# Position measurement of a dipolar scatterer via self-homodyne detection

G. Cerchiari<EMAIL_ADDRESS>Corresponding author  Institut für
Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020
Innsbruck, Austria  L. Dania  Institut für Experimentalphysik, Universität
Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria  D. S. Bykov  Institut
für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020
Innsbruck, Austria  R. Blatt  Institut für Experimentalphysik, Universität
Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria  Institut für
Quantenoptik und Quanteninformation, Österreichische Akademie der
Wissenschaften, Technikerstrasse 21a, 6020 Innsbruck, Austria  T. Northup
Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25,
6020 Innsbruck, Austria

###### Abstract

We describe a technique to measure the position of a dipolar scatterer based
on self-homodyne detection of the scattered light. The method can
theoretically reach the Heisenberg limit, at which information gained about
the position is constrained only by the back-action of the scattered light.
The technique has applications in the fields of levitated optomechanics and
trapped ions and is generally applicable to the position determination of
confined light scatterers.

##  I Introduction

Quantum mechanics is our best predictive theory to describe the microscopic
world. According to this model, elementary particles are described by wave
excitations of fields Peskin and Schroeder (1995), and measurements of
physical quantities on these systems have probabilistic outcomes. This wave
theory strongly contrasts with our everyday experience, in which macroscopic
objects can be described as entities with deterministic measurable properties
using classical theory. At the interface between quantum and classical scales,
physicists challenge the two theories to search for new physics, for example,
to understand the role of gravity in the quantum-mechanical description
Aspelmeyer et al. (2014); Carlesso et al. (2019).

One promising approach in these investigations is to search for quantum-
mechanical properties of the motion of micron- or submicron-sized dielectric
particles levitated in optical, magnetic or electrodynamic traps Aspelmeyer et
al. (2012); Millen et al. (2020). The levitation ensures a high degree of
isolation in space and enables the study of the particles’ wave properties
stemming from the quantization of the confining potential Tebbenjohanns et al.
(2020); Delić et al. (2020); Magrini et al. (2020). In these systems, laser
light is typically used to measure and interact with the levitated particle’s
motion. The standard optical approach to observe single quanta of oscillatory
motion (phonons) is to detect a monochromatic light field scattered by the
particle Bullier et al. (2019); Ranjit et al. (2015); Gieseler et al. (2012).
The scattered field carries information about the location of emission but, at
the same time, perturbs the momentum of the particle via photon recoil Jain et
al. (2016). The trade-off between the information gained and the back action
sets the ultimate limit for detection: the Heisenberg limit. Here, by
Heisenberg limit, we mean that the product of the measurement imprecision in
the determination of the scatterer position and the measurement back action on
the scatterer’s momentum satisfies the lower bound of the Heisenberg
uncertainty relation Clerk et al. (2010). A further challenge is imposed by
the angular distribution of the scattered light, which, in the Rayleigh
approximation, is a dipole pattern that spreads the information over the full
solid angle van de Hulst (1981).

The observation of the scattered light over the entire solid angle is the key
to reaching the Heisenberg limit as proposed in Ref. Tebbenjohanns et al.
(2019a). In the Gedankenexperiment described therein, the following
geometrical configuration is proposed. The scattered field is interfered with
a second dipolar field having the same wavefronts and polarization, allowing
for the reconstruction of the scatterer’s position via homodyne detection. The
secondary dipole field (reference field) must be permanently located at a
reference position which defines the origin of the scatterer’s displacements.

This theoretical configuration poses the challenge of generating the dipolar
reference field, while simultaneously detecting light over the entire solid
angle. This optical problem has been approached in the past by locating the
scatterers in the focus of deep parabolic mirrors to collimate the dipolar
field into a plane wave having a Gaussian $\textrm{TEM}_{01*}$ profile, a so-
called doughnut mode Salakhutdinov et al. (2016); Salakhutdinov (2020). This
solution requires the parabolic mirror to cover a large fraction of the solid
angle around the scatterer to collimate the field and it may be difficult to
implement in experiments with limited optical access. In this work, we propose
that self-homodyne of the field emitted by the particle is a viable
implementation of the ideal configuration. In a symmetric realization of the
self-homodyne technique, the solid angle is split into two parts. One part is
occupied by a hemispherical mirror, the other by a hemispherical pixel
detector. The mirror generates a secondary image of the dipole, which
interferes at the detector with the primary scattered light. Similar
configurations have been explored in the past to obtain nanometric resolution
in microscopy applications Swan et al. (2003); Davis et al. (2007) and in the
context of half-cavity experiments to investigate the spontaneous emission of
a trapped atomic ion Bushev et al. (2006, 2013); Slodička et al. (2012a);
Cerchiari et al. (2020). The previous studies did not address the theoretical
limitations for motion detection or the possibility to extend the technique
beyond the field of trapped ions. Here, we prove that the self-homodyne method
is equivalent to the ideal configuration, and we present how it can be used to
obtain lower imprecision for position detection than obtained in state-of-the-
art experiments in the field of levitated optomechanics.

##  II Self-homodyne detection and the Heisenberg limit

A schematic drawing of the self-homodyne setup is presented in Fig. 1. The
mirror of radius $R_{s}$ and the detector of radius $R_{d}$ are concentric
around the origin. A point-like dipolar emitter of radiation with wavelength
$\lambda$ is located at $\bm{x}_{0}$ near the origin. This configuration is
referred to as a “half-cavity” Dorner and Zoller (2002). We are interested in
deducing the position $\bm{x}_{0}$ of the scatterer with respect to the origin
of the coordinate system by measuring the light intensity impinging on the
detector at different angles. For the discussion, we will refer to
$\bm{x}_{0}$ both as the position or the displacement to be detected.

We assume that the scattered light is reflected and detected in the far field.
This approximation corresponds to the condition under which the modulus of the
displacement $\lVert\bm{x}_{0}\rVert$ from the origin is much smaller than the
distance $R_{d}$ between the particle and the detector and much smaller than
the radius of curvature $R_{s}$ of the mirror. The far-field approximation
allows us to consider the dipolar scatterer as a source of spherical waves
with polarization orthogonal to the radial versor $\hat{\bm{n}}$, which
indicates a generic direction in space departing from the origin. Under these
conditions, we can model the light field’s propagation and interference by
using the Huygens–Fresnel integral in scalar form. Furthermore, the spherical
optical elements enable us to solve the problem without using the paraxial
approximation while disregarding the $1/R$ factor in the Green’s function of
the Huygens–Fresnel integral.

At the location $\bm{r}=\hat{\bm{n}}R_{d}$ on the detector, the expression for
the field’s amplitude is:

$E_{s}\left(\bm{r}\right)=E_{0}\exp{\left(i\frac{2\pi}{\lambda}\hat{\bm{n}}\cdot\bm{x}_{0}\right)}\;,$
(1)

where $E_{0}$ is a constant complex amplitude and “$\cdot$” denotes the scalar
product. The amplitude $E_{0}$ contains the re-scaling and phase pre-factor
$\exp\left(-i2\pi R_{d}/\lambda\right)/R_{d}$, which is constant across the
detector surface. For small displacement in any direction
($\lVert\bm{x}_{0}\rVert\ll R_{s}$), the mirror generates an image of the
emitter at the opposite side of the origin, at $\bm{x}_{1}=-\bm{x}_{0}$. The
polarization and wavefronts of the image are the same as for the primary
dipole. The field of the image has a constant phase shift corresponding to an
optical path of twice the radius $R_{s}$ of the mirror (see Appendix A). The
image field is phase-referenced to the scattered field and can be expressed at
the detector as:

$E_{i}\left(\bm{r}\right)=-\rho
E_{0}\exp{\left(-i\frac{2\pi}{\lambda}\left(\hat{\bm{n}}\cdot\bm{x}_{0}+2R_{s}\right)\right)}\;,$
(2)

where $\rho$ is the reflection coefficient for the electric field at the
mirror. The minus sign in front accounts for the phase difference accumulated
at the reflection.

The intensity measured at $\bm{r}$ is:

$\begin{split}&I\left(\bm{r}\right)\propto\lvert
E_{s}\left(\bm{r}\right)+E_{i}\left(\bm{r}\right)\rvert^{2}\\\ &\propto
1+\rho^{2}-2\rho\cos{\left(\frac{4\pi}{\lambda}\left(\hat{\bm{n}}\cdot\bm{x}_{0}+R_{s}\right)\right)}\;.\end{split}$
(3)

We see that the detected light intensity at any position $\bm{r}$ is
sinusoidal as a function of the position $\bm{x}_{0}$. To detect the position
via an intensity modulation univocally, the modulus of the displacement must
be sub-wavelength: $\lVert\bm{x}_{0}\rVert<\lambda$. The expression
$\lVert\bm{x}_{0}\rVert\ll\lambda$ is normally valid in levitated
optomechanics experiments aiming to observe the oscillations of a scatterer
close to the motional ground state Millen et al. (2020); Chang et al. (2010).

We obtain the maximum sensitivity on the slope of the interference pattern,
i.e., when the equality $4\pi R_{s}/\lambda=2\pi m\pm\pi/2$ holds, with $m$ a
natural number. By simplifying Eq. 3 to first order in
$\lVert\bm{x}_{0}\rVert/\lambda$ under these conditions, we obtain:

$I\left(\bm{r}\right)\propto
1+\rho^{2}\pm\rho\frac{8\pi}{\lambda}\left(\hat{\bm{n}}\cdot\bm{x}_{0}\right)\;.$
(4)

In this formula, either sign in the last term leads to the same sensitivity.
For simplicity, we select the positive sign.

To impose a stable phase of interference for maximal motion sensitivity, the
experimenter can take advantage of the mass difference between the scatterer
and the mirror. As already demonstrated in Refs. Bushev et al. (2006);
Cerchiari et al. (2020), the oscillations of the particle are usually much
faster than the drifts of the mirror position. By averaging the detected light
over several periods of the particle’s oscillation, one can generate a
feedback signal to compensate for the mirror movements without compromising
the information obtained about the particle’s motion.

Figure 1: Schematic representation of the self-homodyne setup. a) Three-
dimensional overview represented in both the Cartesian coordinate system and
the spherical coordinate system used in this article. b) Bi-dimensional
section. A spherical mirror of radius $R_{s}$ images the light emitted by a
dipolar scatterer located at $\bm{x}_{0}$ at the point
$\bm{x}_{1}=-\bm{x}_{0}$. An optical path for the spherical wavefronts is
drawn departing from $\bm{x}_{0}$, reflected at $\bm{x}_{r}$, and imaged at
$\bm{x}_{1}$. The ray travels an optical path given by $t_{0}+t_{1}=2R_{s}$
along the directions defined by the unitary vectors (versors)
$\hat{\bm{v}}_{0}$ and $\hat{\bm{v}}_{1}$. The light emitted by the scatterer
interferes with the scatterer’s image on the detector of radius $R_{d}$ at
position $\bm{r}$. The detector collects all the light within the polar angle
interval $\left[0,\theta_{D}\right]$ from the optical axis bisecting the
mirror and the detector. The numerical aperture of the setup is defined as
$\textrm{NA}=\sin{\theta_{D}}$.

Equation 4 can be compared to the intensity measured for the ideal homodyne
scheme, adapted from Ref. Tebbenjohanns et al. (2019a):

$I_{\textrm{ideal}}\left(\bm{r}\right)\propto
1+\gamma^{2}+\gamma\frac{4\pi}{\lambda}\left(\hat{\bm{n}}\cdot\bm{x}_{0}\right)\;.$
(5)

Here, $\gamma$ is the proportionality coefficient between the reference
electric field and the scattered electric field. We see that $\gamma$
substitutes for the reflection coefficient $\rho$.

The experimenter’s control over the power of the reference field is manifested
in the coefficient $\gamma$. In state-of-the-art experiments adopting homodyne
detection schemes Tebbenjohanns et al. (2021); Magrini et al. (2020), this
control allows one to work in a regime in which $\gamma^{2}+1\sim\gamma^{2}$,
corresponding to adjusting the intensity of the reference field to be much
stronger than that of the direct field. However, this regime is not accessible
for the self-homodyne method because $\rho$ is bounded by the condition
$\rho\leq 1$. Furthermore, we note that the position-dependent terms in Eqs. 4
and 5 differ by a factor of two. The discrepancy stems from the fact that, in
contrast to the ideal detection configuration, the reference field source,
which is the image created by the mirror, is not fixed to the center of the
curvature of the mirror. The displacement of the particle causes a shift of
the image in the opposite direction relative to the center of the curvature,
thus increasing the sensitivity by a factor of two. Also in contrast to the
ideal case, in the self-homodyne method the half solid angle occupied by the
mirror cannot be used for detection and the interference is observed only on
the opposite half of the solid angle. In the remainder of this section, we
calculate the limit for position detection to show that the self-homodyne
setup can be considered equivalent to the ideal configuration notwithstanding
these differences. In Appendix B, we expand our discussion of the self-
homodyne process to highlight the directionality of the interference effect
and its connection to the spontaneous emission of an atomic scatterer-mirror
system for a deeper understanding of the physical phenomenon.

A dipolar scatterer is not a perfect source of spherical waves. In the far-
field approximation, we can correct this model by retaining spherical
wavefronts and by modifying the angular distribution of the radiated light
power. The differential power $dp_{\textrm{dip}}$ per unit solid angle
$d\Omega$ radiated by a dipole depends on the polarization
$\hat{\bm{\epsilon}}$ of the driving light beam according to the far-field
pattern van de Hulst (1981):

$dp_{\textrm{dip}}=\frac{3}{8\pi}P\left(1-\lvert\hat{\bm{n}}\cdot\hat{\bm{\epsilon}}\rvert^{2}\right)d\Omega\;,$
(6)

where $P$ is the total power radiated by an electric field with amplitude
$E_{0}$ (see Eqs. 1 and 2). Since the mirror is in the far field, the
polarization of the image is the same as the polarization of the direct
dipole. Combining the interference term and the angular distribution of the
scattered radiation, we obtain the differential detected power
$dp_{\textrm{det}}$:

$dp_{\textrm{det}}=\left(1+\rho^{2}+\rho\frac{8\pi}{\lambda}\left(\hat{\bm{n}}\cdot\bm{x}_{0}\right)\right)dp_{\textrm{dip}}\;.$
(7)

Any displacement $\bm{x}_{0}$ causes an intensity modulation proportional to

$d\beta=\frac{8\pi\rho}{\lambda}\left(\hat{\bm{n}}\cdot\hat{\bm{x}}_{0}\right)dp_{\textrm{dip}}\;,$
(8)

where $\hat{\bm{x}}_{0}$ is the unitary vector
$\hat{\bm{x}}_{0}=\bm{x}_{0}/\lVert\bm{x}_{0}\rVert$. The sensitivity to
position per unit solid angle, $d\beta$, combines the modulations due to the
homodyning field and the distribution of the scattered light. For small
absolute displacements $\lVert\bm{x}_{0}\rVert\ll\lambda$, the power
$dp_{\textrm{det}}$ described by Eq. 7 is dominated by the term $\rho^{2}+1$.
This constant bias in the signal determines the power spectral density of the
fluctuations $d\sigma$ via the shot noise:

$d\sigma=\frac{\hbar c}{\lambda}\left(R+1\right)dp_{\textrm{dip}}\;,$ (9)

where $R=\rho^{2}$ is the reflection coefficient of the mirror for light
intensity. Equation 9 is valid in experiments in which
$\lVert\bm{x}_{0}\rVert\ll\lambda$, as demonstrated in Ref. Cerchiari et al.
(2020). The imprecision in detecting the position $\bm{x}_{0}$ under a solid
angle $d\Omega$ is defined as Tebbenjohanns et al. (2019a):

$s\left(\hat{\bm{x}}_{0}\right)=\frac{d\sigma}{d\beta^{2}}=\frac{\hbar\lambda
c\left(R+1\right)}{64\pi^{2}R\left(\hat{\bm{n}}\cdot\hat{\bm{x}}_{0}\right)^{2}dp_{\textrm{dip}}}\;.$
(10)

The imprecision captures the interplay between the modulation of the homodyne
interference and the radiated power. In an optomechanics experiment, the
imprecision corresponds to the noise floor of the detected motional spectrum.
Decreasing the imprecision increases the visibility of the thermal mechanical
peak of the particle oscillator above the detection noise Aspelmeyer et al.
(2014). In the context of feedback cooling Genes et al. (2008), a lower
imprecision in the detection of motion is beneficial for reducing the thermal
phonon occupation number. The total imprecision over a detector area $D$ is
defined as Tebbenjohanns et al. (2019a):

$S_{D}\left(\hat{\bm{x}}_{0}\right)=\left(\int_{D}s\left(\hat{\bm{x}}_{0}\right)^{-1}\right)^{-1}\;.$
(11)

Repeating the calculation from Eq. 7 to Eq. 10 for a dipole scatterer measured
with the ideal homodyne configuration, we obtain:

$\frac{s^{\textrm{{ideal}}}\left(\hat{\bm{x}}_{0}\right)}{s\left(\hat{\bm{x}}_{0}\right)}=\frac{4R}{1+R}\;.$
(12)

Due to the disparity in the solid angle available for detection, the ratio of
total imprecision in the two schemes is only half of the value in Eq. 12:

$\frac{S^{\textrm{{ideal}}}_{\Omega}\left(\hat{\bm{x}}_{0}\right)}{S_{\Omega/2}\left(\hat{\bm{x}}_{0}\right)}=\frac{2R}{1+R}\;.$
(13)

Since mirrors with reflectivity $>0.99$ are available on the market, we will
assume $R\sim 1$ in the remainder of this work. This result holds for any
orientation of the half-cavity and shows that for very high mirror
reflectivity, the half-cavity scheme has the same total imprecision as the
ideal configuration in any displacement direction of the dipolar emitter.
Thus, we see that the half-cavity setup is a practical implementation of the
ideal theoretical configuration and, like the ideal configuration, is limited
only by the back action. Note that displacements along different axes are not
necessarily detected with the same imprecision. For example, for linearly
polarized dipoles, the lowest imprecision $S_{\textrm{min}}=5\hbar
c\lambda/(32\pi^{2}P)$ is achieved for scatterer displacements orthogonal to
the polarization direction, which can be compared with the maximal imprecision
$S_{\textrm{max}}=5\hbar c\lambda/(16\pi^{2}P)$ corresponding to displacements
along the polarization axis.

##  III Setup with limited NA and a quadrant detector

In typical levitated optomechanics experiments Millen et al. (2020), the
scatterer is a SiO2 nanoparticle that oscillates in a harmonic potential
generated by optical tweezers or a Paul trap. Each particle is composed of
$\sim 10^{9}$ atoms and has a radius of $\sim 100$ nm. The trapping
frequencies of such particles are $10^{2}-10^{5}$ Hz, and the amplitudes of
single quanta of oscillation of the particles are $\lVert\bm{x}_{0}\rVert\sim
10-100$ pm. These excursions are much smaller than the wavelength $\lambda\sim
500-1500$ nm of the laser field that illuminates the scatterer, validating the
assumption we made in the previous section.

The particle’s position is observed via homodyne detection by locating the
scatterer between a pair of confocal lenses Gittes and Schmidt (1998); Dawson
and Bateman (2019). The first lens focuses an illuminating Gaussian beam on
the particle, and the second lens collimates the scattered light and the
incident beam, which then impinge on a detector. In this way, the Gaussian
beam is the source for the scattered light and can be used as the reference
field for homodyning. This technique is referred to as the “forward” detection
method. Alternatively, the back-scattered light from the particle can be used
to infer the scatterer’s position Tebbenjohanns et al. (2020). In this case,
the returning light is not lost but collimated by the lens that focuses the
Gaussian beam onto the nanoparticle. The returning field is interfered with a
reference beam for the analysis of the scatterer’s position. This technique is
called the “backward” detection method.

We present in Fig. 2 a schematic overview of forward, backward and self-
homodyne setups to illustrate the configurations that are the basis for the
comparison. The forward, backward and self-homodyne detection methods are not
mutually exclusive. For example, the forward and backward methods can be used
together, as depicted in Fig. 2(a), with the optical axis oriented along the
$y$-axis of Fig. 2(b). In this configuration the three techniques can be used
simultaneously on the same setup. In our comparison of these techniques, we
will consider experiments implementing only one technique at a time. This is
equivalent to restricting the measurements to having a single detector.

The self-homodyne approach has three advantages compared to the forward and
backward setups. First, the mirror generates a secondary dipole image that
matches the wavefronts of the primary field, while the Gaussian illuminating
beam of the forward technique does not match the wavefronts of the light
emitted by the dipolar scatterer. Second, in the self-homodyne method, the
coupled system of the mirror and the scatterer emits light in only half of the
solid angle. In contrast, both the forward and backward methods may analyze
only half of the emitted light scattered by the particle over the entire solid
angle. Third, unlike in the forward and backward detection schemes, the
optical axis of the half-cavity may have arbitrary orientation with respect to
the illumination beam.

In this section, we will show how to make use of the three advantages of self-
homodyning to obtain a lower imprecision in the measurement of the scatterer’s
position than is achievable by means of the forward and backward techniques.
In the discussion, we compare the self-homodyne method with the state-of-the-
art forward and backward techniques, considering a possible implementation
adopting limited-NA optical elements and a quadrant photodetector.

Figure 2: Comparison of the forward and backward techniques with the self-
homodyne method. (a) Forward and backward techniques. A Gaussian beam
propagating along the $z$-axis is focused on the nanoparticle. The scattered
light from the particle is detected in the forward and backward directions on
quadrant photodetectors (QPD) to measure the scatterer’s position. (b) Self-
homodyne technique. The half-cavity axis is aligned with the $z$-axis, and the
particle is illuminated by a beam propagating along the y-axis. In the main
text, we compare the techniques under the assumption that the polarization of
the incident light field is aligned with the $x$-axis.

Similarly to the forward and backward methods, the self-homodyne method can be
implemented by locating the scatterer between a pair of confocal lenses in
order to use a flat mirror and a flat detector, as demonstrated in Refs.
Slodička et al. (2012b); Cerchiari et al. (2020). One lens collimates the
scattered radiation on a flat mirror for reflection in order to generate the
image dipole. The second lens focuses the scattered light and the reference
light on the detector. This configuration maps the far-field interference
between the primary field and the reflected field onto the detector, a viable
solution to reproduce the far-field condition while utilizing a millimeter-
size flat detector. In fact, to our knowledge, hemispherical pixel detectors
are not yet available. The lenses introduce a cut over the solid angle, which
can be modeled by reducing the effective NA of the optical spherical
components, as already suggested in Fig.1.

Three directions describe the setup: the versor $\hat{\bm{z}}$ pointing from
the mirror to the detector of the half-cavity system, the displacement of the
scatterer $\hat{\bm{x}}_{0}$ and the polarization $\hat{\epsilon}$ of the
illuminating beam. For the description of directions and positions in space,
we will make use of angular and Cartesian notation according to the map
$(x,y,z)=r(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$ as
depicted in Fig. 1(a). We set the axis of the half-cavity along the $z$-axis
so that the detector $D$ is located between the polar angles
$0<\theta<\theta_{D}$ at any azimuth $0<\phi<2\pi$. The direction
$\hat{\bm{x}}_{0}$ is indicated by the polar angle $\theta_{0}$ and the
azimuthal angle $\phi_{0}$, and the polarization $\hat{\bm{\epsilon}}$ by the
angles $\theta_{\epsilon}$ and $\phi_{\epsilon}$. This configuration can be
interpreted in terms of numerical aperture via the expression
$\textrm{NA}=\sin{\theta_{D}}$. To evaluate our hypotheses and prepare the
results reported here, we have written a script in Wolfram Mathematica v. 11.3
Cerchiari et al. (2021) that calculates the imprecision for arbitrary angles
$\theta_{0}$, $\phi_{0}$, $\theta_{\epsilon}$, $\phi_{\epsilon}$ and
$\theta_{D}$. The solution is cumbersome but useful to evaluate specific
experimental implementations where different constraints are present or
particular optima of operation are desired. Here, we have decided to focus on
the configuration that obtains minimum differential imprecision for a
symmetric lens setup, assuming linearly polarized light. In general, a
superposition of circular or linear polarizations can be used to illuminate
the nanoparticle. However, to minimize the imprecision, linear polarization is
preferred because the distribution of scattered light is less homogenous over
the solid angle, thus allowing higher intensity to be directed onto a limited-
NA detector. In addition, circularly polarized light provides a torque,
causing uncontrolled spinning of the levitated nanoparticles Kane (2010);
Reimann et al. (2018). We include the discussion of arbitrary polarization to
Appendix C because it is relevant for nonlinear dipolar scatterers such as
isolated atoms or molecules.

The configuration with which the minimum differential imprecision $s$ is
obtained is found by minimizing Eq. 10. We note that the minimization of $s$
with respect to the displacement direction $\hat{\bm{x}}_{0}$ is independent
from the minimization over the light polarization $\hat{\bm{\epsilon}}$. The
lowest differential imprecision is found for $\theta_{\epsilon}=\pi/2$ and
$\phi_{\epsilon}=0$, corresponding to polarization along the $x$-axis, and for
$\theta_{0}=0$, corresponding to displacement along the $z$-axis. The
propagation direction of the illuminating beam is fixed in the $y$z$-$plane by
the minimization of $s$. For simplicity, we set the beam orientation to be
along the y-axis. In this configuration, the total imprecision as a function
of the detector aperture $\theta_{D}$ is

$S_{l}=\frac{20\hbar
c\lambda}{P\pi^{2}\left(128-90\cos{\left(\theta_{D}\right)}-35\cos{\left(3\theta_{D}\right)}-3\cos{\left(5\theta_{D}\right)}\right)}\;.$
(14)

In state-of-the-art experiments, the scattered and reference fields are
collimated by a lens on the small detection area of a single detector Bushev
et al. (2006) or a quadrant photodiode Tebbenjohanns et al. (2019b); Dania et
al. (2021). With the help of these detectors, the particle position is
reconstructed by averaging the interference between the scattered and
reference fields over large sections of the solid angle which are imaged on
the sensitive pixels by the lens. In self-homodyne detection, the intensity
common to all quadrants is influenced by the proximity of the scatterer to the
detector, and the modulations between pairs of different pixels are caused by
displacements of the scatterer in directions orthogonal to the light
propagation Gittes and Schmidt (1998). The limited angular resolution of the
pixels prevents a differential inverse-square weighting of the imprecision to
calculate the scatterer’s position from the measured intensity variation, as
suggested in Eq. 11. With a discrete pixel detector, the imprecision must
instead be calculated as the weighted mean over the available detector area
rather than by an integral. This correction has already been presented in Ref.
Tebbenjohanns et al. (2019a) to model the forward and backward techniques, and
we apply a similar analysis for the self-homodyne method.

Figure 3: Comparison of detection efficiency between the half-cavity setup
(self) and the forward (fw) and backward (bw) detection schemes as a function
of the NA. The labels $x$, $y$ and $z$ refer to specific motional axes of the
scatterer. The forward and backward imprecisions are plotted with dashed
colored lines (adapted from Tebbenjohanns et al. (2019a)), and the self-
homodyne imprecisions with solid colored lines. The lines fw and bw for the
$y$ and $x$ axes overlap. Along the $x$ and $z$ axes, the data obtained with
the forward scheme are equal to those obtained with the backward scheme. The
efficiency has been normalized for the maximum efficiency at the Heisenberg
limit $S^{-1}_{\textrm{min}}=32\pi^{2}P/(5\hbar c\lambda)$. The efficiency for
$z$-displacements for the forward method is zero in the selected
configuration, and it is not reported in the plot.

We finally have the tools in hand to compare the forward, backward and self-
homodyne methods for experimental application. In this comparison, we align
the optical axis of the forward and backward detection along the $z$-axis and
the light is linearly polarized along the $x$-axis. For each technique, we
calculate the detection efficiency as the inverse of the imprecision in
detecting the scatterer displacements along the $x$, $y$, and $z$ axes, and we
normalize it with the maximum efficiency obtained at the Heisenberg limit
achievable in the system: $S^{-1}_{\textrm{min}}=32\pi^{2}P/(5\hbar
c\lambda)$. The efficiencies are calculated as being measured by an ideal
quadrant photodetector with unit quantum efficiency. The latter is oriented
with its quadrants being divided by the $x$ and $y$ axes.

Figure 3 shows the detection efficiencies of the three techniques as a
function of the optical NA of the confocal lenses. The horizontal lines mark
the Heisenberg limits for displacements in the $y$z$-$plane and along the
$x$-axis. Comparing data for a given axis, the half-cavity shows higher
efficiency than the forward or backward detection. We attribute the
improvement to both the concentration of the information in only half of the
solid angle and the match of polarization and wavefronts between the reference
and the scattered light fields. In Fig. 3 we also plot the efficiency
corresponding to the ideal pixel detector described by Eq. 14. A comparison of
the ideal detector with the discrete pixel detector, in both cases used in a
self-homodyne measurement of displacement along the $z$-axis, reveals that the
imprecisions corresponding to the two detectors are equivalent up to
$\textrm{NA}\sim 0.8$. Thus, we see that the adoption of a quadrant detector
is not detrimental for practical implementations aiming to reach the lowest
possible imprecision at $\textrm{NA}<0.8$. Further comparisons between the
detection efficiencies for displacements along the principal axes acheivable
and with a quadrant detector with an ideal detector are presented in Appendix
D. Finally, we mention that the freedom of choice in the orientation of the
half-cavity optical axis can further be exploited beyond the current
configuration, for example, to achieve the imprecision that is shown in Fig. 3
for the half-cavity setup along the $z$-axis along any direction in the
$y$z$-$plane.

##  IV conclusion

The self-homodyne detection method implements an ideal setup that is capable
of detecting the motion of a dipolar scatterer at the Heisenberg limit. The
signal is obtained via self-referencing the scattered light by using a
spherical mirror or a combination of a lens and a flat mirror to reflect the
light back in the direction of the scatterer. We have analyzed the setup at
limited NA for a possible realization that combines two confocal lenses, a
flat mirror and a quadrant detector. We have discussed the orientation of the
setup that would minimize the imprecision in detecting the scatterer position
and that can achieve higher efficiency than the state-of-the-art forward and
backward detection methods. Compared to the forward and backward schemes, the
mirror generates an ideal reference field and allows for arbitrary orientation
of the mirror-detector axis for optimal reconstruction of the scatterer’s
position by adopting a single one-sided detector. Compared to a deep parabolic
mirror optical setup, our method offers a scalable approach for detection that
can be adapted in experiments with limited optical access.

In homodyne experiments, the reference may be much stronger than the field
being studied, which leads to an amplification of the interference term over
the shot noise of a real detector. This is not true in the self-homodyne
method, because we expect the primary and the reflected fields to be of
similar intensity. Thus, the self-homodyne method requires the adoption of a
high quantum efficiency detector. For example, quantum efficiencies larger
than 80% are achievable with avalanche photodiodes in the near-infrared range
and even higher efficiencies are possible with superconducting detectors You
(2020).

In particular, the arbitrary orientation of the half-cavity with respect to
the illumination beam constitutes an appealing option for position detection
in complex experimental setups. For example, the self-homodyne apparatus could
be mounted orthogonally to an optical cavity or to a forward-backward
apparatus. Furthermore, the high sensitivity of the half-cavity setup for the
detection of motion at low NAs could be beneficial in levitated optomechanics
systems with limited optical access, such as electromagnetic or magneto-
gravitational traps.

Acknowledgements. We would like to thank Gabriel Araneda for detailed
discussions and suggestions regarding the experimental setup. We thank Lukas
Novotny for suggesting us the connection between our method and previous
microscopy applications and M. Sondermann for indicating us the studies of
dipolar fields in deep parabolic mirrors. G.C. would like to thank Ruggero
Caravita for inspiring discussions about isolated systems and about the
Wigner-Eckart theorem.

This work has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
801110 and the Austrian Federal Ministry of Education, Science and Research
(BMBWF). It reflects only the author’s view, the EU Agency is not responsible
for any use that may be made of the information it contains. This work was
also supported by the Austrian Science Fund (FWF) Project No. Y951. D.B.
acknowledges funding through the ESQ Discovery grant “Sympathetic detection
and cooling of nanoparticles levitated in a Paul trap” of the Austrian Academy
of Sciences. This work has received support by the Institut für
Quanteninformation GmbH.

##  V Appendix A: Position of the image

In this appendix, we derive that an hemispherical mirror produces a secondary
image of a point-like spherical wave emitter on the opposite side of its
curvature center. We remind here that the displacement of the emitter from the
origin of the half-cavity is small enough to describe the light field
propagation by using the Huygens–Fresnel integral. Under this assumption we
can model the dipolar scatterer as an emitter of spherical waves. We will show
that the mirror generates an image in $\bm{x}_{1}=-\bm{x}_{0}$ by
demonstrating that the optical path between $\bm{x}_{1}$ and $\bm{x}_{0}$ is
the same for any part of the outgoing spherical wavefronts which intersect the
mirror.

With reference to Fig. 1, from $\bm{x}_{0}$, we can follow the wavefront along
the generic direction described by the versor $\hat{\bm{v}}_{0}$
($\lVert\hat{\bm{v}}_{0}\rVert=1$) until encountering the surface of the
sphere of radius $R_{s}$ at $\bm{x}_{r}$. There, the wavefront is locally
reflected and the light arrives at $\bm{x}_{1}$ following the direction of the
versor $\hat{\bm{v}}_{1}$. The optical path length between $\bm{x}_{0}$ and
$\bm{x}_{r}$ is $t_{0}$ and between $\bm{x}_{r}$ and $\bm{x}_{1}$ is $t_{1}$.
For any generic direction $\hat{\bm{v}}_{0}$ and length $t_{1}$ the point
$\bm{x}_{1}$ can be found by solving the following system of equations:

$\displaystyle\bm{x}_{r}$ $\displaystyle=\bm{x}_{0}+\hat{\bm{v}}_{0}t_{0}$
(A.1) $\displaystyle\lVert\bm{x}_{r}\rVert^{2}$ $\displaystyle=R_{s}$ (A.2)
$\displaystyle\bm{x}_{1}$ $\displaystyle=\bm{x}_{r}+\hat{\bm{v}}_{1}t_{1}$
(A.3) $\displaystyle\hat{\bm{v}_{1}}$
$\displaystyle=\hat{\bm{v}}_{0}-2\left(\hat{\bm{v}}_{0}\cdot\hat{\bm{x}}_{r}\right)\hat{\bm{x}}_{r}\;;$
(A.4)

where $\hat{\bm{x}}_{r}=\bm{x}_{r}/\lVert\bm{x}_{r}\rVert^{2}$.

The scalar product of Eq. A.1 with $\hat{\bm{v}}_{0}$ allows us to find the
following relation for $t_{0}$:

$t_{0}=-\left(\hat{\bm{v}}_{0}\cdot\bm{x}_{0}\right)+R_{s}\left(\hat{\bm{v}}_{0}\cdot\hat{\bm{x}}_{r}\right)\;.$
(A.5)

Then, we consider the square norm of Eq. A.1:

$R^{2}_{s}=\lVert\bm{x}_{0}\rVert^{2}+t^{2}_{0}+2\left(\hat{\bm{v}}_{0}\cdot\hat{\bm{x}}_{0}\right)t_{0}\;.$
(A.6)

By combining Eq. A.5 and Eq. A.6, we obtain

$\left(\hat{\bm{v}}_{0}\cdot\hat{\bm{x}}_{r}\right)=\sqrt{1-\frac{\lVert\bm{x}_{0}\rVert^{2}-\left(\hat{\bm{v}}_{0}\cdot\bm{x}_{0}\right)^{2}}{R_{s}^{2}}}\;.$
(A.7)

This relation reduces to
$\left(\hat{\bm{v}}_{0}\cdot\hat{\bm{x}}_{r}\right)\sim 1$ to first order in
$\lVert\bm{x}_{0}\rVert/R_{s}$. In this approximation, we find that Eq. A.4
reduces to $\hat{\bm{v}}_{0}\sim\hat{\bm{x}}_{r}\sim-\hat{\bm{v}}_{1}$.

With this simplification, we are now ready to evaluate $\bm{x}_{0}+\bm{x}_{1}$
by combining Eq. A.1, Eq. A.3:

$\bm{x}_{0}+\bm{x}_{1}\sim\hat{\bm{x}}_{r}\left(2R_{s}-t_{0}-t_{1}\right)\;.$
(A.8)

We see that this expression is null if $t_{0}+t_{1}=2R_{s}$. This solution is
describing an image formed at $\bm{x}_{1}=-\bm{x}_{0}$ because it is valid for
any versor $\hat{\bm{v}}_{0}$. The imaged is formed because $\bm{x}_{1}$ is
the point in which the wavefronts interfere with the same phase delay. Thus, a
point-like light source located near the origin is imaged on the opposite side
of the origin with a constant phase delay proportional to twice the radius of
the hemisphere.

##  VI Appendix B: Model description

In the self-homodyne method, the light interference of the directly scattered
and image fields can be interpreted as a variation of the spontaneous emission
rate of the dipolar emitter. The variation of spontaneous emission of the
dipolar emitter in half-cavities has already been described in the literature
for a perfectly conducting mirror ($\rho=1$) by quantizing the electromagnetic
field modes in the presence of a spherical mirror Hétet et al. (2010). In this
section, we will arrive at the same results by taking into account the
interference between the direct field and the reflected field, and we will
extend the results of the previous calculation to arbitrary mirror
reflectivity.

To calculate the variation of the emission of the dipolar scatterer, we
compare the emitted radiation in the presence of the mirror with the emitted
radiation in free space. In the presence of the mirror, we divide the solid
angle in three regions:

  * •

$M$, the angular domain of the mirror,

  * •

$\overline{M}$, the symmetric domain to $M$ obtained by reflection through the
origin,

  * •

$F=4\pi-M-\overline{M}$, the remaining domain.

The ratio of the modified scattering rate $\gamma$ to the free space
scattering rate $\gamma_{0}$ can be calculated as the ratio of emitted powers
in the configurations with and without mirror. The ratio references the light
exiting at all solid angles from the scatterer-mirror composite system to the
light emitted by the scatterer in free space. The expression of the ratio is:

$\frac{\gamma}{\gamma_{0}}=\frac{P\left(M\right)+P\left(\overline{M}\right)+P\left(F\right)}{P}\;,$
(B.1)

where $P(D)$ denotes the power radiated in the solid angle $D$ in presence of
the mirror and $P$ has already been introduced in Eq. 6. The radiated powers
$P(D)$ are

$\displaystyle P\left(F\right)$ $\displaystyle=\int_{F}dp_{\textrm{dip}}\;,$
(B.2) $\displaystyle P\left(M\right)$
$\displaystyle=\int_{M}Tdp_{\textrm{dip}}\;,$ (B.3) $\displaystyle
P\left(\overline{M}\right)$
$\displaystyle=\int_{\overline{M}}\left(1+\rho^{2}-2\rho\cos{\left(\omega\alpha\left(\hat{\bm{n}}\right)\right)}\right)dp_{\textrm{dip}}\;,$
(B.4)

where $T$ is the transmission coefficient for light intensity. For brevity, we
adopt
$\alpha\left(\hat{\bm{n}}\right)=\left(\frac{2}{c}\left(\hat{\bm{n}}\cdot\bm{x}_{0}+R_{s}\right)\right)$
and $\omega=2\pi c/\lambda$. Here, $\rho$ and $T$ can be generic functions of
the direction $\hat{\bm{n}}$, but in any direction,
$\rho\left(\hat{\bm{n}}\right)^{2}+T\left(\hat{\bm{n}}\right)=1$, that is, we
assume the mirror is lossless. Since for any direction in $M$ there is an
opposite in $\overline{M}$, we arrive at:

$\displaystyle\frac{\gamma}{\gamma_{0}}=1-\frac{3}{4\pi}\int_{\overline{M}}\rho\left(-\hat{\bm{n}}\right)\left(1-\lvert\hat{\bm{\epsilon}}\cdot\hat{\bm{n}}\lvert^{2}\right)\cos{\left(\omega\alpha\left(\hat{\bm{n}}\right)\right)}d\Omega\;.$
(B.5)

In the formula, the minus sign in the argument of the $\rho$ function reminds
us of the correct reflectivity coefficient while integrating in the
$\overline{M}$ region. As already pointed out in Ref. Hétet et al. (2010),
this result indicates that the spontaneous emission of the dipolar emitter can
be completely suppressed or doubled in the center of curvature of a
hemispherical mirror. Considering the radiated power, we can interpret the
underlying mechanism of the self-homodyne scheme in the following way.
Assuming perfect reflection, each photon emitted by the scatterer is self-
referenced by the mirror, with the result of a position-dependent enhancement
or suppression of the spontaneous emission of the dipolar scatterer. Every
photon that contributes to the scatterer backaction can only be emitted by the
scatterer-mirror system into the open half of the solid angle, and no
information is lost on the side of the mirror.

The variation of spontaneous emission determines a modulation of the dipole
strength, causing a shift of the energy levels of the dipole. The shift can be
calculated from the variation of the scattering rate by using the Kramers-
Kronig relations. Applying the Kramers-Kronig formula to Eq. B.5 and
approximating the result for a small shift compared to the carrier frequency,
we obtain the formula for the energy shift $\Delta\omega$ already found in
Hétet et al. (2010):

$\frac{\Delta\omega}{\gamma_{0}}=-\frac{3}{8\pi}\int_{\overline{M}}\rho\left(-\hat{\bm{n}}\right)\left(1-\lvert\hat{\bm{\epsilon}}\cdot\hat{\bm{n}}\lvert^{2}\right)\sin{\left(\omega\alpha\left(\hat{\bm{n}}\right)\right)}d\Omega\;.$
(B.6)

Compared to the previous derivation of Eq B.5 and Eq. B.6 reported in Ref.
Hétet et al. (2010), our approach further identifies the coefficient $\rho$ as
the reflection coefficient for the electric field of the mirror.

##  VII Appendix C: Atomic dipole transition

The self-homodyne technique can be used to detect the position of an atom.
This was demonstrated experimentally for motion along the mirror-detector axis
in Refs. Cerchiari et al. (2020); Bushev et al. (2006, 2013); Rotter et al.
(2008). Typically, atoms are driven at a frequency close to resonance with a
transition between a ground state and an excited state in order to generate
light as fluorescence radiation. Under these conditions, the dipolar response
of the atom is nonlinear. Unlike a linear dipole, an atom re-emits radiation
with both linear and circular polarization regardless of the driving field.
Linear polarization corresponds to $\pi$ transitions in which the magnetic
quantum number $m$ is the same in the excited state and in the ground state.
Circular polarization corresponds to $\sigma$ transitions, i.e., the
transition is characterized by a change in quantum number $\Delta m=\pm 1$.

Figure C.1: Comparison of detection efficiencies for different dipolar
scatterer displacements. The labels $\pi$, $\sigma$ and unpol (unpolarized)
refer to the different angular distributions of emitted radiation as explained
in the text. Each curved is also labeled with $x$, $y$ and $z$ according to
the axis of displacement of the atom. Unpolarized $x$ and $y$ curves are
identical. The values are normalized by
$S^{-1}_{\textrm{min}}=32\pi^{2}P/(5\hbar c\lambda)$ as done in the main text.

The light emitted during both $\pi$ and $\sigma$ transitions can be used to
detect the position of the atom in a self-homodyne setup. Position detection
based on $\pi$ transitions is described in the main text. It is equivalent to
the detection based on a linear dipolar scatterer driven with linearly
polarized light. For compatibility with the result reported in the main text,
the quantization axis in an atomic experiment should be aligned along the
$x$-axis of Figs. 1 and 2 in the main text. In this configuration, in Eq. 6,
$\pi$ transitions correspond to light polarized along the axis
$\hat{\bm{\epsilon}}=\hat{\bm{x}}$ and $\sigma$ transitions to light polarized
along the axes combination described by:
$\hat{\bm{\epsilon}}=\left(\hat{\bm{y}}\pm i\hat{\bm{z}}\right)/\sqrt{2}$.
Moreover, the unpolarized decay of an atom, defined as the condition in which
we do not distinguish between the two transition types but integrate over the
possible outcomes, is obtained by the replacement
$\left(1-\lvert\hat{\bm{n}}\cdot\hat{\bm{\epsilon}}\rvert^{2}\right)\rightarrow
2/3$. The last formula can be found by applying the Wigner-Eckart theorem to a
generic dipole transition.

We present in Fig. C.1 the efficiencies for the unpolarized, $\pi$-polarized,
and $\sigma$-polarized fluorescence of an atom as a function of the system NA.
In the plot, we compare the detection efficiencies (inverse imprecisions) for
atom displacements along the principal axes of the reference system. In the
corresponding calculation, we consider an ideal pixel detector and not a
quadradant detector. We see that the detection efficiency depends on the
polarization of the scattered light, which is due to the different radiated
power distributions for the $\pi$ and $\sigma$ transitions. The displacement
of the atom can be detected with highest efficiency along the optical axis of
the half-cavity. In this case, the light emitted due to $\pi$ transitions
leads to a higher efficiency in the detection of the atom’s position than what
is achievable by observing $\sigma$-polarized fluorescence. In fact, the
distribution of the emitted power is more concentrated in the $yz$-plane in
the case of the $\pi$-polarized fluorescence, enabling the reconstruction of
the atom position with higher fidelity.

This difference between $\pi$ and $\sigma$ transitions can be tested in
experiments by selecting photons based on polarization. The asymmetry can be
measured by introducing an external magnetic field along the quantization axis
($x$-axis). If a polarizing beam splitter is used to separate $y$\- and
$x$-polarized fluorescence, the radiation emitted during $\pi$ and $\sigma$
transitions is separated and can be analyzed independently Araneda et al.
(2019). The relevant formulas to reproduce the curves presented in Fig. C.1
can be found by executing the script of Ref. Cerchiari et al. (2021).

##  VIII Appendix D: Quadrant detector

In the main text, we compare the imprecision in the reconstruction of the
scatterer position using the self-homodyne method with the imprecision
obtained by using the forward-backward detection techniques. The comparison
takes into account a discrete detector composed of four quadrants. In this
section, we describe in more detail the calculation with which we obtain the
imprecision of the self-homodyne technique as measured with a quadrant
photodetector (QPD), and how it deviates from the idealized model of a
differential pixel detector. In the discussion, we assume the same
configuration shown in Figs. 1 and 2, in which the QPD occupies the solid-
angle regions $Q$ contained in the polar-angle interval
$0\leq\theta\leq\theta_{D}$. The quadrants regions $Q_{n}$ split the $Q$
region in the azimuthal-angle intervals $(n-1)\pi/2\leq\phi<n\pi/2$, with
$n=\\{1,2,3,4\\}$. The total power impinging on the $n$-th quadrant is

$P^{Q_{n}}=\int_{Q_{n}}\big{(}1+\rho^{2}+\rho\frac{8\pi}{\lambda}(\hat{\bm{n}}\cdot\bm{x}_{0})\big{)}dp_{\textrm{dip}}\;.$
(D.1)

We reconstruct the particle’s displacement $\bm{x}_{0}=(x_{0},y_{0},z_{0})$ by
taking advantage of the symmetries of the integrand. The combinations of the
detected powers over the quadrants that we use to reconstruct the
displacements are Tebbenjohanns et al. (2019a); Gittes and Schmidt (1998):

$\begin{split}x_{0}&=\Big{(}(P^{Q_{1}}+P^{Q_{4}})-(P^{Q_{3}}+P^{Q_{3}})\Big{)}\frac{1}{4B(\frac{\pi}{2},0)},\\\
y_{0}&=\Big{(}(P^{Q_{1}}+P^{Q_{2}})-(P^{Q_{3}}+P^{Q_{4}})\Big{)}\frac{1}{4B(\frac{\pi}{2},\frac{\pi}{2})},\text{~{}and}\\\
z_{0}&=\Big{(}\sum_{n}P^{Q_{n}}-\int_{Q}(1+\rho^{2})dp_{\textrm{dip}}\Big{)}\frac{1}{4B(0,\text{any})}.\end{split}$
(D.2)

Here, $B(\theta_{0},\phi_{0})=\int_{Q_{n}}d\beta$ is the integral of the
sensitivity to displacements along $\hat{\bm{x}_{0}}$ defined in Eq. 8 and is
equal for all quadrants thanks to the detector symmetry. The imprecision
measured with the QPD is calculated as

$S_{QPD}(\hat{\bm{x}}_{0})=\frac{1}{16\,B(\theta_{0},\phi_{0})^{2}}\int_{Q}\sigma
d\Omega,$ (D.3)

where $\sigma$ is the spectral density of the power fluctuations defined in
Eq. 9.

Figure D.1: Comparison of the detection efficiencies of self-homodyne
detection along three axes with an ideal differential pixel detector, in
dashed lines, and using a quadrant photo detector, in solid lines, as a
function of the detector numerical aperture (NA). Values are normalized with
respect to the efficiency at the Heisenberg limit for displacements on the
$\phi_{0}=\pi/2$ plane: $S^{-1}_{\textrm{min}}=32\pi^{2}P/(5\hbar c\lambda)$.

Figure D.1 shows a comparison of $S^{-1}_{QPD}$, the detection efficiency of
the self-homodyne method obtained with the QPD (solid lines), and
$S^{-1}_{D}$, the detection efficiency obtained with the ideal pixel detector
(dashed lines), as a function of the detector numerical aperture
$\text{NA}=\sin(\theta_{D})$. Values are normalized by
$S^{-1}_{\textrm{min}}=32\pi^{2}P/(5\hbar c\lambda)$ as done in the main text.
The expression for the ideal efficiency $S^{-1}_{D}$ is obtained by
integrating Eq. 11 of the main text. We find that detection of the particle’s
motion along the $z$-axis with the QPD is equivalent to the ideal pixel
detector up to $\text{NA}\sim 0.8$, suggesting that differential pixel
weighting only becomes relevant in the limit $\theta_{D}\rightarrow\pi/2$.
Supporting this hypothesis, the imprecision in detecting motion along the $x$
and $y$ axes deviates from the ideal case already at lower NAs. The early NA
deviation of the two curves can be understood based on the angular
distribution of the interference pattern. For $x$ and $y$ displacements, the
intensity is mostly modulated at angles $\theta\sim\pi/2$ and modulated less
at angles $\theta\sim 0$ (Eq. 3), i.e., at polar angles for which the effect
of differential inverse weighting is more pronounced.

## References

  * Peskin and Schroeder (1995) M. E. Peskin and D. V. Schroeder, _An Introduction to Quantum Field Theory_ (Westview Press, 1995), reading, USA: Addison-Wesley (1995) 842 p, URL https://books.google.at/books?id=EVeNNcslvX0C.
  * Aspelmeyer et al. (2014) M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014), URL https://doi.org/10.1103/RevModPhys.86.1391.
  * Carlesso et al. (2019) M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, New J. Phys. 21, 093052 (2019), URL https://doi.org/10.1088/1367-2630/ab41c1.
  * Aspelmeyer et al. (2012) M. Aspelmeyer, P. Meystre, and K. Schwab, Physics Today 65, 29 (2012), URL https://doi.org/10.1063/PT.3.1640.
  * Millen et al. (2020) J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas, Rep. Prog. Phys. 83, 026401 (2020), URL https://doi.org/10.1088/1361-6633/ab6100.
  * Tebbenjohanns et al. (2020) F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, Phys. Rev. Lett. 124, 013603 (2020), URL https://doi.org/10.1103/PhysRevLett.124.013603.
  * Delić et al. (2020) U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, Science 367, 892 (2020), ISSN 0036-8075, URL https://doi.org/10.1126/science.aba3993.
  * Magrini et al. (2020) L. Magrini, P. Rosenzweig, C. Bach, A. Deutschmann-Olek, S. G. Hofer, S. Hong, N. Kiesel, A. Kugi, and M. Aspelmeyer, arXiv 2012.15188 (2020), URL https://arxiv.org/abs/2012.15188.
  * Bullier et al. (2019) N. P. Bullier, A. Pontin, and P. F. Barker, Review of Scientific Instruments 90, 093201 (2019), eprint https://doi.org/10.1063/1.5108807, URL https://doi.org/10.1063/1.5108807.
  * Ranjit et al. (2015) G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, Phys. Rev. A 91, 051805(R) (2015), URL https://doi.org/10.1103/PhysRevA.91.051805.
  * Gieseler et al. (2012) J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, Phys. Rev. Lett. 109, 103603 (2012), URL https://doi.org/10.1103/PhysRevLett.109.103603.
  * Jain et al. (2016) V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, Phys. Rev. Lett. 116, 243601 (2016), URL https://doi.org/10.1103/PhysRevLett.116.243601.
  * Clerk et al. (2010) A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010), URL https://doi.org/10.1103/RevModPhys.82.1155.
  * van de Hulst (1981) H. van de Hulst, _Light Scattering by Small Particles_ , Dover Books on Physics (Dover Publications, 1981), ISBN 9780486642284, URL https://books.google.at/books?id=PlHfPMVAFRcC.
  * Tebbenjohanns et al. (2019a) F. Tebbenjohanns, M. Frimmer, and L. Novotny, Phys. Rev. A 100, 043821 (2019a), URL https://doi.org/10.1103/PhysRevA.100.043821.
  * Salakhutdinov et al. (2016) V. Salakhutdinov, M. Sondermann, L. Carbone, E. Giacobino, A. Bramati, and G. Leuchs, Optica 3, 1181 (2016), URL http://www.doi.org/10.1364/OPTICA.3.001181.
  * Salakhutdinov (2020) V. Salakhutdinov, doctoralthesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) (2020), URL urn:nbn:de:bvb:29-opus4-137383.
  * Swan et al. (2003) A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, and M. S. Unlu, IEEE Journal of Selected Topics in Quantum Electronics 9, 294 (2003), URL http://doi.org/10.1109/JSTQE.2003.814191.
  * Davis et al. (2007) B. J. Davis, M. Dogan, B. B. Goldberg, W. C. Karl, M. S. Ünlü, and A. K. Swan, J. Opt. Soc. Am. A 24, 3762 (2007), URL http://doi.org/10.1364/JOSAA.24.003762.
  * Bushev et al. (2006) P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, Phys. Rev. Lett. 96, 043003 (2006), URL https://doi.org/10.1103/PhysRevLett.96.043003.
  * Bushev et al. (2013) P. Bushev, G. Hétet, L. Slodička, D. Rotter, M. A. Wilson, F. Schmidt-Kaler, J. Eschner, and R. Blatt, Phys. Rev. Lett. 110, 133602 (2013), URL https://doi.org/10.1103/PhysRevLett.110.133602.
  * Slodička et al. (2012a) L. Slodička, G. Hétet, N. Röck, S. Gerber, P. Schindler, M. Kumph, M. Hennrich, and R. Blatt, Phys. Rev. A 85, 043401 (2012a), URL https://doi.org/10.1103/PhysRevA.85.043401.
  * Cerchiari et al. (2020) G. Cerchiari, G. Araneda, L. Podhora, L. Slodička, Y. Colombe, and R. Blatt, arXiv 2009.14098 (2020), URL https://arxiv.org/abs/2009.14098.
  * Dorner and Zoller (2002) U. Dorner and P. Zoller, Phys. Rev. A 66, 023816 (2002), URL https://doi.org/10.1103/PhysRevA.66.023816.
  * Chang et al. (2010) D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, Proceedings of the National Academy of Sciences 107, 1005 (2010), ISSN 0027-8424, eprint https://www.pnas.org/content/107/3/1005.full.pdf, URL https://www.pnas.org/content/107/3/1005.
  * Tebbenjohanns et al. (2021) F. Tebbenjohanns, M. L. Mattana, M. Rossi, M. Frimmer, and L. Novotny, arXiv 2103.03853 (2021), eprint 2103.03853, URL https://arxiv.org/abs/2103.03853.
  * Genes et al. (2008) C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, Phys. Rev. A 77, 033804 (2008), URL https://doi.org/10.1103/PhysRevA.77.033804.
  * Gittes and Schmidt (1998) F. Gittes and C. F. Schmidt, Opt. Lett. 23, 7 (1998), URL https://doi.org/10.1364/OL.23.000007.
  * Dawson and Bateman (2019) C. Dawson and J. Bateman, J. Opt. Soc. Am. B 36, 1565 (2019), URL http://josab.osa.org/abstract.cfm?URI=josab-36-6-1565.
  * Slodička et al. (2012b) L. Slodička, G. Hétet, N. Röck, S. Gerber, P. Schindler, M. Kumph, M. Hennrich, and R. Blatt, Phys. Rev. A 85, 043401 (2012b), URL https://doi.org/10.1103/PhysRevA.85.043401.
  * Cerchiari et al. (2021) G. Cerchiari, L. Dania, and D. S. Bykov, _self-homodyne_ (2021), URL https://doi.org/10.5281/zenodo.4545692.
  * Kane (2010) B. E. Kane, Phys. Rev. B 82, 115441 (2010), URL https://doi.org/10.1103/PhysRevB.82.115441.
  * Reimann et al. (2018) R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, Phys. Rev. Lett. 121, 033602 (2018), URL https://doi.org/10.1103/PhysRevLett.121.033602.
  * Tebbenjohanns et al. (2019b) F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, and L. Novotny, Phys. Rev. Lett. 122, 223601 (2019b), URL https://doi.org/10.1103/PhysRevLett.122.223601.
  * Dania et al. (2021) L. Dania, D. S. Bykov, M. Knoll, P. Mestres, and T. E. Northup, Phys. Rev. Research 3, 013018 (2021), URL https://doi.org/10.1103/PhysRevResearch.3.013018.
  * You (2020) L. You, Nanophotonics 9, 2673 (2020), URL https://doi.org/10.1515/nanoph-2020-0186.
  * Hétet et al. (2010) G. Hétet, L. Slodička, A. Glätzle, M. Hennrich, and R. Blatt, Phys. Rev. A 82, 063812 (2010), URL https://doi.org/10.1103/PhysRevA.82.063812.
  * Rotter et al. (2008) D. Rotter, M. Mukherjee, F. Dubin, and R. Blatt, New Journal of Physics 10, 043011 (2008), URL https://doi.org/10.1088/1367-2630/10/4/043011.
  * Araneda et al. (2019) G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, Nat. Phys. 15, 17 (2019), URL https://doi.org/10.1038/s41567-018-0301-y.


d({\bm{U}}\otimes\mathds{1})|\upphi_{d}\rangle$, where ${\bm{U}}$ is a Haar-
random unitary operator on ${\mathbb{C}}^{d}$. So the vectors
$d|{\bm{\psi}}_{i}\rangle$ have zero mean. We can verify that
$\operatorname{{\mathbb{E}}}|{\bm{\xi}}_{n}\rangle\\!\langle{\bm{\xi}}_{n}|=\mathds{1}$,
so the vectors $d|{\bm{\psi}}_{i}\rangle$ are isotropic. We prove below that
the vector $|{\bm{\xi}}_{n}\rangle$ is sub-gaussian, with sub-gaussian norm at
most a universal constant $\kappa$. So the matrix
${\bm{R}}^{*}=d\sum_{i=1}^{n}|i\rangle\\!\langle\psi_{i}|$ satisfies the
conditions of Theorem 5.4, and we have that for some positive universal
constant $c_{3}$,

$\left\|{\bm{R}}\right\|=\left\|{\bm{R}}^{*}\right\|\quad>\quad\sqrt{n}+c_{3}\kappa^{2}(\sqrt{n}+t_{1})$

with probability at most $2\exp(-t_{1}^{2})$, for all $t_{1}\geq 0$. Let
$t_{1}$ be such that the right hand side above equals $\sqrt{nt}$, i.e.,

$t_{1}\quad\coloneqq\quad\frac{\sqrt{n}}{c_{3}\kappa^{2}}\left(\sqrt{t}-(1+c_{3}\kappa^{2})\right)\enspace.$

Let $c_{1}\coloneqq 4(1+c_{3}\kappa^{2})^{2}$. Whenever $t\geq c_{1}$, we see
that $t_{1}\geq\sqrt{nt}/2c_{3}\kappa^{2}\geq 0$. So

$\Pr\left(\left\|{\bm{R}}\right\|>\sqrt{nt}\right)\quad\leq\quad
2\exp\\!\big{(}-nt/4c_{3}^{2}\kappa^{4}\big{)}\enspace,$

and the theorem holds with $c_{2}\coloneqq 4c_{3}^{2}\kappa^{4}$.

It remains to prove that the sub-gaussian norm $\kappa$ of
$|{\bm{\xi}}_{n}\rangle$ is at most some universal constant. By Lemma 5.3, it
suffices to show that for any unit vector $|u\rangle\in{\mathbb{C}}^{n}$, the
random variable ${\bm{x}}\coloneqq\langle u|{\bm{\xi}}_{n}\rangle$ has sub-
gaussian tails: for a positive universal constant $\kappa_{1}$,

$\Pr(\left\lvert{\bm{x}}\right\rvert\geq t)\quad\leq\quad
2\exp\\!\big{(}-t^{2}/\kappa_{1}^{2}\big{)}\enspace,$ (5.6)

for all $t\geq 0$. We establish this by appealing to Theorem 5.5.

Since $|{\bm{\xi}}_{n}\rangle$ is isotropic,

$\operatorname{{\mathbb{E}}}\left\lvert{\bm{x}}\right\rvert^{2}\quad=\quad\langle
u|\;(\operatorname{{\mathbb{E}}}|{\bm{\xi}}_{n}\rangle\\!\langle{\bm{\xi}}_{n}|)\;|u\rangle\quad=\quad
1\enspace.$

So
$\operatorname{{\mathbb{E}}}\left\lvert{\bm{x}}\right\rvert\leq\big{(}\operatorname{{\mathbb{E}}}\left\lvert{\bm{x}}\right\rvert^{2}\big{)}^{1/2}\leq
1$.

Define the function $f:{\mathsf{U}}({\mathbb{C}}^{d})\rightarrow{\mathbb{C}}$
as $f(U)\coloneqq d\left\lvert\langle
u|(U\otimes\mathds{1})|\upphi_{d}\rangle\right\rvert$. Then
$\left\lvert{\bm{x}}\right\rvert=f({\bm{U}})$. To show that $f$ is Lipschitz,
consider $U,V\in{\mathsf{U}}({\mathbb{C}}^{d})$. Since $|u\rangle$ is a unit
vector and
$\langle\upphi_{d}|(W\otimes\mathds{1})|\upphi_{d}\rangle=\tfrac{1}{d}\operatorname{Tr}(W)$
we have

$\displaystyle\left\lvert f(U)-f(V)\right\rvert\quad$ $\displaystyle\leq\quad
d\left\lvert\langle u|((U-V)\otimes\mathds{1})|\upphi_{d}\rangle\right\rvert$
$\displaystyle\leq\quad
d\left\|((U-V)\otimes\mathds{1})|\upphi_{d}\rangle\right\|$
$\displaystyle=\quad
d\left(\langle\upphi_{d}|((U-V)^{*}(U-V)\otimes\mathds{1})|\upphi_{d}\rangle\right)^{1/2}$
$\displaystyle=\quad\sqrt{d}\left\|U-V\right\|_{2}\enspace,$

where $\left\|\cdot\right\|_{2}$ denotes the Hilbert-Schmidt norm on
${\mathsf{L}}({\mathbb{C}}^{d})$. So the function $f$ is $\sqrt{d}$-Lipschitz
with respect to Hilbert-Schmidt metric. By Theorem 5.5,

$\Pr(\left\lvert{\bm{x}}\right\rvert\geq
1+t)\quad\leq\quad\exp\\!\left(-\frac{(d-2)t^{2}}{24d}\right)$

for every $t>0$. Taking $\kappa_{1}\coloneqq 24$, we may verify that Eq. (5.6)
holds for all $t\geq 0$ whenever $d\geq 3$. ∎

Lemma 5.15 implies that for a large enough constant $\alpha$, the contribution
to the mean $\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}$ outside an
interval $[0,\alpha]$ goes to $0$ as $d\to\infty$. Within this interval, the
contribution to the mean tends to that for
$\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}}$. This helps us derive the
limiting value of the mean.

######  Theorem 5.16.

$\lim_{d\rightarrow\infty}\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}\quad=\quad\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}}\enspace.$

###### Proof.

We formalize the intuition given above by appealing to a weaker property
implied by convergence in distribution, namely that the expectation of any
_bounded_ continuous function $f$ of the random variable ${\bm{\Lambda}}_{d}$
converges to $\operatorname{{\mathbb{E}}}f({\bm{\Lambda}})$.

Fix $\alpha\geq\max\left\\{c_{1},4\right\\}$, where $c_{1}$ is the constant in
the statement of Lemma 5.15 and consider the function $f_{\alpha}$ defined as
follows:

$f_{\alpha}(x)\quad\coloneqq\quad\begin{cases}0&\quad x\leq 0\\\
\sqrt{x}&\quad 0<x\leq\alpha\\\
\sqrt{\alpha}&\quad\alpha<x\enspace.\end{cases}$

Since $f_{\alpha}$ is continuous and bounded, and ${\bm{\Lambda}}\in[0,4]$,

$\lim_{d\to\infty}\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}}_{d})\quad=\quad\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}})\quad=\quad\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}}\enspace.$

On the other hand, ${\bm{\Lambda}}_{d}\geq 0$ and $f_{\alpha}(x)\leq\sqrt{x}$
for all $x\geq 0$. So
$\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}}_{d})\leq\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}$,
and

$\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}}\quad=\quad\lim_{d\to\infty}\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}}_{d})\quad\leq\quad\lim_{d\to\infty}\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}\enspace.$

We prove the reverse inequality using Lemma 5.15. Let $p(x)$ be the
probability density function of ${\bm{\Lambda}}_{d}$. By the definition of
$f_{\alpha}$,

$\displaystyle\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}\quad$
$\displaystyle\leq\quad\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}}_{d})+\int_{x\geq\alpha}\sqrt{x}p(x)\mathop{}\\!\mathrm{d}x\enspace.$
(5.7)

Let $g(\alpha,d)$ denote the second term on the right hand side of Eq. (5.7)
above. This is the contribution to
$\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}$ outside of the interval
$[0,\alpha]$. Using $\alpha\geq 4$, Fubini’s Theorem,
${\bm{\Lambda}}_{d}\leq\left\|{\bm{Q}}\right\|$, and Lemma 5.15, we have

$\displaystyle g(\alpha,d)\quad$
$\displaystyle\leq\quad\int_{x\geq\alpha}xp(x)\mathop{}\\!\mathrm{d}x$
$\displaystyle=\quad\int_{x\geq\alpha}\int_{y\in[0,x]}p(x)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle=\quad\int_{y\geq
0}\int_{x\geq\max\left\\{\alpha,y\right\\}}p(x)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle=\quad\int_{y\in[0,\alpha]}\int_{x\geq\alpha}p(x)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y+\int_{y\geq\alpha}\int_{x\geq
y}p(x)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle=\quad\int_{y\in[0,\alpha]}\Pr({\bm{\Lambda}}_{d}\geq\alpha)\mathop{}\\!\mathrm{d}y+\int_{y\geq\alpha}\Pr({\bm{\Lambda}}_{d}\geq
y)\mathop{}\\!\mathrm{d}y$ $\displaystyle\leq\quad 2\alpha\exp(-\alpha
n/c_{2})+2\int_{y\geq\alpha}\exp(-yn/c_{2})\mathop{}\\!\mathrm{d}y$
$\displaystyle=\quad 2(\alpha+n/c_{2})\exp(-\alpha n/c_{2})\enspace,$

where $c_{2}$ is the universal constant in the statement of Lemma 5.15. Since
$n=d^{2}$, $g(\alpha,d)$ vanishes as $d$ goes to $\infty$. By Eq. (5.7),

$\displaystyle\lim_{d\to\infty}\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}_{d}}\quad$
$\displaystyle\leq\quad\lim_{d\to\infty}\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}}_{d})+\lim_{d\to\infty}g(\alpha,d)$
$\displaystyle=\quad\operatorname{{\mathbb{E}}}f_{\alpha}({\bm{\Lambda}})\quad=\quad\operatorname{{\mathbb{E}}}\sqrt{{\bm{\Lambda}}}\enspace.$

This proves the theorem. ∎

## References

  * [Bai93] Z. D. Bai.  Convergence rate of expected spectral distributions of large random matrices. Part II. sample covariance matrices.  Annals of Probability, 21(2):649–672, April 1993.
  * [BCWdW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf.  Quantum fingerprinting.  Physical Review Letters, 87(16):167902, 2001.
  * [BW92] Charles H. Bennett and Stephen J. Wiesner.  Communication via one-and two-particle operators on einstein-podolsky-rosen states.  Physical review letters, 69(20):2881, 1992.
  * [BZ08] Zhidong Bai and Wang Zhou.  Large sample covariance matrices without independence structures in columns.  Statistica Sinica, 18(2):425–442, 2008.
  * [Cha20] Michael Charezma.  Quantum circuit diagrams.  https://warwick.ac.uk/fac/sci/physics/research/cfsa/people/pastmembers/charemzam/pastprojects, 2006 (accessed October 14, 2020).
  * [Cur79] Paul Joseph Curlander.  Quantum Limitations on Communication Systems.  PhD thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1979.
  * [Kho79] Alexander S. Kholevo.  On asymptotically optimal hypothesis testing in quantum statistics.  Theory of Probability & Its Applications, 23(2):411–415, 1979\.
  * [KR03] Andreas Klappenecker and Martin Rötteler.  Unitary error bases: Constructions, equivalence, and applications.  In Marc Fossorier, Tom Høholdt, and Alain Poli, editors, Proceedings of the 15th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes (AAECC), volume 2643 of Lecture Notes in Computer Science, pages 139–149. Springer, Berlin / Heidelberg, Germany, May12–16, 2003.
  * [Mec19] Elizabeth S. Meckes.  The Random Matrix Theory of the Classical Compact Groups, volume 218 of Cambridge Tracts in Mathematics.  Cambridge University Press, July 2019.
  * [Mon07] Ashley Montanaro.  On the distinguishability of random quantum states.  Communications in Mathematical Physics, 273(3):619–636, August 2007\.
  * [MV16] Benjamin Musto and Jamie Vicary.  Quantum Latin squares and unitary error bases.  Quantum Information and Computation, 16(15-16):1318–1332, November 2016.
  * [NC11] Michael A. Nielsen and Isaac L. Chuang.  Quantum Computation and Quantum Information.  Cambridge University Press, New York, NY, USA, 2011.  10th Anniversary Edition.
  * [ŠB20] Ivan Šupić and Joseph Bowles.  Self-testing of quantum systems: a review.  Quantum, 4:337, 2020.
  * [Tys09] Jon Tyson.  Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo-Curlander bounds.  Journal of Mathematical Physics, 50(3):032106, 2009.
  * [Ver18] Roman Vershynin.  High-Dimensional Probability: An Introduction with Applications in Data Science, volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics.  Cambridge University Press, Cambridge, UK, 2018.
  * [VW00] Karl Gerd H. Vollbrecht and Reinhard F. Werner.  Why two qubits are special.  Journal of Mathematical Physics, 41(10):6772–6782, 2000.
  * [Wat18] John Watrous.  The Theory of Quantum Information.  Cambridge University Press, May 2018.
  * [Wer01] Reinhard F. Werner.  All teleportation and dense coding schemes.  Journal of Physics A: Mathematical and General, 34(35):7081–7094, August 2001.
  * [Wil13] Mark M. Wilde.  Quantum Information Theory.  Cambridge University Press, Cambridge, UK, 2013.
  * [Yas16] Pavel Yaskov.  A short proof of the Marchenko–Pastur theorem. Une courte démonstration du théorème de Marchenko–Pastur.  Comptes Rendus Mathématique, 354(3):319–322, March 2016.


# Book embeddings of graphs and a theorem of Whitney††thanks: see also
https://faculty.georgetown.edu/kainen/pbip3.pdf

Paul C. Kainen
<EMAIL_ADDRESS>
Shannon Overbay
<EMAIL_ADDRESS>
###### Abstract

It is shown that the number of pages required for a book embedding of a graph
is the maximum of the numbers needed for any of the maximal nonseparable
subgraphs and that a plane graph in which every triangle bounds a face has a
two-page book embedding. The latter extends a theorem of H. Whitney and gives
two-page book embeddings for $X$-trees and square grids.

Key Phrases: Hamiltonian planar graph, book thickness, nicely planar graph,
girth, bipartite. This is the full version of Extension of a theorem of
Whitney, Appl Math Lett. 20 (2007) 835–837 by the same authors.

##  1 Introduction

In this paper, we consider finite simple graphs (no loops or parallel edges);
see, e.g.,[11], [19]. Let $bdy(e)=\\{v,w\\}$ denote the boundary vertices
(endpoints) of an edge $e$.

By a closed half-plane, we mean a copy of the complex numbers with imaginary
part $\geq 0$. For $k$ a positive integer, the book $B_{k}$ is the singular
surface formed from the union of $k$ closed half-planes (the pages)
intersecting in a line L (the spine) which is the (topological) boundary of
each of the pages.

In [15] (p. 97), a k-page book embedding of a graph $G=(V,E)$ is defined to be
an embedding of $G$ into $B_{k}$ which carries $V$ to $L$ with the property
that each edge $e\in E$ is mapped into a single page so that $e\cap L=bdy(e)$.
The book thickness, or pagenumber, of $G$, $bt(G)$, is the least number of
pages in which $G$ has a book embedding.

Equivalently, $bt(G)$ is the least number of colors which suffice to color the
edges of $G$ so that no two edges of the same color intersect other than at a
common endpoint when the vertices of $G$ are arranged in order around the
boundary of a circle and the edges are drawn as straightline segments,
minimized over all possible cyclic orderings of the vertices [3]. It is easy
to see that $G$ has book thickness 1 if and only if $G$ is outerplanar and
$bt(G)\leq 2$ if and only if $G$ is a subgraph of a planar Hamiltonian graph
[3].

Book thickness has turned up in rather diverse applications such as fault-
tolerant computing and VLSI (Chung, Leighton, and Rosenberg [5]),
computational complexity and graph separators (Galil, Kannan, and Szemeredi
[9]), software complexity metrics and vehicle traffic engineering (Kainen
[16]), and “bisecondary structures” used for modeling RNA folding energy
states (Gleiss and Stadler [10]). In addition, many theoretical questions
remain open such as the relationship of book thickness to other invariants
[6], [16], and the book thickness of standard and useful families of graphs
(e.g., [17], [13], [12], [8]).

Recall that a graph is planar if it can be embedded in the plane; the actual
embedding is called a plane graph. We call a graph nicely planar if it has a
plane embedding in which each triangle bounds a region.

In this paper, we investigate some connections between book thickness and
planarity. Yannakakis has shown [21] that the book thickness of a planar graph
cannot exceed 4 and examples are known of maximal planar graphs with no
Hamiltonian cycle so 3 (and perhaps 4) pages are certainly necessary for some
planar graphs. (As of 2021, it is known that 4 pages are necessary.)

Our main result is that a nicely planar graph is a subgraph of a Hamiltonian
planar graph. We then apply the result to give simple proofs that some useful
graphs in computer science have two-page book embeddings, previously shown by
direct computations [5]. The proof of our main result requires a lemma of
independent interest: $bt(G)=maxbt(B)$, where $B$ is a maximal nonseparable
subgraph of G.

In section 2, we study book thickness in terms of the block-cutpoint tree.
Section 3 gives the main result, while section 4 has applications.

##  2  Book thickness and block decomposition

A graph is nonseparable if it has no cutpoints; a block is a maximal
nonseparable subgraph [11] (p. 26). A family of subsets of some given set
determines an intersection graph where vertices correspond to subsets and
adjacency holds if and only if the corresponding subsets (are distinct and)
have nonempty intersection. The intersection graph formed by the cutpoint
singletons and the vertex sets of the blocks is a forest, called the block-
cutpoint forest [11] (p. 36). The points of the intersection graph which have
degree 1 must be blocks since every cutpoint belongs to more than one block. A
graph is connected if and only if its block-cutpoint forest is a tree.

######  Theorem 1.

The book thickness of a graph is the maximum of the book thicknesses of its
blocks.

Proof. Let G be a graph. Without loss of generality we assume G is connected.
We prove the result by induction on the number of blocks. The basis case of
one block is trivial.

Consider the block-cutpoint tree of $G$ which by assumption is not trivial.
There is a degree-one vertex corresponding to a block $H$ with a unique
cutpoint $v$ and $G^{\prime}=G\setminus(H-v)$ has one fewer block than $G$. By
the inductive hypothesis, $G^{\prime}$ has book thickness equal to the maximum
of that of its blocks. So we now need only show that $G$ requires no more than
the maximum of the book thicknesses of $H$ and $G^{\prime}$. Take minimum book
thickness embeddings of $H$ and $G^{\prime}$ and, by rotating the vertices
along the spine if necessary, make $v$ the first vertex of each embedding and
place the two embeddings consecutively along the spine of a book, using the
maximum number of pages in the two embeddings - WLOG, take $G^{\prime}$
second. Now pull the copy of $v$ in the second embedding above $H$ and
superimpose it on the first copy of $v$. Hence,
$bt(G)=\max\\{bt(G^{\prime}),bt(H)\\}$, as required. $\blacksquare$

Note that the genus of a graph is the sum of the genera of its blocks (Battle,
Harary, Kodama, and Youngs [2]). For book thickness, maximum replaces sum.
With respect to block decomposition, genus is to book thickness as the
$\ell_{1}$-norm is to the $\ell_{\infty}$-norm.

This type of distinction separates invariants, such as the cyclomatic number,
which are measures of global complexity (additive over the blocks) from
invariants, such as clique size, which are local measures of complexity and
have values obtained by maximizing over the blocks.

We wonder whether there are interesting graph theoretic complexity measures
corresponding to the $\ell_{2}$ (Hilbert) norm. Note that the global
complexity measures given here do not distinguish homeomorphic graphs, while
those of local type may have very different values for homeomorphic graphs;
see the last section

##  3 Extending Whitney’s theorem

A triangulation is a planar graph with a maximal set of edges. Equivalently, a
graph is a triangulation if it is isomorphic to a plane graph in which every
face, including the face which contains infinity, has exactly three edges in
its boundary. A triangle in a graph is a cycle of length 3.

By the Jordan Curve Theorem, in a plane graph, any triangle divides the plane
into an interior and an exterior region. A triangle is bounding if either its
interior or exterior region contains no vertices from the graph. A triangle is
separating if its deletion increases the number of connected components of the
graph. In a triangulation, a triangle is separating if and only if it is not
bounding. If there are no separating triangles in a planar graph, then the
graph is nicely planar, and the converse holds if the graph is 2-connected.

As an example, consider the plane graph $G$ which consists of two connected
components: two disjoint triangles side by side with three additional edges
all attached like a bonnet at the top vertex of the first triangle, pointing
up into the outer region of the first triangle. Removing the first triangle
increases the number of connected components from 2 to 4, the components now
being the second triangle and three isolated vertices. The first triangle is
separating but also bounding and the second triangle also bounds so $G$ is
nicely plane.

A graph is Hamiltonian if it has a cycle through all of the vertices. The
following theorem is due to H. Whitney [20].

######  Theorem 2.

Every triangulation with no separating triangles is Hamiltonian.

A graph is subhamiltonian if it is a subgraph of a planar Hamiltonian graph. A
plane graph $G^{\prime}$ is said to be obtained by stellating a face $F$ of
the plane graph $G$ if $G^{\prime}$ is obtained from $G$ by adding a new
vertex $F^{*}$ in the interior of $F$ and continuous curves joining it to each
vertex in the boundary of the face in such a way that the curves intersect $G$
only at their endpoints and the curves are disjoint except for $F^{*}$. See,
e.g., [3].

######  Theorem 3.

Every 3-connected planar graph with no separating triangles is subhamiltonian.

Proof. Choose a plane embedding of the given graph $G$ and extend it by
stellating any nontriangular faces. The resulting graph $G^{\prime}$ is
clearly a planar triangulation and we claim that it has no separating
triangles. Indeed, suppose $T$ were a separating triangle in $G^{\prime}$.
Then at least one of the vertices is new since $G$ had no separating
triangles. But the only new vertices were added as stellation points so
$T=\\{v,x,y\\}$, where $v$ is a stellation point and $x,y$ are vertices in the
boundary of the face which was stellated by $v$. It is straightforward to
verify that $\\{x,y\\}$ would be a separating set for $G$ so no separating
triangles exist in $G^{\prime}$ [18]. By Theorem 2, $G^{\prime}$ and so $G$
are subhamiltonian. $\blacksquare$

######  Lemma 1.

Let $G$ be a nicely planar block. Then there is a 3-connected planar graph
$G^{\prime}$ with no separating triangles such that $G$ is a subgraph of
$G^{\prime}$.

Proof. We induct on the number of separating 2-sets among the vertices of $G$.
If there are none, then $G$ is already 3-connected so the basis case of the
induction holds with $G^{\prime}=G$.

Suppose $G$ has $k\geq 1$ separating 2-sets and take $A=\\{u,v\\}$ to be one
of these separating 2-sets. Choose a particular plane embedding of $G$. Let
$G-A=G_{1}\cup\cdots\cup G_{n}$ with $G_{j}$ denoting the connected components
obtained by removing $A$, listed in the clockwise order of the edges joining
them to $v$. If $vu$ is an edge, we may renumber the components so that the
edge from v to u precedes all the edges from $v$ to $G_{1}$, and after all
those from $v$ to $G_{n}$. With respect to the same fixed clockwise order at
$v$ there is a last vertex $v_{j}$ in $G_{j}$ and a first vertex $w_{j}$ in
$G_{j+1}$, for $j=1,\ldots,n-1$. Add $n-1$ vertices $y_{j}$ together with the
edges $y_{j}v_{j}$, $y_{j}w_{j}$, and $y_{j}v$, keeping the resulting graph
$G^{\prime}$ plane. Indeed, $G^{\prime}$ is nicely plane since the added
triangles all bound disks.

Hence, it suffices to show that (1) $G^{\prime}$ is 2-connected and (2)
$G^{\prime}$ has fewer separating 2-sets than $G$. The assertion (1) follows
from the “ear” decomposition characterization, e.g., [19] (p. 163). One can
form $G^{\prime}$ by adding paths (ears) in the order
$v_{1},y_{1},w_{1},vy_{1},v_{2},y_{2},w_{2},\ldots,vy_{n}$.

Assertion (2) follows from the claims: (i) No added vertex is a member of a
separating 2-set for $G^{\prime}$, (ii) $A=\\{u,v\\}$ is not a separating
2-set for $G^{\prime}$, and (iii) if $B$ is any separating 2-set for
$G^{\prime}$, then $B$ is a separating 2-set for $G$.

These may be established as follows: For (i), first note that for $B$ any
separating 2-set of $G^{\prime}$, exactly one of the vertices must be new.
Since the old vertex can’t be a cutpoint of the block $G$, one component of
$G^{\prime}-B$ consists of a single new vertex. But this is impossible since
each new vertex is adjacent to 3 old ones. For (ii), let $A=\\{u,v\\}$ as
given above in the definition of $G^{\prime}$. By our construction,
$G^{\prime}-A$ is connected so $A$ is not a separating set for $G^{\prime}$.
Last, for (iii), let $B$ be a separating 2-set for $G^{\prime}$. No component
of $G^{\prime}-B$ is a single new vertex by the same degree argument as above.
Hence, each component contains at least one vertex of $G$, so $B$ is a
separating 2-set for $G$. See [18] for a similar and slightly more detailed
argument. $\blacksquare$

######  Theorem 4.

Every nicely plane graph is subhamiltonian.

Proof. By Lemma 1 and Theorem 3, each block is subhamiltonian and this
suffices by Theorem 1. $\blacksquare$

Note that we cannot always add edges alone to a nicely plane graph in a way to
satisfy the conditions of Whitney’s theorem. For example, a square with one
stellated face is nicely plane but extending to a plane graph by adding one
more edge forces a separating triangle.

Our method adds both vertices and edges to avoid creation of separating
triangles - hence, enabling the use Whitney’s Theorem to obtain a Hamiltonian
cycle. However, once such a cycle is created, one can delete both the added
edges and vertices of the construction, keeping the ordering for the original
vertices. By adding edges joining any nonadjacent pairs of consecutive
vertices, a Hamiltonian planar graph is obtained which contains the original
nicely planar graph as a spanning subgraph. The same argument shows that in
the definition of subhamiltonian graph, one can require that the extension
only involve the inclusion of new edges.

##  4 Applications

An X-tree is the plane graph formed by taking a complete plane binary tree
oriented with the root at the top (with branching down and to left and right,
resp.) and adding to it horizontal paths which join all vertices at the same
distance to the root. Let us call an extended X-tree the result of using an
additional edge for each of the added horizontal paths, joining the endpoints
of the path to form a cycle unless these two vertices were already adjacent.
The following is an immediate consequence of Theorem 4 since an extended
$X$-tree is nicely plane; cf. Chung, Leighton and Rosenberg [5]

######  Corollary 1.

Every extended X-tree (hence every X-tree) is subhamiltonian.

In [5] there is also an explicit proof that the product of two paths (called a
square grid) is subhamiltonian. This follows from the next corollary.

Recall that the girth of a graph is the length of its shortest cycle. Graphs
with girth $>3$ are triangle-free and so if they are planar, then they are
nicely planar. In particular, every bipartite planar graph has a 2-page book
embedding.

######  Corollary 2.

Every planar graph with girth $>3$ is subhamiltonian.

Theorem 4 has implications for two open questions. Barnette asked if every
cubic 3-connected planar bipartite graph is Hamiltonian; see, e.g., [14]. By
Corollary 2 such a graph is at least subhamiltonian. Chartrand, Geller and
Hedetniemi [4] conjectured that every planar graph can be written as the edge-
disjoint union of two outerplanar graphs. By Theorem 4, their conjecture is
true for nicely planar graphs (Goncalves, STOC’05, claims a general proof).

Recall that two graphs are homeomorphic if they have isomorphic subdivisions.
It is well known that every graph is homeomorphic to a graph of book thickness
at most three (see Atneosen [1], Bernhart and Kainen [3], and, according to
Jozef Przytycki, also a dissertation by G. Hotz, a student of Reidemeister).
Applications of three-page embeddings have been made to links by Dynnikov [7].

In contrast, for planar graphs, two pages suffice up to homeomorphism.

######  Theorem 5.

A graph is planar if and only if it is homeomorphic to a graph of book
thickness at most two.

Proof. Let $G$ be any planar graph and let $G^{\prime}$ be any subdivision of
$G$ such that all cycles have even length. For instance, one may take the
first barycentric subdivision obtained by subdividing each edge exactly once.
By Corollary 2, $G^{\prime}$ is subhamiltonian. $\blacksquare$

Enomoto and Miyauchi [8] consider “homeomorphic book embeddings” where edges
can cross the spine (i.e., an edge may use more than one page), and they note
that a graph is planar if and only if it has a homeormorphic book embedding in
two pages. It follows from our results above and some standard topological
graph theory arguments that a planar graph with $p$ vertices has a 2-page
homeomorphic book embedding with at most $p-2$ crossings of the spine.

## References

  * [1] G. Atneosen, On the embeddability of compacta in $n$-books: intrinsic and extrinsic properies, Ph.D. Dissertation, Michigan State University, 1968.
  * [2] J. Battle, F. Harary, Y. Kodama, and J. W. T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc.68 (1962), 569-571.
  * [3] F. Bernhart and P. C. Kainen, The book thickness of a graph,J. Combin. Theory Ser. B 27 (1979) no. 3, 320–331.
  * [4] G. Chartrand, D. P. Geller, and S. Hedetniemi, Graphs with forbidden subgraphs, J. Combin. Theory Ser. B 10 (1971) 12-41.
  * [5] F. R. K. Chung, F. T. Leighton, and A. L. Rosenberg, Embeddinggraphs in books: A layout problem with applications to VLSI design,SIAM J. on Alg. and Disc. Methods 8 (1987) no. 1, 33–58.
  * [6] A. M. Dean and J. P. Hutchinson, Relations among embedding parameters for graphs, in Graph Theory, Combinatorics, and Applications, vol. I, Wiley-Interscience Publ., New York, 1991, pp. 287–296.
  * [7] I. A. Dynnikov. Three-page link presentation and an untangling algorithm, in Proc. of the International Conference LowDimensional Topology and Combinatorial Group Theory, Chelyabinsk, July, 31 - August 7, 1999; Kiev, 2000; pp.112–130.
  * [8] H. Enomoto and M. S. Miyauchi, Embedding graphs into a three page book with O(MlogN) crossings of edges over the spine, SIAM J. Discrete Math. 12 (1999) 337–341.
  * [9] Z.Galil, R.Kannan and E.Szemeredi. On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape Turing machines, J. Computer and System Sci., 38 (1989) no.1, 134–149.
  * [10] P. M. Gleiss and P. F. Stadler, Relevant cycles in biopolymers and random graphs, 4th Slovene International Conference in Graph Theory, Bled, June 28 – July 8, 1999.
  * [11] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.
  * [12] T. Hasunuma and Y. Shibata, Embedding de Bruijn, Kautz and shuffle-exchange networks in books,Discrete Applied Math. 78 (1997) 103–116.
  * [13] L. S. Heath and S. Istrail, The pagenumber of genus $g$ graphs is $O(g)$, J. of the ACM 39 (1992) 479–501.
  * [14] D. Holton, B. Manvel, B. D. McKay, Hamiltonian cycles in cubic 3-connected bipartite planar graphs, J. Combin. Theory Ser. B 38 (1985) 279-297.
  * [15] P. C. Kainen, Some recent results in topological graph theory, in Graphs and combinatorics, R. A. Bari and F. Harary, Eds., Springer Lecture Notes in Math. 406, Berlin, 1974, pp. 76–108.
  * [16] P. C. Kainen, The book thickness of a graph, II, Congr. Numer. 71 (1990) 127-132.
  * [17] D. J. Muder, M. L. Weaver, and D. B. West, Pagenumber of complete bipartite graphs, J. Graph Theory 12 (1988) 469–489.
  * [18] S. Overbay, Generalized book embedings, Ph. D. Dissertation, Colorado State University, Fort Collins, CO, 1998.
  * [19] D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, Upper Saddle River, NJ, 2001.
  * [20] H. Whitney, A theorem on graphs,Ann. of Math. 32 (1931) 378–390.
  * [21] M. Yannakakis, Linear and book embeddings of graphs, in VLSI Algorithms and Architectures, Aegean Workshop on Computing (Loutraki, Greece, July 8-11, 1986), F. Makedon, K. Mehlhorn, T. S. Papatheodorou, P. G. Spirakis (Eds.), Lecture Notes in Computer Science, vol. 227, Springer 1986.