# Levitated optomechanics: A tutorial and perspective

George Winstone  1 Mishkat Bhattacharya  2 Andrew A. Geraci  1,* Tongcang Li
3 Peter J. Pauzauskie  4,5 and Nick Vamivakas6  1Center for Fundamental
Physics and Center for Interdisciplinary Exploration and Research in
Astrophysics, Department of Physics and Astronomy, Northwestern University,
Evanston, IL, 60208
2School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb
Memorial Drive, Rochester, 14623 NY
3Department of Physics and Astronomy and Elmore Family School of Electrical
and Computer Engineering, Purdue University, West Lafayette, IN, USA
4Department of Materials Science and Engineering, University of Washington,
Seattle, WA
5Physical and Computational Sciences Directorate, Pacific Northwest National
Laboratory, Richland, WA
6Department of Physics, University of Rochester, Rochester, NY, 14627
<EMAIL_ADDRESS>
###### Abstract

Optomechanics, the study of the mechanical interaction of light with matter,
has proven to be a fruitful area of research that has yielded many notable
achievements, including the direct detection of gravitational waves in
kilometer-scale optical interferometers. Light has been used to cool and
demonstrate quantum control over the mechanical degrees of freedom of
individual ions and atoms, and more recently has facilitated the observation
of quantum “mechanics” in objects of larger mass, even at the kg-scale.
Levitated optomechanics, where an object can be suspended by radiation
pressure and largely decoupled from its environment, has recently established
itself as a rich field of study, with many notable results relevant for
precision measurement, quantum information science, and foundational tests of
quantum mechanics and fundamental physics. This article provides a survey of
several current activities in field along with a tutorial describing
associated key concepts and methods, both from an experimental and theoretical
approach. It is intended as a resource for junior researchers who are new to
this growing field as well as beginning graduate students. The tutorial is
concluded with a perspective on both promising emerging experimental platforms
and anticipated future theoretical developments.

††journal: osajournal††articletype: Research Article

###### Contents

  1. 1 Introduction
  2. 2 Controlling the dynamics of levitated optomechanical systems
    1. 2.1 Principles of optical trapping
      1. 2.1.1 Physical regimes
      2. 2.1.2 Forces and dynamics within regimes
    2. 2.2 Motion defined by optical forces
    3. 2.3 Trapping configurations
    4. 2.4 Particle sources and materials for optical levitation
      1. 2.4.1 Sources
      2. 2.4.2 Materials and morphologies
    5. 2.5 Optomechanics
    6. 2.6 Detection
      1. 2.6.1 Types of detection method
      2. 2.6.2 Optimal detector placement
      3. 2.6.3 Interpreting data - volts to meters and SI unit conversions
    7. 2.7 Quantum regime, and applications
    8. 2.8 Other levitation schemes
  3. 3 Hybrid and coupled systems with multiple degrees of freedom
    1. 3.1 Levitated multi-degree of freedom optomechanical systems
    2. 3.2 Spin systems
    3. 3.3 Atomic gases and nanoparticles
    4. 3.4 Cold and degenerate atomic gases and optical cavities
  4. 4 Thermodynamics
    1. 4.1 Non-equilibrium thermodynamics of small systems
    2. 4.2 Laser Refrigeration
  5. 5 Sensing
    1. 5.1 Force
    2. 5.2 Acceleration
    3. 5.3 Rotation
    4. 5.4 Torque
    5. 5.5 Electric and magnetic fields
  6. 6 Fundamental Physics
    1. 6.1 Gravitational force
    2. 6.2 High frequency gravitational waves
    3. 6.3 Casimir force and torque
    4. 6.4 Dark Matter
  7. 7 Summary and Future Perspectives: Experiment and Theory
    1. 7.1 Biological/chemistry applications
    2. 7.2 Optical binding and Levitated particle arrays
    3. 7.3 Near field interactions
    4. 7.4 High mass matter wave interferometry
    5. 7.5 Detecting high energy radiation
    6. 7.6 Squeezing and non classical state preparation
      1. 7.6.1  Levitated optomechanical state preparation
    7. 7.7 Tests of quantum foundations - collapse models, entanglement
  8. 8 Conclusion

##  1 Introduction

The field of optomechanics dates back at least to the early 1600s, when
Johannes Kepler suggested the force imparted by solar radiation explained the
direction of comet tails (for a historical account see Ref. [1]). Radiation
pressure was experimentally observed and confirmed in the laboratory by
Nichols and Hull as well as Lebedev [2, 3], only after an erroneous
demonstration decades earlier by Crookes due to radiometric forces [4]. Such
radiometric forces, having to do with momentum imparted to an object by
thermal currents of gas molecules surrounding it when the object is heated due
light absorption, were later shown in some cases to exceed radiation pressure
forces by five orders of magnitude [2]. With the invention of the laser [5],
the potential for the manipulation of particles and objects with radiation
pressure became substantially enhanced. In macroscopic systems, the potential
of optomechanics for quantum-limited precision measurement has been realized
in the context of gravitational wave detection [6, 7, 8, 9]. Pioneered by
Arthur Ashkin and coworkers, early work on optical levitation and manipulation
of dielectric particles led both to the development of optical tweezers (for a
review see Ref. [10]), which has had a profound impact on biology and
biophysics, and eventually to the laser cooling and trapping of atoms and
molecules [11].

Optical trapping of dielectric particles was first studied in a liquid
environment [12], and investigations of trapping at high vacuum were carried
out shortly afterwards [13, 14]. In the late 2000s it was realized that
optical trapping and manipulation of dielectric particles in vacuum is a
promising route to observe quantum mechanical behavior, due to the extreme
environmental isolation made possible by suspending a particle using radiation
pressure [15, 16, 17, 18]. Experimental progress in levitated optomechanics in
vacuum has been quite rapid (see Ref. [19] for a recent review), with quantum
control including ground state cooling [20, 21] of optically trapped
nanoparticles, and precision sensing including force sensing at the
zeptonewton level [22], acceleration sensing at the nano-$g$ level [23], along
with torque sensing at the level [24] and realization of world-record GHz
rotation rate of nano-objects [24, 25, 26]. A variety of trapping
architectures and trapped particles of a wide range of mass and size have been
studied, with notable recent examples illustrated in Fig. 1. More broadly,
levitodynamics, the levitation and control of microscopic objects in vacuum
[27], has emerged as an active field, including not only schemes employing
optical radiation pressure but instead relying on magnetic trapping or radio-
frequency trapping, for example for charged levitated objects.

Intended primarily as an introductory resource for junior researchers, in this
tutorial we describe the working principles behind controlling the dynamics of
levitated optomechanical systems, and discuss recent developments in hybrid
and coupled systems with multiple degrees of freedom. We illustrate several
examples of applications and science opportunities relying on optically
levitated systems including precision sensing, fundamental tests of quantum
mechanics and quantum coherence, searches for exotic new forces, dark matter,
and gravitational waves, and tests of thermodynamics and radiative heat
transfer in novel regimes. We conclude with perspective on future prospects in
both theory and experiment, including biological and chemistry applications,
tests and applications of quantum squeezing and entanglement, and sensing
beyond the standard quantum limit.

Figure 1: Optical trapping geometries employed in recent levitated
optomechanics experiments (clockwise from upper left): A) cavity along
trapping optical axis, reproduced from Ref. [28], B) cavity normal to trapping
optical axis - reproduced from Ref. [20], C) scattering force - gravity
balanced single beam trap, adapted from Ref. [29], D) single objective focus
trap with a single propagation direction, as in Ref. [30], E) two
counterpropagating freespace beams in a non cavity configuration, adapated
from Ref. [31], F) hybrid Optical and Paul trap (for the same trapped species)
as in Ref. [32]. G) Schematic mockup of the MAQRO mission proposal [33].

##  2 Controlling the dynamics of levitated optomechanical systems

###  2.1 Principles of optical trapping

In this section we describe the basic features of optical trapping that allow
the levitation of nanoparticles and their subsequent manipulation. We will
begin by identifying the various regimes of optical trapping in terms of the
length scales involved. For each regime we will identify the forces that are
responsible for the dynamics. Subsequently, a variety of trapping
configurations used in the field will be identified along with particle
sources and materials for optical levitation. Next the basic optomechanics of
levitated nanoparticles will be characterized, with reference to laser
cooling, quantum ground state occupation, regenerative amplification, etc.
Finally, mention will be made of other levitation schemes which use electric
fields, magnetic fields and hybrid systems using field types to define the
potential.

####  2.1.1 Physical regimes

As with many fields of physics, the characteristic behaviour of the system
depends greatly on the length scales it presents. In the case of levitated
optomechanics, the ratio between the wavelength of the trapping light and the
size of the levitated object is critical.

As such the size parameter:

$x=\frac{2\pi R}{\lambda_{0}}$ (1)

is commonly defined, in which $\lambda_{0}$ is the wavelength of the incident
light and $R$ is the radius of the spherical object. The case where the
wavelength is much greater than the trapped object’s circumference is commonly
referred to as the Rayleigh regime. The case in which the object is
approximately equal to the wavelength (i.e. $x\sim 1$) is known as the Mie-
Lorentz regime. The case in which the object is much larger than the
wavelength is the geometric optics regime.

Figure 2: A commonly used architecture for a levitated optomechanical system
employing a horizontally-oriented, single-beam optical tweezer trap. A
spherical dielectric nanoparticle of mass $m$ is levitated at the center of an
optical potential. Each vibrational degree of freedom oscillates about the
potential minimum with center frequency (eigenfrequency) $\Omega_{i}$, where
$i=(x,y,z)$. Image adapted from Ref. [34].

####  2.1.2 Forces and dynamics within regimes

The complete motional dynamics of the levitated object can be obtained from
the self consistent EM field configuration in and around the particle, in
which in which scattered photons represent transient fluctuations.

Within the Rayleigh regime, the gradient and scattering forces can be
considered to be separable, with the gradient force producing a force
proportional to the derivative of the intensity of the EM field and the
scattering force one in proportion to the product of the intensity and the
particle’s scattering cross section. It is important to note that the gradient
force typically overpowers the scattering force within the Rayleigh regime for
focused beams - and creates a restoring force about the focus. The scattering
and gradient forces in the Rayleigh limit along the axis of beam propagation
for a single beam are given by [35, 36]:

$F_{\mathrm{scat}}(r)=\hat{z}\frac{n_{\mathrm{med}}}{c}C_{\mathrm{scat}}I(r)=\hat{z}\frac{128\pi^{5}R^{6}}{3c\lambda_{0}^{4}}\Big{(}\frac{m_{\mathrm{obj}}^{2}-1}{m_{\mathrm{obj}}^{2}+2}\Big{)}^{2}n_{\mathrm{med}}^{5}I(r),$
(2)

and

$\displaystyle F_{\mathrm{grad}}(r)$ $\displaystyle=$
$\displaystyle\nabla[p(r,t)\cdot E(r,t)]$ (3) $\displaystyle=$
$\displaystyle\frac{4\pi
n_{\mathrm{med}}^{2}\epsilon_{0}R^{3}}{c}(\frac{m_{\mathrm{obj}}^{2}-1}{m_{\mathrm{obj}}^{2}+2})\cdot\frac{1}{2}\nabla
E^{2}(r,t)=\frac{2\pi
n_{\mathrm{med}}R^{3}}{c}(\frac{m_{\mathrm{obj}}^{2}-1}{m_{\mathrm{obj}}^{2}+2})\nabla
I(r)$ (4)

where $\hat{z}$ is the displacement vector along the optical axis, $r$
describes the out of plane distance from from the optical axis.
$n_{\mathrm{med}}$ is the refractive index of the medium surrounding the
object while $m_{\mathrm{obj}}\equiv n/n_{\mathrm{med}}$ is the refractive
index of the object relative to the surrounding medium. $p(r,t)$ is the dipole
induced on the object from the electric field $E(r,t)$ of the incident light,
while $I(r)$ is the intensity of the lasers light field. $C_{\mathrm{scat}}$
is the scattering cross section of the object and $\lambda_{0}$ is the
wavelength of the incident light generating the optical potential. $R$ is the
radius of the particle. Note that both the dipole response and forces are time
averaged. The characteristic size scalings (w.r.t the levitated object) of the
two forces are $F_{\mathrm{scat}}(r)\propto R^{6}$ and
$F_{\mathrm{grad}}(r)\propto R^{3}$ in the Rayleigh limit only $(2\pi
R<<\lambda_{0})$, this usually sets the requirement for the scattering force
to be counterbalanced by another field (optical, gravitational etc) in the
direction of beam propagation when the particle radius is a large fraction of
the wavelength. A commonly employed trapping configuration for a Rayleigh-size
particle is a single beam optical tweezer and is illustrated in Fig. 2. The
center of mass motion of the particle can be described by an asymmetric three-
dimensional harmonic potential in the limit of small motional amplitudes.

We also note that the gradient force described by Eq. (4) is not universally
applicable. In particular, dielectric particles sufficiently large compared to
the trap wavelength and with large enough refractive index (e.g. silicon) can
also exhibit strong magnetic dipole responses, with different force
coefficients [37, 38]. Further, the force induced by the imaginary component
of the Poynting vector generally needs to be considered [39, 40]. However for
the low-optical-loss silica particles typically used in levitated
optomechanics experiments, we expect Eqs. (4) and (2) to be a reasonable
approximation.

For most setups, the beam profile is typically Gaussian, or a superposition of
counter - propagating Gaussian beams:

$I_{Gaussian}(x,y,z)=\frac{I_{0}}{\left(\frac{z^{2}}{z_{R}^{2}}+1\right)}e^{\frac{-2x^{2}-2y^{2}}{w_{0}^{2}\left(\frac{z^{2}}{z_{R}^{2}}+1\right)}}$
(5)

For the case of the gradient force, taking the derivative of such a function
leads to a series of high order terms for each axis, providing coupling
between the degrees of freedom for even a perfect sphere. In the limit of
small displacement, these terms are negligible and a Gaussian potential
approximates an $x^{2}$ or linear harmonic oscillator. Since $U=\frac{\chi
I}{c}$, $F(r)=-\frac{dU}{dr}$ and $F(r)=k\cdot r$ quantifying the coupling
rates as a function of displacement is possible.

Figure 3: Scattering cross section of a dielectric sphere as a function of its
size parameter, with regimes labelled. The upper horizontal axis shows the
corresponding radius of a SiO2 sphere when scattering light of wavelength $1$
$\mu$m. Inset: Regime in which the levitated object is much larger than the
interacting wavelength (left), the forces and thus motional dynamics are
dominated by conservation of momentum of the refracted rays. Regime in which
the levitated object is much smaller than the interacting wavelength (right),
the dominating force is gradient force, and to a good approximation the nano-
object acts like a dipole. (Middle) The optical force on an arbitrarily shaped
particle in the Mie-Lorentz regime is obtained by integrating the maxwell
stress tensor (as indicated in Eq. 6) over a surface enclosing the object.

In the Mie-Lorentz regime, in which the optical size parameter is roughly
equal to the wavelength of the trapping light, this approximation breaks down
and the computation of optical forces is achieved by a variety of other means.
Fig. 3 illustrates how the scattering cross section of a dielectric sphere
varies with size parameter across the Rayleigh, Mie-Lorentz, and geometric
optics regimes.

Exact analytic solutions for the EM field configuration within the particle
and therefore the force exerted by the light field only exist for certain
symmetric particle shapes, materials and beam profiles. For example, the Mie
equations describe the forces on spheres [41], spheroids[42] and infinite
cylinders [43] for a limited set of optical potentials. Outside of these
limits approximations exist to extend these solutions for objects of
refractive indices similar to the environment [44] or of limited difference in
optical size parameter to the Rayleigh regime. Furthermore, semi-analytic
solutions, such as the T-matrix[45] method may also be used for objects
similar to those with an exact Mie solution - finite cylinders as opposed to
infinite cylinders. Beyond these limited cases, fully numerical methods are
required to compute the optical forces in this regime.

In the general case, the force apparent upon the particle may be obtained by
solving the Maxwell Stress Tensor :

$T_{ij}=\epsilon_{0}[E_{i}E_{j}+c^{2}B_{i}B_{j}-\frac{1}{2}(|\overrightarrow{E}^{2}|+c^{2}|\overrightarrow{B}|^{2}\delta_{ij}].$
(6)

The force and torque exerted on the object by the optical field can then be
calculated as:

$\overrightarrow{F}=\oint\overleftrightarrow{T}\dot{\hat{n}}dS$ (7)

$\overrightarrow{\tau}=\oint\overleftrightarrow{M}\dot{\hat{n}}dS$ (8)

where the tensor $M$ is given by
$\overrightarrow{M}=-\overleftrightarrow{T}\times\overrightarrow{r}$ and
$\overrightarrow{r}$ is the displacement vector. [46]

This naturally requires the user to know the configuration of optical fields
within the object, for complex objects not matching the cases above often
obtained with an iterative solver. The Discrete Dipole Approximation (DDA)[47]
or Green’s Dyad Method (GDM)[48] being two popular examples. Several other
popular solvers exist in including MEEP (a Finite-Difference Time-Domain
(FDTD) method)[49] and COMSOL[50].

While the Maxwell stress tensor is often quoted as the canonical method,
equivalent integrals - for example over volume rather than area are often
used[51, 52], with different scaling of numerical convergence and scaling of
required computations resources in each case.

It should be noted that while the restoring optical potential in the Mie-
Lorentz regime is still defined by the gradient force, momentum carried in the
propagation direction of the light field in the Mie-Lorentz regime is
significant and creating a stable trap usually requires this force to be
balanced against another optical beam, field, or gravity.

In the geometric optics limit, in which the optical size parameter is much
larger than the wavelength, forces are generally given by conservation of
momentum and 4-matrix formalism. Due to practical limitations of transferring
adequate power to compete against gravity, few levitated optomechanics
experiments take place within this regime[53, 54].

###  2.2 Motion defined by optical forces

At higher pressures (> 50mbar depending on object size and geometry) the
motion is completely dominated by random momentum interactions with the
background gases and undergoes a random walk around the optical potential.

At lower pressures (<10mbar) - where the damping is linear and no longer the
leading term in the equation of motion, the levitated object behaves like a
driven damped harmonic oscillator, manifesting as a Lorentzian peak in
frequency space for each degree of freedom with a width defined by the sum of
its damping processes described in Fig. 4. A more detailed discussion of gas
damping and Brownian motion is also included in Sec. 4.

In the intermediate regime, where the mean free path of the gas molecules is
of order the size of the dielectric particle to be trapped, additional
instabilities are possible due to radiometric forces, which can dominate over
the confining forces due to radiation pressure, as was first studied by Ashkin
and coworkers in the late 1970s [14]. Thus one of the challenges of reaching
high vacuum is to overcome radiometric forces. Of prime importance is to
minimize heating of the particle surface from optical absorption, as
temperature gradients across the surface of the levitated object can drive
drive currents of gas molecules which are de-stabilizing over a range of
pressure spanning orders of magntitude. A detailed study of these effects was
performed in Ref. [55] for trapped silica microspheres. Heating from optical
absorption can be mitigated by choice of material as well as trapping laser
wavelength. As described later in Sec. 4, solid-state laser cooling may enable
negligible heating of the particle surface, or cooling substantially below
room temperaure. To compensate for radiometric forces, laser feedback cooling
at a rate $\Gamma_{\mathrm{fb}}$ can be utilized to restore damping that is no
longer provided by the background gas being evacuated from the trapping
chamber.

For each degree of freedom, the temperature or phonon occupation number is
defined by the ratio of the heating and cooling processes in the system:

$n_{\infty}=\frac{\Gamma_{\mathrm{th}}+\Gamma_{\mathrm{fb}}+\Gamma_{\mathrm{photon}}+\Gamma_{\mathrm{other}}}{\gamma_{\mathrm{th}}+\gamma_{\mathrm{fb}}+\gamma_{\mathrm{photon}}+\gamma_{\mathrm{other}}},$
(9)

where $\Gamma_{\mathrm{th}}$ describes the heating corresponding to
interactions with background gases, $\Gamma_{\mathrm{photon}}$ photon recoil
scattering events, $\Gamma_{\mathrm{fb}}$ heating events corresponding to
phase imprecision in the feedback loop and the lower case gammas $\gamma$
correspond to cooling events from the same physical interaction. Each of these
can be thought as the levitated object’s connection to a heat bath. At higher
pressures the dominating term is the collisions with background gases, and at
lower pressures various random processes defined by the laser dominate (photon
recoil, laser phase noise etc).

In the limit of negligible other technical heating and noise sources, i.e.
$\Gamma_{\mathrm{other}}=\gamma_{\mathrm{other}}=0$, as well as negligible
heating from feedback $\Gamma_{\mathrm{fb}}\approx 0$ and
$\gamma_{\mathrm{photon}}=0$, and where $\Gamma_{\mathrm{photon}}$ is
dominiated by photon recoil heating via e.g. Rayleigh scattering, we arrive at
the theoretical mean steady state phonon number for this limited case

$n_{\infty}\approx\frac{\Gamma_{\mathrm{th}}+\Gamma_{\mathrm{photon}}}{\gamma_{\mathrm{th}}+\gamma_{\mathrm{fb}}}\approx\frac{n_{i}\gamma_{g}+\gamma_{\mathrm{sc}}}{\gamma_{g}+\gamma_{\mathrm{fb}}}$
(10)

where in the last equation we specialize to the case of a nanosphere optically
levitated with trap frequency $\omega_{0}/2\pi$ in contact with a thermal gas
at temperature $T$ with damping rate $\gamma_{g}$ where
$n_{i}=k_{B}T/\hbar\omega_{0}$ with laser cooling at rate
$\gamma_{\mathrm{fb}}$ and photon recoil heating at rate
$\gamma_{\mathrm{sc}}$.

The time required to enter equilibrium (thermalise) with any given heat bath
is related to the heating rate of the process (in Hz) and the frequency of the
degree of freedom of the oscillator, and can be on the order of milliseconds
to seconds depending on the process and frequency, providing an accessible
platform for thermodynamics, as well as slightly non Markovian system noise.

Figure 4: Each single uncoupled, linear degree of freedom of an optomechanical
system will oscillate at a central frequency defined by its potential. When
the same oscillating degree of freedom is immersed in a heat bath (i.e from
background gas collisions or laser light) it will randomly change its
trajectory as a result of the collision. Due to the balance of these heating
and damping processes the amplitude, phase, and energy of the degree of
freedom all fluctuate over time. The characteristic time scale of this process
is inversely proportional to the damping rate in Hz. In the Fourier domain,
the damping of the mode manifests as the mode’s width about its central
frequency, and its ’temperature’ as its area. This transformation is described
by by the Wiener-Khinchin theorem [56].

###  2.3 Trapping configurations

The tweezer intensity profile is typically Gaussian and for this the gradient
force is linear in nanoparticle displacement (for amplitude of motion smaller
than the beam waist), resulting in a harmonic trap to lowest order in
nanoparticle displacement [57].

For low-optical-loss silica nanoparticles, with refractive index $n=1.45$, we
expect Eqs. (2) and (4) to be a reasonable approximation, and expanding about
the equilbrium position for a focused Gaussian beam, it can be seen that a
nanosphere will be confined in an approximately harmonic potential, and
attracted to the point of maximal intensity with a nearly linear restoring
force [10]. In vacuum, the motion of the trapped particle largely corresponds
to a 3-dimensional simple harmonic oscillator in the under-damped regime,
damped by background has collisions and driven by thermal noise. The trap
frequency for oscillations transverse to the direction of propagation of the
laser and in the axial directions is approximately given by

$\omega_{r}=\sqrt[]{\frac{6I_{0}}{c\rho
w_{0}^{2}}\left(\frac{m_{\mathrm{obj}}^{2}-1}{m_{\mathrm{obj}}^{2}+2}\right)}$
(11)

and

$\omega_{z}=\sqrt[]{\frac{3I_{0}\lambda_{0}^{2}}{c\pi^{2}\rho
w^{4}_{0}}\left(\frac{m_{\mathrm{obj}}^{2}-1}{m_{\mathrm{obj}}^{2}+2}\right),}$
(12)

respectively, where $\rho$ is the density of the sphere, assuming a trap in
vacuum ($n_{\mathrm{med}}=1$).

For small amplitudes of motion and a spherical particle, the oscillation along
each Cartesian direction is independent. For large amplitudes of oscillation,
the tweezer displays anharmonicities, resulting in Duffing-type nonlinearities
in the next-to-harmonic order [58]. These nonlinearities can not only make
each oscillation direction anharmonic (due to self-Duffing nonlinearities),
they can couple the various directions (due to cross-Duffing nonlinearities).

A variety of trapping architectures have been presented in the literature. A
selection of these is presented within figure 1. These include single [30],
dual [31] and multi-[59] beam tweezers. For single beam dipole traps, the
required focusing of the tweezing beam has been obtained by using high-
numerical aperture (NA) lenses [57] as well as parabolic reflectors [60].
Counter propagating beams do not require high NA optics due to the self
cancellation of the scattering force - which allows larger particles to be
trapped than single beam configurations. Configurations with [61] and without
[62] optical cavities have been realized. Optical tweezers have also been
combined with other confinement devices such as Paul traps [63].

###  2.4 Particle sources and materials for optical levitation

####  2.4.1 Sources

A variety of methods for loading nanometer to micron sized objects into
optical and hybrid traps exist, such as PZT driven launchers[35],
electrosprays, nebulisers[64] and Laser-Induced Acoustic Desorption (LIAD)
methods [65].

Particle sources are typically matched to the size, shape and material of the
desired levitated species, as well as the type of trap being used. For smaller
objects in the 10’s or 100’s of nm regime, nebulisers have historically been
used [64] which disperse a colloidal solution of suspended nanoparticles into
the trapping region. These require operation at atmospheric pressures and
typically contaminate the chamber with whatever liquid is used for the
colloidal suspension. This problem has been overcome to some degree by
separating the trapping and science chambers.

Larger objects in the 100’s of nm to 10’s of $\mu m$ regime have often been
trapped using PZT driven mechanical launchers, in which the particle’s are
physically detached from a surface by vibration. Ref. [66] describes an
apparatus which uses a glass substrate clamped to a piezoelectric transducer
for in-vacuum loading of dielectric nanoparticles into an optical trap. The
device has been shown to generate accelerations of order $10^{7}$ $g$, which
is sufficient to overcome stiction forces between glass nanoparticles and a
glass substrate for particles as small as $170$ nm diameter, consistent with
theoretical expectations. Using this device, single spherical particles with
sizes ranging from $170$ nm to $3$ $\mu$m [55, 22, 67], as well as clumps of
such particles [68], have been successfully loaded into optical traps at
pressures ranging from 1 bar to 0.6 mbar. Similar work was also recently
reported and described in Ref. [69]. For illustrative purposes, Visualization
1 features a short video clip showing particles being launched downwards
towards a vertically oriented optical levitation trap, resulting in the
capture of a silica microsphere of diameter $3$ $\mu$m.

Pulsed laser launchers directly transfer momentum from a light field to an
area of nanofabricated objects on substrate, kinetically directing them
towards a trapping area. Like the PZT sources, this has a distinct advantage
of being able to to launch at vacuum, however its high launch velocities does
make it better suited to a deep trapping potential. This source also has the
advantage of not requiring an intermediate solution of material to be either
loaded into a nebuliser or prepared on a microscope slide - as is the case for
the PZT method. For the PZT method, the velocity distribution for a variety of
particles launched from the substrate was studied in Refs. [66, 69] and
results indicate promise for direct loading into ultra-high-vacuum with
sufficient laser feedback cooling. In contrast to LIAD, where particles tend
to be launched with speeds well in excess of $1$ m/s [65], the PZT method
produces sufficient flux of particles with initial launch velocity less than
$1$ m/s to in principle allow direct ultra-high-vacuum trapping with the
assistance of a slowing laser beam or other arresting field.

####  2.4.2 Materials and morphologies

Figure 5: A sampled zoology of optically trapped species, different geometries
and materials give different optical couplings, internal states and degrees of
freedom. 1: Nanodiamonds[70], 2: Rayleigh regime spheres [71],3,4:
Nanodumbells, combined spheres and agglometerates [72],5: Silicon nanorods
[73],6: YLF crystals [74],7: NaYF high aspect ratio disk likes [31],8: Mie-
Lorentz Regime spheres [29]. The ratio of the COM frequency and the optical
power is used as a limited metric of the optomechanical coupling.

Many different shapes and materials of the levitated object have been
investigated in optical trapping experiments. Each combination and permutation
providing different tunable properties to match the desired experiment, such
as optical coupling, dispersion relation, a controllable distribution of the
scattered light and additional degrees of freedom. Example of different
investigated geometries include; SiO2 spheres from 10’s of nm [20] [75] to
10’s of $\mu$m [23], NaYF disc like objects [31], prolate spheroids [76],
nanorods[73]. A sampled parameter space of these is given in Figure 5.

Control over the specific material properties is especially and increasingly
important given that such attributes of objects at this scale frequently
deviate from those of the bulk, such as the (lower) density of silicon dioxide
spheres made from the stober process for example [29]. This has spurred
efforts in generating/creating/engineering new levitatable nanomaterials with
bespoke desirable properties. Experiments to date include vaderite spheres
[77], and species with potentially useful internally accessible states:
nanodiamonds[78] for a spin degree of freedom, NaYF[79, 31] and YLF[74]
crystals for optically accessible internal state cooling.

###  2.5 Optomechanics

The dynamics of the light-matter interaction between the tweezer (or other)
beams and the nanoparticle motion are rich and have been investigated in two
main types of configurations. The first of these involves optical cavities
which contain levitated nanoparticles [16], see Fig.. Here the coupling
between the nanoparticle and optical cavity mode is dispersive, meaning the
motion of the nanoparticle only shifts the cavity mode resonance frequency.
This configuration can be described using the standard cavity optomechanics
Hamiltonian [56]

$H=\hbar\omega_{c}a^{\dagger}a+\hbar\omega_{m}b^{\dagger}b+\hbar
ga^{\dagger}a(b^{\dagger}+b)+H_{o}+H_{m},$ (13)

where $\omega_{c}(\omega_{m})$ is the cavity mode (mechanical) frequency,
$a(a^{\dagger})$ and $b(b^{\dagger})$ are the creation and annihilation
operators for the optical mode and nanoparticle motion, respectively, obeying
$[a,a^{\dagger}]=1,[b,b^{\dagger}=1]$, and $g$ is the optomechanical coupling.
It should be mentioned that the to realize the given kind of optomechanical
coupling, linear in the particle position $\sim a^{\dagger}a(b^{\dagger}+b)$,
the nanoparticle has to be positioned between a node and an anti-node of the
standing wave cavity mode (i.e. at an intensity gradient). A quadratic
coupling, proportional to the square of the particle displacement, $\sim
a^{\dagger}a(b^{\dagger}+b)^{2}$, can also be engineered, by positioning the
nanoparticle at the antinode of the cavity mode (i.e. at an intensity
maximum). The applications of the linear and quadratic couplings, which may be
implemented simultaneously using two cavity modes, will be mentioned below.
The additional terms $H_{o}(H_{m})$ describe the coupling to the environment
which is responsible for optical (mechanical) damping at the rate of the
cavity (mechanical) linewidth $\gamma_{o}(\gamma_{m})$ and optical as well as
thermal fluctuations. The Hamiltonian of Eq. (13) applies to experiments where
shot noise, blackbody radiation and particle asphericity are negligible and
the nanosphere’s internal degrees of freedom are decoupled from its center of
mass motion.

In principle, the tweezer-in-a-cavity principle can be used to realize
standard optomechanical effects such as cooling the nanoparticle to its
vibrational ground state, ponderomotive (mechanics-induced) quadrature
squeezing of the optical mode, and quantum state transfer between the
mechanics and optics [56]. Ground state cooling, for example, can be
implemented by trapping the nanoparticle using a quadratic optomechanical
coupling and cooling its motion by using a linearly-coupled optical mode
(quadratic-coupling-based cooling has also been shown [80]). However, in
practice, there are undesirable limitations. For example, cooling requires
large optical intensities, and is limited by co-trapping due to the linearly-
coupled mode, cooling rates fall off with power, and laser phase noise
degrades performance at the nanoparticle oscillation frequencies [81].

These problems can be overcome by a modification of the cavity-based setup,
using the technique of coherent scattering, see Fig. In the standard setup
described above, an external laser is responsible both for driving the cavity
as well as scattering from the nanoparticle. In contrast, the coherent
scattering method is based on a tweezer positionally transverse to the cavity
axis, and close in frequency to a cavity mode. The Hamiltonian for the
configuration can be written as

$H_{\mathrm{coh}}=-\frac{\alpha}{2}|\vec{E}_{tw}|^{2}-\frac{\alpha}{2}|\vec{E}_{cav}|^{2}-\alpha\mathcal{R}\left(\vec{E}_{tw}\cdot\vec{E}_{cav}^{*}\right),$
(14)

where $\alpha$ is the nanoparticle polarizability, $\vec{E}_{tw(cav)}$ is the
tweezer(cavity) field and $\mathcal{R}$ denotes the real part.

The third term in Eq. (14) is responsible for the interference between the
tweezer and cavity fields and thus coherent scattering. Due to this term, the
photons scattered by the nanoparticle are preferentially enter the mode of the
initially empty cavity. The optimum position for the nanoparticle in this
method turns out to be the node of the cavity mode. When located at that node,
the scattering of photons from the particle into the cavity mode is prohibited
by the interference between the mode and the optical tweezer beam. However, as
the nanoparticle vibrates, it is displaced from the node and scatters photons,
which after Doppler-shifted in frequency due to the particle motion. The
Stokes (anti-Stokes) photons are responsible for heating (cooling) the
particle motion. Preferential cooling of the particle can be arranged by
choosing the frequency of the tweezer beam appropriately near a cavity mode.
Although we do not go into the details of the mechanism involved, it should be
noted that the polarization of the tweezer plays an important role in the
dynamics [81]. Coherent scattering techniques have been proposed [82] and
demonstrated to eventually cool all six (vibrational and rotational) degrees
of freedom of an anisotropic nanoparticle optically levitated in a cavity
[83].

In the configuration described by Eq. (13), the cavity provides passive
feedback to the nanoparticle motion by recirculating the photons in the
optical mode, while in the coherent scattering technique modeled by Eq. (14),
the cavity serves to stimulate optical scattering at a desired frequency.
While cavity-based levitated optomechanics has achieved many milestones [56],
there are a number of limitations posed by such configurations. For example,
the nanoparticle is limited to interact only with those optical wavelengths
which are resonant with the cavity, the cavity poses bandwidth limitations,
and, perhaps most importantly, it is not easy to freely engineer the light-
matter interaction beyond what is provided by the optical resonator.

These limitations can be overcome by replacing the cavity with active
feedback. This kind of feedback works typically by using position detection of
the nanoparticle, followed by supplying this information as active feedback
through a transfer function implemented by a device such as an electro-optic
modulator which can appropriately modulate the optical beam trapping the
tweezer, see Fig. 6. The optomechanical theory of the active feedback scheme
is somewhat more involved than standard cavity optomechanics. Initial
experiments were carried out in the classical regime of nanoparticle motion
and can be described using the dynamical equation for a damped oscillator with
feedback [19]

$\ddot{q}(t)+\Gamma_{m}\dot{q}(t)+\omega_{m}^{2}q(t)=A\eta(t)+u_{\mathrm{fb}}(t),$
(15)

where $q(t),\Gamma_{m},$ and $\omega_{q}$ are the position, center-of-mass
damping, and oscillation frequency of the nanoparticle, respectively. The
stochastic noise due to sources such as Brownian fluctuations from the
background gas, optical scattering and feedback backaction is represented by
the random variable $\eta(t)$ of strength $A$. The feedback applied to the
nanoparticle is accounted for by $u_{\mathrm{fb}}(t)$. Various kinds of
feedback have been used. Linear damping proportional to the velocity, with
$u_{fb}(t)=C_{l}\dot{q}(t)$, has been used to cool the particle motion [], as
well as nonlinear damping, with $u_{fb}(t)=C_{nl}q(t)\dot{q}(t)$. The terms
linear and nonlinear are used here with the significance that the particle
damping rate is independent of the energy for linear damping and depends on
energy for nonlinear damping. A comparison of the efficacy of the different
types of feedback for cooling has been carried out [84]. State base Bayesian
models from control theory (such as Kalman filters), have also been
demonstrated experimentally for feedback [85].

Figure 6: Different types of feedback demonstrated in optically levitated
experiments. A) Direct time domain modulation of light intensity via AOM [75]
\- compatible with feedback schemes such as parametric. B) feedback/cooling
via an initially empty off axis cavity populated only with photons emitted
from levitated species, often referred to as coherent scattering[86]. C)
Dissapative optical cavity with a pre-populated cavity mode from a second
laser[86], the wavelength of which is tuned to make the equation of motion of
the levitated species dissapative. D) Electrostatic feedback [87] \- suitable
for linear feedback schemes such as cold damping. Adapted from ref. [86].

The cavityless situations considered above regard only a single vibrational
mode of the levitated nanoparticle. Both experiments and theory have
considered multi-mode optomechanics. These include various combinations of
vibration [88] as well as torsional [89, 60, 90] and rotational [91] modes
[92]. Some of these configurations involve composite nanoparticles, like
dumbbells [91] or nanoplatelets [93]. Even untrapped nanoparticles have been
modified by their transient interaction with a cavity [94]. For these cases,
the mathematical descriptions provided above have to be suitably modified or
substituted.

###  2.6 Detection

Figure 7: Information carried by different quadrature of a light field
scattered from a levitated Rayleigh scale nanosphere and their respective
detection mechanisms. A: Information from light intensity varying over time
recorded on a detector - often a position sensitive detector such as a
quadrant photodiode (QPD). This mechanism is similar to a microscope and is
generally not sensitive to information encoded in the phase of the scattered
light. B: Scattered light interfered local oscillator reference light,
sensitive to information encoded in the phase of the scattered light. In many
“homodyne like” schemes, the reference light is light that traversed through
the trapping zone of the levitated object and did not interact with it -
however, it should be pointed out that this approach does not allow for the
phase of the local oscillator reference field to be varied for homodyne state
tomography(such as in [95], and as such only samples part of the wigner space.
It should be noted that schemes utilizing non optical fields as readouts have
been used successfully, such as eletrostatics and near field interactions
[96].

The central observational modality for levitated optomechanics is the
detection of the nanoparticle’s position. This information can be processed,
while making some reasonable assumptions, to obtain the energy (both transient
as well as steady-state) of the particle. In experiments with cavities, the
mechanical displacement information can be found from the transmission of the
optical cavity [56]. In experiments without cavities, oscillator displacement
may be obtained by interfering the light scattered by the nanoparticle with a
local reference field [57], or by directly observing the scattered light of
the particle in a spatially sensitive detection system [22], as illustrated in
Fig. 7.

####  2.6.1 Types of detection method

At each point in space at which we might place a detector, the light field
carries information in its intensity and phase quadratures. Intensity bound
information can be thought of as a mechanism like a microscope, phase bound
information like an interferometer, figure 7. To mathematize this statement:
For each point in space that a levitated particle scatters light into, the
scattered EM field can be thought of as being a vector in complex space,
$A+iB$. As the scattering object moves in space and time, the vector at each
point in space at which we might place a detector can do two things, it can
change in absolute magnitude (intensity), or rotate in complex space (phase).

The distinction between phase sensitive detection (homodyne and hetrodyne like
schemes) and intensity sensitive detection (typically achieved with something
like a quadrant photodiode) is important experimentally, since without time of
flight information a photodetector typically only measures the intensity of an
inbound EM field and not directly its phase (phase representing the
orientation of the light field’s vector in imaginary notion and intensity its
magnitude), a reference field can be used to transform changes in phase into
changes in intensity by transforming the rotation of the imaginary vector into
a change in its magnitude which is then measurable by a photodetector.

####  2.6.2 Optimal detector placement

Each frequency band of information propagates into space via a its own set of
Bessel functions, meaning that the position of the detector on the optics
bench determines the ratio of information it sees about each degree of freedom
and its resonances. Solutions for optimal placement of the detector with
respect to the trapping location can be found in Ref. [97] for Rayleigh sized
particles, and in Ref. [98] for Mie-Lorentz regime sized spheres. The
computation of these parameters for non spherical particles of arbitrary
geometries is an ongoing element of research.

In terms of practical experiment construction, phase bound information is
recovered by interfering the scattered light from the levitated object with a
local reference field - conditioned for maximum interference and spatial mode
overlap, while in the case of intensity modulated detection the objects motion
is usually literally imaged on a spatially sensitive detector. A review of the
ultimate sensitivity limits of both methods (for Rayleigh regime sized
spheres) may be found in Ref. [97] \- as well as a discussion on optimum
detector placement with respect to the trapping region. An extension of this
discussion on optimum detector placement to particles in the Mie-Lorentz
regime may be found in Ref. [98].

####  2.6.3 Interpreting data - volts to meters and SI unit conversions

Given the complex set of systematic errors which are coupled to the conversion
of photons to volts to nanometers (in terms of particle motion), we stress the
importance of calibrating the sensor under its final operating conditions. For
an example if the placement of the detector upon on the tabletop affects the
proportional of frequency band information available at each detector
(important for the final volts-meters conversion), then thermal drift and
other positional systematics will affect the measurement. Indeed, it is
possible to accidentally fool oneself by pathological placement of a detector
optimising for certain frequency bands at the expense of others. An
experimentalist might trap a nonspherical particle and place a detector in
such a position as to ignore or de-weight the frequency peaks containing
rotational information, and conclude they have trapped a sphere. For
microscopic parameters; experimental pressure and COM temperature dramatically
affect the volts-meters conversion. COM temperature especially - many papers
in the field assuming a linear conversion of volts to meters, it has been
demonstrated that this is a deterministically nonlinear function [59]. While a
complete discussion is beyond the scope of this Tutorial article, references
for the interpretation of detector signals and transforming then into physical
displacement units can be found in Refs. [99, 100]. Finally, we note that in
the limit of the oscillators occupation number being small enough such that
quantum effects become relevant, the center of mass temperature assigned to
the particle’s external degrees of freedom may also be found by sideband
thermometry [101, 102].

###  2.7 Quantum regime, and applications

For levitated optomechanics, the quantum regime is of interest coming from two
directions. The first is the ’top-down’ direction where the system may consist
of a mesoscopic or nanoscopic mechanical object and the aim of the
investigation is to introduce the object into the quantum domain. This is the
motivation behind the preparation of levitated nanoparticles in the ground
state [20] or squeezed states [103, 104, 105]. The second is the ’bottom-up’
direction, where the system may consist of a sample of cold [106] or
degenerate atomic gas [107, 108]. These gases are already at or close to the
ground states of their confining potentials and the object is to use them to
establish optomechanical effects cleanly.

For cavity-based systems, strong-coupling effects have been observed [109],
optically levitated nanoparticles have been prepared in their ground state of
motion using coherent scattering into cavity modes [20]. More recent work has
demonstrated simultaneous ground state cooling of two translational modes of a
levitated nanosphere in a cavity [110].

For systems without cavities, the photon recoil-dominated regime has been
reached [111], ground state preparation has been achieved via active negative
feedback in cryogenic environments [87]; light scattered from such particles
has been squeezed ponderomotively [112, 113]; the mechanical analog of a laser
has been realized with a combination of positive linear feedback and negative
nonlinear feedback [114].

Both experiments [115] and theory [105] are now moving towards considering
multiple simultaneously levitating nanoparticles.

###  2.8 Other levitation schemes

Although it is beyond the scope of this tutorial to describe in detail the
many alternative platforms for levitodynamics, we briefly mention several
systems which have recently been the subject of research. Magnetically
levitated liquid helium droplets [116], superconductors, magnets [117, 118,
119], degenerate gases [120] (Esslinger, Brantut), gravimagnetic trap
(d’Urso), dynamic electric traps ([121]).

One useful metric of comparison between systems is the mass-frequency-Q factor
parameter space, as well as the fundamental limiting noise source and floor.

In general at time of publication, Magnetic and ion trap systems allow for
higher levitated masses and non photon limited noise sources at the expense of
having a significantly lower mechanical frequency. Notably, nanospheres in
Paul traps were recently demonstrated to have quality factors in excess of
$10^{10}$ [122].

Liquid helium systems have also demonstrated extremely high Q factors with
substantial tunability.

##  3 Hybrid and coupled systems with multiple degrees of freedom

This section examines advances that have been made with levitated
optomechanical systems consisting of multiple degrees of freedom. Such systems
present opportunities to explore novel optomechanical dynamics unique to
levitated systems, new paradigms for sensing as well as opportunities for
exploring fundamental physics. What are some examples of these modalities? One
example is an isolated levitated particle that engage multiple modes of
oscillation, torsion and/or rotation. A second example is levitated diamond
nanocrystals containing nitrogen vacancy center defects. The embedded defects
internal degrees of freedom, for example spin, can couple with the particles
center of mass degrees of freedom. Another system described is an atomic gas
that envelopes a levitated nanoparticle and the coupling of these two hybrid
systems. Finally, work with cold and degenerate atomic gases coupled with
optical cavities will be described.

Progress in cooling of levitated nano-objects into the quantum regime can lead
to the development of new hybrid quantum systems where the motion of levitated
optomechanical systems can be coupled with other quantum systems such as
these.In these setups, mechanical oscillators can act as transducers,
providing coupling between photons, spins, and charges via phonons (see for
example Refs. [123, 124]). Such transducers could play an important role in
quantum networks, by allowing coupling between different types of quantum
systems, each with differing advantages.

###  3.1 Levitated multi-degree of freedom optomechanical systems

A unique feature of levitated optomechanical systems is they possess multiple
degrees of freedom - translation, rotation and torsion - that can be coupled.
In this section we review research that has explored the
controlled/intentional coupling of these degrees of freedom. The coupling
provides a new opportunity for controlling the optomechanical system’s
dynamics. One of the earliest works was by Arita et. al. [125] studying how
rotation of a levitated birefringent vaterite microcrystal could couple with
the crystal’s translational motion. The translational motion in the plane
transverse to the optical axis was harmonic and had a fundamental resonance on
the order of $600$ Hz ($f_{\rm{xy}}$). Rotation was accomplished via spin
angular momentum transfer from the trapping light to the birefringent
microcrystal. The circularly polarized light induces a rotation and a rotation
frequency ($f_{\rm{rot}}$) of $5$ MHz was observed at $0.1$ Pa. The control
came through changing the microcrystal’s rotation frequency by varying the
background gas pressure. Figure 8a presents an exemplary data set where
resonant enhancement is observed when the rotation frequency is equal to the
oscillation frequency ($f_{\rm{rot}}\sim$ $f_{\rm{xy}}$) or twice the
oscillation frequency ($f_{\rm{rot}}\sim$ $2f_{\rm{xy}}$ ). The former
enhancement is the result of a driven resonance whereas the latter is a
parametric resonance. Exciting is when the rotation frequency is commensurate
with the translation frequency. When the previous is achieved, a passive
cooling of the microcrystal’s harmonic translational motion was possible
achieving minimum effective temperatures of 40K.

Parametric coupling was also used by Gieseler et. al. [126] as a means to
promote nonlinear mode coupling between the transverse modes of translational
motion. In their experiments, an additional frequency and amplitude modulation
was imparted to the trapping laser with a frequency in close vicinity to the
oscillation frequency. First, the external modulation and one direction of
oscillation can become phase synchronized. The motional degree of freedom’s
linewidth can narrow, exhibit amplification and have its frequency pulled
toward the external modulation. The coupling to the other translational degree
of freedom is mediated by the optical trap’s potential well’s intrinsic
Duffing nonlinearity. For large enough oscillation amplitude the transverse
degrees of translational motion our coupled via the nonlinearity. Here, via
the Duffing nonlinearity, one mode of particle oscillation is controlled by
the external modulation through the mode the external modulation is directly
actuating. As shown in Fig. (8b), the mode locked to the external drive
(y-mode) has its frequency pulled. The nonlinearity leads to the controlled
frequency shift of the x-mode. Similar effects are observed for the y-mode
when the x-mode is phase locked to the external modulation.

Figure 8: a) Position spectral density data as the frequency of rotation is
tuned through the fundamental and second harmonic frequency of the
translational motion. b) Top panel: The parametric modulation frequency is
swept from 285 to 230 kHz to excite both the $y$ and $x$ resonances. Notice
when the y-mode is phase locked with the modulation the nonlinearly coupled
x-mode frequency also has its frequency pulled to lower frequencies. c)
Sympathetic cooling of the $x$ mode via coherent coupling to the $y$ mode.
Energy transferred from $x$ to $y$ is subsequently actively damped out of the
system lowering the $x$ mode’s effective temperature.

Coherent control of a levitated nanoparticle’s motion was implemented by
Frimmer et. al. [127] to sympathetically cool one translational mode of motion
that was coherently coupled to a mode that was being actively cooled. Much of
the theoretical work describing the experiments can be found in an excellent
article in the American Journal of Physics [128]. The key to this work was to
couple two orthogonal transverse modes of oscillation. Such coupling was
achieved by rotating the linear orientation of the trapping laser via an
electro-optic modulator sinusoidally: $\phi(t)=\phi_{0}\cos(\omega t)$. The
small angle harmonic polarization rotation creates an effective coupling
between the $x$ and $y$ directions in the original, unrotated, frame.

Sympathetic cooling was achieved by using Rabi oscillations in the coupled
motion to pump energy out of the uncooled mode via coherent control and then
dissipate the removed energy by active feedback cooling. The measurement is
presented in Fig. (8c). The feedback cooled mode (y) is initialized to $0.01$
$k_{B}T_{0}$ and the mode to be cooled (x) begins at $0.8$ $k_{B}T_{0}$ (a
fluctuation away from $k_{B}T_{0}$). When the polarization coupling is
introduced Rabi oscillations are evident and the energy in both mode’s decay’s
exponentially at a timescale determined by the feedback cooling gain. The
long-time steady energy in each mode $0.01$ $k_{B}T_{0}$ and sympathetic
cooling is achieved. Such a technique would also enable the cooling of dark
modes that might be invisible to the specific measurement technique.

###  3.2 Spin systems

Levitated optomechanical particles that contain intrinsic defects provide a
unique platform to study the coupling of a quantum degree of freedom, such as
a spin, to the particle motion. An exemplary system in this regard are diamond
nano/microcrystals that host the nitrogen vacancy (NV) defect [129]. The left
inset of Fig. 9a) illustrates the NV- defect’s crystal structure, a
substitutional nitrogen adjacent to a vacancy in the diamond lattice. Figure
9a) shows the NV-center photoluminescence spectrum. Notable features in the
spectrum are the zero phonon line at $\sim 637$ nm and the series of phonon-
assisted transitions extending out to $\sim 800$ nm. The neutral NV center has
a zero phonon line at $\sim 575$ nm. All features are evident in the ensemble
NV center PL spectra presented in Fig. 9a). The electronic structure also
consists of a paramagnetic ground state that in bulk diamond has a splitting
between the $m_{s}=0$ and $m_{s}=\pm 1$ of $2.87$ GHz. See the right inset of
Fig. 9a). Important for experiments is the the NV-1 spin active ground state
can be optically prepared, controlled and detected. A number of theoretical
proposals [130, 131] and experiments [132, 133, 134, 135, 136] have explored
this hybrid levitated optomechanical system. Early experiments demonstrated
that it was possible to optically trap diamond nanocrystals con- taining NV
centers and observe the NV center photoluminescence [132]. The previous was a
first step towards using the PL as a channel to monitor the NV center spin
dynamics. Subsequent studies explored the possibility of engaging the NV-1’s
spin. One study focused on how the center of mass dynamics for the nanodiamond
influenced the photophysics and observed optically detected magnetic resonance
[133]. A second work explored how robust the spin properties are to the
gaseous nanodiamond environment. Figure 2b) shows how the NV-1’s ODMR
resonance is influenced by the background gas pressure [134]. The possibility
for coherent spin control was evidenced by imprinting Rabi oscillations into
the time resolved ODMR [135], see Fig. 2c). Spin coherence times of $\sim 100$
ns for the NV center on the levitated nanodiamond were estimated from these
measurements.

More recently, instead of using an optical trap, Delord et. al. [136]
levitated a microdiamond in a Paul trap taking advantage of the native diamond
surface charge. Although this review has focused on optical levitation, the
experiments in the Paul trap are exciting in that motional cooling is observed
dependent on how the spin resonance is actuated. The left panel of Fig. 9c)
illustrates how torsional cooling is realized. First, as a result of diamond
rotation, the microwave frequency detunes from the NV spin resonance. The
dynamic detuning leads to a torque that stiffens or relaxes the Paul trap
confinement dependent on whether the microwave frequency is blue or red
detuned. As a result of the previous and the fact the spin lifetime is
commensurate with torsional oscillation, the diamond torsional motion is
cooled when the microwave frequency is red detuned from the spin resonance.
The right panel of Fig. 9c) presents data , trace (iii), where the torsional
motion is cooled from 300K to 80K [136].

Figure 9: a) The nitrogen-vacancy center photoluminescence spectra. Left
inset: The NV center crystal structure. Right inset: The NV center electronic
structure. b) The NV center electronic structure illustrating the zero-field
split paramagnetic ground state. c) Exemplary NV center en- semble
photoluminescence spectrum. Visible are both the NV-1 and NV0.

###  3.3 Atomic gases and nanoparticles

It has been shown theoretically that cooling of an optically trapped
nanosphere is achievable by sympathetically cooling a cold atomic gas
optically coupled to the nanoparticle. In addition to being a potential method
to cool the particle to its quantum ground state of motion in one dimension,
it allows the dynamics of the hybrid system to be explored with cooling turned
off, in principle enabling observation of strongly-coupled dynamics between
the atoms and nanosphere. In addition, unlike other methods capable of cooling
the center of mass motion of levitated particles to the quantum ground state
such as cavity cooling by coherent scattering [20], such a setup would require
only a modest cavity finesse [137].

Along this direction, recent experimental results have demonstrated
sympathetic cooling of a membrane in a cavity using atoms in a separate vacuum
chamber [138, 139, 140, 141]. By placing a cryogenic membrane in a medium-
finesse optical cavity, the strong coupling regime and ground state cooling of
the membrane is predicted to be achievable via sympathetic cooling of the
atoms [142, 139]. Sympathetic cooling can significantly enhance optomechanical
cooling of mechanical resonators even outside of the resolved sideband regime
[143].

Figure 10: An ensemble of laser-cooled atoms trapped in an optical lattice
(right chamber) could be used to sympathetically cool the center of mass
motion of a trapped nanobead in a separate optical cavity (left chamber). The
left-most mirror of the cavity containing the bead has higher reflectivity to
provide increased coupling of the reflected field to the cold atoms. A dual-
beam optical tweezer traps the sphere at a position of maximal linear opto-
mechanical coupling between the sphere and lattice beam, as discussed in the
text. Adapted from Ref. [137]

.

For either a (tethered) membrane oscillator or a trapped dielectric particle
in an optical cavity, a two-way mechanical link is established between the two
systems. The finesse of the optical cavity acts as a lever to enhance the
coupling between the systems and helps to compensate for the impedance
mismatch that results from the lower mass of the atomic ensemble relative to
the levitated or tethered macroscopic mechanical system. The setup is
illustrated in Fig. 10. Cold atoms are trapped in an optical lattice in one
vacuum chamber and the same lattice beam also drives an optical cavity in a
second vacuum chamber containing the optomechanical element. The
optomechanical coupling between the oscillator and the cavity results in a
phase shift of the light reflected by the cavity if the nanoparticle moves.
This phase shift results in a shift of the position of the optical standing
wave potential which results in a force imparted on the atoms.
Correspondingly, for atoms confined in the one-dimensional standing wave
optical lattice potential formed by the beam reflecting from the optical
cavity, atoms feel a restoring force confining them to the antinodes of the
laser field for a laser that is red-detuned with respect to the atomic
resonance. If the atom moves away from this equilibrium position at the
maximial intensity anti-node, a force is established through absorption and
stimulated emission of trap laser photons which impart momentum on the atoms
to bring them back the trap minimum. Due to conservation of momentum, to
establish this however the intensity balance between the right-moving and
left-moving components of the standing wave must be disturbed. This results in
an intensity fluctuation on the light incident on the optical cavity where the
nanoparticle is trapped. since the nanoparticle is trapped at a location where
there is a linear optomechanical coupling, i.e. at a location where there is a
slope of the intensity of the standing wave inside the cavity, this results in
a force on the bead due to the motion of the atoms. Given this mechanical
link, it is thus possible to reduce the kinetic energy of the bead by
performing laser cooling of the moition of the atoms in their optical trap.

For an atom-membrane coupled system such as that realized in Ref. [139, 140],
ground state cooling can be achieved if the coherent dynamics occur at a
sufficiently fast time scale compared with dissipative effects. The fast
thermalization rate

$\Gamma_{\rm{th}}=\frac{k_{B}T}{\hbar Q},$ (16)

where $T$ and $Q$ are the initial temperature and mechanical quality factor of
the oscillator, requires cryogenic pre-cooling of the membrane [139, 140]. On
the other hand, because their coupling to the thermal environment is very weak
under ultra-high vacuum conditions, the thermalization rate in Eq. (16) for
levitated mechanical systems [144, 19] can be exceedingly low even at room
temperature, as the mechanical quality factors are predicted to exceed
$Q>10^{11}$. In Ref. [137] it was shown that optically trapped nanospheres in
vacuum can theoretically be cooled from room temperature to the ground-state
by sympathetic cooling with atoms in an optical lattice. With $Q\sim
10^{11}-10^{12}$ at room temperature the thermalization rate in Eq. (16)
becomes $\sim 10-100$ Hz. For an ensemble of $5\times 10^{7}$ Rb atoms in an
optical lattice, the opto-mechanical coupling rate can exceed several kHz,
making this system able to attain the strong coupling regime. With a suitable
sympathetic cooling rate of order $\sim 10$ kHz, ground state cooling of a
$300$ nm or smaller diameter sphere is possible.

Recent work has demonstrated a light-mediated coupling between the spins of
cold atoms and the mechanical motion of a membrane in a cavity [141]. By
utilizing a cryogenically precooled membrane,this method is predicted to allow
cooling of the mechanical oscillator close to its quantum mechanical ground
state and permit the preparation of non-classical states. Similar experients
using the spins degrees of freedom of cold atoms could also be applied to
levitated optomechanical systems.

###  3.4 Cold and degenerate atomic gases and optical cavities

The ready availability of samples of cold as well as degenerate samples of
atoms have led to levitated optomechanical studies of such gases coupled to
optical modes of electromagnetic cavities.

Figure 11: An atom cloud coupled to an optomechanical system, reproduced from
Ref. [145]

Typically, the atoms are levitated using the dipole force from the optical
lattice formed by the standing wave optical mode of the cavity. The attraction
for using atoms for cavity optomechanics is based on the strong light matter
interaction strengths achievable using dispersive coupling, that is, by
detuning the cavity optical mode far enough away from the atomic transition
that spontaneous emission has a very low probability of occurring. This
interaction strength is much larger than the dissipation - due to finite
cavity lifetimes and atom loss from traps - in the system, leading to a large
optomechanical cooperativity (which is the ratio of the coherent light-matter
interaction to the product of the optical and mechanical mode linewidths), up
to $\sim 10^{4}$ as can be seen in Fig.12 of [81]. These are about the largest
cooperativities displayed by any optomechanical system, and make effects such
as ground state cooling and radiation pressure shot noise easier to observe
compared to other platforms.

Cold - but not degenerate - atoms were introduced into optical cavities in an
optomechanical context in [146]. In these systems, about $10^{5}$ atoms at
$\sim 0.8\mu$K are localized at the wells formed by the optical wave and
oscillate harmonically about those points, forming a collection of
oscillators. With such setups optomechanical bistability was observed with
intracavity photon numbers far less than unity ($\sim 0.05$), pointing to the
strong light-matter coupling in the system. Signatures of collective atomic
motion were also observed in the cavity transmission in this work. The same
group also reported observation of measurement quantum backaction in the
system [147], followed by the demonstration of both linear and quadratic
optomechanical coupling [148] and ponderomotive squeezing [145]. The ability
of this configuration to detect forces near the standard quantum limit was
then shown in Ref. [149].

Degenerate bosonic atoms - in the form of a Bose-Einstein condensate - were
first coupled experimentally to an optical cavity in [150]. This configuration
coupled the optical lattice formed by a cavity mode to density oscillations in
the condensate created by matter-wave diffraction from the optical field. In
effect, the condensate provided a Kerr-like medium for the cavity optical
field. The density oscillations modulated the cavity transmission, which in
turn provided information about the mode dynamics in the condensate. It is
worth noting that this information is obtained with small backaction to the
condensate; this minimally destructive measurement may be contrasted with the
usual completely destructive measurements made usign absorption imaging on
condensates. Bistability in the system was also observed, for photon numbers
below unity. A large number of works treating the system, both experimental
[[151], [152], [153]] as well as theoretical [154],[155], [156]] have been
published, addressing density waves characterized by their linear [157],
pseudo-spin [[158]] or angular [159]] momentum .

Degenerate fermionic atoms - in the form of a quantum degenerate Fermi gas -
were first coupled experimentally to an optical cavity in [108]. The
configuration had earlier been proposed in [[160]]. Optomechanical response of
a Fermi gas was investigated in [[161]], across the so called BCS-BEC
transition (where BCS refers to a gas of weakly bound Fermi pairs and a BEC to
a Bose-Einstein Condensate). A superradiant transition was demonstrated in
[[162]]. A unitary Fermi gas - a strongly correlated quantum fluid - was
strongly coupled to cavity light, demonstrating the formation of polaritons in
[163].

The high degree of control that the experimentalists have been able to
establish over cold and degenerate samples of gases; the versatility that such
platforms display in offering selection over spin, atomic species, interaction
strengths; the minimally destructive readout offered by the dispersive
interactions; and the amenability of these systems to microscopic yet
tractable theoretical modeling are very great attractions for continuing the
study of such optomechanical systems.

##  4 Thermodynamics

###  4.1 Non-equilibrium thermodynamics of small systems

Non-equilibrium processes of small systems are widespread in the fields of
biology, chemistry, and physics, yet they can be difficult to understand. For
instance, how does a molecular motor work in noisy environments where
fluctuations are prevalent? Unlike macroscopic systems, thermal fluctuations
have a significant impact on microscopic systems. An optically levitated
particle serves as an ideal model for investigating stochastic thermodynamics.
By lowering the air pressure, the levitated particle can transition into an
under-damped regime, as opposed to the over-damped regime typically seen in
liquid-based particles. The same micro or nanoparticle can be stably trapped
in air or vacuum for many days, enabling in-depth examination of its dynamic
behavior.

Figure 12:  (A) When a silica nanosphere is trapped in an optical tweezer
(magenta), a series of laser pulses (green) applies an optical force on the
nanosphere to drive the system out of equilibrium. (B) Examples of measured
trajectories of the nanosphere in the phase space. Figure reproduced from
[164].

A levitated particle will do Brownian motion due to the collisions between
surrounding air molecules and the particle. For small displacements, we can
approximate the optical tweezers as a harmonic trap with a trapping frequency
of $\omega_{m}$. Due to the energy equipartition theorem, the rms amplitude of
the Brownian motion displacement of a levitated particle along one axis is
$x_{rms}=\sqrt{k_{B}T/(m\omega^{2}_{m})}$, where $m$ is the mass of the
particle. At short time scales, the mean square displacement (MSD) of the
Brownian motion scales as $\left<[\Delta x(t)]^{2}\right>=(k_{B}T/m)t^{2}$,
which is very different from Einstein’s law for diffussive Brownian motion of
a free particle: $\left<[\Delta x(t)]^{2}\right>=2Dt$, where $D$ is the
diffusion constant. With sufficient temporal and spatial resolution, we can
measure the instantaneous velocity of the Brownian motion of a levitated
particle. In 2010, with a levitated glass microsphere in air and in vacuum, Li
_et al._ measured the instantaneous velocity of the Brownina motion of a
particle for the first time [165]. The measured velocity of the Brownian
motion of a levitated particle follows the Maxwell-Boltzmann distribution. The
rms velocity of the particle along one direction will be
$v_{rms}=\sqrt{k_{B}T/m}$.

With a measured position trajectory, we can calculate its Fourier transform
and the power spectral density (PSD). The expected value of the PSD of the
Brownian motion of a levitated particle in a harmonic trap is:

$S(\omega)=\frac{2k_{B}T}{m\omega^{2}_{m}}\frac{\omega^{2}_{m}\Gamma_{gas}}{(\omega^{2}_{m}-\omega^{2})^{2}+\omega^{2}\Gamma_{gas}^{2}},$
(17)

where $\Gamma_{gas}$ is the damping rate of the levitated particle due to air
molecules. This equation is often used to calibrate the detection system in
levitated optomechanical experiments. For a spherical particle, the damping
rate at thermal equilibrium is [17]

$\Gamma_{gas}=\frac{6\pi\eta R}{m}\frac{0.619}{0.619+Kn}(1+c_{k}),$ (18)

where $\eta$ is the air viscosity coefficient, $R$ is the radius of the
levitated spherical particle, $Kn=l/R$ is the Knudsen number, $l$ is the mean
free path of air molecules. $c_{k}=(0.31Kn)/(0.785+1.152Kn+Kn^{2})$. This
equation enables us to estimate the radius $R$ of a levitated particle.

Over the past three decades, several thermodynamic relationships related to
nonequilibrium processes, known as fluctuation theorems [166, 167, 168, 169,
170], have been discovered. These fluctuation theorems generalize the second
law of thermodynamics and can be unified into a differential fluctuation
theorem (DFT) [171, 172]. To test DFT and deepen our understanding of
nonequilibrium physics including dissipation and irreversibility, we need
large statistics and the ability to measure the instantaneous velocities of
Brownian motion, which is doable with a levitated particle. Recently, Hoang
_et al._ conducted the experimental test of the DFT using an optically
levitated nanosphere in both underdamped and overdamped scenarios (Fig. 12)
[164]. In the experiment, a 1550-nm optical tweezer levitated the nanosphere
in a vacuum chamber. The nonequilibrium processes were controlled by a 532-nm
laser that applied a time-dependent optical force from the side. With a large
set of experimental data, DFT and several theorems derived directly from DFT,
including a generalized Jarzynski equality applicable to any initial state,
were tested [164]. This work demonstrated the power of using a levitated
particle for studying stochastic energetics.

Recently, Debiossac _et al._ investigated the thermodynamics of an optically
levitated microparticle under continuous, time-delayed feedback control [173].
With a time-delay in the feedback loop, the dynamics becomes non-Markovian.
This work tested the generalized second law under non-Markovian feedback
control. Rademacher _et al._ utilized feedback cooling to achieve rapid and
controlled temperature variations in a levitated particle, and investigated a
fluctuation theorem that included the effects of both mechanical and thermal
control [174]. When a thermalization process is interrupted by a sudden change
of the bath temperature, the Kovacs memory effect will lead to a nonmonotonic
evolution of its energy. It is challenging to control the real temperature of
the environment suddenly. Instead, the thermal bath can be simulated by an
external white Gaussian noise source. By changing the effective thermal bath
temperature much faster than thermal relaxation time fo the system, the Kovas
memory effect was observed with an optically levitated nanoparticle [175].

Many microscopic processes, such as chemical reactions and protein folding,
involve multiple metastable states. An optically levitated particle can also
be used to study thermodynamics involving two or more metastable states. The
required multistable potential profiles can be generated by two or more
optical tweezers [176, 177, 115]. Recently, Rondin _et al._ studied stochastic
processes of a levitated nanoparticle in a bistable optical potential
generated by two optical tweezers [59]. They measured the thermally activated
transition rate between two traps for different damping levels and observed
the Kramers turnover, which predicted that the escape rate reached a maximum
at intermediate damping. Militaru _et al._ added an engineered stochastic
force that mimicked self-propulsion to study active escape dynamics of a
levitated nanoparticle in a bistable potential [178]. They observed an
additional activity-related turnover in the escape rate.

###  4.2 Laser Refrigeration

During experiments in high vacuum a levitated particle no longer thermalizes
with the surrounding gas environment due to an increase in the mean-free-path
between particle collisions. Consequently, the power from optical absorption
no longer can be carried away by the reduced number of molecular collisions,
leading to an increase in the particle’s internal absolute temperature (Tint).
The molecules that do collide with a trapped particle leave the particle’s
surface with a temperature (Tem) greater than the surrounding gas temperature
(Tgas), leading to a new damping term
$\Gamma_{em}/2\pi=1/16\cdot\sqrt{(T_{em}/T_{gas})}\cdot\Gamma_{gas}$. If
optical absorption is high enough, additional effects become significant
including photophoretic forces [179], changes in the trapped particle’s
physical properties (density, refractive index), particle melting, or even
complete particle destruction through evaporation [16] or burning [180].

One potential approach to mitigate the detrimental consequences of internal
particle heating is through solid-state laser refrigeration (SSLR)[181, 182].
SSLR is a process that has been demonstrated to cool the internal temperature
of materials with near-unit external radiative quantum efficiencies,
$\eta_{ext}$, through the efficient emission of anti-Stokes (upconverted)
luminescence. The process is based on the emission of luminescence with an
average photon energy ($h\nu_{ave}$) that is greater than the average photon
energy that is absorbed ($h\nu_{abs}$). In many ways the process is analogous
to operating a continuous-wave laser in reverse. First, coherent, low-entropy
photons are used to create an electronic excited state in a solid. Second, the
excited state extracts phonons from the host matrix with phonon scattering
time scales on the order of picoseconds. Third, the excited state relaxes,
emitting a photon with both a higher energy and a higher entropy relative to
the photon absorbed originally from the excitation laser. The cooling cycle is
consistent with the second law of thermodynamics when considering that the net
entropy of the emitted photons [183] (summing over their phase, polarization,
direction, and energy) is greater than the net entropy of the photons from the
excitation laser. The theoretical thermodynamic efficiency of the process
$\eta=(h\nu_{ave}-h\nu_{abs})/h\nu_{abs}$ is typically on the order of a few
percent.

In 1995 a team led by Richard Epstein at the Los Alamos National Laboratory
reported a seminal experimental observation of SSLR [184] using an amorphous
fluoride (ZBLAN) glass material doped with trivalent ytterbium ions. A
measurement based on photothermal deflection was used to quantify cooling in
vacuum on the order of 1K below room temperature. An excellent recent review
article[181] discusses subsequent advances by Mansoor Sheik-Bahae’s group at
the University of New Mexico which demonstrated that using crystalline hosts
for Yb(III) ions leads to dramatic reductions in the minimum achievable
temperature that can be reached when using a continuous-wave laser to induce
cooling. In comparison with amorphous ZBLAN glass materials, using a
stoichiometric crystalline material such as yttrium lithium tetrafluoride
(YLiF4) significantly increases the magnitude of laser cooling due to a
combination of the near-unit external radiative quantum efficiency of Yb(III)
ions in this crystal, the crystal field splitting of the ytterbium ion’s 4f
orbitals by the crystal’s S4 point group site symmetry, and a corresponding
increase in the optical absorption coefficient for crystal field transitions
between 2F7/2 and 2F5/2 manifolds. A helpful review article has been
published[185] on the use of Judd-Ofelt theory to interpret experimental
spectra arising from interconfigurational 4f <-> 4f transitions within
lanthanide ions that are frequently used for solid state laser refrigeration.
It is possible to reach cryogenic temperatures ($\sim$90K) using laser gain
crystals of yttrium lithium fluoride (YLF) doped with ytterbium (III) ions
[181] which could have significant advantages in all-optical cooling of high
precison silicon reference cavities[186]. Recent reports have demonstrated the
use of SSLR in the optical cooling of focal plane arrays[187] for vibration-
free imaging.

One challenge in the field of SSLR has been the time required to grow bulk
crystals with low background absorption levels using either the classical
Czochralski or Bridgemann growth methods. These processing approaches normally
require large amounts of purified reagents and time scales on the order of
weeks to grow, cut, and polish a bulk crystal. Recently in 2015 it was shown
that a low-cost hydrothermal processing approach can be used to synthesize
crystalline host materials that are capable of laser cooling[188]. These
materials were used to report the first experimental observation of cold
Brownian motion (CBM) [188] using optically-levitated microcrystals of Yb:YLF
in water. This was the first experimental demonstration of CBM since
Einstein’s seminal paper[189] on Brownian motion in 1905. Before this work it
was not clear whether it would be possible to observe SSLR in liquid water due
to the absorption of the pumping laser. It was also not obvious that materials
grown through hydrothermal processing could be laser cooled due to the
potential for background absorption from hydroxyl ions in the resulting
ceramic fluoride microcrystals. The theoretical modeling of CBM is identical
to what has been described recently for hot Brownian motion[190] except the
diffusing particle’s temperature is less than that of its surrounding medium.

CBM has also been reported for YLiF4 (YLF) nanocrystals levitated in
vacuum[74]. The particles of YLF were observed to have irregular shapes
following their preparation through top-down milling of bulk Czochralski
single crystals. The irregular morphology of milled YLF crystals was reported
to create significant shape induced birefringence that caused particle
rotation during levitation, and motivates the synthesis of materials for SSLR
with spherical morphology.

The ability to control the size, shape, composition, and phase of particles
for SSLR via bottom-up, chemical synthetic methods has opened up novel
experimental possibilities using a wide range of nano- to micro- meter size
materials[182, 191]. For example, hydrothermal processing has been used to
investigate non-stoichiometric crystalline materials for SSLR, including the
hexagonal ($\beta$) and cubic ($\alpha$) phases of sodium yttrium fluoride
(NaYF)[192]. The $\beta$ phase of NaYF is of great interest after being
reported to be an efficient emitter of upconverted photons[193]. However, the
growth of large bulk crystals of $\beta$-NaYF has not been reported to date
using Czochralski or Bridgmann methods due to large anisotropic thermal
expansion stresses related to the crystal’s hexagonal Bravais lattice.

The hydrothermal synthesis of crystals with well-defined shapes also has
enabled the use of van der Waals bonding to attach Yb-doped microcrystals for
the refrigeration optomechanical semiconductor sensors[194]. Hydrothermal SSLR
materials have also been used to demonstrate the laser cooling of nanodiamonds
containing the negatively charged nitrogen-vacancy center[195] that are used
for quantum sensing through optically detected magnetic resonance
spectroscopy[196]. The synthesis of spherical nanocrystals from cubic
$\alpha$-phase of NaYF has helped mitigate challenges in controlling both
shape birefringence[74] and optical birefringence during optical levitation in
vacuum[79]. Recent groundbreaking reports of SSLR using amorphous silicon
dioxide glass doped with ytterbium ions[197, 198] has opened up promising
future directions towards the formation of spherical materials for SSLR based
on the processing of high purity optical fibers[199]. Most recently, the
synthesis of two dimensional hexagonal $\beta$-NaYF discs have been
reported[200] that could be used for in future experimental searches for high-
frequency gravitational waves[31] based on how their two-dimensional
morphology minimizes photon recoil heating due to light scattering[201]. The
design and synthesis of new materials that demonstrate SSLR is a promising
future research direction, where the particle’s shape, size, density, mass,
composition, and birefringence can be selected to achieve a number of
experimental objectives in precision sensing and quantum information science,
discussed in more detail below.

##  5 Sensing

Optically levitated dielectric objects in ultra-high vacuum exhibit an
excellent decoupling from their environment, making them highly promising
systems for precision sensing. In particular, the center of mass modes of
optically-trapped silica nanospheres have exhibited high mechanical quality
factors in excess of $10^{7}$ [18] and zeptonewton ($10^{-21}$ N) force
sensing capabilities [202]. Such devices make promising candidates for sensors
of extremely feeble forces [203], accelerations [204, 205, 206], torques
[207], and rotations [24, 25, 26], testing the foundations of quantum
mechanics [208], observing quantum behavior in the vibrational of modes of
mechanical systems [209, 20, 81].

###  5.1 Force

Force sensing has a variety of applications, ranging from force microscopy at
the nanoscale to electromagnetic field sensing to gravitational wave
detection. High force sensitivity resonant sensors have typically consisted of
solid-state micro-fabricated structures [210, 211, 212, 213, 214, 215], for
example cantilever beams or membranes. In these systems, the internal
materials losses and losses from clamping mechanisms are responsible for
limiting the $Q$ factor of the oscillator in high-vacuum environments, to
typically below $Q\sim 10^{8}$. In contrast, in ultra-high vacuum (UHV), the
center of mass motion of micrometer sized dielectric spheres is well decoupled
from the environment, and could exhibit $Q$ factors approaching $10^{12}$,
leading to force sensitivity well below 1 aN/Hz1/2. Force sensing with
levitated particles is typically accomplished by measurement of the
displacement of the trapped particle, given knowledge of the mechanical
susceptibility of the particle, which depends on the spring constant and
dissipation. Gradients in force are also able to be deduced by measuring a
change in the mechanical resonance frequency of the oscillator, where a linear
force gradient manifests as a modification of the effective spring constant.
Our discussion will assume that the response of the trapped particle is
linear. Non-linearities e.g. Duffing terms, have been shown to occur for
nanoparticles trapped in optical traps in vacuum due to the non-harmonic
nature of the focused trap for sufficiently large particle amplitude [58]. By
cooling the motion of the particles, these nonlinear effects can be minimized
in practice. Fig. 13 illustrates a prior demonstration of calibrated
zeptonewton level force sensing with levitated nanoparticles, using
electrostatic calibration [22].

Figure 13:  Adapted from Ref. [22]). Force sensing with 300 nm nanospheres
using charged and uncharged beads (0,1, or 2 excess electrons). A known
applied electric field permits a calibration of the measured force which is
deduced from the displacement spectral density of the particle’s center of
mass motion (inset). Under high vacuum (HV) of $5\times 10^{-6}$ Torr,
background-free force measurement is demonstrated to the $\sim 6$ zN level for
long averaging times.

.

Feedback cooling and dynamical backaction cooling are possible to implement
simultaneously with force sensing, and generally do not improve the force
sensitivity for example from the thermal-noise limit. The cooling does allow
stabilization of the particle under high vacuum conditions as well as control
of the bandwidth of the sensor. For example in ultrahigh vacuum the mechanical
quality factor of trapped particles can exceed $10^{7}$, requiring 100s or
more for perturbations to the system to ring-down. By applying active or
passive feedback cooling, the ratio of the temperature to the damping rate
remains approximately constant, thus maintaining the same force sensitivity,
however the sensing bandwidth can be significantly increased by lowering the
$Q$ factor. The displacement detection of the particle however becomes more
challenging the more the damping is applied, particularly for on-resonance
detection schemes (see also Sec. 2 for further discussion of displacement
sensing).

Minimum detectable force – For force detection, it is desirable to have
minimal dissipation, leading to a high quality factor $Q$ for the center of
mass motion of the object. This can be seen as the minimum detectable force
due to thermal noise scales as $Q^{-1/2}$, and can be expressed as

$F_{\rm{min}}=\sqrt{\frac{4kk_{B}Tb}{\omega_{0}Q}},$ (19)

where $b$ is the bandwidth of the measurement, $T$ is the temperature of the
center of mass motion, $\omega_{0}$ is its resonance frequency, and $k$ is the
spring constant. This result can be derived by considering the equipartition
theorem: for decoupled degrees of freedom and harmonic motion, the root-mean-
square displacement $x_{\mathrm{rms}}$ for each translational mechanical
degree of freedom is related to the thermal energy according to
$k_{B}T=kx_{\mathrm{rms}}^{2}$. Assuming the particle is driven with a white-
noise spectrum due to random thermal motion, writing the frequency-independent
power spectral density of force noise as $S_{\mathrm{FF}}$ we find the
displacement power spectral density as
$S_{\mathrm{XX}}(\omega)=S_{\mathrm{FF}}|\chi(\omega)|^{2}$ where
$\chi(\omega)$ is the mechanical susceptibility of the oscillator. Solving

$<x^{2}>=\frac{1}{k^{2}}\int_{0}^{\infty}S_{\mathrm{FF}}d\omega\frac{\omega_{0}^{2}}{\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+\frac{\omega_{0}^{2}\omega^{2}}{Q^{2}}}}$
(20)

for $x_{\mathrm{rms}}$, we find
$x_{\mathrm{rms}}=\frac{S_{FF}^{1/2}}{k}(\frac{\omega Q}{4})^{1/2}$, which
after applying the equipartition theorem implies Eq. 19, as
$F_{\mathrm{min}}=S_{FF}^{1/2}b^{1/2}$.

Photon Recoil heating. For optically levitated particles, photon recoil
heating represents a second limitation to force sensitivity. The recoil
heating rate for a nanosphere is
$\gamma_{sc}=\frac{2}{5}\frac{\pi^{2}\omega_{0}V}{\lambda^{3}}\frac{(\epsilon-1)}{(\epsilon+2)}$
where $V$ is the volume of the nanosphere and $\lambda$ is the trap laser
wavelength. Including the contributuion from recoil heating, the minimum
detectable force $F_{\rm{min}}$ for a particle with center-of-mass temperature
$T_{\mathrm{CM}}$ is approximately [216]

$F_{\mathrm{lim}}=\sqrt{4k_{B}T_{\mathrm{CM}}\gamma_{g}bM\left[1+\frac{\gamma_{\mathrm{sc}}}{N_{i}\gamma_{g}}\right]},$
(21)

where $N_{i}=k_{B}T_{\mathrm{CM}}/\hbar\omega_{0}$ is the mean initial phonon
occupation number of the center-of-mass motion. In the high-vacuum regime,
$\gamma_{g}=\frac{16P}{\pi\bar{v}\rho r}$ is the gas damping rate at pressure
$P$ with mean gas speed $\bar{v}$ for a nanosphere of radius $r$ and density
$\rho$.

Minimum detectable frequency shift – gradient detection. Another modality of
detection relies on observing the gradient of an applied force, which
manifests as a frequency shift of the oscilator. Such techniques have been
previously employed in solid-state mechanical sensors such as cantilever beams
[217]. In this case by driving the oscillator with a well defined frequency
and phase into a amplitude $z_{\mathrm{rms}}$ which is typically larger than
the thermal motion amplitude, the minimum detectable frequency shift improves.
The minimum detectable frequency shift due to thermal noise is given by

$|\delta\omega_{0}/\omega_{0}|_{\rm{min}}=\sqrt{\frac{k_{B}T_{\rm{CM}}b}{k\omega_{0}Q_{\rm{eff}}z_{\rm{rms}}^{2}}},$
(22)

where $T_{\rm{CM}}$ is the effective temperature of the center of mass motion.
Photon recoil heating and technical noise depending on the setup also
represent limitations in principle.

###  5.2 Acceleration

Depending on the application, the acceleration or force sensitivity of a
levitated sensor, rather than its displacement sensitivity, is typically of
interest, particularly when operated in a regime where the displacement
detection, limited by shot noise, or electronic noise or other technical
noise, is not the limiting factor. Optically levitated particles can serve as
accelerometers both in a trapped configuration [205, 26] and in a free-fall
configuration [206]. For trapped nanoparticles acting as accelerometers, the
minimal detectable acceleration due to thermal noise and photon recoil heating
can be deduced from Eq. (21) by simply dividing by the mass of the trapped
particle. Thus while force sensitivity tends to improve for smaller size
particles, acceleration sensitivity tends to be better for larger objects,
although the bandwidth of the accelerometer may be limited for larger
particles due to correspondingly lower trapping frequencies. Acceleration
sensitivities for trapped $\sim 10$ $\mu$m size microspheres in vacuum have
been demonstrated at the $\sim$100 n$g$/$\mathrm{\sqrt{Hz}}$ level in Ref
[218], with a resonance frequency of order 100 Hz.

For free-fall accelerometers, a sensitivity to static forces of $10$ aN was
shown in Ref. [206]. In this case a particle was prepared in an optical trap
and then released from the trapping potential. During free fall a static
electrostatic force was applied via a known electic field, and the resulting
displacement of the particle was observed upon its recapture up to
approximately $100$ $\mu$s afterwards. Similar cooling and free-fall schemes
may also be possible with matter-wave interferometer based accelerometers, as
described in Sec. 7.

###  5.3 Rotation

Figure 14:  (a) A silica nanoparticle (NP) is levitated by a tightly focused
1550 nm laser in vacuum. The polarization of the 1550 nm laser is controlled
by a quarter waveplate. It is split by a polarizing beam splitter (PBS) and
sent to detectors for monitoring the rotation of the levitated nanoparticle. A
1020 nm laser applies an external torque for testing the sensitivity of torque
detection. Inset: SEM images of a silica nanosphere (left) and a silica
nanodumbbell (right). The scale bar is 200 nm for both images. (b) A measured
power spectral density (PSD) of the rotation of an optically levitated
nanoparticle . The corresponding rotation frequency is 5.2 GHz. Figure adapted
from Ref. [219].

In 1998, a calcite microparticle trapped by optical tweezers in water was
driven to rotate up to 357 Hz by a circularly polarized laser [220]. In 2013,
an optically levitated vaterite microsphere (diameter: 4.4 $\mu$m) was driven
to rotate up to 5 MHz in vacuum [125]. In 2018, two groups reported ultrafast
rotation of silica nanoparticles at about 1 GHz, which was mainly limited by
the bandwidth of the detectors [221, 25]. By using faster detectors and
optimizing other conditions, the rotation speed of a levitated silica
nanoparticle was further increased to more than 5 GHz in vacuum [219, 222].
Figure 14 shows an experimental schematic of driving a levitated nanoparticle
in vacuum to rotate with a circularly polarized laser.

###  5.4 Torque

If the system is limited by thermal noise, the minimum torque that can be
measured is $M_{min}=\sqrt{4k_{B}TI\gamma}/\sqrt{\Delta t}$ [219, 207]. Here
$k_{B}$ is the Boltzmann constant, $T$ is the environmental temperature, $I$
is the moment of inertia of the levitated nanoparticle, $\gamma$ is the
rotational damping rate, and $\Delta t$ is the measurement time. For a 150 nm-
diameter silica nanosphere, a measured torque detection sensitivity of
$(4.2\pm 1.2)\times 10^{-27}$ N m Hz-1/2 at room temperature has been reported
[219]. This sensitivity is several orders better than the best-reported
nanofabricated torque sensor at milikelvin temperatures.

###  5.5 Electric and magnetic fields

Electric fields have been used to calibrate the force sensitivity of levitated
objects [22] [223], to couple the Cartesian degrees of freedom of the
levitated potential [224], to drive motion in non Cartesian degrees of freedom
[225], and has also been used as as a field for actuating feedback cooling
with lower noise properties and greater linearity that optical AOM’s [87].
They also allow for quantification and control of the charge present on a
levitated object [226].

For electric field sensing it is important to note that particle launching
mechanisms often charge the particle [227], that levitated silica spheres
frequently possess a semi-permanent dipole moment [228], and a pressure
dependant complex internal and surface structure that can modify the charge
state of the particle during pumpdown [229]. As such, calibration of any
levitated sensor under its final operating conditions (especially with regards
to pressure and optical power) is essential [99].

##  6 Fundamental Physics

###  6.1 Gravitational force

Experimentally witnessing the quantum nature of gravity has been a long
elusive goal in modern physics. In part due to its extremely feeble nature,
gravity is arguably the least understood fundamental force in nature. The
gravitational attraction between two protons in the nucleus of an atom is
approximately $10^{36}$ times weaker than their electromagnetic repulsion,
reflecting the vast difference in the energy scale of quantum gravity versus
the other Standard model interactions. An explanation for this mysterious
“hierarchy problem” could be obtained by performing precise tests of the
gravitational inverse square law of attraction at sub-millimeter distances.
Deviations from the gravitational inverse square law are typically
parametrized in terms of a Yukawa correction to the Newtonian potential,
[230]:

$V(r)=\frac{G_{N}Mm}{r}(1+\alpha e^{-r/\lambda})$ (23)

where $G_{N}$ is Newton’s constant, $m$ and $M$ are the sphere and attractor
masses (if point-like), $r$ is their separation, $\alpha$ parameterizes the
strength of the new force, and $\lambda$ parameterizes its characteristic
range. For a realistic geometry an integral needs to be performed over the
test and attractor mass volumes. If the deviation from the ISL is mediated by
a new force carrier with mass $m_{\phi}$, then typically
$\lambda=\hbar/(m_{\phi}c$). Precision tests of gravitation require not only
extreme sensitivity, but also the ability to mitigate or subtract out
background forces of non-gravitational origin, including electromagnetic
effects from patch charges or the Casimir effect. Previous experiments have
employed sensitive torsion balances [231] testing the gravitational inverse
square law down to length scales below $50$ microns. Cryogenic
microcantilevers [210] and torsional oscillators [232] have been used to
search for inverse square law violations at even shorter distances. With an
optically-trapped nanosphere as a test mass, several orders of magnitude of
improvement is possible in the search for new gravity-like forces at the
micron distance scale, in a regime where several theories of physics beyond
the Standard Model predict possible new signatures [203, 233]. In the quantum
regime, operating a nanosphere matter-wave interferometer near a surface, as
described further in Sec. 7, may enable future tests of Newtonian gravity down
to the 5-$\mu$m length scale or below, yielding important clues about the
interface of gravity with the other three known forces in nature.

Optically trapped dielectric spheres can function as a test mass when held
using optical radiation pressure near a gravitational source mass. Possible
trapping geometries include confining the nanoparticle in an optical standing
wave formed near the surface of a mirror, either in an optical cavity [203] or
in free-space [67]. Non-Newtonian Gravity-like forces and Casimir forces can
be tested by monitoring the motion of the sphere as a gravitational source
mass is brought behind the mirror. Other approaches involving an optical
levitation trap are also being investigated [234, 235, 236]. Several orders of
magnitude of improvement is possible in the search for new gravity-like forces
at the micron distance scale due to the sensitivity of the technique. Setups
for test masses and attractor masses are shown in Fig. 15. Ref. [233] shows
the potential reach along with theoretical predictions for new fifth forces
that are Yukawa-type corrections to gravity at short distance scales using
spheres of sizes 300 nm and 20 $\mu$m, currently being experimemtally
investigated [22, 234, 67, 236].

Figure 15: Adapted from Ref. [233]. Attractor designs for short-range gravity
measurements using levitated systems proposed to-date. (a) A subwavelength
nanosphere is trapped at the antinode of a standing wave potential, several
hundred nm from a gold coated membrane shield. A patterned mass with different
density materials (green and yellow) is oscillated behind the shield. Figure
reproduced from Ref. [203]. (b) A linear array of Au and Si fingers, each 25
$\mu$m wide, is oscillated at several $\mu$m distance from a 5 $\mu$m diameter
sphere trapped in the center of the surrounding shielding electrodes [235]. An
additional microfabricated shield between the moving fingers and sphere can be
positioned on an independent stage [236]. Figure reproduced from [235, 236].
(c) A Au-coated microfluidic capillary is positioned at several $\mu$m
distance from a $\sim$10-20 $\mu$m diameter sphere, and alternating droplets
of a dense fluid (e.g. a polytungstate salt solution, $\rho\approx 3$ g/cm3)
and carrier fluid (e.g. mineral oil, $\rho\approx 0.8$ g/cm3) flow through the
capillary at $\sim$50 Hz [237].

Advances in sensitivity made possible by pushing the sensitivity of these
sensors into the quantum regime along with improved understanding and
mitigation of systematic effects due to background electromagnetic
interactions such as the Casimir effect and patch potentials will enable
several orders of magnitude of improvement in the search for new physics
beyond the Standard model.

###  6.2 High frequency gravitational waves

The extreme force sensitivity made possible by optical levitation lends itself
to the search for weak astrophysical signals, including feeble strain signals
from Gravitational waves or impulses from passing Dark Matter. One of the most
interesting sources of Gravitational waves in the high-frequency regime arises
from physics Beyond the Standard Model. The QCD axion is a well-motivated dark
matter candidate that naturally solves the strong CP problem in strong
interactions and explains the smallness of the neutron’s electric dipole
moment [238, 239, 240, 241]. The Compton wavelength of the QCD axion with
axion decay constant $f_{a}\sim 10^{16}$ GeV (at the Grand-Unified-Theory
[GUT] energy scale) matches the size of stellar mass BHs and allows for the
axion to bind with the BH “nucleus,” forming a gravitational atom in the sky.
A cloud of axions grows exponentially around the BH, extracting energy and
angular momentum from the BH [242, 243]. Axions in this cloud produce
gravitational radiation through annihilations of axions into gravitons. For
annihilations, the frequency of the produced GWs is given by twice the mass of
the axion: $f=145\ \mathrm{kHz}$, which lies in the optimal sensitivity range
for optically leviated sensors when $f_{a}$ is around the GUT scale. The
signal is coherent, monochromatic, long-lived, and thus completely different
from all ordinary astrophysical sources. The fraction of the BH mass the axion
cloud carries can be as high as ${10}^{-3}$ [243], leading to strain signals
detectable within the sensitivity band of optically levitated sensors [216,
201].

A Michelson interferometer configuration with Fabry-Pérot arms can be employed
as in Ref. [201], where a dielectric object is suspended at an anti-node of
the standing wave inside each Fabry-Pérot arm. A second laser can be used to
read out the position of the object as well as cool it along the cavity axes,
as described for a similar setup in Ref. [216]. The optical potential for this
trap is $U=\frac{1}{c}\int{I(\vec{r})(\epsilon(\vec{r})-1)d^{3}\vec{r}}$ where
$I$ is the laser intensity, $\epsilon$ is the relative dielectric constant,
and the integration is performed over the extent of the dielectric particle.
The trapping frequency along the axis of the cavity is determined by
$\omega_{0}^{2}=\frac{1}{M}\frac{d^{2}U}{dx^{2}}{|}_{x=x_{s}}$ for a sensor of
mass $M$ trapped at equilibrium position $x_{s}$.

A passing GW with frequency $\Omega_{\text{gw}}$ imparts a force on the
trapped particle [216], which is resonantly excited when $\omega_{0}$
$=\Omega_{\text{gw}}.$ Unlike a resonant-bar detector, $\omega_{0}$ is widely
tunable with laser intensity. The second cavity arm permits rejection of
common mode noise, for example from technical laser noise or vibration.

The minimum detectable strain $h_{\rm{limit}}$ for a particle with center-of-
mass temperature $T_{\mathrm{CM}}$ is approximately [216]

$h_{\mathrm{limit}}=\frac{4}{\omega_{0}^{2}L}\sqrt{\frac{k_{B}T_{\mathrm{CM}}\gamma_{g}b}{M}\left[1+\frac{\gamma_{\mathrm{sc}}}{N_{i}\gamma_{g}}\right]}H\left(\omega_{0}\right),$
(24)

where the cavity response function
$H\left(\omega\right)\approx\sqrt{1+4\omega^{2}/\kappa^{2}}$ for a cavity of
linewidth $\kappa$. Here $N_{i}=k_{B}T_{\mathrm{CM}}/\hbar\omega_{0}$ is the
mean initial phonon occupation number of the center-of-mass motion.
$\gamma_{g}=\frac{32P}{\pi\bar{v}\rho t}$ is the gas damping rate at pressure
$P$ with mean gas speed $\bar{v}$ for a disc of thickness $t$ and density
$\rho$, and $b$ is the bandwidth.

The photon recoil heating rate [216, 75]
$\gamma_{sc}=\frac{V_{c}\lambda\omega_{0}}{4L}\frac{1}{\int{dV(\epsilon-1)}}\frac{1}{\mathcal{F}_{\rm{disc}}}$
is inversely proportional to the disc-limited finesse
$\mathcal{F}_{{\rm{disc}}}$, i.e. ($2\pi$) divided by the fraction of photons
scattered by the disc outside the cavity mode. The integral is performed over
the extent of the suspended particle. Here $V_{c}$ is the cavity mode volume
[216]. While for a nanosphere the scattering and recoil is nearly isotropic
[75], for a disc, if the beam size is smaller than the radius of the object
and the wavefront curvature at the surface is small, the scattered photons
acquire a stronger directional dependence and tend to be recaptured into the
cavity mode. This reduces the variance of the recoil direction of the
levitated object caused by the scattered photons.

Both of the damping rates that contribute to sensitivity in Eq. 24 scale
inversely with the thickness of the levitated disc, for thickness smaller than
radius. In the gas-dominated regime, $\gamma_{\rm{sc}}\ll N_{i}\gamma_{g}$,
the sensitivity scales as $\sqrt{1/Mt}$ at fixed frequency. For sufficiently
low vacuum, the sensitivity becomes photon-recoil-limited, and the strain
sensitivity goes as $1/M\sqrt{\mathcal{F_{\mathrm{disc}}}}$.

For the application of GW detection, using high-mass, high-trap-frequency,
disc- or plate-like microparticles is ideal for achieving maximal sensitivity
with this technique. Recent experimental work has demonstrated optical
trapping of Yb-doped $\beta-$NaYF sub-wavelength-thickness high-aspect-ratio
hexagonal prisms with a micron-scale radius [31]. The prisms are trapped in
vacuum using an optical standing wave, with the normal vector to their face
oriented along the beam propagation direction, yielding much higher trapping
frequencies than those typically achieved with microspheres of similar mass.
This plate-like geometry is planned to be used in high frequency gravitational
wave searches in the Levitated Sensor Detector, currently under construction
[201]. Furthermore, the Yb-doped NaYF has previously been shown to exhibit
internal cooling via laser refrigeration, which may enable higher trapping
intensity and thus higher trap frequencies for gravitational wave searches
approaching the several hundred kHz range [31].

###  6.3 Casimir force and torque

According to quantum electrodynamics (QED), vacuum is not empty, but full of
virtual photons due to the zero-point energy. The quantum vacuum fluctuations
will lead to an attractive force between neutral metal plates in vacuum, which
is known as the Casimir force. The Casimir force can dominate the interactions
at nanoscales, and must accounted in searching for new force (such as the non-
Newtonian gravity) at small separations. The Casimir force has been measured
between two objects with different geometries and between three objects.
Besides linear momentum, the virtual photons also have angular momentum, which
can lead to a Casimir torque. So far, there has been only one experimental
report on the Casimir torque, which was measured with a liquid crystal [244].
A levitated nanoparticle in vacuum will an ultrasensitive force and torque
detector. Thus it will be ideal for study the Casimir force and Casimir torque
[245].

###  6.4 Dark Matter

Dark matter can also be detected by observing the interaction of passing
massive particles with the levitated nano-objects. For example, a recent
search has been performed for composite dark matter particles scattering from
an optically levitated nanogram mass, cooled to an effective temperature
$\sim$200 $\mu$K [234]. Similar techniques may allow detection of sufficiently
low momentum transfers that sub-MeV dark matter scattering coherently from 10
nm diameter spheres can be detected[246, 247]. Large arrays of such trapped
objects are possible, and can enable lower cross-sections to be reached [246,
233]. Such detectors are intrinsically sensitive to the direction of the dark
matter scatter, allowing an unambiguous determination of the astrophysical
origin of a signal if detected [234, 246, 233].

##  7 Summary and Future Perspectives: Experiment and Theory

Figure 16:  (a) A one-dimensional array of optically levitated nanospheres.
(b) The optical binding force between two optically levitated nanospheres as a
function of their separation. $\theta$ is the angle between the polarization
of the trapping laser and the $x$ axis. Figure adapted from Ref. [248].

###  7.1 Biological/chemistry applications

Optical levitation has been used in wide range of biophysical experiments
since the first demonstration that bacteria and viruses can be optically
trapped by Ashkin and Dziedzic[249]. One early application of optical
levitation was to select and transfer seed crystals used in structural biology
to prepare samples for single-crystal protein x-ray diffraction[250]. Single-
beam laser tweezers have also enabled several seminal advances in the
measurement of biologically relevant forces with pico-Newton sensitivity[251].
Chemically tethering single-molecule motor proteins to optically trapped
microspheres has enabled fundamental insights into the mechanical properties
of single DNA molecules[252], the equilibrium free energy difference between
distinct molecular conformational states[253], and the asymmetric hand-over-
hand mechanism underlying the motion of kinesin motor proteins[254]. Gold
nanorods have also been used in place of transparent microspheres as molecular
handle, leading to photothermal heating that increases the translational
stepping speed of myosin-V motor proteins[255].

Optically trapped polystyrene microspheres have been used for the sensing of
pathogenic viruses, including the in situ detection of H7N9 influenza virus
DNA based on a sandwich assay using fluorescence from biotinylated quantum
dots[256]. Detection limits of 1 pM have been observed which are two orders of
magnitude lower than conventional methods. More recently, individual PhiX174
viruses with a diameter of just 25 nanometers and have been trapped using
plasmonic double nanohole (DNH) apertures in a gold film[257]. Optomechanical
measurement of the virus’ acoustic radial breathing mode frequency (32 GHz)
were also made using extraordinary acoustic Raman spectrosocpy.

Beyond the characteirzation of biomolecules and viruses in liquid buffers,
both single- and dual-beam optical trapping have been used to open up new
possibilities in the spectroscopic characterization of biological cells and
other bioaerosols[258] in air. For example, chemical information about
optically levitated biological specimens can be acquired through the analysis
of inelastic Raman light scattering[259]. Optical levitation has also been
used to hold biological samples for background-free x-ray fluorescence
spectroscopy using synchrotron radiation[260]. More recently, photophoretic
forces have also been used to deliver aerosolized viruses into the focus of an
x-ray free-electron laser for femtosecond x-ray diffractive imaging[261].

###  7.2 Optical binding and Levitated particle arrays

While most studies in levitated optomechanics in vacuum have been focused on
single particles, there is a growing number of works on multiple particles. An
array of optically trapped ultracold atoms have been used for quantum
simulation and quantum computing. An array of optically levitated
nanoparticles in vacuum will offer new opportunities that do we can not do
with a single levitated nanoparticle [248]. Fig. 16 illustrates the effect of
the laser polarization on optical binding forces for two optically levitated
nanospheres separated by varying distances.

###  7.3 Near field interactions

Several experiments have realized methods for trapping nanoparticles near
surfaces, including silicon and SiN membranes [262, 263] and near photonic
crystal structures [264], and near a metallic mirror [67]. Extending such a
setup to allow systematically scanning the position of the particle with
respect to the surface creates opportunities for three dimensional surface
force microscopy. To date no 2-D image of a surface has been rendered using a
levitated nanoparticle as a probe of the surface potential. Due to the high
degree of force sensitivity and ability to constrain the particles motion to
the zero point level this is a potentially useful future goal for the field.
Furthermore, a greater understanding of the distribution of surface charges
and permanent and induced dipoles over the levitated object itself is an
another near-term goal that can be realized by studying nanoparticles trapped
nearby objects. Another interesting application enabled by near surface
trapping is to study radiative heat transfer at the nanoscale [265].

###  7.4 High mass matter wave interferometry

Wave-particle duality is at the core of quantum mechanics. Wave-like behavior
of massive particles including electrons, neutrons, and atoms has been studied
over the past several decades, and recently even molecules as large as 25,000
atomic mass units have been shown to exhibit wave-like behavior [266]. It is
natural to ask how massive of an object can behave like a wave, i.e. be placed
into a quantum superposition state after undergoing diffraction. Optically
levitated nanoparticles offer a promising route to push the limits of quantum
theory in this manner and to test foundational aspects of quantum theory. In
particular, optically-levitated dielectric objects in ultra-high vacuum
exhibit an excellent decoupling from their environment, making them highly
promising systems for precision sensing. An exciting prospect is to harness
uniquely quantum aspects of sensing in these levitated nanoparticle systems,
taking advantage of wave-particle duality in nanoparticles of unprecedented
size [267, 204]. Given the recent experimental demonstration of quantum ground
state cooling of optically trapped nanoparticles, experiments become
increasingly feasible to demonstrate the wave-like nature of the states of
ultracold nano-objects. Matter wave interference of nanoparticles cooled to
act as a point like source can be realized even for thermal states
significantly above the quantum ground state [204].

Figure 17:  Adapted from Ref. [204]. An example of a nanoparticle matter wave
interference protocol. (left) Schematic of setup. A nanosphere is trapped and
cooled near a surface. The optical trap is turned off and the wave packet
expands as the particle falls. The particle is then diffracted from a light
grating, and its position is measured after an additional free-fall time. The
acceleration of the nanosphere towards the surface from Casimir and
gravitational forces shifts the interference pattern that is generated when
the nanosphere diffracts from the grating. (middle) Density plot of
$|\Psi(x)|^{2}$ following the grating for zero acceleration for releasing the
trap at $\omega_{0}=2\pi\times 100$ Hz from its ground state. (right) As in
middle panel, with $a_{\pi}=4\times 10^{-7}g$ constant acceleration.
Parameters chosen as in Ref. [204] with mass of $10^{6}$ amu and interrogation
time $T_{T}$ of 0.25s.

A possible accelerometer protocol utilizing nanosphere interference is shown
in Fig. 17. Here the center of mass temperature of a nanoparticle is
sufficiently cooled so that the nanoparticle acts as a coherent source. In one
possible scheme, the particle is allowed to free fall for a duration equal to
the Talbot time. At this point a light-pulse grating is applied to diffract
the wave packet of the falling particle, and after another duration of the
Talbot time the location of the particle is measured [268]. After repeated
measurements, an interference pattern appears one particle at a time, with the
locations of the fringe maxima and minima being sensitive to the acceleration
experienced parallel to the grating direction during the free fall time. As
the locations of the interference maxima do not depend on the initial
transverse velocity, such approaches may have inherent sensitivity advantages
when compared to simple free fall schemes, depending on backgrounds and noise
sources [204]. The greatly reduced wave packet expansion of nanoparticles when
compared with atoms used in atom interferometry provide a route to measure
more highly localized forces and interactions, for example in close proximity
to surfaces [204]. Technical challenges that must be overcome to realize
interference protocols which generate an interference pattern one-particle-at-
a-time include a fast on-demand loading mechanism compatible with ultra high
vacuum and potentially cryogenic operation.

In massive quantum-superposition experiments, background gas collisions
arising from imperfect vacuum are a serious limitation to the masses of the
particles that can be placed in quantum superposition and to the associated
coherence times. Furthermore if optically-trapped nanoparticles are diffracted
from a light grating and placed into a superposition state, blackbody emission
from the surface of a room temperature sphere can decohere the quantum
superposition. To overcome these limitations, interferometry with levitated
nanoparticles in a cryogenic, extreme high vacuum chamber may be a route
forwards. In addition cryogenic laser refrigeration of levitated nanoparticles
could assist to mitigate this decoherence [74].

Additionally, quantum point defects such as the negatively-charged nitrogen
vacancy center have been proposed to enable ultra-sensitive tests of quantum
mechanics focused on novel demonstrations of matter wave interferometry using
macroscopic test masses. When optically trapping individual nanodiamonds that
contain intrinsic NV centers, laser heating ultimately limits the achievable
base pressure since residual absorption leads to crystal heating that cannot
be dissipated via thermal conduction. This presents an obstacle to preparing
COM quantum states. Nanoparticle state control, when coupled with the ability
to laser cool the nanodiamond material, will present an opportunity for
pursuing matter wave interference.

###  7.5 Detecting high energy radiation

Levitated optomechanical experiments are naturally highly sensitive to
momentum impulses [233]. Experiments have therefore been designed such that
study of nuclear decays may be possible by directly observing the motion of a
levitated optomechanical sensor embedded with a decaying radioactive species.
As the species decays, the momentum from the outbound high energy ejecta is
offset by an equal and opposite momentum impulse on the levitated sensor.

This, in principle might allow for the study of nuclear processes without
having to transform the outbound nuclear ejecta directly into electrons and
may therefore be sensitive to parameter spaces not covered by existing
detector technologies. For example, heavy sterile neutrinos [269].

Another proposal would be to use a levitated silica sphere as a
platform/substrate to actually modify nuclear decays via the nuclear Purcell
effect changing weak decay rates [270]. In this case a radioactive species is
also embedded with the lattice of the levitated object (silica in this
proposal), which provides a set of cavity like boundary conditions to modify
the low lying energy state of thorium 229.

###  7.6 Squeezing and non classical state preparation

Correlated phase spaces have been reported as "classical squeezing" by several
groups, both in real time [271] and by reconstructed phase spaces sampled over
several hours of experiment time in a pulsed operation[103]. It should be
noted that these examples are completely classical effects that do not
compress an axis of the phase space below any limit defined by an uncertainty
principle. However genuine squeezing of the quantum ground state state of a
mechanical oscillator may well be achieved in the near future due to the
proliferation of ground state capable levitated quantum systems.

####  7.6.1  Levitated optomechanical state preparation

A major catalyst for the interest in levitated optomechanical oscillators was
the expectation that it should be possible to create an environment where the
levitated oscillator can be prepared into its quantum mechanical ground state
[272]. Tantalizing was the suggestion that it could be done at room
temperature. Nearly a decade after such predictions, the first experiments
demonstrating ground state preparation have been achieved. Of course,
mechanical number states are not the only possible states that can be
engineered and in this section we briefly survey some of the non-thermal
levitated mechanical oscillator states that have been reported.

Passive (or free-running) levitated optomechanical oscillators, meaning no
feedback mechanism has been introduced to modulate the oscillator’s dynamics,
naturally exhibit thermal statistics. It is only through engineered feedback,
either passive or active, that the oscillators mechanical state can be
controlled. The first experiments evidencing deviation from thermal statistics
reported correlated phase spaces that exhibit “classical squeezing”. The phase
spaces were reconstructed over several hours and relied on a pulsed experiment
[273]. One example of the deviation from the phase space of a thermal
oscillator is shown in Fig. 18a). A second experiment reconstructed the phase
space in real time to demonstrate classical squeezing of two coupled levitated
oscillators [274]. These experiments are classical in the sense that they do
not compress an axis of the phase space below a limit defined by an
uncertainty principle. However genuine squeezing of the quantum ground state
state of a mechanical oscillator may well be achieved in the near future due
to the proliferation of ground state capable levitated quantum systems.

A second modulation of the free running levitated optomechanical oscillators
thermal statistics has been realized. It was informed by the discovery that
the center-of-mass oscillator dynamics in one direction resembled that of a
single mode phonon laser [275]. By carefully tuning the applied nonlinear
feedback cooling and linear feedback gain for one mode of oscillation the
thermal statistics of the oscillator were converted to the statistics of a
single mode laser. The change in phonon statistics were confirmed by measuring
the phonon number distribution, seeing the change in the zero-delay energy
correlation function of the oscillator dynamics from that of a thermal state
(bunching) to that of a laser (unity) and observing the change in the
oscillator’s phase space distribution, see Fig. 18b).

As mentioned in the introductory paragraph of this section, most recently a
number of groups have reported cooling the oscillator’s center-of-mass into
its quantum ground state [276, 277, 278, 279]. In one experiment, a levitated
oscillator (approximately 140 nm diameter) situated at the node in an optical
cavity was cooled to its ground state by a coherent scattering. The cavity
suppressed both elastic scattering, via the particle spatial position, and
cooled the center-of-mass motion since spectrally the cavity suppressed Stokes
scattering and enhanced anti-Stokes scattering. The experiments achieved a
minimum average phonon number of $\bar{n}=0.43\pm 0.003$ when the optical
tweezer is detuned from the cavity resonance by the levitated oscillator’s
oscillation frequency, see Fig. 18c) [276]. A second experiment was conducted
without a cavity in a cryogenic environment. The reduced gas temperature of
60K leads to cryogenic pumping the gas to a pressure on the order of
$10^{(-9)}$ mbar. A combination of nonlinear feedback cooling and cold damping
via the Coulomb force result in average occupation numbers of $\bar{n}=0.66\pm
0.08$. Stokes and anti-Stokes thermometry, see Fig. 18d) was used to determine
the average phonon number [277]. A third approach to ground state cooling took
advantage of a standing wave trap along one oscillator direction and employed
optical cold damping to achieve the center-of-mass ground state with
$\bar{n}=0.85\pm 0.20$ [278]. Finally, in a recent experiment [279], again
using coherent scattering into an optical cavity mode, two center-of-mass
oscillation directions were cooled to their motional ground state with average
phonon occupation numbers of $\bar{n}_{1}=0.83\pm 0.10$ and
$\bar{n}_{2}=0.81\pm 0.12$.