# A hybrid quantum system formed by trapping atoms in the near-field of a
levitated nanosphere

A. Hopper  P. F. Barker<EMAIL_ADDRESS>Department of Physics and
Astronomy, University College London

(September 3, 2024)

###### Abstract

Near-field, radially symmetric optical potentials centred around a levitated
nanosphere can be used for sympathetic cooling and for creating a bound
nanosphere-atom system analogous to a large molecule. We demonstrate that the
long range, Coulomb-like potential produced by a single blue detuned field
increases the collisional cross-section by eight orders of magnitude, allowing
fast sympathetic cooling of a trapped nanosphere to microKelvin temperatures
using cold atoms. By using two optical fields to create a combination of
repulsive and attractive potentials, we demonstrate that a cold atom can be
bound to a nanosphere creating a new levitated hybrid quantum system suitable
for exploring quantum mechanics with massive particles.

Introduction.— The ability to cool and manipulate the centre-of-mass motion of
isolated nanoparticles levitated in vacuum is seen as an important enabling
technology for exploring macroscopic quantum mechanics at mass scales well
beyond the current state-of-the-art (25$\times$ 103 amu)bigmole . Experiments
have been proposed for studying wavefunction collapse, entanglement,
decoherence, and for probing the quantum nature of gravity of particles in the
$10^{6}-10^{10}$ amu mass rangeoriol1 ; oriol2 ; Bateman1 ; goldwater ;
Rahman_2019 . As a levitated particle in vacuum is much like a large atomic or
molecular system, it can be cooled and trapped using techniques first
developed in atomic physicsritschcavitycooling ; rempefeedback . Over the last
ten years, significant progress has been made towards cooling both the centre-
of-mass motion and the also the internal temperature of levitated
nanoparticlesnovotny1 ; weasel ; refrig . Cavity cooling and feedback cooling
to near the motional ground state has very recently been demonstrated, with
temperatures reaching the 10 $\mu$K range, bringing these systems well into
the quantum regimegroundstate ; PhysRevLett.124.013603 .

Sympathetic cooling of the centre-of-mass motion by elastic collisions with a
cold gas is another route towards creating cold trapped nanoparticles. These
methods have been very successful in creating both ultra-cold atoms and
molecules that cannot be laser-cooledSon2020 , with temperatures recently
reaching 200 nK. Sympathetic cooling via the coupling of the thermal motion of
cold atoms to a levitated nanosphere with a mediating cavity light field has
also been proposedgeracicooling  while more recently sympathetic cooling of
YIG nanosphere via an ultra-cold spin-polarized gas has been
proposedseberson2019optical .

Unlike other quantum optomechanical systems, cold levitated nanoparticles can
also be released from the trap, allowing matter-wave interferometry and the
measurement of small forces, in the absence of perturbing trapping fields.
However, unlike an atomic system, most levitated nanoparticles do not
typically have well-defined internal quantum states that can be coherently
manipulated and used for interferometry. A type of Ramsey interferometry using
a single nanoscale particle has been proposed. This uses the spin of a single
nitrogen vacancy (S=1, N-V-1) as a well-controlled quantum system that is
embedded within a levitated nanodiamondbose1 . However, while this general
scheme is a promising way to evidence non-classicality in the center-of-mass
motion of a massive quantum system, in optical traps it has so far been
plagued by technical issues due in large part to the high internal temperature
inherent in these levitation schemesRahman ; delord . High temperatures lead
to phonon broadening of the internal states as well as both motional and spin
decoherence, which hamper the use of this system for matter-wave
interferometryentanglementgravity . Despite these problems, it has recently
been used to demonstrate spin coolingDelord2020 .

Figure 1: A diagram illustrating the creation of a spherically symmetric
radial optical potential around a nanosphere trapped in a Paul trap. The
potential, which is always centered on the nanosphere, is created by both the
incident and scattered fields from three loosely focused orthogonally
polarized optical fields, shown in blue.

In this letter, we describe a different type of levitated hybrid quantum
system, which utilises the good control over both the internal and external
degrees of freedom of cold atoms when trapped within optical potentials
centered around a nanosphere. The optical potentials are created by both
incident and scattered light fields and can be tuned by controlling laser
intensity and wavelength. This system is attractive because atoms can be
strongly bound to the nanosphere via the optical fields, but, unlike an NV
nanodiamond or quantum dot where the quantum emitter is embedded in the
macroscopic object, they are well isolated from the surface and are therefore
not subject to phonon broadening or lattice efects. In addition, the optical
potentials can be used to increase the collisional cross-section of the
nanoparticle enabling rapid thermalization via collisions with cold atoms.

Radially symmetric optical potentials.— We consider a dielectric nanosphere
that is levitated in a Paul trap in high vacuumBullier_2020 . To create a
radially symmetric optical potential around the nanosphere it is illuminated
with three weakly focused optical fields propagating in orthogonal directions,
as shown in figure 1. When the particle is significantly smaller than the
wavelength of light, the scattered electric field and intensity can be readily
calculated in the Rayleigh approximationnovotnybook . For a polarizable
particle illuminated by an incident optical field $E=E_{0}\exp{i(kx-\omega
t)}$, propagating in the $x$ direction and polarized along the $z$ direction,
the total field due to both the incident and scattered field in the radial and
azimuthal directions using spherical polar co-ordinates is given by

$\begin{split}E_{r}&=CE\cos{\theta}\left[\frac{2}{k^{2}r^{2}}-\frac{2i}{kr}\right]\frac{k^{2}}{r}+E\cos{\theta}+c.c\\\
E_{\theta}&=CE\sin{\theta}\left[\frac{1}{k^{2}r^{2}}-\frac{i}{kr}-1\right]\frac{k^{2}}{r}-E\sin{\theta}+c.c.\\\
\end{split}$ (1)

For this polarization there is no $\phi$ dependence in the field, and
$C=\alpha/4\pi\epsilon_{0}=a^{3}(n^{2}-1)/(n^{2}+2)$ where $\alpha$ is the
polarizability of the nanosphere, $a$ is the radius of the nanoparticle of
refractive index $n$, $k=2\pi/\lambda$ is the wavevector of the light, and
$\lambda$ is its wavelength. The total near-field irradiance or intensity
around the nanosphere from a single plane wave is then given by

$\begin{split}I_{s}&=I_{0}\frac{[4Cr^{3}+r^{6}+4C^{2}(1+k^{2}r^{2})]}{r^{6}}\cos^{2}{\theta}\\\
+&I_{0}\frac{[r^{6}-2Cr^{3}(1-k^{2}r^{2})+C^{2}(1-k^{2}r^{2}+k^{4}r^{4})]}{r^{6}}\sin^{2}{\theta}\\\
\end{split}$ (2)

where $I_{0}=1/2\epsilon_{0}cE_{0}^{2}$. Figure 2a shows a plot of the near
field intensity around the nanoparticle, with radius $a=40$ nm, refractive
index n=1.43 and wavelength 1000 nm. Note that in contrast to the far-field
scattered radiation, the near field scattered intensity is maximized along the
polarization direction. As this simple analytical solution only applies in the
Rayleigh regime when $a<<\lambda$, we confirm that this is a good
approximation for this radius by performing numerical simulations by finite
difference time domain calculations using the Lumerical software
packagelumerical . In order to create a spherically symmetric optical near-
field potential, we illuminate the particle with three fields, with orthogonal
angles of incidence and polarization. When the three optical fields with
angles $\theta$ of $cos^{-1}(x/r)$, $cos^{-1}(y/r)$, $cos^{-1}(z/r)$ are added
incoherently, the total intensity is given by

$\begin{split}I_{T}&=I_{0}\bigg{[}3+\frac{6C^{2}}{r^{6}}+\frac{2C^{2}k^{2}}{r^{4}}+\frac{2C^{2}k^{4}}{r^{2}}+\frac{4Ck^{2}}{r}\bigg{]}\end{split}$
(3)

We choose the wavevectors of the three incident fields such that $k\approx
k_{1}\approx k_{2}\approx k_{3}$. A plot of this combined intensity in the y-x
plane is shown in figure 1b, which illustrates the radial spherical symmetry
around the nanosphere. Note that the radial dependence of the total field is
dependent on the wavelength of light through the wavevector
$k=\frac{2\pi}{\lambda}$, which is dominated at short range by the $1/r^{6}$
term, and by the $1/r$ term at long range.

Figure 2: The total near-field intensity normalized by the incident beam
intensity in the y-z plane. The light of wavelength 1000 nm is incident on a
nanosphere of 40 nm radius with and a refractive index of n=1.43. The field
inside the nanosphere, shown as the circular white region,is not plotted. a)
Is a plot of the intensity when illuminated by a single field propagating in
the $x$ direction and polarized in the $z$ direction. b) A plot of the
intensity when illuminated by three orthogonally propagating fields with
orthogonal polarizations.

The interaction of the optical field with either the atom or the nanosphere
induces an oscillating dipole moment that leads to an optical dipole potential
given by $U_{dip}(r)=-\frac{1}{2}\frac{\alpha I(r)}{\epsilon_{0}c}$, where
$\alpha$ is the polarizability of an atom or nanosphereGRIMM200095 . When
operating close to an atomic resonance the potential is better described by
$U_{dip}(r)=\frac{3\pi c^{2}}{2\omega_{0}^{3}}\frac{\Gamma}{\Delta}I(r)$,
where $\Delta$ is the detuning of the laser from resonance and $\Gamma$ is the
atomic line width. The potential is inversely proportional to the detuning
from an atomic resonance, and such potentials are routinely used to trap cold
atoms in free space and more recently in the evanescent fields around a
tapered optical fiberatomsfiber1 . As the sign of the potential is determined
by its detuning with respect to the atomic resonance or resonances, the
detuning can be used to produce both repulsive and attractive potentials. The
magnitude of the potential can be tuned by controlling the incident light
intensity. The maximum potential at any detuning is eventually limited by the
resonant excitation of atoms to other internal states or by increased
scattering of light, which can lead to recoil heating.

To construct a realistic potential, in addition to the applied optical fields,
we must also take into account the strong short-range attractive interactions
from the Casimir-Polder force and also the charge-induced polarization force
between a charged nanosphere and the neutral atom. For charges distributed
evenly on the surface, the polarization potential is given by
$U=-\frac{1}{(4\pi\epsilon_{0})^{2}}\frac{\alpha_{a}q^{2}}{2r^{4}}$ where $q$
is the total charge on the nanophere, and $\alpha_{a}$ is the static
polarizability of the atom. For the Casimir-Polder interaction we consider the
non-retarded case where the radius of the sphere is significantly less than
the resonance wavelengthscp1 . In this case the potential is given by

$\begin{split}&U_{CP}=-\frac{\hbar}{8\pi^{2}\epsilon_{0}}\sum_{l=0}^{\infty}(2l+1)(l+1)\\\
&\times\frac{a^{2l+1}}{r^{2l+4}}\int_{0}^{\infty}d\zeta\alpha_{a}(i\zeta)\frac{\epsilon(i\zeta)-1}{\epsilon(i\zeta)+[(l+1)/l]}\end{split}$
(4)

where $\zeta$ is the angular frequency, $\alpha_{a}$ is the atomic
polarizability, $\epsilon$ is the permittivity of the nanosphere and $l$ is an
integercp1 . We calculate numerical solutions based on the properties of the
silica and the cold atoms. We find that for less than 100 charges on the
nanosphere, which is consistent with Paul trapping experimentsBullier_2020 ,
the Casimir-Polder (C-P) force is dominant at all radial distances from the
surface of the nanosphere. We therefore only consider the C-P interaction in
all subsequent calculations.

Although in principle any laser cooled atom can be trapped around a nanosphere
we consider interactions between metastable helium atoms that can be laser
cooled in the $2^{3}S_{1}$ electronic ground state and a SiO2 nanosphere of
radius $a=40$ nm. For the metastable helium atoms we need only consider the
few allowed transitions to the $n^{3}P$ states where $n=2-4$alphahe .

Sympathetic cooling.— We first consider sympathetic cooling of a trapped
nanosphere by using a single repulsive optical potential created by three
intersecting beams to increase the nanosphere collision cross-section so that
the nanoparticle can be sympathetically cooled via collisions with surrounding
cold atoms. The repulsive field not only increases the cross-section but also
prevents atoms from directly colliding with the nanosphere surface, which will
remain at 300 K. These types of collision are important to prevent since they
would induce recoil heating of the nanosphere by releasing internal energy
into the motion of the nanosphereheating . By using a magic wavelength, where
the AC stark shift induced by the potential is the same in the upper and lower
electronic states, it is possible to simultaneously laser cool the atoms
within the optical potential. This will allow continuous cooling of the atoms
around the nanosphere. For metastable helium, a repulsive magic wavelength of
318.611 nm has been previously identifiedalphahe .

The full potential for the repulsive optical component $U_{R}(r)$ and the
attractive Casimir-Polder potential is then given by
$U_{TR}(r)=U_{R}(r)+U_{CP}(r)$, where the optical potential is given by the
far off-resonance form $U_{R}(r)=\frac{1}{2}\alpha_{a}I_{T}(r)/\epsilon_{0}c$,
where $\alpha_{a}=1.33\times 10^{-38}$ Cm2V-1 alphahe  for helium atoms at the
magic wavelength. Figure 3 is a plot of the total potential, as a function of
radial distance from the center of the sphere, created by a single beam
intensity of $I_{0}=1\times 10^{5}$ W/cm2. This leads to a well depth of 100 K
for the silica nanosphere, which is orders of magnitude smaller than Paul trap
confining potentialBullier_2020 . For small collision energies with $E$ in the
$E/k_{b}$=10 $\mu$K range, typical of ultra-cold atomic gases, only the $1/r$
component needs to be considered.

The scattering of atoms from this potential is then the same form as a
repulsive Coulomb potential encountered in classical Rutherford scattering.
Without shielding, this type of potential has an infinite total cross-section
due to its long range character. However, as the $1/r$ nature of the optical
potential results from an interference between incident and scattered fields,
and the incidence field has a finite extent, the 1/r range will be limited to
the incident beam size with a spot size of 20 $\mu$m. As we are interested in
sympathetic cooling we determine the finite total cross-section, approximating
the limited range by using a Yukawa form where $U_{Y}\approx
U_{TR}\exp(-\frac{r}{\mu})$ and where $\mu=20\mu$m. This total cross section
is plotted in the inset in figure 3 as a function of collision energy in units
of $\mu$K. At a collision energy of 10 $\mu$K, this corresponds to a cross-
section of $\sigma$ = 6.6$\times$1011 nm2, which is more than 8 orders of
magnitude larger than the physical cross-section of the 40 nm nanosphere.

Figure 3: The radial potential formed by the combination of an attractive
short range potential due to the Casimir-Polder force and a short and long
range potential formed by a repulsive optical dipole potential using near
field light around a nanoparticle of radius 40 nm. The red line is the full
potential and the black line is the Yukawa potential approximation with
$\mu$=20 $\mu$m. The inset figure is the classical momentum transfer cross-
section for metastable helium atoms when the nanosphere is illuminated by
light at the magic wavelength of 318.611 nm.

The collisional cooling rate of the nanosphere induced by a bath of cold atoms
is approximately given by
$\gamma_{C}=\frac{\xi}{\alpha_{c}}N<\sigma(v)>v_{th}$, where the brackets
denote the speed averaged cross-section, $N$ is the number density of the gas,
$v_{th}=\sqrt{\frac{8k_{b}}{\pi}(\frac{T_{a}}{m_{a}}+\frac{T_{n}}{m_{n}})}$ is
the thermal speed of the gas, and $\alpha_{c}$=2.7 is the number of collisions
required to thermalize collision partners with equal massessympcool . For
unequal masses, as in our case, the cooling rate is reduced by the factor
$\xi=\frac{4m_{n}m_{a}}{(m_{n}+m_{a})^{2}}$, where $m_{n}=3.2\times 10^{8}$
amu is the mass of the nanosphere, and $m_{a}=4.0$ amu is the atomic mass.
This large mass difference, which usually makes sympathetic cooling
inefficient is compensated for by the large cross-section created by the
repulsive optical potential. For a silica nanosphere of density $\rho$=2000 kg
m-3 with radius 40 nm and a cold atom gas number density of 5$\times$1012
cm-3, we determine a cooling/damping rate of $2\pi\times 2.3$ kHz at
$T_{a}$=10 $\mu$K, which can be compared with a damping rate of 2$\pi\times$10
$\mu$Hz for the bare sphere. The ultimate temperature that could be reached
will be given by additional technical noise sources, but we estimate an
effective temperature from
$T_{e}=(\gamma_{C}T_{C}+\gamma_{H}T_{H}+\gamma_{R}T_{R})/(\gamma_{C}+\gamma_{H}+\gamma_{R})$,
where $\gamma_{C}$ is the the damping rate due to collisions with cold atoms
of temperature $T_{C}$, $\gamma_{H}$ and $T_{H}$ are the same quantities due
to collision with background atoms, while $\gamma_{R}$ and $T_{R}$ are from
recoil heating from the light used to form the potentialnovotnyrecoil . We do
not include the effect of electrical force noise as this depends on supply,
trap geometry and charge on the sphere. Assuming a background gas pressure of
10-9 mbar of air, we derive a temperature of 14 $\mu$K for a cold gas
temperature of $T_{C}$=10 $\mu$K and 2.6$\mu$K at 1 $\mu$K. Trap frequencies
up to at least 50 kHz appear feasible for these small nanoparticles levitated
in a Paul trap doi:10.1021/nn405920k  where, at a temperature of 1 $\mu$K an
average phonon occupancy approaching unity could be achieved. In such a scheme
ground state cooling may be possible with only small additional feedback
cooling.

Although we have considered sympathetic cooling in a Paul trap we point out
that such a scheme could also be used in an optical trap formed by three
intersecting beams. Here the intensity is orders of magnitude higher than in
the Paul trap and the increased heating rate due to photon recoil would not
allow cooling to below one Kelvin.

Atom-nanosphere bound states.— We now explore the creation of a bound atom-
nanosphere hybrid system using a combination of both a repulsive and an
attractive optical potential given by

$\begin{split}&U_{0}(r)=U_{CP}+3(U_{R}-U_{A})+6\frac{U_{R}C_{R}^{2}-U_{A}C_{A}^{2}}{r^{6}}\\\
&+4\frac{U_{R}C_{R}k_{R}^{2}-U_{A}C_{A}k_{A}^{2}}{r}\\\ \end{split}$ (5)

where $U_{R}$ and $U_{A}$ are the maximum repulsive and attractive well depths
in the absence of the scattered fields, and $U_{C}P$ is due to the Casimir-
Polder interaction as before. Note that the $1/r^{4}$ and the $1/r^{2}$ terms
of equation 3 are not significant and are not included here. Initially
ignoring the effect of the CP term, which is only dominant at very short range
($r<50$ nm), the short range($1/r^{6}$) optical repulsive term constructed
from blue detuned light must exceed the corresponding term for attractive red
detuned light. In addition, its maximum value must be significantly greater
than the energy of the cold atoms to avoid direction collisions with the
surface of the sphere. From equation 5 this condition implies that
$U_{R}C_{R}^{2}>U_{A}C_{A}^{2}$. However, at long range the attractive ($1/r$)
potential created by the red detuned field must be larger than the repulsive
component so that $U_{R}C_{R}k_{R}^{2}<U_{A}C_{A}k_{A}^{2}$. For a small
refractive index variation between the two wavelengths, such that
$C_{R}\approx C_{A}$, then $k_{R}^{2}<k_{A}^{2}$ and the wavelength of light
used to construct the repulsive optical potential must therefore be larger
than that used for the attractive potential
($\lambda_{R}^{2}>\lambda_{A}^{2}$). For a small variation in the refractive
index of the nanosphere, we obtain approximate limits of the ratio of the
magnitude of the repulsive and attractive potentials given by
$1<\frac{U_{R}}{U_{A}}<\frac{\lambda_{R}^{2}}{\lambda_{A}^{2}}$. Using the
$2^{3}S_{1}\rightarrow n^{3}P$ resonances of metastable helium to enhance the
optical well depth, we can choose a blue detuned field (5000 GHz) with respect
to the $2^{3}S_{1}-2^{3}P$ resonance at 1083 nm to create a repulsive
potential, and an attractive potential formed by a field that is red detuned
by 100 GHz with respect to the $2^{3}S_{1}-3^{3}P$ transition at 389.0 nm. Our
choice of wavelength for both potentials limits the ratio of the repulsive to
attractive optical potential in the range $1<U_{R}/U_{A}<7.75$. We explored
the properties of this potential for creating bound states for
$U_{R}/U_{A}=2.47$ using well depths of $U_{R}$=2.7 mK and $U_{A}$= 14.6 mK.
Intensities of approximately $10{{}^{5}}$ Wcm2 are required, leading to atom
trap lifetimes on the order of 10 ms.

Figure 4 is plot of the combined potential, $U_{0}$, which has a well depth of
340 $\mu$ K that is sufficient to trap laser-cooled atoms. We calculate the
bound states of this potential and the wavefunctions and allowed energies
using the Numerov methodALLISON1970378 . These are shown in figure 4 for the
first four states. This bound potential is similar to the optical potentials
created around tapered optical fibers, and the same techniques can in
principle be used to load the atoms into this radially symmetric atom-
nanosphere potentialatomsfiber1 . Sideband-resolved cooling has been
demonstrated around these tapered optical fibers and could also be used to
cool the atoms into the ground state of the optical potential around the
nanosphereatomsfiber2 . As the optical potential can be tuned by controlling
the relative intensity between the repulsive and attractive fields scattering,
resonances could be used to enhance the cross-section and the binding of cold
atoms to the nanosphere. Figure 5 is a calculation of the total cross-section,
as a function collision energy using the log-derivative method showing narrow
resonances due to partial waves of l=3,7 and higherJOHNSON1973445 . The
scattering of cold atoms from these potentials and the measurement of energy
of these higher energy resonances could be used to characterise the Casimir-
Polder potential or to enhance association between the free atom and the
nanosphere. Finally, as the trapping volume is small ($\approx$500 nm3) we
expect that collisional blockade will result in only one atom captured in the
potential, forming a system that is analogous to a massive diatomic molecule.
blockade .

Figure 4: The radial potential formed by the combination of an attractive
short-range Casimir-Polder potential and a long range component formed by a
repulsive optical dipole potential from the scattering of the light from the
nanoparticle. Also shown are the first four wavefunctions and their energies.

Conclusion.— Whilst we have explored interactions with cold metastable helium
atoms, the general scheme outlined for both sympathetic cooling and the
creation of bound states could be also be carried out with most laser-cooled
species. We point out that this scheme only works well for particles where a
strong scattered field is created by relatively large nanoscale polarizable
particles. Similar fields will be created around large macromolecules of
similar size particles, such as a virus which have been trapped in vacuum.
Because they are not spherical, as considered here, a more complicated non-
spherical potential will be created. However, sympathetic cooling should still
be feasible and, as relatively low intensity fields can be used optical damage
by absorption of light can be minimised within a Paul trap.

Figure 5: The total scattering cross-section as function of collision energy
in units of $\mu$K. Scattering resonances from higher order partial waves
could be used to probe the bound state potential and the Casimir-Polder force.
Resonances from l=3,7 are indicated.

The hybrid atom-nanosphere system described here may be used as an alternative
for the Stern-Gerlach Ramsey type interferometry that has been proposed for NV
centres in referencesbose1 ; entanglementgravity . Here, the metastable helium
atom, which is optically bound to the nanosphere, could be placed in a
superposition of spin states (m=-1,0,1) by initially preparing the atom in a
stretched state (m=-1) via optical pumping before creating a superposition of
all spin statesVassen2016 . By placing the trapped particle in a inhomgeneous
magnetic field we can entangle the atomic spin with the motion of the particle
via the Zeeman effect converting the spin superposition into a center-of-mass
superposition. The creation of macroscopic superpositions of a different
optically bound hybrid system will be outlined elsewheretoroscatstate .

Acknowledgements.—The authors would like to acknowledge useful discussions
with M. Toroš, S. Bose, A. Pontin and A. Rahman. The authors acknowledge
funding from the EPSRC Grant No. EP/N031105/1 and the H2020-EU.1.2.1 TEQ
project Grant agreement ID: 766900.

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