# Levitated Nanoparticles for Microscopic Thermodynamics - A Review

Jan Gieseler  Department of Physics, Harvard University, 17 Oxford Street,
Cambridge, MA 02138, USA  James Millen  Faculty of Physics, University of
Vienna, Boltzmanngasse 5, Vienna 1090, Austria  Department of Physics, Kings
College London, Strand, London WC2R 2LS, UK

###### Abstract

In this article, we review the current state of the art in using levitated
nanoparticles to answer questions related to thermodynamics and non-
equilibrium physics. We begin in Section 2 with a summary of the relevant
deterministic and stochastic forces, which determine the particle dynamics and
allow for control of the particle. In Section 3 we give a brief review of the
stochastic (i.e. Brownian) motion of levitated particles, since the Brownian
particle is fundamental to the theory of stochastic thermodynamics. Then we
discuss the stability of the trapped particles and the related Kramers escape
problem in Section 4. After that, we introduce effective potentials for the
energy in Section 5. These potentials are useful to describe the dynamics of
levitated nanoparticles in a time-modulated trap, where the particle is driven
far away from equilibrium. In Section 6 we discuss the dynamics of relaxation
towards equilibrium, before we review the work on fluctuation theorems in
Section 7. Fluctuation theorems are a powerful generalization of the well
known thermodynamic inequalities for systems far from equilibrium. Finally,
Section 8 discusses the potential of constructing new kinds of heat engines
based on nanoparticles levitated in high vacuum.

##  I Introduction

In 1827, botanist Robert Brown noted the erratic movement of tiny particles
emitted from pollen grains in a liquid Brown (1828). This seemingly
unspectacular observation would play a critical role in the development of the
atomistic theory of matter. However, only in 1905 did Albert Einstein’s
theoretical analysis Einstein (1905) of Brown’s observation provide crucial
evidence for the existence of atoms. Einstein surmised that the random motion
of the suspended particles is a consequence of the thermal motion of
surrounding fluid molecules. Ever since the _Brownian Particle_ has been
essential in the development of our theoretical understanding of stochastic
processes in different fields, ranging from the sciences (chemistry, biology
and physics) to economics (e.g., finance).

Einstein further concluded that “ _the velocity and direction of motion of the
particle will be already very greatly altered in an extraordinarily short
time, and, indeed, in a totally irregular manner_ ” and that “ _it is
therefore impossible—at least for ultramicroscopic particles—to ascertain the
instantaneous velocity by observation_ ”.

This changed with the advent of optical tweezers, now a workhorse for studying
thermodynamics and non-equilibrium physics of small systems. The pioneering
experiments on optical forces were carried out by Ashkin and Dziedzic, where
they used optical forces to counteract gravity and thereby suspend micrometer-
scale spheres in vacuum Ashkin (1970, 1971); Ashkin and Dziedzic (1976). Later
Ashkin et al. demostrated the stable 3D optical trapping of micron-scale
particles purely by light Ashkin et al. (1986). Since then there has been an
explosion of research using _optical tweezers_ , to the point that they are an
off-the-shelf tool for physical and biological scientists Jones et al. (2015);
Spesyvtseva and Dholakia (2016). In this system, it is possible to control and
track the motion of mesoscopic objects with astounding precision.

While most of the research on optical tweezers has focused on trapping and
manipulating particles in suspension, there has been a renewed interest in
optical trapping in high vacuum Li et al. (2010); Gieseler et al. (2012);
Kiesel et al. (2013); Millen et al. (2015), motivated by the possibility to
enter the quantum regime Romero-Isart et al. (2010); Chang et al. (2010);
Barker (2010), which has lead to the development of the field of levitated
optomechanics Yin et al. (2013); Vamivakas et al. (2016). Even though there
has been tremendous progress towards entering the quantum regime, with
residual occupations of tens of phonons Jain et al. (2016), achieving the
ground state has been elusive. However, the exquisite control achieved in
these experiments does not only bring us closer to the quantum regime, it also
opens up a wide range of exciting new experiments in the classical domain. In
particular, they allow the study of Brownian motion of a single well-isolated
particle with high temporal and spatial resolution and controllable coupling
to the environment, thereby rebutting Einstein’s original statement Li et al.
(2010); Li and Raizen (2013) and providing new insights into microscale
thermodynamic processes in the underdamped regime. Specifically, thermodynamic
processes of a single particle are stochastic, that is quantities such as
energy, work and entropy are fluctuating quantities where the fluctuations are
of similar magnitude or even larger than the mean. This has profound
implications in the operation and fundamental limitations of microscopic
machines.

Here we review the current state of the art in using levitated nanoparticles
to answer questions related to thermodynamics and non-equilibrium physics. We
begin in Section 2 with a summary of the relevant deterministic and stochastic
forces, which determine the particle dynamics and allow for control of the
particle. In Section 3 we give a brief review of the stochastic (i.e.
Brownian) motion of levitated particles, since the Brownian particle is
fundamental to the theory of stochastic thermodynamics. Then we discuss the
stability of the trapped particles and the related Kramers escape problem in
Section 4. After that, we introduce effective potentials for the energy in
Section 5. These potentials are useful to describe the dynamics of levitated
nanoparticles in a time-modulated trap, where the particle is driven far away
from equilibrium. In Section 6 we discuss the dynamics of relaxation towards
equilibrium, before we review the work on fluctuation theorems in Section 7.
Fluctuation theorems are a powerful generalization of the well known
thermodynamic inequalities for systems far from equilibrium. Finally, Section
8 discusses the potential of constructing new kinds of heat engines based on
nanoparticles levitated in high vacuum.

##  II Forces and Potentials

###  II.1 Deterministic Forces

Most experiments with particles levitated in vacuum use optical forces to
create a stable trap (c.f. Fig 1). This gives a great deal of flexibility
since optical fields can be controlled very well in intensity, position and
time, allowing the creation of almost arbitrary force fields. However, since
light absorption heats Hebestreit et al. (2018) and potentially destroys the
particle, experiments have been limited to low absorption materials like
Silica and Silicon. In addition to the optical forces, the particle is subject
to gravity $\mathbf{F}_{\rm g}=m\mathbf{g}$, electric forces $\mathbf{F}_{\rm
e}=q\mathbf{E}$ if the particle carries a charge $q$ and magnetic forces
$\mathbf{F}_{\rm mag}=\nabla(\boldsymbol{\mu}\cdot\mathbf{B})$ if the particle
has a magnetic dipole moment $\boldsymbol{\mu}$.

Particles typically have a radius of $a\sim$ 100 $\rm nm$ but can also be much
bigger Li et al. (2010); Ashkin (1971). When the radius is much smaller than
the wavelength $\lambda_{\rm L}$ of the trapping laser $ak_{\rm L}\ll 1$,
where $k_{\rm L}=2\pi/\lambda_{\rm L}$, the particle can be treated as a
dipole in the Rayleigh approximation. The polarizability for a particle with
volume $V$ is thereby given by

$\alpha_{\rm 0}=\epsilon_{0}V\boldsymbol{\chi},$ (1)

where the total susceptibility of the particle
$\boldsymbol{\chi}=\boldsymbol{\chi}_{e}\left(1+\boldsymbol{N}\boldsymbol{\chi}_{e}\right)^{-1}$
depends on the material via the material susceptibility
$\boldsymbol{\chi}_{e}$ and on its geometry through the depolarization tensor
$\boldsymbol{N}$, which in general are both rank-2 tensors. However, for
isotropic materials, the material susceptibility simplifies to a scalar
$\chi_{e}$ and similarly for a sphere the depolarization tensor is isotropic
and simplifies to a scalar $N=1/3$. Thus, for a sphere we recover the
Clausius-Mossotti relation $\chi=3(\epsilon_{p}-1)/(\epsilon_{p}+2)$, where we
use $\epsilon_{p}=1+\chi_{e}$. For a particle with a uniaxial anisotropy, the
susceptibility
$\boldsymbol{\chi}=\text{diag}(\chi_{\|},\chi_{\perp},\chi_{\perp})$, has a
component $\chi_{\parallel}$ parallel and a component $\chi_{\perp}$
perpendicular to the symmetry axis. For example, the depolarization tensor of
a cylinder is $\boldsymbol{N}=\text{diag}(0,1/2,1/2)$ in the frame of the
cylinder, where the cylinder axis is along the $x$-axis. Consequently,
$\chi_{\|}=\epsilon_{\rm r}-1$, $\chi_{\perp}=2(\epsilon_{\rm
r}-1)/(\epsilon_{\rm r}+1)$ for a cylinder with isotropic $\epsilon_{\rm r}$.
This means the maximal polarizability of a cylinder is $(\epsilon_{\rm
r}+2)/3$ times higher than for a sphere of the equivalent volume, i.e. a
factor of 2 for silica, and a factor of 4.6 for silicon.

In general, one has to consider the total field to calculate the optical
forces Lukas Novotny and Hecht . The total field is the sum of the incident
and the scattered field and is the self consistent solution to Maxwell’s
equations. For arbitrary shaped particles, the total field has to be
calculated with numerical methods. However, for a spherical particle, the
modified polarizability

$\alpha=\alpha_{\rm 0}\left(1-i\frac{k_{\rm
L}^{3}\alpha_{0}}{6\pi\epsilon_{0}}\right)^{-1},$ (2)

accounts for the radiation reaction of the particle to its own scattered
field, such that the induced polarization due to a field $\mathbf{E}_{0}$ is
$\mathbf{P}=\alpha\mathbf{E}_{0}$. We introduce $\alpha^{\prime}$ and
$\alpha^{\prime\prime}$ to refer to the real and imaginary part of the
polarizability, respectively.

Knowing the polarizability, we can calculate the optical force for sub-
wavelength particles in the Rayleigh approximation. The optical force has
conservative and non-conservative contributions Albaladejo et al. (2009)

$\mathbf{F}_{\rm opt}=\alpha^{\prime}\nabla I_{0}/4+\sigma_{\rm
tot}\left[\mathbf{S}/c+c\nabla\times\mathbf{L}\right],$ (3)

where $\sigma_{\rm tot}=\alpha^{\prime\prime}k_{\rm L}/\epsilon_{0}$,
$\omega_{\rm L}$ is the optical frequency, and
$\mathbf{L}=-i\epsilon_{0}\left<\mathbf{E}\times\mathbf{E}^{*}\right>\left/4\omega_{\rm
L}\right.$, $\langle\dots\rangle$ representing a time average. The total
cross-section $\sigma_{\rm tot}$ is the sum of the absorption and scattering
cross-sections. The first term is a conservative force $\mathbf{F}_{\rm
grad}=\alpha^{\prime}\nabla I_{0}/4$. It pulls particles with a high
refractive index relative to their surroundings toward the region of maximum
light intensity. In optical tweezers, this is the focal volume of the light
beam.

Figure 1: Schematic of optical levitation setup. A nanoparticle is trapped by
a tightly focused laser beam. The translational degrees of freedom of the
nanoparticle are measured with photodetectors and the center-of-mass motion is
cooled by parametric feedback. In addition to feedback, external modulation
allows excitation of the particle to drive it far from equilibrium. The top
inset highlights the dominant forces in a typical optical levitation
experiment, which are the optical gradient and scattering forces and gravity.
The bottom inset shows the temperatures involved in a collision with a heated
sphere: the sphere’s centre-of-mass temperature ($T_{{\rm CM}}$) and surface
temperature ($T_{{\rm int}}$), and the temperatures of the impinging gas
particles ($T_{{\rm gas}}$) and emerging gas particles ($T_{{\rm em}}$) with
$T_{{\rm gas}}\leq T_{{\rm CM}}\leq T_{{\rm em}}\leq T_{{\rm int}}$. The
collision with the air molecules leads to damping $\Gamma_{\rm gas}$, which
depends on the pressure. Main figure taken from Gieseler et al. (2014) with
permission from Physical Review Letters. Inset adapted from Millen et al.
(2014) with permission from Nature Nanotechnology.

The non-conservative scattering force $\mathbf{F}_{\rm scat}=\sigma_{\rm
tot}\left[\mathbf{S}/c+c\nabla\times\mathbf{L}\right]$ has two contributions:
the radiation pressure term, which is proportional to the time averaged
Poynting vector $\mathbf{S}=\left<\mathbf{E}\times\mathbf{H}^{*}\right>$,
$\mathbf{H}$ being the magnetic field and a curl force associated to the non-
uniform distribution of the time averaged spin density of the light field. The
curl force is zero for a plane wave but can be significant for a tightly
focused beam in optical tweezers. However, since
$\alpha^{\prime\prime}/\alpha^{\prime}\propto a^{3}$, the non-conservative
forces vanish for small particles and we will neglect them in the following
discussion.

####  II.1.1 Optical Potential

The conservative force in Eqn. (3) can be expressed as the gradient of a
potential $\mathbf{F}_{\rm opt}\approx\mathbf{F}_{\rm grad}=-\nabla U_{\rm
opt}$. At the bottom of the potential, the centre-of-mass motion is harmonic,
with frequencies

$\displaystyle\Omega_{q}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{\chi}{c\pi\rho}}\frac{\sqrt{P_{\rm
opt}}}{\text{w}_{0}\text{w}_{q}},$ (4)

where $\text{w}_{q}$ denotes the width of the optical intensity distribution
along the three directions ($q=x,y,z$) and
$\text{w}_{0}^{2}=\text{w}_{x}\text{w}_{y}$, $P_{\rm opt}$ is the optical
power, $\rho$ the density of the particle. Note that for tightly focused laser
beams with linear polarization, as commonly used in optical trapping, the
field distribution is slightly elongated along the direction of polarization
of the incident field, which leads to slightly different trapping frequencies
along the two transverse directions. For larger oscillation amplitudes, the
motion becomes anharmonic, and the nonlinear coefficients can be obtained from
higher derivatives of the optical potential Gieseler et al. (2013).

####  II.1.2 Rotation

For anisotropic particles, the light matter interaction is more complicated,
since it depends upon the alignment of the object relative to the polarization
axis of the field Stickler et al. (2016); Kuhn et al. (2017). For linearly
polarized light, the particle experiences an optical torque which aligns the
particle with respect to the polarization axis Geiselmann et al. (2013). For
small deflections from the polarization axis the angular motion is harmonic.
In contrast to linearly polarized light, the polarization axis of circularly
polarized light rotates at the optical frequency. This is too fast for the
particle to follow. Nonetheless, light scattering transfers the angular
momentum of the light to the particle and exerts a torque Kuhn et al. (2017).
The polarization anisotropy can originate from the intrinsic birefringence of
the particle Arita et al. (2013); Bishop et al. (2004); Paterson et al. (2001)
or from the anisotropic shape of the particle, e.g., a cylinder Kuhn et al.
(2015) (c.f. Eqn. (1)). The rotational degree-of-freedom of a levitated
nanoparticle could be used to design microscopic engines in the classical and
quantum regime Roulet et al. (2017). As of yet, there has been no such
experimental implementation.

###  II.2 Stochastic Forces

So far we have only considered the static forces that are responsible for
creating a potential landscape. In addition, stochastic forces excite the
particle motion and lead to stochastic dynamics, which are of particular
interest when studying thermodynamics of individual small particles. The
stochastic forces result from the interaction of the particle with its
environment. These interactions lead to dissipation $\Gamma^{n}$ and are the
source of the different random forces acting on the particle (labelled $n$).
The strength of the random forces is characterized by their power spectral
densities $S_{\rm ff}^{n}$. For most practical purposes, they can be
considered as frequency independent (white noise). After a time $\approx
1/\Gamma_{\rm CM}$, where $\Gamma_{\rm CM}=\sum_{n}\Gamma^{n}$ is the total
damping rate, the particle reaches an effective thermal equilibrium, which is
characterized by the effective temperature through the fluctuation-dissipation
relation:

$T_{{\rm CM}}=\frac{\pi S_{\rm ff}}{{\rm k_{B}}m\Gamma_{\rm CM}},$ (5)

where $S_{\rm ff}=\sum_{n}S_{\rm ff}^{n}$ is the total force spectral density,
$m=V\rho$ is the particle mass, and ${\rm k_{B}}$ is Boltzmann’s constant. In
the following we describe the individual contributions. They are: collisions
with air molecules ($n=$ gas), radiation damping ($n=$ rad), feedback or
cavity damping ($n=$ fb) and external driving ($n=$ drive).

####  II.2.1 Gas Damping

For high pressures, the interaction with the gas is so strong that the
particle motion is heavily damped and its internal temperature $T_{{\rm int}}$
and centre-of-mass temperature $T_{{\rm CM}}$ quickly thermalize with the gas
temperature $T_{{\rm gas}}$. In this regime the damping becomes independent of
pressure $\Gamma_{\rm CM}/2\pi\approx 3a\mu_{v}/m$, as predicted by Stokes
law, where $\mu_{v}$ is the viscosity. However, for strong absorbers, e.g.,
resonantly illuminated plasmonic particles Baffou and Quidant (2012); Baffou
et al. (2010); Jauffred et al. (2015), the particle’s internal temperature can
rise significantly above the temperature of the environment, leading to so
called “hot Brownian motion” Rings et al. (2010, 2012); Falasco et al. (2014).

For decreasing pressure, the mean free path of the gas molecules increases
(e.g., $\bar{l}\sim 60\,\upmu$m at 1 mbar). As a consequence, the particle no
longer thermalizes with the gas since the impinging gas molecules no longer
carry away enough thermal power to balance the optical absorption from the
trapping laser. Due to the increased internal temperature $T_{{\rm int}}$ of
the particle, the average energy of the gas molecules after a collision with
the particle increases. The process by which a surface exchanges thermal
energy with a gas is called accommodation, which is characterized by the
accommodation coefficient $c_{\rm acc}=(T_{{\rm em}}-T_{{\rm gas}})/(T_{{\rm
int}}-T_{{\rm gas}})$, where $T_{{\rm em}}$ is the temperature of the gas
molecules emitted from the particle surface. Accommodation quantifies the
fraction of the thermal energy that the colliding gas molecule removes from
the surface, such that $c_{\rm acc}=1$ means that the molecule fully
thermalizes with the surface. Since the mean free path in a dilute gas is
long, one can safely assume that an emitted molecule will not interact again
with the particle before thermalizing with the environment. Consequently, we
can consider the particles that impinge on the particle surface and those that
leave the surface as being in equilibrium with two different baths with
temperatures corresponding to the temperature of the environment and the
particle surface, respectively Millen et al. (2014). Therefore, we get an
additional contribution to the damping from the emerging hot molecules

$\frac{\Gamma_{\rm em}}{2\pi}=\frac{1}{16}\sqrt{\frac{T_{{\rm em}}}{T_{{\rm
gas}}}}\Gamma_{\rm gas},\quad S_{\rm ff}^{em}=\frac{m{\rm
k_{B}}}{\pi}\left[c_{\rm acc}T_{{\rm int}}+(1-c_{\rm acc}\right)T_{{\rm
gas}}]\Gamma_{\rm em}.$ (6)

For pressures below $P_{\rm gas}^{\prime}=0.57{\rm k_{B}}T_{{\rm
gas}}\left/\sigma_{\text{gas}}a\right.\approx 54.4\,{\rm
mbar}\times(a/\upmu{\rm m})^{-1}$, where the mean free path $\bar{l}={\rm
k_{B}}T_{{\rm gas}}\left/(\sqrt{2}\sigma_{\text{gas}}P_{\rm gas})\right.$ is
much larger than the radius of the particle $a$, the damping is linear in the
pressure $P_{\rm gas}$ and given by

$\frac{\Gamma_{\rm
gas}}{2\pi}=\frac{3}{\pi\sqrt{2}}\frac{\mu_{v}\sigma_{\text{gas}}}{{\rm
k_{B}}T_{{\rm gas}}\rho}\frac{P_{\rm gas}}{a},$ (7)

where $\mu_{v}=2\sqrt{m_{\text{gas}}{\rm k_{B}}T_{{\rm
gas}}}\left/3\sqrt{\pi}\sigma_{\text{gas}}\right.$ is the viscosity of a
dilute gas, $\sigma_{\text{gas}}$ is the cross-section of the air molecules
and $m_{\text{gas}}$ the molecule mass. The total damping due to the hot
particle with the gas environment is $\Gamma_{\rm em}+\Gamma_{\rm
gas}=2\pi{\rm c_{\rm P}}P_{\rm gas}/a$, where typically ${\rm c_{\rm
P}}\approx 50\,\text{Hz}(\mu\text{m}/\text{mbar})$.

So far, we have only considered spherical particles. For anisotropic
particles, e.g., a rod, the friction term is different along each of the axes,
and depends upon the alignment of the particle. As a consequence, the friction
coefficient has to be replaced by a tensor $\boldsymbol{\Gamma}$ and the
damping in a direction $\mathbf{s}$ is given by
$\boldsymbol{\Gamma}\cdot\mathbf{s}$. For a thorough discussion see Ref.
Martinetz et al. (2018).

####  II.2.2 Radiation Damping

At very low pressure ($\leq 10^{-6}\,\rm mbar$), gas damping becomes extremely
small and photon shot noise starts to dominate Jain et al. (2016). Photon shot
noise is a consequence of the particulate nature of light. As a consequence,
photons arrive at discrete times, where the number of photons arriving per
time interval $\Delta t$ is given by $\sqrt{\Delta tP_{\rm
opt}\left/\hbar\omega_{\rm L}\right.}$. The recoil from the fluctuating number
of photons impinging on the nanoparticle can be modeled as an effective bath
with the characteristics Novotny (2017)

$\frac{\Gamma_{\rm rad}}{2\pi}=c_{\rm dp}\frac{P_{\rm scat}}{2\pi
mc^{2}}\quad{\rm and}\quad S_{\rm ff}^{rad}=c_{\rm dp}\frac{\hbar\omega P_{\rm
scat}}{2\pi c^{2}},$ (8)

where $c_{\rm dp}$ depends on the direction of motion of the particle with
respect to the polarization of the laser and is $c_{\rm dp}=2/5$ for motion
along the direction of polarization and $c_{\rm dp}=4/5$ for motion
perpendicular to the polarization. The scattered power is $P_{\rm
scat}=\sigma_{\rm scat}I_{\rm opt}$, where $\sigma_{\rm
scat}=|\alpha|^{2}k_{\rm L}^{4}/6\pi\epsilon_{0}^{2}$ and $I_{\rm opt}$ is the
laser intensity. The effective temperature of this bath can be calculated via
Eqn. (5).

####  II.2.3 Artificial Damping and Heating

The noise processes described so far are present in any experiment with
optically levitated nanoparticles in high vacuum. In addition, random forces
and damping can be introduced through external fields that are under
experimental control. Importantly, since energy can be injected or extracted
from the particle, i.e., it is not in thermal equilibrium, the fluctuation-
dissipation relation does not have to hold and the effective damping and
temperatures can be controlled independently. Feedback cooling damps the
particle motion at a rate $\Gamma_{\rm fb}$ without adding any fluctuating
forces, thus $S_{\rm ff}^{fb}=0$ and it is therefore referred to as cold
damping Gieseler et al. (2012); Li et al. (2011). Note that this simplified
picture assumes that the feedback signal is perfect and that it does not
feedback any noise, which in general is not true. Similarly, cavity cooling
up-converts the particle energy to optical frequencies, which are effectively
at zero temperature because $\hbar\omega\gg{\rm k_{B}}T_{{\rm env}}$ in a room
temperature environment Kiesel et al. (2013); Chang et al. (2010); Romero-
Isart et al. (2010). Conversely, fluctuations of the trapping or control
fields only add fluctuating forces without providing damping. Hence,
$\Gamma_{\rm drive}=0$ and $S_{\rm ff}^{drive}=q^{2}S_{\rm qq}$, where $q$ is
the coupling parameter to the control field and $S_{\rm qq}$ its spectral
density. This can be realized for example with fluctuating electric fields,
where $q$ corresponds to the charge on the particle Frimmer et al. (2017);
Mestres et al. (2014); Martínez et al. (2013).

Generally, one has to be careful to define a temperature for a system out of
equilibrium Casas-Vázquez and Jou (2003). However, the situation we present
here is somewhat simple due to it being steady-state and for many practical
purposes the effective bath model that is characterized by an effective
damping/temperature gives a good description. However, one can also create
situations where this is no longer true. For instance, by parametric feedback
damping, the temperature alone is not sufficient to give a full description of
the bath Gieseler et al. (2014).

##  III Brownian Motion

As discussed in the introduction, the Brownian particle serves as an exemplary
model to describe a variety of stochastic processes in many fields, including
physics, finance and biology. Therefore, a particle trapped in optical
tweezers is paradigmatic since it is a direct experimental realization of the
idealized Brownian particle. Brownian motion in nonequilibrium systems is of
particular interest because it is directly related to the transport of
molecules and cells in biological systems Gnesotto et al. (2017). Important
examples include Brownian motors, active Brownian motion of self-propelled
particles, hot Brownian motion, and Brownian motion in shear flows Bechinger
et al. (2016). Recent theoretical studies also found that the inertia of
particles and surrounding fluids can significantly affect the Brownian motion
in nonequilibrium systems Nicolis and De Decker (2017).

In this section we will discuss the basics of Brownian motion. We will mainly
treat the aspects that are necessary for understanding the following
discussion of thermodynamics with levitated nanoparticles. For details on the
theory of Brownian motion we refer the reader to the work of Ornstein,
Uhlenbeck and Wang Uhlenbeck and Ornstein (1930); Wang and Uhlenbeck (1945)
and for a recent review on Brownian motion in the underdamped regime we refer
the reader to Ref. Li and Raizen (2013).

###  III.1 Harmonic Brownian Motion

Under the influence of trapping forces, the particle will be localized about
its equilibrium position. For small displacements, the trap can be
approximated by a three-dimensional harmonic potential. The three motional
degrees of freedom are largely decoupled and we limit the discussion to a
single coordinate $q(t)$ ($q=x,y,z$). The equation of motion for a
harmonically trapped Brownian particle is Wang and Uhlenbeck (1945)

$\ddot{q}\,+\,\Gamma\\!\;\dot{q}+\Omega_{0}^{2}q=\;\sqrt{2{\rm k_{B}}T_{{\rm
CM}}\,\Gamma_{\rm CM}/m}\,\Xi(t).$ (9)

The particle oscillates in the trap at the characteristic frequency
$\tilde{\Omega}=\sqrt{\Omega_{0}^{2}-\Gamma_{\rm CM}^{2}/4}$. For the optical
potential the trap frequency $\Omega_{0}$ is given by Eqn. (4). We distinguish
between three cases, the overdamped ($\Omega_{0}\ll\Gamma_{\rm CM}$), the
critically damped ($\Omega_{0}\approx\Gamma_{\rm CM}$) and underdamped case
$\Omega_{0}\gg\Gamma_{\rm CM}$. This stochastic equation of motion has been
studied in detail by Ornstein and Uhlenbeck Uhlenbeck and Ornstein (1930) and
we summarize their results here. The variance of the position of a Brownian
particle in an under-damped harmonic trap is

$\sigma_{q}^{2}(t)=\frac{2{\rm k_{B}}T_{{\rm
CM}}}{m\Omega_{0}^{2}}\left[1-e^{-\frac{1}{2}\Gamma_{\rm
CM}t}\left(\cos(\tilde{\Omega}t)+\frac{\Gamma_{\rm
CM}}{2\tilde{\Omega}}\sin(\tilde{\Omega}t)\right)\right].$ (10)

In the over-damped harmonic trap, set $\tilde{\Omega}\to i\tilde{\Omega}$. In
a critically damped harmonic trap, set $\tilde{\Omega}\to 0$. The position
autocorrelation function is related to the variance as follows

$\langle q(t)q(0)\rangle=\frac{{\rm k_{B}}T_{{\rm
CM}}}{m\Omega_{0}^{2}}-\frac{1}{2}\sigma_{q}^{2}(t),$ (11a) and the velocity
autocorrelation and the position-velocity correlation function are given by

$\displaystyle\langle v(t)v(0)\rangle$ $\displaystyle=\frac{{\rm k_{B}}T_{{\rm
CM}}}{m}e^{-\frac{1}{2}\Gamma_{\rm
CM}t}\left(\cos(\tilde{\Omega}t)-\frac{\Gamma_{\rm
CM}}{2\tilde{\Omega}}\sin(\tilde{\Omega}t)\right),$ (11b)
$\displaystyle\langle q(t)v(0)\rangle$ $\displaystyle=\langle
v(t)q(0)\rangle=\frac{{\rm k_{B}}T_{{\rm
CM}}}{m\tilde{\Omega}}e^{-\frac{1}{2}\Gamma_{\rm CM}t}\sin(\tilde{\Omega}t).$
(11c)

For a long time it was believed that the timescale at which these correlations
exist is too fast to be observable in experiment Einstein (1905). The first
experimental observation was first achieved in vacuum Li et al. (2010) and
later in liquid Kheifets et al. (2014), demonstrating that levitated
nanoparticles indeed allow one to attain an entirely new parameter regime to
study thermodynamics of individual particles. Fig. 2 shows the experimental
results from Li et al. Li et al. (2010).

Figure 2: First experimental observation of the instantaneous velocity of a
Brownian particle. (a) The mean-square displacement for short times is
proportional to $t^{2}$, a signature of ballistic motion. (b) The normalized
velocity autocorrelation functions for different pressures in perfect
agreement with Eqn. (11b). Figures taken from Li et al. (2010) with permission
from Science.

###  III.2 Power Spectral Density and Calibration

According to the Wiener-Khinchin theorem, the position autocorrelation
function is the Fourier transform of the power spectral density
$S_{qq}(\Omega)=\int_{-\infty}^{\infty}\langle q(t)q(0)\rangle e^{i\Omega
t}{\rm d}t$, which for Eqn. (9) is given by

$S_{qq}(\Omega)=|\chi(\Omega)|^{2}S_{\rm ff}(\Omega)=\frac{\Gamma_{\rm CM}{\rm
k_{B}}T_{{\rm CM}}\left/\pi
m\right.}{(\Omega^{2}-\Omega_{0}^{2})^{2}+\Gamma_{\rm CM}^{2}\Omega^{2}},$
(12)

where $\chi(\Omega)=m^{-1}\left[\Omega^{2}-\Omega_{0}^{2}+i\Gamma_{\rm
CM}\Omega\right]^{-1}$ is the response function or susceptibility of a
harmonic oscillator. In the underdamped regime, the frequency spectrum of the
autocorrelation function is strongly peaked around the trap frequency
$\Omega_{0}$, whereas when overdamped the frequency spectrum is broad. For an
example of the power spectral density in the underdamped regime see Fig. 3a.

Figure 3: Feedback cooling of a levitated nanoparticle. (a) Power spectral
density under phase locked feedback cooling at three different pressures and
constant feedback gain. The area under the power spectral densities is a
measure for the effective center-of-mass temperature. (b) The effective
temperature expressed in terms of the phonon occupation as a function of gas
pressure. Figures reproduced from Physical Review Letters Jain et al. (2016).

The power spectrum in Eqn. (12) is valid for a harmonic oscillator. However,
the actual trapping potential is nonlinear. For a symmetric potential the
lowest order nonlinear term is a cubic or Duffing nonlinearity. In an optical
trap the symmetry is broken along the direction of propagation of the trapping
laser and along the vertical direction due to the scattering force and
gravity, respectively. However, the symmetry breaking tends to be small and
therefore the quadratic nonlinearity is typically neglected. Because of the
Duffing nonlinearity, the oscillation frequency along an axis $i=x,y,z$
becomes a function of the oscillation amplitude and is red shifted by Gieseler
et al. (2013) $\Delta\Omega_{\rm
i}=\frac{3}{8}\Omega_{i}\sum_{j}\xi_{ij}A_{\rm j}^{2}$ where $A_{j}$ is the
instantaneous amplitude of mode $j$ and the $\xi_{ij}\sim 1/\text{w}_{j}^{2}$
are the Duffing terms, where $\text{w}_{j}$ is the width of the optical
potential along the $j$ direction. The frequency shift due to changes in the
oscillation amplitudes is also known as self-phase modulation ($j=i$) and
cross-phase modulation ($j\neq i$). The frequency shift can be neglected as
long as it is much smaller than the linewidth $\Gamma_{\rm CM}$. This is the
case for high pressure $\simeq$1 mbar, but for low pressures the amplitude
fluctuations of the particle lead to significant frequency fluctuations and
the power spectral density becomes distorted. This nonlinear Brownian motion
typically does not play a role in nanomechanical systems because the amplitude
fluctuations are small. However, levitated nanoparticles have a low mass and a
high motional quality-factor and, therefore, nonlinear Brownian motion can be
observed in these systems Gieseler et al. (2013).

The power spectral density is a useful tool in experiments with harmonic
oscillators, since the dynamics of the oscillator can be separated from
(spectrally distant) noise. In addition, the analysis of the power spectral
density allows one to extract the center-of-mass temperature of the oscillator
and the damping rate Li et al. (2010); Gieseler et al. (2012); Millen et al.
(2014); Hebestreit et al. (2018). However, to avoid miss-calibration due to
the above mentioned nonlinearities, one should use the velocity power spectral
density $S_{\dot{q}\dot{q}}=\Omega^{2}S_{qq}$ instead of (12) Hebestreit et
al. (2017).

###  III.3 Quantum Brownian Motion

In the quantum regime, when the center-of-mass temperature is of the order of
a single quantum of motion ${\rm k_{B}}T_{{\rm CM}}\approx\hbar\Omega_{0}$,
the position autocorrelation Eqn. (11a) contains the product of the Heisenberg
time-evolved operators $\hat{q}(t)$, $\hat{q}(0)$, which do not commute. As a
result, the spectrum Clerk et al. (2010)

$S_{Q}(\Omega)=\frac{\hbar/\pi}{1-\exp\left(-\frac{\hbar\Omega}{{\rm
k_{B}}T_{{\rm CM}}}\right)}{\rm Im}\chi(\Omega)=\frac{\hbar\Omega m\Gamma_{\rm
CM}/\pi}{1-\exp\left(-\frac{\hbar\Omega}{{\rm k_{B}}T_{{\rm
CM}}}\right)}|\chi(\Omega)|^{2},$ (13)

is asymmetric in frequency and the PSD at positive frequencies is a factor
$\exp(\hbar\Omega_{0}\left/{\rm k_{B}}T_{{\rm CM}}\right.)$ higher than the
PSD at negative frequencies. The positive-frequency part of the spectral
density is a measure of the ability of the oscillator to absorb energy, while
the negative-frequency part is a measure of the ability of the oscillator to
emit energy. Therefore, we can understand the positive frequency part of the
spectral density as being related to stimulated emission of energy into the
oscillator, while the negative-frequency part is related to the emission of
energy by the oscillator.

Typically, the motional frequencies of a levitated particle are $\sim$100 kHz.
Hence, the ground-state temperature is a few micro-kelvin and therefore out of
reach for cryogenic techniques, and one has to resort to active cooling
methods. Recent experiments using feedback cooling have already attained
motional occupations of a few tens of phonons Jain et al. (2016). However, a
measurement of the sideband asymmetry in a homodyne measurement, as observed
in other nano-mechanical systems Safavi-Naeini et al. (2012); Weinstein et al.
(2014); Underwood et al. (2015); Peterson et al. (2016); Kampel et al. (2017),
is still elusive.

##  IV Trap Stability and Kramers Turnover

For a particle to be trapped in optical tweezers, the axial component of the
gradient force must exceed the destabilizing effects of the scattering force
and gravity. The scattering force is negligible for small particles but
increases quickly with particle size such that large particles are pushed away
from the focal volume. In addition, the ratio between the scattering and
gradient forces scales with the refractive index contrast Spesyvtseva and
Dholakia (2016). This places an upper limit on the maximum particle size and
materials that can be trapped, even if they experience very little optical
absorption. The destabilizing effect from the scattering force can be
circumvented by using a configuration with counter-propagating beams Divitt et
al. (2015); Li et al. (2010). However, in this case polarization fluctuations
translate into intensity fluctuations, which can also destabilize particles in
the trap. Besides, radiometric forces can play a role in the stability
condition for larger particles ($\sim$$\upmu$m), where non-uniform heating
leads to temperature gradients across the particle Millen et al. (2014) and
the resulting forces might destabilize the trap Ranjit et al. (2015).

Overcoming the destabilizing effects from the scattering force is not
sufficient to guarantee that particles can be trapped. As we discussed
earlier, the particle is subject to fluctuating forces from the environment.
The energy of a thermal bath follows a Maxwell-Boltzmann distribution with
mean value of ${\rm k_{B}}T_{{\rm CM}}$. Since the tail of the distribution
extends to high energies, the potential depth should be at least $\approx
10{\rm k_{B}}T_{{\rm CM}}$ Ashkin et al. (1986) to make particle escape
through thermal excitation unlikely (the likelihood of finding the particle
with energy $\approx 10{\rm k_{B}}T_{{\rm CM}}$ is less than $0.02\%$). Hence,
there is a finite probability that the particle will gain enough energy to
escape the potential, even when it is confined by a potential much deeper than
${\rm k_{B}}T_{{\rm CM}}$, in a process known as Kramers escape. This form of
“ _classical tunneling_ ” appears in a diverse range of physical systems,
including chemical reaction rates, protein folding, atomic transport in
optical lattices and molecular diffusion at solid-liquid interfaces. The
Kramers escape rate is given by an Arrhenius law

${\rm R}_{K}={\rm R}_{0}\exp\left(-\frac{U_{\rm barrier}}{{\rm k_{B}}T_{{\rm
CM}}}\right)$ (14)

where ${\rm R}_{0}$ is the attempt frequency and $U_{\rm barrier}$ is the
barrier height. From the Boltzmann factor in Eqn. (14) it follows that such a
transition is exponentially suppressed if the potential is much deeper that
the thermal energy $U_{\rm opt}\gg{\rm k_{B}}T_{{\rm CM}}$. Kramers found
Kramers (1940) that in the underdamped regime, the transition rate increases
with _increasing_ friction, and that in the overdamped regime the transition
rate increases with _decreasing_ friction, with the transition region labelled
the turnover. Fifty years later, a theory was developed that linked the two
regimes Mel’nikov (1991); Hänggi et al. (1990).

Closely related to Kramers escape is the Kramers turnover problem. It
describes the transitioning between two local potential minima as the friction
is varied. This is often more relevant in physical situations, describing the
transitions between two protein configurations, for example. It is also much
more convenient to study experimentally, since the particle is not lost after
the transition but instead recaptured in the other well. In particular,
levitated particles are well suited to studying the Kramers escape and
recently led to its first quantitative observation Rondin et al. (2017). The
double well potential can be created by using two tightly focused laser beams.
The intensity and exact relative position of the two foci determines the
height of the barrier. The hopping rates between the two wells is determined
by the local curvatures of the potential at the extrema and by the interaction
strength with the environment. In addition, the interaction strength can be
varied over many orders of magnitude through a change in the gas pressure
$P_{\rm gas}$.

Fig. 4 shows the experimental data from Rondin et al., which for the first
time measured the Kramers rate across the turnover Rondin et al. (2017), using
an optically levitated nanoparticle. In addition, the figure includes the
limiting cases in the high and low damping regime, and the full solution for
arbitrary damping.

Figure 4: Measurement of the Kramers turnover with a levitated nanoparticle.
Data illustrating the first experimental observation of Kramers turnover,
taken from Rondin et al. (2017). The full theory from Mel’nikov (1991) (solid
line) is shown as a solid blue line together with the limiting cases as
predicted by Kramers Kramers (1940) (dot-dashed lines). The red dashed line
highlights the expected turnover point as predicted from the measured shape of
the double well potential and is in excellent agreement with the experimental
observations.

##  V Effective Potentials in the Steady State

At constant trapping laser power, the trapping potential is static in time
$U_{\rm opt}(\mathbf{r})$. However, through modulation of the trapping beam
intensity, the optical potential becomes time-dependent. This is particularly
useful when studying non-equilibrium dynamics and for engineering effective
baths in the context of nano heat engines. From Eqn. (4) it follows that a
change in optical power $\delta P_{\rm opt}(t)$ changes the trap frequency by
$\Omega(t)=\Omega_{0}(1+\zeta(t)/2)$, where $\zeta(t)=\delta P_{\rm
opt}(t)\left/\bar{P}_{\rm opt}\right.$ and $\bar{P}_{\rm opt}$ is the mean
optical power. Energy is most effectively exchanged between the trapping laser
and the particle if the modulation $\zeta(t)=\zeta_{0}\cos(\Omega_{\rm mod}t)$
occurs at twice the trapping frequency $\Omega_{\rm mod}\approx 2\Omega_{0}$.
The flow of energy is thereby determined by the relative phase $\phi_{\rm
mod}$ between the particle oscillation and the laser intensity modulation
(note that $\phi_{\rm mod}$ does not appear in $\zeta(t)$, since the
modulation serves as the time reference and $\phi_{\rm mod}$ is the phase of
the particle with respect to the modulation). If the modulation is in-phase,
energy is extracted (cooling), while the motion is excited when the modulation
is out-of-phase (heating). Without active stabilization of the modulation
phase with respect to the particle motion, the relative phase is random.
However, since the out-of-phase motion is amplified and the in-phase motion is
damped, the out-of-phase motion quickly dominates and the particle motion
synchronizes or entrains with the parametric modulation Gieseler et al.
(2014).

Therefore, to achieve cooling the phase needs to be actively stabilized, for
instance with a phase-locked loop Jain et al. (2016). Alternatively, a
parametric feedback modulation of the form
$\zeta_{\text{fb}}(t)=-(\eta/\Omega_{0})q(t)\dot{q}(t)$ Gieseler et al.
(2012), where $\eta$ parameterizes the feedback strength, leads to a
modulation at the parametric resonance condition, ensuring a phase that is
optimized for extracting energy from the mechanical mode. In contrast to
feedback with a phase-locked loop, where the modulation amplitude is constant,
here the modulation amplitude is proportional to the particle energy. As a
consequence, the particle feels a nonlinear friction force with $\Gamma_{\rm
NL}\propto E$. This kind of friction has also be observed in carbon based
nanomechanical resonators Eichler et al. (2011) and leads to non-equilibrium
steady states that can no longer be described by a thermal distribution as we
will discuss in the following.

###  V.1 Effective Potential for the Energy

The main physics of the particle motion under parametric modulation and
feedback is captured by a one dimensional equation of motion along each axis:

$\ddot{q}\,+\,\Gamma\\!\;\dot{q}+\Omega_{0}^{2}\left[1+\zeta_{0}\cos(\Omega_{\rm
mod}t)+\xi q^{2}+\Omega_{0}^{-1}\eta q\dot{q}\right]q=\;\sqrt{2{\rm
k_{B}}T_{{\rm CM}}\Gamma_{\rm CM}/m}\,\xi(t),$ (15)

where $q=(x,y,z)$. The total energy of a single degree-of-freedom is given by

$E(q,p)=\frac{1}{2}m\Omega_{0}^{2}q^{2}+\frac{p^{2}}{2m}+\frac{1}{4}\xi
m\Omega_{0}^{2}q^{4},$ (16)

where $p(t)=m\dot{q}$ is the momentum. The energy obeys the stochastic
differential equation Gieseler et al. (2015)

${\rm d}E=\left[-\Gamma_{\rm CM}(E-{\rm k_{B}}T_{{\rm
CM}})-\frac{\eta\Omega_{0}E^{2}}{2m\Omega^{2}}-\frac{E\zeta\Omega_{0}^{2}\sin(2\phi_{\rm
mod})}{2\Omega}\right]{\rm d}t+\sqrt{2E\Gamma_{\rm CM}{\rm k_{B}}T_{{\rm
CM}}}{\rm d}W.$ (17)

From (17) one derives the probability distribution for the energy

$P_{E}(E)=\frac{1}{Z}\exp\left\\{-\beta H(E)\right\\},$ (18)

where $Z=\int P_{E}(E){\rm d}E$. Thus, the energy distribution is that of an
equilibrium system with effective energy

$H_{\text{eff}}=\left[1+\frac{\zeta_{0}\Omega_{0}^{2}\sin(2\phi_{\rm
mod})}{2\Gamma_{\rm CM}\Omega}\right]E+\frac{\eta\Omega_{0}}{4m\Gamma_{\rm
CM}\Omega^{2}}E^{2}.$ (19)

While the term proportional to $E^{2}$ is caused by the feedback cooling, the
term proportional to $E$ is affected only by the parametric modulation. Note
that the Duffing term is included in the energy $E$ on the right-hand side of
the above equation (c.f. Eqn. (16)).

Eqn. (17) is quite general, as it captures the dynamics of the slowly varying
energy under parametric heating without active stabilization, phase-locked
loop feedback cooling and parametric feedback cooling. However, one has to
bear in mind that the oscillation frequency $\Omega$ is not necessarily the
same as the frequency $\Omega_{0}$ of the unperturbed harmonic oscillator. For
instance, for strong modulation the particle motion entrains with the
modulation and $\Omega\approx\Omega_{\rm mod}/2$ Gieseler et al. (2014), while
for weak modulation $\Omega\approx\Omega_{0}$. The weak and strong regime are
determined by the threshold condition $\zeta_{0}>2{\rm Q}^{-1}\sqrt{1+{\rm
Q}^{2}\left(2-\Omega_{\rm mod}/\Omega_{0}\right)^{2}}\approx 2{\rm Q}^{-1}$,
the approximation being exact at parametric resonance $\Omega_{\rm
mod}=2\Omega_{0}$ and $Q=\Omega_{0}/\Gamma_{\rm CM}$ is the quality factor.
Above threshold, the effective temperature diverges and the motion transitions
from a thermal state to a coherent oscillation, similar to the lasing
condition of an optical oscillator Gieseler et al. (2014).

Note that the evolution of the position of a _real_ Brownian particle in the
underdamped regime and in a _time-dependent_ optical potential is determined
by Eqn. (15). In contrast, Eqn. (17) describes the evolution of its energy or
amplitude. This evolution can be interpreted as the evolution of an
underdamped, albeit _fictitious_ , Brownian particle in a _static_ potential
Gieseler et al. (2015). Interestingly, this _fictitious_ Brownian particle can
exhibit dynamics similar to what we have seen for the _real_ Brownian
particle, such as transitions between two local minima that are described by
Kramers’ theory Ricci et al. (2017).

###  V.2 Effective Temperature

Without non-linear parametric feedback ($\eta=0$), the energy distribution is
that of a harmonic oscillator with effective temperature

$T_{{\rm CM}}^{\prime}=T_{{\rm
CM}}\left(1+\frac{\zeta_{0}\Omega_{0}^{2}\sin(2\phi_{\rm mod})}{2\Gamma_{\rm
CM}\Omega}\right)^{-1}.$ (20)

Eqn. (20) states that parametric modulation of the trapping potential results
in an effective temperature change of the environment, where the particle
centre-of-mass temperature changes from $T_{{\rm CM}}$ to $T_{{\rm
CM}}^{\prime}$. For $-\pi/2<\phi_{\rm mod}<0$, $T_{{\rm CM}}^{\prime}>T_{{\rm
CM}}$, that is the particle motion is heated, while for $0<\phi_{\rm
mod}<\pi/2$, $T_{{\rm CM}}^{\prime}<T_{{\rm CM}}$ and the particle motion is
cooled. The rate at which the particle thermalizes with this effective bath is
$\Gamma_{\rm CM}^{\prime}=\Gamma_{\rm CM}\left(T_{{\rm CM}}/T_{{\rm
CM}}^{\prime}-1\right)$, where the largest rates are achieved at $\phi_{\rm
mod}=-\pi/4$ and $\phi_{\rm mod}=\pi/4$, for heating and cooling respectively.
If the relative phase between the particle motion and the modulation
$\phi_{\rm mod}$ is not stabilized actively, the particle motion will self-
lock to $\phi_{\rm mod}=-\pi/4$. Thus, an effective hot bath can be
implemented easily by a simple modulation of the trapping laser at
$\Omega_{\rm mod}\approx 2\Omega_{0}$. Fig. 3 shows the effective temperature
or occupation number for a particle under high vacuum as a function of
pressure (i.e., $\Gamma_{\rm CM}$) and constant feedback strength
($\zeta_{0}$).

##  VI Relaxation

In the steady-state, a trapped particle samples the distribution Eqn. (18),
which depends on experimental parameters, such as the average power of the
trapping laser, and the strength and frequency of the modulation of any
potential modulation. Hence, under a non-adiabatic change of the parameters,
the systems relaxes into a new steady state. The Fokker-Planck equation that
describes the time for the evolution probability density function
$P_{E}(E,t)$, including feedback and modulation, is given by Gieseler et al.
(2015)

$\frac{\partial P_{E}(E,t)}{\partial t}=\frac{\partial}{\partial
E}\left[\Gamma_{\rm CM}(E-{\rm k_{B}}T_{{\rm
CM}})+\frac{\eta\Omega_{0}E^{2}}{2m\Omega^{2}}+\frac{E\zeta_{0}\Omega_{0}^{2}\sin(2\phi_{\rm
mod})}{2\Omega}\right]P_{E}(E,t)+{\Gamma_{\rm CM}{\rm k_{B}}T_{{\rm
CM}}}\frac{\partial^{2}}{\partial E^{2}}EP_{E}(E,t).$ (21)

In general it is non-trivial to find an analytic solution to Eqn. (21).
Amazingly, in the absence of feedback cooling ($\eta=0$), the equation of
motion for the energy corresponds to the Cox-Ingersoll-Ross model for interest
rates, for which the exact analytical solution is given by the Noncentral Chi-
squared distribution Salazar and Lira (2016)

$P_{E}(E|E_{0},t)=c_{t}e^{-c_{t}(E+E_{0}e^{-\Gamma_{\rm
CM}t})}I_{0}\left(2c_{t}\sqrt{EE_{0}e^{-\Gamma_{\rm CM}t}}\right),$ (22)

where $c_{t}=\beta\left(1-e^{-\Gamma_{\rm CM}t}\right)^{-1}$, $I_{0}(x)$ is
the modified Bessel function of the first kind and $E_{0}$ is the initial
energy, i.e., $P_{0}(E|E_{0})=\delta(E-E_{0})$. As expected, the equilibrium
distribution $P_{\infty}(E|E_{0})=\beta\exp(-\beta E)$ does not depend on the
initial conditions and is given by the Maxwell-Boltzmann distribution at
temperature $T_{{\rm CM}}=1/({\rm k_{B}}\beta)$. If the system is initially
prepared at $t=0$ in a steady state with energy distribution $P_{0}(E_{0})$,
the energy distribution after time $t$ is

$P_{E}(E,t)=\int_{0}^{\infty}P_{E}(E|E_{0},t)P_{0}(E_{0}){\rm d}E_{0}.$ (23)

For an initial Maxwell-Boltzmann distribution, corresponding to a thermal
equilibrium distribution at temperature $T_{{\rm 0}}$, the energy distribution
at time $t$ is also a Maxwell-Boltzmann distribution

$P^{\text{MB}}_{E}(E,t)=\beta(t)e^{-\beta(t)E},$ (24)

with time dependent temperature

$T_{{\rm CM}}(t)=T_{{\rm\infty}}+(T_{{\rm 0}}-T_{{\rm\infty}})e^{-\Gamma_{\rm
CM}t}.$ (25)

Note that the initial temperature $T_{{\rm 0}}$ and final temperature
$T_{{\rm\infty}}$ can be controlled in the experiment by modulation of the
trapping laser, as discussed earlier. Explicitly, a levitated nanoparticle can
be cooled via feedback to a centre-of-mass temperature $T_{{\rm CM}}$ far
below the ambient temperature. Once the feedback modulation is switched off,
the particle will thermalize with the environment (in general via collisions
with surrounding gas), at an average rate $\Gamma_{\rm CM}$, which can be
controlled by varying the gas pressure. Fig. 5 shows the relaxation from a
non-equilibrium state towards thermal equilibrium. The rate at which the
particle reaches equilibrium can be accelerated using time-dependent
potentials. This has been demonstrated recently with a colloidal particle
Martínez et al. (2016) and a similar strategy has been proposed for
underdamped systems Chupeau et al. (2018).

Figure 5: Relaxation from a non-equilibrium steady state. (a) Individual
trajectories of the energy as the system relaxes toward equilibrium (b)
Position distribution during the relaxation process. The energy distribution
is given by Eqn. (23). (c) Energy distribution in the steady state ($t\leq 0$)
in agreement with Eqn. (18). The deviation from a thermal state due to the
nonlinear feedback is clearly visible. (d) Experimental verification of the
detailed fluctuation theorem (c.f. Eqn. (26)). All figures taken from Gieseler
et al. (2014) with permission from Nature Nanotechnology.

##  VII Fluctuation Theorems

As a system relaxes to a thermal equilibrium the dynamics satisfy detailed
balance with respect to the equilibrium distribution. The time reversibility
of the underlying dynamics implies that the Crooks-like Crooks (1999); Collin
et al. (2005) transient fluctuation theorem Seifert (2012); Gieseler et al.
(2014)

$\frac{P(-\Delta\mathcal{S})}{P(\Delta\mathcal{S})}=e^{-\Delta\mathcal{S}},$
(26)

for the relative entropy change
$\Delta\mathcal{S}=\beta\mathcal{Q}+\Delta\Phi$ (or Kullback-Leibler
divergence) holds. The quantity $\Delta\Phi=\Phi(t)-\Phi(0)$ is the difference
in trajectory dependent entropy $\Phi(t)=-\ln P_{0}(u(t))$ between the initial
and the final states of the trajectory and $\mathcal{Q}$ is the heat absorbed
by the bath at reciprocal temperature $\beta$. Here, $u(t)$ denotes an entire
trajectory of length $t$ including position and momentum of the oscillator and
$u^{*}(t)$ denotes the trajectory that consists of the same states visited in
reverse order with inverted momenta. Because no work is done on the system,
the heat $\mathcal{Q}$ exchanged along a trajectory equals the energy lost by
the system, $\mathcal{Q}=-[E(t)-E(0)]$, where $E(0)$ and $E(t)$ are the energy
at the beginning and at the end of the stochastic trajectory.

The fluctuation theorem holds for any time $t$ at which $\Delta\mathcal{S}$ is
evaluated, and it is not required that the system has reached the equilibrium
distribution at time $t$. In general, the steady distribution $P_{0}(u(t))$
necessary to compute $\Delta\Phi$ is unknown. However, from the distribution
derived for our model Eqn. (19), we find that for relaxation from a non-
equilibrium steady state generated by nonlinear feedback of strength $\eta$
and parametric modulation of strength $\zeta_{0}$, the relative entropy change
is given by Gieseler et al. (2015); Gieseler (2014)

$\Delta\mathcal{S}=\beta\Delta H_{\text{fb}}$ (27)

where $\Delta H_{\text{fb}}=H_{\text{fb}}(t)-H_{\text{fb}}(0)$ and
$H_{\text{fb}}=H_{\text{eff}}-E$ is the contribution to the effective energy
(19) from the modulation of the trapping laser. Thus, our stochastic model
(c.f. Eqn. (27)) allows us to express the relative entropy change during a
relaxation trajectory in terms of the energy at the beginning and the end of
that trajectory. Note that $H_{\text{fb}}$ can also be negative, e.g. when we
cool the particle motion. For example, setting $\phi_{\rm mod}=\pi/4,\eta=0$
in Eqn. (19) amounts to cooling with phase looked loop feedback Jain et al.
(2016). In this case,
$H_{\text{fb}}(t)=\left(\zeta\Omega_{0}\left/2\Gamma_{\rm
CM}\right.\right)E(t)=(\beta_{\text{eff}}-\beta)E(t)/\beta$ and consequently
$\Delta\mathcal{S}=(\beta-\beta_{\text{eff}})Q$. Since here the relative
entropy change is proportional to the heat $\mathcal{Q}$, it is intuitively
clear that $\Delta\mathcal{S}$ is a measure for the dissipation during the
relaxation process. This dissipation function was measured in the underdamped
regime with optical tweezers and colloidal particles and was one of the very
first experimental demonstrations of the validity of a fluctuation theorem
Wang et al. (2002). In contrast, for parametric feedback, where $\zeta=0$,
$\eta>0$, we find that $\Delta\mathcal{S}\propto E^{2}(t)-E^{2}(0)$ is no
longer proportional to $\mathcal{Q}$. In this case, $\Delta\mathcal{S}$ still
is a measure for the reversibility of the relaxation process. However, it is
no longer a simple function of the exchanged heat. This was verified in the
underdamped regime using a levitated nanoparticle by Gieseler et al., Gieseler
et al. (2014), when starting from a variety of non-equilibrium steady states
(c.f. Fig. 5). It is important to note that the relative entropy change
defined here is a special case of an infinite class of quantities $R$, which
have been introduced in their general form in a classic paper by Seifert
Seifert (2005) and has been called the dissipation function $\Omega_{t}$ by
Evans and Searles Evans and Searles (2002). Because it depends on the
stochastic trajectory it is commonly referred to as “stochastic entropy”.

Later Hoang et al., Hoang et al. (2018) experimentally demonstrated another
differential fluctuation theorem with levitated nanoparticles

$\frac{P\left(-W,u^{*}(t)\right)}{P\left(W,u(t)\right)}=e^{-\beta(W-\Delta
F)},$ (28)

which determines the probabilities that the system performs work
$W=-\int_{0}^{\tau}\dot{f}(t)q(t){\rm d}t$ against an external force $f(t)$.
In the experiment, the force is ramped from $f_{\text{off}}$ to
$f_{\text{on}}$ at a rate that is much faster than the velocity and position
relaxation times, such that when the ramp finishes, the system is far from
thermal equilibrium. The free energy difference between the equilibrium states
at the beginning and at the end of the ramp is given by $\Delta
F=-(f_{\text{on}}^{2}-f_{\text{off}}^{2})/(2m\Omega^{2})$.

Note that the differential fluctuation theorems can be integrated to yield
integral fluctuation theorems $\langle\exp(-R)\rangle=1$, such as the
Jarzynski equality Jarzynski (1997); Liphardt et al. (2002) and its refined
version, the Hummer-Szabo relation Hummer and Szabo (2001); Gupta et al.
(2011), which allows the reconstruction of free energy potentials. Thus, by
verifying the underlying differential fluctuation theorem, the validity of the
integral fluctuation theorem is implied and consequently also the “second law”
inequality $\langle R\rangle\geq 1$ Seifert (2012). Importantly, the
fluctuation theorems are valid for arbitrarily-far-from-equilibrium processes.
Both detailed and integral fluctuation theorems allow the estimation of
equilibrium free energy changes from nonequilibrium protocols and have found
applications in determining the free energies of DNA molecules Liphardt et al.
(2002); Collin et al. (2005). For a detailed review see Refs. Seifert (2012);
Martinez et al. (2017); Ciliberto (2017); Ritort (2006).

##  VIII Heat Engines

In the previous section we saw that in microscopic systems, thermodynamic
quantities such as the work against an external force and the heat exchanged
with the environment, become stochastic quantities due to the underlying
fluctuating trajectories through phase space. Yet work, heat and efficiency
can be rigorously defined within the framework of stochastic thermodynamics,
yielding the respective ensemble quantities after averaging Seifert (2012);
Sekimoto (2010). As a consequence, the output of a microscopic engine will be
fluctuating with the possibility of it running “ _in reverse_ ”.
Interestingly, one can draw analogies between a particle in an optical trap
and an ideal gas inside a piston, where the trap stiffness is analogous to the
inverse of an effective volume while the variance of the trajectory of the
particle can be seen as an effective pressure. By monitoring the motion of a
particle as it undergoes the cyclic heat engine, one can extract the work
statistics. This is the basic idea behind the following stochastic heat
engines. We leave a full discussion of the heat and entropy statistics to
other sources, for example Spinney & Ford Spinney and Ford (2012).

Schmiedl & Siefert gave the first full description of a colloidal stochastic
heat engine Schmiedl and Seifert (2008) and the first experimental realization
was by Blickle & Bechinger Blickle and Bechinger (2011), who locally heated
the liquid (water) surrounding an optically trapped particle through laser
absorption (c.f. Fig. 6a). Thereby, they realized temperature changes of 70 ∘C
in 10 ms, and their data agreed well with theoretical predictions. This study
was of particular importance, since it clearly demonstrated the fluctuating
nature of the work statistics and observed that for _individual_ cycles
sometimes the engine operated in reverse, due to the fluctuating position
statistics, thereby demonstrating that microscopic heat engines behave
fundamentally differently from their macroscopic counterparts. The
fluctuations of the power can be accounted for in a power fluctuation theorem
Martinez et al. (2017) and a stochastic definition of efficiency is given by
the ratio of the stochastic work extracted in a cycle and the stochastic heat
transferred from the hot bath to the system.

The efficiency of the Stirling engine is fundamentally limited by the
isochoric steps, where heat is transferred between the system and the heat
baths that are at different temperatures, making the cycle inherently
irreversible. The Carnot engine overcomes this limit by replacing the
isochoric steps with adiabatic changes, during which no heat is exchanged with
the environment. The second law of thermodynamics imposes a maximum (Carnot)
efficiency $\eta_{C}=1-T_{{\rm C}}/T_{{\rm H}}$ that can be reached by any
heat engine operating between two baths at temperature $T_{{\rm C}}$ and
$T_{{\rm H}}$, respectively Carnot .

However, it is commonly believed that the realization of an adiabatic change
requires that the control to the system is applied extremely slowly and
therefore in a Markovian process the Carnot efficiency can only be achieved in
the limit of zero power output Shiraishi et al. (2016). For practical
applications, however, one is interested in the efficiency at maximum power,
which has lead to the birth of finite-time thermodynamics Andresen et al.
(1984). For the ideal case, the efficiency at maximum power is limited by
Novikov-Curzon-Ahlborn efficiency Novikov ; Curzon and Ahlborn
$\eta^{*}=1-\sqrt{T_{{\rm C}}/T_{{\rm H}}}$, which is smaller than the Carnot
efficiency $\eta^{*}<\eta_{C}$. Later it was found that it is possible to
generate a shortcut to adiabaticity Berry (2009); Demirplak and Rice (2003).
In these protocols the evolution of the system mimics the adiabatic dynamics
without the requirement of slow driving by introducing a counterdiabatic
driving term, raising the question whether the Novikov-Curzon-Ahlborn
efficiency can be surpassed Schmiedl and Seifert (2008). As a consequence,
optimal protocals that lead to shortcuts to adiabaticity have received much
attention recently Deffner et al. (2014); Jarzynski (2013); del Campo (2013);
Tu (2014), both for their experimental relevance and as an interesting
theoretical problem in its own right.

Recent studies have also asked whether single parameter bounds such as the
Novikov-Curzon-Ahlborn bound is the best metric, discussing instead “trade-
offs” between efficiency and power under different experimental conditions and
under non-equilibrium operation Brandner et al. (2015). Levitated
nanoparticles could test these relations over a large parameter space. For
instance, one can explore non-Markovian dynamics Rosinberg et al. (2015) due
to feedback and operation under periodic temperature variations Brandner et
al. (2015) which could be achieved through modulation of the laser beam as we
discussed before. Besides, the full over- to under-damped regime is easily
accessible whilst dynamically varying all of the relevant thermodynamics
quantities (such as trapping volume and temperature). In addition, the
tantalizing potential to operate in the quantum regime could enable
exploration of constraints on the efficiency and power production of non-
Markovian quantum engines Shiraishi and Tajima (2017).

To overcome the limitations of the Stirling cycle, Martinez et al. implemented
a Carnot cycle with an optically trapped colloidal particle (c.f. Fig. 6b).
The adiabatic ramp was thereby realized by changing the temperature and trap
stiffness together such that the ratio $T^{2}/{\rm k}$ remained constant
Martínez et al. (2015); Tu (2014). This protocol required a precise control
over the bath temperature that is synchronized with the change of the trap
stiffness. This is not possible with heating of the surrounding water as done
by Blickle & Bechinger Blickle and Bechinger (2011). Instead they produced an
effective hot temperature bath with fluctuating electromagnetic fields as we
discussed earlier Martínez et al. (2013). For slow driving, their Carnot
engine attained the fundamental limit given by the Carnot efficiency and the
efficiency at maximum power was in excellent agreement with the Novikov-
Curzon-Ahlborn efficiency. In addition, they showed that the Carnot bound can
be surpassed for a small number of non-equilibrium cycles Martínez et al.
(2015). For a detailed discussion of microscopic heat engines in the
overdamped regime we refer to Refs. Dinis et al. (2016); Ciliberto (2017);
Martinez et al. (2017).

The implementations discussed so far have been realized with colloidal
systems, where the motion of the particle is heavily damped due to the close
contact with the surrounding liquid. Under these conditions, a measurement of
the momentum distribution is very challenging, although not impossible Huang
et al. (2011); Kheifets et al. (2014). Martinez et al., Martínez et al. (2015)
circumvented this challenge by extrapolating the instantaneous velocity from
the mean-squared time-averaged velocity. However, from a fundamental
standpoint it is desirable to have direct access to the instantaneous position
and momentum of the particle. The instantaneous momentum of the particle can
be measured easily by operating under high vacuum conditions Li et al. (2010);
Gieseler et al. (2012). In addition, the investigation of much more isolated
systems provides a path toward the future realization of quantum heat engines
and it has been suggested that super Carnot efficiencies can be attained by
clever bath engineering Roßnagel et al. (2014).

From our previous discussion it is clear that optically levitated particles
are well suited to implement a microscopic heat engine in the underdamped
regime. Analogous to the experiments with colloidal particles Martínez et al.
(2015); Blickle and Bechinger (2011), the volume of the engine is controlled
through the power of the trapping laser (c.f. Eqn. (4)). However, due to the
weak interaction with the environment, the effective temperature can be
controlled with much higher precision (c.f. Eqn. (20)) through a combination
of gas pressure, external driving and feedback cooling and even allows to
create non-thermal baths Gieseler et al. (2014, 2015); Rashid et al. (2016)
that could lead to surpassing the Carnot efficiency Roßnagel et al. (2014).
Such an all-optical heat engine was proposed by Dechant et al., Dechant et al.
(2015). In this proposal, cooling is realized through the interaction with an
optical cavity instead of active feedback cooling. The all-optical control
provides flexibility in optimizing the heat engine for maximizing its
performance. In the overdamped case the analytic treatment of optimal
protocols is possible because the dynamics can be described by a simplified in
terms of the slow position variable Schmiedl and Seifert (2008). In contrast,
this is not possible in the underdamped case, where the position and velocity
variables cannot be separated Gomez-Marin et al. (2008); Dechant et al.
(2017), and numerical methods must be used. An important note is that these
optimal protocols increase the power output and the efficiency of the engine
by introducing rapid changes in the trapping frequency. Being able to realize
this kind of fast control experimentally is a distinct advantage of the all-
optical nature of the heat engine. In particular, a heat engine realized with
a fast cavity response or a cavity-free setup could prove advantageous, since
the control is not limited by the finite cavity response time.

Figure 6: Single particle engines. (a) Realization of a Stirling engine by
Blickle & Bechinger Blickle and Bechinger (2011) using laser absorption to
change the temperature of the environment. (b) Martinez et al., Martínez et
al. (2015) realized a microscopic Carnot engine. The adiabatic steps of the
Carnot engine requires to change the temperature and the trap stiffness
synchronously. Figures reproduced from Nature Physics.

##  IX Conclusions

In this review we have explored the potential for levitated nanoparticles to
address questions in stochastic thermodynamics and non-equilibrium physics on
the single particle level in the underdamped regime.

In the overdamped regime, micron-sized colloidal particles in liquid have
already been used extensively to construct micro-engines and to study the
statistical properties of their power and efficiency, largely motivated by
trying to understand biological systems. A better understanding of how nature
builds machines and motors at the molecular level then allows scientists to
build their own molecular devices Browne and Feringa (2006); Erbas-Cakmak et
al. (2015). Although these objects operate in overdamped environments, the
timescale of their operation requires one to account for inertial
contributions Gomez-Marin et al. (2008). As discussed, measuring the
instantaneous momentum in real time is not straightforward in overdamped
systems. In contrast, the momentum relaxation in underdamped systems is much
slower due to the weak interaction with the environment, which allows for
obtaining a complete picture of the dynamics. Hence, levitated particles may
offer new insights into the molecular world.

In levitated systems, deterministic forces and stochastic forces are well
controlled experimentally, thus giving access to new parameter regimes. For
example, levitated nanoparticles have led to the first observation of
ballistic Brownian motion Li et al. (2010) and to the first quantitative
measurement of Kramers turnover Rondin et al. (2017) in addition to
demonstrating general fluctuation theorems in the underdamped regime Gieseler
et al. (2014); Hoang et al. (2018).

The underdamped regime is of fundamental interest, since the underlying
equations of motion contain inertia which in overdamped systems is typically
ignored. It also allows one to make the connection to the even more
fundamental unitary evolution of quantum mechanical systems. Therefore, future
experiments with levitated nanoparticles will help to characterize the sources
of irreversibility in micro-engines and give new insight into the statistical
properties of their efficiencies that could inspire new strategies in the
design of efficient nano-motors. In addition, rapid progress in cooling the
center-of-mass motion will enable the operation in the quantum regime, thereby
realizing a textbook quantum Brownian particle.

In the quantum regime, the information we can extract from a system is limited
by the Heisenberg uncertainty principle Heisenberg (1927). The fact that
information is physical is also well established in thermodynamics through
Landauer’s principle, which asserts that there is a minimum possible amount of
energy required to erase one bit of information development et al. (1961).
This link between information theory and thermodynamics was verified
experimentally with a colloidal system where the information was obtained by
light scattering from the particle Lutz and Ciliberto (2015); Bérut et al.
(2012). In ultra-high vacuum, the interaction with the optical light field is
the dominant interaction of the levitated particle with its environment Jain
et al. (2016). Therefore, the quantum back-action of the measurement Braginsky
and Khalili (1992); Purdy et al. (2013) starts to play a role in this regime.
The impact of the measurement process in the operation of heat engines and
work extraction Talkner and Hänggi (2016) is a very active field of research
that combines information theory, the quantum measurement process and
thermodynamics. Levitated particles in high vacuum are already exploring
thermodynamics in the underdamped regime and are poised for venturing into the
quantum regime. However, definite theoretical proposals for the realization of
thermodynamic protocols with levitated nanoparticles in the quantum regime are
still lacking. We hope that this review changes this by making the
fundamentals of levitated nanoparticles easily accessible to the community of
quantum thermodynamics while also raising awareness of this exciting field
among researchers working in levitation.

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