# Backaction suppression in levitated optomechanics using reflective
boundaries

Rafał Gajewski  James Bateman<EMAIL_ADDRESS>Department of
Physics, Faculty of Science and Engineering, Swansea University, SA2 8PP UK

###### Abstract

We show theoretically that the noise due to laser induced backaction acting on
a small nanosphere levitated in a standing-wave trap can be considerably
reduced by utilising a suitable reflective boundary. We examine the spherical
mirror geometry as a case study of this backaction suppression effect,
discussing the theoretical and experimental constraints. We study the effects
of laser recoil directly, by analysing optical force fluctuations acting on a
dipolar particle trapped at the centre of a spherical mirror. We also compute
the corresponding measurement imprecision in an interferometric, shot-noise-
limited position measurement, using the formalism of Fisher information flow.
Our results show that the standing-wave trapping field is necessary for
backaction suppression in three dimensions, and they satisfy the Heisenberg
limit of detection.

##  I Introduction

In recent years, experiments which utilise nano-sized particles trapped at the
focus of a laser beam emerged as a promising experimental platform for
detecting ultrasmall forces [1, 2], gravitational waves [3], searching for
dark matter [4], and potentially probing quantum behaviours of large masses in
wide and untested parameter regime [5]. These levitated optomechanical systems
constitute high quality mechanical oscillators, whose position is probed by
phase-sensitive measurement of the light scattered by the particle. The
interferometric position readout and extreme isolation from thermal
dissipation [6], reached due to the lack of physical contact, enabled
levitated optomechanical systems to cool the particle motion to the ground
state. [7, 8, 9].

However, in a continuous measurement by the laser, fluctuations of the
electromagnetic field introduce random perturbations to the motion of the mass
known as measurement backaction [10, 11, 12, 13]. This backaction noise is
linked to the minimum achievable imprecision in the position measurement by
the Heisenberg limit of detection [12, 14], which keeps the product of the two
quantities fixed. Although cooling of the particle to its motional ground
state has been demonstrated [7, 8, 9], the coherence times of the prepared
motional states are limited by excessive recoil heating imposed by the optical
measurement process. This limits access to the envisioned applications [6] and
therefore finding ways to mitigate the effects of backaction is of vital
importance.

The degree to which the backaction process can be controlled in levitated
optomechanics without turning off the trapping light has not yet been fully
explored. Recently, a backcation suppression method has been proposed based on
illumination by squeezed light [15], enabling control over the information
encoded in particle scatter. A different method relies on trapping hexagonal
plates instead of spheres [16, 17], because a different geometry of the
trapped object alters its radiation pattern, thereby confining the solid angle
in which backaction noise can act.

Backaction noise in levitated optomechanics can be understood as a result of
interference of the laser field with the local vacuum field fluctuations [11,
18, 19, 20]. In general, the vacuum fluctuations are function of the
environment, depending on its material properties and geometry [21, 22]. It is
known that surfaces can lead to backaction different than would be expected in
free-space [19]. In this work, we investigate an experimental scheme to
suppress backaction by trapping the particle at the centre of a spherical
mirror. We show that a spherical geometry in conjunction with a standing wave
optical trapping potential can lead to a substantial reduction in mechanical
noise, limited mainly by the mirror reflectivity.

Previous QED analysis of the spherical mirror geometry found that the local
density of optical states is strongly modify at the centre [23, 24] which
allows for full suppression of spontaneous emission from an atom. In the
context of levitated optomechanics, optimal position measurement of a particle
trapped at the centre of a spherical mirror was recently analysed [25, 26] and
the authors show that this can be used to realise an ideal reference field for
self-homodyne detection which is theoretically able to reach the Heisenberg
limit of detection by observation from only half of the solid angle.

A similar setup was also recently proposed for the purpose of backaction
suppression [27]. In contrast to the findings presented in this recent paper,
we find that suppression of scattered power does not correspond to the
suppression of backaction. Instead, we find that the linear position read-out
becomes inaccessible in an interferometric measurement of the far-field
scatter when the scattered power is maximally enhanced by the spherical mirror
and the full effect in three dimensions is only realised in a standing wave
trapping potential. We show that our results satisfy the Heisenberg limit of
detection, and therefore constitute a trade-off between measurement
imprecision and backaction without altering the illuminating laser power. This
work has general implications for recoil noise reduction in a structured
environment, offers a feasible experimental implementation and may aid in the
search for other useful configurations.

##  II Problem statement

We consider the setup shown in figure 1 as a case study.

Figure 1: Spherical mirror trapping configuration. A laser beam polarised out
of the page is focused close to the centre of the spherical mirror, such that
the beam retroreflects from the mirror surface and generates a standing wave
trap. A particle is trapped at the central maximum with displacement from the
centre $|\mathbf{r}_{0}|\ll\lambda$.

In this setup, the beam is incident upon the mirror surface and forms a
standing wave trap. We assume that the beam parameters are selected such that
the beam is retroreflected and an intensity maximum appears at the mirror
centre (see appendix C) which we also take to be the origin of the coordinate
system. The particle is trapped at the intensity maximum appearing at the
centre of a spherical mirror, and is located at $\mathbf{r}_{0}$ (with
$|\mathbf{r}_{0}|\ll\lambda$) which corresponds to the displacement due to the
motion about the trap centre. The particle is modelled as a point electric
dipole [22] (radius $r_{p}\ll\lambda$) with isotropic polarisability $\alpha$
acted on by a stochastic force $\delta\mathbf{F}$. We assume that we can
ignore the contribution of residual gas and that the force fluctuations are
dominated by the contribution from laser recoil. Throughout this analysis, we
work in the quasistatic limit, ignoring any effects the motion has on the
imprecision-backaction characteristics. Assuming that the mirror is not too
large, this is permissible since the particle motion is significantly slower
than time delay of the mirror-particle round trip.

We assume that the beam is weakly-focused and that the central maximum is
sufficiently close to the beam waist such that the standing wave field
resembles a field of two counter-propagating plane waves,

$\mathbf{E}_{0}(\mathbf{r},t)=\text{Re}\left[E_{0}\cos(Akz)e^{-i\omega_{0}t}\right]\hat{\mathbf{x}}.$
(1)

In equation (1), $k=\omega_{0}/c=2\pi/\lambda$ is the free-space wavenumber of
the laser beam and the Gouy factor $A$ increases the effective wavelength of
the field, depending on the degree of focusing [14].

In the next section, we compute measurement backaction by employing a
stochastic electrodynamics description of the local field fluctuations, [20,
18, 28, 29, 30] and computing the spectral noise density of the recoil force
in a standing wave formed by reflection from the spherical mirror.

##  III Measurement backaction

In this section we quantify the measurement backaction by computing the
spectral density of the optical force fluctuations at the mechanical frequency
of the trapped particle. The particle interacts with the laser field and a
fluctuating background field,

$\mathbf{E}(\mathbf{r}_{0},t)=\mathbf{E}_{0}(\mathbf{r}_{0},t)+\delta\mathbf{E}(\mathbf{r}_{0},t).$
(2)

In equation (2), $\delta\mathbf{E}$ represent zero-point or thermal field
fluctuations that are modelled as a statistically stationary process with zero
average [20], and $\mathbf{E}_{0}$ is the trapping laser field. In general,
the particle also interacts with reflections of its own scatter, which lead to
an infinite series of additional terms in equation (2) scaling in powers of
$k^{3}\alpha$. Since $k^{3}\alpha\sim(r_{p}/\lambda)^{3}\ll 1$ this series
converges quickly, and since we are interested in computing the dominating
contribution to backaction, we truncate at zeroth order in $\alpha$. Treating
the particle as a point electric dipole located at $\mathbf{r}_{0}$, the field
in equation (2) induces a dipole moment $\mathbf{p}$. In this dipole
approximation, the particle experiences an optical force given by, [22]

$\mathbf{F}=\sum_{i}p_{i}\nabla
E_{i}+\frac{d}{dt}(\mathbf{p}\times\mathbf{B})$ (3)

where $i=x,y,z$. Since we are only interested in computing the contribution of
laser recoil, we ignore the intrinsic fluctuations of the dipole moment. In
the frequency domain in the limit $\omega\ll\omega_{0}$, the force is well
approximated by the first term in Eq. (3),

$\displaystyle\tilde{\mathbf{F}}(\mathbf{r}_{0},\omega)$
$\displaystyle=\sum_{i}\left(\tilde{p_{i}}\ast\nabla\tilde{E_{i}}\right)(\mathbf{r}_{0},\omega)$
$\displaystyle\approx\tilde{\mathbf{F}}_{0}(\mathbf{r}_{0},\omega)+\delta\tilde{\mathbf{F}}(\mathbf{r}_{0},\omega)$
(4)

where the quantities with a tilde denote the respective Fourier transforms and
the ‘$\ast$’ symbol denotes a convolution in frequency. In the second line of
equation (4) we chose $\tilde{\mathbf{F}}$ to represent the deterministic
force and $\delta\tilde{\mathbf{F}}$ to represent force fluctuations,
approximated with terms containing fluctuations in the fields up to linear
order,

$\delta\tilde{\mathbf{F}}(\mathbf{r}_{0},\omega)=\left(\alpha\tilde{E_{0}}\ast\nabla\delta\tilde{E}\right)(\mathbf{r}_{0},\omega)+\left(\alpha\delta\tilde{E}\ast\nabla\tilde{E_{0}}\right)(\mathbf{r}_{0},\omega),$
(5)

where we used $\tilde{\mathbf{p}}=\alpha\tilde{\mathbf{E}}$ and have assumed
that $\mathbf{E}_{0}$ is $\hat{\mathbf{x}}$ polarised, which allows us to drop
the subscript ‘$x$’ from the field components $\delta E_{x}$ and $E_{0x}$ for
clarity. Assuming that $\delta\tilde{\mathbf{F}}$ is a stationary stochastic
process with zero ensemble average, we find the corresponding spectral density
of the force fluctuations using the expression, [31]

$S^{F}_{ii}(\omega)=\int_{\mathbb{R}}\left\langle\delta\tilde{F}_{i}^{*}(\mathbf{r}_{0},\omega)\delta\tilde{F}_{i}(\mathbf{r}_{0}^{\prime},\omega^{\prime})\right\rangle\operatorname{d}\\!{\omega}^{\prime}\bigg{|}_{\mathbf{r}_{0}=\mathbf{r}^{\prime}_{0}},$
(6)

where the angled brackets denote an ensemble average. We can evaluate the
field correlations arising in equation (6) by applying the fluctuation-
dissipation theorem for the field fluctuations given in equation (26). Upon
substitution of equation (5) and (26) into equation (6), we find that for a
monochromatic driving field of the form
$\mathbf{E}(\mathbf{r},t)=\text{Re}\left[\tilde{E}(\mathbf{r})e^{-i\omega_{0}t}\right]\hat{\mathbf{x}}$,
the spectral density of force fluctuations in the limit $\hbar\omega_{0}\gg
k_{B}T$ can be approximated as,

$\displaystyle S^{F}_{ii}(\omega)$
$\displaystyle=\frac{\hbar\omega_{0}^{2}}{4\pi
c^{2}\epsilon_{0}}\alpha^{2}\Big{(}$
$\displaystyle\phantom{+}\tilde{E}_{0}^{*}(\mathbf{r}_{0})\tilde{E}_{0}(\mathbf{r}_{0}^{\prime})\partial_{i}\partial_{i}^{\prime}\text{Im}\left[G_{xx}(\mathbf{r}_{0},\mathbf{r}_{0}^{\prime},\omega_{0})\right]$
$\displaystyle+\partial_{i}\tilde{E}_{0}^{*}(\mathbf{r}_{0})\partial_{i}^{\prime}\tilde{E}_{0}(\mathbf{r}_{0}^{\prime})\text{Im}\left[G_{xx}(\mathbf{r}_{0},\mathbf{r}_{0}^{\prime},\omega_{0})\right]$
$\displaystyle+\tilde{E}_{0}^{*}(\mathbf{r}_{0})\partial_{i}^{\prime}\tilde{E}_{0}(\mathbf{r}_{0}^{\prime})\partial_{i}\text{Im}\left[G_{xx}(\mathbf{r}_{0},\mathbf{r}_{0}^{\prime},\omega_{0})\right]$
$\displaystyle+\tilde{E}_{0}(\mathbf{r}_{0}^{\prime})\partial_{i}\tilde{E}_{0}^{*}(\mathbf{r}_{0})\partial_{i}^{\prime}\text{Im}\left[G_{xx}(\mathbf{r}_{0},\mathbf{r}_{0}^{\prime},\omega_{0})\right]\Big{)}_{\mathbf{r}_{0}=\mathbf{r}_{0}^{\prime}}$
(7)

Figure 2: Backaction $S_{ii}^{F}(\omega)$ against spherical mirror spanning
angle found by solving Eq. (9). Filled lines indicate backaction for
$kr_{m}=n\pi$ while dotted lines for $kr_{m}=n\pi\pm\pi/2$. Dashed lines
indicate free-space levels in a standing wave, which in our model corresponds
to $\theta_{m}=0$.

where we introduced the dyadic Green’s function of the system (see appendix
section A), defined the free-space scattered power
$P_{0}=E_{0}^{2}\alpha^{2}k^{4}c/(12\pi\epsilon_{0})$ and the partial
derivatives $\partial_{i}\equiv\partial/\partial x_{i}$ and
$\partial_{i}^{\prime}\equiv\partial/\partial x_{i}^{\prime}$ with respect to
the first and second argument of the Green’s function respectively. As long as
the aforementioned approximations apply, equation (7) represents the
dominating term of backaction noise experienced by the particle in a driving
laser field polarised along $\hat{\mathbf{x}}$ and an arbitrary environment
described by $G_{xx}$. For the case of a standing wave in equation (1), at an
intensity maximum, where the particle is trapped, the expression in equation
(7) can be further approximated as,

$S^{F}_{ii}(\omega)\approx\frac{3\hbar
P_{0}}{k^{2}c}\partial_{i}\partial_{i}^{\prime}\text{Im}\left[G_{xx}(\mathbf{r}_{0},\mathbf{r}_{0}^{\prime},\omega_{0})\right]_{\mathbf{r}_{0}=\mathbf{r}_{0}^{\prime}}.$
(8)

In general, the gradient of the driving field in equation (7) gives rise to
additional terms in the force spectral density. However, in the case of a
particle trapped about an intensity maximum of a standing-wave, with the
assumption that $|\mathbf{r}_{0}|\ll\lambda$, the gradient of the field does
not contribute up to second order in particle position [32]. In the case of a
particle trapped at the focus of a single focused beam in free-space, we
recover the expressions for optimal position detection found in [14] including
the terms which contain the factor $A$ corresponding to the gradient of the
focused field. The remaining term in equation (8) corresponds to the spectral
density of the field gradient fluctuations. Within our approximations, our
expression agrees with the expression for the decoherence rate due to laser
recoil [19] for a model of a particle trapped in front of a flat plane, in the
framework of macroscopic QED [33, 34]. In that context, the dominant
contribution to the noise arises from the driven-Casimir–Polder interaction
term in the Hamiltonian of the trapped particle dynamics. This term is a
result of interference of the classical driving field with medium-assisted
vacuum fields.

We find the effect of the mirror on the force spectral density by substituting
into equation (8) the Green’s function which characterises the spherical
mirror geometry, as discussed in appendix A. The resulting total Green’s
function is given by equation (28), which upon substitution into equation (8)
yields,

$S^{F}_{ii}(\omega)=\frac{\hbar kP_{0}}{2\pi
c}\left[\frac{1}{5}(2-\delta_{ix}\delta_{xi})-2\cos(2kr_{m})\int_{(\text{a})}\operatorname{d}\\!{S}_{ii}\right]$
(9)

which represents the modified backaction noise of a particle trapped in a
standing-wave at the centre of a spherical mirror. In equation (9) we
introduced,

$\operatorname{d}\\!{S}_{ii}=(\hat{r}_{i})^{2}\rho_{x}(\theta,\phi)\operatorname{d}\\!{\Omega}$
(10)

and the radiation pattern of an $\hat{\mathbf{x}}$ oriented dipole,

$\rho_{x}(\theta,\phi)=\frac{3}{8\pi}[1-(\hat{\mathbf{x}}\cdot\hat{\mathbf{r}})^{2}].$
(11)

where the radial unit vector is given by
$\hat{\mathbf{r}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The
first term in equation (9) represents the free-space backaction in a standing-
wave, while the second represents the contribution of the mirror. The
integration in the second term runs over the mirror surface as depicted in
figure 3. For a full hemispherical mirror, the result in equation (9)
simplifies to,

$S^{F}_{ii}(\omega)=\frac{2}{5}\frac{\hbar kP_{0}}{2\pi
c}\sin^{2}(kr_{m})(2-\delta_{ix}\delta_{xi})$ (12)

which shows that for $kr_{m}=n\pi$ with integer $n$ the dominating
contribution to backaction noise vanishes. This backaction suppression
condition coincides with the condition on the mirror radius needed for
trapping at the mirror centre (see appendix section C) and is also the
condition giving maximum enhancement of scattered power (see appendix section
B). This is in contrast to recent findings for the spherical mirror system
[27] which predict suppression of backaction when the emission (or the
scattered power) is inhibited. In addition, our result shows that the full
backaction suppression in three dimensions is only possible in a standing-wave
trapping field, in which case the equation (8) is valid and the phase gradient
of the driving field does not lead to additional contributions to backaction
[32]. While the spherical geometry facilitates suppression of field gradient
fluctuations, a standing wave trapping configuration is necessary to suppress
residual field amplitude fluctuations. Our result can be intuitively explained
by considering the measurement imprecision achievable in an ideal
interferometric measurement of the particle’s position, which we consider in
the next section.

In figure 2 we also show how the backaction noise in (9) varies with the
mirror polar angle $\theta_{m}$, and find that the rate at which backaction
decreases with $kr_{m}=n\pi$ is faster for $S_{zz}^{F}$ than for $S_{xx}^{F}$
and $S_{yy}^{F}$. This can be understood by considering the distribution of
position information radiated into the far-field, which we introduce in the
next section.

##  IV Measurement imprecision

Figure 3: Diagram denoting different measurement domains. For a spherical
mirror that is perfectly reflecting, no field gets transmitted and dipole
emission is not accessible for measurement within domain (a). Because dipole
scatter is retroreflected by the mirror surface, the mirror only affects the
detected emission in domain (c), while domain (b) corresponds to free-space
dipole emission. In domains (b) and (c), the dipole emission fields are given
by equations (16) and (17) respectively.  Figure 4: Angular spread of position
information, here defined as the integrand of equation (18) normalised to the
free-space imprecision in a standing wave $S_{ii}^{\text{imp}}(\theta_{m}=0)$.
The mirror is perfectly reflecting, has its axis of symmetry along
$\hat{\mathbf{z}}$ as in figure 1 and was chosen to have $\theta_{m}=\pi/3$.
The white region corresponds to domain (c) in figure 3, in which the scatter
is affected by the mirror. The information in the white region of the top
half-space (domain (a)) is never accessible because this portion of the solid
angle is covered by the mirror. At $kr_{m}=n\pi$ the position information is
inaccessible in the white region of both half-spaces (a) and (b). The colored
region corresponds to free-space information in a standing wave.

We can formally quantify minimum measurement imprecision in the far-field,
shot-noise-limited detection of particle position using the recently developed
formalism of Fisher information (FI) flow [35], which places a lower bound on
the variance in the measured quantity. Specifically, the formalism introduces
a local quantity of Fisher information flow, akin to the Poynting vector which
describes the flow of electromagnetic energy. In the time average, the
reciprocal of this quantity then places a bound on the average rate at which
the variance of a variable can be reduced in a given bandwidth of shot-noise-
limited measurement. The treatment is equivalent to treating the problem of an
ideal interference experiment with a strong, mode-matching reference field
[35, 14]. In a 1 Hz bandwidth we have,

$\text{Var}(x_{i})\geq\left(2\pi\int\mathbf{S}^{\text{FI}}_{ii}\cdot\hat{\mathbf{r}}\operatorname{d}\\!{A}\right)^{-1}2\pi[\text{Hz}]$
(13)

where,

$\mathbf{S}^{\text{FI}}_{ii}=\frac{2}{\hbar\omega_{0}}\text{Re}\left[\partial_{i}\mathbf{E}_{s}\times\partial_{i}\mathbf{H}_{s}^{*}\right]$
(14)

defines the time-averaged FI flux for the far-fields $\mathbf{E}_{s}$ and
$\mathbf{H}_{s}$ with $\partial_{i}\equiv\partial/\partial x_{i}$, and the
quantity in parentheses in equation (13) corresponds to the flat spectral
density of minimum imprecision noise,

$S^{\text{imp}}_{ii}=\left(2\pi\int\mathbf{S}^{\text{FI}}_{ii}\cdot\hat{\mathbf{r}}\operatorname{d}\\!{A}\right)^{-1}$
(15)

in the measurement of $x_{i}$. To obtain an optimal estimate of the particle’s
position for general mirror spanning angle, we need to find
$\mathbf{S}^{\text{FI}}_{ii}$ in each angular domain at some large distance
$r_{d}$ over the angular element
$\operatorname{d}\\!{A}=r_{d}^{2}\operatorname{d}\\!{\Omega}$, as depicted in
figure 3. In domain (b), the dipole emission is unaffected by the mirror and
hence the electric field is equal to the free-space dipole emission field
driven by the field in equation (1),

$\mathbf{E}_{s}^{(b)}(\mathbf{r}_{d},\mathbf{r}_{0})=\mathbf{E}_{s}(\mathbf{r}_{d})\cos(Akz_{0})e^{-ik\hat{\mathbf{r}}\cdot\mathbf{r}_{0}}$
(16)

where we expanded $r^{\prime}=|\mathbf{r}_{d}-\mathbf{r}_{0}|\approx
r_{d}-\hat{\mathbf{r}}\cdot\mathbf{r}_{0}$ and where
$\hat{\mathbf{r}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The
fields in equation (16) $\mathbf{E}_{s}^{(b)}(\mathbf{r}_{d},\mathbf{r}_{0})$
and $\mathbf{E}_{s}(\mathbf{r}_{d})$ represent the complex field amplitudes in
domain (b) from a dipole oriented along $\hat{\mathbf{x}}$ and located at
position $\mathbf{r}_{0}$ and the origin respectively (see also equation
(25)). In domain (c), the fields are a sum of the free-space dipole emission
and its reflection from the spherical mirror. For a perfectly reflecting
mirror and small displacement of the dipole from the mirror centre, we find
the field in domain (c) to be,

$\mathbf{E}_{s}^{(c)}(\mathbf{r}_{d})=\mathbf{E}_{s}(\mathbf{r}_{d})e^{ikr_{m}}\cos(Akz_{0})\cos(k[r_{m}+\hat{\mathbf{r}}\cdot\mathbf{r}_{0}]).$
(17)

Note that in the field in equation (17) the free-space dipole emission and its
reflection are in-phase. This is because the reflected field undergoes a
change of sign upon propagating through the focus from the mirror surface to
domain (c), akin to the Gouy shift of a focused Gaussian beam [36].

Substituting these fields into equation (15) and evaluating the derivatives
about the mirror centre yields the final result for a general spanning angle
of the mirror,

$S^{\text{imp}}_{ii}=\left(\frac{8\pi kP_{0}}{\hbar
c}\left[\int_{(\text{b})}\operatorname{d}\\!{S}_{ii}+4\sin(kr_{m})^{2}\int_{(\text{c})}\operatorname{d}\\!{S}_{ii}\right]\right)^{-1}$
(18)

where $\operatorname{d}\\!{S}_{ii}$ is given by equation (10). In the absence
of the mirror (vanishing of domain (c)), the expression in equation (18)
recovers the expressions for free-space from [14] with the exception of the
term proportional to the factor $A$ along the beam propagation direction. This
is because in a standing-wave field, the field gradient does not carry linear
position information. For a full-hemisphere, the result in equation (18)
reduces to,

$S^{\text{imp}}_{ii}=\frac{5}{8}\frac{\hbar c}{\pi
k}\frac{1}{P_{0}}\sin(kr_{m})^{-2}(1+\delta_{ix}\delta_{xi}).$ (19)

Equation (19) diverges at the backaction suppression condition found in
previous section, suggesting that the emission from the particle available for
measurement contains no first-order position information. Indeed, when
$kr_{m}=n\pi$ the total emission field in the available solid angle,

$\mathbf{E}_{s}^{(c)}(\mathbf{r}_{d},\mathbf{r}_{0})\approx
2\mathbf{E}_{s}(\mathbf{r}_{d})$ (20)

is independent of the source position to first order; the free-space dipole
emission is in-phase with its image, making it impossible to distinguish
between them to first order.

Let us also remark on the distribution of available position information in
the context of the angular information patterns which are employed in the
literature [14, 15]. In figure 4 we show the distribution of radiated position
information for a spherical mirror with an illustrative spanning angle of
$\theta_{m}=\pi/3$. The shapes of the corresponding information patterns are
unchanged from free-space patterns in a standing-wave, but the amount of
information changes discontinuously between angular domains depicted in figure
4. Note that the distribution of the $\mathcal{I}_{z}(\theta,\phi)$
information differs from that given in [14]; in a standing wave field, the
information about the position along $\hat{\mathbf{z}}$ is radiated
symmetrically about the $xy$ plane. For a perfectly reflecting mirror at
$kr_{m}=n\pi$, the position information is inaccessible in domains (a) and
(c). In addition, figure 4 shows that at $\theta_{m}=\pi/3$ a significantly
larger portion of the $\mathcal{I}_{z}$ solid angle is affected by the mirror
compared with $\mathcal{I}_{x}$ and $\mathcal{I}_{y}$. The spherical mirror
with its axis of symmetry aligned along the axis of the standing wave, is
particularly well-suited for suppression of position information along the
axis. This is also visible in the corresponding plot of backaction noise
against $\theta_{m}$ in figure 2, which shows that the noise along
$\hat{\mathbf{z}}$ changes more rapidly with $\theta_{m}$ than along
$\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$.

##  V Discussion

We find that the product of the imprecision in Eq. (19) and the backaction in
Eq. (12) satisfies the Heisenberg limit of detection, [14, 12]

$S_{ii}^{F}S_{ii}^{\text{imp}}=\left(\frac{\hbar}{2\pi}\right)^{2}$ (21)

for any choice of mirror radius. The limit is also satisfied by the solutions
in equation (18) and (9) for an arbitrary choice of mirror spanning angle
$\theta_{m}$. This results suggests that by utilising a spherical mirror one
can make a trade-off between measurement imprecision and backaction by
choosing an appropriate mirror radius. For a perfectly reflecting hemisphere
at $kr_{m}=n\pi$, the detected scatter is maxiamlly enhanced, but contains no
linear position information. Correspondingly, the dominating term of
backaction noise vanishes. This reinforces the fact that the flow of energy
does not necessarily carry information about the environment with which it has
interacted [35, 14].

The suppression effect could be tested experimentally by measuring the
reheating rates via trapped particle trajectories, as was done in free space
[13]. The necessary linear measurement of particle position could be accessed
with illumination in the $xy$ plane for a mirror with a spanning angle
$\theta_{m}<\pi/2$, or using a detection beam which does not satisfy the
suppression condition for the mirror radius ($kr_{m}\neq n\pi$). In addition,
we note that second-order position information should be accessible under the
suppression condition, which is sufficient for the implementation of
parametric feedback cooling [37] or temperature measurement.

The largest contributions to residual backaction noise are likely to arise
from technical limitations in the experimental realisation. According to
equation (29), imperfect reflectivity of a metallic mirror affects the
suppression result in equation (9) linearly, and would limit the suppression
of backaction noise to about $10^{-2}$ of its free-space magnitude. A
realisation scheme will also require surface quality and long-term mirror
radius stability $\delta r_{m}\ll\lambda/2\pi$. Considering the coefficient of
linear expansion of aluminium $\alpha_{\text{Al}}=2.3\times
10^{-5}\text{K}^{-1}$ and a mirror with radius of about 1 mm, sufficient
radius stability can be achieved with thermal stability $\delta T<1\text{K}$.
A recent experimental proposal utilising a spherical mirror for similar
application suggests that such temperature stabilisation and the quality of
the fabricated mirror surface should be achievable [38].

##  VI Conclusion

In summary, we have found that a suitable structured environment can yield
significant reduction of mechanical noise due to backaction. We have studied a
spherical mirror as a particular case, and found that it can be used to
inhibit the backaction force noise in three dimensions. The suppression scheme
also requires the particle to be trapped in a standing wave trapping field. We
have found that under a suitable condition on the mirror radius, the mirror
significantly attenuates the attainable acquisition rate of linear position
information, at the same time suppressing the largest backaction noise term.
In the case that the mechanical noise experienced by the particle is dominated
by backaction and the detection is shot-noise limited, we have shown that the
imprecision and backaction noise satisfy the Heisenberg limit of detection for
any choice of mirror radius $r_{m}$ and spanning half-angle
$\theta_{m}\leq\pi/2$. Experimental considerations show that the suppression
scheme should be achievable in a realistic setting; the amount by which recoil
noise can be suppressed using our method will either be limited by
experimental factors (such as mirror reflectivity, thermal stability, or
surface quality) or smaller residual backaction noise terms unaccounted for in
our calculation.

Counter-intuitively, our results show that the suppression condition is also
the condition for which the scattered power, usually associated with recoil
noise, is maximally enhanced by the spherical mirror. This is in contrast to
recent findings [27], which suggest that suppression of emission leads to
suppression of backaction noise. Our result can be understood in the context
of the position information content in the outgoing particle scatter; at the
suppression condition, the image dipole appears identical to the real particle
to first order in particle position about the mirror centre, preventing
localisation in a position tracking experiment. For all experimental
conditions considered, we have shown that this measurement imprecision and the
corresponding backaction satisfy the Heisenberg limit of detection. The method
presented here shows how an appropriate reflective surface geometry can be
used to mitigate backaction, and may enable development of more sophisticated
schemes for particles of different sizes and shapes in future experiments. In
addition, a wider range of geometries may be suitable, if one wishes to
suppress the backaction noise only in one or two directions.

The main result shows a strong dependence of the imprecision-backaction
characteristics on the surrounding mirror geometry. The spherical mirror
geometry is uniquely suited for point-like particles as the mirror reflection
can perfectly match dipole emission away from the mirror. Other geometries may
present similar useful properties in future investigations, such as a plane
mirror-lens system. Other practical arrangements suitable for the scattering
properties of particles with different and more complex morphologies may be
found by adapting existing analysis tools, used for tailoring the optical
properties of nanoscale emitters [39, 40]. We once again note that although
the spherical mirror geometry plays a necessary role for the suppression
effect, it is not sufficient for full suppression in three dimensions; a
standing wave trapping potential is also required to remove the contribution
of the local phase gradient, which would be present in a single-beam
configuration [14].

###### Acknowledgements.

R.G. was supported by the UK Engineering and Physical Sciences Research
Council (EPSRC) through Standard Research Studentship (DTP) EP/T517987/1. No
new data were generated or analysed during this study.

##  Appendix A System’ dyadic Green’s function

Green’s functions are solutions to inhomogeneous differential equations with
the inhomogeneity given by a delta function. In homogeneous space, a point
dipole gives a singular inhomogeneity in the Helmholtz equation. Fields
radiated by a point-like electric dipole are determined by the dyadic Green’s
function $\overline{\overline{\mathbf{G}}}$; a tensor of rank two which solves
the equation, [22]

$\nabla\times\nabla\times\overline{\overline{\mathbf{G}}}(\mathbf{r},\mathbf{r}_{0},\omega)-k^{2}\overline{\overline{\mathbf{G}}}(\mathbf{r},\mathbf{r}_{0},\omega)=\overline{\overline{\mathbf{I}}}\delta(\mathbf{r}-\mathbf{r}_{0})$
(22)

with wavenumber $k=\omega/c$, unit dyad $\overline{\overline{\mathbf{I}}}$,
source position $\mathbf{r}_{0}$ and observation point $\mathbf{r}$. In free
space, the solution for the Green’s function is, [22]

$\overline{\overline{\mathbf{G}}}_{0}(\mathbf{r},\mathbf{r}_{0},\omega)=\left[\overline{\overline{\mathbf{I}}}+\frac{1}{k^{2}}\nabla\nabla\right]\frac{\exp(ikr^{\prime})}{4\pi
r^{\prime}}$ (23)

where we wrote $r^{\prime}=|\mathbf{r}^{\prime}|=|\mathbf{r}-\mathbf{r}_{0}|$
and a direct product of two vectors corresponds to a dyadic product. In the
subsequent calculations, we will make particular use of the part of
$\overline{\overline{\mathbf{G}}}_{0}$ which scales as $(r^{\prime})^{-1}$,

$\overline{\overline{\mathbf{G}}}_{\text{ff}}(\mathbf{r},\mathbf{r}_{0},\omega)=\frac{\exp(ikr^{\prime})}{4\pi
r^{\prime}}\left[\overline{\overline{\mathbf{I}}}-\frac{\mathbf{r}^{\prime}\mathbf{r}^{\prime}}{(r^{\prime})^{2}}\right]$
(24)

which is gives rise to the far-field free-space dipole emission given by,

$\mathbf{E}_{s}(\mathbf{r},\mathbf{r}_{0})=\frac{k^{2}}{\epsilon_{0}}\overline{\overline{\mathbf{G}}}_{\text{ff}}(\mathbf{r},\mathbf{r}_{0},\omega)\cdot\mathbf{p}$
(25)

where $\mathbf{p}$ is the electric dipole moment. We can also use the system’s
dyadic Green’s function element corresponding to the dipole orientation to
evaluate the correlation functions of the field fluctuations in the framework
of stochastic electrodynamics [20, 22]. At thermal equilibrium at temperature
$T$ this fluctuation-dissipation theorem for the $\hat{\mathbf{x}}$ component
of the field fluctuations reads,

$\left\langle\delta\tilde{E}^{*}(\mathbf{r},\omega)\delta\tilde{E}(\mathbf{r}^{\prime},\omega^{\prime})\right\rangle=f(\mathbf{r},\mathbf{r}^{\prime};\omega,T)\delta(\omega-\omega^{\prime}),$
(26)

where,

$f(\mathbf{r},\mathbf{r}^{\prime};\omega,T)=\frac{\hbar\omega^{2}}{2\pi
c^{2}\epsilon_{0}}\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\text{Im}\left[G_{xx}(\mathbf{r},\mathbf{r}^{\prime},\omega)\right].$
(27)

In the main text, we use equation (26) to model the spectral density of force
fluctuations due to backaction.

In the presence of reflective boundaries, the total Green’s function can be
decomposed as,

$\overline{\overline{\mathbf{G}}}(\mathbf{r},\mathbf{r}_{0},\omega)=\overline{\overline{\mathbf{G}}}_{0}(\mathbf{r},\mathbf{r}_{0},\omega)+\overline{\overline{\mathbf{G}}}_{s}(\mathbf{r},\mathbf{r}_{0},\omega)$
(28)

where the first term corresponds to free-space emission, and the second term
corresponds to the reflected field and depends on the boundary. For a dipole
emitter located at at or near the centre of a large spherical mirror (with
radius $r_{m}\gg\lambda$), only the far-fields contribute to the reflected
fields near the source position. Modelling the reflected fields using Fresnel
coefficients and the angular spectrum representation for focused far-fields
[22, 31], at an observation point $\mathbf{s}$ at or near the centre of the
mirror we find the expression,

$\overline{\overline{\mathbf{G}}}_{s}(\mathbf{s},\mathbf{r}_{0},\omega)=-\rho\frac{ikr_{m}e^{ikr_{m}}}{2\pi}\int_{(\text{a})}\overline{\overline{\mathbf{G}}}_{\text{ff}}(\mathbf{r},\mathbf{r}_{0},\omega)e^{-ik\hat{\mathbf{r}}\cdot\mathbf{s}}d\Omega$
(29)

where $k=\omega/c$, $\rho$ denotes the Fresnel reflection coefficient of the
mirror at normal incidence, $\mathbf{r}$ denotes a vector from the origin to a
point on the mirror surface, and the integration of the solid angle element
$\operatorname{d}\\!{\Omega}$ runs over the solid angle of the mirror (denoted
as domain (a) in figure 3)

##  Appendix B Dipole emission near boundaries

The scattering rate of a point dipole emitter depends on its environment. For
a point dipole emitter, the average scattered power can be expressed as, [41,
22]

$\left\langle P\right\rangle=\frac{6\pi
P_{0}}{k}\mathbf{n}_{p}\cdot\text{Im}\left[\overline{\overline{\mathbf{G}}}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{0})\right]\cdot\mathbf{n}_{p}$
(30)

where the dyadic Green’s function is evaluated at the location of the dipole
in both arguments, $P_{0}$ denotes he free-space scattered power, and the
vector $\mathbf{n}_{p}$ denotes the dipole orientation.

Figure 5: Modified scattered power, expressed as a ratio with the free-space
scattering rate $P/P_{0}$, evaluated at the mirror radius condition
$kr_{m}=n\pi$. The modified scattered power for a displaced source is found by
numerically solving the scattering Green’s function in equation (29).

By substituting equation (28) into (30) and using equations (23) and (29), we
find that at the centre of a perfectly reflecting ($\rho=-1$) hemispherical
mirror the scattered power is given by,

$\left\langle P\right\rangle=2\cos^{2}(kr_{m})P_{0}.$ (31)

In agreement with the QED result found for a spontaneous emission from an atom
at the centre of a hemispherical mirror [23]; the condition $kr_{m}=n\pi$
corresponds to a two-fold enhancement of the total scattered power. In figure
5 we also show that the scattered power varies as the location of the dipole
is varied about the centre of the mirror.

##  Appendix C Retro-reflection from a spherical mirror

In this section we find the necessary conditions for retro-reflection of a
laser beam from the surface of a spherical mirror, and the corresponding
mirror radius which allows trapping at the mirror centre. The beam is retro-
reflected when its radius of curvature matches that of the mirror. Modelling
the laser as a paraxial Gaussian beam with radius of curvature $R(z)$, this
condition is expressed as, [42]

$r_{m}=R(z-z_{f})=z-z_{f}+\frac{z_{R}^{2}}{z-z_{f}}$ (32)

where $z_{f}$ is the axial position of the focus along the mirror axis. The
condition in equation (32) has two solutions for the focus position,

$z_{f}^{\pm}=\frac{r_{m}}{2}\left(1\pm\sqrt{1-4\frac{z_{R}^{2}}{r_{m}^{2}}}\right).$
(33)

Since we want the laser intensity at the trapping position to be as large as
possible, we choose the negative sign solution. Therefore, we can achieve
maximum modulation when the beam is focused a distance $z_{R}^{2}/r_{m}$ (for
$z_{R}\ll r_{m}$) from the mirror centre towards its surface. Note that since
$z_{R}\approx\lambda/\pi(\text{NA})^{2}$ for a beam of numerical aperture NA,
the tighter the focus of the beam, the closer the solution $z_{f}^{-}$ is to
the centre of the mirror.

To find a condition on the mirror radius which yields an intensity maximum at
the centre of the mirror, we consider the round-trip phase about the mirror
centre. For a plane wave reflecting from a perfect plane mirror a distance
$r_{m}$ away this round-trip phase would simply be equal to $2kr_{m}+\pi$. For
a Gaussian beam, there is an additional contribution arising from the Gouy
phase shift $\phi_{g}(z)=\arctan(z/z_{R})$ [36]. With the focus at $z_{f}^{-}$
we find that,

$\Delta\phi=2kr_{m}+\pi+\underbrace{2\left[\phi_{g}(z_{f})+\phi_{g}(r_{m}-z_{f})\right]}_{\pi}=2kr_{m}.$
(34)

That is, the contribution of the Gouy phase shift to the round-trip phase is
equal to exactly $\pi$ when the beam is retroreflected, such that the total
phase is fixed only by the round-trip distance. To ensure that the incident
beam is in-phase with its reflection at the centre, the mirror radius has to
be restricted to $kr_{m}=n\pi$, for integer $n$. This condition corresponds to
the maximum enhancement of scattered power from a trapped particle. As we
discussed in the main text, this condition is exactly the condition needed to
achieve suppression of backaction noise acting on a trapped particle. This
coincidental feature ensures that the recoil noise suppression condition is
matched when the particle is brought to the centre of the spherical mirror in
a standing-wave, formed by reflection from the mirror.

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