Casimir nanoparticle levitation in vacuum with broadband perfect magnetic conductor metamaterials Adri an E. Rubio L opez1and Vincenzo Giannini2 3 4

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Casimir nanoparticle levitation in vacuum with broadband perfect magnetic

conductor metamaterials

Adri´an E. Rubio L´opez1, ∗and Vincenzo Giannini2, 3, 4

1Birck Nanotechnology Center, School of Electrical and Computer Engineering,

Purdue University, West Lafayette, IN 47907, USA

2Technology Innovation Institute, P.O. Box 9639,

Building B04C, Masdar City, Abu Dhabi, United Arab Emirates

3Instituto de Estructura de la Materia (IEM-CSIC),

Consejo Superior de Investigaciones Cient´ıﬁcas, Serrano 121, 28006 Madrid, Spain

4Centre of Excellence ENSEMBLE3 Sp.zo.o., Wolczynska Str. 133, 01-919, Warsaw, Poland†

(Dated: June 29, 2023)

The levitation of nanoparticles is essential in various branches of research. Casimir forces are

natural candidates to tackle it but the lack of broadband metamaterials precluded repulsive forces

in vacuum. We show sub-micron nanoparticle levitation in vacuum only based on the design of

a broadband metamaterial perfect magnetic conductor surface, where the Casimir force is mostly

given by the (quantum) zero-point contribution and compensates the nanoparticle’s weight. In

the harmonic regime, the volume-independent characteristic frequency depends linearly on Planck’s

constant ℏ.

Levitation is an intriguing physical phenomenon that

could majorly impact our daily life; a typical example

is magnetically levitated trains. Currently, diﬀerent ap-

proaches for levitating objects of diﬀerent shapes, sizes

and materials, and also under a broad variety of sce-

narios were investigated [1–8]. Some approaches exploit

the repulsive electric forces perceived by charges of the

same sign, while others are based on employing optical

potentials or tweezers. Because of its high controllability

and hybrid properties, levitated nanoparticles in highly

isolated scenarios are objects of high interest since its im-

pact in both technological applications and fundamental

science. Given their rich phenomenology, nanoparticles

are sensitive to ﬂuctuation phenomena, such as Casimir

forces. The latter were broadly studied as a possible

advantageous levitation mechanism [9–15], highlighting

particularly the pioneer work of Ref.[16] in connection

to the present work. But strong limitations were found

on narrow bandwidths of the materials involved (either

for ordinary materials or metamaterials) [17–21], or the

necessity of liquid immersion of the interacting bodies

[22–24]. Nevertheless, a successful realization in vacuum

may lead to a new generation of experiments and appli-

cations characterized by minimalistic setups and extreme

nanoparticle’s isolation.

In this Letter we demonstrate the levitation of

nanoparticles in vacuum at arbitrary temperature on a

sub-micron distance by simply exploiting Casimir inter-

actions with a broadband perfect magnetic conductor

(PMC) metamaterial plane surface. We also suggest a

possible way to realize such metamaterial.

Taking advantage from the natural repulsive inter-

action between point objects and a PMC surface, we

show that even when the PMC property is restricted

to an enough-broad bandwidth, excluding high and low

frequencies, the force remains repulsive while its mag-

nitude presents modest variations with respect to the

full-bandwidth PMC surface. By opposing this force

to the nanoparticle’s weight, we show sub-micron sta-

ble levitation for diﬀerent nanoparticle’s materials. For

small nanoparticles, the levitation dynamics is volume-

independent. The Casimir force is mostly given by the

(quantum) zero-temperature contribution, so the levita-

tion mechanism results robust to thermal eﬀects. The

resulting asymmetric potential gives anharmonic motion

for energies well above the potential’s minima. Harmonic

dynamics are obtained for low-energies (close to the min-

ima), characterized by a volume-independent frequency

with linear dependence on Planck’s constant ℏ, showing

the quantum nature of the phenomenon.

We consider a spherical nanoparticle of radius Rand

mass m=ρV , being ρthe density and V= 4πR3/3

the volume. Nanoparticles are well described as electric

point-dipoles of polarizability α(ω) = V ξ(ω) (Clausius-

Mossotti formula), with ξ(ω) = 3[ε(ω)−1]/[ε(ω) + 2],

and ε(ω) the nanoparticle’s material permittivity. The

nanoparticle is placed at a distance zfrom a plane surface

with ˆnthe normal direction (see inset Fig.1a).

The physical intuition about the interaction with a

PMC plane comes from basic objects in the static case.

Schemes with image charges are shown in Figs.1b and c.

While for a PEC the boundary conditions are ˆn×E= 0,

for the PMC we have ˆn×B= 0. This implies (for any

incident angle and frequency) that for a PEC (PMC)

we have rs=−rp=∓1. A striking consequence is

that while a positive charge interacts with a PEC sur-

face with a negative mirror-charge, for the PMC the

mirror-charge is positive; so the force between the PEC

(PMC) and a charge is attractive (repulsive). The same

intuition is valid for electric dipoles, giving an insight

on nanoparticles, although the latter are ﬂuctuating ob-

jects. For nanoparticle levitation the spectral broadness

arXiv:2210.12094v2 [quant-ph] 27 Jun 2023

FIG. 1. a,b) Schemes showing the physical intuition for the interaction of basic objects with a PEC (PMC) surface. For

the PEC (PMC) case, the charge has an image-charge with the opposite (equal) sign. The force is attractive (repulsive) for

the PEC (PMC) case. c) Scheme of the scenario, a quasi-PMC metamaterial surface with a z−dependent dielectric constant

according to Eq.(1), i.e. a gradient index material. At a distance zabove the surface, the nanoparticle of radius Ris located.

d) Reﬂection coeﬃcients as a function of {ck0, ck∥}for the quasi-PMC with permittivity according to Eq.(2). It is observed

that for most of the values rs≃1 (PMC behavior), while rs=−1 (PEC behavior) is obtained in a small region around the

light-cone.

of the PMC property is the key-point. Previous works

have proved that magnetic resonant metamaterials are

not enough [17, 19, 20]. The conclusion was that the mag-

netic properties of resonant metamaterials do not have

enough spectral broadness, precluding a concrete realiza-

tion. The metamaterials inherit their component’s res-

onant nature, implying that the magnetic behavior was

limited to a narrow frequency region not enough for lev-

itation eﬀects. Another limiting factor they found is the

losses in the metamaterial [19].

Here we take another approach. We design a long

wavelength metamaterial, i.e., a material with the de-

sired properties when the photon wavelength λis much

larger than the characteristic spatial scales of the meta-

materials (for example, the unit cell in a periodic system).

This way, we avoid having a functional material working

only on a narrow frequency band. These ideas have been

recently applied to obtain metallic transparent metama-

terials [25, 26]. Thus, we can obtain a quasi-PMC, i.e.,

a metamaterial behaving as a PMC in a broad range of

frequencies.

First, we suggest two possible ways to fabricate such a

system. Second, we elucidate the broadband properties

of a metamaterial to behave as a PCM.

Our main objectives are to show that a broadband

PMC is possible, demonstrating none fundamental re-

strictions against it; and also that is enough for realizing

nanoparticle levitation.

We aim to obtain a surface with rs≈+1 in a broad

range of frequencies and k−vectors. The ﬁrst idea relies

on subwavelength gradient materials [27, 28] and the ex-

istent exact analytical solutions for some refractive index

proﬁles describing light scattering [27, 29]. Such materi-

als have been studied for a long time [27]. In addition,

thanks to the continuously improvement of nanofabrica-

tion techniques [30], the nanoscale design of these ma-

terials opens a high-interest alternative. A more tradi-

tional way to design proﬁles includes controlling regimes

of doping, molecular beam epitaxy, nanoscale porosity

variations, fabrication of graded metal-dielectric compos-

ites, physical vapor deposition of multiple materials, ion

implantation etching, and photolithography.

The ﬁrst non-trivial proﬁle with an analytical solution

was found by Rayleigh a long time ago, in the 1880 [31].

After that, many other works have contributed to this

topic. Here we are interested in an inverse quadratic vari-

ation of the refractive index that leads to Bessel functions

to solve the scattering problem [29]. Let us assume the

following proﬁle for the dielectric constant for the posi-

tive semispace z > 0:

ε(z) = ε1−b2

(z+L)2,(1)

where ε1, b and Lare positive constant in our case. Such

an inverse square proﬁle has a transition length given by

band L. The negative semispace, z < 0, is assumed to

be the vacuum (see ﬁgure 1c). The exact solution at such

problem for s-polarization is given by [29]:

rs=k⊥0−s

k⊥0+s,(2)

where k⊥0=qk2

0−k2

∥is the component of the k-vector

in free space perpendicular to the plane, k∥is the parallel

component (conserved) of the k-vector and k0=ω/c

while sis given by:

s="H(2)′

ν(βL)

H(2)

ν(βL)+1

2βL #ik0β, (3)

where H(2)

νand H(2)′

νare the Hankel function of the

second kind and its derivative, β=qε1k2

0−k2

∥and

ν=qk2

0b2+1

By inspection of Eq. (2) and Eq. (3) we can see that

if we have ε1≫1 and b2≈ε1L2≫1 this means that

ν≈βL ≫1. In this regime, the ratio of Hankel functions

and 1/(2βL) go fast to zero, giving rs≈+1. In order

to mimic a PMC, we need to go slowly from lower to

high permittivity. It is not complex to ﬁnd high permit-

tivity materials; for example, with self-assembled metal

nanoparticles, we can easily get ε∼100 [25, 26] or with

a composite material value around ε∼105are possi-

ble [32, 33].

For example, choosing ε1= 100, b = 103nm, and L=

120 nm, from Eqs.(2) and (3) we obtain that rs≈+1

in a broad range of frequencies and k-vectors, as shown

in Fig.1d. It turns out that only for a irrelevant sharp

region near the light cone (k⊥0= 0) we have rs=−1.

Another possible solution to mimic a PMC could be

found in future advances in magnetic nanomaterial com-

posites [34, 35]. Such materials present strong mag-

netic eﬀects up to the far infrared but with promis-

ing extensions to the near-IR. This can be easily seen

from the reﬂection coeﬃcient between two materials (vac-

uum/magnetic composite):

rs=µ1k⊥0−k⊥1

µ1k⊥0+k⊥1

.(4)

Having large values of the magnetic permeability of the

nanocomposite (µ1≫1) implies rs≈+1.

We want to highlight that there are probably other

possible solutions for a broadband PMC but no fun-

damental reason against it. We hope that more re-

searchers will explore this phenomenon. Now we show

that nanoparticle levitation is possible with a PMC be-

havior over a broad region of frequencies and k-vectors,

while full-spectrum is not necessary.

The force on a nanoparticle arises from the interac-

tion between the surrounding EM ﬁeld, the plane sur-

face and the nanoparticle considered as a point dipole.

A detailed derivation is shown in Sect.I of the Suppl.

Mat. Following Refs.[36, 37], the force over the dipole

to the lowest order is F(r)≈ ⟨ˆ

d(ind)

i(t)∇ˆ

E(ﬂ)

i(r, t)⟩+

⟨ˆ

d(ﬂ)

i(t)∇ˆ

E(ind)

i(r, t)⟩, where ris the nanoparticle’s po-

sition (summation over subscripts is implicit). The ﬁrst

term describes the ﬂuctuations of the ﬁeld that corre-

late with the corresponding induced dipole, while the

second involves dipole ﬂuctuations and the ﬁeld they in-

duce. In principle, each entity have its own temperature,

{TEM, TS, TNP}. The force over the nanoparticle at a dis-

tance zfrom the surface for a general scenario results:

Fz(z) = F0(z)+FR(z, TEM, TNP)+FT(z, TEM, TS),(5)

where F0stands for the contribution of the zero-point

ﬂuctuations, depending on the surface’s reﬂection coeﬃ-

cients {rs,p};FRstands for the contribution associated

to the surrounding EM ﬁeld and the nanoparticle radi-

ation also depending on {rs,p}, while FTrelates to the

surface’s radiation and depends exclusively on its trans-

mission coeﬃcients {ts,p}. A metamaterial surface may

present frequency cutoﬀs, having restricted the values of

(ω, k∥) where rs= +1 and rp=−1. A full-bandwidth

PEC (PMC) surface has rs=∓1, rp=±1, while ts,p= 0

for every (ω, k∥). The latter implies that FT→0 regard-

less on the temperatures. In agreement to the intuitive

picture of Fig.1, in Sect.II of the Suppl. Mat. we show

the striking feature F(PMC)

z=−F(PEC)

z. In principle,

this theoretically guarantees the levitation of a nanopar-

ticle provided the full-bandwidth PMC property is ef-

fective. However, in general metamaterial will present

PMC properties on ﬁnite bandwidth. We now analyze

its impact on the Casimir-Polder force.

For a nanoparticle of R= 50nm, the weight

mg ∼10−17N for common materials such as SiC, Au

and Si. The levitation takes place where the Casimir

force compensate the weight, as we show below, this

occurs for z < 1µm. In the short-distance regime, for

which kBTMinz/[ℏc]≪1 (with TMin = min[TEnv, TNP]),

the Casimir force is given by the zero-temperature

(fully quantum) contribution [see Eq.(S.43) of the Suppl.

Mat.]:

Fz(z)≈F0(z).(6)

This implies that the conclusions obtained from now

are robust to thermal eﬀects and relies on the (quan-

tum) zero-point ﬂuctuations (see Sect.IIIA of the Suppl.

Mat.). For the full-bandwidth PMC, this contribution

reads:

F0(z)→F(PMC)

0(z)=3ℏV I0(z)/(8πz4),(7)

having I0(z)≡R+∞

dω

2πξ(iω)A(iω, z)e−2ω

cz, with

A(iω, z) = P3

n=0 1

n!2ω

czn. In Fig. 2 we show the

Casimir force acting on a SiC nanoparticle of R= 50nm

for diﬀerent upper and lower frequency/k-vectors cutoﬀs

combinations numerically obtained by employing Eq.(2)

for the surface (dashed-red and dashed-yellow curves), as

well as the exact analytical and numerical cases for case

the full-bandwidth PMC given by Eq.(7) (blue solid and

dashed-green curves). For SiC we employed a permittiv-

ity model εSiC(ω) = ε∞(ω2

L−ω2−iγω)/(ω2

T−ω2−iγω),

where ωL= 18.253 1013s−1,ωT= 14.937 1013s−1,

γ= 8.966 1011s−1and ε∞= 6.7. According to the basic

physical intuition conveyed in Fig. 1, a repulsive force

acts on the nanoparticle even in the broadband PMC

case, i.e. the not full-bandwidth case (dashed red and

yellow lines). A broader interval in frequencies increases

the repulsion at every distance from the surface. The

maximum repulsion is achieved for the full-bandwidth

PCM (blue solid and dash green lines). Furthermore,

in Sect.III of the Suppl. Mat., we show that for a SiC

nanoparticle the force by a full-bandwidth ideal PMC

surface, F(PMC)

0(blue solid curve), can be approximated

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CasimirnanoparticlelevitationinvacuumwithbroadbandperfectmagneticconductormetamaterialsAdri´anE.RubioL´opez1,∗andVincenzoGiannini2,3,41BirckNanotechnologyCenter,SchoolofElectricalandComputerEngineering,PurdueUniversity,WestLafayette,IN47907,USA2TechnologyInnovationInstitute,P.O.Box9639,BuildingB04C,...

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Casimir nanoparticle levitation in vacuum with broadband perfect magnetic conductor metamaterials Adri an E. Rubio L opez1and Vincenzo Giannini2 3 4

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