# Fluctuation-induced Forces on Nanospheres in External Fields

Clemens Jakubec<EMAIL_ADDRESS>Vienna Center for Quantum
Science and Technology (VCQ), Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria  Pablo Solano  Departamento de
Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción,
Concepción, Chile  Uroš Delić  Vienna Center for Quantum Science and
Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090 Vienna, Austria  Kanu Sinha<EMAIL_ADDRESS>Wyant College of Optical
Sciences and Department of Physics, University of Arizona, Tucson, AZ USA
85721

###### Abstract

We analyze the radiative forces between two dielectric nanospheres mediated
via the quantum and thermal fluctuations of the electromagnetic field in the
presence of an external drive. We generalize the scattering theory description
of fluctuation forces to include external quantum fields, allowing them to be
in an arbitrary quantum state. The known trapping and optical binding
potentials are recovered for an external coherent state. We demonstrate that
an external squeezed vacuum state creates similar potentials to a laser,
despite its zero average intensity. Moreover, Schrödinger cat states of the
field can enhance or suppress the optical potential depending on whether they
are odd or even. Considering the nanospheres trapped by optical tweezers, we
examine the total interparticle potential as a function of various
experimentally relevant parameters, such as the field intensity, polarization,
and phase of the trapping lasers. We demonstrate that an appropriate set of
parameters could produce mutual bound states of the two nanospheres with
potential depth as large as $\sim 200$ K. Our results are pertinent to ongoing
experiments with trapped nanospheres in the macroscopic quantum regime, paving
the way for engineering interactions among macroscopic quantum systems.

##  I Introduction

Bringing massive systems to the quantum regime is an essential step towards
understanding the quantum-to-classical transition, a goal of foundational
importance [1]. Remarkable progress in the control of atomic, molecular, and
optical (AMO) systems has made the macroscopic quantum regime increasingly
accessible to experiments [2, 3, 4]: The Schrödinger’s cat has been realized
by photons, atoms and mechanical resonators [5, 6, 7, 8], molecular clusters
as massive as $\sim 25000$ amu have been shown to exhibit quantum interference
[9], and millimeter-sized objects have been cooled down to their quantum
ground states [10].

Optically levitated dielectric nanospheres are among the most promising
experimental platforms for realizing large superpositions of massive objects
[11]. First pioneered by Ashkin in 1970 [12], they bring together the
advantages of optical trapping and cooling methods in terms of control, while
being well-isolated from the environment without mechanical clamping, thus
minimizing decoherence. Recent experiments have brought dielectric nanospheres
as massive as $\sim 10^{8}$ amu to their motional quantum ground states [13,
14, 15, 16, 17, 18]. This finds application in quantum sensing and metrology
while also providing an avenue for investigating foundational questions such
as quantum-to-classical transition and gravitational interaction of quantum
systems. More recent experiments have achieved tunable interactions between
two silica nanospheres trapped via optical tweezers to create non-conservative
and non-reciprocal interparticle potentials [19]. The ability to precisely
control and engineer the interactions between two dielectric nanospheres paves
the way for realizing correlated macroscopic quantum systems.

Figure 1: Schematic representation of two nanospheres (A and B) interacting
via the quantum and thermal fluctuations of the EM field and externally
applied arbitrary quantum fields. The spheres are located at
${\mathbf{r}}_{A}$ and ${\mathbf{r}}_{B}$, separated along the $x$-axis, with
a polarizability $\alpha\left(\omega\right)$ each.

When considering interactions between two nanospheres in the near-field
regime, one will inevitably encounter fluctuation-induced forces resulting
from the quantum and thermal fluctuations of the electromagnetic (EM) field
[20]. Such fluctuation-induced forces exist even when the EM field is in the
vacuum state and are typically attractive in nature, thus imposing fundamental
constraints on how close can two particles be stably trapped near each other
and limiting the realization of macroscopic quantum states of such particles
[21] as well as influencing the decoherence of such states. It is thus
critical to develop experimentally amenable ways to control fluctuation-
induced forces between nanoparticles in the near-field regime. Furthermore,
the role of fluctuations in internal temperature sensing of levitated
nanospheres has been investigated [22].

Previous works have shown that fluctuation-induced forces can be substantially
modified in the presence of external drives [23, 24, 25, 26, 27]. Such drive-
induced modifications to the vacuum forces can be significant compared to its
pure fluctuation-induced counterpart, and even repulsive in character. Turning
the strong short-ranged Casimir attraction to repulsion can allow for
levitation and trapping particles at nanoscales [24, 28] and mitigate stiction
in nano- and micro-mechanical devices [29]. Therefore, control over external
drives can open new avenues for tailoring fluctuation forces in a system of
two or more nanospheres.

In this work we explore fluctuation-induced forces in a system of two
dielectric nanospheres in the presence of an externally applied field in a
general quantum state. We describe the radiative interactions between the
nanospheres mediated via the total field by summing over the various
scattering processes to obtain the potentials seen by each sphere for a
general quantum state of the external field to second-order in the particle
polarizabilities. Analyzing these potentials for specific states of the
external field, we show that in addition to the first and second order
Casimir-Polder (CP) potential: (1) for a coherent state of the external field
one recovers the well-known single particle trapping and interparticle optical
binding potentials; (2) a squeezed vacuum state of the external field can
generate an equivalent potential to the trapping and optical binding
potentials, even in the absence of a coherent amplitude; (3) a cat state of
the external field can allow one to tune the trap and optical binding
potentials via the phase between the two superposed coherent states. We
further analyze the scaling behavior of the various contributions to the total
potential for nanospheres trapped in optical tweezers for different regimes of
the interparticle separation. Our results show that combining the near-field
fluctuation forces with optical binding can create mutual bound states of the
two nanospheres with a potential depth as large as $\sim 200$ K. We study such
potentials as a function of various drive parameters – intensity, polarization
and the relative optical phase of the tweezer fields.

The paper is organized as follows. First, in Sec. II, we present our model to
describe the system up to second-order in the particle polariazbility
$(\alpha(\omega))$, which we use to derive the interparticle potentials for
general states of the external field in Sec. III. We then study the obtained
potentials in the presence of tweezer fields in Sec. IV and analyze their
parameter dependence in the light of finding bound states of two nanospheres,
presenting a summary and outlook of our work in Sec. VI.

##  II Model

Figure 2:  Electric field and induced dipole moment at ${\mathbf{r}}_{A}$ to
the lowest orders in the particle polarizabilities $\alpha(\omega)$ as denoted
by $\hat{{\mathbf{E}}}^{(0,1)}\left({\mathbf{r}_{A}}\right)$ and
$\hat{{\mathbf{P}}}^{(1,2)}\left({\mathbf{r}_{A}}\right)$ at the position of
sphere A.

We consider a system of two nanospheres at positions ${\mathbf{r}}_{A}$ and
${\mathbf{r}}_{B}$, optically trapped by external fields, as shown in Fig. 1.
The Hamiltonian of the system is given by
$\hat{H}=\hat{H}_{\mathrm{F}}+\hat{H}_{\mathrm{int}}$, where:

$\displaystyle\hat{H}_{F}=$ $\displaystyle\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega\hbar\omega\hat{{\mathbf{f}}}_{\lambda}^{\dagger}({\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}({\mathbf{r}},\omega),$
(1)

is the Hamiltonian of the quantized EM field, with
$\hat{{\mathbf{f}}}_{\lambda}^{(\dagger)}\left({\mathbf{r}},\omega\right)$ as
bosonic operators for the EM field in the macroscopic QED formalism [30, 31].
These operators obey the canonical commutation relations
$\left[\hat{\mathbf{f}}_{\lambda}\left({\mathbf{r}},\omega\right),\hat{\mathbf{f}}_{\lambda^{\prime}}^{\dagger}\left({\mathbf{r}}^{\prime},\omega^{\prime}\right)\right]=\delta(\omega-\omega^{\prime})\delta\left({\mathbf{r}}-{\mathbf{r}}^{\prime}\right)\delta_{\lambda\lambda^{\prime}}$.
The dipole interaction Hamiltonian

$\displaystyle\hat{H}_{\mathrm{int}}=$
$\displaystyle-\sum_{i=A,B}\frac{1}{2}\hat{{\mathbf{P}}}({\mathbf{r}}_{i},t)\cdot\hat{{\mathbf{E}}}({\mathbf{r}}_{i},t)$
(2)

represents the interaction between the induced dipole moments of the
individual nanospheres $\hat{\mathbf{P}}\left({\mathbf{r}}_{i},t\right)$ and
the electric field $\hat{\mathbf{E}}\left({\mathbf{r}}_{i},t\right)$ in the
point-particle approximation. The induced dipole moment is defined as
${\mathbf{P}}=\alpha{\mathbf{E}}$.

The total electric field seen by sphere $i$ is
$\hat{\mathbf{E}}({\mathbf{r}}_{i},t)=\hat{{\mathbf{E}}}_{\mathrm{f}}({\mathbf{r}}_{i},t)+\hat{{\mathbf{E}}}_{\mathrm{ex}}({\mathbf{r}}_{i},t)$,
where $\hat{{\mathbf{E}}}_{\mathrm{f}}({\mathbf{r}}_{i},t)$ refers to field
fluctuations and
$\hat{\mathbf{E}}_{\mathrm{ex}}\left({\mathbf{r}}_{i},t\right)$ represents an
external electric field, such as that of a laser. Since the particles act as
scatteres, light can bounce between the spheres multiple times. Each time the
field will pick up an additional factor of the polarizability $\alpha$.
Consequently, we can describe both the fluctuation field and the external
field, including all their scattered components, as an expansion in $\alpha$.
We express each of these fields up to first order in the particle
polarizability $\alpha(\omega)$, including the components scattered off the
nanospheres:

$\displaystyle\hat{{\mathbf{E}}}_{\mathrm{f,{ex}}}({\mathbf{r}}_{i},t)=\hat{{\mathbf{E}}}_{\mathrm{{f},{ex}}}^{(0)}({\mathbf{r}}_{i},t)+\hat{{\mathbf{E}}}_{\mathrm{f},\mathrm{ex}}^{(1)}({\mathbf{r}}_{i},t)+O(\alpha^{2}),$
(3)

where the superscripts $k$ in
$\hat{\mathbf{E}}^{(k)}_{\mathrm{f},\mathrm{ex}}$ refer to the order of
polarizability $\alpha^{k}(\omega)$. The zeroth order field

$\displaystyle\hat{{\mathbf{E}}}^{(0)}_{\mathrm{f}}({\mathbf{r}}_{i})=\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega
G^{\lambda}({\mathbf{r}}_{i},{\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}(\mathrm{r},\omega)$
$\displaystyle+\mathrm{H.c.}$ (4)

refers to the quantum and thermal fluctuations of the free electric field, as
depicted in Fig. 2 (a). The coefficients
$G_{\lambda}({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)$ are proportional to
the Green’s tensor $G\left({\mathbf{r}},{\mathbf{r}}^{\prime},\omega\right)$
of the EM field, such that [30]:

$\displaystyle\sum_{\lambda}\int\mathrm{d}^{3}{\mathbf{r}}^{\prime}{G}_{\lambda}\left({\mathbf{r}}_{1},{\mathbf{r}}^{\prime},\omega\right)\cdot{G}^{\dagger}_{\lambda}\left({\mathbf{r}}_{2},{\mathbf{r}}^{\prime},\omega\right)=$
$\displaystyle\frac{\hbar\mu_{0}}{\pi}\omega^{2}\mathrm{Im}{G}\left({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega\right),$
(5)

where $\lambda$ characterizes the source of the quantum noise polarization
$(\lambda=e)$ or magnetization $(\lambda=m)$. The Green’s tensor
${G}\left({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega\right)$ describes the
propagation of a photon at frequency $\omega$ between positions
${\mathbf{r}}_{1}$ and ${\mathbf{r}}_{2}$ and is obtained as a solution to the
inhomogeneous Helmholtz equation (see Appendix A for details) [32]. The
expression

$\displaystyle\hat{{\mathbf{E}}}_{\mathrm{f}}^{(1)}({\mathbf{r}}_{i})=$
$\displaystyle\sum_{j=A,B}\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega\alpha(\omega)\omega^{2}\mu_{0}$
$\displaystyle G({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)\cdot
G^{\lambda}({\mathbf{r}}_{j},{\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}(\mathrm{r},\omega)+\mathrm{H.c.}$
(6)

stands for the electric field at the position ${\mathbf{r}}_{i}$ of sphere
$i$, sourced by the dipole moment induced in sphere $j$ by the fluctuation
field, as shown in Fig. 2 (c) and (e). We assume a real polarizability
$\alpha\left(\omega\right)$ of the nanospheres such that there is no internal
dissipation.

Similarly the incident and scattered external fields
$\hat{{\mathbf{E}}}_{\mathrm{ex}}({\mathbf{r}},t)$ seen by sphere $i$ are
given by:

$\displaystyle\hat{\mathbf{E}}_{\mathrm{ex}}^{(0)}({\mathbf{r}}_{i},t)=$
$\displaystyle\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,{\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{i},{\mathbf{k}},\omega)\hat{a}_{\sigma}({\mathbf{k}})e^{-i\omega
t}+\mathrm{H.c.}$ (7)
$\displaystyle\hat{\mathbf{E}}_{\mathrm{ex}}^{(1)}({\mathbf{r}}_{i},t)=$
$\displaystyle\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,\mu_{0}\omega^{2}\alpha(\omega)G({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)\cdot$
$\displaystyle\quad\quad\quad\quad{\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{j},{\mathbf{k}},\omega)\hat{a}_{\sigma}({\mathbf{k}})e^{-i\omega
t}+\mathrm{H.c.}$ (8)

Here
${\mathbf{\Phi}}_{\sigma}({\mathbf{r}},{\mathbf{k}},\omega)=\sqrt{\hbar\omega/2\pi\epsilon_{0}(2\pi)^{3}}\tilde{{\mathbf{\Phi}}}_{\sigma}({\mathbf{r}},{\mathbf{k}},\omega)$
where $\tilde{{\mathbf{\Phi}}}_{\sigma}({\mathbf{r}},{\mathbf{k}},\omega)$ is
the mode function of the field normalized to one at its maximum with
wavevector ${\mathbf{k}}$, angular frequency $\omega$ and polarization
$\sigma$.

The fields $\hat{\mathbf{E}}^{(k)}_{\mathrm{f,ex}}({\mathbf{r}}_{i},t)$ at
sphere $i$ can each induce a dipole moment in sphere $i$ given by
$\hat{\mathbf{P}}_{\mathrm{f,ex}}^{(k+1)}({\mathbf{r}}_{i},t)$:

$\displaystyle\hat{{\mathbf{P}}}_{\mathrm{f,ex}}({\mathbf{r}}_{i})=\hat{{\mathbf{P}}}_{\mathrm{f,ex}}^{(1)}({\mathbf{r}}_{i})+\hat{{\mathbf{P}}}_{\mathrm{f,ex}}^{(2)}({\mathbf{r}}_{i})+O(\alpha^{3}).$
(9)

The first term in the above equation represents the dipole moment induced by
the EM field fluctuations (f) or the incident external field (ex), as shown in
Fig. 2 (b). The second term refers to the dipole moment induced by the
electric field scattered off one of the spheres, as in Fig. 2 (d) and (d).

The fluctuating dipole moment up to second order in $\alpha(\omega)$ is

$\displaystyle\hat{{\mathbf{P}}}_{\mathrm{f}}^{(1)}({\mathbf{r}}_{i})=$
$\displaystyle\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega\alpha(\omega)G^{\lambda}({\mathbf{r}}_{i},{\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}({\mathbf{r}},\omega)+\mathrm{H.c.}$
(10)
$\displaystyle\hat{{\mathbf{P}}}_{\mathrm{f}}^{(2)}({\mathbf{r}}_{i})=\sum_{j=A,B}\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega\alpha(\omega)^{2}\omega^{2}\mu_{0}$
$\displaystyle\quad\quad\quad\quad
G({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)\cdot
G^{\lambda}({\mathbf{r}}_{j},{\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}({\mathbf{r}},\omega)+\mathrm{H.c.},$
(11)

The dipole moment induced in sphere $i$ by the external field
$\hat{\mathbf{E}}_{\mathrm{ex}}({\mathbf{r}}_{i},t)$ can be similarly obtained
as:

$\displaystyle{\mathbf{P}}_{\mathrm{ex}}^{(1)}({\mathbf{r}}_{i},t)=\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,\alpha(\omega){\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{i},{\mathbf{k}},\omega)\hat{a}_{\sigma}({\mathbf{k}})e^{-i\omega
t}$ $\displaystyle+\mathrm{H.c.}$ (12)
$\displaystyle{\mathbf{P}}_{\mathrm{ex}}^{(2)}({\mathbf{r}}_{i},t)=$
$\displaystyle\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,\alpha(\omega)^{2}\mu_{0}\omega^{2}G({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)\cdot$
$\displaystyle\quad\quad\quad\quad{\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{j},{\mathbf{k}},\omega)\hat{a}_{\sigma}({\mathbf{k}})e^{-i\omega
t}+\mathrm{H.c.}$ (13)

We can thus combine the contributions to second order in polarizability
$\alpha$ and write the interaction Hamiltonian in Eq. (14) as:

$\displaystyle\begin{split}\hat{H}_{\mathrm{int}}=\hat{H}_{\mathrm{int}}^{(1)}&+\hat{H}_{\mathrm{int}}^{(2)}+O(\alpha^{3}),\end{split}$
(14)

where the first-order interaction Hamiltonian

$\displaystyle\hat{H}_{\mathrm{int}}^{(1)}\equiv$
$\displaystyle\sum_{i=A,B}-\frac{1}{2}\hat{{\mathbf{P}}}^{(1)}({\mathbf{r}}_{i},t)\cdot\hat{{\mathbf{E}}}^{(0)}({\mathbf{r}}_{i},t),$
(15)

represents the interaction between the electric field at the position of the
nanospheres and the dipole moment it induces, as shown by Fig. 2 (a) and (b).
We note that the contributing processes only pertain to a single sphere, thus
$H^{(1)}_{\mathrm{int}}$ does not contribute to the interparticle potential.

The second-order Hamiltonian is given by:

$\displaystyle\hat{H}_{\mathrm{int}}^{(2)}\equiv\sum_{i=A,B}-\frac{1}{2}\hat{{\mathbf{P}}}^{(1)}({\mathbf{r}}_{i},t)\cdot\hat{{\mathbf{E}}}^{(1)}({\mathbf{r}}_{i},t)$
$\displaystyle-\frac{1}{2}\hat{{\mathbf{P}}}^{(2)}({\mathbf{r}}_{i},t)\cdot\hat{{\mathbf{E}}}^{(0)}({\mathbf{r}}_{i},t),$
(16)

where the first term corresponds to the interaction between the dipole moment
induced by the fluctuation or external field in sphere $i$ (Fig. 2 (b)) and
the electric field scattered off either of the particles at position
${\mathbf{r}}_{i}$ (Fig. 2 (c) and (e)). The second term corresponds to the
interaction between the dipole moment induced in particle $i$ by the field
scattered off of one of the particles (Fig. 2 (d) and (f)) and the fluctuation
or external fields at ${\mathbf{r}}_{i}$ (as in Fig. 2 (a)).

##  III Potentials for general states of the field

We now focus on the potential that arises from the fields and interactions
described in the previous section. We denote the total potential seen by
sphere $i$ as

$\displaystyle
U_{i}({\mathbf{r}}_{i},{\mathbf{r}}_{j})=U_{i}^{(1)}({\mathbf{r}}_{i})+U_{i}^{(2)}({\mathbf{r}}_{i},{\mathbf{r}}_{j}),$
(17)

where
$U_{i}^{(1)}({\mathbf{r}}_{i},{\mathbf{r}}_{j})=\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}\hat{H}_{\mathrm{int}}^{(1)}\right]$
and
$U_{i}^{(2)}({\mathbf{r}}_{i},{\mathbf{r}}_{j})=\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}\hat{H}_{\mathrm{int}}^{(2)}\right]$
represent the first and second-order potentials in the particle polarizibility
calculated in first-order perturbation theory. It suffices to include first-
order shifts because the Hamiltonians Eq. (15) and (II) include all scattering
processes to second-order in the particle polarizability. We analyze these
potentials below for various quantum states of total field denoted by
$\rho_{\mathrm{F}}=\rho_{\mathrm{th}}\otimes\rho_{\mathrm{ex}}$, with
$\rho_{\mathrm{th}}=\mathrm{exp}(-\hat{H}_{\mathrm{F}}/k_{B}T)/Z$ being the
thermal state of the fluctuation field, with $Z$ as the partition function,
and $\rho_{\mathrm{ex}}$ being a general state of the external field.

###  III.1 First-order potential

We find the first-order potential seen by the sphere A as
$U^{(1)}_{A}({\mathbf{r}}_{A})=U_{A,\mathrm{f}}^{(1)}({\mathbf{r}}_{A})+U_{A,\mathrm{ex}}^{(1)}({\mathbf{r}}_{A})$,
where

$\displaystyle
U_{A,\mathrm{f}}^{(1)}({\mathbf{r}}_{A})=-\frac{1}{2}\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}\hat{\mathbf{P}}_{\mathrm{f}}^{(1)}({\mathbf{r}}_{A},t)\cdot\hat{\mathbf{E}}_{\mathrm{f}}^{(0)}({\mathbf{r}}_{A},t)\right]$
(18)

represents the contribution from quantum and thermal fluctuations of the field
and

$\displaystyle
U_{A,\mathrm{ex}}^{(1)}({\mathbf{r}}_{A})=-\frac{1}{2}\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}:\hat{\mathbf{P}}_{\mathrm{ex}}^{(1)}({\mathbf{r}}_{A},t)\cdot\hat{\mathbf{E}}_{\mathrm{ex}}^{(0)}({\mathbf{r}}_{A},t):\right]$
(19)

is the potential induced by the external field. We note that the cross-
coupling terms between the fluctuation field and the external field that are
linear in
$\left\\{\hat{\mathbf{f}}_{\lambda},\hat{\mathbf{f}}^{\dagger}_{\lambda}\right\\}$,
vanish in first order perturbation theory, as
$\mathrm{Tr}\left[\hat{\rho}_{\mathrm{F}}\hat{{\mathbf{f}}}^{(\dagger)}_{\lambda}({\mathbf{r}},\omega)\right]=0$
for diagonal states like the thermal state, but can be non-zero for other
states. Furthermore, the Hamiltonian $H_{\mathrm{int}}^{(1)}$ is normal
ordered with respect to the operators $\hat{a}_{\sigma}({\mathbf{k}})$ and
$\hat{a}^{\dagger}_{\sigma}({\mathbf{k}})$ of the external field. In our
description fluctuation effects arise solely from
$\hat{{\mathbf{E}}}_{\mathrm{f}}({\mathbf{r}}_{i},t)$. Nevertheless,
$\hat{{\mathbf{E}}}_{\mathrm{ex}}({\mathbf{r}}_{i},t)$ is a quantum field,
which exhibits ground state fluctuations. Consequently, the normal ordering is
necessary to prevent the overcounting of ground state effects. Substituting
the field $\hat{\mathbf{E}}_{\mathrm{f}}^{(0)}$ (Eq. (II)) and dipole moment
$\hat{\mathbf{P}}_{\mathrm{f}}^{(1)}$ (Eq. (II)) in Eq. (18) we obtain the
first-order fluctuation-induced potential:

$\displaystyle U_{A,\mathrm{f}}^{(1)}({\mathbf{r}}_{A})=$
$\displaystyle-\frac{\hbar\mu_{0}}{2\pi}\int
d\omega~{}\alpha(\omega)\omega^{2}(2n(\omega)+1)\mathrm{Tr}\left[\mathrm{Im}\
G_{AA}(\omega)\right],$ (20)

where $n(\omega)=\frac{1}{e^{{\hbar\omega}/\left(k_{B}T\right)}-1}$ is the
average thermal photon number and we have defined the shorthand notation
$G_{ii}(\omega)\equiv G({\mathbf{r}}_{i},{\mathbf{r}}_{i},\omega)$. If there
are no surfaces present in the system, then the Green’s tensor is given by the
free-space Greens tensor. The imaginary part of the free-space Green’s tensor,
which in the coincidence limit is given by $\mathrm{Im}\
G_{\mathrm{free}}({\mathbf{r}}_{i},{\mathbf{r}}_{i},\omega)=\mathbb{1}k/6\pi$,
yields a constant potential given by:

$\displaystyle
U_{A,\mathrm{f}}^{(1)}({\mathbf{r}}_{A})=-\frac{\hbar\mu_{0}}{4\pi^{2}c}\int
d\omega~{}\alpha(\omega)\omega^{3}(2n(\omega)+1)$ (21)

However, if there is a boundary present in the system, the Green’s tensor can
be split into a free part and a scattering part,
$G({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)=G_{\mathrm{free}}({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)+G_{\mathrm{sc}}({\mathbf{r}}_{i},{\mathbf{r}}_{j},\omega)$.
In the coincidence limit, the scattering part depends on the distance between
the nanosphere and the surface. Consequently, the single particle potential
will only give a distance-dependent energy, or a force, if there is a boundary
present in the system, corresponding to the usual thermal Casimir-Polder
potential [31].

Similarly, we obtain the first-order potential induced by the external field
by substituting Eq. (7) and (II) in (19):

$\displaystyle U_{A,\mathrm{ex}}^{(1)}({\mathbf{r}}_{A})=-\int
d^{3}{\mathbf{k}}\int d^{3}{\mathbf{k}}^{\prime}\alpha(\omega)$
$\displaystyle\mathrm{Re}\
\left\\{\sum_{\sigma\sigma^{\prime}}{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot{\mathbf{\Phi}}_{\sigma}^{A}\left\langle\hat{a}^{\dagger}_{\sigma}({\mathbf{k}})\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\right\rangle
e^{-i(\omega-\omega^{\prime})t}\right\\},$ (22)

where we have used the abbreviations
${\mathbf{\Phi}}_{\sigma}^{i}\equiv{\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{i},{\mathbf{k}},\omega)$,
${\mathbf{\Phi}}_{\sigma^{\prime}}^{i}\equiv{\mathbf{\Phi}}_{\sigma^{\prime}}({\mathbf{r}}_{i},{\mathbf{k}^{\prime}},\omega^{\prime})$
. If the external field is a single-mode field, the above potential is
proportional to the expectation number of photons in that mode.

###  III.2 Second-order potential

The second-order potential seen by sphere A can be obtained from Eq. (II) as
$U_{A}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=U_{A,\mathrm{f}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})+U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})$,
where:

$\displaystyle
U_{A,\mathrm{f}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-\frac{1}{2}\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}\left\\{\hat{\mathbf{P}}^{(1)}_{\mathrm{f}}({\mathbf{r}}_{i},t)\cdot\hat{\mathbf{E}}^{(1)}_{\mathrm{f}}({\mathbf{r}}_{i},t)\right.\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.\left.+\hat{\mathbf{P}}^{(2)}_{\mathrm{f}}({\mathbf{r}}_{i},t)\cdot\hat{\mathbf{E}}^{(0)}_{\mathrm{f}}({\mathbf{r}}_{i},t)\right\\}\right]$
(23)

represents the fluctuation-induced component, and

$\displaystyle
U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-\frac{1}{2}\mathrm{Tr}_{\mathrm{F}}\left[\rho_{\mathrm{F}}\left\\{:\hat{\mathbf{P}}^{(1)}_{\mathrm{ex}}({\mathbf{r}}_{i},t)\cdot\hat{\mathbf{E}}^{(1)}_{\mathrm{ex}}({\mathbf{r}}_{i},t):\right.\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\left.\left.+:\hat{\mathbf{P}}^{(2)}_{\mathrm{ex}}({\mathbf{r}}_{i},t)\cdot\hat{\mathbf{E}}^{(0)}_{\mathrm{ex}}({\mathbf{r}}_{i},t)\right\\}:\right],$
(24)

the component induced by the external field. Substituting the fields (Eq. (II)
and (II)) and induced dipole moments (Eq. (II) and (II)) in Eq. (III.2),
yields the fluctuation-induced second-order potential for sphere A:

$\displaystyle U_{A,\mathrm{f}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\frac{\hbar\mu_{0}^{2}}{\pi}\int
d\omega~{}\alpha(\omega)^{2}\omega^{4}(2n(\omega)+1)$
$\displaystyle\left[\mathrm{Tr}\left[\mathrm{Im}\
G_{AA}(\omega)\cdot\mathrm{Re}\ G_{AA}(\omega)\right]\right.$
$\displaystyle\left.+\mathrm{Tr}\left[\mathrm{Im}\
G_{AB}(\omega)\cdot\mathrm{Re}\ G_{BA}(\omega)\right]\right],$ (25)

where the first term represents the modification to the single-sphere
potential arising from the self-interaction terms and can be ignored in the
absence of external boundary conditions. The second term represents the
interparticle thermal CP potential to second order in the particle
polarizability [33]. As before, if the substitute the free-space Green’s
tensor, thus neglecting the self-interactions, we get:

$\displaystyle
U_{A,\mathrm{f}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-\frac{c^{4}\hbar\mu_{0}^{2}}{16\pi^{3}r^{6}}\int
d\omega~{}(2n(\omega)+1)\left[\text{sin}(2kr)\right.$
$\displaystyle\left.\left(-3+5(kr)^{2}-(kr)^{4}\right)+\text{cos}(2kr)\left(-6+2(kr)^{3}\right)\right],$
(26)

where $r=\left|{\mathbf{r}}_{A}-{\mathbf{r}}_{B}\right|$. Similarly the
externally-induced second order potential can be obtained from substituting
the external fields (Eqs. (7) and (II)) and the corresponding induced dipole
moments (Eqs. (II) and (II)) in Eq. (III.2):

$\displaystyle U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\int d^{3}{\mathbf{k}}\int
d^{3}{\mathbf{k}}^{\prime}\,\mu_{0}\omega^{2}\alpha(\omega)\alpha(\omega^{\prime})\mathrm{Re}\
\left\\{\sum_{\sigma\sigma^{\prime}}\left({\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G^{\dagger}_{AA}(\omega^{\prime})\cdot{\mathbf{\Phi}}^{A}_{\sigma}+{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G_{AA}(\omega)\cdot{\mathbf{\Phi}}^{A}_{\sigma}\right)\left\langle\hat{a}^{\dagger}_{\sigma}({\mathbf{k}})\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\right\rangle
e^{-i(\omega-\omega^{\prime})t}\right\\}$ $\displaystyle-\int
d^{3}{\mathbf{k}}\int
d^{3}{\mathbf{k}}^{\prime}\,\mu_{0}\omega^{2}\alpha(\omega)\alpha(\omega^{\prime})\mathrm{Re}\
\left\\{\sum_{\sigma\sigma^{\prime}}\left({\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G^{\dagger}_{AB}(\omega^{\prime})\cdot{\mathbf{\Phi}}^{A}_{\sigma}+{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G_{AB}(\omega)\cdot{\mathbf{\Phi}}^{A}_{\sigma}\right)\left\langle\hat{a}^{\dagger}_{\sigma}({\mathbf{k}})\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\right\rangle
e^{-i(\omega-\omega^{\prime})t}\right\\}$ (27)

The first and second terms in Eq. II correspond to the modification of the
single-sphere and the interparticle potentials resulting from the external
field. Both the single-particle and interparticle potential between two
dielectric spheres for a general state of the external EM field are determined
by the expectation value
$\left\langle\hat{a}_{\sigma}({\mathbf{k}})^{\dagger}\hat{a}_{\sigma^{\prime}}({\mathbf{k}}^{\prime})\right\rangle$.
In the following we analyze particular cases of external field states.

###  III.3 Coherent state

We first consider the external field to be in a single-mode coherent state,
corresponding to the case of optical tweezers. To this end, we apply the
displacement operator
$\hat{D}=e^{\beta_{\sigma}({\mathbf{k}})\hat{a}_{\sigma}^{\dagger}({\mathbf{k}})-\beta_{\sigma}^{*}\hat{a}_{\sigma}({\mathbf{k}})}$
to the vacuum state of the total field, giving an expectation value
$\left\langle\hat{a}_{\sigma}({\mathbf{k}})^{\dagger}\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\right\rangle=\beta_{\sigma}({\mathbf{k}})^{*}\beta_{\sigma}({\mathbf{k}}^{\prime})$
.

Since the mode function describes the spatial part of the electric field, we
can map it to the electric field, thus showing that the coherent state
potential reduces to the optical trap and optical binding potential. The
details of this calculation are presented in Appendix B. As a consequence, we
find that the first-order external potential in this case reduces to the well-
known trap potential created by the optical tweezers after the application of
a single-mode approximation at frequency $\omega_{0}$:

$\displaystyle
U_{A,\mathrm{coh}}^{(1)}({\mathbf{r}}_{A})=-\frac{1}{4}\alpha(\omega_{0})\left|{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right|^{2}$
(28)

Similarly, it can be seen that the second-order potential created by the
tweezers corresponds to the interparticle optical binding potential [34, 35,
36] (see Appendix B):

$\displaystyle U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\frac{1}{2}\mu_{0}\omega_{0}^{2}\alpha(\omega_{0})^{2}\mathrm{Re}\
\left\\{{\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot
G({\mathbf{r}}_{A},{\mathbf{r}}_{A},\omega_{0})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right\\}$
$\displaystyle-\frac{1}{2}\mu_{0}\omega_{0}^{2}\alpha(\omega_{0})^{2}\mathrm{Re}\
\left\\{{\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot
G({\mathbf{r}}_{A},{\mathbf{r}}_{B},\omega_{0})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{B})\right\\},$
(29)

Using the free-space Green’s tensor as before and ignoring the divergent self-
interaction term, the optical binding potential yields:

$\displaystyle
U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=\frac{c^{2}\mu_{0}\alpha(\omega_{0})^{2}}{8\pi
r^{3}}\mathrm{Re}\left[h(kr){\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{B})\right.$
$\displaystyle\left.-f(kr)\sum_{ij}{\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})_{i}\frac{r_{i}r_{j}}{r^{2}}{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{B})_{j}\right]$
(30)

where $h(kr)=e^{ikr}\left[1-ikr-(kr)^{2}\right]$ and
$f(kr)=e^{ikr}\left[3-3ikr-(kr)^{2}\right]$.

###  III.4 Squeezed vacuum state

We now consider the case where the external field is in a squeezed vacuum
state for a single mode. The effect of squeezed light in a single particle
system has been investigated in [37]. The external field state is obtained by
applying the single-mode squeezing operator
$\hat{S}=e^{\frac{1}{2}(\xi_{\sigma}({\mathbf{k}})\hat{a}_{\sigma}({\mathbf{k}})^{2}-\xi_{\sigma}({\mathbf{k}})\hat{a}_{\sigma}({\mathbf{k}})^{\dagger
2})}$ to the ground state, with
$\xi_{\sigma}({\mathbf{k}})=r_{\sigma}({\mathbf{k}})e^{i\Phi_{\sigma}({\mathbf{k}})}$
as the squeezing parameter, where $\xi_{\sigma}({\mathbf{k}})$ is defined such
that it is non-zero only for a single mode ${\mathbf{k}}_{0}$. Together with
the expectation value
$\left\langle\hat{a}_{\sigma}({\mathbf{k}})^{\dagger}\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\right\rangle=\delta({\mathbf{k}}-{\mathbf{k}}_{0})\delta({\mathbf{k}}^{\prime}-{\mathbf{k}}_{0})\delta_{\sigma\sigma^{\prime}}{\text{sinh}^{2}r_{\sigma}({\mathbf{k}})}$,
this leads to the first and second-order potentials for sphere A as:

$\displaystyle
U_{A,\mathrm{sq}}^{(1)}({\mathbf{r}}_{A})=-\alpha(\omega){\mathbf{\Phi}}_{\sigma}^{A\dagger}\cdot{\mathbf{\Phi}}_{\sigma}^{A}\text{sinh}^{2}r_{\sigma}({\mathbf{k}}_{0})$
(31) $\displaystyle
U^{(2)}_{A,\mathrm{sq}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-\mu_{0}\omega^{2}\alpha(\omega)^{2}2\text{sinh}^{2}r_{\sigma}({\mathbf{k}}_{0})$
$\displaystyle\left(\mathrm{Re}\
\left\\{{\mathbf{\Phi}}_{\sigma}^{A\dagger}\cdot
G_{AA}(\omega)\cdot{\mathbf{\Phi}}^{A}_{\sigma}\right\\}+\mathrm{Re}\
\left\\{{\mathbf{\Phi}}_{\sigma}^{A\dagger}\cdot
G_{AB}(\omega)\cdot{\mathbf{\Phi}}^{B}_{\sigma}\right\\}\right)$ (32)

As a striking result, we observe in this case that the squeezed vacuum state
creates: (1) a single particle potential similar to that created by a tweezer
field (Eq. (B)) and (2) an interparticle potential similar to the optical
binding potential (Eq. (B)). To quantitatively compare the two, e.g., to see
how much squeezing of the vacuum is necessary to produce an interaction of the
same strength as the tweezer or the coherent optical binding potentials, we
consider the ratio of the potentials
$\frac{U_{A,\mathrm{sq}}^{(1)}}{U_{A,\mathrm{coh}}^{(1)}}=\frac{U_{A,\mathrm{sq}}^{(2)}}{U_{A,\mathrm{coh}}^{(2)}}$.
As mentioned above, we assume the tweezer field to be single-mode and have a
flat intensity profile such that only the $\omega_{0}$ mode of the vacuum is
squeezed and only the $\omega_{0}$ mode is displaced, with no spatial
dependence. Because the spatially dependent part under these approximations is
the same for both potentials, we define the following ratio of the two
potentials:
$\frac{U_{\mathrm{Sq}}}{U_{\mathrm{OB}}}=\frac{2\text{sinh}^{2}r\left|\Phi_{\mathrm{sq}}\right|^{2}}{\left|\beta({\mathbf{k})}\right|^{2}\left|\Phi_{\mathrm{coh}}\right|^{2}}=\frac{\text{sinh}^{2}r\hbar\omega_{0}c}{(2\pi)^{4}I}$,
where $I$ is the intensity of the tweezer and we used that
$\tilde{\Phi}_{\mathrm{sq}}=1$. Using $I=10^{-2}W/\mu m^{2}$ and
$\lambda_{0}=1064$ nm, the squeezing parameter would need to be $r\approx 27$,
which amounts to a squeezing of $\approx 240$ dB for the squeezed state
potential to be comparable to the coherent state potential.

###  III.5 Cat state

Another example we present is the case where the external field is in a
Schrödinger cat state. Cat states are defined as superpositions of coherent
states given by
$\left|C\right\rangle=N\left(\left|\beta\right\rangle+e^{i\theta}\left|-\beta\right\rangle\right)$,
where $\left|\beta\right\rangle$ is a coherent state with coherent amplitude
$\beta$ and
$N=\left[2+2\mathrm{exp}(-2\beta^{2})\mathrm{cos}\theta\right]^{\frac{1}{2}}$
is the normalization factor. We consider a single-mode cat state, meaning that
one mode $\omega_{0}$ is in a cat state and all other modes remain in the
ground state. Using the single-mode cat state $\left|C\right\rangle$ as
defined above, we find $\left\langle
C\right|\hat{a}_{\sigma}({\mathbf{k}})^{\dagger}\hat{a}_{\sigma^{\prime}}({\mathbf{k}^{\prime}})\left|C\right\rangle=N({\mathbf{k}})N({\mathbf{k}}^{\prime})\beta^{*}({\mathbf{k}})\beta({\mathbf{k}}^{\prime})\left(2-e^{i\theta({\mathbf{k}})}-e^{-i\theta({\mathbf{k}}^{\prime})}\right)\delta({\mathbf{k}}-{\mathbf{k}}_{0})\delta({\mathbf{k}}^{\prime}-{\mathbf{k}}_{0})$.
Substituting these expectation values into Eqs. (III.1) and (III.2) and
evaluating both integrals yields:

$\displaystyle U_{A,\mathrm{ex}}^{(1)}({\mathbf{r}}_{A})=$
$\displaystyle-\alpha(\omega_{0})\mathrm{Re}\
\left\\{{\mathbf{\Phi}}^{A\dagger}\cdot{\mathbf{\Phi}}^{A}N^{2}({\mathbf{k}}_{0})\left|\beta({\mathbf{k}}_{0})\right|^{2}2\left(1-\mathrm{cos}\theta\right)\right\\}$
(33)

and

$\displaystyle
U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-4\mu_{0}\omega_{0}^{2}\alpha(\omega_{0})^{2}$
$\displaystyle\mathrm{Re}\
\biggl{\\{}{\mathbf{\Phi}}^{A\dagger}\cdot\left(G_{AA}^{\dagger}(\omega_{0})+G_{AB}^{\dagger}(\omega_{0})\right)\cdot{\mathbf{\Phi}}^{A}N^{2}({\mathbf{k}}_{0})\biggr{.}$
$\displaystyle\biggl{.}\left|\beta({\mathbf{k}}_{0})\right|^{2}(1-\mathrm{cos}\theta)\biggr{\\}}$
(34)

If we compare this with the coherent state potentials for a single mode, we
see that the cat state potential differs by a factor of
$N^{2}({\mathbf{k}}_{0})(1-\mathrm{cos}\theta)$. Since $1-\mathrm{cos}\theta$
takes values between $0$ and $2$, it is possible to turn off the interaction
potential as well as the trap potential by choosing a suitable $\theta$. This
means that for an even cat state ($\theta=2n\pi$) there will be no potential
in spite of the presence of an external laser field. While for an odd cat
state ($\theta=(2n+1)\pi$) the potential is enhanced by a factor of two
compared to a coherent state with the same average photon number.

Furthermore, the potential created by a general statistical mixture
$\hat{\rho}=p_{1}\left|\beta\right\rangle\left\langle\beta\right|+p_{2}\left|-\beta\right\rangle\left\langle-\beta\right|$
of coherent states with $p_{1}+p_{2}=1$ is equivalent to the potential created
by a coherent state $\left|\beta\right\rangle$. This means that the potential
of a single-mode cat state differs from that of a simple statistical mixture.
The fact that both the first order and second order potentials are enhanced or
suppressed for cat states can thus be solely attributed to the quantum
coherence between the superposed states.

##  IV Engineering sphere-sphere potential

In this section we will focus on engineering the interparticle potential for
the case where two silica nanospheres are trapped via optical tweezers. We
examine the optical potential by tuning various parameters: polarization,
relative optical phase and intensity of the tweezers.

###  IV.1 Scaling regimes

Figure 3: Comparison of all regimes for the optical binding and thermal
Casimir-Polder potentials of two nanospheres. The optical binding potential
(OB) scales as $\propto\mathrm{cos}(k_{0}r\pm\Delta\phi)/r^{3}$ in the near-
field $(r\ll\lambda_{0})$ and as $\propto\mathrm{cos}(k_{0}r\pm\Delta\phi)/r$
in the far-field $(r\gg\lambda_{0})$. The Casimir-Polder potential (CP) scales
as $\propto 1/r^{6}$ in the near-field $(r<\lambda_{t_{2}})$, as $\propto
1/r^{7}$ in the intermediate regime $(\lambda_{t_{2}}<r<\lambda_{T})$ and as
$\propto 1/r^{6}$ in the far-field $(r>\lambda_{T})$.The length scales
$\lambda_{0},\lambda_{t_{2}}$ and $\lambda_{T}$ are defined in Sec. IV.1.

Before we consider the total potential in more detail, it is instructive to
look at the scaling of the thermal Casimir-Polder and the optical binding
potentials across various regimes of the sphere-sphere separation. A summary
of the different scaling regimes is presented in Fig. 3.

####  IV.1.1 Thermal Casimir-Polder potential

The thermal Casimir-Polder potential exhibits three different scaling regimes,
given by the characteristic length scales $\lambda_{t_{2}}$ and $\lambda_{T}$,
where $\lambda_{t_{2}}=11.3\ \mathrm{nm}$ is the dominant transition
wavelength of silica, which appears in the Drude-Lorentz permittivity [38, 39,
40].

$\displaystyle\epsilon(\omega)=1+\frac{\omega^{2}_{p_{1}}}{\omega_{t_{1}}^{2}-\omega^{2}-i\gamma_{1}\omega}+\frac{\omega^{2}_{p_{2}}}{\omega_{t_{2}}^{2}-\omega^{2}-i\gamma_{2}\omega}$
(35)

in the form of a transition frequency $\omega_{t_{2}}$. The other parameters
are the plasma frequencies $\omega_{p_{1}}=1.75\times 10^{14}$ Hz,
$\omega_{p_{2}}=2.96\times 10^{16}$Hz, the transition frequencies
$\omega_{t_{1}}=1.32\times 10^{14}$ Hz, $\omega_{t_{2}}=2.72\times 10^{16}$ Hz
and the damping coefficients $\gamma_{1}=4.28\times 10^{13}$ Hz,
$\gamma_{2}=8.09\times 10^{15}$ Hz [41]. $\lambda_{T}=\frac{\hbar c}{k_{B}T}$
is the thermal length scale. The permittivity enters the polarizability
$\alpha(\omega)$ via the Clausius-Mossotti relation
$\alpha(\omega)=4\pi\epsilon_{0}R^{3}\frac{\epsilon(\omega)-1}{\epsilon(\omega)+2}$,
$R$ being the radius of the nanospheres.

Introducing the dimensionless polarizability
$\tilde{\alpha}(\omega)=\frac{\epsilon(\omega)-1}{\epsilon(\omega)+2}$, we can
write the thermal Casimir-Polder potential approximated in the near-field
($r\ll\lambda_{t_{2}}$) as [38]:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx-4k_{B}T\left(\frac{R}{r}\right)^{6}\tilde{N}_{T}-\frac{\hbar\omega_{t_{2}}}{\pi}\left(\frac{R}{r}\right)^{6}\tilde{N}_{0}$
(36)

where the dimensionless quantities $\tilde{N}_{T}$ and $\tilde{N}_{0}$ are
$\tilde{N}_{T}=\sum_{j}\tilde{\alpha}(i\tilde{\xi}_{j})^{2}$ and
$\tilde{N}_{0}=\int_{0}^{\infty}d\tilde{\omega}\tilde{\alpha}(\tilde{\omega})^{2}$,
with the sum being evaluated over the dimensionless Matsubara frequencies
$\tilde{\xi}_{j}=2\pi j\frac{k_{B}T}{\hbar\omega_{t_{2}}}$. The first term
corresponds to the thermal contribution with the thermal energy scale
$k_{B}T$, whereas the second terms corresponds to the ground state
contribution with energy scale $\hbar\omega_{t_{2}}$.

In the intermediate regime ($\lambda_{t_{2}}<r<\lambda_{T}$) the thermal
Casimir-Polder potential can be approximated as [38]:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx-\frac{23}{4}\frac{\hbar
c}{r\pi}\left(\frac{R}{r}\right)^{6}\tilde{\alpha}(0)^{2},$ (37)

with the characteristic energy scale as $\frac{\hbar c}{r}$ and in the far-
field regime ($r\gg\lambda_{T}$) as:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx-6k_{B}T\left(\frac{R}{r}\right)^{6}\tilde{\alpha}(0)^{2}$
(38)

with the thermal energy scale $k_{B}T$.

In the limit of $T\rightarrow 0$, the thermal length scale $\lambda_{T}$ goes
to infinity. The intermediate regimes will thus extend to infinity, revealing
the two scaling regimes of the ground state Casimir-Polder potential between
two dielectric spheres [38].

Figure 4:  (a) Total potential for $\Delta\phi=0$ (green) and $\Delta\phi=\pi$
(blue) at intensity $I=10^{-2}~{}\text{W}/\mu\text{m}^{2}$, $T=300$ K and with
both lasers polarized along the y-axis. The spheres have radius $R=100$ nm.
For these relative optical phases the system is conservative. The potential
thus represents the joint potential of both spheres.The inset shows the
deepest potential well around $r=\lambda_{0}$. The plateau of the potential at
large distances is formed by the flat tweezer potential. (b) Total potential
seen by sphere A (maroon) and sphere B (black) at $\Delta\phi=\pi/2$. Since
they now exhibits non-conservative forces, there is no joint potential, but
sphere A will see an optical binding potential with $\Delta\phi=\pi/2$ and
sphere B with $\Delta\phi=-\pi/2$. The length scales
$\lambda_{0},\lambda_{t_{2}}$ and $\lambda_{T}$ are defined in Sec. IV.1.

####  IV.1.2 Optical binding potential

Under the assumption that the drive is comprised of a single mode, we can use
the single-mode expression for optical binding potential in Eq. (III.3). The
characteristic length scale of the optical binding potential is determined by
the tweezer wavelength $\lambda_{0}$. If we also assume a flat tweezer
profile, then, the potential in the near-field regime ($r\ll\lambda_{0}$) is:

$\displaystyle
U^{\mathrm{A/B}}_{\mathrm{OB}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx
4\pi\frac{\sqrt{I_{A}I_{B}}}{c}R^{3}\left(\frac{R}{r}\right)^{3}\tilde{\alpha}(\omega_{0})^{2}\text{cos}(\Delta\phi)$
(39)

where $I_{A/B}$ is the intensity of the tweezers A and B, $\Delta\phi$ is the
relative optical phase between the two tweezers and $\omega_{0}$ is the drive
frequency.

Figure 5:  The total potential for different polarizations of the tweezer: (a)
yy-polarization, (b) xy-polarization and (c) xx-polarization, in the $z=0$
plane. The dashed blue line indicates the potential plotted in Fig. 4 (a).
Regarding interactions along the x-axis, as shown in Fig 4, it can be seen
that going from yy-polarization to xx-polarization enables one to turn off the
optical binding interaction between the spheres. This is due to the radiation
pattern of a dipole, which radiates perpendicular to the induced polarization.
We assume the intensity to be $I=10^{-2}~{}\text{W}/\mu\text{m}^{2}$ and
$\Delta\phi=0$. In the intermediate arrangement, at xy-polarization, there is
weak interaction along both the x and the y-axis, due to the interference of
the two dipole fields.

In the far-field $(r\gg\lambda_{0})$ we get:

$\displaystyle
U^{\mathrm{A/B}}_{\mathrm{OB}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})$
$\displaystyle\approx-4\pi\frac{\sqrt{I_{A}I_{B}}}{c}R^{3}(k_{0}R)^{2}\left(\frac{R}{r}\right)\tilde{\alpha}(\omega_{0})^{2}\text{cos}(k_{0}r\pm\Delta\phi),$
(40)

which scales with the square root of the radiation pressure $\sim\sqrt{I/c}$
on either sphere. It is worth pointing out, that in the radiative regime when
$r\approx\lambda_{0}$, the potential scales as
$\sim\text{sin}(k_{0}r\pm\Delta\phi)/r^{2}$, meaning that it exhibits a phase
shift of $\pi/2$ compared to the near-field and far-field regimes.

The two spheres, A and B, can see different optical binding potentials for
$\Delta\phi\neq 0$ or $\pi$, resulting in non-conservative forces between the
trapped nanospheres. As illustrated in the next section, the difference
between potentials is the largest for $\Delta\phi=\pi/2$ and vanishes for
$\Delta\phi=0$ or $\pi$. Such non-conservative optical binding forces were
experimentally observed by Rieser et al. in [19].

Utilizing the different scaling behavior of the interparticle potentials along
with the fact that the optical binding and trapping potentials can be readily
tuned via the tweezer field one can realize various sphere-sphere potential
landscapes. In the next section we will investigate how this can be used to
produce a bound state potential of the two-nanospheres.

###  IV.2 Bound state of nanospheres

Figure 6: The total potential as a function of relative optical phase
$\Delta\phi$ at intensity $I=10^{-2}~{}\mathrm{W}/\mu\mathrm{m}^{2}$, $T=300$
K and with both lasers polarized along the y-axis by sphere (a) A and (b) B.
The spheres have radius $R=100$ nm. The dashed coloured lines coincide with he
potentials plotted in Fig. 4. (c) Stability analysis along the x-axis of the
two nanosphere system. The dotted lines show the minima of the potential of
sphere A (teal) and sphere B (orange) as presented in Fig. 6. The dashed lines
correspond to the potentials in Fig. 4. Due to the non-conservative nature of
the optical binding interaction, stable bound states along the x-axis can only
be found at $\Delta\phi=0$ and $\Delta\phi=\pi$, since only then both spheres
can be at a potential minimum simultaneously.

The interplay between the different radiative potentials gives rise to a
tunable potential landscape. While the analogue of the Earnshaw’s theorem for
fluctuation-induced forces fundamentally constrains the possibility of
creating stable equilibria for objects interacting purely via quantum
fluctuations [21], driven systems can overcome this limitation. The drive thus
allows one to create repulsive and tunable optical binding potentials, which
can be modified by changing the optical tweezers intensity $I$, relative
optical phase $\Delta\phi$ and polarizations.

Fig. 4 shows the interparticle potential at $T=300$ K for a drive intensity
$I=10^{-2}~{}\text{W}/\mu\text{m}^{2}$, with both tweezer fields polarized
along the y axis for four different relative optical phases. Fig. 4 (a) shows
the potentials for $\Delta\phi=0$ (green) and $\Delta\phi=\pi$ (blue). For
these two relative optical phases the forces are purely conservative and a
joint potential of both spheres can be written down. It can be seen from the
inset in (a) that for $\Delta\phi=0$, there is a $\approx 200$ K deep well
around $x=\lambda_{0}$. In the flat tweezer approximation, this potential well
– and other wells at a greater distance – are created by the oscillatory
nature of the optical binding potential. Their depth therefore increases
linearly with the tweezer intensity, which is a readily tunable parameter in
the experiment. Fig. 4 (b) shows the potential seen by either sphere at
$\Delta\phi=\pi/2$. In that case the system is no longer conservative and each
sphere sees a different potential. Consequently, for the values of the sphere-
sphere separation $x$ where sphere A sees a potential minimum, sphere B sees a
maximum. Such a scenario will lead to an unstable configuration of the two
spheres, wherein they would eventually attract each other in the near-field
and stick together.

Fig. 5 shows the total potential for three different polarization
configurations of the two tweezer fields applied at ${\mathbf{r}}_{A}$ and
${\mathbf{r}}_{B}$ (yy, xy and xx) with
$I=10^{-2}~{}\mathrm{W}/\mu\mathrm{m}^{2}$ and $T=300$ K. It can be seen that
the potential reflects the interference of the dipole radiation pattern from
the two nanospheres, thus permitting one to turn off the optical binding
interaction between between the spheres.

In order to study the existence of stable minima in more detail – taking into
account the non-conservative nature of the interaction – one must consider the
potential of either sphere and its dependence on $\Delta\phi$. Fig. 6 (a) and
(b) present the potentials seen by either sphere as a function of relative
optical phase $\Delta\phi$. In the single mode approximation of the tweezer
the optical binding potential of spheres A and B satisfy
$U^{\mathrm{A}}_{\mathrm{OB}}(\Delta\phi)=U^{\mathrm{B}}_{\mathrm{OB}}(-\Delta\phi)$,
such that the potentials are equivalent, but reflected around
$\Delta\phi=\pi$. The system is stable if both spheres experience a potential
minimum at a distance x. To search for such positions, we numerically find the
minima of both potentials in Fig. 6 (a) and (b); the results are shown in Fig.
6 (c). The dotted lines in teal and orange represent the minima of the
potential of sphere A and sphere B respectively. Since stability is only
possible whenever the two dotted lines cross – indicating that both spheres
are at a potential minimum – it can be seen that bound states only exist at
$\Delta\phi=0$ and $\Delta\phi=\pi$, i.e. when there are no non-conservative
forces acting on the spheres.

##  V Summary

In this work we have presented a description of the radiative forces between
two dielectric nanospheres interacting via the quantum and thermal
fluctuations of the EM field, as well as an external drive in a general
quantum state. We analyze the interaction between the total field, comprising
the field fluctuations and externally applied fields, and the dipole moments
induced in the nanospheres by summing over the light scattering processes up
to second order in the particle polarizabilities. Considering the two
nanospheres to be trapped by a laser, we demonstrate that the tweezer
intensity, polarizations and the relative optical phase between the tweezers
provides us with the means to control the interparticle potential.

##  VI Discussion and Outlook

Analyzing the interaction potential between the spheres to second-order in the
particle polarizabilities, we recover the known expressions for the trapping,
optical binding and thermal Casimir-Polder potentials when the external field
is in a coherent state and the fluctuation field in a thermal state (Sec.
III.3). For a coherent state of the external field, the two spheres see a
mutually bound trap potential as deep as $\approx 200$ K at a separation of
$x\approx 1~{}\mu$m for a tweezer intensity of $I=10^{-2}$ W/$\mu$m2 and phase
$\Delta\phi=0$ (Fig. 4). The relative optical phase between the tweezers
permits the creation of non-conservative and non-reciprocal forces. Utilizing
the tweezer polarizations, we can selectively turn-off the optical binding
interaction between the spheres.

Tailoring quantum statistics of light can have exciting applications in many
optical phenomena [37, 42, 43]. We show that an external field in a single-
mode squeezed vacuum state creates a potential similar to the optical trapping
and optical binding potentials, remarkably in the absence of any coherent
field amplitude (Sec. III.4). The squeezing required for creating a strong
enough potential to trap a nanosphere of radius $\sim 100$ nm, however, is
substantially large ($\sim 240$ dB). For the external field in a cat state,
the resulting potential differs from the coherent state by a factor of
$\sim(1-\cos\theta)$, where $\theta$ refers to the phase between the
superposed coherent states (Sec. III.5).

Understanding and controlling the radiative interactions between optically
trapped nanospheres is crucial for future experimental studies of macroscopic
quantum systems. Such levitated nanoparticles in the quantum regime provide an
ideal testbed for exploring gravitational interactions between quantum
systems, paving the way for exploring the potential role of gravity in
engendering decoherence and bringing about the quantum-to-classical
transition. In such systems it becomes imperative to control the
electromagnetic interactions that are fundamentally far stronger than
gravity111For reference, the ratio of electric to gravitational forces between
two electrons is $e^{2}/(4\pi\epsilon_{0}Gm_{e}^{2})\sim 10^{42}$, where
$m_{e}$ is the electron mass., in order to delineate gravitational effects. It
will be pertinent to extend the present work to include finite-size effects
such that one can prepare bound states of the nanospheres at shorter distances
or larger nanospheres and analyze gravitational interactions between the
particles in such a regime.

Having a full description of the radiative interactions will allow one to
engineer the interparticle potential in more detail, including other
interactions such as Coulomb or magnetic interactions. As we have illustrated,
one can realize mutually trapped bound states of nanospheres, with the
potential depths comparable to $\approx 200$ K. Such trap depths can be
readily increased by increasing the tweezer intensity, enabling one to realize
such bound states of nanoparticles at room temperatures. Such systems can be
excellent sensors of external electromagnetic fields, forces and have been
used in developing probes for detection of new physics beyond the standard
model.

Furthermore, it would be interesting to extend the present results to consider
the spheres in a delocalized superposition of their center-of-mass positions,
or with their centers-of-mass being prepared in entangled states. In such a
scenario one can ask how quantum fluctuations provide a fundamental limit to
decoherence of macroscopic quantum systems. On the other hand, it would be
relevant to consider the effect of the various quantum states of center-of-
mass of the two nanospheres on their individual and mutual quantum fluctuation
effects – such as fluctuation forces and decoherence.

##  VII Acknowledgements

We are grateful to Pierre Meystre and Paulo A. Maia Neto for helpful
discussions and feedback on the manuscript. We thank Markus Aspelmeyer for his
support. K.S. acknowledges support from the National Science Foundation under
Grant No. PHY-2309341, and by the John Templeton Foundation under Award No.
62422. C.J., U.D., and K.S. acknowledge support by the John Templeton
Foundation under Award No. 63033. U.D. acknowledges support from the FWF
(Austrian Science Fund, Project No. I 5111-N). This work was supported by NSF
Grant No. PHY-1748958.

##  Appendix A Green’s Tensor

The Green’s tensor is the solution to the inhomogeneous Helmholtz equation:

$\displaystyle\left[\nabla\times\frac{1}{\mu({\mathbf{r}},\omega)}\nabla\times-\frac{\omega^{2}}{c^{2}}\epsilon({\mathbf{r}},\omega)\right]G({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)=\delta({\mathbf{r}}-\mathrm{r}^{\prime}),$
(41)

where $\mu({\mathbf{r}},\omega)$, $\epsilon({\mathbf{r}},\omega)$ are the
spatially dependent relative permeability and relative permittivity
respectively that include the presence of any media.

The electric field sourced by a noise current density
${\mathbf{j}}({\mathbf{r}}^{\prime},\omega)$ for the given boundary conditions
is given by:

$\displaystyle{\mathbf{E}}({\mathbf{r}},\omega)=i\mu_{0}\omega\int
d^{3}{\mathbf{r}}\,G({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)\cdot{\mathbf{j}}({\mathbf{r}}^{\prime},\omega)$
(42)

The noise current density can be related to the noise polarization and
magnetization as
$\hat{{\mathbf{j}}}_{N}=-i\omega\hat{{\mathbf{P}}}_{N}+\nabla\times\hat{{\mathbf{M}}}_{N}$,
which describe the source of the fluctuation field [30, 31]. Thus, the
electric field is:

$\displaystyle\hat{{\mathbf{E}}}_{\mathrm{f}}({\mathbf{r}},\omega)=i\mu_{0}\omega\int
d^{3}{\mathbf{r}}\,G({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)\cdot\left(-i\omega\hat{{\mathbf{P}}}_{N}+\nabla\times\hat{{\mathbf{M}}}_{N}\right)$
(43)

Defining the bosonic operators
$\hat{{\mathbf{f}}}_{\lambda}({\mathbf{r}},\omega)$ associated with the noise
polarization $(\lambda=e)$ and magnetization $(\lambda=m)$ as:

$\displaystyle\hat{{\mathbf{P}}}_{N}({\mathbf{r}},\omega)$
$\displaystyle=i\sqrt{\frac{\hbar\epsilon_{0}}{\pi}\mathrm{Im}\
\epsilon({\mathbf{r}},\omega)}\hat{{\mathbf{f}}}_{e}({\mathbf{r}},\omega)$
(44) $\displaystyle\hat{{\mathbf{M}}}_{N}({\mathbf{r}},\omega)$
$\displaystyle=\sqrt{\frac{\hbar}{\pi\mu_{0}}\frac{\mathrm{Im}\
\mu({\mathbf{r}},\omega)}{\left|\mu({\mathbf{r}},\omega)\right|}}\hat{{\mathbf{f}}}_{m}({\mathbf{r}},\omega)$
(45)

and substituting into Eq. (43) permits us to define modified Green’s tensors:

$\displaystyle G_{e}({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)=$
$\displaystyle
i\frac{\omega^{2}}{c^{2}}\sqrt{\frac{\hbar\epsilon_{0}}{\pi}\mathrm{Im}\
\epsilon({\mathbf{r}},\omega)}G({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)$
(46) $\displaystyle G_{m}({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)=$
$\displaystyle
i\frac{\omega}{c}\sqrt{\frac{\hbar}{\pi\mu_{0}}\frac{\mathrm{Im}\
\mu({\mathbf{r}},\omega)}{\left|\mu({\mathbf{r}},\omega)\right|}}\left[\nabla^{\prime}\times
G({\mathbf{r}}^{\prime},{\mathbf{r}},\omega\right]^{T}$ (47)

This, in turn, we can use to write the fluctuation field as we have done in
Eq. (II) as:

$\displaystyle\hat{{\mathbf{E}}}_{\mathrm{f}}({\mathbf{r}}_{i})=\sum_{\lambda=e,m}\int
d^{3}{\mathbf{r}}\int_{0}^{\infty}d\omega\,G^{\lambda}({\mathbf{r}}_{i},{\mathbf{r}},\omega)\cdot\hat{{\mathbf{f}}}_{\lambda}(\mathrm{r},\omega)+\mathrm{H.c.}$
(48)

##  Appendix B Radiative Forces in optical tweezers

In Sec. III.3 we have calculated the potential seen by two nanospheres when
the external field is in a coherent state. Here we elaborate on those results
and show how they connect to more commonly known aspects of optical trapping
and optical binding of nanospheres. To make this comparison we will first map
the external field in a coherent state onto the tweezer field and the
scattered tweezer field as follows. We define the complex fields as:

$\displaystyle{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})=\left\langle\beta\right|\hat{{\mathbf{E}}}^{(0)}_{\mathrm{ex}}({\mathbf{r}}_{A})\left|\beta\right\rangle=$
$\displaystyle=\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,{\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{A},{\mathbf{k}},\omega)\beta_{\sigma}({\mathbf{k}})e^{-i\omega
t}$ (49)

and similarly the dipole moment induced by the tweezer field as:

$\displaystyle{\mathbf{P}}_{\mathrm{tw}}({\mathbf{r}}_{A})=\left\langle\beta\right|\hat{{\mathbf{P}}}^{(1)}_{\mathrm{ex}}({\mathbf{r}}_{A})\left|\beta\right\rangle=$
$\displaystyle=\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,\alpha(\omega){\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{A},{\mathbf{k}},\omega)\beta_{\sigma}({\mathbf{k}})e^{-i\omega
t}.$ (50)

The scattered field, namely the fields scattered scattered off sphere A or
sphere B and seen by sphere A, we can thus write as:

$\displaystyle{\mathbf{E}}^{\mathrm{A/B}}_{\mathrm{sc}}({\mathbf{r}}_{A})=\left\langle\beta\right|\hat{{\mathbf{E}}}^{(1)}_{\mathrm{ex}}({\mathbf{r}}_{A})\left|\beta\right\rangle=$
$\displaystyle=\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,G({\mathbf{r}}_{A},{\mathbf{r}}_{A/B},\omega){\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{A/B},{\mathbf{k}},\omega)\beta_{\sigma}({\mathbf{k}})e^{-i\omega
t}$ (51)

Where the superscript in
${\mathbf{E}}^{\mathrm{A/B}}_{\mathrm{sc}}({\mathbf{r}}_{A})$ denotes whether
the tweezer field is scattered off sphere A or sphere B. Once again, this
implies that the dipole moment induced by the scattered tweezer field is:

$\displaystyle{\mathbf{P}}^{\mathrm{A/B}}_{\mathrm{sc}}({\mathbf{r}}_{A})=\left\langle\beta\right|\hat{{\mathbf{P}}}^{(2)}_{\mathrm{ex}}({\mathbf{r}}_{A})\left|\beta\right\rangle=\frac{1}{2}\sum_{\sigma}\int
d^{3}{\mathbf{k}}\,\alpha(\omega)$ $\displaystyle
G({\mathbf{r}}_{A},{\mathbf{r}}_{A/B},\omega){\mathbf{\Phi}}_{\sigma}({\mathbf{r}}_{A/B},{\mathbf{k}},\omega)\beta_{\sigma}({\mathbf{k}})e^{-i\omega
t}$ (52)

Using these definitions allows us to write the coherent state potentials:

$\displaystyle U_{A,\mathrm{coh}}^{(1)}({\mathbf{r}}_{A})=$
$\displaystyle-\sum_{\sigma\sigma^{\prime}}\int d^{3}{\mathbf{k}}\int
d^{3}{\mathbf{k}^{\prime}}\,\alpha(\omega)$ $\displaystyle\mathrm{Re}\
\left\\{{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot{\mathbf{\Phi}}_{\sigma}^{A}\beta_{\sigma^{\prime}}^{*}({\mathbf{k}}^{\prime})\beta_{\sigma}({\mathbf{k}})e^{-i(\omega-\omega^{\prime})t}\right\\}$
(53)

and

$\displaystyle U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\int d^{3}{\mathbf{k}}\int
d^{3}{\mathbf{k}}^{\prime}\,\mu_{0}\omega^{2}\alpha(\omega)\alpha(\omega^{\prime})\mathrm{Re}\
\left\\{\sum_{\sigma\sigma^{\prime}}\left({\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G^{\dagger}_{AA}(\omega^{\prime})\cdot{\mathbf{\Phi}}^{A}_{\sigma}+{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G_{AA}(\omega)\cdot{\mathbf{\Phi}}^{A}_{\sigma}\right)\beta_{\sigma}({\mathbf{k}})^{*}\beta_{\sigma}({\mathbf{k}}^{\prime})e^{-i(\omega-\omega^{\prime})t}\right\\}$
$\displaystyle-\int d^{3}{\mathbf{k}}\int
d^{3}{\mathbf{k}}^{\prime}\,\mu_{0}\omega^{2}\alpha(\omega)\alpha(\omega^{\prime})\mathrm{Re}\
\left\\{\sum_{\sigma\sigma^{\prime}}\left({\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G^{\dagger}_{AB}(\omega^{\prime})\cdot{\mathbf{\Phi}}^{A}_{\sigma}+{\mathbf{\Phi}}_{\sigma^{\prime}}^{A\dagger}\cdot
G_{AB}(\omega)\cdot{\mathbf{\Phi}}^{A}_{\sigma}\right)\beta_{\sigma}({\mathbf{k}})^{*}\beta_{\sigma}({\mathbf{k}}^{\prime})e^{-i(\omega-\omega^{\prime})t}\right\\}$
(54)

as

$\displaystyle
U_{A,\mathrm{coh}}^{(1)}({\mathbf{r}}_{A})=-\frac{1}{4}{\mathbf{P}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})$
(55)

and

$\displaystyle U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\frac{1}{2}\sum_{i=A,B}\mathrm{Re}\left\\{{\mathbf{P}}^{\mathrm{i}}_{\mathrm{sc}}({\mathbf{r}}_{A})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right\\}$
$\displaystyle-\frac{1}{2}\sum_{i=A,B}\mathrm{Re}\left\\{{\mathbf{E}}^{\mathrm{i}}_{\mathrm{sc}}({\mathbf{r}}_{A})\cdot{\mathbf{P}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right\\}$
(56)

If we assume that the tweezer comprises only one single mode $\omega_{0}$ then
we can write these potentials in a more familiar form, showing that we
correctly recover the tweezer and the optical binding potential [34, 35].

$\displaystyle
U_{A,\mathrm{coh}}^{(1)}({\mathbf{r}}_{A})=-\frac{1}{4}\alpha(\omega_{0})\left|{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right|^{2}$
(57)

and

$\displaystyle U_{A,\mathrm{ex}}^{(2)}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=$
$\displaystyle-\frac{1}{2}\mu_{0}\omega_{0}^{2}\alpha(\omega_{0})^{2}\mathrm{Re}\
\left\\{{\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot
G({\mathbf{r}}_{A},{\mathbf{r}}_{A},\omega_{0})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{A})\right\\}$
$\displaystyle-\frac{1}{2}\mu_{0}\omega_{0}^{2}\alpha(\omega_{0})^{2}\mathrm{Re}\
\left\\{{\mathbf{E}}^{*}_{\mathrm{tw}}({\mathbf{r}}_{A})\cdot
G({\mathbf{r}}_{A},{\mathbf{r}}_{B},\omega_{0})\cdot{\mathbf{E}}_{\mathrm{tw}}({\mathbf{r}}_{B})\right\\},$
(58)

where Eq. (57) is the trapping potential created by the tweezer and Eq. (B) is
the optical binding potential seen by sphere A.

##  Appendix C Free space Green’s tensor and approximate potentials

The free space Green’s tensor between points ${\mathbf{r}}_{1}$ and
${\mathbf{r}}_{2}$ is given by:

$\displaystyle{G}_{\mathrm{free}}\left({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega\right)=\frac{e^{ikr}}{4\pi
k^{2}r^{3}}\left\\{f(kr)\mathbb{1}-h(kr){\mathbf{e}}_{r}\otimes{\mathbf{e}}_{r}\right\\}$
(59)

where $f\left(x\right)\equiv 1-ix-x^{2}$, $h\left(x\right)\equiv 3-3ix-x^{2}$,
$r=\left|{\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right|$, and
${\mathbf{e}}_{r}={\mathbf{r}}/r$ is the unit vector connecting the positions
of the two spheres.

###  C.1 Thermal CP-potential

Utilizing this form of the Green’s tensor, we will now briefly outline the
derivations of the approximations in Sec. IV.1. We begin by writing the
thermal CP-potential as:

$\displaystyle U({\mathbf{r}}_{A},{\mathbf{r}}_{B})=-\frac{\hbar
c}{16\pi^{3}\epsilon_{0}^{2}r^{7}}\int_{0}^{\infty}dx\alpha(xc/r)^{2}\mathrm{Im}\
\left\\{e^{-2ix}\left[3+6ix+5\left(ix\right)^{2}+2\left(ix\right)^{3}+\left(ix\right)^{4}\right]\right\\}\text{coth}\left(\frac{\hbar
cx}{2k_{B}Tr}\right)$ (60)

where we have used that $\omega/k=c$, $k=\tilde{k}k_{t_{2}}$ and subsequently
set $\tilde{k}k_{t_{2}}r=x$. Here $k_{t_{2}}=2\pi/\lambda_{t_{2}}$ is the
wavenumber corresponding to the dominant transition wavelength
$\lambda_{t_{2}}$ of silica as defined in Sec. IV.1.1. In the regime where
($r\gg\lambda_{t_{2}}$) the polarisability is well approximated by the static
polarisability $\alpha(0)$. Utilizing this approximation, we can see that
within this regime there is now another regime defined by the thermal length
scale $\lambda_{T}\equiv\frac{\hbar c}{k_{B}T}$. We thus distinguish between
the two cases $\lambda_{T}\gg r$ (far-field regime) and $\lambda_{T}\ll r$
(intermediate regime). We discuss these two cases below:

  1. 1.

For $\lambda_{T}\gg r$, we can set
$\text{coth}\left(\frac{\lambda_{T}x}{2r}\right)\approx 1$, which allows us to
Wick rotate the integral in Eq. 60 and evaluate it to get:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx-\frac{\hbar
c}{16\pi^{3}\epsilon_{0}^{2}r^{7}}\alpha(0)^{2}\frac{23}{4},$ (61)

exhibiting a $r^{-7}$ scaling. Using the dimensionless quantities defined in
Sec. IV.1.1, we recover Eq. 37 in the main text.

  2. 2.

For $\lambda_{T}\ll r$, we extend the integral down to $-\infty$ by splitting
up the imaginary part and using the fact that $\text{coth}(x)$ is an odd
function. We note that the distance between the poles of the integrand in Eq.
60, which are given by the Matsubara frequencies $\omega_{n}=\frac{2\pi
k_{B}T}{\hbar}n$, is bigger than $c/r$, which is the decay length of the
exponential in the integrand. This means that because of the decaying
exponential, only the first residue at the origin will contribute to the sum.
This gives:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx--\frac{2k_{B}T}{16\pi^{3}\epsilon_{0}^{2}r^{6}}\alpha(0)^{2}3\pi,$
(62)

which, after substitution of the dimensionless quantities, yields Eq. 38. Thus
the potential returns to a $r^{-6}$ scaling [31].

In the near-field ($r\ll\lambda_{t_{2}}$) regime we can approximate the
potential as:

$\displaystyle
U({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx-\frac{3\hbar}{16\pi^{3}\epsilon_{0}^{2}r^{6}i}\int_{-\infty}^{\infty}d\omega\alpha(\omega)^{2}\text{coth}\left(\frac{\hbar\omega}{2k_{B}T}\right).$
(63)

Where we have again extended the integral by breaking up the imaginary part
before neglecting the last higher order terms. The integral can be simplified
by writing $\text{coth}\left(\frac{\hbar\omega}{2k_{B}T}\right)=2n(\omega)+1$
and using the same residues as before and a semi-circle contour around the
upper half plane. We therefore get:

$\displaystyle
U_{\mathrm{CP}}^{\text{ts}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx$
$\displaystyle-\frac{\hbar}{16\pi^{3}\epsilon_{0}^{2}r^{6}}\left\\{2\pi\frac{2k_{B}T}{\hbar}\sum_{j}\alpha(i\xi_{j})^{2}+\int_{0}^{\infty}d\omega\alpha(\omega)^{2}\right\\}$
(64)

The first term represents the near field approximation to the thermal part of
the potential whereas the second term represents the approximation to the
ground state potential. Once again using the dimensionless quantities in Sec.
IV.1 we end up with Eq. 36.

###  C.2 Optical Binding Potential

Next we outline the derivation of the approximation to the optical binding
potential as given in Sec. IV.1.2. We work under two assumtptions. Firstly,
that the two spheres are confined to move along the x-axis and secondly, that
the external field is y-polarized. The coordinate system is defined as in Fig.
1. Under these assumptions, we can write the OB-potential as given in Eq.
III.3 is:

$\displaystyle
U_{OB}^{\mathrm{A/B}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})=\frac{4\pi
R^{6}\sqrt{I_{A}I_{B}}\tilde{\alpha}(\omega)^{2}}{cr^{3}}$
$\displaystyle\left\\{\text{cos}(kr-\Delta\phi)+\text{sin}(kr-\Delta\phi)kr-\text{cos}(kr-\Delta)(kr)^{2}\right\\},$
(65)

where the dimensionless polarizability $\tilde{\alpha}$ is defined in Sec.
IV.1.1. In the near-field regime $(r\ll\lambda_{0})$ only the lowest order
term in $kr$ contributes, which means that the potential is

$\displaystyle
U^{\mathrm{A/B}}_{\mathrm{OB}}({\mathbf{r}}_{A},{\mathbf{r}}_{B})\approx
4\pi\frac{\sqrt{I_{A}I_{B}}}{c}R^{3}\left(\frac{R}{r}\right)^{3}\tilde{\alpha}(\omega_{0})^{2}\text{cos}(\Delta\phi)$
(66)

as in Eq. 39. Contrastingly, in the far-field regime $(r\gg\lambda_{0})$ only
the highest order term in $kr$ contributes, giving us the approximate
potential in Eq. IV.1.2.

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