt{\rm Hz}}$, respectively, at zero and room
temperature. Moreover, by considering the feasible experimental frequency,
$\omega_{a(m)}$, the thermal noises at zero and room temperature are
respectively $\bar{n}_{0}^{\rm bRWA}=0$ and $\bar{n}_{\rm room}^{\rm
bRWA}=166$. As is evident from Eq. (53), $\sqrt{\kappa_{m}}/\eta$ (with
$\eta:=\frac{\gamma}{2}\sqrt{5N}$) plays the key role for determination of the
sensitivity in the optimized condition. Thus, by increasing the number of
spins in the YIG sphere through increasing its diameter during the fabrication
process, and decreasing the magnon decay rate, $\kappa_{m}$, one may achieve
to a more sensitive magnetometer beyond the RWA.

Following the same procedure, one can obtain the sensitivity of our sensor
under the RWA as follows:

$\mathcal{S}^{\rm{RWA}}(\omega)=\frac{\sqrt{\kappa_{m}}}{\eta}[(\bar{n}_{m}+\frac{1}{2})+n^{\prime}_{\mathrm{add}}(\omega)]^{\frac{1}{2}}.$
(54)

The experimental parameters corresponding to the RWA condition ($g_{0}\ll\
\omega_{a(m)}$) are $\omega_{m}/2\pi\simeq\omega_{a}/2\pi=7.875\rm GHz$,
$g_{0}/2\pi=3.1\rm MHz$, $\kappa_{m}/2\pi=19\rm MHz$ and
$\kappa_{a}/2\pi=2.09\rm MHz$ which has been realized by placing a YIG sphere
with diameter of $0.36\rm mm$ inside a microwave cavity with an inner
dimension $(43.0\times 21.0\times 9.6)\rm{mm}^{3}$ and also applying a bias
magnetic field $B_{b}=281\rm{mT}$ to the YIG sphere 17 . These experimental
parameters lead to cooperativity near to the optimized value for magnetic
sensing under the RWA,
$C^{\prime}=\frac{4g_{0}^{2}}{\kappa_{m}\kappa_{a}}\big{|}^{\rm{RWA}}=0.97\simeq
1$. Here, by a $0.36\rm mm$ diameter YIG sphere, the total number of spins is
$N=1.031\times 10^{17}$ 17 , which leads to a smaller coupling strenth,
$g_{0}/2\pi=3.1\rm MHz$, compared to the previous example with a $2.5mm$
diameter YIG sphere leading to the total spin number and coupling strength
respectively as $N=3.5\times 10^{19}$ and $g_{0}/2\pi=2.5\rm GHz$. Therefore,
the sensitivity of our theoretical proposed magnetometer under the RWA near to
on-resonance frequency is obtained as $\mathcal{S}_{0}^{\rm{RWA}}=1.22\times
10^{-16}\frac{\rm{T}}{\sqrt{\rm Hz}}$ and $\mathcal{S}_{\rm
room}^{\rm{RWA}}=4.87\times 10^{-15}\frac{\rm{T}}{\sqrt{\rm Hz}}$ at zero and
room temperature, respectively. Under the RWA, the thermal noises
corresponding to the zero and room temperature are, respectively,
$\bar{n}_{0}^{\rm{RWA}}=0$ and $\bar{n}_{\rm{room}}^{\rm{RWA}}=793$. Similar
to the sensor beyond the RWA, increasing the number of spins in the YIG, $N$,
by choosing a bigger sphere and decreasing the magnon decay rate,
$\kappa_{m}$, lead to more sensitivity.

Note that the bias magnetic field $B_{b}$ is usually a static strong
background field, for example, $B_{b}=1.34\rm{T}$ and $B_{b}=281\rm{mT}$
respectively for the case of beyond and under the RWA, while the signal
magnetic field that we would like to detect is a time-dependent magnetic
field. Hence, this strong background can be distinguished and filtered using
some experimental filtering technique such as Kalman filtering, which is a
well-known technique in engineering and recently has been developed to be used
in force sensing in cavity optomechanics kalman filtering . For example,
similar to our proposed scheme, there is a strong static bias field
$B_{b}=0.4~{}\rm T$ in an experimental cavity magnon polariton-based
magnetometer in Ref. phasemodulatedmagnetometry . Nevertheless, the
possibility of weak magnetic field measurement by this magnetometer with a
sensitivity of the order of $2.0\frac{\rm{pT}}{\sqrt{\rm Hz}}$ has been
reported phasemodulatedmagnetometry  by using the background canceling. Thus,
this scheme and the used experimental techniques to distinguish the signal and
background can guarantee that our proposed theoretical scheme with a similar
strong bias magnetic field $B_{b}=1.34\rm T$ ($B_{b}=281\rm{mT}$) can
similarly measure weak magnetic field with sensitivity in the order of
$1.08\times 10^{-16}\frac{\rm T}{\sqrt{\rm Hz}}$ ($4.87\times
10^{-15}\frac{\rm T}{\sqrt{\rm{Hz}}}$) and $5.90\times 10^{-18}\frac{\rm
T}{\sqrt{\rm{Hz}}}$ ($1.22\times 10^{-16}\frac{\rm T}{\sqrt{\rm{Hz}}}$) beyond
(under) the RWA, at room and zero temperature, respectively. Note that
although the high sensitivities of our theoretical proposed magnetometer are
obtained based on the reported experimental parameters 17 , it is still an
ideal result which needs to be proved and verified experimentally.

For different set of experimental parameters, the sensitivity of the sensor
beyond the RWA is much more than that under the RWA. But, it might be
mentioned that by choosing a YIG sphere with the same size considered for the
system beyond the RWA or even bigger, one can obtain the sensitivity in the
order of the obtained sensitivity beyond the RWA but still a little smaller.
Here, the important point is that for the same sensitivity in both cases, the
sensor beyond the RWA is still much better for quantum magnetic sensing since
it not only suppresses the noise but also amplifies the magnetic-signal
response as an quantum amplifier. Therefore, one can conclude that the optimum
regime for operation of the proposed sensor is the USC or DSC regime
corresponding to system beyond the RWA.

As has been explained in theoretical results of Sec. IV, in this optimized
regime quantum noise suppression below the SQL
($n_{\mathrm{add}}^{\mathrm{SQL}}=\frac{1}{2}$), which is required for precise
magnetometry, is possible even by weak or intermediate cooperativities. But,
according to Eq. (53), by suppressing the quantum noise below the SQL, what
plays the important role for the sensitivity is the factor
$\frac{\sqrt{\kappa_{m}}}{\eta}$. It implies a low magnon mode dissipation
rate and a high total spin number. Since the magnon-photon coupling strength
is as $g=\frac{\gamma B_{0}}{2}\sqrt{2Ns}$, it yields a larger coupling
strength $g$, which is necessary for the ultraprecise magnetic field
measurement. As is evident, both decrease of the magnon mode decay rate and
increase of the total spin number lead to higher cooperativity. To clarify
this dependency on the cooperativity, by considering $g=\frac{\gamma
B_{0}}{2}\sqrt{2Ns}$ and also $\eta=\frac{\gamma}{2}\sqrt{2Ns}$, one can
easily rewrite the factor $\frac{\sqrt{\kappa_{m}}}{\eta}$ as

$\displaystyle\frac{\sqrt{\kappa_{m}}}{\eta}=\frac{2B_{0}}{\sqrt{C\kappa_{a}}},$
(55)

in which $B_{0}$ indicates the magnetic field amplitude of the electromagnetic
field. On the other hand, cavity enhancement, which is proportional to the
increase of the coupling strength $g$ and decrease of the cavity and magnon
mode dissipation rates $\kappa_{a}$, and $\kappa_{m}$, is analogous to the
increase of cooperativity $C=\frac{4g^{2}}{\kappa_{a}\kappa_{m}}$. Thereby,
although the quantum noise suppression is possible even in the weak or
moderate cooperativity regimes, the strong cooperativity regime or cavity
enhancement is necessary for ultraprecise magnetometry. Surprisingly, these
conditions, i.e., ultradense spin ensemble and cavity enhancement, which yield
ultrahigh sensitivity, currently have been experimentally realized in Ref 17 ,
and thus it guarantees our estimate of ultrahigh sensitivity in our proposed
magnetometer since we have used these realized experimental values.

Consequently, the ultrahigh sensitivity originates from both ultradense spin
ensemble and cavity enhancement. Let us discuss the feasibility of the
proposed method in the integrated systems such as coplanar microwave systems.
As we know, ultrahigh cooperativity of electromagnonic interaction based on
coplanar microwave waveguides has not been reported yet, but finding or
fabricating a ferromagnetic material with an ultradense spin number and low
magnon dissipation rate can lead to a cavity-enhanced and consequently an on-
chip integrated ultraprecise magnetometer.

Let us explain more. In addition to microwave cavities, which can provide
photon-magnon interaction conditions as has been reported in Refs. 16 ; 17 ,
according to some experimental results reported in Refs. Highcooperativity ;
CPW 2 ; CPW 3 ; CPW 4 , the coplanar microwave waveguides can also provide a
platform for magnon-photon interaction. Magnon-photon interaction in these
systems is possible not only in the strong coupling regime CPW 2 ; CPW 3 ; CPW
4 , but also (regarding the experimental results of Ref. Highcooperativity )
in the ultrastrong coupling regime, where, here, we have shown is necessary
for simultaneous signal amplification and noise reduction in order to achieve
a highly sensitive magnetometer. However, there are some limitations and
challenges with this kind of resonator. The coplanar waveguides used for
magnon-photon interaction are mostly made from superconducting materials that
work at low temperatures of the order of $\mathrm{mK}$–$\mathrm{K}$, and so
there is need for cryogenic cooling Highcooperativity ; CPW 2 ; CPW 3 ; CPW 4
. Therefore, these waveguides cannot be used for precise measurement at room
temperature, while our proposed system can work precisely at room temperature.
Moreover, the ferromagnetic materials used for magnon-photon interaction in
coplanar microwave waveguides have a small size, yielding a smaller total spin
number, for example, about three orders smaller Highcooperativity  than our
proposed system, which leads to less cavity enhancement and limits
ultraprecise magnetometry.

For example, the experimental results of Ref. Highcooperativity  show the
possibility of magnon-photon interaction using a ferromagnet yttrium iron
garnet (YIG) doped with gallium (YIG: Ga) on a superconducting coplanar
microwave resonator made from Nb. In this case, the resonator structure is
patterned into a $100~{}\mathrm{nm}$-thick Nb film deposited onto an intrinsic
silicon substrate using photolithography and reactive ion etching. The volume
of the commercial YIG: Ga crystal, which is placed in the center of the
resonator with a frequency of $\omega_{a}/2\pi=5.90~{}\rm{GHz}$ is around
$0.7~{}\mathrm{mm^{3}}$. The total spin number coupled to the resonator is
around $N\simeq 4.5\times 10^{16}$, which is about three orders of magnitude
smaller than our proposed macroscopic system, which has a total spin number of
about $N=3.5\times 10^{19}$. By considering the total spin number of
$N=4.5\times 10^{16}$ in waveguide systems, the magnon-photon coupling
strength in this integrated system is around $g/2\pi=450~{}\rm{MHz}$ and
magnon dissipation and resonator decay rates are respectively as
$\kappa_{m}/2\pi=50~{}\rm{MHz}$ and $\kappa_{a}/2\pi=3~{}\rm{MHz}$, which all
together lead to a very large cooperativity as
$C=\frac{4g^{2}}{\kappa_{a}\kappa_{m}}=5400$. Although it has not been
mentioned in Ref. Highcooperativity , by considering
$\frac{g}{\omega_{a(m)}}\simeq 0.08$, one says that these experimental results
that have been obtained at millikelvin temperatures show the ultrastrong
coupling regime of magnon-photon interaction, which based on our proposal is
the necessary regime for ultrasensitive magnetometry. In this regime, which
guarantees the simultaneous signal amplification and noise reduction, by
considering the optimized detuning, then the sensitivity of the integrated
microwave system with a cooperativity of about $C=5400$ and a factor
$\frac{\sqrt{\kappa_{m}}}{\eta}\simeq 4.25\times 10^{-16}\frac{\rm
T}{\sqrt{\rm Hz}}$ is of the order of $3.01\times 10^{-16}\frac{\rm
T}{\sqrt{\rm Hz}}$ near to that of on-resonance frequency at millikelvin
temperature. In comparison, based on the reported experimental results of Ref.
17 , our proposed macroscopic magnetometer including a microwave cavity with a
dimension of $(7.0\times 5.0\times 3.2)~{}\mathrm{mm^{3}}$ and a frequency of
$\frac{\omega_{a}}{2\pi}=37.5~{}\rm{GHz}$ and also containing a
$8.2~{}\mathrm{mm^{3}}$ YIG sphere with total spin number of $N=3.5\times
10^{19}$ has photon-magnon coupling strength and magnon and cavity mode
dissipation rates respectively of $g/2\pi=2.5\rm{GHz}$,
$\kappa_{m}/2\pi=15\rm{MHz}$ and $\kappa_{a}/2\pi=33\rm{MHz}$ , which all lead
to a cooperativity of $C\simeq 50000$, which shows a large cavity enhancement
that has been experimentally realized 17 . By considering the ultrastrong
coupling regime beyond the RWA, the ratio in our case is
$\frac{\sqrt{\kappa_{m}}}{\eta}\simeq 8.34\times 10^{-18}\frac{\rm
T}{\sqrt{\rm Hz}}$ and then the sensitivity in our case is obtained of the
order of $5.90\times 10^{-18}\frac{\rm T}{\sqrt{\rm{Hz}}}$ at nearly zero
temperature and in on-resonance frequency. As is clear, our magnetometer would
be two orders of magnitude more sensitive than the case of coplanar microwave
waveguide. Comparing the magnon mode and cavity decay rates, which has the
same order in these two systems, one can conclude that the main reason of high
difference between cooperativities and also magnetic field sensitivity in
these two systems is the total spin number. The total spin number in the
microwave cavity is about three orders of magnitude larger than the coplanar
waveguides, which leads to a higher cooperativity and hence more sensitive
magnetometer. Therefore, we emphasize that precise magnetic field measurement
requires a small magnon mode decay rate and large total spin number, which
both lead to larger cooperativity. Nevertheless, as we know, although a large
factor of $\frac{\sqrt{\kappa_{m}}}{\eta}$ and hence high cooperativity or
significant cavity enhancement has not been achieved yet for magnon-photon
interaction based on coplanar microwave waveguides, but finding or fabricating
a ferromagnetic material with high spin density and low magnon mode
dissipation rate for overcoming these limitations and accessing on-chip
integrated precise magnetometer can be promising.

Let us now compare our proposed magnetometer with other ones, especially
current state-of-the-art commercial magnetometers such as SQUIDs and atomic-
based magnetometers. The SQUIDs and atomic-based magnetometers have
sensitivities in the order of $(0.1$–$1)~{}{\rm fT}/\sqrt{\rm Hz}$ 12 ; 13 ;
14 ; five . Although they are so sensitive, there are two important challenges
with them 13 ; 15 : SQUID-based magnetometers cannot work sensitively at room
temperature. In fact, they need cryogenic systems in order to be cooled
because as the temperature increases their sensitivity decreases 15 . On the
other hand, the sensitivity of the atomic-based magnetometers decreases at
lower frequencies 13 . This means that they can work well at high frequencies
and, thus, they cannot work over a wide range of frequencies. Besides, it has
been shown 10  that the quantum optomechanical systems with no cooling or
cryogenic system can be engineered to operate as a sensitive magnetometer in a
wide range of frequencies $2~{}\rm{Hz}$-$1~{}\rm{kHz}$ with sensitivity in the
order of $150~{}{\rm nT}/\sqrt{\rm Hz}$ 10 , which is appropriate for some
significant applications including magnetic anomaly detection 40 , geophysical
surveys 41 , and magnetoencephalography 42 . Also, the sensitivities of the
order of $200$, $131$ and $26~{}{\rm pT}/\sqrt{\rm Hz}$ respectively at
frequencies about $17\rm MHz$, $127\rm kHz$ and $\rm 11MHz$ were achievable in
optomechanic-based magnetometers 10 ; four ; highestsensitive optomechanics .
Although the proposed optomechanical-based magnetometers can beat the
mentioned challenges 10 ; four , their sensitivity is still less than the
SQUIDs and atomic-based magnetometers. In addition to the cavity
optomechanical systems, recently, some precise magnetometers based on cavity
magnon–polariton have been proposed one ; 8 . It has been shown that they have
the capability to achieve sensitivity on the order of $1~{}{\rm fT}/\sqrt{\rm
Hz}$ at room temperature 8 . Moreover, the sensitivity in the order of
$2.4~{}{\rm pT}/\sqrt{\rm Hz}$ in some magnetometers based on cavity
magnon–polariton has been estimated one . Beside these systems, phase
estimation algorithm-based magnetometer with sensitivity of the order of
$20.7~{}\rm{pT}/\sqrt{\rm Hz}$ has also been reported in Ref. magnetometry
Sorin .

It is also worth comparing the sizes of different magnetometers. Our proposed
cavity electromagnonic magnetometer, with its requirement of high spin
density, has a large size, especially compared to other magnetometers such as
SQUIDs and cavity optomechanics, which have sizes in 1D respectively on the
order of $1$ size of SQUID  and $10~{}\rm{\mu m}$ 10  . Although our proposed
magnetometer has a larger size in 1D, on the order of $1$–$10~{}mm$, it has
some advantages such as higher sensitivity (on the order of $5.90\times
10^{-18}\frac{\rm T}{\mathrm{\sqrt{Hz}}}$) than the other devices and its
ability to work at room temperature and in a wider frequency band (up to
$\rm{MHz}$). These can motivate their use in some practical applications. In
other words, there is a trade-off between the higher sensitivity with the
larger size in this scheme and the integrated size with smaller sensitivities
in other devices, i.e., higher sensitivity at the cost of larger size. It is
worth emphasizing that to avoid and decrease the large size in our system, a
ferromagnetic material with higher spin density and lower magnon decay rate
should be found or fabricated, which is a challenging issue. Although our
proposed magnetometer has a relatively large size with a volume of about
$112~{}\rm{mm^{3}}$ and sensitivity of the order of $5.90\times
10^{-18}\frac{\rm T}{\mathrm{\sqrt{Hz}}}$, based on the reported results of
Ref. 13 , it is still around four times smaller and 27 times more sensitive
than the current commercial atomic-based magnetometers, such as one of the
most popular precise magnetometers, with volume and sensitivity respectively
on the order of $0.45~{}\rm{cm^{3}}$and $160\times 10^{-18}\frac{\rm
T}{\sqrt{\rm Hz}}$.

Surprisingly, although it has a relatively large size, according to the
theoretical estimated sensitivities, our electromagnetic-based magnetic
amplifier sensor, notably beyond the RWA, can operate as a very high-sensitive
magnetometer with high sensitivity in the order of
$10^{-18}~{}\rm{T}/\sqrt{\rm Hz}$ at zero temperature which is two or three
orders of magnitude better than the current SQUIDs and atomic-based
magnetometers. Note that at room temperature our sensor still operates with
high sensitivity of the order of $0.1~{}\mathrm{fT/\sqrt{\rm Hz}}$ which is in
the same order of the SQUIDs and atomic-based magnetometers. More
interestingly, our proposed magnetic sensor can surpass the current challenges
to the SQUIDs and atomic systems because it can operate with very high
sensitivity at room temperature and also can work in a very wide range of
frequencies $\omega\leq\ \kappa_{m}\sim 20\rm MHz$ without cooling compared to
the optomechanics, SQUIDs, and atomic systems. This proposed magnetometer is
even more sensitive than the recent proposed ones based on cavity
magnon–polariton and also can work in a wider frequency range. In fact, by
considering the available facilities, there is a trade-off between the higher
sensitivity in a wider frequency range with the larger size in the present
scheme and the miniaturized sized with smaller sensitivities in other magnetic
field sensors. Therefore, these features of the presented magnetic sensor open
another platform for a commercial magnetometer as well as a device for
researching the fundamental physics issues like dark matter axions one ; two ;
three  in the future.

##  VI  SUMMARY, CONCLUSION, and outlooks

In the present contribution, a scheme of a magnetic sensor based on the cavity
electromagnonic has been proposed in which magnons and photons interact via
the magnetic dipole interaction. It has been shown that beyond the RWA by
choosing the suitable system parameters, one can simultaneously achieve signal
amplification and added noise suppression below the SQL in order to precise
magnetic measurement. The advantages of the proposed sensor is that our
magnetometer can work in a wide range of frequencies at the room temperature
with high sensitivity, which is even better than the current SQUIDs and
atomic-based magnetometers that need cooling and work at high-frequency
regimes, respectively. These properties means that the proposed sensor can be
competitive compared to the SQUIDs and atomic-based systems and might be
commercial in the future, or used for fundamental applications such as
searching the dark matter axion.

Instead of magnetic sensing using the output phase quadrature of cavity mode,
the single-quadrature magnetic sensing, one can use the generalized rotated
output cavity quadrature to exploit the ponderomotive squeezing in order to
enhance noise reduction and signal response amplification. Furthermore,
considering the squeezed vacuum injection instead of thermal noise for the
microwave mode may lead to optimization and more controllability in order to
achieve more noise suppression and signal amplification. Moreover, in real
applications, one needs to consider the classical laser phase noises as
presented in Ref. Review of force sensing . Finally, the physical
interpretation of the proposed scheme can be deeply analyzed using the
introduced approach of Green’s function in Ref. aliGreen .

##  VII AUTHOR CONTRIBUTIONS

AMF proposed and developed the primary idea of quantum magnetic-sensing and
MSE organized the idea in the cavity quantum electromagnonics. All
calculations have been performed and checked by MSE and AMF, respectively. All
authors contributed to prepare the manuscript. MSE and AMF had equal
contributions to write the manuscript. MSE and AMF revised and answered
equally to the referees’ comments during the referee process. All numerical
calculations and graphics have been done by MSE. MBH is group supervisor and
supervisor of MSE’s PhD thesis, and also AMF is advisor of MSE’s thesis.

##  VIII acknowledgments

The authors thank Prof. Yasunobu Nakamura and also Prof. David Vitali for
reading the manuscript and providing helpful suggestions. They would also like
to express their gratitude to the referees whose valuable comments have
substantially improved the article. A.M.F. wishes to thank the Office of
Graduate Studies of the University of Isfahan (OGSUI) and ICQTs for their
support.

##  APPENDIX: MAGNETOMETRY under the RWA

Under the RWA, the Hamiltonian of the system is given by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\omega_{a}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{m}\hat{m}^{\dagger}\hat{m}+\hbar
g_{0}(\hat{a}^{\dagger}\hat{m}+\hat{a}\hat{m}^{\dagger})$ (56)
$\displaystyle+i\hbar\epsilon_{L}(\hat{a}^{\dagger}e^{-i\omega_{L}t}-\hat{a}e^{i\omega_{L}t})+i\hbar\epsilon_{d}(\hat{m}^{\dagger}e^{-i\omega_{d}t}-\hat{m}e^{i\omega_{d}t})$
$\displaystyle-\hbar\eta B(t)(\hat{m}+\hat{m}^{\dagger}).$

By rewriting the above Hamiltonian in the frame rotating at frequencies
$\omega_{L}$ and $\omega_{d}$, we get

$\displaystyle\hat{H}^{(\mathrm{rot})}$ $\displaystyle=$
$\displaystyle\hbar\Delta(\hat{a}^{\dagger}\hat{a}+\hat{m}^{\dagger}\hat{m})+\hbar
g_{0}(\hat{a}^{\dagger}\hat{m}+\hat{a}\hat{m}^{\dagger})+i\hbar\epsilon_{L}(\hat{a}^{\dagger}-\hat{a})$
(57) $\displaystyle+i\hbar\epsilon_{d}(\hat{m}^{\dagger}-\hat{m})-\hbar\eta
B(t)(\hat{m}e^{-i\omega_{d}t}+\hat{m}^{\dagger}e^{i\omega_{d}t}),\ $

in which we have assumed $\omega_{L}=\omega_{d}$, and since
$\omega_{a}\simeq\omega_{m}$, thus $\Delta_{a}\simeq\Delta_{m}\equiv\Delta$.
Following the method used for the system beyond the RWA, one can obtain the
output phase quadrature under the RWA in the Fourier space as follows:

$\displaystyle\hat{P}^{\mathrm{out}}_{a}(\omega)$ $\displaystyle=$
$\displaystyle\mathcal{A}^{\prime}(\omega)\hat{X}^{\prime\mathrm{in}}_{m}(\omega)+\mathcal{B}^{\prime}(\omega)\hat{P}^{\prime\mathrm{in}}_{m}(\omega)$
(58)
$\displaystyle+\mathcal{C}^{\prime}(\omega)\hat{X}^{\mathrm{in}}_{a}(\omega)+\mathcal{D}^{\prime}(\omega)\hat{P}^{\mathrm{in}}_{a}(\omega)$

where $\hat{X}^{\prime\mathrm{in}}_{m}(\omega)$ and
$\hat{P}^{\prime\mathrm{in}}_{m}(\omega)$ are defined as in Eqs. (18) and (19)
and $\mathcal{A}^{\prime}(\omega)$, $\mathcal{B}^{\prime}(\omega)$,
$\mathcal{C}^{\prime}(\omega)$, and $\mathcal{D}^{\prime}(\omega)$ are defined
as

$\displaystyle\mathcal{A}^{\prime}(\omega)$ $\displaystyle=$
$\displaystyle\psi(\omega)\Bigg{\\{}\frac{g_{0}\Delta\chi_{m}(\omega)\chi_{a}(\omega)\sqrt{\kappa_{a}\kappa_{m}}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}-\frac{g_{0}\chi_{m}(\omega)\sqrt{\kappa_{a}\kappa_{m}}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{\\}},$
(59) $\displaystyle\mathcal{B}^{\prime}(\omega)$ $\displaystyle=$
$\displaystyle\psi(\omega)\Bigg{\\{}\frac{g_{0}\Delta\chi_{a}(\omega)\sqrt{\kappa_{a}}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}\Bigg{[}-g_{0}^{2}\chi_{m}^{3}(\omega)\chi_{a}(\omega)\sqrt{\kappa_{m}}\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}+\Delta\chi_{m}^{2}(\omega)\sqrt{\kappa_{m}}\Bigg{]}-g_{0}\chi_{m}(\omega)\chi_{a}(\omega)\sqrt{\kappa_{a}\kappa_{m}}$
(60) $\displaystyle(1+\Delta^{2}\chi_{m}^{2}(\omega)-\Delta
g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}-\frac{g_{0}\sqrt{\kappa_{a}}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}-g_{0}^{2}\chi_{m}^{3}(\omega)\chi_{a}(\omega)\sqrt{\kappa_{m}}\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}+\Delta\chi_{m}^{2}(\omega)\sqrt{\kappa_{m}}\Bigg{]}\Bigg{\\}},$
$\displaystyle\mathcal{C}^{\prime}(\omega)$ $\displaystyle=$
$\displaystyle\psi(\omega)\Bigg{[}-g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}^{2}(\omega)\Delta\kappa_{a}\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}-\kappa_{a}\chi_{a}(\omega)\Bigg{(}1+\Delta^{2}\chi_{m}^{2}(\omega)-\Delta
g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}\Bigg{)}$
(61)
$\displaystyle+g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)\kappa_{a}\Bigg{]},$
$\displaystyle\mathcal{D}^{\prime}(\omega)$ $\displaystyle=$
$\displaystyle\psi(\omega)\Bigg{\\{}\frac{-g_{0}^{2}\Delta^{2}\chi_{m}^{2}(\omega)\chi_{a}^{2}(\omega)\kappa_{a}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa}{2}\Bigg{)}\Bigg{]}+\frac{\kappa_{a}-\Delta\chi_{a}(\omega)\kappa_{a}}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}1+\Delta^{2}\chi_{m}^{2}(\omega)-\Delta
g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)\Bigg{(}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{)}\Bigg{]}$
(62)
$\displaystyle+\frac{\Delta\kappa_{a}g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{\\}}-1,$

where we have defined $\psi(\omega)$, $\chi_{a}(\omega)$, and
$\chi_{m}(\omega)$ as follows

$\displaystyle\psi(\omega)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}1+\Delta^{2}\chi_{m}^{2}(\omega)-\Delta
g_{0}^{2}\chi_{m}^{2}(\omega)\chi_{a}(\omega)\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}$
$\displaystyle-\frac{\Delta
g_{0}^{2}\chi_{m}(\omega)\chi_{a}(\omega)}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{[}1+\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}+\frac{g_{0}^{2}\chi_{m}(\omega)}{(i\omega+\frac{\kappa_{a}}{2})}\Bigg{\\}}^{-1},$
$\displaystyle\chi_{a}(\omega)$ $\displaystyle=$
$\displaystyle\Delta\Bigg{[}\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}^{2}+\Delta^{2}+g_{0}^{2}\chi_{m}(\omega)\Bigg{(}i\omega+\frac{\kappa_{a}}{2}\Bigg{)}\Bigg{]}^{-1},$
(64) $\displaystyle\chi_{m}(\omega)$ $\displaystyle=$
$\displaystyle\Bigg{[}i\omega+\frac{\kappa_{m}}{2}\Bigg{]}^{-1}.$ (65)

Therefore, analogous to Eq. (28), the spectrum of the optical output phase
quadrature can be obtained as follows:

$\displaystyle S_{P^{\mathrm{out}}_{a}}(\omega)$ $\displaystyle=$
$\displaystyle[|\mathcal{A}^{\prime}(\omega)|^{2}+|\mathcal{B}^{\prime}(\omega)|^{2}](\bar{n}_{m}+\frac{1}{2})$
(66)
$\displaystyle+[|\mathcal{C}^{\prime}(\omega)|^{2}+|\mathcal{D}^{\prime}(\omega)|^{2}](\bar{n}_{a}+\frac{1}{2}),$

which can be rewritten as

$S_{P^{out}_{a}}(\omega)=R_{m}^{\prime}(\omega)[(\bar{n}_{m}+\frac{1}{2})+n_{add}^{\prime}(\omega)],$
(67)

where

$\displaystyle
R_{m}^{\prime}(\omega)=[|\mathcal{A}^{\prime}(\omega)|^{2}+|\mathcal{B}^{\prime}(\omega)|^{2}]$
(68)

and

$\displaystyle
n_{add}^{\prime}(\omega)=(\bar{n}_{a}+\frac{1}{2})\frac{[|\mathcal{C}^{\prime}(\omega)|^{2}+|\mathcal{D}^{\prime}(\omega)|^{2}]}{[|\mathcal{A}^{\prime}(\omega)|^{2}+|\mathcal{B}^{\prime}(\omega)|^{2}]}$
(69)

are the magnonic response and the added noise, respectively. By choosing the
zero detuning for external drive fields and considering the resonance
frequency, one can obtain the magnonic response and the added noise as given
in the main text in Eqs. (36) and (37).

## References

  * (1) M. Tsang and C. M. Caves, “Coherent quantum-noise cancellation for optomechanical sensors,”Phys. Rev. Lett. 105, 123601 (2010).
  * (2) F. Acernese, M. Agathos, L. Aiello, A. Ain, A. Allocca, A. Amato, S. Ansoldi, S. Antier, M. Ar‘ene, N. Arnaud, et al., “Quantum Backaction on kg-Scale Mirrors: Observation of Radiation Pressure Noise in the Advanced Virgo Detector,” Phys. Rev. Lett. 125, 131101 (2020).
  * (3) X. Xu, and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
  * (4) B. Levitan, A. Metelmann, and A. Clerk, “Optomechanics with two-phonon driving,”New J. Phys. 18, 093014 (2016).
  * (5) A. Motazedifard, F. Bemani, M. Naderi, R. Roknizadeh, and D. Vitali, “Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection,”New J. Phys. 18, 073040 (2016).
  * (6) A. Motazedifard, A. Dalafi, F. Bemani, and M. Naderi, “Force sensing in hybrid Bose-Einstein-condensate optomechanics based on parametric amplification,”Phy. Rev. A 100, 023815 (2019).
  * (7) A. Motazedifard, A. Dalafi, and M. Naderi, “Ultraprecision quantum sensing and measurement based on nonlinear hybrid optomechanical systems containing ultracold atoms or atomic Bose–Einstein condensate,” AVS Quantum Science 3, 024701 (2021).
  * (8) J. Lenz and S. Edelstein, “Magnetic sensors and their applications,”IEEE Sensors journal 6, 631-649 (2006).
  * (9) D. Budker and M. Romalis, “Optical magnetometry,”Nat. Phys. 3, 227-234 (2007).
  * (10) K.-M. C. Fu, G. Z. Iwata, A. Wickenbrock, and D. Budker, “Sensitive magnetometry in challenging environments,” AVS Quantum Sci 2, 044702 (2020).
  * (11) N. Crescini, C. Braggio, G. Carugno, A. Ortolan, and G. Ruoso, “Cavity magnon polariton based precision magnetometry ,”Applied Physics Letters 117, 144001 (2020).
  * (12) P.-H. Chu, Y. J. Kim, and I. Savukov, “Search for an axion-induced oscillating electric dipole moment for electrons using atomic magnetometers,”Phys. Rev D 99, 075031 (2019).
  * (13) A. V. Gramolin, D. Aybas, D. Johnson, J. Adam, and A. O. Sushkov,“Search for axion-like dark matter with ferromagnets,” Nat. Phys. 17, 79 (2021).
  * (14) N. Crescini, G. Carugno, and G. Ruoso, “Phase-modulated cavity magnon polaritons as a precise magnetic field probe,” arXiv:2010.00093v2.
  * (15) Y. Cao, and P. Yan, “Exceptional magnetic sensitivity of PT-symmetric cavity magnon polaritons,”Phys. Rev. B 99, 214415 (2019).
  * (16) S. Forstner, S. Prams, J. Knittel, E. Van Ooijen, J. Swaim, G. Harris, A. Szorkovszky, W. Bowen, and H. Rubinsztein-Dunlop,“Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
  * (17) S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. Rubinsztein‐Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,”Adv. Mater. 26, 6348-6353 (2014).
  * (18) B.-B. Li, J. Bilek, U. B. Hoff, L. S. Madsen, S. Forstner, V. Prakash, C. Schäfermeier, T. Gehring, W. P. Bowen, and U. L. Andersen, “Quantum enhanced optomechanical magnetometry,”Optica 5, 850-856 (2018).
  * (19) C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. Rubinsztein-Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. Applied 5, 044007 (2016).
  * (20) I. Kominis, T. Kornack, J. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,”Nature (London) 422, 596-599 (2003).
  * (21) H. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,”Appl. Phys. Lett. 97, 151110 (2010).
  * (22) J. A. Binefa and J. Kolodynski, “Noisy atomic magnetometry in real time,” arXiv preprint arXiv:2103.12025 (2021).
  * (23) R. Kleiner, D. Koelle, F. Ludwig, and J. Clarke, “Superconducting quantum interference devices: State of the art and applications,”Proceedings of the IEEE 92, 1534 (2004).
  * (24) J. Gallop, “SQUIDs: Some limits to measurement,”Supercond. Sci. Technol. 16, 1575 (2003).
  * (25) F. Baudenbacher, L. Fong, J. Holzer, and M. Radparvar, “Monolithic low-transition-temperature superconducting magnetometers for high resolution imaging magnetic fields of room temperature samples ,”Appl. Phys. Lett. 82, 3487 (2003).
  * (26) Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,”Phys. Rev. Lett. 113, 083603 (2014).
  * (27) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,”Phys. Rev. Lett. 113, 156401 (2014). note that the experimental parameters for obtaining the sensitivity under the RWA are choosen from the version on arxiv (https://arxiv.org/abs/1405.7062).
  * (28) M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,”Phys. Rev. Applied. 2, 054002 (2014).
  * (29) T. Liu, X. Zhang, H. X. Tang, and M. E. Flatté, “Optomagnonics in magnetic solids,”Phys. Rev. B 94, 060405(R) (2016).
  * (30) S. V. Kusminskiy, H. X. Tang, and F. Marquardt, “Coupled spin-light dynamics in cavity optomagnonics,”Phys. Rev. A 94, 033821 (2016).
  * (31) S. Sharma, Y. M. Blanter, and G. E. Bauer, “Light scattering by magnons in whispering gallery mode cavities,”Phys. Rev. B 96, 094412 (2017).
  * (32) X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, “Optomagnonic whispering gallery microresonators,”Phys. Rev. lett. 117, 123605 (2016).
  * (33) A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, “Cavity optomagnonics with spin-orbit coupled photons,”Phys. Rev. lett. 116, 223601 (2016).
  * (34) A. Serga, A. Chumak, and B. Hillebrands, “YIG magnonics,” J. Phys. D: Appl. Phys. 43, 264002 (2010).
  * (35) A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, “Magnon spintronics,”Nat. Phys. 11, 453-461 (2015).
  * (36) D. D. Stancil, and A. Prabhakar, Spin waves (Springer, Berlin, 2009).
  * (37) S. P. Wolski, D. Lachance-Quirion, Y. Tabuchi, S. Kono, A. Noguchi, K. Usami, and Y. Nakamura, “Dissipation-based quantum sensing of magnons with a superconducting qubit,”Phys. Rev. Lett.125.117701 (2020).
  * (38) D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. You,“Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Information 1, 1 (2015).
  * (39) H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. Goennenwein,“High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids,” Phys. Rev. Lett. 111, 127003 (2013).
  * (40) D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, “Hybrid quantum systems based on magnonics,”Appl. Phys. Expr 12, 070101 (2019).
  * (41) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,”Science advances 2, e1501286 (2016).
  * (42) C. Potts, E. Varga, V. Bittencourt, S. V. Kusminskiy, and J. Davis,“Dynamical backaction magnomechanics,” arXiv preprint arXiv:2104.11218 (2021).
  * (43) R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, “Bidirectional conversion between microwave and light via ferromagnetic magnons”Phys. Rev. B 93, 174427 (2016).
  * (44) J. Li, S.-Y. Zhu, and G. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,”Phys. Rev. Lett. 121, 203601 (2018).
  * (45) M. Yu, H. Shen, and J. Li, “Magnetostrictively induced stationary entanglement between two microwave fields,”Phys. Rev. Lett. 124, 213604 (2020).
  * (46) M. Yu, S.-Y. Zhu, and J. Li, “Macroscopic entanglement of two magnon modes via quantum correlated microwave fields,” J. Phys. B: At. Mol. Opt. Phys. 53, 065402 (2020).
  * (47) J. Li and S. Groeblacher, “Entangling the vibrational modes of two massive ferromagnetic spheres using cavity magnomechanics,”Quantum Sci. Technol. 6, 024005 (2021).
  * (48) J. Li and S.-Y. Zhu,“Entangling two magnon modes via magnetostrictive interaction”New J. Phys. 21, 085001 (2019).
  * (49) J. M. P. Nair, and G. S. Agarwal, “Quantum drives produce strong entanglement between YIG samples without using intrinsic nonlinearities,”arXiv preprint arXiv:1905.07884 (2019).
  * (50) Z. Zhang, M. O. Scully, and G. S. Agarwal, “Quantum entanglement between two magnon modes via Kerr nonlinearity driven far from equilibrium,”Phys. Rev. Research 1, 023021 (2019).
  * (51) H. Tan and J. Li,“‘Einstein-Podolsky-Rosen entanglement and asymmetric steering between distant macroscopic mechanical and magnonic systems,”Phys. Rev. Research 3, 013192 (2021).
  * (52) F. Bemani, R. Roknizadeh, A. Motazedifard, M. H. Naderi, and D. Vitali,  Phys. Rev. A 99, 063814 (2019).
  * (53) J. Li, Y.-P. Wang, J. You, and S.-Y. Zhu, “Squeezing microwave fields via magnetostrictive interaction,” arXiv:2101.02796.
  * (54) J. Li, S.-Y. Zhu, and G. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,”Phys. Rev. A 99, 02180(R) (2019).
  * (55) M.-S. Ding, L. Zheng, and C. Li, “Phonon laser in a cavity magnomechanical system,”Sci. Rep. 9, 1 (2019).
  * (56) C. Potts, V. A. Bittencourt, S. V. Kusminskiy, and J. Davis, “Magnon-phonon quantum correlation thermometry,”Phys. Rev. Applied 13, 064001 (2020).
  * (57) Z.-X. Liu, H. Xiong, and Y. Wu, “Magnon blockade in a hybrid ferromagnet-superconductor quantum system,” Phys. Rev. B 100, 134421 (2019).
  * (58) J.-k. Xie, S.-l. Ma, and F.-l. Li, “Quantum-interference-enhanced magnon blockade in an yttrium-iron-garnet sphere coupled to superconducting circuits,”Phys. Rev. A 101, 042331 (2020).
  * (59) Q. Cai, J. Liao, B. Shen, G. Guo, and Q. Zhou, “Microwave quantum illumination via cavity magnonics,”arXiv:2011.04301.
  * (60) S.-f. Qi and J. Jing, “Magnon-assisted photon-phonon conversion in the presence of the structured environments,”Phys. Rev. A 103, 043704 (2021).
  * (61) B. Sarma, T. Busch, and J. Twamley,“Cavity magnomechanical storage and retrieval of quantum states,”New J. Phys. 23 043041 (2021).
  * (62) Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Quantum magnonics: The magnon meets the superconducting qubit,”Compt. Ren. Phys. 17, 729-739 (2016).
  * (63) P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, “Ultrastrong coupling regimes of light-matter interaction,”Rev. Mod. Phys 91, 025005 (2019).
  * (64) A. Motazedifard, A. Dalafi, M. Naderi, and R. Roknizadeh, “Strong quadrature squeezing and quantum amplification in a coupled Bose–Einstein condensate—optomechanical cavity based on parametric modulation,”Annals of Physics 405, 202 (2019).
  * (65) A. Motazedifard, M. H. Naderi, and R. Roknizadeh, “Analogue model for controllable Casimir radiation in a nonlinear cavity with amplitude-modulated pumping: Generation and quantum statistical properties,” J. Opt. Soc. Am. B 32, 1555 (2015).
  * (66) A. Motazedifard, M. H. Naderi, and R. Roknizadeh, “Dynamical Casimir effect of phonon excitation in the dispersive regime of cavity optomechanics,” J. Opt. Soc. Am. B 34, 642 (2017).
  * (67) A. Motazedifard, A. Dalafi, M. H. Naderi, and R. Roknizadeh,“Controllable generation of photons and phonons in a coupled Bose-Einstein condensate-optomechanical cavity via the parametric dynamical Casimir effect,” Ann. Phys. 396, 202 (2018).
  * (68) O. Di Stefano, A. Settineri, V. Macrì, A. Ridolfo, R. Stassi, A. F. Kockum, S. Savasta, and F. Nori, “Interaction of mechanical oscillators mediated by the exchange of virtual photon pairs,” Phys. Rev. Lett. 122, 030402 (2019).
  * (69) W. Qin, V. Macrì, A. Miranowicz, S. Savasta, and F. Nori, “Experimentally feasible dynamical Casimir effect in parametrically amplified cavity optomechanics,” arXiv:1902.04216v1 (2019).
  * (70) A. Settineri, V. Macrì, L. Garziano, O. Di Stefano, F. Nori, and S. Savasta, “Conversion of mechanical noise into correlated photon pairs: Dynamical Casimir effect from an incoherent mechanical drive,”Phys. Rev. A 100, 022501 (2019).
  * (71) A. Pontin, M. Bonaldi, A. Borrielli, L. Marconi, F. Marino, G. Pandraud, G. A. Prodi, P. M. Sarro, E. Serra, and F. Marin, “Dynamical two-mode squeezing of thermal fluctuations in a cavity optomechanical system,” Phys. Rev. Lett. 116, 103601 (2016).
  * (72) A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,”Nat. Rev. Phys. 1, 19 (2019).
  * (73) B. Gong, D. Dong, W. Su, and W. Cui, “Force tracking in cavity optomechanics with a two-level quantum system by Kalman filtering,” arXiv:1903.01283 (2019).
  * (74) Y. Li, T. Polakovic, Y. L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W. K. Kwok, Z. Xiao, V. Novosad, A. Hoffmann, and W. Zhang, “Strong coupling between magnons and microwave photons in on-chip ferromagnet-superconductor thin-film devices,”Phys. Rev. Lett 123, 107701 (2019).
  * (75) J. T. Hou and L. Liu,“Strong coupling between microwave photons and nanomagnet magnons,” Phys. Rev. Lett 123, 107702 (2019).
  * (76) R. Morris, A. Van Loo, S. Kosen, and A. Karenowska, “Strong coupling of magnons in a YIG sphere to photons in a planar superconducting resonator in the quantum limit,”Sci. Rep. 7, 1 (2017).
  * (77) J. Li, W. Quan, B. Zhou, Z. Wang, J. Lu, Z. Hu, G. Liu, and J. Fang,“SERF Atomic Magnetometer–Recent advances and applications: A review,” IEEE Sensors J.18, 8198 (2018).
  * (78) J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland, “Detection of pico-Tesla magnetic fields using magneto-electric sensors at room temperature,”Appl. Phys. Lett. 88, 062510 (2006).
  * (79) H. G. Meyer, R. Stolz, A. Chwala, and M. Schulz, “SQUID technology for geophysical exploration,”phys. Stat. Solidi (C) 2, 1504 (2005).
  * (80) H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. Romalis, “Magnetoencephalography with an atomic magnetometer,” Appl. Phys. Lett. 89, 211104 (2006).
  * (81) B.-B. Li, G. Brawley, H. Greenall, S. Forstner, E. Sheridan, H. Rubinsztein-Dunlop, and W. P. Bowen,“Ultrabroadband and sensitive cavity optomechanical magnetometry,” Photonics Research 8, 1064 (2020).
  * (82) S. Danilin, A. V. Lebedev, A. Vepsäläinen, G. B. Lesovik, G. Blatter, and G. Paraoanu, “Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom,”npj Quantum Information 4, 1 (2018).
  * (83) Ali Motazedifard, A. Dalafi, M. H. Naderi, J. Phys. A: Math. Theor. 54, 215301 (2021).