# Quantum degeneracy in mesoscopic matter: Casimir effect and Bose-Einstein
condensation.

I. Todoshchenko  M. Kamada  J.-P. Kaikkonen  Y. Liao  A. Savin  E. Kauppinen
E. Sergeicheva  P.J. Hakonen  Low Temperature Laboratory, Deptartment of
Applied Physics, Aalto University University, 00076 AALTO, Finland
<EMAIL_ADDRESS>
###### Abstract

The ground-state phonon pressure is an analogue to the famous Casimir pressure
of vacuum produced by zero-point photons. The acoustic Casimir forces are,
however, many orders of magnitude weaker than the electromagnetic Casimir
forces, as the typical speed of sound is 100 000 times smaller than the speed
of light. Because of its weakness, zero-point acoustic Casimir pressure was
never observed, although the pressure of artificially introduced sound noise
on a narrow aperture has been reported. However, the magnitude of Casimir
pressure increases as $1/L^{3}$ with the decrease of the sample size $L$, and
reaches picoNewtons in the sub-micron scales. We demonstrate and measure the
acoustic Casimir pressure induced by zero-point phonons in solid helium
adsorbed on a carbon nanotube. We have also observed Casimir-like ”pushing
out” thermal phonons with the decreasing temperature or the length. We also
show that all thermodynamic quantities are size-dependent, and therefore in
the mesoscopic range $L\lesssim\hbar{c}/(k_{B}T)$ quadruple points are
possible on the phase diagram where four different phases coexist. Due to the
smallness of solid helium sample, temperature of Bose-Einstein condensation
(BEC) of vacancies is relatively high, $10-100$ mK. This allowed us to
experimentally discover the BEC in a system of zero-point vacancies, predicted
more than 50 years ago.

## Introduction

Casimir effect is one of the most counter-intuitive phenomenon of various
manifestations of quantum mechanics. It was first formulated by Lars Casimir
[1] as a pressure of vacuum, i.e. of empty space containing no material
particles. According to Casimir, zero-point fluctuations related to
electromagnetic modes between two parallel plates lead to attractive force
between the plates. Equivalently, zero-point fluctuations for other
excitations, e.g. phonons, can lead to Casimir-like effects.

In general, there exists three different types of Casimir-like effects. First,
the ordinary Casimir force which is a quantum mechanical effect due to zero-
point photons/phonons. The predictions for electromagnetic Casimir effect were
verified in beautiful experiments in 1997 [2]. Second, Casimir effect in which
thermal excitation of modes plays an important role, and the Casimir force
scales with temperature [3]. Third, Casimir-like phenomena which are generated
by broad band noise that is not able to propagate to sections of interest in
the studied geometry. Acoustic Casimir effect has so far been only
demonstrated using external noise [4]. In fact, this kind acoustic noise
Casimir pressure has been found almost 200 years ago by noticing the
attraction of two closely placed ships [5]. However, genuine acoustic analogue
of the zero-point Casimir pressure has never been observed prior to our work.
Our measurements indicate Casimir pressure due to zero-point motion of quasi
one-dimensional phonons in solid 3He on a carbon nanotube.

The acoustic zero-point Casimir pressure is many orders of magnitude tinier
than the electromagnetic Casimir pressure because of the smallness of speed
sound compared to speed of light. However, the Casimir forces rapidly increase
with the decrease of the length of the sample. Fortunately, helium adsorbed on
a carbon nanotube (CNT) represents a sample which is small enough for Casimir
forces to be significant, and still large enough to be measured.

##  I Experimental

We have studied mechanical resonances of a suspended carbon nanotube (CNT)
with adsorbed sub-monolayer of 3He. The CNTs were synthesized in the gas phase
with the floating catalyst chemical vapour deposition growth method (FC-CVD)
followed by the direct thermophoretic deposition onto prefabricated chips [6].
Radius of the tube $r_{0}=0.8$ nm and length $L=700$ nm has been measured with
electron microscope. Electro-mechanical scheme of the measurements described
in the Appendix. After a detailed electrical characterization of the suspended
CNT along with its mechanical resonance properties in vacuum, 3He atoms were
gradually added to the sample chamber. As a single nanotube can adsorb only
very little amount of 3He, 1-10 thousands atoms, we used a grafoil ballast [7]
with the surface area 10 m2. Using grafoil, we were able to control the
coverage of the adsorbed 3He with the accuracy of 0.03 $\text{nm}^{-2}$.

After each addition of 3He atoms, we waited for 10-20 hours and, subsequently,
performed temperature sweeps up and down at a rate of 10 … 30 mK$/$h.
Typically, the $T$ sweep at each coverage was repeated at least once,
sometimes several times. The coverage of helium $\rho$ was determined by
measuring the shift of the resonance frequency when helium was in fluid phase
at high enough temperature, see Fig. 6.

In constructing our coverage scale

$\frac{\Delta F}{F_{0}}=-\frac{M_{He}}{2M_{C}}=-\frac{\rho_{He}}{8\rho_{C}}$
(1)

we have assumed, that the change in the elastic state (stiffness) of the tube
can be neglected in fluid phases at high temperatures, and that the frequency
shift is governed by the total mass of helium atoms ${M_{He}}$ on the CNT
compared with the mass of the carbon lattice ${M_{C}}$. At high $T$, besides
being in the fluid phase, the distance between helium atoms and the substrate
is the largest, which minimizes the influence of helium atoms on the elastic
properties of the CNT. Upon localization of the atoms when entering solid
phases, the free surface energy becomes lowered, which leads to a change in
the spring constant of the tube as observed in the experiments as a frequency
shift [8].

##  II Spectroscopy of phonons: Casimir effect and the fate of thermodynamics
in mesoscopic matter

Similarly to the pressure of vacuum due to zero-point photons, there is a
pressure in condensed matter due to zero-point phonon modes. As in the Casimir
effect, this pressure depends on the size of the sample, because the larger
the sample, the longer wavelengths are allowed. Owing to the presence of long-
wave photons (phonons), which do not fit the smaller volume, the larger volume
has a higher pressure compared with the small one. Helium adsorbed on the
nanotube is a perfect model system to investigate the size dependence of zero-
point pressure, a full analog of the Casimir force. The lack of the phonon
zero-point energy in the sample of the length $L$ is the energy of the phonons
with the wavelength $\lambda>L$,
$\Delta{E}_{0}=-m(\hbar/2)\sum_{k_{n}=\pi/\mathcal{L}}^{\pi/L}{c}k_{n}=-m(\hbar{c}/2)\sum_{n=1}^{\mathcal{L}/L}(\pi/\mathcal{L})n=-m\hbar{c}\pi\mathcal{L}/4L^{2}$,
where the long wavelength cutoff $\mathcal{L}\approx 10$ cm is the
characteristic size of helium layer outside the nanotube. As helium on
nanotube is not a true 1D system but has 20-25 atoms on its cross-sectional
circumference, we include a corresponding multiplicity $m$ for each
longitudinal phonon mode. The acoustic Casimir effect differs from the
traditionally considered electromagnetic Casimir effect because of the
existence of the short wavelength cutoff (interatomic distance), and the long
wavelength cutoff (size of the sample) for phonons. This results in
convergence for the total sum of phonon energies, in contrast to the photon
case and, consequently, in a different analytical form of the Casimir energy.

At $T=0$ the Casimir force acting on the tube is found simply as
$\mathcal{F}_{\rm
C}=\partial\Delta{E}_{0}/\partial{L}=m\hbar{c}\pi\mathcal{L}/2L^{3}$. The
power $L^{-3}$ instead of the $L^{-2}$ in ordinary 1D Casimir force [3] is due
to the finite cutoffs in the case of phonons, while there is no cutoffs for
photons. 1D pressure $P$ of helium adsorbed on the tube, which is the force
acting on the circumference at the boundary, contributes directly to the
tension $\mathcal{F}_{0}$, $\mathcal{F}=\mathcal{F}_{0}+P$. Indeed, 1D
pressure of adsorbed helium is the line energy density, $P=dE_{He}/dL$, whilst
the line tension of the tube is the elastic energy density,
$\mathcal{F}_{0}=dE_{CNT}/dL$. Physically, it can be understood as the force
produced on the tube by reflection of individual (quasi-) particles from
boundaries. Casimir force reduces the pressure of helium on the tube and,
accordingly, the tension,
$\mathcal{F}=\mathcal{F}_{0}+P-\mathcal{F}_{\rm{C}}$. By measuring the
resonant frequency $F_{0}=(1/2L)\sqrt{\mathcal{F}/\mu}$ we can therefore
detect the Casimir force. Due to the smallness of the tube, the phonon modes
are mostly in the ground state up to temperature
$T_{ZP}(L)=\pi\hbar{c}/L\approx 20$ mK for $L=700$ nm, and up to higher
temperatures for smaller solid fractions.

At lower average densities $\rho<\rho_{1/3}$ the solid phase occupies only a
part of the tube with the length
$L_{1/3}=L(\rho-\rho_{L})/(\rho_{1/3}-\rho_{L})$, while the rest is liquid
with low density $\rho_{L}$. This configuration is energetically more
favorable than the alternation of many small phases ”solid-liquid-solid…”, the
so-called Domain wall solid, suggested in Refs. 9, 10, 11 because in the
latter case there appear many interfaces between the regions of different
phases, which have additional energy due to the surface tension.

Historically, the term ”acoustic Casimir effect” was introduced for the effect
of mutual force between two closely placed plates under the sound noise
generated in the chamber by a microphone [12, 4]. Depending on the spectrum of
the generated noise, the force could be attractive or repulsive. In contrast,
in our experiment we deal with a complete analogue of the Casimir pressure:
the pressure is due to acoustic vacuum, to which only ground-state phonons
with energy $\hbar\omega/2$ contribute. The contribution of thermal phonons in
the tube is negligible at $T\lesssim 10$ mK. Pressure of thermal phonons in
the surrounding bath $F_{Bath}=\pi{m}k_{B}^{2}T^{2}/12\hbar{c}\simeq 10^{-17}$
N is also negligible. The effect is achieved by reducing the phonon phase
space using a small sample, which is exactly along the lines of experiments
targeting the original Casimir pressure [2].

Figure 1:  Resonance frequency shift $\Delta F_{0}$ due to 1D pressure (force)
of zero-point phonons ($T<20$ mK) as a function of atomic coverage per nm2
(lower axis) and the corresponding length occupied by the 1/3 solid,
$L_{1/3}=L(\rho-\rho_{L})/(\rho_{1/3}-\rho_{L})$ (higher axis). Curve displays
the contribution of the Casimir force, $\mathcal{F}_{\rm
C}=50\hbar{c}\pi\mathcal{L}/L^{3}$, see text for details.

Fig. 1 shows the dependence of the central frequency $F_{0}$ on the density of
helium at the lowest $T=10$ mK. With the increase of density, the helium layer
transforms from liquid below $\rho=4.1$ nm-2 to the so-called soft 2/5 solid,
through a mixture of solid 1/3 phase and liquid upto $\rho_{1/3}=6.4$ nm-2.
There are two major effects of adsorbed helium on the resonance frequency of
the tube: increase due to elastic stiffening and pressure, and decrease due to
additional mass. After subtracting the linear trend in frequency due to the
increase of helium mass, we are left with contributions of helium-induced
stiffness and pressure. In the liquid phase, the pressure is obviously zero as
it is in contact with exponentially diluted gas, as shown in the leftmost
insert in Fig. 1. Indeed, the length of the 1/3 solid shown on the top axis
has a dramatic effect on phonon pressure, which is illustrated by the solid
curve $\mathcal{F}_{\rm C}=m\hbar{c}\pi\mathcal{L}/L^{3}$ using the
multiplicity $m=50$ and an offset corresponding to the additional tension
produced by solid helium in the limit of large samples, $L\rightarrow\infty$.
This offset tension of long samples, at densities close to
$\rho\rightarrow\rho_{1/3}=6.4$ nm-2, is set by the attraction of the adsorbed
helium atoms to neighboring carbon atoms [8]. The opposite limit,
$L\rightarrow 0$ is not realized physically as it corresponds to a negative
effective pressure due to the Casimir force (see above,
$P-\mathcal{F}_{\rm{C}}$ becomes $<0$), which is not stable. Liquid phase at
$\rho<\rho_{L}=4.1$ nm-2 is in contact with strongly rarefied gas, and thus
has zero pressure. Therefore, when 1/3 solid starts to form on the tube, it
has zero pressure at $\rho=\rho_{L}\simeq 4.8$ nm-2 which increases at larger
average densities. In the range 4.1 nm${}^{-2}<\rho<4.8$ nm-2, there appears a
gap in the phase diagram which can be assigned to the described negative
pressure instability. Indeed, during the experiment we could not fill this
domain of coverages, while we were able to fill neighboring areas with steps
in coverage less than 0.05 nm-2.

The abrupt drop at $\rho=\rho_{1/3}$ indicates a quantum phase transition to
the soft 2/5 solid ($\rho_{2/5}=7.6$ nm-2) which does not support phonons [8].
Mobile defects have very low zero-point energy, of the order of
$\hbar^{2}/(2mL^{2})\sim 1$ $\mu$K, with vanishing contribution to pressure at
low temperatures. Zero-point pressure of a commensurate solid phase is thus
only due to zero-point phonons which are absent in the 2/5 phase whose
characteristic pressure is therefore nearly zero. Hence, at $\rho>\rho_{1/3}$
a mixture of 2/5 phase and incommensurate solid is realized. As pressure must
be constant over the tube, density of the incommensurate phase tunes by the
unique pressure of the 2/5 phase.

At finite temperatures thermal phonons are created, in addition to zero-point
phonons. However, the small size of the sample restricts greatly the amount of
thermal phonons. In the limit $L\ll\pi\hbar{c}/T$ the free energy $F$, entropy
$S$, energy $E$, and pressure $P$ of 1D phonons with multiplicity $m$ of each
mode are written as

$\displaystyle
F=\frac{mLk_{B}T}{2\pi{c}}\int_{\pi{c}/L}^{\infty}\ln{[1-e^{-\hbar\omega/k_{B}T}]}d\omega=-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar
c}\int_{\pi\hbar c/Lk_{B}T}^{\infty}\ln{[1-e^{-x}]}dx=$
$\displaystyle-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar{c}}\int_{\pi\hbar{c}/Lk_{B}T}^{\infty}\frac{xdx}{e^{x}-1}\simeq-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar{c}}\int_{\pi\hbar{c}/Lk_{B}T}^{\infty}xe^{-x}dx=-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar{c}}\Big{(}1+\frac{\pi\hbar{c}}{2Lk_{B}T}\Big{)}e^{-\pi\hbar{c}/Lk_{B}T}$
$\displaystyle S=-\frac{dF}{dT}\Big{|}_{L}\simeq
m\Big{(}\frac{Lk_{B}^{2}T}{\pi\hbar{c}}+\frac{3k_{B}}{4}+\frac{\pi\hbar
c}{4LT}\Big{)}e^{-\pi\hbar{c}/Lk_{B}T}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2) $\displaystyle E=F+TS\simeq
m\Big{(}\frac{Lk_{B}^{2}T^{2}}{2\pi\hbar{c}}+\frac{k_{B}T}{2}+\frac{\pi\hbar{c}}{4L}\Big{)}e^{-\pi\hbar{c}/Lk_{B}T}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle P=-\frac{dF}{dL}\Big{|}_{T}\simeq
m\Big{(}\frac{k_{B}^{2}T^{2}}{2\pi\hbar{c}}+\frac{k_{B}T}{2L}+\frac{\pi\hbar{c}}{4L^{2}}\Big{)}e^{-\pi\hbar{c}/Lk_{B}T}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

As expected, at $Lk_{B}T\ll\hbar{c}$ all thermodynamic potentials are
exponentially small. Practically, this means that there are no phonons except
for zero-point ones, and energy and pressure in the sample are temperature-
independent.

We demonstrate vanishing of phonons in our solid helium samples by measuring
temperature dependence of pressure in the low temperature domain below the
melting transition, see Fig. 2b. Indeed, the pressure is zero in small
samples, and it appears only when the product $LT$ exceeds the onset condition
given by $\pi\hbar{c}/k_{B}$. Note that, to reach a similar depletion of
thermal phonons in a sample of 1 mm size, a cool down to microKelvin
temperatures would be required, which is not possible yet.

Figure 2:  (a) Pre-factor $A$ of the fit of the resonant frequency shift
$\Delta{F_{0}}=AT^{2}$ as a function of coverage/length of 1/3 phase. Red
curve is a quadratic fit. (b) Smoothed data on pressure of thermal phonons (in
terms of frequency shift $\Delta F_{0}$ with color scale on the right) as a
function of the length/coverage (horizontal axes) and temperature. Quadratic
fits $P=AT^{2}$ were used, with the pre-factor $A$ fitted as in panel (a). Red
hyperbola is the onset of thermal excitation of phonons,
$T_{ph}=\pi\hbar{c}/k_{B}L$. Note that thermal phonons contribute by order of
magnitude less than zero-point phonons shown in Fig. 1

With increasing temperature (or, length), phonons start to enter the sample
one by one, and thermodynamics should be revised, as number of phonons is so
small. For instance, the pressure, which is the derivative of energy over the
length, is essentially zero at most of the lengths except for special points
$L=\pi\hbar{c}/k_{B}T,2\pi\hbar{c}/k_{B}T,3\pi\hbar{c}/k_{B}T$ etc., where the
pressure becomes formally infinite. In small samples, obviously, energy can
not be considered as a good thermodynamic potential, as its density is size-
dependent. Moreover, the energy is not additive: when two samples join
together, total energy becomes more than the sum of energies of the original
samples, due to the appearance of additional long wave phonon modes. With
further increase in length (temperature), thermodynamics reaches its classical
form, $P=A{T^{2}}$.

Lacking better representation, we assume that the low temperature dependence
$P(T)$ is quadratic, $P=AT^{2}$, with the pre-factor $A$ dependent on the
length $L$. Accordingly, we have fitted temperature dependence of the central
frequency as $\Delta{F_{0}}=A(L)T^{2}$, and the results of the fit are shown
in the Fig. 2a. As one can see, the pre-factor $a$ is increasing by many
orders of magnitude with the increase of length of the 1/3 phase. We interpret
this as an expansion of the phase space of phonons.

##  III Size-dependent thermodynamics

In Eqs. II, we have estimated thermodynamic quantities in the domain
$Lk_{B}T\ll\pi\hbar{c}$ and shown that they are exponentially vanishing in
this regime. Here we calculate exact expressions for the principal
thermodynamic functions in the whole range of $L$ and $T$. We start from the
statistical sum of a single phonon mode $k$,

$z_{k}=\sum_{v}e^{-E_{v}/(k_{B}T)}=\sum_{v}e^{-\hbar\omega_{k}(v+1/2)/(k_{B}T)}=\frac{e^{-\hbar\omega_{k}/(2k_{B}T)}}{1-e^{-\hbar\omega_{k}/(k_{B}T)}},$
(3)

where $v$ is the occupation quantum number of $k$th mode [13]. Free energy of
a single mode is thus
$f_{k}=-k_{B}T\ln{z_{k}}={\hbar\omega_{k}}/{2}+k_{B}T\ln{(1-e^{\hbar\omega_{k}/{k_{B}T}})}$,
and the whole free energy of the sample supporting $m$ modes is

$\displaystyle
F=\frac{m\hbar}{2}\sum_{k=\pi/L}^{\pi/a}\omega_{k}+mk_{B}T\sum_{k=\pi/L}^{\pi/a}\ln{(1-e^{\hbar\omega_{k}/k_{B}T})}=$
$\displaystyle\frac{m\hbar{c}}{2}\sum_{k=\pi/L}^{\pi/a}k+\frac{mk_{B}TL}{\pi}\int_{\pi/L}^{\pi/a}\ln{(1-e^{\hbar{c}k/k_{B}T})}dk\cong$
$\displaystyle\frac{m\pi\hbar{c}L}{4a^{2}}-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar
c}\int_{\pi\hbar c/Lk_{B}T}^{\infty}\frac{tdt}{e^{t}-1},~{}~{}~{}~{}~{}~{}$
(4)

where $a$ is the interatomic distance that sets up a minimum possible
wavelength of phonons $\lambda_{min}=2a$.

In macroscopic thermodynamics, no more than three phases can co-exist. It
follows from the equalities of chemical potentials for the phases
$\mu_{1}(P,T)=\mu_{2}(P,T)=\mu_{3}(P,T)$ which are two equations with two
unknowns, $P$ and $T$. One cannot add another phase to the coexistence point
because there will be three equalities and only two unknowns, which can be
satisfied only accidentally. However, in the mesoscopic domain,
$T\lesssim\pi\hbar{c}/L$, thermodynamics becomes more rich because here
emerges an additional thermodynamic parameter, the length (volume) of the
sample. Chemical potential, as all other thermodynamic potentials, is now a
function of $L$, $\mu=\mu(P,T,L)$. As we add one more variable, three
equalities can be satisfied simultaneously,
$\mu_{1}(P,T,L)=\mu_{2}(P,T,L)=\mu_{3}(P,T,L)=\mu_{4}(P,T,L)$, and
correspondingly four different phases may co-exist at a quadruple point.

The differential of the chemical potential should now be written as $d\mu=vdP-
sdT+v(\partial{P}/\partial{L})dL$ where the term $\partial{P}/\partial{L}$ is
found as

$\displaystyle
S=-\frac{\partial{F}}{\partial{T}}=\frac{mLk_{B}^{2}T}{\pi\hbar{c}}\int_{u}^{\infty}\frac{tdt}{e^{t}-1}+\frac{mk_{B}}{2}\frac{u}{e^{u}-1},$
$\displaystyle
E=F+TS=\frac{\pi{m}\hbar{c}L}{4a^{2}}+\frac{mk_{B}T}{2}\frac{u}{e^{u}-1}-\frac{mLk_{B}^{2}T^{2}}{2\pi\hbar{c}}\int_{u}^{\infty}\frac{tdt}{e^{t}-1},$
$\displaystyle
P=-\frac{\partial{F}}{\partial{L}}=\frac{mk_{B}T}{2L}\frac{u}{e^{u}-1}+\frac{mk_{B}^{2}T^{2}}{2\pi\hbar{c}}\int_{u}^{\infty}\frac{tdt}{e^{t}-1},$
(5)
$\displaystyle\frac{\partial{P}}{\partial{L}}=\frac{m\pi\hbar{c}}{2L^{3}}~{}\frac{e^{u}(u-1)+1}{(e^{u}-1)^{2}}$

where $u\equiv\pi\hbar{c}/(Lk_{B}T)$ is the universal thermodynamic factor
reflecting mesoscopic effects. At $u\rightarrow 0$ ($L\rightarrow\infty$ or
$T\rightarrow\infty$), the temperature-dependent free energy in Eq. III
simplifies to the standard macroscopic 1D expression $F=-(\pi
Lk_{B}^{2}T^{2})/(12\hbar c)$. However, the pressure $P$ acquires an
additional term proportional to temperature, which is yet a small correction.
It is easy to show that $e^{u}(u-1)+1>0$ at any positive $u$, and therefore
the derivative $\partial{P}/\partial{L}$ is always positive. Note that the
pressure calculated in Eq. III is not the Casimir pressure calculated in the
first section but the pressure inside the sample, while the Casimir pressure
is the difference between the pressure of the surrounding bath and that of the
sample.

Figure 3:  Possibility of a quadruple point on the phase diagram. (a) A
segment of the phase diagram of 3He on a nanotube [8] including liquid, 1/3
solid, 2/5 mobile solid, and high density (HD) solid. (b) Sketch of three-
dimentional phase diagram. Increase of the sample length shifts the liquid–HD
solid coexistence curve towards HD solid due to increasing pressure in the
solid, see text for details. At a certain length the melting curve reaches the
1/3 – 2/5 mobile solids coexistence curve forming a quadruple point (red
diagram). Our measured phase diagram corresponds to the black diagram on the
sketch.

Taking into account the additional thermodynamic parameter $L$, the phase
diagram becomes three-dimensional as illustrated in the Fig. 3(b). To be
specific, we show phases observed in helium on nanotube: 1/3 solid, 2/5 mobile
solid with delocalized vacancies, high density (HD) solid, and liquid [8].
With the increase of the length of the tube, the pressure in HD solid phase
increases as discussed above. Therefore, the chemical potential of HD solid
phase also increases, and the melting curve shifts to the right towards higher
densities where pressure in the liquid is also larger. At certain length, the
melting curve reaches the 1/3 – 2/5 mobile solid coexistence line forming a
quadruple point.

##  IV Bose-Einstein condensation

At large densities, $\rho>6.4$ nm-2, the 2/5 solid starts to form. This solid
has been proposed to consist of bosonic dimers [8] and its lattice structure
has topologically stable vacancies which arise owing to mismatch between the
most symmetric helium structures and the carbon lattice. These vacancies exist
even at zero-point and should Bose-condense at low enough temperatures [14,
15, 16, 17].

The transversal motion of an individual vacancy around the tube has a
characteristic energy $\varepsilon_{\phi}=\hbar^{2}l^{2}/(2Mr^{2})$ where $l$
is the angular momentum quantum number. The first excited state with $l=1$ and
$M\sim 2m_{3}$ has an energy of the order of 100 mK, and therefore at lower
temperatures the transversal motion is frozen, and the system of vacancies
becomes truly one-dimensional. One can estimate the Bose-Einstein condensation
temperature of a system of vacancies by adapting the standard approach for
ideal bosonic gas (see, for instance Ref. 18) to the one-dimensional case. The
number of particles will be given by a Bose-distribution integral with zero
chemical potential,

$\displaystyle
N=\frac{L}{2\pi\hbar}\int\,\frac{\mathrm{d}p}{\exp{\varepsilon/T}-1}=\frac{L\sqrt{MT}}{\sqrt{8}\pi\hbar}\int_{\varepsilon_{min}/T}^{\infty}\frac{\mathrm{d}z}{\sqrt{z}(\exp{z}-1)},$
(6)

where $L$ is the length of the sample. Note that, in contrast to the 3D case,
the integral diverges at low energies as $z^{-1/2}$, and the integration
should start from the cutoff $\varepsilon_{min}/T=\hbar^{2}(2\pi/L)^{2}/(2M)$.
The integral
$I=\int_{\varepsilon_{min}/T}^{\infty}z^{-1/2}\mathrm{d}z/(\exp{z}-1)$ in Eq.
6 can be crudely estimated as follows: if $z\ll 1$, it can be approximated as
$\int_{\varepsilon_{min}/T}^{\infty}z^{-3/2}\mathrm{d}z=(1/2)\sqrt{T/\varepsilon_{min}}$.
However, when $z$ is of the order of 1 and larger, the integrand is
significantly smaller than $z^{-3/2}$, and we may take
$I\sim\sqrt{T/\varepsilon_{min}}=\sqrt{2MT}L/(2\pi\hbar)$, which yields

Figure 4:  (a) Evidence for the Bose-Einstein condensation of vacancies. The
dissipation increases substantially when temperature of mobile solid decreases
below 50 mK. Data taken with coverage 8.8 nm-2 corresponding to the length of
the 2/5 mobile solid phase
$L_{2/5}=L(\rho_{inc}-\rho)/(\rho_{inc}-\rho_{2/5})=460$ nm (density of the
incommensurate phase is $\rho_{inc}=11.0$ nm-2. (b) Possible onset temperature
$T^{*}$ of Bose-condensation of delocalized vacancies as a function of the
length $L_{2/5}$ occupied by the 2/5 solid phase, see text for details. Solid
curve represents Eq. 7 with mass $M=2m_{3}$ for the theoretical estimation of
BEC transition. Inserts: sketches of the 2/5 phase (on the right) and of the
incommensurate phase (on the left); horizontal bar illustrate partial
occupation of the surface of the tube by the 2/5 and incommensurate solid.

$T_{BEC}\sim\frac{4\pi^{2}\hbar^{2}n_{vac}}{ML},$ (7)

where $n_{vac}=N/L$ is the linear density of vacancies. For the 2/5 solid
shown in Fig. 4(b), $n_{vac}=1/(3a/2)=4.7$ nm-1. With the length of our system
$L=700$ nm, the condensation temperature would be in the range 10–20 mK
depending on the effective mass $M$ of the vacancies. We emphasize that Bose-
Einstein condensation is possible in one-dimentional sample only due to the
finiteness of the length $L$: in infinitely long 1D sample temperature of the
condensation is zero.

At an arbitrary 3He coverage, a mixture of different phases is realized.
However, two different commensurate phases cannot co-exist as they have
different internal pressure. Indeed, density of commensurate phase cannot be
changed at given temperature, therefore, pressure also is fixed, and
mechanical equilibrium is not possible. At coverages 4.0 nm${}^{-2}<\rho<6.4$
nm${}^{-2}=\rho_{1/3}$ the commensurate 1/3 solid is coexisting with fluid. At
higher coverages the 2/5 solid appears and coexists either with fluid or with
incommensurate solid, depending on the filling factor. Clusters of different
co-existing phases have smaller length than the length $L$ of the nanotube. In
such reduced-size clusters of gliding solid, delocalized vacancies will Bose-
condense even at higher temperatures, according to Eq. 7, forming a
supersolid.

The expected BEC transition would manifest itself by even stronger dissipation
in the gliding solid phase at the lowest temperatures. Indeed, most of the
measured $\Delta{F}(T)$ traces show significant increase of the dissipation
$\Delta{F}$ when lowering temperature of the gliding solid (see Fig. 4).

The length of the 2/5 ”gliding”solid phase in contact with the incommensurate
phase can be calculated using the fact that pressure remains constant, as
density of commensurate phase cannot be changed with filling. Therefore,
density of incommensurate phase also doesn’t change with filling and equals to
maximum first layer coverage, $\rho_{inc}=\rho_{max}\simeq 11.0$ nm-2 [19, 8].
With the known density of both phases it is easy to calculate the lengths of
both phases for a particular filling. For example, 1:3 mixture of 2/5
”gliding” solid and incommensurate solid is realized at the coverage
$\rho=\rho_{2/5}\rho_{inc}/[(x(\rho_{inc}-\rho_{2/5})+\rho_{2/5}]=9.9$ nm-2
($\rho_{2/5}=7.64$ nm-2, $x=0.25$). The cluster of 2/5 solid will have a
length $L_{2/5}=x\rho_{inc}L/[(x(\rho_{inc}-\rho_{2/5})+\rho_{2/5}]=240$ nm,
and the Bose-Einstein condensation temperature of 45 mK.

Due to a lag of helium motion behind the oscillating nanotube, friction occurs
resulting in an additional dissipation. This additional dissipation is
proportional to the mobility $\mu$ of helium, similarly to an electrical
circuit in which the current $I$ and the power dissipation $P$ are
proportional to the conductance $G$, $I=GU$, $P=GU^{2}$. The high dissipation
in the low temperature phase at elevated coverages is one of the strongest
arguments for the model of ”gliding” solid [8]. However, when mobility
increases further, to infinity in the supersolid case, dissipation should drop
again because the relaxation time will become much longer than the oscillation
period. Indeed, the oscillating tube drags helium along by the drag force
$f_{d}=(\dot{x}_{CNT}-\dot{x}_{He})/\mu=m_{He}\ddot{x}_{He}$ where
$x_{CNT}=X\exp{i\omega{t}}$ and $x_{He}=x\exp{i\omega{t}}$, and $m_{He}$ is
effective mass of helium. Amplitude of oscillations of adsorbed helium turns
out to be always smaller than the amplitude of the tube,
$x=X/(1+i\omega{m}_{He}\mu)$. Average dissipation due to the drag is found as

$\displaystyle\overline{P}=\overline{f_{d}\cdot(\dot{x}-\dot{X}})=m\overline{\ddot{x}(\dot{x}-\dot{X}})=\frac{1}{2}\frac{m_{He}^{2}X^{2}\omega^{4}\mu}{1+(\omega{m}_{He}\mu)^{2}}.$

Energy of the oscillating nanotube is
${E}={E_{kin}+E_{pot}}=2\overline{E_{kin}}=(1/2)MX^{2}\omega^{2}$, and the
dissipation rate is therefore

$\Gamma=2\pi\Delta{F}=\overline{P}/{E}=(m_{He}/M)\cdot\omega^{2}\tau/(1+\omega^{2}\tau^{2})$
(8)

where $\tau=m_{He}\mu$ is the relaxation time of helium on nanotube. The case
of short $\tau$ corresponds to a solid phase in which helium is frozen to the
carbon lattice. Dissipation due to mutual motion of helium and nanotube is
minimal as there is no or very small mutual friction. When helium melts to a
fluid, the relaxation time increases together with the dissipation rate. More
interesting is the case of mobile zero-point vacancies in the range $\rho>6.4$
nm-2 where three regimes are possible. At temperatures above 0.1 K vacancies
are localized [8], and solid helium behaves as normal stiff crystal which
oscillates in phase with the tube and has therefore low dissipation
($\tau\rightarrow 0$), see Fig. 4. When temperature is lowered below 0.1 K,
vacancies becomes delocalized providing a finite mobility and relaxation time
of solid. This so-called ”gliding solid” is lagging behind the nanotube, which
provides even higher dissipation $\Gamma$ than in the liquid. On further
cooling, dissipation increases even more which we attribute to partial Bose-
Einstein condensation of vacancies. Indeed, vacancies condensed into the
ground state are superfluid, and effective mobility increases. However, at
$T\rightarrow 0$ where the condensation of vacancies is nearly complete,
mobility and relaxation time go to infinity, and, according to Eq. 8, the
dissipation also goes to zero; in this regime vacancies are not moving at all.

From the above analysis it follows that there must be a maximum of dissipation
rate $\Gamma$ at $\omega\tau=\omega{m}\mu=1$ where helium and the nanotube
oscillate in counter-phase. Therefore, the increase of dissipation at $T<40$
mK seen in Fig. 4 should change to a decrease when significant part of
vacancies will be in the ground state. Observation of such a bump in
temperature dependence of the dissipation is an evidence of the Bose-Einstein
condensation of vacancies.

Generally, in 1D-case the number of vacancies in the excited state is
proportional to temperature, $n_{ex}/n_{vac}=T/T_{BEC}$. In the normal state,
$T\geq{T_{BEC}}$, the relaxation time $\tau$ is finite and determined by the
friction coefficient $m_{0}/\tau_{0}$ of individual vacancies:
$m_{vac}/\tau_{norm}=N_{vac}m_{0}/\tau_{0}$. Below $T_{BEC}$, vacancies in the
condensate don’t contribute to the friction, and friction coefficient
decreases correspondingly
$1/\tau=N_{ex}/\tau_{0}=(N_{ex}/N_{vac})N_{vac}/\tau_{0}=(T/T_{BEC})/\tau_{norm}$.
After substituting this scaling relation to Eq. 8, we obtain the desired
expression for the width of the resonance below $T_{BEC}$:

$2\pi[\Delta{F}(T)-\Delta{F}_{b}]=\omega\frac{m_{vac}}{M}\frac{\omega\tau_{norm}(T_{BEC}/T)}{1+\omega^{2}\tau_{norm}^{2}(T_{BEC}/T)^{2}},$
(9)

where $\Delta{F}_{b}$ is the width of the resonance of the bare nanotube. In
the Fig. 5 we show the temperature dependence of the linewidth at density 6.74
nm-2 ($L_{2/5}=206$ nm). The fit of the experimental data with the Eq. 9 is
excellent and gives reasonable values of the relaxation time, $\tau=0.22$ ns,
and of the relative mass of vacancies $m_{vac}/M=5.1\cdot 10^{-4}$.

Deep in the BEC-state, vacancies decouple solid helium from the tube, and the
effective mass of the resonator decreases with a corresponding increase in the
central frequency, $\Delta F_{0}=(F_{0}/2)(m_{He}/M)$. The shift of the
resonance at $\rho=6.74$ nm-2 is about 0.55 MHz which corresponds to relative
decoupled mass $m_{He}/M=3.2\cdot 10^{-3}$. As 2/5 solid consists of dimers,
we assume that the mass of single vacancy is twice the mass of helium atom
which gives the concentration of vacancies in the dimer 2/5 solid as
$n_{vac}=(m_{vac}/M)/(m_{He}/M)=16$ %. Assuming one vacancy per period of the
2/5 solid along the tube, and 4-5 periods perpendicular to the axis of the
tube (see insert to Fig. 4b), we have one vacancy per 8-10 dimer sites which
gives the concentration of zero-point vacancies about 10–15 %, perfectly
matching the value obtained from the fits 111The helium lattice period along
the tube is two rows, and, assuming one vacancy per elementary cell, we have
one vacancy per 8-10 dimer sites which gives the concentration of zero-point
vacancies about 10–15 %. The mass of decoupled helium is also reliable, as the
length of the sample is about 30 % of the total length of the tube, and for
2/5 solid the number of helium atoms is 4/10 of number of carbon atoms (we
ignore number of vacancies) and taking into account atomic masses we find
$m_{2/5}/M=1.3\cdot 10^{-2}$. This value differs by only four times from the
value estimated from line shift, and this difference can be explained by the
non-symmetric position of the 2/5 phase on the tube..

Figure 5: Dissipation (open circles) and central frequency (filled circles) of
the resonance with the sample of the length $L_{2/5}=210$ nm
($\rho_{2/5}=6.74$ nm-2). The curve is of fit of dissipation-induced
$\Delta{F}$ using Eq. 9 and
$m_{vac}/M=2(\Delta{F}_{max}-\Delta{F}_{b})/F_{0}=5.1\cdot 10^{-4}$ (see
text). Increase of the central frequency at lowest temperatures by 0.55 MHz
corresponds to the relative mass decoupling $m_{He}/M=3.2\cdot 10^{-3}$.

The BEC of vacancies and the acoustic Casimir effect have very much in common.
Indeed, both particles obey Bose-statistics and have certain minimum thermal
energy due to the restricted geometry. Both systems therefore become
degenerate, when the length of the sample decreases. In the case of vacancies,
Bose-Einstein condensation temperature is inversely proportional to the length
$L$, see Eq. 7. On their part, phonons condense in the ground state at
temperature $T_{C}\approx\pi\hbar{c}/L$ which differs, besides numerical
factors, by the ratio $c/(\hbar{n_{vac}}/M)$. Note that, the combination
$\hbar{n_{vac}}/M$ has a meaning of average zero-point velocity of vacancy.
Indeed, according to Heisenberg uncertainty principle, characteristic momentum
of the particle is $p\sim\hbar/\delta{x}$ where uncertainty of the coordinate
is of the order is an inverse linear density, $\delta{x}\sim 1/n_{vac}$. After
dividing by mass of vacancy $M$ we end up with the velocity of vacancy. Thus,
the degeneracy in both systems, phonons and vacancies, has inherently the same
origin: it happens when temperature decreases below lowest possible frequency
in the system, $T<\hbar\omega_{min}\sim\hbar({v}/L)$.

Indeed quite analogously, there exists zero-point energy of vacancies and
zero-point energy of phonons in our experimental object. The statistics of the
two systems is identical, and the number of (quasi-)particles is conserved in
both systems: number of vacancies is set by the topological invariant of the
mismatch between the carbon lattice and helium superlattice [8], while the
number of phonon modes is set by the number of helium atoms and the geometry
of the tube.

Helium on nanotube is therefore a unique setting, which is small enough to
manifest mesoscopic thermodynamics already at temperatures below 100 mK which
are available in a regular dilution cryostat. It would be extremely
interesting to cool the 3He-on-nanotube system further towards lower
temperatures to investigate the fully condensed state of zero-point vacancies.
Intriguingly, the second layer of helium on CNT promises various new phases as
it is very weakly coupled to the carbon lattice. The proposed length-dependent
quadruple point could be demonstrated for the first time using 3He on
nanotube. Furthermore, subatomic layers of 4He on carbon nanotube could prove
very interesting objects to study as zero-point vacancies in this case are
lighter than in the dimer 3He solid, so that their Bose-condensation is
expected to take place at even higher temperatures.

## Appendix: electromechanical scheme and measurements of coverage.

Figure 6:  Determination of helium coverage on the suspended carbon nanotube
at 0.25 K. The shift of the resonance is due to increase in the mass of the
oscillating nanotube, while the additional stiffness can be ignored because
helium is in a fluid phase.

When RF voltage $\Delta U=U_{RF}\cos{\Omega t}$ is applied to the tube, it
starts to oscillate because of the oscillating electrical field $\Delta
E=\Delta U/H$, where $H$ denotes the distance between back gate and the CNT.
With gate-induced charge $Q$, the oscillating electrical force $F=Q\cdot\Delta
E$ will produce mechanical motion $h=H-H_{\mathrm{eq}}$ of the tube from the
equilibrium position $H_{\mathrm{eq}}$, governed by the equation of motion
$CU_{gate}U_{RF}\cos{\Omega t}/(r_{0}\log{2H/r_{0}})=M\partial^{2}h/\partial
t^{2}+\gamma\partial h/\partial t+kh$, where
$C=2\pi\varepsilon_{0}L/\log{(2H/r)}$ is the capacitance of the suspended
tube, $U_{gate}$ is the DC gate voltage, and $M$ is the effective mass of the
tube. The conductance $G(E)$ of the tube also oscillates at the same
frequency, as it is field-dependent. As a result, the current through the tube
$I=\Delta UG\propto\cos^{2}{\Omega t}$ has a constant term which is
proportional to the amplitude of the mechanical oscillations [20]. If the RF
frequency is modulated by a much lower frequency $\omega\sim 1$ kHz,
$\Omega=\Omega+\delta\cos{\omega t}$, the current will have a component at low
frequency,

$<I>_{\omega}\propto\frac{d}{d\Omega}~{}\frac{1}{\Omega_{0}^{2}-\Omega^{2}+2\pi
i{\it\Gamma}\Omega},$ (10)

which can be detected using a low-frequency lock-in amplifier. Here
$\Omega_{0}=2\pi{F_{0}}$ is the central circular frequency and
$\Gamma=2\pi\Delta{F}$ is the dissipation. In the method described in Ref. 20,
only the real part of the differentiated Lorentzian in Eq. (10) is considered.
In the more general case, due to finite capacitance of the tube and the
electrodes, there is a phase shift between the voltage and conductance, and
the uncertainty in the phase of the Lorentzian response needs to be taken into
account.

## Acknoledgements

We are grateful to Vladimir Eltsov for fruitful discussions. This work was
supported by the Academy of Finland (AF) projects 341913 (EFT). The research
leading to these results has received funding from the European Union’s
Horizon 2020 Research and Innovation Programme, under Grant Agreement No.
824109 (EMP). The experimental work benefited from the Aalto University
OtaNano/LTL infrastructure. We are grateful to Petri Tonteri from Densiq Ltd.
(90620 Oulu, Finland) for providing us with the ultra-pure grafoil [7].

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