# Quantum Vacuum Sagnac Effect

Guilherme C. Matos1, Reinaldo de Melo e Souza2, Paulo A. Maia Neto1, and
François Impens1 1 Instituto de Física, Universidade Federal do Rio de
Janeiro, Rio de Janeiro, RJ, 21941-972, Brazil  2 Instituto de Física,
Universidade Federal Fluminense, Niterói, RJ, 24210-346, Brazil

###### Abstract

We report on the quantum electrodynamical analogue of a Sagnac phase induced
by the fast rotation of a neutral nanoparticle onto atomic waves propagating
in its vicinity. The quantum vacuum Sagnac phase is a geometric Berry phase
proportional to the angular velocity of rotation. The persistence of a
noninertial effect into the inertial frame is also analogous to the Aharonov-
Bohm effect. Here, a rotation confined to a restricted domain of space gives
rise to an atomic phase even though the interferometer is at rest with respect
to an inertial frame. By taking advantage of a plasmon resonance, we show that
the magnitude of the induced phase can be close to the sensitivity limit of
state of the art interferometers. The quantum vacuum Sagnac atomic phase is a
geometric footprint of a dynamical Casimir-like effect.

The rotation of a frame attached to an interferometer with respect to an
inertial frame induces an interferometric phase proportional to the angular
frequency of the rotation and to the area enclosed by the interferometer. This
phenomenon, known as the Sagnac effect Sagnac13 , has several technological
applications such as optical-based Sagnac sensors which have been embarked on
aircraft for decades. Three decades ago, it has been extended to matter waves
Borde89 . Since then, cold-atom gyrometers based on the Sagnac effect
Gustavson96 ; Pritchard97  have been constantly improved to outperform their
optical equivalents Canuel06 ; RMPCronin09 ; Sackett20 ; Ryu2020 ;
Schubert2021  (see Ref. Geiger2020  for a recent review).

We propose here to investigate the following closely related scheme – what if,
instead of rotating the interferometer as a whole, one simply spins a neutral
nanoparticle placed between its arms? Such question naturally arises as
rotation speeds beyond $5\,{\rm GHz}$ are achieved with optically levitated
nanoparticles Ahn2020 . As in the Casimir effect, we show that the interaction
between the atom and the nanoparticle mediated by the quantum electromagnetic
field leads to an atomic phase induced by the particle’s spinning. We
introduce the quantum vacuum Sagnac phase (QVSP) as an atom-interferometry
footprint of noninertial effects in the quantum vacuum. The QVSP is thus
analogous to the dynamical Casimir effect Dalvit2011 ; Dodonov2020 , but no
real photons are emitted in the configuration discussed hereafter. Indeed,
here the relative motions of the interfering wave packets with respect to the
sense of the particle’s rotation plays a crucial role in the atom-particle
interaction mediated by the quantum vacuum field. While dynamical Casimir
photons are too scarce to be measurable MaiaNeto96 ; Dodonov2021 ; Dalvit2021
even when considering a cavity resonance Lambrecht96 ; Dodonov96 ;
Schutzhold98 ; Crocce2001 , we show that the magnitude of the QVSP is close to
the sensitivity limit of state-of-the-art atom interferometers Kumiya2016 ;
Geiger2020  when taking into account the record rotation frequencies recently
demonstrated with optically levitated nanoparticles Reimann2018 ; Ahn2018 ;
Ahn2020 .

The emission of dynamical Casimir photons out of the quantum vacuum state by a
dielectric sphere undergoing a non-uniform rotation was considered in Ref.
Barton96 . The electromagnetic field at finite temperature was predicted to
exert a quantum friction torque on a neutral dispersive microsphere spinning
at a constant rotation frequency Manjavacas2010 ; ManjavacasPRL2010 . The QVSP
imprinted on the atomic center of mass (CM) provides an additional insight on
how the rotation modifies the surrounding quantum electromagnetic field even
when no real photons are emitted.

The QVSP is also a consequence of the motion of atomic CM with respect to the
nanoparticle. In the context of atom interferometry, a nonlocal phase
associated with pairs of paths (rather than with individual ones) was shown to
result from the field-mediated interaction between a moving atom and a
material surface Nonlocal1 ; Nonlocal2 ; Nonlocal3 . Dynamical Casimir
emission of photons Souza2018 ; Lo2018 ; Farias2019 ; Dolan2020 ; Agusti2021 ,
decoherence Scheel2012 ; Souza2016  and quantum friction Scheel09 ;
Pieplow2013 ; Reiche2020 ; Fosco2021 ; Donaire2016 ; Farias2020 ; Lombardo21
also result from the coupling between a moving atom and the quantum
electromagnetic field. Given their high sensitivity, atom interferometers are
candidates for the first experimental demonstration of motional effects in
Casimir physics, and the QVSP would be particularly appealing for that
purpose.

The QVSP is also related to the Aharonov-Bohm effect Aharonov1959 . Such
connection is well understood in the case of the standard Sagnac effect (see
Ref.ReviewSagnacBerryArxiv21  for a recent review), as a rotating referential
emulates the presence of magnetic fields thanks to the similarity between
Coriolis and Lorentz forces Dalibard11 . This analogy has enabled the
production of artificial effective magnetic fields in neutral cold-atom gases
set into rotation Cornell04 . Like the standard Sagnac phase, the QVSP is a
geometric phase that can be cast in terms of an effective magnetic field. In
addition, the QVSP can be seen as the Aharonov-Bohm-like counterpart of the
Sagnac effect. Indeed, in the Aharonov-Bohm experiment, a magnetic field
confined to a solenoid imprints a phase in a region free of magnetic fields.
Here, we show that a rotation confined to a domain of space imprints a phase
on matter waves probing quantum vacuum fluctuations outside the rotating
region.

For simplicity, we consider a nanoparticle rotating around an axis of symmetry
with constant angular velocity $\Omega.$ In this case, the modification of the
surrounding quantum field arises from the frequency dependence of the particle
dielectric constant. We consider a 2-level atom in the ground state
interacting with the quantum vacuum field. The atom CM is in a quantum
superposition of two wave packets that propagate in the vicinity of the
spinning nanoparticle as indicated in Fig. 1. We show that the resulting QVSP
is geometric, i.e. independent of the atomic velocity Miniatura92 ; Gauguet13
in the limiting case of very narrow wave packets. Furthermore, we express the
QVSP as the circulation of a geometric vector field, analog to the vector
potential in the Aharonov-Bohm effect, along the interferometer paths. The
effect can be enhanced by considering nanoparticles with a plasmon resonance
PlasmonMetallic2012  in order to optimize the material dispersion at the
atomic transition frequency.

Figure 1: Scheme of the quantum vacuum Sagnac interferometer. The CM of a
ground-state atom propagates as a quantum superposition of two wave packets
around a spinning neutral nanoparticle (angular frequency $\Omega$).

Motional van der Waals (vdW) atomic phase. We consider a moving atom
interacting with the rotating nanoparticle between the initial/final times
$t=\mp T/2$. The atomic waves acquire a phase associated with the dipolar
interaction $\hat{H}_{\rm dip}=-\hat{\mathbf{d}}\cdot\hat{\mathbf{E}},$ with
$\hat{\mathbf{d}}$ representing the atomic dipole moment operator. The
electric field operator $\hat{\mathbf{E}}$ is taken at the instantaneous
average atomic position
$\mathbf{r}_{k}(t)=\langle\hat{\mathbf{r}}(t)\rangle_{k}$ for each wave packet
$k.$ We evaluate the phase difference $\Delta\phi_{12}$ accumulated by the
coherent superposition state of two narrow atomic wave packets following the
two distinct paths $\mathcal{P}_{1}=[\mathbf{r}_{1}(t)],$
$\mathcal{P}_{2}=[\mathbf{r}_{2}(t)].$ Up to second order in perturbation
theory, this phase difference reads Nonlocal3

$\displaystyle\Delta\phi_{12}$ $\displaystyle=$
$\displaystyle\varphi_{11}-\varphi_{22}+\varphi_{12}-\varphi_{21}$ (1)
$\displaystyle\mathcal{\varphi}_{kl}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\iint_{-\frac{T}{2}}^{\frac{T}{2}}dt\,dt^{\prime}\left[g^{H}_{\hat{\mathbf{d}}}(t,t^{\prime})\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r}_{k}(t),t;\mathbf{r}_{l}(t^{\prime}),t^{\prime})\right.$
(2)
$\displaystyle\left.\qquad\qquad+g^{R}_{\hat{\mathbf{d}}}(t,t^{\prime})\mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r}_{k}(t),t;\mathbf{r}_{l}(t^{\prime}),t^{\prime})\right]$

The contributions $\mathcal{\varphi}_{kl}$ for $k=l$ and $k\neq l$ correspond
to local and non-local phases, respectively. In the concrete applications
discussed later on, the local phases $\phi_{k}\equiv\varphi_{kk}$ will play a
more important role. We have used the trace of the retarded Green’s function
for the scattered electric field
$\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},t;\mathbf{r}^{\prime},t^{\prime})={\rm
Tr}\left[\boldsymbol{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},t;\mathbf{r}^{\prime},t^{\prime})\right]$,
which captures how electrodynamical propagation is modified by the presence of
the nanoparticle (scatterer) placed at the origin. Likewise, the trace
$\mathcal{G}^{H,S}_{\hat{\mathbf{E}}}$ of the Hadamard Green’s function
represents the change in the field fluctuations induced by the presence of the
nanoparticle. The retarded Green’s function of a vectorial operator
$\hat{\mathbf{O}}(t)$ is defined as the averaged commutator
$\boldsymbol{G}_{\hat{\mathbf{O}}\>ij}^{R}(t,t^{\prime})=\frac{i}{\hbar}\Theta(t-t^{\prime})\langle[\hat{O}_{i}(t),\hat{O}_{j}(t^{\prime})]\rangle$
with $\Theta(\tau)$ denoting the Heaviside function. The Hadamard Green’s
function corresponds to the average value of the anti-commutator
$\boldsymbol{G}_{\hat{\mathbf{O}}\>ij}^{H}(t,t^{\prime})=\frac{1}{\hbar}\langle\\{\hat{O}_{i}(t),\hat{O}_{j}(t^{\prime})\\}\rangle.$

The first term on the r.-h. s. of (2) accounts for the electric field response
to dipole fluctuations, while the second one corresponds to the dipole
response to vacuum fluctuations modified by the presence of the nanoparticle.
The dipole Hadamard Green’s function is isotropic and has the analytical form
$g^{H}_{\hat{\mathbf{d}}\>ij}(t,t^{\prime})=\alpha_{0}^{\rm
A}\omega_{0}\cos\omega_{0}(t-t^{\prime})\delta_{ij}$ for a 2-level model.
Here, $\alpha_{0}^{\rm A}$ represents the static polarizability and
$\omega_{0}$ is the transition frequency. We focus on the nonretarded vdW
regime, for which the atom-particle distance $r(t)$ is much smaller than the
transition wavelength $\lambda_{0}=2\pi c/\omega_{0}$. As shown below, the
QVSP is maximized in the immediate vicinity of the rotating nanoparticle,
which turns the vdW regime more interesting for experimental implementations.

We now consider the retarded Green’s function for the scattered electric field
$\boldsymbol{G}^{R,(S)}_{\hat{\mathbf{E}}\>ij}(\mathbf{r},t;\mathbf{r}^{\prime},t^{\prime}).$
This function corresponds to the $i$-th component of the electric field at
position $\mathbf{r}$ and time $t$ induced by an instantaneous point dipole
oriented along the $j$-th direction at position $\mathbf{r}^{\prime}$ and time
$t^{\prime}$ after scattering at the nanoparticle at some intermediate time
$t^{\prime\prime}$ such that $t^{\prime}<t^{\prime\prime}<t$. From now on, we
assume that the nanoparticle is very small and neglect multipolar
contributions beyond the electric dipolar one. The retarded Green’s function
in the frequency domain can then be expressed in terms of the electric
polarizability tensor $\boldsymbol{\alpha^{\Omega}}(\omega)$ of the rotating
nanoparticle as

$\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\boldsymbol{G}^{0}(\mathbf{r},\mathbf{0},\omega)\cdot\boldsymbol{\alpha^{\Omega}}(\omega)\cdot\boldsymbol{G}^{0}(\mathbf{0},\mathbf{r}^{\prime},\omega)$
(3)

The free-space retarded Green’s function for the electric field HeitlerBook
becomes frequency-independent in the nonretarded vdW regime
$\boldsymbol{G}^{0}_{ij}(\mathbf{r},\mathbf{r}^{\prime},\omega)\approx(3R_{i}R_{j}/R^{2}-\delta_{ij})/(4\pi\epsilon_{0}R^{3})$
with $\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}$. In the absence of rotation,
any direction orthogonal to the symmetry axis of the nanoparticle is a
principle axis of the polarizability tensor $\boldsymbol{\alpha}^{0}(\omega)$
with an eigenvalue denoted by $\tilde{\alpha}(\omega)$. Rotation around the
symmetry axis leads to a nondiagonal correction
$\delta\boldsymbol{\alpha^{\Omega}}(\omega)_{lm}=\boldsymbol{\alpha^{\Omega}}_{lm}(\omega)-\boldsymbol{\alpha}^{0}_{lm}(\omega)\approx
i\tilde{\alpha}^{\prime}(\omega)\sum_{n}\epsilon_{lmn}\Omega_{n}$
Manjavacas2010  which lies at the heart of the QVSP. We have assumed a
nonrelativistic rotation with angular velocity $\boldsymbol{\Omega}$,
$\tilde{\alpha}^{\prime}(\omega)$ is the frequency derivative of the
polarizability eigenvalue, and $\epsilon_{lmn}$ denotes the Levi-Civitta
tensor components.

QVSP induced by a rotating nanoparticle. We provide first a heuristic
discussion of the QVSP derived below. This phase arises from the scattering of
virtual photons on the spinning nanoparticle, which carry back to the atom a
trace of the particle rotation. The QVSP captures the noninertial footprint on
the quantum electromagnetic field as the atom probes fluctuations in the
vicinity of the particle. In this sense, it constitutes a dynamical Casimir-
like effect. However, as a nanoparticle spinning at constant velocity produces
no radiation Barton96 ; Manjavacas2010 , the QVSP does not rely on the
presence of real dynamical Casimir photons – nor does it rests on open-quantum
system dynamics Lombardo06  responsible for quantum friction Farias2020 ;
Lombardo21 .

The problem under consideration involves very different time scales, listed
below from the slowest to the fastest in the vdW regime: the time-of-flight
$T$ of the atomic CM in the vicinity of the rotating particle and the period
of rotation (a fraction of nanosecond), the inverse of the atomic transition
frequency $2\pi/\omega_{0}$, the response time of the rotating particle due to
dispersion, and finally the light travel time between the moving atom and the
particle, $r/c$. Such hierarchy of time scales allows us to take several
approximations. The dominant contribution in Eq. (2) comes from intervals
$t-t^{\prime}$ of the order of the response time of the nanoparticle, enabling
us to neglect the CM acceleration and take
$\mathbf{r}_{k}(t^{\prime})\simeq\mathbf{r}_{k}(t)-(t-t^{\prime})\mathbf{v}_{k}(t)$
in Eqs. (2) and (6).

In addition, the displacement of the atomic CM during $t-t^{\prime}$ is much
smaller than the wavelengths of field modes contributing to the electric field
Green’s functions. Thus, we Taylor expand the latter around the position
$\mathbf{r}_{k}(t)$. We start by deriving the local contribution to the QVSP
and define $\phi^{\rm\Omega}_{k}$ as the $\Omega-$dependent contribution to
the phase $\phi_{k}$ in Eq. (2). We write it as the sum
$\phi^{\rm\Omega}_{k}=\phi^{\rm\Omega}_{{\rm qs},k}+\phi^{\rm\Omega}_{{\rm
mot},k}$ of a quasistatic and of a motional contribution. The former is
obtained by taking identical arguments for the retarded and advanced positions
($\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)$) in the electric Green’s
function, while the latter involves instead the gradient of the Green’s
functions and the instantaneous atomic CM velocity.

The quasistatic contribution $\phi^{\rm\Omega}_{{\rm qs},k}$ vanishes by
symmetry considerations Supplementary . The QVSP thus arises exclusively from
the atomic motion during the electrodynamical delay time $t-t^{\prime}$
associated with the exchange of virtual photons between the atom and the
nanoparticle. We write $\phi^{\rm\Omega}_{k}=\phi^{\rm\Omega}_{{\rm
mot},k}=\phi^{\rm\Omega,dip}_{k}+\phi^{\rm\Omega,f}_{k}$ as the sum of
contributions from dipole and field fluctuations, respectively. They
correspond to the first and second terms in Eq. (2). We use the condition
$\omega_{0}T\gg 1$ to derive

$\phi^{\rm\Omega,dip}_{k}=-\frac{\omega_{0}\alpha_{0}^{\rm
A}}{4}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\partial_{\omega}\nabla_{\mathbf{r}^{\prime}}\mbox{Im}\left[\delta\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]$
(4)

where the spatial and frequency derivatives are taken at
$\mathbf{r}^{\prime}=\mathbf{r}=\mathbf{r}_{k}(t)$ and $\omega=\omega_{0},$
respectively. Here, $\delta\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}$ represents
the $\boldsymbol{\Omega}-$dependent contribution to the scattered Green
function, which is obtained by taking
$\delta\boldsymbol{\alpha^{\Omega}}(\omega)$ instead of the full tensor
$\boldsymbol{\alpha}(\omega)$ in Eq. (6). The frequency derivative captures
the time delay associated with the nanoparticle response to the field (in the
form of an induced dipole) during the atomic motion. On the other hand, the
delay associated with the light propagation time from the atom to the
nanoparticle is negligible within the vdW approximation. Accordingly, we take
the nonretarded vdW approximation for the free-space Green’s function
$\boldsymbol{G}^{0}(\mathbf{r},\mathbf{r}^{\prime},\omega)$ when deriving the
scattered field propagator
$\delta\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)$
from Eq. (6).

The contribution from field fluctuations $\phi^{\rm\Omega,f}_{k}$ is given by
an expression similar to (4) in terms of the retarded Green’s function for the
dipole (polarizability) and the Hadamard Green’s function for the field. Using
the fluctuation-dissipation theorem
$\nabla_{\mathbf{r}^{\prime}}\mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)=2\,{\rm
sgn}(\omega)\,\mbox{Im}[\nabla_{\mathbf{r}^{\prime}}\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)],$
with ${\rm sgn}(\omega)$ denoting the sign function, we find a contribution to
the local QVSP identical to the dipolar one. We write the final result for the
local QVSP as a geometric integral Supplementary , which is the main result of
this Letter:

$\displaystyle\phi^{\rm\Omega}_{k}=\frac{9}{2}\,\frac{\omega_{0}\alpha_{0}^{\rm
A}\tilde{\alpha}_{R}^{\prime\prime}(\omega_{0})}{(4\pi\epsilon_{0})^{2}}\int_{\mathcal{P}_{k}}d\mathbf{r}\>\cdot\>\frac{\mathbf{\Omega}\times\mathbf{r}}{r^{8}},$
(5)

where $\tilde{\alpha}_{R}(\omega_{0})$ is the real part of the nanoparticle’s
polarizability eigenvalue. The integral is performed along the interferometer
path $\mathcal{P}_{k}$ delimited by the initial/final positions
$\mathbf{r}_{k}(\mp T/2).$ As in the standard Sagnac effect Dalibard11 ;
Cornell04 , the QVSP given by Eq. (5) is a geometric phase that can be cast as
the line integral of an effective vector potential proportional to the angular
velocity $\Omega.$ The QVSP (5) possess all the distinctive features of a
geometric phase: it is independent of the velocity magnitude, but changes sign
when the direction of propagation is reversed.

As an important insight in its geometric nature, one can show Supplementary
that the QVSP is indeed a Berry phase Berry1 ; Berry2  of the full quantum
system “2-level atom+field” undergoing a unitary and adiabatic quantum
evolution steered by the atomic position. Because of the dipole interaction,
the instantaneous ground state of this quantum system changes continuously as
the atom propagates nearby the spinning nanoparticle, following a quantum
trajectory which depends on the interferometer path. By integrating the
corresponding Berry connection, one retrieves exactly the local QVSP (5).

The correction associated to eletrodynamical retardation is given by
Supplementary  $\phi^{\rm\Omega}_{(c),k}=3\omega_{0}\alpha_{0}^{\rm
A}\tilde{\alpha}_{R}^{\prime}(\omega_{0})/[(4\pi\epsilon_{0})^{2}c^{2}]\int_{\mathcal{P}_{k}}d\mathbf{r}\>\cdot\>\frac{\mathbf{\Omega}\times\mathbf{r}}{r^{6}}$
and is negligible for the example discussed below. In the case of a finite
temperature $\theta,$ the local QVSP is multiplied by the factor
$\coth(\hbar\omega_{0}/2k_{B}\theta)$ ($k_{B}=$ Boltzmann constant), which is
very close to one for any realistic example of atomic transition. In other
words, the contribution from thermal fluctuations is negligible and the QVSP
is a genuine quantum vacuum effect.

Examples of interferometer designs. To illustrate our findings, we calculate
the phase for two specific geometries: either a circle of radius $R$ centered
on the nanoparticle, or two parallel straight lines enclosing the particle. In
both cases we assume that ${\bf\Omega}=\Omega\,\mathbf{\hat{z}}$ is orthogonal
to the plane containing the trajectories. The first arrangement is a textbook
example of a Sagnac interferometer, whereas the latter corresponds to a more
realistic situation in atom interferometry Cronin04 ; Vigue09 ; Vigue11 . One
finds $\phi^{\rm\Omega}_{\\{r=R\\}}=9\pi\ell^{6}_{\Omega}/R^{6}$ for the
circular trajectory when the senses of rotation of atom and nanoparticle
coincide; and $\phi^{\rm\Omega}_{\\{x=x_{1},-L/2\leq
y\leq+L/2,z=z_{1}\\}}\approx 45\pi\ell^{6}_{\Omega}\,x_{1}/(32r_{\perp}^{7})$
with $r_{\perp}=(x_{1}^{2}+z_{1}^{2})^{1/2}$ for a straight segment of length
$L$ satisfying the condition $|x_{1}|,|z_{1}|\ll L/2\ll\lambda_{0}$ for
consistency with the vdW approximation. We have introduced the characteristic
length scale
$\ell_{\Omega}=[\omega_{0}\alpha_{0}\alpha_{R}^{\prime\prime}(\omega_{0})\Omega/(4\pi\epsilon_{0})^{2}]^{1/6}.$

Non-local contributions to the QVSP difference. We now discuss the non-local
QVSP contribution $\varphi^{\Omega}_{12}-\varphi^{\Omega}_{21}$ corresponding
to the $\Omega$-dependent part of the non-local phases in Eq. (1). In a
different context, cross-talks of the interferometer paths can influence
significantly the motional phases Nonlocal1 ; Nonlocal2 ; Nonlocal3  or
decoherence rates Mazzitelli03 ; Scheel2012  induced by the quantum vacuum.
While the nonlocal QVSP difference vanishes for circular trajectories, it is
nonzero when considering two parallel straight line trajectories with the
nanoparticle at the midpoint ($x_{2}=-x_{1}$ and $z_{1}=z_{2}=0$). In this
case, the non-local QVSP contribution has an opposite sign with respect to the
local one, thus reducing the total QVSP difference to
$\Delta\phi^{\Omega}_{12}=63\pi\ell^{6}_{\Omega}\,{\rm
sgn}(x_{1})/(32x_{1}^{6}).$ For this geometry, the non-local contribution
represents a sizable part of the total QVSP difference.

Estimation of the QVSP for finite-width atomic wave packets and a spherical
nanoparticle. We estimate the magnitude of the QVSP in a practical
interferometer implementation. Specifically, we consider a Mach-Zehnder
configuration where path $1$ flies near a rotating nanosphere of radius $a$,
and path $2$ evolves far away from the nanosphere. A similar interferometer
geometry was used in Cronin04 ; Vigue09 ; Vigue11  for the measurement of
quasistatic vdW phases. In this setup, the QVSP receives only a local
contribution from path $1$, namely
$\Delta\phi_{12}^{\Omega}=\phi_{1}^{\Omega}$. In order to enhance the QVSP, we
investigate materials for which the polarizability exhibits a sharp frequency
dependence at the atomic transition frequency.

For this purpose, we use a plasmon resonance PlasmonMetallic2012  and seek
metals for which the polarizability
$\tilde{\alpha}(\omega)=(4\pi\epsilon_{0})a^{3}[\epsilon(\omega)-1]/[\epsilon(\omega)+2]$
is maximized by reducing $\epsilon(\omega)+2$ at the atomic transition
frequency $\omega_{0}.$ Within the Drude model, the dielectric constant reads
$\epsilon(\omega)=1-\omega_{P}^{2}/[\omega(\omega+i\gamma)],$ where
$\omega_{P}$ is the plasma frequency and $\gamma$ is the inverse of the
electronic relaxation time. The plasma resonance in the dipole approximation
is at $\omega_{\rm res}=\omega_{P}/\sqrt{3}.$ Since we want to maximize
$\tilde{\alpha}_{R}^{\prime\prime}(\omega)$ rather than
$\tilde{\alpha}^{\prime}(\omega),$ we need the atomic transition frequency to
be slightly shifted with respect to $\omega_{\rm res}.$ We consider Na atoms,
for which the static polarizability is $\alpha_{0}^{\rm
A}/(4\pi\epsilon_{0})=2.4\times 10^{-29}{\rm m^{3}}$ and the dominant
transition ($3s_{1/2}-3p_{3/2}$) has a frequency $\omega_{0}=3.198\times
10^{15}\>{\rm rad/s}$ Cronin10 . We take the plasma frequency of the
nanosphere to be $\omega_{\rm P}=5.549\times 10^{15}{\rm rad/s}$ so as to
maximize $\tilde{\alpha}_{R}^{\prime\prime}(\omega_{0}).$ Such fine tuning can
be achieved, for instance, from the size dependence of the plasmon resonance
in nanospheres Derkachova2016 ; Karimi2019 . Our value for $\omega_{\rm P}$ is
also very close, within less than $1\%,$ to the bulk plasma frequency of
potassium Blaber09 . Accordingly, we take the relaxation frequency of
potassium $\gamma=2.795\times 10^{13}\,{\rm rad/s}$ in our numerical
estimation (Fig. 2).

Figure 2: Average QVSP versus (a) velocity and (b) width of the Gaussian
atomic beam used in the interferometer. Parameters: (a) $a=50\>{\rm nm}$,
$w=100\>{\rm nm}$. (b) $a=35\>{\rm nm}$, $v=3\>{\rm km/s}$ (dotted line),
$v=4\>{\rm km/s}$ (dash-dotted line) and $v=5\>{\rm km/s}$ (solid line). In
(a,b) we have considered a 2-level Na atom with the static polarizability
$\alpha_{0}^{\rm A}=(4\pi\epsilon_{0})\times 2.4\times 10^{-29}{\rm m}^{3}$
and transition frequency $\omega_{0}=3.198\times 10^{15}\>{\rm rad/s}$. We
have taken a nano-sphere of Potassium spinning at the angular velocity
$\Omega=2\pi\times 5\>{\rm GHz}$.

We average the atomic phase over the transverse wave packet widths following
the procedure of Refs. Cronin04 ; Vigue09 ; Vigue11  for (quasistatic) vdW
phases. The phase acquired by atoms flying along the trajectory
$\\{x^{\prime}=x,-L/2\leq y^{\prime}\leq L/2,z^{\prime}=z\\}$ at a constant
velocity $\mathbf{v}=v\>\hat{\mathbf{y}},$ is the sum of the vdW phase
$\phi^{\rm vdW}(x,z,v),$ as derived by integration of the instantaneous vdW
potential along the trajectory, and the QVSP $\phi^{\Omega}(x,z)$ :
$\phi(\Omega,x,z,v)=\phi^{\rm vdW}(x,z,v)+\phi^{\Omega}(x,z).$ The known
result Cronin04  for the vdW phase follows from Eqs. (2) and (6) in the
quasistatic limit: $\phi^{\rm vdW}(x,z,v)\simeq
9\pi\alpha_{0}\omega_{0}\tilde{\alpha}_{R}(\omega_{0})/[4.(4\pi\epsilon_{0})^{2}vr_{\perp}^{5}]$
with $r_{\perp}=(x^{2}+z^{2})^{1/2}$. There are two important differences
between the vdW phase and the QVSP. The former is dynamical and thus inversely
proportional to the velocity, while the latter is geometric and thus velocity-
independent. On the other hand, the vdW phase is unaffected by the
nanosphere’s spinning, while the QVSP is proportional to the angular velocity
$\Omega.$

The experimentally accessible phase reads $\overline{\phi}(\Omega,v)={\rm
Arctan}\left[\overline{\sin(\phi(\Omega,x,z,v))}/\overline{\cos(\phi(\Omega,x,z,v))}\right],$
where the averaging is performed over the transverse coordinates $x,z$. We
consider a Gaussian atomic wave packet of transverse width $w$ and
longitudinal velocity $v$. In practice, one may gradually increase the
nanosphere rotation in order to isolate the average QVSP as the
$\Omega-$dependent part of the total average phase:
$\overline{\phi}^{\Omega}(\Omega,v)\equiv\overline{\phi}(\Omega,v)-\overline{\phi}(0,v).$
In spite of the geometric nature of $\phi^{\Omega}(x,z)$, the presence of the
dynamical phase $\phi^{\rm vdW}(x,z,v)$ together with the averaging procedure
involving trigonometric functions turn $\overline{\phi}^{\Omega}(\Omega,v)$
velocity-dependent in the case of finite-width wave packets.

We take the angular velocity $\Omega=2\pi\times 5\>{\rm GHz}$ recently
achieved with optically-levitated nanoparticles Ahn2020 . We also consider
thin atomic beams of width $w\leq 100\,{\rm nm}$ centered at the edge of the
spinning particle. Such collimation may be obtained by using diffraction
through a nano-grating Vigue11  placed in the vicinity of the spinning
particle, or by tight focusing techniques considered for atom lithography
Prentiss92 . Prospective focusing techniques Martin21a ; Martin21b  show
indeed that atomic beam widths $w\simeq 8\,{\rm nm}$ may be attained. We plot
$\overline{\phi}^{\Omega}(\Omega,v)$ versus velocity in Fig. 2a for a
nanosphere of radius $a=50\,{\rm nm}$ and an atomic wave packet of width
$w=100\,{\rm nm}.$ Fig. 2b presents the variation of
$\overline{\phi}^{\Omega}(\Omega,v)$ with the wave packet width, for
$a=35\,{\rm nm}$ and for different velocities of a few km/s. The averaged QVSP
tends to increase with the atomic velocity and is enhanced by atomic beam
focusing. However, $\overline{\phi}^{\Omega}(\Omega,v)$ may be significantly
attenuated around specific velocities or width values under the influence of
the quasistatic vdW phase. To avoid the detrimental effect of the vdW
potential, we consider fast atomic beams, with velocities comparable to those
of Ref. Vigue11 . Fig. 2b shows that an average QVSP
$\overline{\phi}^{\Omega}(\Omega,v)\simeq 0.1\,{\rm mrad}$ is attained for
$w=8\>{\rm nm}$. Such value is close to the current phase sensitivity limit in
atom interferometry Geiger2020 ; Salvi2018 .

Conclusions. We have shown that the fast rotation of a nanoparticle imprints a
geometric phase analogous to a Sagnac phase on a ground-state atom propagating
in its vicinity. The persistence of a noninertial effect beyond the region
where the rotation actually occurs is reminiscent of the Aharonov-Bohm effect,
with the vector potential yielding a finite atomic phase in a region free of
magnetic field. We have assumed the interferometer to be at rest with respect
to an inertial frame. Thus, the noninertial effects are exclusively mediated
by the quantum vacuum field through the scattering on a rotating nanoparticle.
The resulting QVSP is a dynamical Casimir-like modification of the atomic
phase, whose observation might be more at hand than the detection of dynamical
Casimir photons. Quantum dipolar and field fluctuations contribute equally to
the QVSP, which can be enhanced by using materials exhibiting a plasmon
resonance near the atomic transition frequency. The QVSP might become within
reach of experimental observation given the state of the art in atom
interferometry and nanorotors.

###### Acknowledgements.

F.I. thanks Jean Dalibard for stimulating initial discussions, and A. Z.
Khoury, M. B. da Silva Neto for discussions on geometric phases. F.I. and
P.A.M.N. are grateful to Ryan O. Behunin and Claudio Ccappa Ttira for a
previous collaboration on non-local atomic phases. We acknowledge funding from
the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) (409994/2018-9), Coordenação de Aperfeiçoamento de Pessoal
de Nível Superior (CAPES), and Fundação de Amparo à Pesquisa do Estado do Rio
de Janeiro (FAPERJ) (202.874/2017, 210.242/2018 and 210.296/2019). P.A.M.N.
also thanks partial finantial support from Instituto Nacional de Ciência e
Tecnologia de Fluidos Complexos (INCT-FCx) and Fundação de Amparo à Pesquisa
do Estado de São Paulo (FAPESP) (2014/50983-3).

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## SUPPLEMENTAL MATERIAL

In this Supplemental Material, we provide additional details on the derivation
of the Quantum Vacuum Sagnac Phase (QVSP) arising from the interaction between
a 2-level atom in the ground state and a spinning nano-particle. We provide an
alternative derivation of the local QVSP based on the Berry connection of the
full quantum system “2-level atom+field” steered by the atomic position. We
show the consistency of our model with the well-known van-der-Waals potential
in the quasistatic regime. We also obtain the non-local contributions to the
quantum vacuum Sagnac phase, and detail the averaging procedure used for
finite atomic wave-packets.

We remind here for convenience a few notations of the main text. The scattered
electric-field Green’s function
$\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\tau)=\frac{1}{2\pi}\int
d\omega\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\omega)e^{-i\omega\tau}$
is obtained from the following tensor product

$\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\boldsymbol{G}^{0}(\mathbf{r},\mathbf{0},\omega).\boldsymbol{\alpha}^{\Omega}(\omega).\boldsymbol{G}^{0}(\mathbf{0},\mathbf{r}^{\prime},\omega)$
(6)

The free-field Green’s function is given by

$\boldsymbol{G}^{0}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\frac{e^{ikR}}{4\pi\epsilon_{0}R^{3}}\boldsymbol{T}(\mathbf{R},\omega),$
(7)

with
$T_{ij}(\mathbf{R},\omega)=(k^{2}R^{2}+ikR-1)\delta_{ij}-(k^{2}R^{2}+3ikR-3)R^{-2}R_{i}R_{j}$
and $k=\omega/c$,$\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}.$ We shall often
use the van-der-Waals (vdW) approximation of the free-field Green’s function

$\boldsymbol{G}^{\rm
vdW}(\mathbf{r},\mathbf{r}^{\prime})=\frac{1}{4\pi\epsilon_{0}R^{3}}\boldsymbol{T}^{0}(\mathbf{R})\qquad{\rm
with}\qquad T^{0}_{ij}(\mathbf{R})=3\hat{R}_{i}\hat{R}_{j}-\delta_{ij}$ (8)

and the unit vector $\hat{\mathbf{R}}=\mathbf{R}/R$.

The free-field Green’s function
$\boldsymbol{G}^{0}(\mathbf{r},\mathbf{r}^{\prime},\tau)$ obtained from (7) is
a retarded propagator, as well as the nano-particle polarizability response
$\mathbf{\alpha}^{\Omega}(\tau)$. By Eq. (6), the scattered Green’s function
$\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\tau)$
is a convolution of these functions, and is thus also a retarded propagator.

In practice, we use only the trace of scattered electric-field Green’s
functions
$\mathcal{G}_{\hat{\mathbf{E}}}^{R(H),S}(\mathbf{r},\mathbf{r}^{\prime},\tau)={\rm
Tr}\left[\boldsymbol{G}_{\hat{\mathbf{E}}}^{R(H),S}(\mathbf{r},\mathbf{r}^{\prime},\tau)\right].$
We assume from now on thermal equilibrium and the zero-temperature limit. The
Fluctuation-Dissipation Theorem (FDT) yields:

$\displaystyle\mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega)\\!$
$\displaystyle\\!=\\!$ $\displaystyle\\!2{\rm
sgn}(\omega)\mbox{Im}\left[\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega)\right]$
(9)

${\rm sgn}(\omega)$ is a sign function, i.e. ${\rm sgn}(\omega)=1$ (-1) for
$\omega>0$ ($\omega<0$). A similar equality holds for the dipole Green’s
functions.

##  I Obtention of the dispersive phase in the quasistatic approximation

We first use our model to obtain the dispersive phase associated to the
interaction between a moving atom and a nano-particle (spinning or at rest).
We take here the quasistatic approximation, namely we neglect the effects of
the atomic motion, one retrieves the expected vdW phase. Beyond this
consistency check, this derivation is used in Section V to estimate the
average Quantum Vacuum Sagnac Phase (QVSP) for finite atomic wave-packets.

We start by writing the local phases of Eq.(1) in the main text [noted here
$\phi_{{\rm qs}}$ to emphasize the quasistatic approximation]:

$\displaystyle\phi_{{\rm
qs},k}=\\!\frac{1}{4}\int_{-T/2}^{T/2}dt\int_{-T/2}^{T/2}dt^{\prime}\left[\frac{}{}g_{\hat{d}}^{H}(t,t^{\prime})\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t);t-t^{\prime})+g_{\hat{d}}^{R}(t,t^{\prime})\>\mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t);t-t^{\prime})\frac{}{}\right]$
(10)

where we have taken the quasistatic approximation by setting
$\mathbf{r}_{k}(t^{\prime})\simeq\mathbf{r}_{k}(t)$. By Parseval’s theorem, in
the long-time limit $\omega_{0}T\gg 1$ this phase reads:

$\phi_{{\rm
qs},k}=\\!\frac{1}{4(2\pi)}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\int_{-\infty}^{+\infty}\\!\\!\\!\\!\\!\\!d\omega\left[\frac{}{}g_{\hat{d}}^{H}(-\omega)\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t);\omega)+g_{\hat{d}}^{R}(-\omega)\>\mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t);\omega)\frac{}{}\right]$
(11)

The first and second term on the r.h.s. correspond respectively to the
quasistatic contributions of dipole and field fluctuations. To simplify
notations, we omit in the rest of this Section the path label and take simply
the phase along $\mathcal{P}\equiv[\mathbf{r}(t)]$.

###  I.1 Contribution of dipole fluctuations to the dispersive phase

Using the expression of the dipole Hadamard Green’s function of the dipole in
frequency

$g^{H}_{\hat{d}}(\omega)=\pi\alpha^{\rm
A}_{0}\omega_{0}[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]\,,$ (12)

one can write the phase $\phi^{\rm dip}_{{\rm qs}}$ related to dipole
fluctuations as

$\phi^{\rm dip}_{{\rm qs}}=\\!\frac{\alpha^{\rm
A}_{0}\omega_{0}}{4}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt{\rm
Re}\left[\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}(t),\mathbf{r}(t);\omega_{0})\right]$
(13)

For the considered van-der-Waals regime, we substitute in Eq. (6) the free-
field propagator $\mathbf{G}^{0}(\mathbf{r},\mathbf{r}^{\prime},\omega)$ by
its unretarded approximation (8):

$\phi^{\rm dip}_{{\rm qs}}=\\!\frac{\alpha_{0}^{\rm
A}\omega_{0}}{4(4\pi\epsilon_{0})^{2}}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\frac{1}{r(t)^{6}}\sum_{l,m,n}\left(\delta_{lm}-3\frac{r_{l}(t)r_{m}(t)}{r^{2}(t)}\right)\boldsymbol{\alpha}_{R\>mn}(\omega_{0})\left(\delta_{nl}-3\frac{r_{n}(t)r_{l}(t)}{r^{2}(t)}\right)$
(14)

where the summation is done over the three space coordinates $\\{x,y,z\\}$).
The static polarizability $\boldsymbol{\alpha}^{0}(\omega)$ is given in terms
of its dispersive and dissipative part as
$\boldsymbol{\alpha}(\omega)=\boldsymbol{\alpha}_{R}(\omega)+i\boldsymbol{\alpha}_{I}(\omega).$
For a non-rotating object, the polarizability tensor is diagonal, so that
$\boldsymbol{\alpha}_{R,mn}(\omega_{0})=\tilde{\alpha}_{R}(\omega_{0})\delta_{mn}.$
The summation can be done as

$\displaystyle\sum_{l,m,n}\left(\delta_{lmn}-3\frac{r_{l}(t)r_{m}(t)}{r^{2}(t)}\right)\delta_{mm}\left(\delta_{nl}-3\frac{r_{n}(t)r_{l}(t)}{r^{2}(t)}\right)$
$\displaystyle=$ $\displaystyle\left(3-2\times 3+3\times 3\right)=6$ (15)

Finally, the quasistatic phase is given by the time integral as the atom moves
along the path $\mathcal{P}=[\mathbf{r}(t)]$

$\phi^{\rm dip}_{{\rm qs}}=\\!\frac{3\omega_{0}\alpha^{\rm
A}_{0}\tilde{\alpha}_{R}(\omega_{0})}{2(4\pi\epsilon_{0})^{2}}\int_{-T/2}^{T/2}\frac{dt}{r(t)^{6}}$
(16)

###  I.2 Dispersive phase associated to the electric field fluctuations

The phase contribution arising from the field fluctuations reads, by virtue of
the FDT (9)

$\phi^{\rm f}_{{\rm
qs}}=\\!\frac{1}{2(2\pi)}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\int_{-\infty}^{+\infty}\\!\\!\\!\\!\\!\\!d\omega\>{\rm
sgn}(\omega)g_{\hat{d}}^{R}(-\omega)\>\mbox{Im}\left[\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r}(t),\mathbf{r}(t);\omega)\right]$
(17)

The retarded Green’s functions
$\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega),g^{R}_{\hat{\mathbf{d}}}(\omega)$
have real/imaginary parts which are even/odd functions of the frequency
$\omega$. By parity, only the real part ${\rm
Re}[g_{\hat{d}}^{R}(\omega)]=\frac{1}{2}(g_{\hat{d}}^{R}(\omega)+g_{\hat{d}}^{R}(-\omega))$
contributes to the integral (17). The retarded dipole Green’s function reads
$g^{R}_{\hat{d}}(\tau)=\alpha_{0}^{A}\omega_{0}\Theta(\tau)\sin\omega_{0}\tau$.
Using the relation

$\int_{0}^{+\infty}d\tau
e^{i(\omega-\omega_{0})\tau}=i\>\mathcal{P}\left(\frac{1}{\omega-\omega_{0}}\right)+\pi\delta(\omega-\omega_{0})$
(18)

where $\mathcal{P}$ is the principal value, we obtain

$g^{R}_{\hat{d}}(\omega)=\frac{1}{2}\alpha_{0}^{\rm
A}\omega_{0}\left[\mathcal{P}\left(\frac{1}{\omega+\omega_{0}}\right)-\mathcal{P}\left(\frac{1}{\omega-\omega_{0}}\right)\right]+\frac{i\pi}{2}\alpha_{0}^{\rm
A}\omega_{0}[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0}))].$ (19)

As a consistency check, one has
$g^{R}_{\hat{d}}(-\omega)=g^{R}_{\hat{d}}(\omega)^{*}$. Taking the vdW
approximation in the scattered field propagator, namely

$\displaystyle{\rm
Im}\left[\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]={\rm
Tr}\left[\boldsymbol{G}^{\rm
vdW}(\mathbf{r},\mathbf{0}).\boldsymbol{\alpha}_{I}(\omega).\boldsymbol{G}^{\rm
vdW}(\mathbf{0},\mathbf{r}^{\prime})\right]$ (20)

and using the isotropy of the polarizability tensor
$\boldsymbol{\alpha}_{I,ij}(\omega)=\tilde{\alpha}_{I}(\omega)\delta_{ij},$ we
arrive at the following expression for the quasistatic phase

$\phi^{\rm f}_{{\rm qs}}=\\!\frac{3\alpha_{0}^{\rm
A}\omega_{0}}{4\pi(4\pi\epsilon_{0})^{2}}\left[\int_{-\infty}^{+\infty}d\omega\>{\rm
sgn}(\omega)\tilde{\alpha}_{I}(\omega)\mathcal{P}\left(\frac{1}{\omega+\omega_{0}}-\frac{1}{\omega-\omega_{0}}\right)\right]\int_{-T/2}^{T/2}\frac{dt}{r(t)^{6}}$
(21)

where we have used again the identity (15). The frequency integral can be
simplified by parity arguments. Introducing

$\overline{\alpha_{I}}(\omega_{0})=\frac{1}{\pi}\mathcal{P}\left[\int_{-\infty}^{+\infty}d\omega\>{\rm
sgn}(\omega)\frac{\tilde{\alpha}_{I}(\omega)}{\omega+\omega_{0}}\right]$ (22)

Summing these contributions, we obtain the total quasistatic dispersive phase

$\phi_{{\rm qs}}=\\!\frac{3\omega_{0}\alpha_{0}^{\rm
A}(\tilde{\alpha}_{R}(\omega_{0})+\overline{\alpha_{I}}(\omega_{0}))}{2(4\pi\epsilon_{0})^{2}}\int_{-T/2}^{T/2}\frac{dt}{r(t)^{6}}$
(23)

In the numerical examples considered below
$\overline{\alpha_{I}}(\omega_{0})\approx\tilde{\alpha}_{R}(\omega_{0})$, so
that field and dipole fluctuations contribute almost equally to the
quasistatic phase.

###  I.3 Consistency with the vdW potential

Finally, we retrieve the vdW atom-atom interaction by replacing the scatterer
polarizability with the retarded atom Green’s function (19). Precisely, in
this Section only, one sets $\tilde{\alpha}(\omega)=g_{\hat{d}}^{R}(\omega)$.
As $\mathcal{P}\int
dx\frac{\delta(x+x_{0})}{x+x_{0}}=\lim_{\epsilon\rightarrow
0^{+}}\left[\int_{-\infty}^{-x_{0}-\epsilon}dx+\int_{-x_{0}+\epsilon}^{+\infty}dx\right]\left(\frac{\delta(x+x_{0})}{x+x_{0}}\right)=0,$
one finds $\overline{\alpha}_{I}(\omega_{0})=\frac{1}{4}\alpha_{0}^{A}$.
Similarly, $\tilde{\alpha}_{R}(\omega_{0})=\int
d\omega\delta(\omega-\omega_{0}){\rm
Re}[g_{\hat{d}}^{R}(\omega)]=\frac{1}{4}\alpha_{0}^{A}.$ Thus, the quasistatic
dispersive phase reads in this case

$\phi_{{\rm qs},k}^{\rm at.-at.}=\\!\frac{3\omega_{0}\alpha_{0}^{\rm
A2}}{4(4\pi\epsilon_{0})^{2}}\int_{-T/2}^{T/2}\frac{dt}{r(t)^{6}}$ (24)

This is consistent with the expected vdW phase, given by the temporal
integration $\phi_{{\rm qs},k}^{\rm
at.-at.}=-\frac{1}{\hbar}\int_{-T/2}^{T/2}V_{\rm vdW}(\mathbf{r}(t))$ of the
vdW potential between two ground-state atoms ThiruBook  $V_{\rm
vdW}(\mathbf{r})=-3\hbar\omega_{0}\alpha_{0}^{\rm
A2}/[4(4\pi\epsilon_{0})^{2}r^{6}].$

##  II Obtention of the Quantum Vacuum Sagnac Phase

The local QVSP $\phi^{\rm\Omega}_{k}$ corresponds to a local phase
[$\varphi_{kk}$, Eq.(1) of the main text]:

$\displaystyle\phi^{\rm\Omega}_{k}=\\!\frac{1}{4}\int_{-T/2}^{T/2}dt\int_{-T/2}^{T/2}dt^{\prime}\left[\frac{}{}g_{\hat{d}}^{H}(t,t^{\prime})\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t^{\prime});t-t^{\prime})+g_{\hat{d}}^{R}(t,t^{\prime})\>\delta\mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t^{\prime});t-t^{\prime})\frac{}{}\right]$
(25)

in which one retains only the scattered electric-field Green’s functions
$\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R(H),S}(\mathbf{r}_{k}(t),\mathbf{r}_{k}(t^{\prime});t-t^{\prime})$
associated to the $\Omega$-dependent polarizability tensor

$\delta\alpha^{\Omega}_{lm}(\omega)=i\sum_{n}\epsilon_{lmn}\Omega_{n}\tilde{\alpha}^{\prime}(\omega)$
(26)

$\epsilon_{klm}$ is the Levi-Civitta tensor. The $\Omega$-dependent retarded
electric-field Green’s function reads

$\delta\boldsymbol{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\boldsymbol{G}^{0}(\mathbf{r},\mathbf{0},\omega).\delta\boldsymbol{\alpha}^{\Omega}(\omega).\boldsymbol{G}^{0}(\mathbf{0},\mathbf{r}^{\prime},\omega)$
(27)

and the corresponding Hadamard Green’s function
$\delta\boldsymbol{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r},\mathbf{r}^{\prime},\omega)$
is obtained from the FDT (9) in the low-temperature limit.

In order to account for motional effects, we write the CM atomic position as:
$\mathbf{r}_{k}(t^{\prime})\simeq\mathbf{r}_{k}(t)-(t-t^{\prime})\mathbf{v}_{k}(t)$
and use a perturbative expansion of the Green’s tensor. The QVSP then admits
the formal expression $\phi^{\rm\Omega}=\phi^{\rm\Omega}_{\rm
qs}+\phi^{\rm\Omega}_{\rm mot}$ where $\phi^{\rm\Omega}_{\rm qs}$ and
$\phi^{\rm\Omega}_{\rm mot}$ represent respectively a quasistatic and a
motional contribution.The latter reads in the long-time limit $\omega_{0}T\gg
1$

$\displaystyle\phi^{\rm\Omega}_{\rm
mot,k}=\\!\frac{-1}{4}\int_{-T/2}^{T/2}\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\int_{-\infty}^{+\infty}\\!\\!\\!\\!d\tau\left[\frac{}{}\tau
g_{\hat{d}}^{H}(\tau)\left.\nabla_{\mathbf{r}^{\prime}}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\tau)\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}+\tau
g_{\hat{d}}^{R}(\tau)\left.\nabla_{\mathbf{r}^{\prime}}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r},\mathbf{r}^{\prime};\tau)\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}\frac{}{}\right]$
(28)

###  II.1 Cancellation of the quasistatic contribution to the QVSP

The quasistatic dipole contribution to the Sagnac phase is given by an
expression analog to Eq. (13), up the replacement of the scattered electric
field propagator by its $\Omega$-dependent contribution
$\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}(t),\mathbf{r}(t),\omega)$.
Nevertheless, by using Eqs. (26,27), one sees that
$\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r}(t),\mathbf{r}(t),\omega)=0$
by contraction of the antisymmetric Levi-Civitta tensor $\epsilon_{klm}$ with
the symmetric tensor
$G^{0}_{jk}(\mathbf{r}(t),\mathbf{0},\omega)G^{0}_{lj}(\mathbf{0},\mathbf{r}(t),\omega).$
As a result, the Sagnac phase has no quasistatic dipole contribution. The
quasistatic field contribution to the Sagnac phase, analog to Eq. (17),
vanishes by the same argument. Thus, there is no quasistatic contribution to
the QVSP, and $\phi^{\rm\Omega}_{k}=\phi^{\rm\Omega}_{\rm mot,k}.$

###  II.2 General expressions for the QVSP associated to dipole and field
fluctuations

####  II.2.1 Contribution of dipole fluctuations to the QVSP

From Eq. (28), one obtains the quantum Sagnac phase associated to dipole
fluctuations:

$\displaystyle\phi^{\rm\Omega,dip}_{k}=\frac{-1}{4}\int_{-T/2}^{T/2}\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\int_{-\infty}^{+\infty}\\!\\!\\!\\!d\tau\tau
g_{\hat{d}}^{H}(\tau)\left.\nabla_{\mathbf{r}^{\prime}}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\tau)\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}$
(29)

Using $\int d\tau\tau
e^{-i(\omega+\omega^{\prime})\tau}=(2\pi)i\delta^{\prime}(\omega+\omega^{\prime}),$
one can recast this phase contribution as a frequency integral

$\displaystyle\phi^{\rm\Omega,dip}_{k}=\frac{i}{4(2\pi)}\int_{-T/2}^{T/2}\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\int_{-\infty}^{+\infty}\\!\\!\\!\\!d\omega
g_{\hat{d}}^{H}(-\omega)\left.\nabla_{\mathbf{r}^{\prime}}\frac{d}{d\omega}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}$
(30)

We use again parity considerations, precisely that the real/imaginary part of
the electric field Green’s function are even/odd functions of the frequency
$\omega$. The Hadamard dipole Green’s function is of even parity. Thus, only
the imaginary part of the electric field Green’s function contributes, and by
using Eq. (12) one obtains:

$\displaystyle\phi^{\rm\Omega,dip}_{k}=\frac{-\omega_{0}\alpha_{0}^{\rm
A}}{4}\int_{-T/2}^{T/2}\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot{\rm
Im}\left[\left.\nabla_{\mathbf{r}^{\prime}}\frac{d}{d\omega}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t),\omega=\omega_{0}}$
(31)

####  II.2.2 Contribution of field fluctuations to the QVSP

The corresponding phase contribution

$\phi^{\rm\Omega,f}_{k}=\\!\frac{i}{4(2\pi)}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\int_{-\infty}^{+\infty}\\!\\!\\!\\!\\!\\!d\omega\>\>g_{\hat{d}}^{R}(-\omega)\>\left.\nabla_{\mathbf{r}^{\prime}}\frac{d}{d\omega}\delta\mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}$
(32)

can be written, by virtue of the FDT (9), as:

$\phi^{\rm\Omega,f}_{k}=\\!\frac{i}{2(2\pi)}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\int_{-\infty}^{+\infty}\\!\\!\\!\\!\\!\\!d\omega\>\>{\rm
sgn}(\omega)\>g_{\hat{d}}^{R}(-\omega)\>\left.\nabla_{\mathbf{r}^{\prime}}\frac{d}{d\omega}\mbox{Im}\left[\delta\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t)}$
(33)

The dipole and field retarded Green’s functions have real/imaginary parts
which are respectively of even/odd parity in the frequency. Taking into
account the sign function and the frequency derivative, one sees that only the
imaginary part of the retarded dipole Green’s function contributes. Using Eq.
(19), one obtains an expression of $\phi^{\rm\Omega,f}_{k}$ identical to (31),
i.e. $\phi^{\rm\Omega,f}_{k}=\phi^{\rm\Omega,dip}_{k}$. Thus the total QVSP
reads

$\phi^{\rm\Omega}_{k}=\\!\frac{-\omega_{0}\alpha_{0}^{\rm
A}}{2}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\>\mathbf{v}_{k}(t)\cdot\>\>{\rm
Im}\left[\left.\nabla_{\mathbf{r}^{\prime}}\frac{d}{d\omega}\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]\right|_{\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{k}(t),\omega=\omega_{0}}$
(34)

###  II.3 Expression of the QVSP as a geometric integral

The frequency derivative of the $\Omega$-dependent scattered Green’s function
yields three separate terms in the QVSP (34), associated respectively to the
delay of the material response and to the light travel time between the moving
atom and the nano-particle. To leading-order in the small parameter
$\omega_{0}r/c$, one has:

$\displaystyle\left.\frac{d}{d\omega}{\rm
Im}\left[\delta\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]\right|_{\omega=\omega_{0}}=\sum_{j,l,m,n}\epsilon_{lmn}\Omega_{n}$
$\displaystyle\left[\frac{}{}G_{jl}^{\rm
vdW}(\mathbf{r},\mathbf{0})\tilde{\alpha}_{R}^{\prime\prime}(\omega_{0})G_{mj}^{\rm
vdW}(\mathbf{0},\mathbf{r^{\prime}})\right.$ $\displaystyle\left.+G^{\prime
0}_{jl}(\mathbf{r},\mathbf{0},\omega_{0})\tilde{\alpha}_{R}^{\prime}(\omega_{0})G^{\rm
vdW}_{mj}(\mathbf{0},\mathbf{r}^{\prime})+G_{jl}^{\rm
vdW}(\mathbf{r},\mathbf{0})\tilde{\alpha}_{R}^{\prime}(\omega_{0})G^{\prime
0}_{mj}(\mathbf{0},\mathbf{r}^{\prime},\omega_{0})\frac{}{}\right]$

The first line term yields the leading QVSP contribution associated to the
material response time [noted $\phi^{\rm\Omega}_{{\rm vdW},k}$], while the
second line contributions capture corrections to the QVSP arising from
retardation effects [noted $\phi^{\rm\Omega,dip}_{(c),k}$].

From Eqs. (34,II.3), one finds the non-retarded QVSP contribution arising from
the nano-particle response time:

$\displaystyle\phi^{\rm\Omega}_{{\rm vdW},k}=\frac{\omega_{0}\alpha_{0}^{\rm
A}\tilde{\alpha}_{R}^{\prime\prime}(\omega_{0})}{4(4\pi\epsilon_{0})^{2}}\sum_{j,l,m,n}\epsilon_{lmn}\Omega_{n}\int_{\mathcal{P}_{k}}\frac{d\mathbf{r}}{r^{6}}\>\cdot
T^{0}_{jl}(\mathbf{r})\nabla_{\mathbf{r}}T^{0}_{mj}(\mathbf{r})$ (36)

We have used the vdW propagator (8) and $d\mathbf{r}=\mathbf{v}_{k}(t)dt$ in
Eq. (31) in order to obtain a geometric integral along the path
$\mathcal{P}_{k}$. We also use $\frac{\partial}{\partial
r_{i}}T^{0}_{lj}(\mathbf{r})=3(\delta_{il}\hat{r}_{j}+\delta_{ij}\hat{r}_{l}-2\hat{r}_{i}\hat{r}_{j}\hat{r}_{l})/r.$
Contributions to the differential
$d\mathbf{r}\cdot\sum_{j}T_{jl}^{0}(\mathbf{r})\nabla_{\mathbf{r}}T^{0}_{mj}(\mathbf{r})$
symmetric with respect to the indices $l,m$ actually disappear by contraction
with the Levi-Civitta tensor $\epsilon_{lmn}.$ Thus, the unretarded QVSP is
given by the integral along the path $\mathcal{P}_{k}$

$\displaystyle\phi^{\rm\Omega}_{{\rm vdW},k}=\frac{9\omega_{0}\alpha_{0}^{\rm
A}\tilde{\alpha}_{R}^{\prime\prime}(\omega_{0})}{2(4\pi\epsilon_{0})^{2}}\int_{\mathcal{P}_{k}}d\mathbf{r}\>\cdot\>\frac{\mathbf{\Omega}\times\mathbf{r}}{r^{8}}$
(37)

We now proceed to estimate the contribution to the QVSP
$\phi^{\rm\Omega,dip}_{(c),k}$ associated to retardation. We first write the
derivative of the free propagator, keeping only the leading-order in the small
parameter $kR\ll 1$:

$G^{\prime
0}_{\rm\>ij}(\mathbf{r},\mathbf{r}^{\prime},\omega)\simeq\frac{\omega}{c^{2}R}\left(\hat{R}_{i}\hat{R}_{j}+\delta_{ij}\right)$
(38)

From Eqs. (31,II.3), one obtains the contribution to the QVSP arising from
retardation effects

$\phi^{\rm\Omega}_{(c),k}=\frac{3\omega_{0}^{2}\alpha_{0}^{\rm
A}\tilde{\alpha}_{R}^{\prime}(\omega_{0})}{(4\pi\epsilon_{0})^{2}c^{2}}\int_{\mathcal{P}_{k}}d\mathbf{r}\>\cdot\>\frac{\mathbf{\Omega}\times\mathbf{r}}{r^{6}}$
(39)

In the example discussed below retardation effect represent a very small
correction, so that in practice
$\phi^{\rm\Omega}_{k}\simeq\phi^{\rm\Omega}_{{\rm vdW},k}.$

##  III Expression of the quantum vacuum Sagnac phase as a Berry phase.

To establish the connection between the motional phase and the Berry phase, we
consider the full quantum system “2-level atomic sytem+EM field” undergoing a
unitary evolution in presence of the spinning nano-particle and use a
perturbative treatment of the dipolar interaction.

For the considered two-level quantum emitter propagating in the quantum
vacuum, one notes $\hat{H}_{0}=\hat{H}_{A}+\hat{H}_{F}$ the interaction-free
Hamiltonian, with $\hat{H}_{A},\hat{H}_{F}$ are the atomic and the free-field
Hamiltonians respectively.The dipole interaction potential reads
$\hat{V}(\mathbf{r})=-\hat{\mathbf{d}}\cdot\hat{\mathbf{E}}(\mathbf{r})$ in
the Schrödinger picture, so that the full Hamiltonian is given by
$\hat{H}=\hat{H}_{0}+\hat{V}(\mathbf{r})$. Switching to the interaction
picture, the dipolar interaction becomes time-dependent

$\hat{V}(\mathbf{r},t)=e^{\frac{i}{\hbar}\hat{H}_{0}t}\hat{V}(\mathbf{r})e^{-\frac{i}{\hbar}\hat{H}_{0}t}.$
(40)

The system is assumed to be initially in the ground atomic state atom and with
a field in the vacuum state, so that the global quantum state is noted
$|0\rangle=|g\rangle_{A}\otimes|0\rangle_{F}.$ This initial state is coupled
through the dipolar interaction to the one-photon states
$|n\rangle=|e\rangle_{A}\otimes|1_{n}\rangle_{F}$ associated to each possible
mode $n$ of the EM field in presence of the boundary condition (the spinning
nano-particle). These are eigenstates of the free Hamiltonian $\hat{H}_{0}$,
i.e. $\hat{H}_{0}|n\rangle=\hbar\omega_{n}|n\rangle.$ The index $n$ represents
indeed a continuous parameter corresponding typically to a wave-vector and a
polarization. We use the short-hand notation $\sum_{n}$ for a sum over all
possible one-photon quantum states - this sum corresponds actually to a
continuous integral. It is convenient to use these notations, as we do not
need to evaluate the sum in order to establish the desired correspondence
between the QVSP and a Berry phase.

###  III.1 Expression of the QVSP in terms of transition amplitudes to one-
photon states

We first write the dispersive phase $\phi_{k}$ along a path $k$ (including
motional effects) as a second-order term in the Dyson series Nonlocal3

$\phi_{k}={\rm
Re}\left[\frac{{i}}{\hbar^{2}}\int_{-T/2}^{T/2}dt\int_{-T/2}^{t}dt^{\prime}\langle
0|\hat{V}(\mathbf{r}_{k}(t),t)\hat{V}(\mathbf{r}_{k}(t^{\prime}),t^{\prime})|0\rangle\right]$
(41)

with the dipolar potential $\hat{V}(\mathbf{r},t)$ in the interaction picture
(40). Eq.(41) is indeed equivalent to the starting point of the QVSP
derivation in the main text [Eq.(1) of the Letter], by virtue of the following
relation ${\rm Re}\left[\frac{i}{\hbar^{2}}\langle
0|(-\hat{d}(t)\cdot\mathbf{E}(\mathbf{r}_{k}(t),t))(-\hat{d}(t^{\prime})\cdot\mathbf{E}(\mathbf{r}_{k}(t^{\prime}),t^{\prime}))|0\rangle\right]=\frac{1}{4}\left[g^{R}_{\hat{\mathbf{d}}}(t,t^{\prime})\mathcal{G}_{\hat{\mathbf{E}}}^{R}(\mathbf{r}_{k}(t),t;\mathbf{r}_{k}(t^{\prime}),t^{\prime})+g^{R}_{\hat{\mathbf{d}}}(t,t^{\prime})\mathcal{G}_{\hat{\mathbf{E}}}^{H}(\mathbf{r}_{k}(t),t;\mathbf{r}_{k}(t^{\prime}),t^{\prime})\right]$.
As the phase is real, one can indifferently write

$\phi_{k}=\frac{{i}}{2\hbar^{2}}\int_{-T/2}^{T/2}dt\int_{-T/2}^{t}dt^{\prime}\left(\langle
0|\hat{V}(\mathbf{r}_{k}(t),t)\hat{V}(\mathbf{r}_{k}(t^{\prime}),t^{\prime})|0\rangle-h.c.\right)$
(42)

Following the derivation in the Letter, the motional phase correction is
obtained by performing a perturbative expansion in the retarded position
$\mathbf{r}_{k}(t^{\prime})=\mathbf{r}_{k}(t)-\tau\mathbf{\dot{r}}_{k}(t)$.
This motional phase correction corresponds to the QVSP (by symmetry, for the
considered trajectories, these motional corrections would vanish for a nano-
particle at rest)

$\phi_{k}^{\Omega}=\frac{i}{{2}\hbar^{2}}\int_{-T/2}^{T/2}dt\>\dot{\mathbf{r}}_{k}(t)\cdot\int_{0}^{t+T/2}d\tau\>\tau\>\left(\langle
0|\hat{V}(\mathbf{r}_{k}(t),t)\nabla_{\mathbf{r}}\hat{V}(\mathbf{r}_{k}(t),t-\tau)|0\rangle-h.c.\right)$
(43)

where we have done the variable change $\tau=t-t^{\prime}$. We then take the
long-time limit $T\gg{\rm min}(1/\omega_{n})$, use a closure relation, and
switch back to the Schrödinger picture with Eq. (40). The QVSP can thus be
written as

$\phi_{k}^{\Omega}=\frac{i}{{2}\hbar^{2}}\int_{\mathcal{P}_{k}}d\mathbf{r}\cdot\left(\sum_{n\neq
0}\langle 0|\hat{V}(\mathbf{r})|n\rangle\langle
n|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|0\rangle\int_{0}^{+\infty}d\tau\>\tau\>e^{-i(\omega_{n}-\omega_{0})\tau}-h.c.\right)$
(44)

We have used $d\mathbf{r}=dt\dot{\mathbf{r}}_{k}(t)$ to obtain a geometric
integral along the path $\mathcal{P}_{k}$. The sum has been restricted to
$n\neq 0$ without loss of generality as $\langle
0|\hat{V}(\mathbf{r})|0\rangle=0$. The integral over $\tau$ can be formally
expressed as

$\int_{0}^{+\infty}d\tau\>\tau\>e^{-i(\omega_{n}-\omega_{0})\tau}=-i\mathcal{P}\frac{1}{(\omega_{n}-\omega_{0})}\left(i\mathcal{P}\frac{1}{(\omega_{n}-\omega_{0})}+\pi\delta(\omega_{n}-\omega_{0})\right)\,,$

where $\mathcal{P}$ is the principal part. The contribution of the Dirac
distribution can be safely ignored as all the quantum states ${|n\rangle,n>0}$
have an energy strictly above that initial state, with an energy difference
corresponding at least to the two-level frequency. Finally, the QVSP can be
written as:

$\phi_{k}^{\Omega}=\frac{i}{{2}\hbar^{2}}\int_{\mathcal{P}_{k}}d\mathbf{r}\cdot\sum_{n\neq
0}\frac{\left(\langle 0|\hat{V}(\mathbf{r})|n\rangle\langle
n|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|0\rangle-\langle
0|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|n\rangle\langle
n|\hat{V}(\mathbf{r})|0\rangle\right)}{(\omega_{n}-\omega_{0})^{2}}$ (45)

###  III.2 Expression of the Berry phase of the full quantum system
“atom+field”

The full quantum system (internal atomic degrees of freedom+EM field) is
steered as the atomic position evolves along the path $\mathcal{P}_{k}$, and
acquires a geometric Berry phase resulting from the continuous modification of
the ground-state $|\psi_{0}(\mathbf{r})\rangle$:

$\phi_{k}^{\rm
Berry}=i\int_{\mathcal{P}_{k}}d\mathbf{r}\cdot\langle\psi_{0}(\mathbf{r})|\nabla_{\mathbf{r}}\psi_{0}(\mathbf{r})\rangle$
(46)

The atomic motion does not produce by itself no excitation of the internal
atomic level for the considered non-relativistic atomic velocities: thus, the
evolution is completely adiabatic. For the considered closed atom
interferometers where the two trajectories ($\mathcal{P}_{1},\mathcal{P}_{2}$)
split at the initial time and recombine at the final time, the geometric phase
contribution to the phase difference corresponds to the integration of the
parameter $\mathbf{r}$ over a cycle.

We derive below this Berry phase to second-order in the interaction potential.
Consistently with our calculation of the dispersive phase, we use standard-
perturbation theory in order to express the instantaneous, position-dependent
eigenstates in the interaction-free basis:

$\displaystyle|\psi_{0}(\mathbf{r})\rangle$ $\displaystyle=$
$\displaystyle|0\rangle+\sum_{n\neq 0}\frac{\langle
n|\hat{V}(\mathbf{r})|0\rangle}{\hbar(\omega_{n}-\omega_{0})}|n\rangle-\frac{1}{2}|0\rangle\sum_{n\neq
0}\frac{|\langle
n|\hat{V}(\mathbf{r})|0\rangle|^{2}}{\hbar^{2}(\omega_{n}-\omega_{0})^{2}}$
(47) $\displaystyle+\sum_{n\neq 0,n^{\prime}\neq n}|n\rangle\frac{\langle
n|\hat{V}(\mathbf{r})|n^{\prime}\rangle\langle
n^{\prime}|\hat{V}(\mathbf{r})|0\rangle}{\hbar^{2}(\omega_{0}-\omega_{n})(\omega_{n}-\omega_{n^{\prime}})}-\sum_{n\neq
0}|n\rangle\frac{\langle n|\hat{V}(\mathbf{r})|0\rangle\langle
0|\hat{V}(\mathbf{r})|0\rangle}{\hbar^{2}(\omega_{n}-\omega_{0})^{2}}+O\left(\hat{V}(\mathbf{r})^{3}\right)$

For the considered dipolar interaction the terms of the second line on the
right-hand side actually vanish, as the dipole operator elements are null on
the diagonal (at least two of the three system eigenstates
$\\{|n\rangle,|n^{\prime}\rangle,|0\rangle=|g\rangle_{A}\otimes|0\rangle_{F}\\}$)
involve the same atomic state).

One obtains:

$|\nabla_{\mathbf{r}}\psi_{0}(\mathbf{r})\rangle=\sum_{n\neq 0}\frac{\langle
n|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|0\rangle}{\hbar(\omega_{n}-\omega_{0})}|n\rangle-\frac{1}{2}|0\rangle\sum_{n\neq
0}\frac{\langle 0|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|n\rangle\langle
n|\hat{V}(\mathbf{r})|0\rangle+\langle 0|\hat{V}(\mathbf{r})|n\rangle\langle
n|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|0\rangle}{(\omega_{n}-\omega_{0})^{2}}+O\left(\hat{V}(\mathbf{r})^{3}\right)$
(48)

yielding the Berry connection

$i\langle\psi_{0}(\mathbf{r})\
|\nabla_{\mathbf{r}}\psi_{0}(\mathbf{r})\rangle=\frac{i}{2\hbar^{2}}\sum_{n\neq
0}\left(\frac{\langle 0|\hat{V}(\mathbf{r})|n\rangle\langle
n|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|0\rangle-\langle
0|\nabla_{\mathbf{r}}\hat{V}(\mathbf{r})|n\rangle\langle
n|\hat{V}(\mathbf{r})|0\rangle}{(\omega_{n}-\omega_{0})^{2}}\right)+O\left(\hat{V}(\mathbf{r})^{3}\right)$
(49)

This shows that the QVSP (45) corresponds to the Berry phase (46), i.e.
$\phi_{k}^{\Omega}=\phi_{k}^{\rm Berry}$.

##  IV Nonlocal Quantum Vacuum Sagnac phase

###  IV.1 General expression of the non-local QVSP.

We obtain in this Section the non-local QVSP, which depends simultaneously on
two paths $\alpha,\beta,$ and can be written as

$\phi_{\alpha\beta}^{\Omega}=\\!\frac{-\omega_{0}\alpha_{0}^{\rm
A}}{2}\int_{-T/2}^{T/2}\\!\\!\\!\\!\\!\\!dt\>\mathbf{v}_{\beta}(t)\cdot\>\>\nabla_{\mathbf{r}^{\prime}}\left.\frac{d}{d\omega}\mbox{Im}\left[\delta\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\right]\right|_{\mathbf{r}=\mathbf{r}_{\alpha}(t),\mathbf{r}^{\prime}=\mathbf{r}_{\beta}(t),\omega=\omega_{0}}$
(50)

for $\alpha\neq\beta$. We neglect retardation effects, so that the non-local
QVSP reads:

$\displaystyle\phi^{\rm\Omega}_{\alpha\beta}=\frac{\alpha_{0}^{A}\tilde{\alpha}^{\prime\prime}_{R}(\omega_{0})}{2(4\pi\epsilon_{0})^{2}}\int_{-T/2}^{T/2}dt\frac{\boldsymbol{\xi}_{\alpha\beta}(t)\cdot\mathbf{\Omega}}{r_{\alpha}(t)^{3}r_{\beta}(t)^{4}}$
(51)

with
$\mathbf{\xi}_{\alpha\beta\>m}(t)=r_{\beta}(t)\sum_{i,j,k,l}\epsilon_{klm}v_{\beta
i}(t)T^{0}_{jk}(\mathbf{r}_{\alpha}(t))\partial_{i}T^{0}_{lj}(\mathbf{r}_{\beta}(t)).$
We omit the explicit time dependence in the positions and velocities in order
to simplify notations. From Eq.(8), the quantity
$\boldsymbol{\xi}_{\alpha\beta}(t)$ can be expressed as

$\xi_{\alpha\beta\>m}(t)=3\sum_{i,j,k,l}\epsilon_{klm}v_{\beta
i}\left(3\hat{r}_{\alpha j}\hat{r}_{\alpha
k}-\delta_{jk}\right)\left(\delta_{ij}\hat{r}_{\beta
l}+\delta_{il}\hat{r}_{\beta j}-2\hat{r}_{\beta i}\hat{r}_{\beta
j}\hat{r}_{\beta l}\right)$

which can be put, after summation, in the simpler geometric form

$\boldsymbol{\xi}_{\alpha\beta}(t)=\left[1-2(\hat{\mathbf{r}}_{\alpha}\cdot\hat{\mathbf{r}}_{\beta})\right](\hat{\mathbf{r}}_{\alpha}\cdot\mathbf{v}_{\beta})\hat{\mathbf{r}}_{\alpha}\times\hat{\mathbf{r}}_{\beta}+(\hat{\mathbf{r}}_{\alpha}\cdot\hat{\mathbf{r}}_{\beta})\hat{\mathbf{r}}_{\alpha}\times\mathbf{v}_{\beta}$
(52)

Eqs. (51,52) constitute the general geometric form of the non-local QVSP.

###  IV.2 Non-local QVSP for parallel linear trajectories

Let us derive the non-local QVSP in a particular geometry of two linear paths
$\alpha,\beta$ equidistant to the spinning particle. We thus take
$\mathbf{r}_{1,2}(y)=\pm x_{1}\hat{\mathbf{x}}+y\hat{\mathbf{y}}$ with
$x_{1}>0$ and $\mathbf{v}_{1,2}=v\mathbf{y}.$ One has
$r_{1,2}(y)=r(y)=(x_{1}^{2}+y^{2})^{1/2}$,
$\hat{\mathbf{r}}_{\alpha}\cdot\hat{\mathbf{r}}_{\beta}=(y^{2}-x_{1}^{2})/r(y)^{2},$
$\hat{\mathbf{r}}_{\alpha}\cdot\hat{\mathbf{v}}_{\beta}=vy/r(y),$
$\hat{\mathbf{r}}_{\alpha}\times\mathbf{v}_{\beta}=(-1)^{\alpha+1}x_{1}v/r(y)\mathbf{z}$,
$\hat{\mathbf{r}}_{\alpha}\times\hat{\mathbf{r}}_{\beta}=(-1)^{\alpha+1}2x_{1}y/r(y)^{2}\hat{\mathbf{z}.}$
Taking the limit $vT/2=L/2\gg x_{1},$ one obtains

$\displaystyle\phi^{\rm\Omega}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{9(-1)^{\alpha+1}\alpha_{0}^{A}\tilde{\alpha}^{\prime\prime}_{R}(\omega_{0})\Omega}{2(4\pi\epsilon_{0})^{2}}\int_{-\infty}^{+\infty}\frac{dy}{(x_{1}^{2}+y^{2})^{7/2}}\left[\frac{(3x_{1}^{2}-y^{2})2x_{1}y^{2}}{(x_{1}^{2}+y^{2})^{5/2}}+\frac{(y^{2}-x_{1}^{2})x_{1}}{(x_{1}^{2}+y^{2})^{3/2}}\right]$
(53) $\displaystyle=$
$\displaystyle-\frac{27\pi(-1)^{\alpha+1}\alpha_{0}^{A}\tilde{\alpha}^{\prime\prime}_{R}(\omega_{0})\Omega}{64(4\pi\epsilon_{0})^{2}x_{1}^{6}}$

##  V Quantum Vacuum Sagnac phase for finite atomic wave packets

We estimate the QVSP for finite atomic wave packets propagating at a constant
velocity nearby the spinning nano-particle. We take for the atomic wave
function $\psi(x,y,z,t)=\psi_{\perp}(x,z)\psi_{/\\!/}(y,t)$ with
$\psi_{\perp}(x,z)=\frac{1}{\pi
w^{2}}e^{-\frac{(x-a)^{2}}{2w^{2}}}e^{-\frac{z^{2}}{2w^{2}}}.$ and
$\psi_{/\\!/}(y,t)=\frac{1}{\pi^{1/2}w_{y}}e^{-\frac{(y-vt)^{2}}{2w_{y}^{2}}+\frac{i}{\hbar}mvy},$
where $m$ is the Sodium mass. This corresponds to a CM atomic motion at a
constant velocity $v$ along a line $\\{x=a,y=vt,z=0\\}$ passing at the edge of
the spinning nano-particle, and with a constant transverse profile. This
approximation is legitimate as the time of flight is in practice too small to
yield a noticeable change in the wave-packet shape. Indeed for the considered
transition wave-length $\lambda_{0}$ and velocity $v$, the time-of flight is
approximately $T\sim\lambda_{0}/v\sim 10^{-10}\rm{\>s.}$ Even for the smallest
considered transverse width $w\sim 8{\rm nm}$, the transverse velocity
dispersion $\Delta v_{\perp}=\hbar/mw\sim 0.4\>{\rm m/s}$ results in a
negligible increase of the wave packet size during $T$. The experimentally
accessible phase is obtained through a WKB approach and an averaging procedure
as exposed in Refs. Cronin04 ; Vigue11 . Here, one proceeds to an averaging
over the bundle of linear trajectories
$\\{\mathbf{r}(t)=x\hat{\mathbf{x}}+vt\hat{\mathbf{y}}+z\hat{\mathbf{z}}\\}$
with a weight given by the transverse wave-function profile. The resulting
phase is given by

$\overline{\phi}(\Omega,v)={\rm
Arctan}\left[\frac{\overline{\sin(\phi(\Omega,x,z,v))}}{\overline{\cos(\phi(\Omega,x,z,v))}}\right]\,.$
(54)

Here $\phi(\Omega,x,z,v)$ is the sum of the quasistatic vdW and Sagnac phases,
namely $\phi(\Omega,x,z,v)=\phi^{\rm vdW}(x,z,v)+\phi^{\Omega}(x,z,v).$ The
averaging is defined as $\overline{f(x,z,...)}=\int
dxdzf(x,z,...)|\psi_{\perp}(x,z)|^{2}$. The wave-packet dispersion in the
longitudinal coordinate $y$ does not affect the dispersive phase. The QVSP
acquired along a line $\\{x^{\prime}=x,-L/2\leq y^{\prime}\leq
L/2,z^{\prime}=z\\}$ reads
$\phi^{\Omega}(x,z,v)=45\pi\ell^{6}_{\Omega}\,x/(32r_{\perp}^{7})$ with
$r_{\perp}=(x^{2}+z^{2})^{1/2}$ and
$\ell_{\Omega}=[\omega_{0}\alpha_{0}\alpha_{R}^{\prime\prime}(\omega_{0})\Omega/(4\pi\epsilon_{0})^{2}]^{1/6}.$
The quasistatic vdW phase depends on the velocity $v$ and corresponds to
($\overline{\alpha}_{I}(\omega_{0})\simeq\tilde{\alpha}_{R}(\omega_{0})$
within a $1\%$ approximation)

$\phi^{\rm
vdW}(x,z,v)=\frac{9\pi\alpha_{0}\omega_{0}(\tilde{\alpha}_{R}(\omega_{0})+\overline{\alpha}_{I}(\omega_{0}))}{8(4\pi\epsilon_{0})^{2}vr_{\perp})^{5}}\simeq\frac{9\pi\alpha_{0}\omega_{0}\tilde{\alpha}_{R}(\omega_{0})}{4(4\pi\epsilon_{0})^{2}vr_{\perp}^{5}}$