# Nanoparticle acts as a light source to make atom interferometers

M<EMAIL_ADDRESS>School of Physical Science and Technology,
Southwest Jiaotong University, Chengdu 610031, China

###### Abstract

Atomic Kapitza-Dirac (KD) scattering in the classical standing wave of lights
is widely used to make the laser-pulsed atom interferometers. In this
theoretical work, we show that the dielectric nanoparticle can be used as a
weak light source to generate atomic KD scattering and make atom
interferometer. We consider a cavity system consisting of atoms and a single
nanoparticle, where the nanoparticle is illuminated by the external laser
pulses. Atoms and the cavity mirrors are not illuminated by the laser beam, so
nanoparticle acts as a pulsed source to excite the cavity mode which non-
resonantly excites atomic internal state and generates a weak KD scattering of
atomic external motion. We use twice such scattering to split atomic path and
consequently use two classical laser pulses to recombine the separated paths
for generating the Ramsey-Bordé atom interferometer. This atom interferometer
is entangled with the centre-of-mass motion of nanoparticle and could be
developed to search the small nonclassicality of motional nanoparticle,
because the cavity mode excitation is associated with nanoparticle’s position
relative to the node or antinode of that optical mode.

##  I Introduction

The nanometer-size dielectric particle (nanoparticle) consisting of millions
of atoms has been proposed to test quantum gravity and the spontaneous
collapse models of wave function Romero-Isart-PRL ; Romero-Isart-collapse ;
Quantum gravity1 ; Quantum gravity2 ; RMP ; NP-decoherence ; NP-Gdecoherence ;
CQG-decoherence . The advantage on quantum gravity testing can be easily seen
from the Newtonian gravitational potential energy, $Gm_{1}m_{2}/r$, where $r$
is the distance between two massive spheres $m_{1}$ and $m_{2}$, and
$G=6.67\times 10^{-11}{\rm N}\cdot{\rm m}^{2}\cdot{\rm kg}^{-2}$ the
gravitational constant. This energy is equivalent to that of a microwave
photon $\hbar\omega_{m}$ of frequency $\omega_{m}=Gm_{1}m_{2}/(\hbar r)$.
Using atoms as the test particles, we have $\omega_{m}=6.4\times 10^{-22}$ Hz,
with the mass $m_{1}=m_{2}=10^{-25}$ kg and the distance $r=10\,\,\mu{\rm m}$
between atoms. It should be extremely difficult to measure such a microwave
photon. Note that, the test particles can not be too close for avoiding the
electromagnetic interactions, i.e., the Casimir-Polder interaction between
neutral particles. Considering if two molecules of mass $m_{1}=m_{2}=1.6\times
10^{-23}$ kg molecular , the frequency $\omega_{m}=1.6\times 10^{-17}$ Hz is
still small. However, if the test particles are two nanoparticles of more mass
$m_{1}=m_{2}=10^{-15}$ kg, the frequency $\omega_{m}=0.064$ Hz, or say the
phase shift $\omega_{m}t$ with an effective interaction time $t$, seems to be
experimental detectable by using the matter wave interferometry of
nanoparticles Quantum gravity1 ; Quantum gravity2 . Basically, the
gravitational interaction energy is proportional to the quadratic mass Diosi ;
Penrose ; arXivGD , so the large particles seem to be essential for
demonstrating the gravity-induced entanglement Quantum gravity1 ; Quantum
gravity2  and the gravitational decoherence RMP ; NP-decoherence ; NP-
Gdecoherence ; CQG-decoherence .

The matter wave interferometers of nanoparticles have been not realized yet,
but great progress has been made both in theories and experiments Barker2010 ;
Barker2012 ; EXP-PNAS2010 ; EXP-PRL2012 ; EXP-NJP2013 ; EXP-PRL2016 ; EXP-
PRL2019-1 ; EXP-PRL2019-2 ; EXP-Science ; EXP-Nature ; Rep2020 ; Radim-Filip ;
Quantum-Sci-Techol ; arXiv , focused on nanoparticles loading, trapping,
cooling, and measurement. The experiment EXP-Science  shows that the single
nanoparticle can be trapped as a harmonic oscillator by the optical tweezer
and can be cooled into the motional quantum ground state by using the cavity
QED. In that experiment, the position and momentum uncertainties of the cooled
nanoparticle are on the orders of $\Delta_{x}=\sqrt{\hbar/(2m\omega)}\approx
0.25\,\,{\rm\AA}$ and $\Delta_{p}=\sqrt{m\hbar\omega/2}=m\times 13\,\,\mu{\rm
m}\cdot{\rm s}^{-1}$, with the vibrational frequency $\omega=2\pi\times 80$
kHz and the $m=10^{8}$ atomic mass units (amu) of harmonic oscillator.
Switching off optical trap instantaneously, the nanoparticle will be released
as a free fall MAQRO2012 ; MAQRO2015 ; NC2021 ; free-N-KD . This provides
opportunities of making the matter wave interferometers of nanoparticles
Romero-Isart-PRL ; Romero-Isart-collapse .

A major difficulty of making nanoparticle interferometers is the split-
recombination of nanoparticle’s wave packet. According to Wigner equation
Wigner1932 ; MiaoPRD , i.e., the Schrödinger equation in (position, momentum)
phase space, the dynamical nonclassicality of centre-of-mass motion of
nanoparticle is generated by the high-order gradient potential that the
particle underwent. Laser can provide the considerable high-order nonlinear
potential for test particles due to the small wave length of light, and thus
has been widely used to make atom interferometers. However, this seems
insufficient for coherently splitting and recombining nanoparticle’s wave
packet. Using the atomic mass $m_{a}=86.9$ amu D2  and the wave number
$k=2\pi/\lambda=2\pi/(780\,{\rm nm})$ in the usual atom interferometers, one
can easily find that the single photon recoil velocity $\hbar k/m\approx
5\,\,{\rm nm}\cdot{\rm s}^{-1}$ of nanoparticle is far smaller than that
$\hbar k/m_{a}\approx 5.8\,\,{\rm mm}\cdot{\rm s}^{-1}$ of atom, because the
nanoparticle of mass $m=10^{8}$ amu is overweight. The long time free
evolutions of nanoparticle in microgravity laboratory MAQRO2012 ; MAQRO2015 ;
NC2021 , such as $10$ s, may generate the experimentally detectable
interference fringes, but shall face the obstacles of decoherence, such as the
collision with residual air molecules and the blackbody radiation Romero-
Isart-collapse . The multi-photon scattering of highly transparent
nanoparticle would be useful NC2021 ; free-N-KD . Similar to the Kapitza-Dirac
(KD) scatterings of atom RMP-AKD ; KD-E ; ZHU-ATOM  and molecules KD-molecules
; KD-molecules2019  in the non-resonant standing lights, the multi-photon KD
scattering of dielectric nanoparticle would results in many paths of
nanoparticle, and thus needs also the high spatial resolution measurements,
for example, by using the tightly focused ultraviolet (UV) laser pulse
MAQRO2015 ; MAQRO2012 . Different from the above traditional approaches,
Romero-Isart et al. Romero-Isart-PRL ; Romero-Isart-collapse  propose using
the feature of wave collapse to generate nanoparticle interference. Based on
the entanglement between nanoparticle and photons in a cavity, they measure
the output photons for collapsing nanoparticle’s state.

In this article, we design an atom interferometer to search the small
nonclassicality of centre-of-mass motion of nanoparticle. The atomic matter
wave interferometers have advantages not only on the measurements of
gravitational acceleration ChuG  but also on the detections of some weak
interactions between atom and electromagnetic fields, such as the He-McKellar-
Wilkens phase HMW  and the relativistic Aharonov-Casher effect AC  of motional
atoms. The atomic matter waves may be also very sensitive to the weak lights
radiated by the dielectric nanoparticle AN-PRA ; ZJ-PRA ; NWJ-OC ; NWJ-PRA ;
AN-Paul . For that, we consider a hybrid cavity system consisting of an atom
and a nanoparticle. The input external laser beam is the pulsed one and
directed at the nanoparticle, does not illuminate atom and the cavity mirrors.
So nanoparticle acts as a pulsed source to excite cavity modes and
consequently generate the atomic KD scattering. This scattering is explained
by the position dependent ac Stark effect of atom in the non-resonant standing
wave. Atomic internal state is not excited, and so the present KD scattering
is also effective for the case that many atoms (atomic cloud) are
simultaneously in the cavity. This allowed us to make an atom interferometer,
for example, the symmetric Ramsey-Bordé atom interferometer. This atom
interferometer has several uses, demonstrating the cavity QED itself,
detecting the classical motions of nanoparticle, and searching the small
nonclassicality of nanoparticle, because the cavity mode excitation is related
to the centre-of-mass motion of nanoparticle.

The article is organized as follows. In Sec. II, we use the well-known Jaynes-
Cummings (JC) model JC  to describe atom-cavity interaction and use the widely
recognized theories in ref. Romero-Isart2011  to describe the laser-
nanoparticle-cavity interaction. Considering atomic internal transition is
detuning a litter with the input laser, we use several unitary transformations
to derive atomic ac Stark effect (called also two-photon Stark effect). The
obtained Stark shift is directly coupled with the centre-of-mass motion of
nanoparticle and is the crucial one for generating the nanoparticle dependent
KD scattering of atoms. Based on the obtianed KD scattering, in Sec. III, we
design a symmetric Ramsey-Bordé atom interferometer to measure the possible
nonclassicality of motional nanoparticle. The time-evolution operators method
is used to derive the signal of the considered atom interferometer. Finally,
we present our conclusion in Sec. IV.

##  II The weak KD scattering of atom

Considering an atom and a nanoparticle being inside a cavity, as shown in Fig.
1, where the nanoparticle is illuminated by a classical laser pulse which
propagates along the perpendicular direction of cavity axis. Nanoparticle acts
a light source to nearly-resonantly excite one of the cavity modes and
consequently generate the atomic ac Stark effect when the transition frequency
between two atomic levels is detuning a little with that cavity mode. In this
section, we use the unitary transformation approach to derive the considered
ac Stark effect. The obtained ac Stark effect depends not only on the atomic
position but also the nanoparticle’s position, and thus results in the atom-
nanoparticle coupled KD scattering. We temporarily consider the single atom
and the single nanoparticle inside the cavity. In the later, we will easily
find that the single atom theory is also effective to describe the practical
case that many atoms are simultaneously inside the interaction cavity.

Figure 1: Nanoparticle acts as a light source to generate atomic KD
scattering. An external classical laser beam (propagating along $z$ direction)
is directed at the nanoparticle, does not illuminate atom and the cavity
mirrors, so nanoparticle is used as a light source to nearly-resonantly excite
one of the cavity modes and generates atomic KD scattering. This scattering is
coupled with the centre-of-mass motion of nanoparticle, as the mode excitation
is associated with the position of nanoparticle relative to the node or
antinode of that mode.

###  II.1 The Hamiltonian

The Hamiltonian of the cavity system Fig. 1 is written as

$\displaystyle\hat{H}=\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{2}}{2m}+\hat{H}_{\rm
AC}+\hat{H}_{\rm NF}+\hat{H}_{\rm CF}\,.$ (1)

The $\textbf{p}_{a}^{2}/(2m_{a})$ and $\textbf{p}^{2}/(2m)$ are the non-
relativistic kinetic energies of atom and nanoparticle, respectively. The
Hamiltonian $\hat{H}_{\rm AC}$ describes the interaction between atom and the
cavity mode. The $\hat{H}_{\rm NF}$ describes the interactions between
nanoparticle and the electromagnetic fields of the cavity modes, the input
classical laser, and the free environment. The $\hat{H}_{\rm CF}$ is the
Hamiltonian of cavity modes interacting with the free fields in environment.
The interaction between atom and the free fields is negligible, because the
atom is not excited throughout the paper.

We use the well-known JC model JC  to describe the atom-cavity interaction,
i.e.,

$\displaystyle\hat{H}_{\rm
AC}=\hbar\Omega_{a}(\hat{a}\hat{\sigma}_{+}e^{i\delta_{ac}t}+\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i\delta_{ac}t})\,,$
(2)

where $\delta_{ac}=\omega_{a}-\omega_{c}$ is a small detuning between atom and
a selected standing wave between the cavity mirrors. The frequencies of cavity
standing waves are on the order of $\omega_{n}=nc/(2L)\approx n\times 7.5$
GHz, with a cavity length $L=2$ cm, for example. The $7.5$ GHz means that,
only one cavity mode could be effective to generate the considerable effect of
atom, because the detunings between atom and the other cavity modes are too
large. This is why we say the selected cavity mode. In the above JC model, the
$\hat{a}^{\dagger}$ and $\hat{a}$ are respectively the bosonic creation and
annihilation operators of that cavity mode. The
$\hat{\sigma}_{+}=|e\rangle\langle g|$ and $\hat{\sigma}_{-}=|e\rangle\langle
g|$ are called usually raising and lowering operators, which generate the
transition between atomic internal states. The resonant Rabi frequency
$\Omega_{a}$ of atom-cavity interaction is related to the atomic position
$(x_{a},y_{a},z_{a})$, for example, $\Omega_{a}=\Omega_{a0}\cos(kx_{a})$ with
the plane wave approximation, and where $\Omega_{a0}$ is the maximum Rabi
frequency at the antinode of the standing wave. The $\cos(kx_{a})$ is the key
to generate atomic KD scattering, with the optical $k$.

The Hamiltonian of nanoparticle-lights interaction is given by refs. Romero-
Isart2011 ; Romero-Isart2019 ,

$\displaystyle\hat{H}_{\rm NF}$
$\displaystyle=-\frac{\epsilon_{c}\epsilon_{0}}{2}\int_{V_{N}}\textbf{E}(\textbf{x}+\textbf{x}^{\prime})\cdot\textbf{E}(\textbf{x}+\textbf{x}^{\prime}){\rm
d}^{3}\textbf{x}^{\prime}\approx-\frac{\epsilon_{c}\epsilon_{0}V_{N}}{2}\textbf{E}(\textbf{x})\cdot\textbf{E}(\textbf{x})\,,$
(3)

with the permittivity $\epsilon_{0}$ of vacuum, and where
$\epsilon_{c}=3(\epsilon_{r}-1)/(\epsilon_{r}+2)$, with $\epsilon_{r}$ being
the relative dielectric constant of nanoparticle. The x is the center of mass
of the dielectric sphere (with volume $V_{N}$), and $\textbf{x}^{\prime}$ is
the relative coordinate of the volume-element ${\rm d}^{3}\textbf{x}^{\prime}$
in the dielectric sphere. Considering the radius $R$ of nanoparticle is far
smaller than the wavelength of lights, so the integral has been solved by
using the point-particle approximation. The electric field can be written as

$\displaystyle\textbf{E}(\textbf{x})=\textbf{E}^{\prime}(\textbf{x},t)+\textbf{E}_{c}\cos(kx)(\hat{a}^{\dagger}e^{i\omega_{c}t}+\hat{a}e^{-i\omega_{c}t})+\textbf{E}_{l}[e^{-i(\omega_{l}t-k_{l}z)}+e^{i(\omega_{l}t-k_{l}z)}]\,,$
(4)

where $\textbf{E}^{\prime}(\textbf{x},t)$ is used to explain the nanoparticle
radiated lights in free space and the possible excitations of some other
standing waves (different from $\omega_{c}$) between the cavity mirrors. The
$\textbf{E}_{c}=\textbf{e}_{c}\sqrt{\hbar\omega_{c}/(2\epsilon_{0}V_{c})}$ is
the electric field strength of the cavity mode of wave number
$k=\omega_{c}/c$, and where $V_{c}$ is the volume of cavity. The
$\textbf{e}_{c}$ is the polarization vector of electric field perpendicular to
the cavity axis. The $\textbf{E}_{l}$ is the electric field strength of the
input classical laser beam, which propagates along $z$ direction, with wave
number $k_{l}=\omega_{l}/c$ and frequency $\omega_{l}$. The plane waves
approximation is used, i.e., considering the positional uncertainties of
particles are far smaller than the waists of laser and the cavity, so the
electric fields $\textbf{E}_{c}(y,z)\approx\textbf{E}_{c}$ and
$\textbf{E}_{l}(x,y)\approx\textbf{E}_{l}$ are approximately invariant during
the short interaction time (of laser pulse).

Inserting electric field (4) into Eq. (3), we write the interaction
Hamiltonian of nanoparticle as

$\displaystyle\hat{H}_{\rm NF}$
$\displaystyle=-\hbar\Omega_{c}\hat{a}^{\dagger}\hat{a}-\hbar(\Omega_{l}\hat{a}^{\dagger}e^{i\delta_{cl}t}+\Omega_{l}^{*}\hat{a}e^{-i\delta_{cl}t})+\widetilde{H}_{\rm
NF}\,,$ (5)

with the laser-cavity detuning $\delta_{cl}=\omega_{c}-\omega_{l}$, the two-
photon Rabi frequencies

$\displaystyle\Omega_{c}=\frac{\epsilon_{c}\epsilon_{0}V_{N}\textbf{E}_{c}\cdot\textbf{E}_{c}}{2\hbar}\cos^{2}(kx)\,,$
(6)

and

$\displaystyle\Omega_{l}=\frac{\epsilon_{c}\epsilon_{0}V_{N}\textbf{E}_{c}\cdot\textbf{E}_{l}e^{ik_{l}z}}{2\hbar}\cos(kx)\,.$
(7)

The $\widetilde{H}_{\rm NF}$ contains all the other terms in (3), for example,
the interaction between nanoparticle and the other cavity modes. Nevertheless,
the excitations about the other cavity modes should be weak because the input
laser is large detuning with those cavity modes. According to Eqs. (1) and
(5), we can write the Hamiltonian of the total system as

$\displaystyle\hat{H}=$ $\displaystyle\hat{H}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{2}}{2m}+\hbar\Omega_{a}(\hat{a}\hat{\sigma}_{+}e^{i\delta_{ac}t}+\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i\delta_{ac}t})$
(8)
$\displaystyle-\hbar\Omega_{c}\hat{a}^{\dagger}\hat{a}-\hbar(\Omega_{l}\hat{a}^{\dagger}e^{i\delta_{cl}t}+\Omega_{l}^{*}\hat{a}e^{-i\delta_{cl}t})\,,$

with $\hat{H}_{\rm NCF}=\hat{H}_{\rm CF}+\widetilde{H}_{\rm NF}$. The second
line in Hamiltonian (8) describes the non-resonant excitation of the desired
cavity mode $\omega_{c}$ which is used to generate the ac Stark effect of
atom.

###  II.2 The unitary transformations approach for atomic ac stark effect

In order to clearly show the ac Stark effect of atom, we use several unitary
transformations to change Hamiltonian (8). Firstly, we perform an unitary
transformation $\exp(i\delta_{cl}t\hat{a}^{\dagger}\hat{a})$ to the total
system, then Hamiltonian (8) becomes

$\displaystyle\hat{H}_{1}=$ $\displaystyle\hat{H}^{\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{2}}{2m}+\hbar\Omega_{a}(\hat{a}\hat{\sigma}_{+}e^{i\delta_{al}t}+\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i\delta_{al}t})+\hat{H}_{D}\,,$
(9)

with the driving cavity mode

$\displaystyle\hat{H}_{D}=\hbar\delta_{cl}^{\prime}\hat{a}^{\dagger}\hat{a}-\hbar(\Omega_{l}\hat{a}^{\dagger}+\Omega_{l}^{*}\hat{a})\,,$
(10)

and the detunings $\delta_{al}=\omega_{a}-\omega_{l}$ and
$\delta_{cl}^{\prime}=\delta_{cl}-\Omega_{c}$. The Hamiltonian
$\hat{H}^{\prime}_{\rm NCF}$, with apostrophe, is that after the above unitary
transformation, i.e., $\hat{H}^{\prime}_{\rm
NCF}=\exp(-i\delta_{cl}t\hat{a}^{\dagger}\hat{a})\hat{H}_{\rm
NCF}\exp(i\delta_{cl}t\hat{a}^{\dagger}\hat{a})$, and the similar
presentations will be used in follows for the more unitary transformations.

Using the second unitary evolution $\hat{D}=\exp(-it\hat{H}_{\rm D}/\hbar)$ of
the driving cavity mode, we compute the so-called BCH formula Zassenhaus-2012
; Zassenhaus-1967

$\displaystyle\hat{D}^{\dagger}\hat{a}^{\dagger}\hat{D}$
$\displaystyle=\hat{a}^{\dagger}+\frac{it}{\hbar}[\hat{H}_{D},\hat{a}^{\dagger}]+\frac{1}{2!}(\frac{it}{\hbar})^{2}[\hat{H}_{D},[\hat{H}_{D},\hat{a}^{\dagger}]]+\cdots$
(11)
$\displaystyle=\hat{a}^{\dagger}+it\delta_{cl}^{\prime}(\hat{a}^{\dagger}-\frac{\Omega_{l}^{*}}{\delta_{cl}^{\prime}})+\frac{(it\delta_{cl}^{\prime})^{2}}{2!}(\hat{a}^{\dagger}-\frac{\Omega_{l}^{*}}{\delta_{cl}^{\prime}})+\cdots$
$\displaystyle=\hat{a}^{\dagger}e^{i\delta_{cl}^{\prime}t}-\frac{\Omega_{l}^{*}}{\delta_{cl}^{\prime}}(e^{i\delta_{cl}^{\prime}t}-1)\,,$

and which can be checked by $\hat{D}^{\dagger}\hat{H}_{D}\hat{D}=\hat{H}_{D}$
with
$(\hat{D}^{\dagger}\hat{a}^{\dagger}\hat{D})^{\dagger}=\hat{D}^{\dagger}\hat{a}\hat{D}$.
Consequently, Hamiltonian (9) becomes

$\displaystyle\hat{H}_{2}=$ $\displaystyle\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m}+\hbar\Omega_{a}(\hat{a}\hat{\sigma}_{+}e^{i\Delta
t}+\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i\Delta t})$ (12)
$\displaystyle-\hbar\Omega_{a}\left[\eta\hat{\sigma}_{+}(e^{i\Delta
t}-e^{i\delta_{al}t})+\eta^{*}\hat{\sigma}_{-}(e^{-i\Delta
t}-e^{-i\delta_{al}t})\right]\,,$

with

$\displaystyle\Delta=\omega_{a}+\Omega_{c}-\omega_{c}=\delta_{al}-\delta^{\prime}_{cl}\,,$
(13)

and

$\displaystyle\eta=\frac{\Omega_{l}}{\delta_{cl}^{\prime}}=\frac{\Omega_{l}}{\omega_{c}-\omega_{l}-\Omega_{c}}\,.$
(14)

The parameter $\eta$ is controlled by $\Omega_{l}$ (i.e., the power of input
classical laser), so we consider some values in the regime of $|\eta|\geq 1$.
Furthermore, we assume the large detuning condition
$\Omega_{a}\ll\delta_{al}\ll\Delta$ is satisfied, which implies
$\Omega_{l}\gg\omega_{a}-\omega_{l}=\delta_{al}\gg\Omega_{a}$. Limited within
these conditions, the terms associated with $\exp(\pm i\Delta t)$ in
Hamiltonian (12) are negligible, and thus

$\displaystyle\hat{H}_{2}\approx$ $\displaystyle\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m}+\hbar\Omega_{a}\left(\eta\hat{\sigma}_{+}e^{i\delta_{al}t}+\eta^{*}\hat{\sigma}_{-}e^{-i\delta_{al}t}\right)\,.$
(15)

The last term describes a driving atom but decouples with the cavity mode. In
order to clearly show the effect of that term, we take the third unitary
evolution $\exp(i\delta_{al}t|e\rangle\langle e|)$ to change Hamiltonian (15),

$\displaystyle\hat{H}_{3}=$ $\displaystyle\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m}+\hbar\delta_{al}|e\rangle\langle
e|+\hbar\Omega_{a}\left(\eta\hat{\sigma}_{+}+\eta^{*}\hat{\sigma}_{-}\right)\,.$
(16)

The last term in Hamiltonian (16) describes the resonant excitation of atom,
but is suppressed by the detuning $\hbar\delta_{al}|e\rangle\langle e|$. Thus,
we take the unitary transformation $\exp(-i\hat{R})$ with a weak resonant
operator $\hat{R}=\mu\hat{\sigma}_{+}+\mu^{*}\hat{\sigma}_{-}$, and where
$\mu=-i\eta\Omega_{a}/\delta_{al}$ is the small complex number and time
independent. This unitary transformation can exactly eliminate the resonant
coupling
$\hbar\Omega_{a}\left(\eta\hat{\sigma}_{+}+\eta^{*}\hat{\sigma}_{-}\right)$ in
Eq. (16), such that the standard Stark effect of atom appears. The computation
is based on the following two-order BCH formulas of atom,

$\displaystyle e^{i\hat{R}}\hat{\sigma}_{+}e^{-i\hat{R}}$
$\displaystyle=\hat{\sigma}_{+}+i[\hat{R},\hat{\sigma}_{+}]+\frac{i^{2}}{2!}[\hat{R},[\hat{R},\hat{\sigma}_{+}]]+\mathcal{O}(|\mu|^{3})$
(17)
$\displaystyle\approx\hat{\sigma}_{+}-i\mu^{*}\hat{\sigma}_{z}+\mu^{*}\mu^{*}\hat{\sigma}_{-}-\mu^{*}\mu\hat{\sigma}_{+}$
$\displaystyle=\hat{\sigma}_{+}+\frac{\eta^{*}\Omega_{a}}{\delta_{al}}\hat{\sigma}_{z}+\mu^{*}\mu^{*}\hat{\sigma}_{-}-\mu^{*}\mu\hat{\sigma}_{+}\,,$

and

$\displaystyle e^{i\hat{R}}|e\rangle\langle e|e^{-i\hat{R}}$
$\displaystyle=|e\rangle\langle e|+i[\hat{R},|e\rangle\langle
e|]+\frac{i^{2}}{2!}[\hat{R},[\hat{R},|e\rangle\langle
e|]]+\mathcal{O}(|\mu|^{3})$ (18) $\displaystyle\approx|e\rangle\langle
e|+i(\mu^{*}\hat{\sigma}_{-}-\mu\hat{\sigma}_{+})-|\mu|^{2}\hat{\sigma}_{z}$
$\displaystyle=|e\rangle\langle
e|-\frac{\eta^{*}\Omega_{a}}{\delta_{al}}\hat{\sigma}_{-}-\frac{\eta\Omega_{a}}{\delta_{al}}\hat{\sigma}_{+}-|\frac{\eta\Omega_{a}}{\delta_{al}}|^{2}\hat{\sigma}_{z}\,.$

Note that,
$\exp(i\hat{R})\hat{\sigma}_{-}\exp(-i\hat{R})=[\exp(i\hat{R})\hat{\sigma}_{+}\exp(-i\hat{R})]^{\dagger}$,
and the following commutators are used,

$\displaystyle[\hat{R},\hat{\sigma}_{z}]=2(\mu^{*}\hat{\sigma}_{-}-\mu\hat{\sigma}_{+})\,,\,\,\,\,\,\,\,\,[\hat{R},|e\rangle\langle
e|]=\mu^{*}\hat{\sigma}_{-}-\mu\hat{\sigma}_{+}\,,$ (19)
$\displaystyle[\hat{R},\hat{\sigma}_{+}]=-\mu^{*}\hat{\sigma}_{z}\,,\,\,\,\,\,\,\,\,[\hat{R},\hat{\sigma}_{-}]=\mu\hat{\sigma}_{z}\,,\,\,\,\,\,\,\,\,\hat{\sigma}_{z}=|e\rangle\langle
e|-|g\rangle\langle g|\,.$

Following the above BCH formulas, Hamiltonian (16) is transformed to be

$\displaystyle\hat{H}_{4}=$ $\displaystyle\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m}+\widetilde{H}(|\eta\Omega_{a}\mu|^{2}\hat{\sigma}_{\pm})+\hbar\delta_{al}|e\rangle\langle
e|+\hbar\frac{|\eta|^{2}\Omega_{a}^{2}}{\delta_{al}}\hat{\sigma}_{z}\,\,$ (20)

with

$\displaystyle e^{i\hat{R}}(\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m})e^{-i\hat{R}}\approx\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime 2}}{2m}\,.$
(21)

The last term in (20) is the standard Stark effect of atom. The term
$\widetilde{H}(|\eta\Omega_{a}\mu|^{2}\hat{\sigma}_{\pm})$ refers to the high
order small quantity $|\mu|^{2}$, and is negligible. The unitary
transformation $\exp(-i\hat{R})$, with
$\hat{R}=\mu\hat{\sigma}_{+}+\mu^{*}\hat{\sigma}_{-}$, made for producing
Hamiltonian (20) would excite atom, but the probability is small (with
$|\mu|\rightarrow 0$), so we list the approximate Eq. (21). Considering atom
retaining on its internal ground state, then Hamiltonian (20) reduce to

$\displaystyle\hat{H}_{4}=$ $\displaystyle\hat{H}^{\prime\prime}_{\rm
NCF}+\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{\prime
2}}{2m}-\hbar\Omega_{\rm eff}(\hat{x},\hat{x}_{a})\,,$ (22)

with the effective Rabi frequency $\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})=|\eta|^{2}\Omega_{a}^{2}/\delta_{al}$, which couples
the external motions of nanopartcile and atom, because $\eta$ and $\Omega_{a}$
are associated with nanoparticle’s position $x$ and atomic position $x_{a}$,
repectively.

###  II.3 The short time approximation

We now take the last unitary transformation

$\displaystyle\hat{U}_{\rm{NA}}(t)=\exp\left[\frac{-it}{\hbar}\left(\frac{\textbf{p}_{a}^{2}}{2m_{a}}+\frac{\textbf{p}^{2}}{2m}-\hbar\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})\right)\right]$ (23)

to the total system. The Hamiltonian (22) becomes

$\displaystyle\hat{H}_{5}(t)=$ $\displaystyle\hat{H}^{\prime\prime\prime}_{\rm
NCF}+\frac{\textbf{p}^{\prime\prime 2}}{2m}-\frac{\textbf{p}^{2}}{2m}\,,$ (24)

with $\hat{H}^{\prime\prime\prime}_{\rm
NCF}=\hat{U}_{\rm{NA}}^{\dagger}(t)\hat{D}^{\dagger}(t)\exp(-i\delta_{cl}t\hat{a}^{\dagger}\hat{a})\hat{H}_{\rm
NCF}\exp(i\delta_{cl}t\hat{a}^{\dagger}\hat{a})\hat{D}(t)\hat{U}_{\rm{NA}}(t)$
and
$\hat{\textbf{p}}^{\prime\prime}=\hat{U}_{\rm{NA}}^{\dagger}(t)\hat{D}^{\dagger}(t)\hat{\textbf{p}}\hat{D}(t)\hat{U}_{\rm{NA}}$.
Recalling $\hat{H}_{\rm NCF}=\widetilde{H}_{\rm NF}+\hat{H}_{\rm CF}$, where
$\widetilde{H}_{\rm NF}$ is the Hamiltonian of nanoparticle interacting with
some other cavity modes and the free fields, and $\hat{H}_{\rm CF}$ the
Hamiltonian of cavity modes interacting with the free fields. Therefore, the
Hamiltonian $\hat{H}_{5}$ arise decoherence to the motional particles, because
the operator $\hat{D}(t)$ of driving cavity is associated with nanoparticle’s
position and the bosonic operators of photon. The $\hat{U}_{\rm{NA}}(t)$
depends not only on nanoparticle’s position but also on atomic position.
Nevertheless, it is reasonable to assume $\hat{H}_{5}\rightarrow 0$ within the
weak driving optical system. This can be explained by the case of single mode
cavity, i.e., $\hat{D}(t)\rightarrow 1$ and
$\hat{U}_{\rm{NA}}(t)\rightarrow\hat{U}_{0}(t)$ with $t\Omega_{l}\rightarrow
0$. The $\hat{U}_{0}(t)=\exp(-it\hat{H}_{K}/\hbar)$ is the free evolution
operator of the motional atom and nanoparticle when the input laser is absent,
and $\hat{H}_{K}=\textbf{p}_{a}^{2}/(2m_{a})+\textbf{p}/(2m)$ the kinetic
energy of two particles.

According to the several unitary evolutions made above, the time evolution
operator of the laser-manipulated cavity system can be written in a general
form of

$\displaystyle\hat{U}_{L}(t)$
$\displaystyle=\hat{D}(t)\hat{U}_{\rm{NA}}(t)\hat{U}_{5}(t)\,.$ (25)

The $\hat{U}_{\rm NA}(t)$ is the leading term about atom, and coupling with
nanoparticle, so we use the subscript NA to mark it. The $\hat{U}_{5}(t)$ is
the time-evolution operator of Hamiltonian $\hat{H}_{5}(t)$. Considering the
laser pulse is sufficiently short, i.e., $t=\tau\rightarrow 0$, then the
evolution operator (25) can be approximately written as

$\displaystyle\hat{U}_{L}(\tau)$ $\displaystyle=\hat{D}(\tau)\hat{U}_{\rm
NA}(\tau)\hat{U}_{5}(\tau)$ (26)
$\displaystyle\approx\hat{D}(\tau)\hat{U}_{5}(\tau)\hat{U}_{\rm NA}(\tau)$
$\displaystyle\approx\hat{D}(\tau)\hat{U}_{5}(\tau)e^{\frac{-i\tau}{\hbar}\hat{H}_{K}}e^{i\tau\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})}$
$\displaystyle\approx\hat{U}_{0}(\tau)\widetilde{U}(\tau)e^{i\tau\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})}\,,$

with a perturbation operator
$\widetilde{U}(\tau)=\hat{U}_{0}^{\dagger}(\tau)\hat{D}(\tau)\hat{U}_{5}(\tau)\hat{U}_{0}(\tau)$
of the cavity system. In the second line of Eq. (26), we have used the
approximation $\hat{U}^{\dagger}_{5}(\tau)\hat{U}_{\rm
NA}(\tau)\hat{U}_{5}(\tau)\approx\hat{U}_{\rm NA}(\tau)$. In the third line of
Eq. (26), we have used the approximation $\hat{U}_{\rm
NA}(\tau)\approx\exp(-i\tau\hat{H}_{K}/\hbar)\exp(i\tau\Omega_{\rm eff})$, by
neglecting the terms of $\tau^{2}[\Omega_{\rm eff},\hat{H}_{K}]/(2\hbar)$. The
present two approximations are both limited within the short time interaction,
similar to that of the so-called Raman-Nath in atom optics ZHU-ATOM .
Certainly, the short time interaction decreases also the desired signal of
$\tau\Omega_{\rm eff}$, so we consider using atom interferometry to measure
it. Once using atom as the detector, we can simply regard
$\widetilde{U}(\tau)=1$, because $\tau\Omega_{\rm eff}$ is already a small
quantity. As a consequence, we can directly use the leading term

$\displaystyle\hat{U}_{L}(\tau)\approx\hat{U}_{0}(\tau)e^{i\tau\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})}$ (27)

to compute the Ramsey-Bordé atoms interferometer in Sec. III.

###  II.4 The data estimations for the effective Rabi frequency

According to Eq. (14), we rewrite the effective Rabi frequency as

$\displaystyle\Omega_{\rm
eff}(\hat{x},\hat{x}_{a})=\frac{|\eta|^{2}\Omega_{a}^{2}}{\delta_{al}}=\Omega_{\rm
effm}\cos^{2}(kx)\cos^{2}(kx_{a})\,,$ (28)

with the nanoparticle’s position $x$ and atomic position $x_{a}$. Here,
$\Omega_{\rm effm}=\eta_{0}^{2}\Omega_{a0}^{2}/\delta_{al}$ is maximum
coupling frequency, with

$\displaystyle\eta_{0}=\frac{\epsilon_{0}\epsilon_{c}V_{N}\textbf{E}_{c}\cdot\textbf{E}_{l}}{2\hbar(\omega_{c}-\omega_{l}-\Omega_{c})}\approx\frac{\epsilon_{0}\epsilon_{c}V_{N}\textbf{E}_{c}\cdot\textbf{E}_{l}}{2\hbar(\omega_{c}-\omega_{l})}\,.$
(29)

The weak coupling $\Omega_{c}\ll\omega_{c}-\omega_{l}$ is considered, and
where $\Omega_{c}=\Omega_{c0}\cos^{2}(kx)$ is given by Eq. (6). We use the
two-photon Rabi frequency
$\Omega_{c0}=\epsilon_{0}\epsilon_{c}V_{N}\textbf{E}_{c}\cdot\textbf{E}_{c}/(2\hbar)=\epsilon_{c}V_{N}\omega_{c}/(4V_{c})$
of nanoparticle-cavity interaction as the criterion to take the numerical
estimations, and where
$\textbf{E}_{c}=\textbf{e}_{c}\sqrt{\hbar\omega_{c}/(2\epsilon_{0}V_{c})}$ is
the vacuum fluctuation of electric field strength of cavity mode $\omega_{c}$.
In terms of $\Omega_{c0}$, Eq. (29) reads

$\displaystyle\eta_{0}=\frac{\Omega_{c0}}{\omega_{c}-\omega_{l}}\frac{(\textbf{e}_{c}\cdot\textbf{e}_{l})E_{l}}{E_{c}}\,,$
(30)

and where $\textbf{e}_{c}$ and $\textbf{e}_{l}$ are the unit vectors of cavity
field and the laser field, respectively.

Considering the wavelength of input laser is approximate $\lambda=780$ nm, and
the effective volume of cavity is $V_{c}\approx\pi(w/2)^{2}L$, with the waist
$w=40\,\,\mu{\rm m}$, and the length $L=1$ cm. Here the $\lambda=780$ nm is
different from 1064 nm in the usual nanoparticle experiment EXP-Science ,
because we need a light which is not so large detuning with the ${\rm D}_{2}$
transition of ${\rm Rb}$ atom D2 ; 102 ; g12  for generating the considerable
effects. Furthermore, considering a volume $V_{N}=4\pi R^{3}/3$ of
nanoparticle with radius $R=150$ nm and the relative dielectric constant
$\epsilon_{r}=2.1$ EXP-NJP2013 , we have the two-photon Rabi frequency of
nanoparticle-cavity interaction, $\Omega_{c0}\approx 1.4$ MHz with
$\epsilon_{c}=3(\epsilon_{r}-1)/(\epsilon_{r}+2)$. The single photon Rabi
frequency of atom-cavity interaction is computed by
$\Omega_{a0}=\textbf{d}_{a}\cdot\textbf{E}_{c}/\hbar=(\textbf{d}_{a}\cdot\textbf{e}_{c})\sqrt{\omega_{c}/(2\epsilon_{0}V_{c}\hbar)}\approx
12$ MHz, with the atomic transition dipole moment
$(\textbf{d}_{a}\cdot\textbf{e}_{c})\approx 3.6\times 10^{-29}\,\,{\rm
C}\cdot{\rm m}$ D2 .

The above $\Omega_{\rm c0}\approx 1.4$ MHz and $\Omega_{a0}\approx 12$ MHz are
too large because the waist $w=40\,\,\mu{\rm m}$ of cavity is small (used
originally in the nanoparticle cooling experiments) EXP-NJP2013 . This would
be not practical for generating the coupling between the freely flying atom
and nanoparticle (as they should be inside the cavity at the same time). A
more larger cavity is needed, for example, $w=1\,\,{\rm mm}$ and $L=2$ cm. As
a result, we have the two-photon Rabi frequency $\Omega_{c0}\approx 1.1$ kHz
of nanoparticle and the single-photon Rabi frequency $\Omega_{a0}\approx 0.33$
MHz of atom. We further consider the detuning $\delta_{al}\approx
10\times\Omega_{a0}\approx 3.3$ MHz between atom and the external laser, and
then the two-photon Rabi frequency of atom is
$\Omega_{a0}^{2}/\delta_{al}\approx 33$ KHz. Moreover, considering
$\eta_{0}=5$ with
$\omega_{c}-\omega_{l}\approx(\textbf{e}_{c}\cdot\textbf{e}_{l}E_{l}/E_{c})\times
0.22$ KHz and $(\textbf{e}_{c}\cdot\textbf{e}_{l}E_{l}/E_{c})>1.5\times
10^{3}$, we have the value of the effective Rabi frequency $\Omega_{\rm
effm}=\eta_{0}^{2}\Omega_{a0}^{2}/\delta_{al}\approx 0.83$ MHz. The Raman-Nath
approximation works within the regime $kv\tau\ll 1$ with $v$ being the
characteristic velocity of particle. The velocities of cold nanoparticle and
cold atom are on the orders of $v=13\,\mu{\rm m}/{\rm s}$ EXP-Science  and
$v_{a}=2\,{\rm mm}/{\rm s}$ KD-E , respectively. As a consequence,
$kv\tau=2\pi v\tau/\lambda\approx 1.05\times 10^{-5}\lll 1$ and
$kv_{a}\tau=2\pi v_{a}\tau/\lambda\approx 1.6\times 10^{-3}\ll 1$ with
$\lambda=780$ nm and a short time $\tau=0.1\,\,\mu{\rm s}$. Using such an
interaction time, we estimate the coupling strength $\tau\Omega_{\rm
effm}=0.083$.

##  III The Ramsey-Bordé atom interferometer

Following Eq. (27), we expand the atomic term as

$\displaystyle e^{i\tau\Omega_{\rm eff}(\hat{x},\hat{x}_{a})}$
$\displaystyle\approx
1+i\xi\cos^{2}(k\hat{x})\left(2+e^{i2k\hat{x}_{a}}+e^{-i2k\hat{x}_{a}}\right)+\mathcal{O}(\xi^{2})\,,$
(31)

by neglecting the probability-amplitude of nonlinear $\xi^{2}=(\Omega_{\rm
effm}\tau/4)^{2}$ throughout the paper. This operator can be regarded as a
coupled KD scattering of atom and nanoparticle, and where $\exp(\pm
2k\hat{x}_{a})$ changes $x$ directional momentum of atom in terms of $\pm
2\hbar k$, resulting in three distinguishable paths, because $\hbar k\approx
m_{a}\times 5.8\,\,{\rm mm}\cdot{\rm s}^{-1}$ is larger than the initial
momentum uncertainty of cold atom KD-E . This is why atoms can be used to make
a matter wave interferometer, but nanoparticle is difficult.

###  III.1 The freely expanding Gaussian wave packet of nanoparticle

According to Eq. (31), perhaps the most direct way of nanoparticle detections
is to observe how many atoms are scattered. The probability of finding the
scattered atoms reads

$\displaystyle\xi^{2}\langle
N|\cos^{4}(kx)|N\rangle\approx\frac{3\xi^{2}}{8}+\frac{\xi^{2}}{4}\langle
N|e^{i2kx}|N\rangle+\frac{\xi^{2}}{16}\langle N|e^{i4kx}|N\rangle+{\rm
c.c.}\,,$ (32)

where the ${\rm c.c.}$ is the conjugate complex number of all the non-real
numbers in the right hand of the equation. For the state of nanoparticle, we
consider the simplest case, that the freely expanding Gaussian wave packet
along horizontal direction,

$\displaystyle|N\rangle=e^{\frac{-it}{\hbar}\frac{\hat{p}^{2}}{2m}}|G\rangle,$
(33)

with

$\displaystyle|G\rangle=\int_{-\infty}^{+\infty}\phi(p)|p\rangle{\rm
d}p=(2\pi\Delta_{p}^{2})^{-1/4}\int_{-\infty}^{+\infty}e^{\frac{-p^{2}}{4\Delta_{p}^{2}}}|p\rangle{\rm
d}p\,,$ (34)

and where $\Delta_{p}$ is the momentum uncertainty of initial nanoparticles.
This state can be generated by a released nanoparticle which was initially
confined as a harmonic oscillator in the optical trap and cooled at the
motional quantum ground state EXP-Science . Using the Heisenberg operator of
uniform rectilinear motion,

$\displaystyle
e^{\frac{it}{\hbar}\frac{\hat{p}^{2}}{2m}}\hat{x}e^{\frac{-it}{\hbar}\frac{\hat{p}^{2}}{2m}}=\hat{x}+\frac{\hat{p}}{m}t=\hat{x}+\hat{v}t\,,$
(35)

we write one of the terms in Eq. (32) as

$\displaystyle\langle N|e^{ink\hat{x}}|N\rangle=\langle
G|e^{ink(\hat{x}+\hat{v}t)}|G\rangle=e^{\frac{i\hbar n^{2}k^{2}t}{2m}}\langle
G|e^{ink\hat{x}}e^{ink\hat{v}t}|G\rangle\,,$ (36)

with $n=2,4$. The phase $\hbar n^{2}k^{2}t/(2m)=n^{2}kv_{k}t/2$, with the
single photon recoil velocity $v_{k}=\hbar k/m$ of nanoparticle, should be the
nonclassical one, because it came from the canonical quantization
$[\hat{x},\hat{v}]=i\hbar/m$ of nanoparticle.

Unfortunately, the phase $\hbar n^{2}k^{2}t/(2m)$ is exactly eliminated by the
following Gaussian integral

$\displaystyle\langle G|e^{ink\hat{x}}e^{ink\hat{v}t}|G\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}e^{\frac{inkpt}{m}}\phi^{*}(p+n\hbar
k)\phi(p){\rm d}p$ (37)
$\displaystyle=(2\pi\Delta_{p}^{2})^{-1/2}\int_{-\infty}^{+\infty}e^{\frac{inkpt}{m}}e^{\frac{-p^{2}}{4\Delta_{p}^{2}}}e^{\frac{-(p+n\hbar
k)^{2}}{4\Delta_{p}^{2}}}{\rm d}p$
$\displaystyle=\exp\left(-\frac{n^{2}\hbar^{2}k^{2}}{8\Delta_{p}^{2}}\right)\exp\left[-\frac{1}{2}\left(\frac{nk\Delta_{p}t}{m}\right)^{2}\right]\exp\left(-i\frac{\hbar
n^{2}k^{2}t}{2m}\right)\,.$

Thus, the $\cos^{n}(k\hat{x})$ applied to measure the freely expanding
Gaussian wave packet does not produce nonclassical phenomena. Nevertheless,
the $\cos^{n}(k\hat{x})$ may be useful for measuring the momentum uncertainty
of nanoparticle. Numerically, considering the momentum uncertainty is that of
ground state harmonic oscillator EXP-Science , i.e., $\Delta_{p}=m\times
13\,\,\mu{\rm m}\cdot{\rm s}^{-1}$ introduced in Sec. I., we find
$\exp\left[-\hbar^{2}n^{2}k^{2}/(8\Delta_{p}^{2})\right]\approx 1$, because
the single photon recoiled momentum $\hbar k\approx m\times 5\,\,{\rm
nm}\cdot{\rm s}^{-1}$ is far smaller than the initial momentum uncertainty
$\Delta_{p}$ of nanoparticle. The value of
$\exp\left[-(nk\Delta_{p}t/m)^{2}/2\right]$ is considerable, if
$\Delta_{p}t/m<1/(2k)=\lambda/(4\pi)$, i.e., $t<5$ ms.

One may suppose a $x$ directional “ gravity” acceleration $g_{x}$, then the
familiar phase $nkg_{x}t^{2}/2$ in Mach-Zehnder atom interferometer ChuG  is
obtained by replacing Heisenberg’s operator (35) with
$\hat{x}+\hat{v}t+g_{x}t^{2}/2$. However, this phase should be not regarded as
the nonclassical effect of nanoparticle. The Eq. (37) indicates that a long
time evolution of Gaussian wave packet results in a small signal due to the
spatial symmetry of function $\cos^{n}(kx)$. This means that a more smaller
momentum uncertainty $\Delta_{p}$ of the initial nanoparticle is needed. One
may use the locality of light spot to measure nanoparticle. However, the waist
of laser is much larger than its wave length, so the resolution ratio of
particle’s position would be not significantly high by using this approach. In
this paper, we focus on the effects from the short wave length
$\lambda=2\pi/k$ of light. Although the above measurement does not generate
the signal about nanoparticle’s motions (after a long time free evolution of
Gaussian wave packet), the first term $3\xi^{2}/8$ in Eq. (32) is still useful
for extracting the nonclassical signals of nanoparticle in the following
Ramsey-Bordé atom interferometer.

Figure 2: Sketch for the conceptual experiment. Inside the interaction cavity,
nanoparticle is illuminated by two $z$-directional laser pulses at $t_{1}$ and
$t_{2}$, generates twice atomic KD scattering (31) to realize the $+x$
directional Ramsey-Bordé atom interferometer showed in Fig. 3. Inside the
interaction cavity, the nanoparticle’s velocity $v_{y}$ and atomic velocity
$v_{ay}$ are not so large that the particles will pass through the cavity
before the two laser pulses are completed. For example, $v_{ay}<v_{y}<w/\delta
t\approx 1\,\,{\rm m\cdot s}^{-1}$, with $w\approx 1\,\,{\rm mm}$ being the
waists of cavity and the laser beam, and $\delta t=t_{2}-t_{1}\approx 1$ ms
the time interval between two laser pulses. The $-x$ direction scattered atoms
are not used to make the Ramsey-Bordé atom interferometer, but used to measure
the values of $\xi_{j}$ based on Eq. (61), allowing us to extract the desired
nonclassical signal in Ramsey-Bordé atom interferometer. Figure 3: Sketch for
the $+x$ directional Ramsey-Bordé atom interferometer, with
$t_{2}-t_{1}=t_{4}-t_{3}$. Two weak standing waves at $t_{1}$ and $t_{2}$ are
due to the laser-illuminated nanoparticle, which split the $x$ directional
paths of atom. Two classical laser pulses at $t_{3}$ and $t_{4}$ are used to
recombine atomic paths for completing the matter wave interferometer. The
large distance $2v_{ak}\times(t_{2}-t_{1})$ between the red and blue lines is
the key to generate the visible atom interference, for example,
$2v_{ak}\times(t_{2}-t_{1})\approx 12\,\,\mu{\rm m}$ may be large enough, with
$t_{2}-t_{1}\approx 1$ ms and the single photon recoil velocity $v_{ak}\approx
5.8\,\,{\rm mm}\cdot{\rm s}^{-1}$ of atom.

###  III.2 The time-evolution operator approach for Ramsey-Bordé atom
interferometer

We design a symmetric Ramsey-Bordé atom interferometer SRB  to measure the
small nonclassicality of nanoparticle. This atom interferometer is made by
using twice interaction (31) at times $t_{1}$ and $t_{2}$ to split atomic path
along horizontal $x$ direction, and consequently, using two classical laser
pulses at times $t_{3}$ and $t_{4}$ to recombine atomic paths, as shown in
Fig. 2 and Fig. 3.

The KD scatterings for making atom interferometers will generate many
distinguishable paths of freely falling atom 102 ; g12 . Nevertheless, we can
consider just only two selected paths of atoms, because the atoms moving along
the other paths do not arrive at the small detection zone of atoms. We use
$\alpha$ and $\beta$ to denote the probability-amplitudes of atoms which came
from the two selected paths, i.e., the red line and blue line in Fig. 3, and
write the final state as

$\displaystyle|\psi\rangle$ $\displaystyle=\alpha|{\rm Red}\rangle+\beta|{\rm
Blue}\rangle+\cdots$ (38)
$\displaystyle=\hat{R}|A(t_{1})\rangle|N(t_{1})\rangle+\hat{B}|A(t_{1})\rangle|N(t_{1})\rangle+\cdots\,.$

The $|{\rm Red}\rangle$ and $|{\rm Blue}\rangle$ are the states of the total
system, with atom moving along the red and the blue lines, respectively. The
two states are further denoted by the path-operators $\hat{R}$ and $\hat{B}$,
with the initial state $|A(t_{1})\rangle|N(t_{1})\rangle$ at $t_{1}$, i.e.,
the moment of the first laser pulse of Ramsey-Bordé atom interferometer.

According to Fig. 3, we can write the path-operators as

$\displaystyle\hat{R}=$ $\displaystyle
i\xi_{1}\alpha_{l}\hat{U}_{0}(t_{4}-t_{3})e^{-i2k\hat{x}_{a}}\hat{U}_{0}(t_{3}-t_{1})\cos^{2}(k\hat{x})e^{i2k\hat{x}_{a}}\,,$
(39)

and

$\displaystyle\hat{B}=i\xi_{2}\beta_{l}e^{-i2k\hat{x}_{a}}\hat{U}_{0}(t_{4}-t_{2})\cos^{2}(k\hat{x})e^{i2k\hat{x}_{a}}\hat{U}_{0}(t_{2}-t_{1})\,.$
(40)

Here, the probability-amplitudes $\alpha_{l}$ and $\beta_{l}$ are due to the
classical laser pulses used for recombining atomic paths. The
$\hat{U}_{0}(t)=\hat{U}_{\rm N0}(t)\hat{U}_{\rm A0}(t)$ is the free evolution
operator of particles (with negligible interaction energy between them), and
where $\hat{U}_{\rm N0}(t)$ is the free evolution operator of nanoparticle.
The scattering operator $\exp(\pm i2k\hat{x}_{a})$ changes atomic momentum
instantaneously, resulting in two distinguishable paths due to the free
evolution operator $\hat{U}_{\rm A0}(t)$ of atom. The
$i\xi_{j}\cos^{2}(k\hat{x})$, with $j=1,2$, is due to the Eq. (31), and can be
also regarded as the probability-amplitudes of atomic scattering, if
nanoparticle is the classical one.

Using the property of unitary operator
$\hat{U}_{0}^{\dagger}(t)\hat{U}_{0}(t)=\hat{U}_{0}(t)\hat{U}_{0}^{\dagger}(t)=1$
and the commutator $[\hat{U}_{\rm A0}(t),\hat{U}_{\rm N0}(t)]=0$ of atom and
nanoparticle, we further write the states (39) and (40) as

$\displaystyle\hat{R}=$ $\displaystyle
i\xi_{1}\alpha_{l}\hat{U}_{0}(t_{4}-t_{1})e^{-i2k\hat{x}_{a}(t_{3}-t_{1})}e^{i2k\hat{x}_{a}}\cos^{2}(k\hat{x})\,,$
(41)

and

$\displaystyle\hat{B}=$ $\displaystyle
i\xi_{2}\beta_{l}\hat{U}_{0}(t_{4}-t_{1})e^{-i2k\hat{x}_{a}(t_{4}-t_{1})}e^{i2k\hat{x}_{a}(t_{2}-t_{1})}\cos^{2}[k\hat{x}(t_{2}-t_{1})]\,.$
(42)

Here, $\hat{x}_{a}(t)=\hat{x}_{a}+\hat{v}_{a}t$ and
$\hat{x}(t)=\hat{x}+\hat{v}t$ are the Heisenberg’s operators of atom and
nanoparticle moving along the horizontal $x$ directions, see also Eq. (35).

Eq. (41) and (42) can be written in more clear, i.e., $\hat{R}=\hat{R}_{\rm
A}\hat{R}_{\rm N}$ and $\hat{B}=\hat{B}_{\rm A}\hat{B}_{\rm N}$, with atomic
parts $\hat{R}_{\rm A}$, $\hat{B}_{\rm A}$, and nanoparticle’s parts
$\hat{R}_{\rm N}$ and $\hat{B}_{\rm N}$. The four operators are given by
follows,

$\displaystyle\hat{R}_{\rm A}=$
$\displaystyle\hat{U}_{A0}(t_{4}-t_{1})e^{-i2k\hat{x}_{a}(t_{3}-t_{1})}e^{i2k\hat{x}_{a}}\,,$
(43) $\displaystyle\hat{B}_{\rm A}=$ $\displaystyle\hat{U}_{\rm
A0}(t_{4}-t_{1})e^{-i2k\hat{x}_{a}(t_{4}-t_{1})}e^{i2k\hat{x}_{a}(t_{2}-t_{1})}\,,$
(44) $\displaystyle\hat{R}_{\rm N}=$ $\displaystyle
i\alpha_{l}\xi_{1}\hat{U}_{\rm N0}(t_{4}-t_{1})\cos^{2}(k\hat{x})\,,$ (45)
$\displaystyle\hat{B}_{\rm N}=$ $\displaystyle i\beta_{l}\xi_{2}\hat{U}_{\rm
N0}(t_{4}-t_{1})\cos^{2}[k\hat{x}(t_{2}-t_{1})]\,.$ (46)

Considering $t_{3}-t_{1}=t_{4}-t_{2}=T$, i.e., $t_{4}-t_{3}=t_{2}-t_{1}$, the
atomic operators reduce to

$\displaystyle\hat{R}_{\rm A}=\hat{B}_{\rm
A}=\hat{U}_{A0}(t_{4}-t_{1})e^{-i2\hbar k^{2}T/m_{a}}e^{-i2k\hat{v}_{a}T}\,,$
(47)

and here the so-called Zassenhaus formula Zassenhaus-2012 ; Zassenhaus-1967
is used,

$\displaystyle
e^{i2k[\hat{x}_{a}(t_{i})-\hat{x}_{a}(t_{j})]}=e^{-i2k\hat{x}_{a}(t_{j})}e^{i2k\hat{x}_{a}(t_{i})}e^{-2k^{2}[\hat{x}_{a}(t_{j}),\hat{x}_{a}(t_{i})]}\,,$
(48)

with $[\hat{x}_{a}(t_{j}),\hat{x}_{a}(t_{i})]=i\hbar(t_{i}-t_{j})/m_{a}$.

According to Eq. (47), the superposition state (38) reads

$\displaystyle|\psi\rangle$ $\displaystyle=\left[\hat{R}_{\rm
N}|N(t_{1})\rangle+\hat{B}_{\rm
N}|N(t_{1})\rangle\right]|A(t_{4})\rangle+\cdots\,,$ (49)

with atomic state $|A(t_{4})\rangle=\hat{R}_{\rm
A}|A(t_{1})\rangle=\hat{B}_{\rm A}|A(t_{1})\rangle$ at $t_{4}$. As a result,
the probability density of atom reads $P_{a}(x_{a})=|\langle
x_{a}|\psi\rangle|^{2}=|\langle x_{a}|A(t_{4})\rangle|^{2}\times P$, with

$\displaystyle P$ $\displaystyle=\langle N(t_{1})|(\hat{R}_{\rm
N}+\hat{B}_{\rm N})^{\dagger}(\hat{R}_{\rm N}+\hat{B}_{\rm
N})|N(t_{1})\rangle\,.$ (50)

The $P$ is the desired signal. According to Eqs. (45) and (46), we have

$\displaystyle\hat{R}^{\dagger}_{\rm N}\hat{R}_{\rm
N}\approx|\alpha_{l}|^{2}\xi_{1}^{2}\cos^{4}(k\hat{x})+\mathcal{O}(\xi^{3})\,,$
(51) $\displaystyle\hat{B}^{\dagger}_{\rm N}\hat{B}_{\rm N}$
$\displaystyle\approx|\beta_{l}|^{2}\xi_{2}^{2}\cos^{4}[k\hat{x}(t_{2}-t_{1})]+\mathcal{O}(\xi^{3})\,,$
(52) $\displaystyle\hat{R}^{\dagger}_{\rm N}\hat{B}_{\rm N}$
$\displaystyle\approx\alpha_{l}^{*}\beta_{l}\xi_{1}\xi_{2}\cos^{2}(k\hat{x})\cos^{2}[k\hat{x}(t_{2}-t_{1})]+\mathcal{O}(\xi^{3})\,.$
(53)

The higher orders than $\xi^{2}$ are neglected.

###  III.3 The long time expanding Gaussian wave packet of nanoparticle

In order to compute Eq. (50), we should know the “ initial” state of
nanoparticle at $t_{1}$. The nanoparticle’s state at $t_{0}$ is that at the
moment of nanoparticle releasing from the optical harmonic oscillator
potential, which can be regarded as a Gaussian wave packet
$|N(t_{0})\rangle=|G\rangle$. Thus,
$|N(t_{1})\rangle=\exp[-i(t_{1}-t_{0})p^{2}/(2m\hbar)]|G\rangle$, if
nanoparticle in stage $t_{1}-t_{0}$ is really a free fall, and therefore Eq.
(50) becomes

$\displaystyle P=$ $\displaystyle|\alpha_{l}|^{2}\xi_{1}^{2}\langle
G|\cos^{4}[k\hat{x}(\Delta t_{1})]|G\rangle+|\beta_{l}|^{2}\xi_{2}^{2}\langle
G|\cos^{4}[k\hat{x}(\Delta t_{2})]|G\rangle$ (54)
$\displaystyle+\alpha_{l}^{*}\beta_{l}\xi_{1}\xi_{2}\langle
G|\cos^{2}[k\hat{x}(\Delta t_{1})]\cos^{2}[k\hat{x}(\Delta
t_{2})]|G\rangle+{\rm c.c.}\,,$

with $\Delta t_{j}=t_{j}-t_{0}$ and $j=1,2$.

In order to compute Eq. (54), we introduce the following Gaussian integral,

$\displaystyle\langle G|e^{in_{1}k\hat{x}}e^{in_{2}k\hat{v}\Delta
t_{j}}|G\rangle=$
$\displaystyle\exp\left(-\frac{\hbar^{2}n_{1}^{2}k^{2}}{8\Delta_{p}^{2}}\right)\exp\left[-\frac{1}{2}\left(\frac{n_{2}k\Delta_{p}\Delta
t_{j}}{m}\right)^{2}\right]$ (55)
$\displaystyle\times\exp\left(-i\frac{n_{1}n_{2}kv_{k}\Delta
t_{j}}{2}\right)\,,$

with $n_{1}$ and $n_{2}$ being arbitrary real numbers, and $v_{k}=\hbar k/m$
the single photon recoil velocity of nanoparticle (not atom). This integral is
a generalization of Eq. (37), and where $\Delta_{p}=m\times 13\,\,\mu{\rm
m}\cdot{\rm s}^{-1}$ is the momentum uncertainty of the initially cooled
nanoparticle, and $k=2\pi/(780\,{\rm nm})$ the wave number of the cavity mode.

Considering a long time free fall before the Ramsey-Bordé atom interferometer,
e.g., $t_{1}-t_{0}\approx 0.1$ s, and a short time between the laser pulses of
Ramsey-Bordé atom interferometer, e.g., $\delta t=t_{2}-t_{1}=1$ ms, then
$n_{2}\Delta_{p}\Delta t_{j}/m\gg 1/(2k)$ with $n_{2}=2,4$, and the Gaussian
integral (55) is approximately equal to zero, see the discussion below Eq.
(37). This greatly reduces the results of Eq. (54), as follows,

$\displaystyle\langle G|\cos^{4}[k\hat{x}(\Delta
t_{1})]|G\rangle\approx\langle G|\cos^{4}[k\hat{x}(\Delta
t_{2})]|G\rangle\approx\frac{3}{8}\,,$ (56)

and

$\displaystyle\langle G|\cos^{2}[k\hat{x}(\Delta
t_{1})]\cos^{2}[k\hat{x}(\Delta t_{2})]|G\rangle$ (57)
$\displaystyle\approx\frac{1}{4}+\frac{1}{4}\langle G|\cos[2k\hat{x}(\Delta
t_{1})]\cos[2k\hat{x}(\Delta t_{2})]|G\rangle$
$\displaystyle\approx\frac{1}{4}+\frac{1}{16}\langle
G|\left[e^{i2k\hat{x}(\Delta t_{1})}e^{-i2k\hat{x}(\Delta
t_{2})}+e^{-2ik\hat{x}(\Delta t_{1})}e^{i2k\hat{x}(\Delta
t_{2})}\right]|G\rangle$
$\displaystyle=\frac{1}{4}+\frac{1}{16}e^{i\theta_{q}}\langle
G|\left[e^{-i2k\hat{v}(t_{2}-t_{1})}+e^{2ik\hat{v}(t_{2}-t_{1})}\right]|G\rangle$
$\displaystyle=\frac{1}{4}+\frac{1}{8}e^{i\theta_{q}}G(\delta t)\,.$

In Eq. (57), the Gaussian function

$\displaystyle G(\delta
t)=\exp\left[-\frac{1}{2}\left(\frac{2k\Delta_{p}\delta
t}{m}\right)^{2}\right]\,,$ (58)

and the phase

$\displaystyle\theta_{q}=\frac{2\hbar k^{2}\delta t}{m}=2kv_{k}\delta t\,,$
(59)

are the signals of motional nanoparticle. Specially, the phase $\theta_{q}$ is
the quantum mechanical one because it is generated by the commutators
$[\hat{x}(\Delta t_{1}),\hat{x}(\Delta t_{2})]=i\hbar(t_{2}-t_{1})/m$ of
nanoparticle, and where $v_{k}=\hbar k/m$ is the single photon recoil velocity
of nanoparticle. According Eqs. (56) and (57), the probability (54) is solved,

$\displaystyle
P=\frac{3(\xi_{1}^{2}|\alpha_{l}|^{2}+\xi_{2}^{2}|\beta_{l}|^{2})}{8}+\frac{\xi_{1}\xi_{2}\alpha_{l}^{*}\beta_{l}}{4}\left[1+\frac{1}{2}e^{i\theta_{q}}G(\delta
t)\right]+{\rm c.c.}\,.$ (60)

Considering the time-interval $\delta t=t_{2}-t_{1}=1.2\,\,{\rm ms}$, the term
$G(\delta t)\approx 1$, and the phase $\theta_{q}=2kv_{k}\delta t\approx
10^{-4}$, with $v_{k}=\hbar k/m\approx 5\,\,{\rm nm}\cdot{\rm s}^{-1}$,
$k=2\pi/(780\,{\rm nm})$, and the mass $m=10^{8}$ amu of nanoparticle. The
phase $\theta_{q}$ can be enlarged by increasing the time interval $\delta
t=t_{2}-t_{1}$ between the two laser pulses. However, a considerable value of
Gaussian function $G(\delta t)$ needs the small momentum uncertainty $\Delta
p$ of initial nanoparticle, i.e., a more colder nanoparticle is much-needed.
Using the data of experiments EXP-Science , the $\delta t$ is limited within
$5$ ms, see the explanation below Eq. (37). In addition to the interest of
$\theta_{q}$, the Eq. (60) can be useful for testing the Gaussian function
$G(\delta t)$, i.e., the initial momentum uncertainty of nanoparticle.
Moreover, the general Eq. (50) of the present Ramsey-Bordé atom interferometer
could be also useful for measuring the non-Gaussian state at $t_{1}$, prepared
by the strong KD scattering of nanoparticle in the classical laser pulse
before $t_{1}$ NC2021 ; free-N-KD .

###  III.4 Discussion

On the $\theta_{q}$ detection, the parameter $\xi_{j}$ has great uncertainty
due to the imprecise positions of atom and nanoparticle relative to the waists
of external laser and the cavity. Thus, the releasing times of particles and
their initial velocities are needed to be exactly controlled. According to the
symmetry of KD scattering, the $-x$ direction scattered atoms in Fig. 2 are
also related to the parameter $\xi_{j}$. The probability of finding those
atoms reads

$\displaystyle P_{\rm ref}$
$\displaystyle=\frac{3(\xi_{1}^{2}+\xi_{2}^{2})}{8}\,,$ (61)

and can be measured before the $+x$ directional Ramsey-Bordé atom
interferometer is completed. The probability (61) can be used as a reference
to select the signal of $\theta_{q}$ in Eq. (60). Therefore, the $\alpha_{l}$
and $\beta_{l}$ become the main noises source due to the imperfect laser
pulses at $t_{3}$ and $t_{4}$, as well as that in the usual atom
interferometers. In addition to the noises of laser manipulations, there
exists environmental noises in the free fall stage of nanoparticle due to the
collisions with residual gas particles and the blackbody radiation. This issue
has been well studied by Romero-Isart et al. Romero-Isart-collapse . In
principle, the environmental noises can be suppressed by using the ultrahigh
vacuum chamber with the cryogenic temperatures Romero-Isart-Science .
Launching satellite, for example the MAQRO project MAQRO2012 ; MAQRO2015 ;
NC2021 , should be the most effective way to realize the long time free fall
and suppress the environmental noises. Throughout the paper, we consider just
only the single nanoparticle, so need the nanoparticle reloading. Similar to
the magneto-optical trap in cold atoms system, the nanoparticle reloading
could be realized by using the optical tweezer together with the gradient
electric or magnetic fields AN-Paul ; CONTEMPORARY-PHYSICS ; Romero-Isart-
Science .

Finally, we discuss the question of many atoms being simultaneously inside the
cavity. According to the single atom scattering operator (27), the operators
of $n$ atoms scattered by two laser pulses can be written as
$\hat{U}_{L1}=\Pi_{j=1}^{n}\exp(i\hat{\xi}_{j})$ and
$\hat{U}_{L2}=\Pi_{j=1}^{n}\exp(i\hat{\xi}^{\prime}_{j})$, with
$\hat{\xi}_{j}=4\xi_{1}\cos^{2}(k\hat{x})\cos^{2}(k\hat{x}_{aj})$ and
$\hat{\xi}^{\prime}_{j}=4\xi_{2}\cos^{2}(k\hat{x})\cos^{2}(k\hat{x}_{aj})$.
The $\hat{x}_{aj}$ is the horizontal position of $j$-th atom. After the two
scatterings (at $t_{1}$ and $t_{2}$), the state is prepared in

$\displaystyle|\psi(t_{2})\rangle$
$\displaystyle=\hat{U}_{L2}\hat{U}_{0}(t_{2}-t_{1})\hat{U}_{L1}|\psi(t_{1})\rangle=\hat{U}_{0}(t_{2}-t_{1})|\psi^{\prime}(t_{1})\rangle\,,$
(62)

with $\hat{U}_{0}(t)=\hat{U}_{N0}(t)\Pi_{j=1}^{n}\hat{U}_{j0}(t)$ being the
free evolution operator of the nanoparticle and the many atoms. In this
presentation, the “ initial” state reads

$\displaystyle|\psi^{\prime}(t_{1})\rangle$ $\displaystyle=\hat{U}_{L2}(\delta
t)\hat{U}_{L1}|\psi(t_{1})\rangle$ (63)
$\displaystyle=[e^{i\hat{\xi}^{\prime}_{1}(\delta
t)}e^{i\hat{\xi}^{\prime}_{2}(\delta t)}\cdots
e^{i\hat{\xi}^{\prime}_{n}(\delta
t)}][e^{i\hat{\xi}_{1}}e^{i\hat{\xi}_{2}}\cdots
e^{i\hat{\xi}_{n}}]|\psi(t_{1})\rangle$
$\displaystyle\approx[e^{i\hat{\xi}^{\prime}_{1}(\delta
t)}e^{i\hat{\xi}_{1}}][e^{i\hat{\xi}^{\prime}_{2}(\delta
t)}e^{i\hat{\xi}_{2}}]\cdots[e^{i\hat{\xi}^{\prime}_{n}(\delta
t)}e^{i\hat{\xi}_{n}}]|\psi(t_{1})\rangle\,,$

with $\hat{\xi}^{\prime}_{j}(\delta t)=\hat{U}_{0}^{\dagger}(\delta
t)\hat{\xi}^{\prime}_{j}\hat{U}_{0}(\delta t)$ and $\delta t=t_{2}-t_{1}$, and
where

$\displaystyle|\psi(t_{1})\rangle=|A_{1}(t_{1})\rangle|A_{2}(t_{1})\rangle\cdots|A_{n}(t_{1})\rangle|N(t_{1})\rangle\,,$
(64)

is the state of $n+1$ particles at $t_{1}$, i.e., $n$ atoms and one
nanoparticle.

In the last line of Eq. (63), we have used the approximate Zassenhaus formula

$\displaystyle e^{i\hat{\xi}^{\prime}_{i}(\delta t)}e^{i\hat{\xi}_{j}}\approx
e^{i\hat{c}_{ij}}e^{i\hat{\xi}_{j}}e^{i\hat{\xi}^{\prime}_{i}(\delta
t)}\approx e^{i\hat{\xi}_{j}}e^{i\hat{\xi}^{\prime}_{i}(\delta t)}\,,$ (65)

and where

$\displaystyle\hat{c}_{ij}=i16\xi_{1}\xi_{2}[\cos^{2}(k\hat{x}(\delta
t)),\cos^{2}(k\hat{x})]\cos^{2}(k\hat{x}_{ai}(\delta
t))\cos^{2}(k\hat{x}_{aj})$ (66)

is due to nanoparticle’s commutator $[\cos^{2}(k\hat{x}(\delta
t)),\cos^{2}(k\hat{x})]=[\cos^{2}(k\hat{x}+k\hat{v}\delta
t),\cos^{2}(k\hat{x})]\neq 0$, which generates the atom-atom coupling.
However, the probability-amplitudes are related to the quadratic small
quantity $\xi^{2}$. In this paper, the probability-amplitudes are limited
within the linear $\xi$, so we can neglect the terms of $\hat{c}_{ij}$ in the
last line of Eq. (63).

Following Eq. (63), the prepared state by nanoparticle light sources can be
written as

$\displaystyle|\psi^{\prime}(t_{1})=$
$\displaystyle|A^{\prime}_{1}(t_{1})\rangle|A^{\prime}_{2}(t_{1})\rangle\cdots|A^{\prime}_{n}(t_{1})\rangle|N(t_{1})\rangle\,,$
(67)

with the normalized state of $j$-th atom,

$\displaystyle|A^{\prime}_{j}(t_{1})\rangle=e^{i\hat{\xi}^{\prime}_{j}(\delta
t)}e^{i\hat{\xi}_{j}}|A_{j}(t_{1})\rangle\,.$ (68)

This presentation shows that atoms are identical and independent from each
others. Thus, the single atom model in this paper is valid to describe the
practical system of many atom being inside the interaction cavity
simultaneously.

##  IV conclusion

We have shown that nanoparticle can be used as a light source to make Ramsey-
Bordé atoms interferometer. The proposal is based on the cavity QED system
which consists of a single atom and a single nanoparticle. The nanoparticle is
illuminated by the external laser pulses, and then nearly-resonantly excite
one of the cavity modes. This cavity mode generates atomic ac Stark effect,
and then the atomic KD scattering if the input laser is the pulsed one. The
unitary transformations approach is used to derive such a KD scattering. It is
proved that the present single atom model is also valid for the case that many
atoms are simultaneously inside the interaction cavity, because the atomic
internal state is not excited. Atomic KD scattering in the classical standing
wave of lights is widely used to make the laser-pulsed atom interferometers.
Thus, the present cavity system could be also feasible to generate atom
interferometers. A slight difference is that the present atom interferometer
is coupled with the centre-of-mass motion of nanoparticle, and thus could be
developed as a tool to measure the classical or nonclassical motions of freely
falling nanoparticle. Once the motion of nanoparticle is exactly detectable,
the freely falling nanoparticle (in the stage of $t_{1}-t_{0}$) may be further
developed for the other uses Romero-Isart-Science , for example, studying the
frequency-mixed effects of nanoparticle’s nonlinear optics. In addition to
nanoparticle detections, this study provides an alternative way to measure the
classical or nonclassical motions of some other condensed particles, such as
the macromolecule, which could be also used as the light source to make atom
interferometers.

Acknowledgments: This work was supported partly by the National Natural
Science Foundation of China, Grant No. 12047576.

## References

  * (1) O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large Quantum Superpositions and Interference of Massive Nanometer-Sized Objects ”, Phys. Rev. Lett. 107, 020405 (2011).
  * (2) O. Romero-Isart, “Quantum superposition of massive objects and collapse models”, Phys. Rev. A 84, 052121 (2011).
  * (3) S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, “Spin Entanglement Witness for Quantum Gravity ”, Phys. Rev. Lett. 119, 240401 (2017).
  * (4) C. Marletto, and V. Vedral, “Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity ”, Phys. Rev. Lett. 119, 240402 (2017).
  * (5) A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function collapse, underlying theories, and experimental tests”, Rev. Mod. Phys. 85, 471 (2013).
  * (6) M. Arndt, and K. Hornberger, “Testing the limits of quantum mechanical superpositions”, Nat. Phys. 10, 271 (2014).
  * (7) S. Donadi, K. Piscicchia, C. Curceanu, L. Diósi, M. Laubenstein, A. Bassi, “Underground test of gravity-related wave function collapse”, Nat. Phys. 17, 74 (2021).
  * (8) A. Bassi, A. Großardt, and H. Ulbricht, “Gravitational decoherence”, Class. Quantum Grav. 34, 193002 (2017).
  * (9) S. Eibenberger, S. Gerlich, M. Arndt, M. Mayor, and J. Tüxen, “Matter-wave interference of particles selected from a molecular library with masses exceeding 10 000 amu ”, Phys. Chem. Chem. Phys. 15, 14696 (2013).
  * (10) L. Diósi, “Gravitation and quantum-mechanical localization of macro-objects”, Phys. Lett. A 105, 199 (1984).
  * (11) R. Penrose, “On Gravity’s Role in Quantum State Reduction ”, Gen. Rel. Gravit. 28, 581 (1996).
  * (12) C. Anastopoulos and B. L. Hu, “Gravitational Decoherence: A Thematic Overview ”, arXiv:2111.02462 (2021).
  * (13) P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle”, Phys. Rev. A 81, 023826 (2010).
  * (14) G. A. T. Pender, P. F. Barker, F. Marquardt, J. Millen, and T. S. Monteiro, “Optomechanical cooling of levitated spheres with doubly resonant fields ”, Phys. Rev. A 85, 021802(R) (2012).
  * (15) D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere ”, PNAS 107, 1005 (2010).
  * (16) J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin Parametric Feedback Cooling of a Laser-Trapped Nanoparticle ”, Phys. Rev. Lett. 109, 103603 (2012).
  * (17) T. S. Monteiro, J Millen, G. A. T. Pender, F. Marquardt, D. Chang, and P. F. Barker, “Dynamics of levitated nanospheres: towards the strong coupling regime ”, New J. Phys. 15, 015001 (2013).
  * (18) V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, “Direct Measurement of Photon Recoil from a Levitated Nanoparticle ”, Phys. Rev. Lett. 116, 243601 (2016).
  * (19) U. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, and M. Aspelmeyer, “Cavity Cooling of a Levitated Nanosphere by Coherent Scattering ”, Phys. Rev. Lett. 122, 123602 (2019).
  * (20) D. Windey, C. Gonzalez-Ballestero, P. Maurer, L. Novotny, O. Romero-Isart, and R. Reimann, “Cavity-Based 3D Cooling of a Levitated Nanoparticle via Coherent Scattering ”, Phys. Rev. Lett. 122, 123601 (2019).
  * (21) U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, “Cooling of a levitated nanoparticle to the motional quantum ground state ”, Science 367, 892 (2020).
  * (22) F. Tebbenjohanns, M. L. Mattana, M. Rossi, M. Frimmer, and L. Novotny, “Quantum control of a nanoparticle optically levitated in cryogenic free space ”, Nature (London) 595, 378 (2021).
  * (23) J. Millen, T. S Monteiro2, R. Pettit, and A N. Vamivakas, “Optomechanics with levitated particles”, Rep. Prog. Phys. 83, 026401 (2020).
  * (24) A. K. Chauhan, O. Černotík, and R. Filip, “Stationary Gaussian entanglement between levitated nanoparticles”, New J. Phys. 22, 123021 (2020).
  * (25) I. Brandão, D. Tandeitnik, and T. Guerreiro, “Coherent scattering-mediated correlations between levitated nanospheres”, Quantum Sci. Technol. 6, 045013 (2021).
  * (26) M. Nikkhou, Y. Hu, J. A. Sabin, and J. Millen, “Direct and clean loading of nanoparticles into optical traps at millibar pressures”, Photonics 8, 458 (2021)
  * (27) R. Kaltenbaek, et al., “Macroscopic quantum resonators (MAQRO)”, Exp. Astron. 34, 123 (2012).
  * (28) R. Kaltenbaek, et al., “Macroscopic Quantum Resonators (MAQRO): 2015 update”, EPJ Quantum Technology 3, 5 (2016).
  * (29) G. Gasbarri, A. Belenchia, M. Carlesso, S. Donadi, A. Bassi, R. Kaltenbaek, M. Paternostro, and H. Ulbricht, “Testing the foundation of quantum physics in space via Interferometric and non-interferometric experiments with mesoscopic nanoparticles”, Commun. Phys. 4, 155 (2021).
  * (30) J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source”, Nat. Commun. 5, 4788 (2014).
  * (31) E. Wigner, “On the quantum correction for thermodynamic equilibrium”, Phys. Rev. 40, 749 (1932).
  * (32) M. Zhang and Y. Wang, “Nonclassical effects in geodesic motion”, Phys. Rev. D 101, 026005 (2020).
  * (33) D. A. Steck, “Rubidium 87 D Line Data ”, available online at http://steck.us/alkalidata (revision 2.2.2, 9 July 2021).
  * (34) A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules ”, Rev. Mod. Phys. 81, 1051 (2009).
  * (35) R. E. Sapiro, R. Zhang, and G. Raithel, “Atom interferometry using Kapitza-Dirac scattering in a magnetic trap”, Phys. Rev. A 79, 043630 (2009).
  * (36) H. Müller, S. W. Chiow, and S. Chu, “Atom-wave diffraction between the Raman-Nath and the Bragg regime: Effective Rabi frequency, losses, and phase shifts”, Phys. Rev. A 77, 023609 (2008).
  * (37) S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, “A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules”, Nat. Phys. 3, 711 (2007).
  * (38) Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, “Quantum superposition of molecules beyond 25 kDa”, Nat. Phys. 15, 1242 (2019).
  * (39) A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms”, Nature (London) 400, 849 (1999).
  * (40) S. Lepoutre, A. Gauguet, G. Trénec, M. Büchner, and J. Vigué, “He-McKellar-Wilkens Topological Phase in Atom Interferometry”, Phys. Rev. Lett. 109, 120404 (2012).
  * (41) K. Sangster, E.A. Hinds, S. M. Barnett, and E. Riis, “Measurement of the Aharonov-Casher phase in an atomic system”, Phys. Rev. Lett. 71, 3641 (1993).
  * (42) G. Ranjit, C. Montoya, and A. A. Geraci, “Cold atoms as a coolant for levitated optomechanical systems ”, Phys. Rev. A 91, 013416 (2015).
  * (43) J. Zhang, T. Zhang, and J. Li, “Probing spontaneous wave-function collapse with entangled levitating nanospheres”, Phys. Rev. A 95, 012141 (2017).
  * (44) A. Chen, W. Nie, L. Li, W. Zeng, Q. Liao, X. Xiao, “Steady-state entanglement in levitated optomechanical systems coupled to a higher order excited atomic ensemble”, Opt. Commun. 403, 97 (2017).
  * (45) G. Li, W. Nie, Y. Wu, Q. Liao, A. Chen, and Y. Lan, “Manipulating the steady-state entanglement via three-level atoms in a hybrid levitated optomechanical system”, Phys. Rev. A 102, 063501 (2020).
  * (46) M. Toroš, S. Bose, and P. F. Barker, “Creating atom-nanoparticle quantum superpositions ”, Phys. Rev. Research 3, 033218 (2021).
  * (47) E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser”, Proc. IEEE 51, 89 (1963).
  * (48) O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: Theory and protocols”, Phys. Rev. A 83, 013803 (2011).
  * (49) C. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, “Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach”, Phys. Rev. A 100, 013805 (2019).
  * (50) F. Casas, A. Murua, and M. Nadinic, “Efficient computation of the Zassenhaus formula”, Computer Physics Communications 183, 2386 (2012).
  * (51) R. M. Wilcox, “Exponential Operators and Parameter Differentiation in Quantum Physics”, Journal of Mathematical Physics 8, 962 (1967).
  * (52) S. W. Chiow, T. Kovachy, H. C. Chien, and M. A. Kasevich, “102$\hbar k$ Large Area Atom Interferometers ”, Phys. Rev. Lett. 107, 130403 (2011).
  * (53) P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, “Atom-Interferometric Test of the Equivalence Principle at the $10^{-12}$ Level ”, Phys. Rev. Lett. 125, 191101 (2020).
  * (54) M. Cadoret, E. D. Mirandes, P. Cladé, F. Nez, L. Julien, F. Biraben, and S. G.-Khélifa, “Atom interferometry based on light pulses: Application to the high precision measurement of the ratio $h/m$ and the determination of the fine structure constant ”, Eur. Phys. J. Special Topics 172, 121 (2009).
  * (55) C. G.-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, O. Romero-Isart, “Levitodynamics: Levitation and control of microscopic objects in vacuum ”, Science 374, 168 (2021).
  * (56) J. Millen, and B. A. Stickler, “Quantum experiments with microscale particles ”, Contemporary Physics 61, 155 (2020).