# Spin-mechanics with nitrogen-vacancy centers and trapped particles

Maxime Perdriat  Clément Pellet-Mary  Paul Huillery  Laboratoire De Physique
de l’École Normale Supérieure, École Normale Supérieure, PSL Research
University, CNRS, Sorbonne Université, Université de Paris , 24 rue Lhomond,
75231 Paris Cedex 05, France  Loïc Rondin  Université Paris-Saclay, CNRS, ENS
Paris-Saclay, CentraleSupélec, LuMIn, 91190, Gif-sur-Yvette, France  Gabriel
Hétet<EMAIL_ADDRESS>Laboratoire De Physique de l’École Normale
Supérieure, École Normale Supérieure, PSL Research University, CNRS, Sorbonne
Université, Université de Paris , 24 rue Lhomond, 75231 Paris Cedex 05, France

###### Abstract

Controlling the motion of macroscopic oscillators in the quantum regime has
been the subject of intense research in recent decades. In this direction,
opto-mechanical systems, where the motion of micro-objects is strongly coupled
with laser light radiation pressure, have had tremendous success. In
particular, the motion of levitating objects can be manipulated at the quantum
level thanks to their very high isolation from the environment under ultra-low
vacuum conditions. To enter the quantum regime, schemes using single long-
lived atomic spins, such as the electronic spin of nitrogen-vacancy (NV)
centers in diamond, coupled with levitating mechanical oscillators have been
proposed. At the single spin level, they offer the formidable prospect of
transferring the spins’ inherent quantum nature to the oscillators, with
foreseeable far-reaching implications in quantum sensing and tests of quantum
mechanics. Adding the spin degrees of freedom to the experimentalists’ toolbox
would enable access to a very rich playground at the crossroads between
condensed matter and atomic physics. We review recent experimental work in the
field of spin-mechanics that employ the interaction between trapped particles
and electronic spins in the solid state and discuss the challenges ahead. Our
focus is on the theoretical background close to the current experiments, as
well as on the experimental limits, that, once overcome, will enable these
systems to unleash their full potential.

##  I Introduction

The unique control offered by single quantum systems, such as atoms or ions,
has enabled an immense boost in the development of quantum technologies.
Extending these technologies to larger masses is important both for
fundamental questions on the nature of quantum physics at larger scales, and
for the development of innovative devices such as ultra-high precision force
sensors and accelerometers Aspelmeyer et al. (2014).

Electro-, magneto- or opto- mechanically controlled levitating objects in
vacuum are fascinating in this regard and have been at the focus of intense
recent research activity Millen et al. (2020). This attention is supported by
the exquisite control that one can exert over the levitated objects. These
platforms indeed allow addressing of numerous degrees of freedom, easy tuning
of the trapping potential, as well as enabling free-fall experiments
Hebestreit et al. (2018), as in atomic physics.

Initial work proposed to coupling levitated silica nano-spheres, and even
viruses, to the optical modes of a high finesse cavities Chang et al. (2010);
Romero-Isart et al. (2010); Barker and Shneider (2010). The promises of this
schemes have been supported by recent experiments that reported trapped
particles cooled to the quantum regime Delić et al. (2019); Tebbenjohanns et
al. (2020); Magrini et al. (2020); Tebbenjohanns et al. (2021). Active
development of force sensors are in progress and tests of various models of
fundamental physics have also been proposed using various platforms Moore and
Geraci (2020); Xiong et al. (2020); Vinante et al. (2020). To push these
developments further and to enable operate the mechanical oscillator in the
quantum regime, coupling the dynamics of the levitated system to a single
intrinsically quantum system, such as ions, atoms, or artificial atoms, has
been envisioned Rabl et al. (2009); Treutlein et al. (2014). Towards this
goal, amongst all condensed matter system, the negatively charged nitrogen-
vacancy center (NV- center for short) in diamond centers stands out because of
the ease with which one can optically polarize and read-out its electronic
spin under ambient conditions. Most of the proposals initially designed for
clamped oscillators coupled with NV- centers Rabl et al. (2009); Arcizet et
al. (2011); Kolkowitz et al. (2012) can then be carried over to trapped
diamonds Yin et al. (2015) which are, in all current experiments, operating
close to room temperature. Ensembles of NV- centers coupled identically to
mechanical oscillators can also exhibit magnetic phase transitions, paving the
way towards nano-scale magnetism with long-lived and controllable spins in a
trapped particle Ma et al. (2017); Wei et al. (2015).

This growing research field will drive advances in quantum metrology via
quantum enhanced gyroscopy and matter-wave interferometry Ma et al. (2017);
Scala et al. (2013); Pedernales et al. (2020). The spin–mechanical coupling
also brings important advantages for quantum sensing and metrology by
providing enhanced measurement sensitivity. At the single spin level it also
features additional non-linearity and/or control that could be helpful to
build non classical states of motion Yin et al. (2015). Finally, they could
also serve as transducers between optical and RF signals via the mechanical
mode Rabl et al. (2010).

In this review, we describe the state-of-the-art levitation systems that
involve NV- centers and describe their specificities and limitations. The goal
is not to draw a comparison between the performance of existing systems.
Instead, we focus on the important results that have been accomplished and the
remaining hurdles on the way towards operating in the quantum regime with
these platforms.

In Section II, we briefly present the existing levitation platform and the
equations of motion describing the different mechanical modes. In Section III,
we present a rapid overview of the ancillary quantum system used to interact
with the mechanical modes, namely the nitrogen-vacancy center in diamond. The
Hamiltonian of the total coupled spin–mechanical system is derived in Section
IV. Section V reviews experiments that demonstrate read-out of the mechanical
motion of levitated particles using the NV spin. Section VI and Section VII
provide a classical analytical treatment of spin-bistability, spin-spring and
spin-cooling in the adiabatic limit. Finally, Section VIII presents the
current challenges in spin-mechanics with trapped particles.

##  II Trapping Crystals

The basic idea behind particle levitation is to hold a particle under
atmospheric conditions or in vacuum against gravity. Here, after presenting
the classical theoretical framework for harmonic motion analysis, we discuss
current methods for trapping crystals in vacuum.

###  II.1 Center of Mass Harmonic Motion

The center of mass dynamics of a levitated particle along a direction
parametrized by the coordinate $q$, can be described by a Langevin equation.
For a stably trapped particle, one can linearize the trapping force so that
the particle dynamics can, to a good approximation, be described by the
equation of a harmonic oscillator

$m\frac{\mathrm{d}^{2}q}{\mathrm{d}t^{2}}+m\gamma_{q}\frac{\mathrm{d}q}{\mathrm{d}t}+m\omega_{q}^{2}q=F_{\mathrm{L}}(t)\,,$
(1)

where $\gamma_{q}$ is the translational damping rate due to collisions with
gas molecules and $F_{\mathrm{L}}$ is the Langevin fluctuating force induced
by the interaction between the particle and the gas molecules, $m$ is the
particle mass, and $\omega_{q}$ is the trapping frequency. Rarefied gas can be
described as a Markovian thermal bath with a white noise spectrum so that the
fluctuating force satisfies $\langle F_{\mathrm{L}}\rangle=0$ and $\langle
F_{\mathrm{L}}(t)F_{\mathrm{L}}(t^{\prime})\rangle=2m\gamma_{q}kT\delta(t-t^{\prime})$
at temperatures such that $kT\gg\hbar\omega_{q}$.

$\gamma_{q}$ depends on the exact shape of the particle Martinetz et al.
(2018) and is proportional to the residual gas pressure at low pressures. This
property makes levitating platforms attractive, since the thermal noise can be
made arbitrary small by reducing the gas pressure inside the vacuum chamber.
Note that in practice, the dynamics of the particle may appear more complex
than a simple harmonic oscillator at high temperatures because of
instabilities induced by the non-linearity of the potential Gieseler et al.
(2013).

###  II.2 Angular Confinement: The Librational Mode

When considering the angular degree of freedom, two characteristic motional
behaviours can be observed: pure rotation and libration, namely oscillation of
the particle angle about a mean angular position. We will see that confining
the angle so that the libration can be described by a harmonic oscillator is
also of crucial importance when dealing with NV- centers. The condition for
librational confinement is that the total energy in the angular mode must not
be larger than the angular potential depth. Typically, angular potentials are
$\pi$ or $\pi/2$-periodic so that when the standard deviation of the angle is
in that range, the particle angle may jump from one angular well to another.
This can be due to collisions with gas molecules for instance.

Once confined close to an angle $\theta=0$, the angle follows this equation of
motion

$\displaystyle
I\frac{d^{2}\theta}{dt^{2}}+I\gamma_{\theta}\frac{d\theta}{dt}+I{\omega_{\theta}}^{2}\theta=\tau_{L}(t).$
(2)

$\tau_{L}(t)$ are Langevin fluctuation torques satisfying
$\langle\tau_{\mathrm{L}}\rangle=0$ and
$\langle\tau_{\mathrm{L}}(t)\tau_{\mathrm{L}}(t^{\prime})\rangle=2\gamma_{\theta}IkT\delta(t-t^{\prime})$.
$\gamma_{\theta}$ is the damping rate for the angular degree of freedom due to
collisions with gas molecules and $I$ the particle moment of inertia. As for
translational modes, the damping rate for libration is proportional to the gas
pressure at low pressures and strongly depends on the particle shape.
According to the fluctuation dissipation theorem, the standard deviation of
the angle $\theta$ is
$\sqrt{\langle\theta^{2}\rangle}=\sqrt{kT/(I\omega_{\theta}^{2})}$. One can
thus obtain an approximate criterion for stable angular confinement requiring
that $\sqrt{\langle\theta^{2}\rangle}$ is bounded by $\approx\pi/10$ in order
to ensure small angular deviations. We then obtain a condition on the angular
rigidity $K_{t}=I\omega_{\theta}^{2}$ such that $K_{t}\gtrsim 10\times kT$.
Below this value, libration is not guaranteed, and angular deviations may be
too large for efficient control of the NV electronic spin.

###  II.3 Trapping Platforms

The original proposals for trapping particles suggested use of optical forces.
Under laser illumination, small dielectric particles can indeed become
polarized. The induced dipole is then attracted to the highest intensity
region where particles are stably trapped in three dimensions Ashkin (1997),
with typical trapping frequencies in the 100 kHz to 1 MHz range. Reference
Jones et al. (2016) presents a broad overview of optical tweezers. Particles
can be trapped and cooled either at the focus of a laser beam as in Gieseler
et al. (2012) (Figure 1a), at the node of a cavity field Kiesel et al. (2013)
or in the near-field of a photonic crystal Magrini et al. (2018). Recent
results in opto-mechanics with optical tweezers and cavities are described in
a recent review Millen et al. (2020).

Another means to trap particles, described in Figure 1b, is to use a Paul trap
Paul (1990). The radio-frequency modulation of a high voltage electric field
applied to the trap electrode ensures three-dimensional confinement of charged
particles with typical frequencies ranging from 100 Hz to 10 kHz. The center
of mass motion of silica nano-particles Dania et al. (2021), graphene flakes
Nagornykh et al. (2015) and nanodiamonds Conangla et al. (2018) in Paul traps
have been cooled to low temperatures under ultra-high vacuum levels using
parametric feedback cooling.

Another way to trap micro-particles is to use magnetic fields. In magnetic
traps, a magnetic object is levitating above a diamagnetic/superconductor
material Gieseler et al. (2020). Alternatively, a diamagnetic particle—such as
diamond—is levitating above magnets (see Figure 1c). Trapping frequencies in
the hundreds of hertz range are typically observed. The latter trapping
approach was efficiently employed to demonstrate cooling of the center of mass
motion of trapped diamonds using feedback cooling Hsu et al. (2015); O’Brien
et al. (2019).

Figure 1: Different approaches to particle levitation. (a) Optical tweezers.
Picture of an infrared laser focused through a high numerical aperture (NA)
objective (shown on the left) allowing trapping a nanodiamond at its focus.
The efficient scattering of the green laser used for NV excitation enables one
to see the nanodiamond with the naked eye. (b) Paul trap: a charged diamond
particle is held in vacuum by electric field gradients. (c) A magnetic
particle is levitated above a superconductor. NV- centers implanted in a
diamond slab above the levitated particle can be coupled with its motion
through the magnetic field gradient. Adapted from ref Gieseler et al. (2020).
(d) A diamond particle is trapped by large magnetic field gradients thanks to
its diamagnetism.

As already discussed, the other important degree of freedom for NV spin-
mechanics is the particle angle. Trailblazing experiments with particles in
optical tweezers have shown controlled rotation of silica spheres Arita et al.
(2013); Reimann et al. (2018); Monteiro et al. (2018); Ahn et al. (2018). Very
fast rotation frequencies up to a few GHz have also recently been observed
using dumbbell-shaped silica nanoparticles under ultra-high vacuum Reimann et
al. (2018); Ahn et al. (2018). When using nanodiamonds, such rotational
dynamics could open the door to observations of the Barnett effect on long-
lived electronic spins Barnett (1935) and to coupling spins to the
gyroscopically stabilized angular motion. Importantly, stable angular
confinement and the resulting librational motion of micro-particles was
observed recently by several groups Vinante et al. (2020); Hoang et al.
(2016); Delord et al. (2020). Angular confinement can be the result of the
shape anisotropy and birefringence of the particles Hoang et al. (2016), or
asymmetries of both charged particles and traps Delord et al. (2017). For
micro-crystals in Paul traps, typical librational frequencies are in the kHz
range. As discussed in Section II.2, the moment of inertia must thus be larger
than $\approx 10^{-27}$ N$\cdot$m for Brownian motion not to make trapping
unstable at 300 K. Assuming a 1:2 ellipsoidal particle aspect ratio implies
operation with particles with a greater axis diameter $d\gtrsim 1~{}\upmu$m.
In optical traps, trapping frequencies can be as large as 1 MHz, implying that
particles must have a moment of inertia larger than $10^{-33}$ N$\cdot$m.
Particles with $d\gtrsim 50$ nm must thus be employed to harmonically confine
the angle at room temperature. The upper bound to the particle size will
ultimately be limited by gravity, and by the specificities of the employed
trapping mechanism. Observing such libration on small particles is an
important step forward, because it will facilitate torque sensing with
individual spins (see Section VI).

##  III Coupling to an Ancillary Quantum System: The Special Case of the NV-
Center

As discussed in the introduction, coupling a levitated particle to an
ancillary quantum system is an exiting way to extend its capabilities.
Coupling the internal electronic state of atoms or ions to trapped mechanical
oscillators is intensively studied Ranjit et al. (2015); Bykov et al. (2019)
and was successfully demonstrated with clamped oscillators Treutlein et al.
(2014); Karg et al. (2020); Thomas et al. (2020). Colloidal quantum dots
Salakhutdinov et al. (2020), rare-earth ions in a solid matrix Rahman and
Barker (2017), or color centres in semi-conductor particles Pettit et al.
(2017); Delord et al. (2017); Hoang et al. (2016) have all been levitated in
optical tweezers. These objects feature defects that behave as atoms which are
hosted in a solid-state matrix, so these experiments are important steps
forward. One important extra criterium is the coupling strength between the
ancillary systems’ quantum state and the levitated particle dynamics. The
coupling strength quantifies how much the motion of the object affects the
atomic state, as well as how much the change in the quantum system impacts the
motion. We will describe this coupling more quantitatively in Section IV using
the NV- center.

###  III.1 The NV- Center

The NV- color centers in diamond are of particular interest for spin-
mechanics. First, thanks to the intense research activities around these color
centers, triggered by their potentials in quantum information processing and
the development of innovative sensors, their internal electronic and nuclear
spin properties are very well understood. For the same reason, the control of
diamond materials has progressed significantly over the past decade. Lastly,
the electronic spin of the NV- center can easily be coupled with the diamond
host matrix’s motion using a magnetic field Rabl et al. (2009); Ma et al.
(2017); Delord et al. (2017); Lee et al. (2017); Ge and Zhao (2018);
Ovartchaiyapong et al. (2014); Teissier et al. (2014).

The physics and applications of the NV- centers have already been discussed in
recent reviews Rondin et al. (2014); Doherty et al. (2013). We simply recall
here the basics of NV- physics required for understanding the coupling to
trapped particles.

The NV color center is a point defect inside the diamond matrix consisting of
a substitutional nitrogen atom (N) combined with a vacancy (V) in one of the
nearest neighboring sites of the diamond crystal lattice as depicted in Figure
2a. This defect behaves as an artificial atom hosted by the diamond matrix. It
combines unique luminescence and spin properties. Indeed, because its energy
levels are well within the large band gap of diamond, the NV center has an
extremely stable luminescence in the near-infrared. It can be found in two
different charge states: the NV- and NV0. The NV- zero-phonon line is at
around $\lambda_{\mathrm{ZPL}}=637$ nm. It is associated with broad phonon
sidebands that extend up to $\approx$750 nm. The NV- is the most interesting
for spin–mechanical interactions as we will see.

The NV- luminescence can readily be accessed using standard confocal
microscopy, and can straightforwardly be adapted for levitation platforms. The
photoluminescence from NV- centers in trapped diamonds under atmospheric
conditions or in vacuum was observed by several groups Kuhlicke et al. (2014);
Neukirch et al. (2015); Pettit et al. (2017); Neukirch et al. (2013); Delord
et al. (2017); Hsu et al. (2016). Figure 2b shows the photoluminescence from
NV- centers in optical tweezers observed in Neukirch et al. (2013). Different
pulsing sequences were used in order to mitigate PL quenching from the 1064 nm
trapping laser. Note that, alternatively, photoluminescence (PL) quenching can
be greatly reduced using other trapping laser wavelengths Hoang et al. (2016).
Further studies demonstrated trapping of nanodiamonds containing single NV-
centers Conangla et al. (2018); Neukirch et al. (2015); Pettit et al. (2017).
Figure 2c shows the autocorrelation function of the PL from a single NV-
center inside a nanodiamond trapped in a Paul trap Conangla et al. (2018). The
antibunching dip at zero delay, below 0.5 is a proof for the presence of a
single NV- center inside the trapped diamond, which opens a path towards non-
gaussian spin-mechanics using the NV- spin.

Figure 2: (a) Schematic of the crystalline structure of NV defect in diamond.
The axis $N-V$ is the quantization axis. (b) Photoluminescence from NV-
centers in an optical tweezer using three different excitation conditions.
Adapted with permission from Neukirch et al. (2013). © The Optical Society.
(c) Autocorrelation function of photon emission from a single NV- center in a
nanodiamond levitated in a Paul trap. Adapted with permission from Conangla et
al. (2018). Copyright 2020 American Chemical Society

###  III.2 The NV- Center Electronic Spin

In addition to the possibility of observing stable PL at ambient conditions,
the NV- center also carries a spin that can be manipulated at room
temperature. The electronic spin of the NV- center in the ground state is a
spin triplet $S=1$, with a quantization axis $\textbf{u}_{\mathrm{NV}}$
enforced by the crystalline field around the defect. The spin projections
along this axis are labelled with the quantum number $m_{s}$. Due to a
spin–spin interaction between the two electrons in the ground state the
$\ket{m_{s}=\pm 1}$ states are split from the $\ket{m_{s}=0}$ by the zero-
field splitting $D\approx 2.88$ GHz (see Figure 3a). These states are purely
magnetic states, implying a very long longitudinal decay time $T_{1}$, on the
order of milliseconds in typical diamond samples Jarmola et al. (2012). Note
that in strained diamond, or in the presence of local electric fields Mittiga
et al. (2018), the states $\ket{m_{s}=\pm 1}$ can also be split, by a
parameter often denoted $2E$. This splitting is typically a few MHz in
nanodiamonds. In practice, a magnetic field bias larger than $E$ is employed
in order to reach large spin–mechanical couplings, so we neglect this zero-
field splitting in the following.

Figure 3: NV- center level structure. (a) Simplified electronic structure of
the NV- center. (b) Example of optically detected magnetic resonance on a
single NV- center in a bulk diamond. The magnetic field is applied with an
angle $\theta=74^{\circ}$ with respect to the NV axis. (c) ESR frequencies
from (b) (gray dots) compared with the full theory (red lines) obtained by
computing the NV Hamiltonian eigenvalues (Equation (3)). Figure (b,c) are
adapted from Tetienne et al. (2012).

One interesting aspect of the NV- center resides in the possibility to
optically initialize and read-out its electronic spin state. Indeed, under
green laser irradiation the NV- center is polarized in the $\ket{m_{s}=0}$
state, while the photoluminescence level depends on the populated spin state.
The $\ket{m_{s}=0}$ state can be up to 30 % brighter than the $\ket{m_{s}=\pm
1}$ states Doherty et al. (2013). When a microwave tone is resonant with one
of the transitions from $\ket{m_{s}=0}$ to $\ket{m_{s}=\pm 1}$, a drop of
luminescence can then observed. This forms the basis of _Optically Detected
Magnetic Resonance_ (ODMR). The energy of these transitions can be found from
the Hamiltonian of the electronic spin of the NV- center in the ground state,
which reads

$\hat{\mathcal{H}}/\hbar=D\hat{S}_{z}^{2}+\gamma_{e}\hat{\textbf{S}}\cdot\textbf{B},$
(3)

where $\gamma_{e}$ is the gyromagnetic factor of the electron. Because of the
zero-field splitting $D$ and the spin 1 character of the NV- center, the
eigenstates of the Hamiltonian depend upon the angle between the magnetic
field and the NV- center when $\gamma_{e}B<D$.

Typical single NV- ODMR spectra with fixed diamonds are presented in Figure 3b
showing the transitions from the $|0\rangle$ to $|\pm 1\rangle$ states as a
drop of the PL. For this experiment, the NV- is at an angle of 74∘ with
respect to the magnetic field. Figure 3c shows the change in the frequency of
the two transitions as a fonction of magnetic field as well as a numerical
calculation using Equation (3).

The width of the observed electron spin resonance (ESR) dip is here limited by
the dipolar coupling between the spin of NV- centers to other paramagnetic
impurities in the diamond. The most dominant impurity is the substitutional
nitrogen (P1 centers), with concentrations ranging from 100 to 500 ppm in
high-pressure-high-temperature grown diamonds (see Section VIII.1). Dipolar
coupling between the NV- center and P1 centers typically lead to an
inhomogeneous dephasing time $\Gamma_{2}^{*}$ on the order of 5 to 10 MHz
Bauch et al. (2020). Ultimately, the linewidth of the ESR is of crucial
importance, since it limits the measurement sensitivity as well as impacting
the spin coupling strength to the mechanics. Nevertheless, as discussed in the
Sec. VII, diamond growth allows fine control over impurities.

The frequency shifts of the spin resonances with magnetic field and their
dependence with the magnetic field angle are important ingredients for
coupling the center of mass and the angle of levitated particles to the NV-
spin states, as we now discuss.

##  IV Hamiltonian of the Spin–Mechanical System

In the following, we consider a diamond containing a single NV- center in the
presence of a homogeneous magnetic field as well as in a magnetic field
gradient. We will consider the case of an NV- center in a trapped diamond but
this analysis can be carried over to studies where an NV- center is coupled
with a distant trapped magnet Huillery et al. (2020); Gieseler et al. (2020).

When the genuine quantum nature of single spins is not of interest, the
following treatment can also straightforwardly be adapted to $N$ spins. If
correlations between the NV spins inside the diamond from direct dipolar
interactions Bachelard et al. (2011); PANAT and LAWANDE (2002); Prasanna
Venkatesh et al. (2018); Juan et al. (2017), or resonator mediated
interactions Ma et al. (2017); Wei et al. (2015); Schuetz et al. (2017) are
neglected, the coupling strength can simply be multiplied by the number of
spins. To be able to directly carry a single-spin analysis over to the
ensemble case, one must also assume that the NV centers all have the same
orientation in the diamond. In practice this is not the case since the NV-
centers will typically be found with the same probability along the four
$[111]$ diamond directions. However, in the presence of a magnetic field that
does not broaden the ESR width, the microwave can select only one of the eight
resulting ESR transitions so that one can treat this problem using $N$
effectively spin 1/2 systems in a single orientation with ESR frequencies that
are within the inhomogeneous broadening $\Gamma_{2}^{*}$.

We will study the coupling between the spin and the two main degrees of
freedom of a levitating mechanical oscillator: the center of mass motion (CoM)
and the libration. We will assume that there is no coupling between these two
degrees of freedom so that we can treat them separately. Note that this may
not be true generally. For instance, in Paul traps, if the charge distribution
has a non-zero dipole component, the center of mass and the angle may become
coupled Martinetz et al. (2020). Similarly, shape anisotropy or birefringence
of optically trapped particles can also induce a mode coupling Arita et al.
(2020); Trojek et al. (2012). These mechanisms are neglected here, but can be
of interest for transferring the quantum state of one spin-coupled mode to
another one.

###  IV.1 Coupling to the Center of Mass

The center of mass motion (CoM) of the trapped particle can be coupled with
the NV- spin using a magnetic field gradient Rabl et al. (2009). Let us reduce
the study of the spin-CoM coupling to a 1D problem. We assume that the NV axis
is along the magnetic field. This greatly simplifies the problem by leaving
aside mixing between the NV eigenstates and coupling between the CoM and the
libration. When the magnetic field gradient is in the $z$ direction:
$\textbf{B}=B_{0}\textbf{e}_{z}+\frac{\partial B_{z}}{\partial
z}z\textbf{e}_{z}$ to first order in the position. Under those assumptions,
the Hamiltonian of the spin–mechanical system reads $\hat{\mathcal{H}}_{\rm
com}=\hat{\mathcal{H}}_{\rm mecha}+\hat{\mathcal{H}}_{\rm NV}$, where
$\hat{\mathcal{H}}_{\rm
mecha}=\frac{p_{z}^{2}}{2m}+\frac{1}{2}m\omega_{z}^{2}z^{2}$ and
$\hat{\mathcal{H}}_{\rm NV}=\hbar
D\hat{S}_{z}^{2}+\hbar\gamma_{e}(B_{0}+\frac{\partial B_{z}}{\partial
z}z)\hat{S}_{z}$ .

Due to the Zeeman term in $\hat{\mathcal{H}}_{\rm NV}$, the magnetic energy is
linearly proportional to the particle position. A magnetic force directly
related to the magnetic field gradient and the chosen spin polarisation can
thus be applied to the particle (see Section VI).

In the presence of an oscillating magnetic field $B_{\mu W}$ that is linearly
polarized along the $x$ direction, the spin part of the Hamiltonian reads

$\displaystyle\hat{\mathcal{H}}_{{\rm NV}+\mu\rm w}=\hbar
D\hat{S}_{z}^{2}+\hbar\gamma_{e}(B_{0}+\frac{\partial B_{z}}{\partial
z}z)\hat{S}_{z}+\hbar\Omega\cos({\omega t})\hat{S}_{x}.$ (4)

where $\Omega=\gamma_{e}B_{\mu W}$. Moving to the rotating frame at the
microwave frequency $\omega$ through the unitary transform $\hat{U}=e^{i\omega
t\hat{S}_{z}^{2}}$ and making a rotating wave approximation, we get

$\displaystyle\hat{\mathcal{H}}_{{\rm NV}+\mu\rm
w}^{\prime}=\hbar(D-\omega)\hat{S}_{z}^{2}+\hbar\gamma_{e}(B_{0}+\frac{\partial
B_{z}}{\partial z}z)\hat{S}_{z}+\hbar\frac{\Omega}{2}\hat{S}_{x}.$ (5)

If $\gamma_{e}B_{0}$ is greater than both $2E$ and $\Gamma_{2}^{*}$ and if the
optical pumping process is stronger than the relaxation rate $T_{1}$ (see
Section III), resonantly tuning the microwave frequency to the
$\ket{0}\leftrightarrow\ket{+1}(resp.\ket{-1})$ transition enables the
$\ket{-1}$ (resp. $\ket{+1}$) state to be safely neglected. Choosing a
microwave signal tuned to the transition $\ket{0}\leftrightarrow\ket{+1}$, we
finally obtain the Hamiltonian of a two-level system coupled with a mechanical
oscillator:

$\displaystyle\hat{\mathcal{H}}_{\rm total}=\hat{\mathcal{H}}_{\rm
mecha}+\hat{\mathcal{H}}_{\rm Spin}+\hat{\mathcal{H}}_{\rm Spin-mecha},$ (6)

where $\hat{\mathcal{H}}_{\rm
Spin}=-\hbar\Delta\ket{1}\bra{1}+\hbar\frac{\Omega}{2}(\ket{0}\bra{1}+\ket{1}\bra{0})$
with the microwave detuning $\Delta=\omega-D-\gamma_{e}B_{0}$ and
$\hat{\mathcal{H}}_{\rm Spin-mecha}=\hbar G_{z}z\ket{1}\bra{1}$ with the
coupling constant $G_{z}=\gamma_{e}\frac{\partial B_{z}}{\partial z}$. Note
that the latter has been redefined a $\Omega\equiv\frac{\Omega}{\sqrt{2}}$ to
ensure normalisation of the $\hat{S}_{x}$ operator.

Before studying this Hamiltonian and the related experiments in more details,
we will discuss the spin-coupling to the librational degree of freedom.

###  IV.2 Coupling to the Libration

The theory behind spin-coupling to the libration has been presented in several
papers Ge and Zhao (2018); Ma et al. (2017); Delord et al. (2017). Here, we
derive a simplified Hamiltonian in the $\gamma_{e}B\ll D$ limit. Let us use
the vectors $(0,\textbf{e}_{x},\textbf{e}_{y},\textbf{e}_{z})$ to specify the
orientation of the laboratory frame and the vectors
$(0,\textbf{e}_{x^{\prime}},\textbf{e}_{y^{\prime}},\textbf{e}_{z^{\prime}})$
to specify the particle frame. We choose $\textbf{e}_{z}^{\prime}$ as the
anisotropy axis of the NV- center in the crystalline structure of the diamond.
The three Euler angles operators $({\hat{\phi}},{\hat{\theta}},{\hat{\psi}})$
describing the angular position of the diamond are chosen in the
$(zy^{\prime}z^{\prime\prime})$ convention. The magnetic field is supposed to
be homogeneous and its direction is fixed in the laboratory frame. It is
chosen along the $z$ direction, so that $\textbf{B}=B\textbf{e}_{z}$.

The Hamiltonian of the spin–mechanical system in the laboratory frame reads

$\displaystyle\hat{\mathcal{H}}_{\rm
lib}=\frac{{\hat{\textbf{L}}}^{2}}{2I}+U(\hat{\phi},\hat{\theta},\hat{\psi})+\hbar
D\hat{S}_{z^{\prime}}^{2}+\hbar\gamma_{e}B\hat{S}_{z},$ (7)

where ${\hat{\textbf{L}}}$ is the angular momentum operator of the diamond in
the laboratory frame, $U(\hat{\phi},\hat{\theta},\hat{\psi})$ is the angular
confining potential and $\hat{S}_{z}$,$\hat{S}_{z^{\prime}}$ are NV- center
spin operators. Contrary to when studying the coupling of the NV center to the
center of mass, the NV direction is not necessarily fixed in the laboratory
frame. This implies that the $\hat{S}_{z^{\prime}}$ operator depends on
angular operators $({\hat{\phi}},{\hat{\theta}},{\hat{\psi}})$ which do not
commute with the diamond angular momentum operator $\hat{\textbf{L}}$.
Consequently, the commutator
$\commutator{\hat{\textbf{L}}^{2}}{\hat{S}_{z^{\prime}}^{2}}\neq 0$ Rusconi
and Romero-Isart (2016).

In the following, we will restrict the study to one librational mode that is
assumed to be in the $(zz^{\prime})$ plane formed by the magnetic field and
spin direction. The diamond angular motion is parametrized by the nutation
angle operator $\hat{\theta}$. $\theta^{\prime}$ represents the equilibrium
angular position of the diamond.

The Hamiltonian of the simplified system reads

$\displaystyle\hat{\mathcal{H}}_{\rm
lib}=\frac{{\hat{p}}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2}+\hbar
D\hat{S}_{z^{\prime}}^{2}+\hbar\gamma_{e}B\hat{S}_{z},$ (8)

where ${\hat{p}}_{\theta}$ is the angular momentum operator along the $y$
axis. Moving to the particle frame through the unitary transformation
$\hat{U}=e^{i{\hat{\theta}}\hat{S}_{y}}$ changes the Hamiltonian to

$\displaystyle\hat{\mathcal{H}}^{\prime}_{\rm
lib}=\frac{({\hat{p}}_{\theta}-\hbar\hat{S}_{y})^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2}+\hbar
D\hat{S}_{z}^{2}+\hbar\gamma_{e}B\left(\cos{{\hat{\theta}}}\hat{S}_{z}-\sin{{\hat{\theta}}}\hat{S}_{x}\right).$
(9)

In this frame, ${\hat{p}}_{\theta}$ is changed to
${\hat{p}}_{\theta}-\hbar\hat{S}_{y}$, which is the total angular momentum of
the spin–mechanical system along the $y$ axis. It is the sum of the particle
and the NV-spin angular momentum. This modified angular momentum could cause
precession of the particle. Such a precession is predicted to be observable in
levitated hard ferromagnetic nano-particles Jackson Kimball et al. (2016);
Rusconi et al. (2017), with important applications in gyroscopy. For nano- or
micro- particles containing a smaller amount of spins, the total angular
momenta correction $\hbar N\hat{S}_{y}$ is negligible. In order to estimate
the relevance of this term in current experiments, one can note that
$\left\langle{\hat{p}}_{\theta}\right\rangle\approx\sqrt{kTI}$ is far above
$\hbar N$ even when using a highly-doped ($\approx$10 ppm) micron-sized
diamond at temperatures above $\mu$K. This condition will thus be verified for
all smaller particles, since the maximal density of NV centres scales with the
particle volume. In this considered temperature regime, we can thus treat the
angle as a scalar and neglect the contribution from the angular spin momentum,
bearing in mind that this approximation drops when using an ultra-cold
oscillator.

The remaining step is to diagonalize the spin part of the Hamiltonian. Apart
from in ref. Perdriat et al. (2021), experiments currently operate in the
regime $\gamma_{e}B\ll D$. We thus assume this condition to be fulfilled in
the following. We also assume small mixing between the two NV- excited states,
which is satisfied under the condition
$\sin^{2}{\theta^{\prime}}\ll\cos{\theta^{\prime}}$. Lastly, we change
variable and shift the angle ${\theta}$ to the equilibrium position
$\theta^{\prime}$. Under those assumptions, the Hamiltonian reads (see
Appendix A or the detailed calculation):

$\displaystyle\hat{\mathcal{H}}^{\prime\prime\prime}_{\rm
lib}\simeq\frac{{{p}}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}{\theta}^{2}+\hbar(\omega_{+1}({\theta})\ket{+1^{\prime}}\bra{+1^{\prime}}+\omega_{0}({\theta})\ket{0^{\prime}}\bra{0^{\prime}}+\omega_{-1}({\theta})\ket{-1^{\prime}}\bra{-1^{\prime}}).$
(10)

The expression of the new eigenstates $\ket{\pm 1^{\prime}}$ and
$\ket{0^{\prime}}$ is listed in Appendix A. The frequencies
$\omega_{i}({\theta})=\omega_{i}+\beta_{i}{\theta}$ of the spin resonance are
plotted as a function of the angle $\theta$ for two different magnetic field
values in Figure 4a.

Figure 4: (a) Electronic spin transition energies of the NV- center as a
function of the angle $\theta$ for magnetic fields of 50 G (brown lines) and
300 G (blue lines). The black dots indicate the optimum angles for the largest
spin-torques. b) Optimum coupling strength $G_{\theta}$ on the $\ket{0}$ to
$\ket{-1}$ transition as a function of the external magnetic field. The shaded
area corresponds to the magnetic field values for which the optimized coupling
strength is very close to $\gamma_{e}B$. In both plots, numerical calculations
without approximations are shown by dashed lines.

It was shown in Tetienne et al. (2012) that the pumping process is not
modified to first order in the transverse magnetic field. We can thus neglect
the mixing contribution from the optical pumping process of the NV- center.
This is also manifest in Figure 3b where the contrast of the ODMR drops
significantly only with magnetic field values above $\approx$10 mT.

Similar to the calculation for the center of mass (CoM), we obtain the
Hamiltonian (see Appendix A):

$\displaystyle\hat{\mathcal{H}}^{\prime}_{{\rm NV}+\mu\rm
w}/\hbar\simeq-\Delta_{+1}({\theta})\ket{+1^{\prime}}\bra{+1^{\prime}}-\Delta_{0}({\theta})\ket{0^{\prime}}\bra{0^{\prime}}-\Delta_{-1}({\theta})\ket{-1^{\prime}}\bra{-1^{\prime}}+\frac{\Omega}{2}\hat{S}_{x},$
(11)

where $\Delta_{+1}({\theta})=\omega-\omega_{+1}({\theta})$,
$\Delta_{-1}({\theta})=\omega-\omega_{-1}({\theta})$ and
$\Delta_{0}({\theta})=-\omega_{0}({\theta})$. We redefined
$\Omega\equiv\Omega\cos{\theta^{\prime}}$. The eigenstates are listed in
Appendix A. Choosing a microwave tuned close to resonance with the
$\ket{0^{\prime}}$ to $\ket{+1^{\prime}}$ transition allows us to restrict the
study to the Hamiltonian of a two-level system

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{{\rm NV}+\mu\rm
w}/\hbar\simeq\left(-\Delta+\beta_{1}{\theta}\right)\ket{1^{\prime}}\bra{1^{\prime}}+\beta_{0}{\theta}\ket{0^{\prime}}\bra{0^{\prime}}+\frac{\Omega}{2}(\ket{0}\bra{1^{\prime}}+\ket{1^{\prime}}\bra{0}).$
(12)

Here $\Delta=\Delta_{+1}-\Delta_{0}$ and $\Omega\equiv\frac{\Omega}{\sqrt{2}}$
to take into account the $\sqrt{2}$ factor in the $\hat{S}_{x}$ operator of
the spin-1 system. We have fixed the energy reference to be the energy of the
$\ket{0^{\prime}}$ state.

The second term $\beta_{0}{\theta}\ket{0^{\prime}}\bra{0^{\prime}}$ operates
as a small shift of the angular position when the NV- ground state is
populated and can be interpreted as a consequence of van Vleck paramagnetism.
The mixing between the ground and excited states induced by the transverse
magnetic field indeed generates a non-zero magnetic moment in the NV- center
Pellet-Mary et al. (2021); Perdriat et al. (2021). This term gives rise to a
new equilibrium position in experiments because the $\ket{0^{\prime}}$ is
populated by the green laser.

At this point, a word of caution is in order if one wishes to translate the
above description to ensembles of NV centers. We indeed need to check for the
consistency of our initial assumptions about independent NV directions in the
presence of strong van Vleck paramagnetism. This paramagnetism is present in
the absence of microwave excitation, so all NV orientations contribute to give
a spin-torque in the ground state, the extent of which depends upon the
transverse magnetic field amplitude for each orientation. When including the
four orientations, the first-order effect is a slight reduction of the van
Vleck paramagnetic susceptibility that can be estimated for a single
orientation. This is due to an overall spatial averaging of the magnetizations
from the four NV orientations, which should be recast in $\beta_{0}$. The
reader can find more information in Pellet-Mary et al. (2021).

Let us redefine the center of the mechanical resonator to be around the
equilibrium angular position when the population is in the $\ket{0^{\prime}}$
state by including the van Vleck shift in $\Delta$. Then, by defining
$G_{\theta}=\beta_{1}-\beta_{0}$ to be the single-spin mechanical constant,
the Hamiltonian of the system can finally be written as the simple two-level
atom Hamiltonian:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{{\rm NV}+\mu\rm
w}/\hbar=\begin{pmatrix}-\Delta(\theta)&\Omega/2\\\ \Omega/2&0\end{pmatrix},$
(13)

where $\Delta(\theta)=\Delta-G_{\theta}{{\theta}}$ is the frequency difference
between the states $\ket{0^{\prime}}$ and $\ket{1^{\prime}}$. The Hamiltonian
for the librational mode is thus the same as for the center of mass (Equation
(6)), when replacing the coupling constant $G_{z}$ by $G_{\theta}$.

The optimum coupling strength $G_{\theta}$ on the $\ket{0}$ to $\ket{-1}$
transition is plotted in Figure 4b. Ones finds that
$G_{\theta}\approx\gamma_{e}B$ when $\gamma_{e}B<200$ MHz. There is a good
match between analytical numerical calculations in this range of magnetic
fields. Slight deviations between analytical treatment and numerical
calculations are visible when $\gamma_{e}B>200$ MHz due to stronger state
mixing as $\gamma_{e}B$ approaches $D$.

##  V Sensing the Motion of a Trapped Particle Using NV- Centers

As we just described, the ESR frequencies depend strongly on the angle of the
homogeneous magnetic field with respect to the NV axis, and on the center of
mass in a magnetic field gradient. In the presence of a microwave drive on the
diamond, the change in the photoluminescence from the NV- centers can then be
used as a marker of the center of mass motion of diamonds or distant
ferromagnetic particles, as pioneered in experiments with tethered oscillators
Arcizet et al. (2011); Kolkowitz et al. (2012). With trapped diamond
particles, the angular dependence of the NV center energy levels also means
that it is possible to detect their full rotation and even their librational
motion.

Figure 5a shows an ODMR obtained from optically trapped diamonds in water
featuring broad lines due to significant angular Brownian motion in the
presence of a magnetic field Horowitz et al. (2012). Similar ODMR shapes were
observed in Delord et al. (2017); Neukirch et al. (2015). These results show
that NV centers are an efficient tool to measure the rotation of particles.
Further, the fully rotating regime may also shed new light on geometric phases
acquired by NV centers as well as offering perspectives for efficient sensing
of magnetic fields in the rotating frame Wood et al. (2018).

Figure 5:  (a) ODMR for small ensembles of nano-diamond trapped in water,
under increasing magnetic fields. Adapted form Horowitz et al. (2012). (b)
ODMR spectrum for a small ensemble of NV- centers in a levitated micro-diamond
in a magnetic field of about 50 G. Eight peaks corresponding to the four
orientations of the NV- centers, are observed. Reprinted figure with
permission from Delord et al. (2018). Copyright 2018 by the American Physical
Society.

Note that, additionally, control of the trapping laser polarisation allows
tuning the NV angles Geiselmann et al. (2013). In Delord et al. (2017, 2018),
ODMR from diamond particles in Paul traps was also reported on angularly
stable micro-diamonds (with typical librational frequencies in the kHz range).
Figure 5b shows the PL of NV- centers from a diamond containing about 1000 NV-
centers. Eight ODMR lines were observed due to the four projections of the
magnetic field onto the four possible NV axes in the diamond cristalline
structure and thanks to the efficient particle angular confinement. Assuming
thermalization with a gas at a temperature close to T=300 K, the angular
standard deviation is $\Delta\theta\approx$ 1 mrad (see Section II). The
expected extra broadening of the ODMR lines in a magnetic field $B\approx 50$
G is on the order of $G_{\theta}\Delta\theta\approx\gamma B\Delta\theta=150$
kHz. This value is smaller than the width given by the dipolar coupling
between the NV and P1 centers (see Section III).

Two recent experiments have even shown read out of the harmonic motion of a
trapped particle using NV centers Huillery et al. (2020); Gieseler et al.
(2020). In the first experiment Huillery et al. (2020), a hybrid diamond-
nickel particle was levitated in a Paul trap. The NV- centers in the diamond
were employed to read-out the librational motion. In the second experiment
Gieseler et al. (2020), a magnet was levitating on top of superconducting
sheet and a diamond containing a single NV- center was brought in the vicinity
of the magnet to read-out its center of mass motion.

In order for NV centers to efficiently detect the center of mass motion,
Gieseler et al. Gieseler et al. (2020) used broadband magnetic noise to excite
the motion of the trapped micro-magnet (diameter $\approx 15~{}\upmu$m). The
magnetic field thermal noise at the NV location, generated by the 100
$~{}\upmu$m distant driven trapped magnet, was then observed in the power
spectral density (PSD) of the NV PL evolution. Thanks to the high quality
factor of the oscillator ($Q\approx 10^{6}$), tuning a microwave to the blue
side of the ODMR signal, as shown in the top of Figure 6a, resulted in a sharp
peak at the mechanical oscillator frequency.

Similarly, Huillery et al. Huillery et al. (2020) reported NV-based detection
of the motion of a $10~{}\upmu$m hybrid ferromagnet/diamond particle. The
latter was levitated in a Paul trap and the libration was detected in the time
domain after parametric excitation of the magnetically confined librational
mode. Figure 6b shows the ODMR (top trace) and the ring down of the
librational mode (below) detected both using scattered light and the NV PL.
Notably, the PL change was delayed with respect to the instantaneous particle
motion due to a time lag between the motion and the magnetization.

These studies open a path towards the NV read-out of the Brownian motion of
trapped harmonic oscillators. To reach this limit, more involved dynamical
decoupling (DDC) techniques Kolkowitz et al. (2012) could be used, with
perspectives for attaining the zero-point motion sensitivity of the
oscillator. Rabi oscillations, Ramsey and spin echoes from NV centers in a
trapped diamond have in fact been observed already without significant
deterioration of the $T_{2}$ from either the charge noise or the angular
Brownian motion Delord et al. (2018). Rabi oscillations were also observed in
Pettit et al. (2017) using an optically trapped nanodiamond under weak
magnetic fields. DDC would, however, require to enter the regime where the
trapping frequency of the trapped mechanical oscillator exceeds the
decoherence rate $\Gamma_{2}^{*}$ of the NV spins. We discuss ways to achieve
this in Section VIII.1 .

Figure 6: NV- sensing of the motion of trapped particles. (a) Top trace: ODMR
on a $|m_{s}=0\rangle$ to $|m_{s}=+1\rangle$ transition from a fixed single
NV- center, located 100 $~{}\upmu$m away from the levitating magnet. Below:
PSD of the NV fluorescence signal when a microwave is tuned to the blue side
of the ODMR peak. The right narrow peak shows the magnet motion when the
microwave is modulated at 175 Hz. Adapted from Gieseler et al. (2020). (b) Top
trace: ODMR on a $|m_{s}=0\rangle$ to $|m_{s}=-1\rangle$ transition from
several NV- centers inside a hybrid ferro-diamond particle. Bottom trace: Time
trace showing the librational ring-down from the hybrid structure. The
detection was realized using both the laser scattered light and the PL.
Reprinted figure with permission from Huillery et al. (2020). Copyright 2020
by the American Physical Society.

##  VI Magnetic Forces and Torques on a Trapped Particle from the Spin of NV-
Centers

We have discussed experiments where the trapped particle motion was read-out
by NV spin states. In order to strongly couple a mechanical oscillator to
spins, however, not only should the mechanical motion affect the spin state,
but the spin should also alter the motion.

In this section, we review work on the reverse process where NV centers act on
the motion. We start by a discussion on optimum sensing of the force and the
torque induced by NV centers.

###  VI.1 Force and Torque Sensitivity

Presently, direct optical read-out of the motion of trapped particles using
scattered light is more efficient than using embedded NV centers. Most
experiments that use optical read-out are currently in the regime where the
influence of collisions with the background gas dominates. The detection noise
is thus determined by the resulting Brownian motion of the levitating object.
Note that the ultimate sensitivity should be given by the quantum back-action
of the measurement under low vacuum. This limited was recently reached in
experiments with trapped nano-spheres van der Laan et al. (2020); Jain et al.
(2016).

The most sensitive way to detect an external torque, with current technology,
is to use optical interferometric detection and to modulate the torque at the
mechanical frequency of the oscillator. Modulation of the amplitude of the NV
induced spin-torque can be done straightforwardly by modulating the microwave
tone that excites them. The minimum torque $\tau^{\rm min}_{s}$ that can be
detected in a time $\delta T$ is then obtained by balancing the resulting
signal amplitude and the standard deviation of the Brownian motion noise. One
finds

$\displaystyle\tau_{s}^{\rm min}\sqrt{\delta
T}=\sqrt{4kT\gamma_{\theta}I}=\sqrt{\frac{4kTK_{t}}{Q\omega_{\theta}}},$ (14)

where $Q=\omega_{\theta}/\gamma_{\theta}$ is the quality factor of the
mechanical oscillator libration, and $K_{t}$ is the trap rigidity, namely
$I\omega_{\theta}^{2}$.

Tethered torque sensors with sensitivities in the $10^{-23}$
N$\cdot$m/$\sqrt{\rm Hz}$ range are realized nowadays with state-of-the-art
nano-fabricated oscillators Kim et al. (2016). The largest sensitivities with
levitating systems have been achieved using levitating silica nanospheres that
are attached to form a dumbbell Ahn et al. (2020). The authors reached a
record sensitivity of $10^{-28}$ N$\cdot$m/$\sqrt{\rm Hz}$. Trapped
cristalline particles currently feature a lower torque sensitivity than
trapped amorphous particles, partly because of their currently lower quality
factor. A sensitivity of $10^{-23}$ N$\cdot$m/$\sqrt{\rm Hz}$ was attained in
Huillery et al. (2020) using soft ferromagnetic particles at room temperature
and at pressure levels in the $10^{-2}$ mbar range, very close to the state-
of-the-art torque sensing obtained at dilution fridge temperatures Kim et al.
(2016).

A similar formula can be obtained for the smallest detectable force acting on
the center of mass mode, by replacing $Q$ with $\omega_{z}/\gamma_{z}$,
$K_{t}$ by $m\omega_{z}^{2}$ and $I$ by $m$. Sensitivities in the
zeptonewton$/\sqrt{\rm Hz}$ range were reported experimentally Gieseler et al.
(2012); Hempston et al. (2017); Ranjit et al. (2016). Very large sensitivities
are also predicted for a magnet levitating on top of a punctured
superconductor sheet in the Meissner state Prat-Camps et al. (2017). Amongst
all spin–mechanical systems, trapped magnets currently feature record
sensitivities in the $10^{-18}$N$/\sqrt{\rm Hz}$ range Gieseler et al. (2020);
Vinante et al. (2020). Note that at the low pressure levels employed in these
experiments ($\approx 10^{-5}$ mbar), damping is not determined by collisions
with the background gas so the above model may not apply directly.

These last achievements not only show the capabilities of levitating systems,
close to the sensing capabilities of MEMS, but also offer immediate
perspectives for entering the quantum regime by coupling magnets to distant NV
centers.

###  VI.2 Observing NV Static Spin-Dependent Torque and Force

In this section, we discuss the parameters required for observing spin-
dependent forces and torques on a trapped particle. We focus on the expected
static shifts, assuming that the NV- centers have a lifetime that is greater
than the typical time required to shift the angle or the center of mass. The
latter is typically on the order of the period of the potential.

####  VI.2.1 Angular Displacement Using NV- Centers

Using Equation (10), one finds that the torque applied to the diamond in the
magnetic state $|-1^{\prime}\rangle$ reads

$\displaystyle\tau_{s}=-\left\langle\frac{\partial\hat{\mathcal{H}}^{\prime\prime\prime}_{\rm
lib}}{\partial\theta}\right\rangle=-\hbar N\beta_{-1}.$ (15)

The optimum shift will be found for $\theta=\pi/2$, where $\tau_{s}=-\hbar
N\gamma_{e}B$ so the largest single spin torque can be $\approx 10^{-25}$
N$\cdot$m.

Let us consider a particle with a diameter of 15 $~{}\upmu$m undergoing
Brownian motion in a harmonic trap under 0.1 mbar. Stable harmonic confinement
is readily attainted at 300 K in a Paul trap with these parameters Delord et
al. (2020), enabling a sensitivity of $\approx 10^{-21}$ N$\cdot$m/$\sqrt{\rm
Hz}$. This sensitivity is far from enabling single spin torque detection in a
reasonable amount of time. Reference Delord et al. (2020) reported the
observation of spin-torque in this parameter regime, albeit using $10^{9}$
spins that were all identically coupled with the libration. Figure 7-b
demonstrates mechanical detection of the $\approx 10^{-19}$ N$\cdot$m spin
torque averaged over a few minutes with a large signal to noise ratio (taken
from Delord et al. (2020)), revealing a novel efficient method to probe ESR
from NV centers.

In order to approach the single spin torque level, particles with much smaller
moment of inertia must be used. A levitating particle with a diameter of 100
nm and a modest $Q$ factor of $10^{4}$ would be sufficient to reach
sensitivities in the $10^{-25}$ N$\cdot$m/$\sqrt{\rm Hz}$ range. Single-spin
torque could then be discerned within one second in this experimentally
achievable regime.

Figure 7: (a) Sketch showing a NV-doped diamond in a Paul trap where the probe
laser reflection angle from the diamond surface depends on the spin state. (b)
Mechanically detected magnetic resonance observed by scanning the microwave
tone about the NV transitions. (c) ODMR in similar experimental conditions.
Adapted from Delord et al. (2020).

####  VI.2.2 Center of Mass Displacement Using NV- Centers

A similar discussion can be made to estimate the magnitude of the magnetic
force on the CoM of a trapped diamond subjected to a magnetic field gradient

$F_{s}=-\left\langle\frac{\partial\hat{\mathcal{H}}_{\rm com}}{\partial
z}\right\rangle=-\hbar N\gamma_{e}\frac{\partial B_{z}}{\partial z}.$ (16)

Under a gradient of $10^{5}$ T/m, one finds a single-spin force of $10^{-19}$
N. These large gradients typically require trapped micron-size magnets a few
hundreds of nanometers from the NV center. Although less attractive for
quantum applications, ensembles of spins could be used to bypass this
technical difficulty.

The spin-dependent force has not been observed thus far using NV- centers to
the best of our knowledge, even with large ensembles of spins. Even if strong
magnetic field gradients can be achieved, it may induce a large inhomogeneous
broadening when using ensembles, which will forbid efficient microwave
excitation of the whole ensemble of NV- centers in the selected orientation.
The magnetic field gradient will thus be bounded by the microwave excitation
Fourier width $\delta\nu$ in a pulsed excitation scheme, so that $\partial
B_{z}/\partial z|_{\rm max}=\Delta\nu/\gamma_{e}d$, where $d$ is the diameter
of the probed spin-ensemble. Note that $\Delta\nu$ can reach more than 100 MHz
using high power amplifiers and dedicated fast electronics Fuchs et al.
(2009).

One further difficulty with NV centers coupled with trapped particles is to
distinguish the center of mass motion from spin-induced torques. Indeed, a
magnetic field offset is required in order to magnetize the NV- centers with a
microwave which will rotate the trapped particle. The spin-induced angular
displacements may thus contribute at the same level as the center of mass
shifts, which complicates measurement analyses. One solution is to align the
magnetic field along the $[111]$ direction of the diamond, where no torque
should be applied to the particle. This is a notoriously difficult task when
using levitating particles where the angle between the diamond crystalline
axes and the main trap axes cannot not always be controlled.

Let us conclude by adding that single-spin forces were measured using Magnetic
Resonance Force Microscopy (MRFM) in Rugar et al. (2004). Acquisition times of
many hours and carefully engineered modulation techniques were employed. It is
likely that significant progress can be made by applying these techniques to
spin-mechanics with NV centers, with fascinating prospects for controlling the
motion of micro-objects using single long-lived qubits.

##  VII Dynamical Resonant Spin–Mechanical Interaction

Spin–mechanical systems can show richer physics than just static torques and
forces. When calculating the torque and force in Sections V and VI, it was
implicitly assumed that the spin population does not change after applying the
microwave excitation. This is not the case in the strong spin–mechanical
coupling regime. Indeed, after the microwave excites the spin and gives rise
to a torque (or to a force), the particle angle (or position) changes, which
in turn changes the microwave detuning with respect to the ESR, hence the
magnetization. We discuss the resulting bistable and spin-spring effects
below.

Further, when the spin-torque resulting from the combined laser and microwave
induced magnetization is lagging behind the motion, exchange of heat between
the spin and the mechanical oscillator can take place. Experiments have
recently entered this regime Delord et al. (2020).

We provide a simplified analytical theory of such spin dynamical back-action.
We focus on the angular degree of freedom, but the calculation can be
straightforwardly carried out for the center of mass by making the replacement
$(\theta,G_{\theta})\rightarrow(z,G_{z})$.

###  VII.1 Bistability and Spin-Spring Effect

As in opto-mechanics, bistability and modified spring constants can occur. We
first describe these two effects, in the limit where the microwave
magnetization and the laser polarisation rates are faster than the mechanical
oscillator frequency. This means that no energy exchange between the motion
and the spin can take place.

####  VII.1.1 Spin-Bistability

The torque $\tau_{s}$ exerted on the particle due to the spin–mechanical
coupling can be evaluated using Equation (13). We obtain

$\displaystyle\hat{\tau}_{s}=-\frac{\partial\hat{\mathcal{H}}^{\prime\prime}_{{\rm
NV}+\mu\rm w}}{\partial\theta}=-\hbar
NG_{\theta}\ket{1^{\prime}}\bra{1^{\prime}}.$ (17)

In the dispersive limit where $\Omega\ll\Delta(\theta)$, the spins are mostly
in the lowest energy eigenstate $\ket{-}$ of
$\hat{\mathcal{H}}^{\prime\prime}_{{\rm NV}+\mu\rm w}$ where the torque reads:

$\displaystyle\tau_{s}=\bra{-}\hat{\tau}_{s}\ket{-}=-\hbar
NG_{\theta}|\bra{-}\ket{1^{\prime}}|^{2}\approx-\hbar
NG_{\theta}\left(\frac{\Omega}{\Delta(\theta)}\right)^{2}.$ (18)

This spin torque can be added to the restoring torque of the levitating
system. The new angular stable position can be found by solving the equation
$\tau_{s}+\tau_{\rm trapping}=0$. It gives rise to a third degree polynomial
equation for $\theta$

$\displaystyle\hbar
NG_{\theta}\Omega^{2}+I\omega_{\theta}^{2}\theta(\Delta+G_{\theta}\theta)^{2}=0.$
(19)

In the very same way as in opto-mechanics, two stable solutions for $\theta$
can be found when $\Delta<0$. Angular bistability can then occur when the
microwave is swept across the ESR transition.

####  VII.1.2 Dynamical Backaction: Spin-Spring Effect

Linearizing about an equilibrium position $\theta_{0}$, and introducing the
detuning $\bar{\Delta}=\Delta-G_{\theta}\theta_{0}$, we get

$\displaystyle\bra{-}\hat{\tau}_{s}\ket{-}\approx\tau_{s,0}+K_{s}(\theta-\theta_{0}),$
(20)

where

$\displaystyle\tau_{s,0}=\hbar
NG_{\theta}\left(\frac{\Omega}{\bar{\Delta}}\right)^{2}~{}~{}{\rm
and}~{}~{}K_{s}=-2\hbar NG_{\theta}^{2}\frac{\Omega^{2}}{\bar{\Delta}^{3}},$
(21)

in the limit of small angle shifts. This expression predicts a restoring
torque in the limit where $\bar{\Delta}>0$ (blue detuned with respect to the
spin transition).

In the presence of the green laser light, transitions from the dressed spin
states will alter this predicted shift. We will estimate it together with the
spin-cooling effect using a density matrix formulation in the following.

###  VII.2 Spin-Cooling

In the above estimation, we assumed that the spins react immediately to a
change in the motion. When the spin torque lags behind the motion, friction
forces can alter the motional temperature and lead to spin-cooling, as
calculated in Ge and Zhao (2018) and observed in Delord et al. (2020). In
order to evaluate the dynamical back-action from the spins to the mechanical
oscillator with retardation, one will include the dissipation of the
electronic spin. Lastly, since most experiments are operating in the so-called
adiabatic limit where the frequency of the mechanical oscillator is much
smaller than the spin dephasing rate, we will consider this regime for
simplicity and discuss the corresponding limits to spin-cooling.

####  VII.2.1 Equations of Motion

We will again assume that the microwave is tuned to the $\ket{0^{\prime}}$ to
$\ket{1^{\prime}}$ transition. When the longitudinal decay time $T_{1}\approx$
ms is longer than the time $1/\gamma_{\rm las}\ll 100~{}\mu$s it takes for
laser polarizing the NV spin, the magnetic state $\ket{-1^{\prime}}$ is not
populated. This can be ensured quite straightforwardly experimentally, using
laser powers in the hundreds of $\upmu$W range with standard microscope
objectives. We will assume that this is the case here. We can thus reduce the
study to the two level system described in Equation (13).

The von Neumann equation for the reduced two-by-two spin density matrix
$\hat{\rho}$ reads

$\displaystyle\frac{\partial\rho_{10}}{\partial t}$
$\displaystyle=(-\Gamma_{2}^{*}+i\Delta(\theta))\rho_{10}+i\frac{\Omega}{2}(2\rho_{11}-1)$
(22) $\displaystyle\frac{\partial\rho_{11}}{\partial t}$
$\displaystyle=-\gamma_{\rm
las}\rho_{11}+i\frac{\Omega}{2}(\rho_{10}-\rho_{10}^{*}),$ (23)

where $\Gamma_{2}^{*}$ is the inhomogeneous broadening of the NV- center and
$\gamma_{\rm las}$ is the optical pumping rate to the $\ket{0^{\prime}}$
state. We also assumed $\gamma_{\rm las}/2\ll\Gamma_{2}^{*}$, which is largely
satisfied in practice. Note that these equations are valid when the broadening
is purely homogeneous and of a Markovian nature. In general, the NV- centers
couple to slowly fluctuating spin baths that generally implies Gaussian ESR
lineshapes.

The above equations are coupled with the equation of motion of the particle
via

$\displaystyle I\frac{\partial^{2}\theta}{\partial
t^{2}}+I\gamma\frac{\partial\theta}{\partial
t}+I{\omega_{\theta}}^{2}\theta=\langle\hat{\tau}_{s}\rangle_{\mathcal{B}}+\tau_{L},$
(24)

where $\langle\hat{\tau}_{s}\rangle_{\mathcal{B}}=-\hbar NG_{\theta}\rho_{11}$
and $\mathcal{B}$ accounts for incoherent laser excitation to the ground
state, as well as pure dephasing due to dipolar coupling to the P1 centers.
Because of the $\theta$ dependency in $\Delta$, this system of equations is
nonlinear. Here, we study the system dynamics around a steady-state, which
will linearize the set of equations.

####  VII.2.2 Stationary Solutions

We introduce the steady-state quantities
$\rho_{11}^{0}=\langle\rho_{11}\rangle$,
$\rho_{10}^{0}=\langle\rho_{10}\rangle$, and
$\theta_{0}=\langle\theta\rangle$, where $\langle.\rangle$ denotes time
averaging. Writing the incoherent pumping rate to the magnetic state at the
angle $\theta_{0}$ as
${\Gamma_{0}}=\frac{\Omega^{2}\Gamma_{2}^{*}}{{\Gamma_{2}^{*}}^{2}+{\bar{\Delta}}^{2}}$,
we get:

$\displaystyle\rho_{11}^{0}=\frac{1}{2}\frac{\Gamma_{0}}{\gamma_{\rm
las}+\Gamma_{0}}.$ (25)

Using Equation (24), one finds the steady state solution for the angle to be

$\displaystyle
I{\omega_{\theta}}^{2}\theta_{0}=\langle\hat{\tau}_{s}\rangle^{0}_{\mathcal{B}}=-\hbar
NG_{\theta}\rho_{11}^{0}.$ (26)

This last equation gives a third degree polynomial equation for $\theta_{0}$.
Depending on the microwave detuning, there can either be one or two stable
solutions for $\theta_{0}$.

####  VII.2.3 Effective Susceptibility

Writing each spin and angle parameters in Equation (22)–(24) as
$f(t)=f^{0}+\delta f(t)$ and transforming them to the Fourier domain, these
equations can be recast into the compact expression
$\delta\theta(\omega)=\chi_{\rm eff}(\omega)\delta\tau_{L}(\omega),$ where the
susceptibility $\chi_{\rm eff}(\omega)$ reads

$\displaystyle\chi_{\rm eff}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{I(\omega_{\theta}^{2}-\omega^{2}-i\omega\gamma)-K_{s}(\omega)}.$
(27)

The quantity $K_{s}(\omega)$ is a dynamical spin-rigidity which quantifies the
response of the particle angle to a change in the spin-torque. The real part
of $K_{s}(\omega)$ gives rise to a shift of the mechanical oscillator
frequency while the imaginary part gives rise to a damping of the mechanical
motion. We can rewrite the susceptibility in the more condensed form

$\displaystyle\chi_{\rm eff}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{I(\tilde{\omega}_{\theta}^{2}-\omega^{2}-i\omega\tilde{\gamma})},$
(28)

with

$\displaystyle\tilde{\omega}_{\theta}=\omega_{\theta}\big{[}1-\frac{\real(K_{s}(\omega_{\theta}))}{2K_{t}}\big{]}\quad{\rm
and}\quad\tilde{\gamma}=\gamma\big{[}1+Q\frac{\imaginary(K_{s}(\omega_{\theta}))}{K_{t}}\big{]}.$
(29)

The modified damping and frequency of the mechanical oscillator have been
estimated in the limit $K_{t}\gg\real(K_{s}(\omega_{\theta}))$, where
$K_{t}=I\omega_{\theta}^{2}$ is the trap rigidity. $Q=\omega_{\theta}/\gamma$
is the quality factor of the trapped particle.

####  VII.2.4 Dynamical Spin-Rigidity in the Adiabatic Limit

In the adiabatic limit $|\partial\rho_{10}/\partial
t|\ll|(-\Gamma_{2}^{*}+i\Delta(\theta))\rho_{10}|$, we have

$\displaystyle\frac{\partial\rho_{11}}{\partial t}=-\gamma_{\rm
las}\rho_{11}-\frac{\Omega^{2}}{2\Gamma_{2}^{*}}\mathcal{L}(\theta)(2\rho_{11}-1),~{}~{}{\rm
where}~{}~{}\mathcal{L}(\theta)=\frac{1}{1+(\Delta(\theta)/\Gamma_{2}^{*})^{2}}.$
(30)

We now linearize this equation around the steady states and move to the
Fourier space. We find:

$\displaystyle K_{s}(\omega)=\hbar
N\bar{\Delta}\frac{(\alpha\tau)^{2}}{1+i\omega\tau},~{}~{}{\rm
where}~{}~{}\alpha=G_{\theta}\sqrt{\frac{\gamma_{\rm
las}\Gamma_{0}^{2}}{{\Gamma_{2}^{*}}\Omega^{2}}}~{}~{}{\rm
and}\quad\tau=(\gamma_{\rm las}+\Gamma_{0})^{-1}.$ (31)

Finally, using Equation (29), we obtain

$\displaystyle\tilde{\omega}_{\theta}=\omega_{\theta}\big{[}1+\frac{\hbar
N}{2K_{t}}\frac{(\alpha\tau)^{2}}{1+(\omega_{\theta}\tau)^{2}}\bar{\Delta}\big{]}~{}~{}{\rm
and}~{}~{}\tilde{\gamma}=\gamma\big{[}1-Q\frac{\hbar
N(\alpha\tau)^{2}(\omega_{\theta}\tau)}{K_{t}(1+(\omega_{\theta}\tau)^{2})}\bar{\Delta}\big{]}.$
(32)

As manifest in Equation (32), when $\omega_{\theta}\tau$ is close to $1$ and
$\overline{\Delta}<0$, the oscillator motion can be damped, and thus cooled
down. As in cavity opto-mechanics, the cooling is due to the retarded nature
of the torque / force. Here, the delay comes from the finite time $\tau$ it
takes to re-polarize the spins in the magnetic state after the oscillator is
displaced from equilibrium, as depicted in Figure 8a.

The change in damping induces a change in the oscillator temperature. Using
$\delta\theta(\omega)=\chi_{\rm eff}(\omega)\delta\tau_{L}(\omega),$ one finds
that the variance of the angle is

$\displaystyle S_{\theta}(\omega)=|\chi_{\rm eff}(\omega)|^{2}S_{T},$ (33)

where $S_{T}=2kTI\gamma$ is the two-frequency correlation of the Langevin
torques. It is estimated at the temperature $T$ of the gas molecules
surrounding the particle, in the regime where $\hbar\omega_{\theta}\ll kT$.
Using the fluctuation dissipation theorem, one obtains a simple relation
between the temperature $T_{f}$ in the presence of the NV- spins and the
temperature $T$ of the gas. One indeed has
$T_{f}=\frac{\gamma}{\tilde{\gamma}}T$.

For large quality factors and negative detunings, that is with a microwave
tuned to the red, a pronounced cooling can take place. This effect is strongly
analogous to Raman cooling or cavity cooling and was observed in Delord et al.
(2020) using diamonds levitated in a Paul trap. Figure 8b shows the power-
spectral density of two libration modes of a trapped diamond undergoing spin-
cooling and spin-heating. The limit to the cooling efficiency will ultimately
be given by back-action noise from either the radiation pressure from the
laser beam that is used to polarise the spins or by the atomic spin noise. As
in cavity opto-mechanics, the latter can be mitigated in the sideband resolved
regime (SRR), where the trapping frequency is larger than the spin dephasing
rate $\Gamma_{2}^{*}$.

The SRR regime has not been attained so far using trapped particles coupled
with NV- centers in any platform, to the best of our knowledge. Several
solutions are envisioned to increase the trapping frequency and/or reduce the
spin linewidth. One of them consists of coupling the NV electronic spins to
the nuclear spins of the nitrogen atoms Huillery et al. (2020) to make use of
the very long nuclear spin lifetime. Another solution is to employ distant
coupling schemes between NV centers in ultra-pure diamonds and strongly
confined ferromagnetic particles Huillery et al. (2020). If all other heat
sources occur at a rate smaller than the spin-cooling in the SRR, ground state
spin-cooling of the libration or the center of mass motion can then become a
reality, offering great perspectives for controlling the quantum state of the
motion with the NV spin.

Figure 8: (a) Spin-cooling mechanism described in the adiabatic limit. The two
angular potential wells in the spin states $|0\rangle$ and $|1\rangle$ are
offset from one another by $\delta\theta$ due to the spin-torque. They are
coupled via the laser and microwave tone at the rates $\gamma_{\rm las}$ and
$\Gamma$, respectively, see text for explanations. Here the microwave is tuned
to the red, enabling cooling. (b) PSD of two librational modes when the
microwave is tuned to the blue (trace i) to resonance (trace ii) and to the
red (trace iii) of the spin-resonance, respectively. Adapted from Delord et
al. (2020).

##  VIII Challenges Ahead for Levitated Spin-Mechanics

Levitated spin-mechanics offers prospects for a wide range of applications in
quantum science. The negatively charge NV- center in diamond is a system of
choice because of its robustness, which partly explains its increasing use in
present quantum technologies. Nevertheless, recent experimental
implementations have raised some technical difficulties. First and foremost,
there is poor levitation stability when decreasing the gas pressure. To date,
optical trapping of diamond has been reported only above mbar pressures
Frangeskou et al. (2018), while electrodynamics and diamagnetic traps operate
in the $10^{-3}$ to $10^{-4}$ mbar range Conangla et al. (2018); Hsu et al.
(2015). To go further and operate close to the standard quantum limit for
instance, the collisions with the background gas must be suppressed.
Furthermore, the control of diamond in terms of shape, size, and properties is
not as straightforward as for silica, for example, where chemical processes
allow the production of mono-disperse spherical particles. In the present
section, we highlight some of the issues that can be tackled in the next
generation of levitated spin–mechanical systems.

###  VIII.1 Production of Diamond

The diamonds that are employed for research purposes are almost always of
artificial origin. They are produced mainly using three methods, detailed in
the Appendix B. The most promising technique for achieving flawless diamonds
is chemical vapour deposition (CVD). In the different growth methods, the NV
concentration is controlled using different strategies that depend on the
initial nitrogen content in the diamond. When the sample contains large
nitrogen concentrations, it can be irradiated by electrons or alpha particles
to create vacancies. Diamond annealing then allows vacancy migration until
being stably associated with a nitrogen atom to form a NV- center. The
irradiation dose dictates the concentration of NV- centers per nano-diamond.
Typically, with this technique, a 100 nm diamond can be doped to contain one,
up to thousands of NV- centers. It has been shown that a similar result can
also be obtained using laser irradiation, with the benefit of controlling the
position (and the number) of the produced NV- centers Chen et al. (2017). When
starting with pure CVD diamonds, nitrogen is first implanted and converted to
NV- centers, with a few percent yield, during an annealing process.

###  VIII.2 Control of Diamond Shape and Properties

As discussed previously, thanks to the development of CVD grown diamonds and
implantation techniques, it is possible to finely control the purity, and the
number of hosted NV- centers in the diamond. Over the last decades, nano and
micro-fabrication of diamond has enabled the realization of increasingly
complex diamond structures Rani et al. (2021). Pillars of a controlled aspect-
ratio can be nano-fabricated, and even more advanced structures are achievable
using reactive ion etching technics. This is of particular interest for
trapping diamonds with large librational frequencies Delord et al. (2017).

The development of coated diamond initially intended for biology applications
may also increase the achievable control of the properties of trapped diamond.
A typical example is silica embedded diamonds, which allows obtaining
spherical particles, and that have been shown to favor the NV- centers
luminescence stability in optically levitated nano-diamonds Neukirch et al.
(2015).

Spin-mechanics with trapped objects will thus benefit from the tremendous
progress in diamond material science and continuously improved knowledge about
diamond and NV- centers. A potential drawback is that the production quantity
of diamonds may be limited. Besides, one could be interested in levitating a
specific diamond particle with physical properties that have been well-
characterized beforehand. In both cases, statistical trapping procedures that
start from a sprayed colloidal particle solution, and rely on random trapping
events are not ideal. The different recent approaches proposed for in situ
trapping, using a piezo shacking, a laser impulsion Kuhn et al. (2015); Bykov
et al. (2019), or trapping particles embedded in a polymer thin film Hsu et
al. (2015), are very encouraging for the developments of on-demand trapping.
Coupling these technics with an in situ characterization using a confocal
microscope could solve these issues.

###  VIII.3 Increasing the NV- Concentration

We have seen that increasing the number of NV- centers that couple to the
oscillator motion is a viable route towards observing strong spin–mechanical
effects. While the optimal density of NV- centers is generally a compromise
between sensitivity and coherence time $T_{2}^{*}$, other more exotic effects
start to appear when the spin density reaches a critical point.

For concentrations that are larger than $\sim$1 ppm (corresponding to a mean
distance between spins of $\approx$15 nm or a dipolar coupling strength of
$\approx$15 kHz), the dipolar coupling amongst NV spins plays an important
role. One important effect is the modification of the spin lifetime $T_{1}$
through dipolar coupling with other short lived NV- centers Jarmola et al.
(2012); Mrózek et al. (2015); Choi et al. (2017). This particular effect has
been used with a levitating diamond in a Paul trap to observe a resonant
change in the spins’ magnetic susceptibility Pellet-Mary et al. (2021).

Other collective effects between NV- centers include the cooperative
enhancement of the NV- centers’ dipole interaction, a phenomenon similar to
that of super-radiance described in Bachelard et al. (2011); PANAT and LAWANDE
(2002); Prasanna Venkatesh et al. (2018), and observed with a levitating
diamond in an optical tweezer by Juan et al. Juan et al. (2017).

###  VIII.4 Internal Temperature of Levitated Diamonds

Numerous studies have reported an increase in levitated diamond internal
temperature under vacuum conditions Delord et al. (2017); Hoang et al. (2016);
Pettit et al. (2017); Rahman et al. (2016). This heating is detrimental for
practical reasons since it may lead to the burning or melting of the levitated
particle. Internal temperature also induces extra quantum decoherence
channels, which may prevent the observation of macroscopic quantum effects and
impact the contrast of the spin resonance.

This heating was shown to be induced by laser absorption by the particle. The
final temperature is the result of a competition between absorption, heat
conduction to the surrounding residual gas, and black-body radiative exchange.
Figure 9a shows the expected internal temperature for different diamond
materials using the model detailed in references Frangeskou et al. (2018);
Chang et al. (2010). It can be seen that the expected final temperature can
vary by one order of magnitude depending on the purity of the diamond
material.

Figure 9: Internal heating of levitated diamonds. (a) Expected internal
temperature of optically levitated nano-diamonds of different purity as a
function of background gas pressure. The temperature dependence is shown for
standard commercial diamond (–$\cdot$–) up to the best expected grade (dotted
line). The red-line corresponds to silica. Adapted from Frangeskou et al.
(2018). (b) Electron spin resonance from optically levitated nano-diamonds at
different pressures. A clear shift of the central ESR frequency demonstrates
diamond heating. Adapted from Hoang et al. (2016). (c) Internal temperature of
an optically levitated nano-diamond hosting a single NV- center as a function
of trapping laser power. Adapted with permission from Pettit et al. (2017). ©
The Optical Society.

Studies conducted on high purity CVD bulk diamonds have shown that a very low
laser absorbance is achievable Webster et al. (2015), even below the silica
absorption levels. Currently however, most experiments with levitated diamonds
observe orders of magnitude higher heating rates. Any defects from the diamond
matrix may indeed worsen the absorption First, NV- color centers may
contribute to the absorption due to non-radiative pathways of the electron
population (dashed lines in Figure 9a). Most importantly, other diamond matrix
defects will also play their role, from isolated nitrogen atoms to graphite on
the diamond surface, through grain boundary or other atomic impurities Walker
(1979).

The spin of the NV- centers is actually an invaluable tool to estimate the
diamond temperature Acosta et al. (2010); Hoang et al. (2016); Delord et al.
(2017) because the zero-field splitting $D$, between $m_{s}=0$ and $m_{s}=\pm
1$ states is temperature dependent, and well characterized Toyli et al.
(2012). One can measure the internal temperature of a levitated diamond by
measuring the ESR of the NV- centers it hosts Hoang et al. (2016); Delord et
al. (2017), see Figure 9b,c which shows heating measurements from trapped
diamonds under vacuum.

Note that instead of improving the properties of trapped diamonds that are
destined to be trapped under a magnetic field produced by a fixed magnet, the
roles of the diamond and the magnet can be reversed. Schemes already discussed
in this review Huillery et al. (2020); Gieseler et al. (2020) involving
trapped magnets were recently proposed. In these proposals, the diamond
containing NV- centers is attached to a large heat sink, so it does not heat
up significantly and can readily be made from CVD for improved spin-
properties. Furthermore, no strong laser needs to be shone onto the trapped
magnet.

###  VIII.5 Beyond NV- Centers and Diamond

The present review focused on NV- centers in diamond, which is by far the most
studied system for levitated spin-mechanics. However, our discussions may be
applied to other color centers with similar behavior, and also solve some of
the issues discussed in this section. Typically, over recent decades, color
centers in SiC have been shown to own an optically addressable spin resonance
Koehl et al. (2011). SiC can benefit from silicon-like technologies, which
offer a good level of control over the material and its nano fabrication.
Ultimately, with the recent isolation of single color centers Redjem et al.
(2020), even silicon may become an excellent platform for spin-mechanics.

##  IX Conclusions

In this review, we presented recent levitated spin-mechanics experiments,
focusing specifically on NV- centers in diamonds. We introduced a formalism
describing the spin–mechanical interactions in these experiments and
highlighted the advantages and limitations of this interaction scheme. We
discussed the technical challenges that remain towards taking the full benefit
of levitated spin-mechanics. The common goals and emulation from atomic and
solid-state physic, material science, and the levitation scientific
communities is a cornerstone for success in this field. This will undoubtedly
lead to innovative and exciting experiments and applications.

This research was funded by the Île-de-France Region in the framework of DIM
SIRTEQ and by the Investissements d’Avenir of LabEx PALM
(ANR-10-LABX-0039-PALM).
The authors would like to acknowledge fruitful discussions with Samuel
Deléglise, Itsik Cohen, Cosimo Rusconi, Oriol Romero-Isart and Benjamin
Stickler and thank James Millen for proof reading the manuscript.

Authors Contributions : Conceptualization, G.H., L.R. and M.P.; methodology,
G.H., M.P., P.H. and L.R.; formal analysis, G.H. and M.P; writing—review and
editing, G.H., L.R., C.P.-M., M.P. All authors have read and agreed to the
published version of the manuscript.

##  Appendix A Derivation of a Simplified Hamiltonian

In this section, we detail the calculation of the simplified spin-libration
Hamiltonian Equation (13). Several unitary transformations will be performed
on the Hamiltonian. For the sake of clarity, we will divide our Hamiltonian
into three different parts to do the calculation and see the effect of each
transformation on the different parts. We write

$\displaystyle\hat{\mathcal{H}}=\hat{\mathcal{H}}_{\rm
mecha}+\hat{\mathcal{H}}_{\rm NV}+\hat{\mathcal{H}}_{\mu\rm w},$ (34)

where

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}=\frac{\hat{p}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2},$
(35) $\displaystyle\hat{\mathcal{H}}_{\rm NV}=\hbar
D\hat{S}_{z^{\prime}}^{2}+\hbar\gamma_{e}B\hat{S}_{z},$ (36)

and

$\displaystyle\hat{\mathcal{H}}_{\mu\rm w}=\hbar\Omega\cos({\omega
t})\hat{S}_{x}.$ (37)

A few approximations have been made to derive this Hamiltonian. We suppose
that $\gamma_{e}B$ is smaller than the zero-field splitting $D$ of the NV-
center. We suppose the transverse magnetic field $B_{\perp}$ and the
longitudinal magnetic field $B_{\parallel}$ to be of the same order of
magnitude. We also suppose that the angular momentum of the particle, given by
the mean value $\left\langle\hat{p}_{\theta}\right\rangle$, is a few order of
magnitude larger than the typical spin momenta $\hbar
N\left\langle\hat{S}_{y}\right\rangle$. This assumption is valid for micron-
sized particles. Finally, we consider that $\Omega\ll\gamma_{e}B$.

###  A.1 Moving to the Particle Frame

We move to the particle frame by performing the unitary transformation
$\hat{U}=e^{i{\hat{\theta}}\hat{S}_{y}}$. This frame is relevant because the
eigenstates of the new spin operator $\hat{S}_{z}$ are now the ones where the
optical pumping process of the green laser takes place. In this frame, the NV-
Hamiltonian reads:

$\displaystyle\hat{\mathcal{H}}^{\prime}_{\rm NV}=\hbar
D\hat{S}_{z}^{2}+\hbar\gamma_{e}B\left(\cos{{\hat{\theta}}}\hat{S}_{z}-\sin{{\hat{\theta}}}\hat{S}_{x}\right).$
(38)

The mechanical part of the Hamiltonian becomes

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}^{\prime}=\frac{(\hat{p}_{\theta}-\hbar\hat{S}_{y})^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2}.$
(39)

One of the assumption that we have made is to neglect the spin contribution to
the total angular momenta of the system, which means $\hbar
N\left\langle\hat{S}_{y}\right\rangle\ll\left\langle\hat{p}_{\theta}\right\rangle$.
Thus, we have:

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}^{\prime}\simeq\frac{\hat{p}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2}.$
(40)

The microwave part of the Hamiltonian simply becomes

$\displaystyle\hat{\mathcal{H}}_{\mu\rm w}^{\prime}=\hbar\Omega\cos({\omega
t})\left(\cos{{\hat{\theta}}}\hat{S}_{x}+\sin{{\hat{\theta}}}\hat{S}_{z}\right).$
(41)

###  A.2 Diagonalization of the NV- Hamiltonian

The NV- part of the Hamiltonian is diagonalized in the perturbative limit
$\frac{\gamma_{e}B}{D}\ll 1$. We introduce the operators
$\hat{u}_{\perp}=\frac{\gamma_{e}B}{D}\sin{\hat{\theta}}$ and
$\hat{u}_{\parallel}=\frac{\gamma_{e}B}{D}\cos{\hat{\theta}}$. Thus, we have
$\left\langle\hat{u}_{\perp}\right\rangle\ll 1$ and
$\left\langle\hat{u}_{\parallel}\right\rangle\ll 1$. Using these operators,
the Hamiltonian of the NV reads:

$\displaystyle\hat{\mathcal{H}}^{\prime}_{\rm NV}=\hbar D\hat{S}_{z}^{2}+\hbar
D(\hat{u}_{\parallel}\hat{S}_{z}-\hat{u}_{\perp}\hat{S}_{x}).$ (42)

We can treat the second part of this Hamiltonian as a perturbation and move to
the basis where this Hamiltonian is diagonal to second order in
$\frac{\gamma_{e}B}{D}$. We consider the unitary transformation:

$\displaystyle\hat{U}^{\prime\prime}=\begin{pmatrix}1-\frac{\hat{u}_{\perp}^{2}}{4}&\frac{\hat{u}_{\perp}}{\sqrt{2}}(1-\hat{u}_{\parallel})&-\frac{\hat{u}_{\perp}^{2}}{4}\\\
-\frac{\hat{u}_{\perp}}{\sqrt{2}}(1-\hat{u}_{\parallel})&1-\frac{\hat{u}_{\perp}^{2}}{2}&-\frac{\hat{u}_{\perp}}{\sqrt{2}}(1+\hat{u}_{\parallel})\\\
-\frac{\hat{u}_{\perp}^{2}}{4}&\frac{\hat{u}_{\perp}}{\sqrt{2}}(1+\hat{u}_{\parallel})&1-\frac{\hat{u}_{\perp}^{2}}{4}\end{pmatrix}.$
(43)

The new states we are considering by applying this transformation are defined
by

$\displaystyle\ket{+1^{\prime}}$
$\displaystyle=\left(1-\frac{\hat{u}_{\perp}^{2}}{4}\right)\ket{+1}-\frac{\hat{u}_{\perp}}{\sqrt{2}}(1-\hat{u}_{\parallel})\ket{0}-\frac{\hat{u}_{\perp}^{2}}{4}\ket{-1}$
(44) $\displaystyle\ket{0^{\prime}}$
$\displaystyle=\frac{\hat{u}_{\perp}}{\sqrt{2}}(1-\hat{u}_{\parallel})\ket{+1}+\left(1-\frac{\hat{u}_{\perp}^{2}}{2}\right)\ket{0}+\frac{\hat{u}_{\perp}}{\sqrt{2}}(1+\hat{u}_{\parallel})\ket{-1}$
(45) $\displaystyle\ket{-1^{\prime}}$
$\displaystyle=-\frac{\hat{u}_{\perp}^{2}}{4}\ket{+1}-\frac{\hat{u}_{\perp}}{\sqrt{2}}(1+\hat{u}_{\parallel})\ket{0}+\left(1-\frac{\hat{u}_{\perp}^{2}}{4}\right)\ket{-1}.$
(46)

We have
$\hat{U}^{\prime\prime\dagger}\hat{U}^{\prime\prime}=\hat{I}d+o(||(\hat{u}_{\perp},\hat{u}_{\parallel})||^{2})$
which means that $\hat{U}^{\prime\prime}$ is unitary up to second order in
$||(\hat{u}_{\perp},\hat{u}_{\parallel})||$. Under this transformation, the NV
Hamiltonian reads

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{\rm NV}\simeq\hbar
D\begin{pmatrix}1+\hat{u}_{\parallel}+\frac{\hat{u}_{\perp}^{2}}{2}&0&\frac{\hat{u}_{\perp}^{2}}{2}\\\
0&-\hat{u}_{\perp}^{2}&0\\\
\frac{\hat{u}_{\perp}^{2}}{2}&0&1-\hat{u}_{\parallel}+\frac{\hat{u}_{\perp}^{2}}{2}\end{pmatrix},$
(47)

which can be written as:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{\rm NV}\simeq\hbar
D\begin{pmatrix}1+\frac{\gamma_{e}B}{D}\cos{\hat{\theta}}+\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}&0&\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}\\\
0&-\left(\frac{\gamma_{e}B}{D}\right)^{2}\sin{\hat{\theta}}^{2}&0\\\
\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}&0&1-\frac{\gamma_{e}B}{D}\cos{\hat{\theta}}+\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}\end{pmatrix}.$
(48)

Furthermore, we suppose that the angle satisfies
$\left\langle\hat{u}_{\perp}\right\rangle\simeq\left\langle\hat{u}_{\parallel}\right\rangle$
which implies that
$\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\left\langle\sin{\hat{\theta}}\right\rangle^{2}}{2}\ll\frac{\gamma_{e}B}{D}\left\langle\cos{\hat{\theta}}\right\rangle$.
We can then neglect non diagonal terms in this regime and we obtain a diagonal
Hamiltonian:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{\rm NV}\simeq\hbar
D\begin{pmatrix}1+\frac{\gamma_{e}B}{D}\cos{\hat{\theta}}+\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}&0&0\\\
0&-\left(\frac{\gamma_{e}B}{D}\right)^{2}\sin{\hat{\theta}}^{2}&0\\\
0&0&1-\frac{\gamma_{e}B}{D}\cos{\hat{\theta}}+\left(\frac{\gamma_{e}B}{D}\right)^{2}\frac{\sin{\hat{\theta}}^{2}}{2}\end{pmatrix}.$
(49)

Let us apply this transformation to the mechanical part of the Hamiltonian.
The unitary transformation $\hat{U}^{\prime\prime}$ only depends on
$\hat{\theta}$ so it commutes with it. This transformation can be written
$\hat{U}^{\prime\prime}=\hat{I}d+u\hat{V}(\hat{\theta})+o(|u|)$ with
$u=\frac{\gamma_{e}B}{D}$ and
$\hat{V}(\hat{\theta})=\hat{A}\sin{\hat{\theta}}$ where
$\hat{A}=\frac{1}{\sqrt{2}}(\ket{+1}\bra{0}-\ket{0}\bra{+1}+\ket{-1}\bra{0}-\ket{0}\bra{-1})$.
Furthermore, we have $\hat{V}(\hat{\theta})^{\dagger}=-\hat{V}(\hat{\theta})$.
We have:

$\displaystyle\hat{U}^{\prime\prime\dagger}\hat{p}_{\theta}\hat{U}^{\prime\prime}=\left(\hat{I}d-u\hat{V}(\hat{\theta})+o(|u|)\right)\hat{p}_{\theta}\left(\hat{I}d+u\hat{V}(\hat{\theta})+o(|u|)\right)$
(50)
$\displaystyle\hat{U}^{\prime\prime\dagger}\hat{p}_{\theta}\hat{U}^{\prime\prime}=\hat{p}_{\theta}+u\commutator{\hat{p}_{\theta}}{\hat{V}(\hat{\theta})}+o(|u|)$
(51)
$\displaystyle\hat{U}^{\prime\prime\dagger}\hat{p}_{\theta}\hat{U}^{\prime\prime}=\hat{p}_{\theta}+u\hat{A}\commutator{\hat{p}_{\theta}}{\sin{\hat{\theta}}}+o(|u|)$
(52)
$\displaystyle\hat{U}^{\prime\prime\dagger}\hat{p}_{\theta}\hat{U}^{\prime\prime}=\hat{p}_{\theta}-i\hbar
u\hat{A}\cos{\hat{\theta}}+o(|u|).$ (53)

We can safely neglect the second term under the initial assumption
$\left\langle\hat{p}_{\theta}\right\rangle\gg\hbar Nu$. Thus, we get:

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}^{\prime\prime}\simeq\frac{\hat{p}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}(\hat{\theta}-\theta^{\prime})^{2}.$
(54)

Furthermore, this transformation does not affect the microwave Hamiltonian to
first order in $\frac{\gamma_{e}B}{D}$ so we get:

$\displaystyle\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime}\simeq\hbar\Omega\cos({\omega
t})\left(\cos{{\hat{\theta}}}\hat{S}_{x}+\sin{{\hat{\theta}}}\hat{S}_{z}\right).$
(55)

###  A.3 Equilibrium Position of the Paul Trap

We apply the unitary transformation
$\hat{U}^{\prime\prime\prime}=e^{i\theta^{\prime}\hat{p}_{\theta}/\hbar}$
which redefines $\hat{\theta}$ as $\hat{\theta}-\theta^{\prime}$. This
transformation does not affect the eigenstates of the spin and we obtain to
first order in $\hat{\theta}$:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime\prime}_{\rm
NV}\simeq\hbar(\omega_{+1}+\beta_{+1}\hat{\theta})\ket{+1^{\prime}}\bra{+1^{\prime}}+\hbar(\omega_{0}+\beta_{0}\hat{\theta})\ket{0^{\prime}}\bra{0^{\prime}}+\hbar(\omega_{-1}+\beta_{-1}\hat{\theta})\ket{-1^{\prime}}\bra{-1^{\prime}},$
(56)

with

$\displaystyle\omega_{+1}$
$\displaystyle=D+\gamma_{e}B\cos(\theta^{\prime})+\frac{(\gamma_{e}B)^{2}}{D}\frac{\sin{(\theta^{\prime})}^{2}}{2}$
(57) $\displaystyle\omega_{0}$
$\displaystyle=-\frac{(\gamma_{e}B)^{2}}{D}\sin{(\theta^{\prime})}^{2}$ (58)
$\displaystyle\omega_{-1}$
$\displaystyle=D-\gamma_{e}B\cos(\theta^{\prime})+\frac{(\gamma_{e}B)^{2}}{D}\frac{\sin{(\theta^{\prime})}^{2}}{2}$
(59) $\displaystyle\beta_{i}$
$\displaystyle=\frac{\partial\omega_{i}}{\partial\theta^{\prime}}.$ (60)

This transformation shifts the equilibrium position of the mechanical
oscillator by $\theta^{\prime}$ and we get:

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}^{\prime\prime\prime}\simeq\frac{\hat{p}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}\hat{\theta}^{2}.$
(61)

The microwave Hamiltonian reads

$\displaystyle\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime\prime}\simeq\hbar\Omega\cos({\omega
t})\left(\cos({\theta^{\prime}+\hat{\theta}})\hat{S}_{x}+\sin({\theta^{\prime}+\hat{\theta}})\hat{S}_{z}\right).$
(62)

###  A.4 Rotating Frame of the Micro-Wave

The last unitary transformation is to move to the microwave frame by
performing the unitary transformation
$\hat{U}^{\prime\prime\prime\prime}=e^{i\omega t\hat{S}_{z}^{2}}$. This
transformation is diagonal so it commutes with the NV- center Hamiltonian
which is also diagonal. As it is a time-dependent transformation, this will
add a shift in energy to both the $\ket{+1^{\prime}}$ and $\ket{-1^{\prime}}$
states which gives the Hamiltonian:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime\prime\prime}_{\rm NV}$
$\displaystyle\simeq$
$\displaystyle\hbar(-\Delta_{+1}+\beta_{+1}\hat{\theta})\ket{+1^{\prime}}\bra{+1^{\prime}}+\hbar(-\Delta_{0}+\beta_{0}\hat{\theta})\ket{0^{\prime}}\bra{0^{\prime}}$
(63) $\displaystyle+$
$\displaystyle\hbar(-\Delta_{-1}+\beta_{-1}\hat{\theta})\ket{-1^{\prime}}\bra{-1^{\prime}},$
(64)

by defining $\Delta_{+1}=\omega-\omega_{+1}$, $\Delta_{-1}=\omega-\omega_{-1}$
and $\Delta_{0}=-\omega_{0}$.

The mechanical Hamiltonian is not affected by this transformation as it
commutes with it. Finally, under the rotating wave approximation and by
neglecting first order terms in $\hat{\theta}$ which are negligible since
$\Omega\ll\gamma_{e}B$, we obtain:

$\displaystyle\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime\prime\prime}\simeq\hbar\frac{\Omega}{2}\cos({\theta^{\prime}})\hat{S}_{x}.$
(65)

Redefining $\Omega=\Omega\cos({\theta^{\prime}})$, we get:

$\displaystyle\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime\prime\prime}\simeq\hbar\frac{\Omega}{2}\hat{S}_{x}.$ (66)

Finally, the Hamiltonian can be simply written as:

$\displaystyle\hat{\mathcal{H}}^{\prime\prime\prime\prime}=\hat{\mathcal{H}}_{\rm
mecha}^{\prime\prime\prime\prime}+\hat{\mathcal{H}}_{\rm
NV}^{\prime\prime\prime\prime}+\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime\prime\prime},$ (67)

where

$\displaystyle\hat{\mathcal{H}}_{\rm
mecha}^{\prime\prime\prime\prime}=\frac{\hat{p}_{\theta}^{2}}{2I}+\frac{1}{2}I\omega_{\theta}^{2}\hat{\theta}^{2}.$
(68) $\displaystyle\hat{\mathcal{H}}^{\prime\prime\prime\prime}_{\rm
NV}\simeq\hbar(-\Delta_{+1}+\beta_{+1}\hat{\theta})\ket{+1^{\prime}}\bra{+1^{\prime}}+\hbar(-\Delta_{0}+\beta_{0}\hat{\theta})\ket{0^{\prime}}\bra{0^{\prime}}+\hbar(-\Delta_{-1}+\beta_{-1}\hat{\theta})\ket{-1^{\prime}}\bra{-1^{\prime}}$
(69)

and

$\displaystyle\hat{\mathcal{H}}_{\mu\rm
w}^{\prime\prime\prime\prime}\simeq\hbar\frac{\Omega}{2}\hat{S}_{x}.$ (70)

##  Appendix B Diamond Synthesis

There are three main diamond synthesis methods:

  * •

The _HPHT_ process (high pressure, high temperature): A carbon precursor is
brought under conditions of high pressure (typically > 5 GPa) and high
temperature (T $\approx$ 2000$~{}^{\circ}$C) in order to create diamond. While
this approach has been known since the 1950s, the control of impurities in the
diamond is not straigthforward. The diamonds produced are often rich in
nitrogen impurities, typically around 200 ppm. Most recent works on diamond
levitation used HPHT diamonds due to their ease of use and commercial
availability.

  * •

The _CVD_ growth (chemical vapor deposition). A reactor is used to deposit
carbon atoms from a methane gas, layer by layer on a diamond substrate. It is
then possible to finely control the impurities present in the diamond. It is
the method of choice to create diamonds with very high purity. The
concentration of paramagnetic species such a nitrogen or silicon can indeed be
reduced below the detection level, and the concentration of ${}^{13}C$ atoms
below natural abundance. Importantly, CVD growth also enables NV- center
doping at any time during the growth Achard et al. (2020).

  * •

_Detonation_ nanodiamonds are obtained by an explosive reaction from a carbon
precursor. This approach provides very small nanodiamonds, typically <10 nm,
which are often highly graphitized Bradac et al. (2010). Such diamonds are
thus not suited for the applications discussed in the present review.

Although recent efforts towards micro or nano-diamond creation have been made
towards direct CVD Feudis et al. (2020); Tallaire et al. (2019) and HPHT
Mindarava et al. (2020) synthesis, the general strategy is to mill bulk
diamonds in order to obtain nano (and even micro) diamonds. The size of the
diamonds is then selected by centrifugation.

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