The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article)
We present a brief overview of crucial historical stages in creation of superfluidity theory and of the current
 state of the microscopic theory of superfluid
 ⁴He. We pay special attention to the role of Bose-Einstein condensates
 (BECs) in understanding of physical mechanis...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) / S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, A.V. Chumachenko // Физика низких температур. — 2013. — Т. 39, № 9. — С. 937–957. — Бібліогр.: 155 назв. — англ. |
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| author | Vilchynskyy, S.I. Yakimenko, A.I. Isaieva, K.O. Chumachenko, A.V. |
| author_facet | Vilchynskyy, S.I. Yakimenko, A.I. Isaieva, K.O. Chumachenko, A.V. |
| citation_txt | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) / S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, A.V. Chumachenko // Физика низких температур. — 2013. — Т. 39, № 9. — С. 937–957. — Бібліогр.: 155 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We present a brief overview of crucial historical stages in creation of superfluidity theory and of the current
state of the microscopic theory of superfluid
⁴He. We pay special attention to the role of Bose-Einstein condensates
(BECs) in understanding of physical mechanisms of superfluidity and identification of quantum mechanical
structure of
⁴He superfluid component below -point, in particular — the possibility that at least two types of
condensates may appear and coexist simultaneously in superfluid
⁴He. In this context we discuss the properties
of the binary mixtures of BECs and types of excitations, which may appear due to intercomponent interaction in
such binary mixtures of condensates. We also discuss current status of investigations of persistent currents in toroidal
optical traps and present an outlook of our recent findings on this subject.
|
| first_indexed | 2025-12-07T18:08:37Z |
| format | Article |
| fulltext |
© S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9, pp. 937–957
The nature of superfuidity and Bose-Einstein condensation:
from liquid
4
He to dilute ultracold atomic gases
(Review Article)
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska Str., Kyiv 01601, Ukraine
E-mail: sivil@ukr.net
Received March 4, 2013
We present a brief overview of crucial historical stages in creation of superfluidity theory and of the current
state of the microscopic theory of superfluid
4
He. We pay special attention to the role of Bose-Einstein conden-
sates (BECs) in understanding of physical mechanisms of superfluidity and identification of quantum mechani-
cal structure of
4
He superfluid component below -point, in particular — the possibility that at least two types of
condensates may appear and coexist simultaneously in superfluid
4
He. In this context we discuss the properties
of the binary mixtures of BECs and types of excitations, which may appear due to intercomponent interaction in
such binary mixtures of condensates. We also discuss current status of investigations of persistent currents in to-
roidal optical traps and present an outlook of our recent findings on this subject.
PACS: 03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vor-
tices, and topological excitations;
03.75.Mn Multicomponent condensates; spinor condensates.
Keywords: superfluidity theory, Bose-Einstein condensates, binary mixtures.
Contents
1. Introduction ............................................................................................................................................................. 937
2. Theory of superfluidity: historical aspects and current state .................................................................................... 939
3. Self-consistent microscopic theory of superfluidity
4
He for the case = 0T K ...................................................... 943
4. Conventional and nonconventional Bose-Einstein condensations. Binary mixtures of Bose-Einstein condensates ....... 946
5. Conclusions and future challenges........................................................................................................................... 945
Appendix ..................................................................................................................................................................... 953
References ................................................................................................................................................................... 954
1. Introduction
The phenomenon of superfluidity in liquid
4
He, which is
a Bose-liquid, and in liquid
3
He, which is a Fermi-liquid, as
well the phenomenon of superconductivity in the electron
Fermi-liquid in metals, are fundamental properties of quan-
tum liquids. These phenomena are unique because they are
manifestation of quantum laws at the macroscopic level.
The essence of the superfluidity and superconductivity
phenomena is the emergence of a certain macroscopic
quantity — the complex order parameter, which is the
wave function of bosons or Cooper pairs of fermions that
occupy the same quantum state. Thus, the so-called cohe-
rent condensate appears.
In the Bose-systems such condensate appear due to a
direct grouping of bosons in the ground state and in the
Fermi-systems — due to the formation of Cooper pairs,
which are also bosons. Obviously, the phenomena of su-
perfluidity and superconductivity may occur in Bose-
systems or, more precisely, in systems, which obey Bose-
statistics. Therefore, it seems very attractive to explain
superfluidity and superconductivity on the basis of the
phenomenon of Bose-Einstein condensation (BEC) — the
macroscopic accumulation of bosons in the ground state
of the perfect ideal (noninteracting!) gas, which was de-
scribed first by Einstein [1] in 1925 on the basis of the
combination of Bose-statistics and classical expression
for the density of states in phase space.
Really, very shortly after discovering the phenomenon
of superfluidity (by Kapitsa [2], Allen and Misener [3]) but
before creation of the phenomenological theory of super-
mailto:sivil@ukr.net
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
938 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
fluidity (by Landau [4,5]) London[6,7] developed the con-
cept of macroscopic occupation of the ground state and
showed that it implies long-range coherence properties of
the Bose-Einstein condensate and together with Tisza [8]
in 1938 suggested that transition to superfluidity in liquid
4
He might be an example of BEC. Basis for their argu-
ments was following: atoms of
4
He are bosons and transi-
tion of normal liquid He (He I) to superfluid phase (He II)
takes place at the temperature T = 2.17 K; if atoms of
liquid
4
He are treated as a perfect Bose-gas, its temperature
of BEC would be very close to Tb = 3.14 K. The main idea
of London and Tisza was that the condensation in the per-
fect Bose-gas corresponds to a macroscopic occupation of
the ground state, which is related to collective coherent
properties of the condensate in superfluid
4
He. Moreover,
first microscopic theory of superfluidity, which was pro-
posed by Bogolyubov [9], is based on the fundamental
hypothesis of the existence in the superfluid systems of a
BEC in one specific mode!
As it is well known, notion of the Bose-Einstein con-
densate of bosons has to be reexamined when we consider
interacting systems. Therefore, it is appropriate to clarify
this notion, since atoms in liquid helium are an example of
the interacting system. Intuitively we understand that BEC
is a fraction of particles which does not move — frozen in
momentum space with = 0.q Onsager and Penrose
[10,11] in 1956 were the first who tried to work out a defi-
nition of condensate in the case of interacting system and
they have proposed to identify condensation with an off-
diagonal long-range order, related to asymptotic of single-
particle density matrix. In this approach it was shown that
mean-value of particle operator for the mode = 0q can
still be used as a characterization of a BEC and were ob-
tained 8% as an estimate of the fraction of particles in liq-
uid
4
He that have = 0q at = 0T K. Hohenberg and
Platzman [12] proposed to use deep-inelastic neutron scat-
tering for experimental observation of the Bose-Einstein
condensate in
4
He, then Cowley and Woods [13] in 1968
observed about 17% of condensate at = 1.1T K. After
that Dubna–Obninsk groups improved experiments and for
the first time not only reexamined the fraction of the con-
densate (3.6 1.4)% at = 1.2T K [14], but also measured
its temperature dependence and obtained (2.2 0.2)% for
the condensate at = 0T K and (2.24 0.04)% for the
condensate at critical temperature [15].
After these pioneering attempts, different groups re-
peated and improved experiments in this direction, and
from the middle of 90th of the last century up to nowadays
it is universally agreed by ―the superfluid helium commu-
nitee‖ that the best estimate of the Bose-Einstein conden-
sate fraction is about 9% at = 0T K [16–18].
The experimental evidence of the existence of Bose-
Einstein condensate in the superfluid liquid
4
He bolsters
one of the main hypotheses of London and Tisza, who tried
to show that the superfluidity is closely related to the mo-
tion of the BEC, which moving as whole.
So, if we want to clarify the physical mechanisms of the
superfluidity and superconductivity phenomena (from the
modern point of view superconductivity is nothing but
superfluidity, which occurs in a charged system), we
should understand how these phenomena are connected
with the phenomenon of macroscopic accumulation of bo-
sons in the ground state of the noninteracting Bose-system,
which was described by Einstein, nowadays known as
BEC. At this point we touch one of the most fascinating
problems of physics — would it be possible for BEC to be
present in interacting systems? From this point of view the
experimental realization of Bose-Einstein condensates of
dilute atomic gases has opened new opportunities for in-
vestigation of this question.
The main subject of the present review is the discussion
of these issues from the both side of view — theoretical
and experimental.
First experimental demonstration of a BEC of alkali-
metal atoms of rubidium [19] and sodium [20] has opened
new possibilities for exploring superfluidity at a much
higher level of control. The low-temperature atomic con-
densates can be prepared with essentially all atoms being
in the state of Bose condensate. Because of specific fea-
tures the atomic BECs differ significantly from the helium
BEC: liquid helium usually is uniform while the trapping
potential that confines a vapor BEC yields significantly
nonuniform density; unlike spinless
4
He atoms, alkali
atoms have nonzero hyperfine spins, and various forms of
spin-dependent effects are most pronounced in spinor
BECs [21]. For the weakly interacting BECs, a relatively
simple Gross–Pitaevskii equation (a variant of nonlinear
Schrödinger equation) gives basically good description of
the atomic condensates and their dynamics at low tempera-
tures. It is remarkable that the strength of interaction can
be tuned using Feshbach resonance [22], and different
geometries of the trapping potential provide the possibility
to study a 2D and even a 1D system. Now atomic BEC is
widely used for investigation of a superfluidity allowing
for quantitative tests of microscopic theories using the
tools and precision of atomic physics experiments.
Many phenomena, previously observed in liquid helium
below the -point, have found their counterpart with ultra-
cold alkali-metal gases. As it is well known, superfluid liq-
uid are distinguished from normal fluids by their ability to
support dissipationless flow. Such persistent currents are
intimately related to the existence of quantized vortices,
which are localized phase singularities with integer topolog-
ical charge. The superfluid vortex is an example of a topo-
logical defect that is well known in liquid helium and in
superconductors [23]. After many efforts such quantized
vortices, and also arrays of vortices, were observed in atom-
ic condensates [24–26]. Observation of the first sound
[27,28], scissor modes [29] or the critical velocity [30]
The nature of superfuidity and Bose-Einstein condensation: from liquid
4
He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 939
beyond which the superfluid flow breaks down are examples
of the manifestation of this spectacular macroscopic quan-
tum phenomenon in trapped ultracold atomic systems [31].
We do not intend to discuss in this review the vast sub-
ject of BEC in atomic gases and refer to the standard mo-
nographs and reviews [32–34]. We shall concentrate just
on the crucial historical stages of creation of the theory of
superfluidity and on the current state of the microscopic
theory of superfluity of
4
He (it will be done in Sec. 2 and
partially in Sec. 3). In Sec. 3 and Sec. 4 we pay special
attention to the role of the BECs in understanding of the
physical mechanisms of superfluidity and possibility that at
least two types of condensates may appear and coexist si-
multaneously in
4
He. In this context in the second part of
Sec. 4 we discuss the properties of the binary mixtures of
BECs and types of excitations, which may appear due to
intercomponent interaction in such binary condensate mix-
tures. Section 5 will be devoted to conclusions and future
challenges, namely, persistent currents in toroidal trapped
spinor BECs and vortex ring-on-line structures in BECs.
There we will present an outlook of our recent findings on
stability of superflow in ring spinor BECs. In the Appendix
we will present a derivation of the general form of Gross–
Pitaevskii equation for spinor condensates.
2. Theory of superfluidity: historical aspects and
current state
As it is well known, superfluidity is the phenomenon of
dissipationless mass transfer in macroscopic quantum sys-
tems. It was discovered in 1938 by Kapitsa [2] and bit later
independently by Allen and Misener [3] during the re-
search of liquid helium. It was established that at tempera-
tures below T = 2.17 K helium can flow through thin ca-
pillaries and slits without viscosity.
A phenomenological theory of superfluidity was
created in 1941 by Landau [4,5,35]. On the contrary to the
hypotheses of London and Tisza, Landau was confident
that the physical reason of superfluidity is connected with
the spectrum of ―elementary excitations‖ in a liquid he-
lium, while BEC had nothing to do with this effect. Theory
of Landau (which was named the two-fluid hydrodynam-
ics) was based on the assumption that below T helium
consists of two components — normal and superfluid
(should be noted that Tisza was the first, who suggested
this idea publicly [8,36] in the frame of his naive theory of
superfluidity, and was the first, who gave a correct qualita-
tive explanation of the behavior of superfluid helium, but
his articles did not contain any proper two-fluid equations).
Each component has its own field of velocities and den-
sity, so the total density is a sum of densities of the com-
ponents. It is important to note that superfluid helium is not
a mixture of two different substances, so it is impossible to
separate normal and superfluid components from each oth-
er. It is more precise to talk about a coexistence of two
motions — normal and superfluid. Two-fluid model as-
sume that normal motion has all the usual properties of the
motion of a viscous liquid, whereas the superfluid motion
is responsible for the phenomenon of superfluidity and its
associated effects.
So, Landau phenomenology supposes that in quantum
Bose-liquid (like
4
He, which at a low temperatures is a
weakly excited macroscopic quantum system) collective
(coherent) behavior dominates over the individual move-
ment of atoms.
Theory of Landau allowed to achieve a high level of
understanding of the properties of nonrelativistic superflu-
id state in a Bose-liquid
4
He. For instance, existence of
different types of sound waves in superfluid
4
He were pre-
dicted on the basic of two-fluid hydrodynamics [37] and
then observed in the experiment [38,39]. Moreover, calcu-
lations of thermal conductivity and viscosity of superfluid
helium, obtained according to this theory, are in good
agreement with experiment. Also, Landau two-fluid hy-
drodynamics allows to give a very nice description of the
properties of
4
He–
3
He solutions [40].
One of the greatest achievements of Landau's pheno-
menological approach is that, basing on the temperature
dependence of heat capacity of superfluid
4
He, Bose-
statistics and on the basis of his superfluidity criteria for
the quantum liquids, he predicted the form of the elementa-
ry excitation spectrum of superfluid helium.
Very important that properties of superfluid liquids
can be entirely described by this spectrum of collective
excitations, which has two branches: the ―phonon‖ — for
long-wavelength excitations and the ―roton‖ — for the
relatively short-wavelength collective excitations. The
form of the energy spectrum of elementary excitations in
superfluid helium, which characterized by the linear dis-
persion relation at low momenta and so-called roton min-
imum at 0q was confirmed later by Woods and Hen-
shaw [41] in the experiments on scattering of slow
neutrons in liquid helium. Since then many other precise
experiments confirmed and refined the shape of this
curve [42–47]. Now we are well established: at long wa-
velength, the quasiparticles are phonons with linear rela-
tion = | |c q between energy and momentum q and
c is the (first) sound velocity; at larger momenta the su-
perfluid
4
He quasiparticle spectrum is given by the well-
known maxon–roton dispersion relation (see Fig.1).
However, there are some data points that cause difficul-
ties in explanation of these observations in the frame of
Landau two-fluid hydrodynamics. First difficulty is that
application of the Landau criterion of superfluidity to the
spectrum of elementary excitations gives 60cv m/s for
the critical velocity. Whereas it was well established expe-
rimentally that superfluidity in capillaries disappears when
velocity is of the order of few cantimeter/second and its
very sensitive to the diameter of the channel. Furthermore,
whereas superfluidity will be destroyed for the tempera-
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
940 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
tures > ,T T Landau criterion still gives > 0.cv Next
contradiction is that in the frame of Landau theory is im-
possible to find the answer why the spectrum of elementa-
ry excitations does not change drastically when tempera-
ture crosses from = 0T to = 2.17K.T
But it is expected since Landau two-fluid theory of su-
perfluidity gives only phenomenological explanation of the
phenomenon. Really, Landau phenomenological theory for
the strongly interacting Bose-systems asserts that low-lying
excitations may be conveniently represented by the noninte-
racting quasiparticles. It seems to be incredible and myste-
rious that this assumption describes many experimental ob-
servations successfully. We should understand how this
extremely successful phenomenological description can be
understood from an underlying field theory. Of course, this
is a task for microscopic theory of superfluidity of Bose-
liquids to provide a underground scene of this phenomenon.
However, theoretical investigation of the properties of
the superfluid phase of
4
He at microscopic level meets
with a set of fundamental problems. Most of them con-
nected with strong interaction between bosons and com-
plex quantum-mechanical structure of the effective cohe-
rent condensate. This condensate in the same way as
single-particle BEC in case of ideal Bose-gas forms the
basis of the superfluid component.
The big important steps towards a field theory of
strongly interacting bosons were made in the 1950's by
Bogoliubov [9,49–52], Beliaev [53,54], Hugenholtz and
Pines [55] and later by Gavoret and Nozières [56].
The first microscopic theory of superfluidity, which
stemming from a model of a weakly nonideal Bose-gas,
was proposed by Bogoliubov and the concept of the BEC
is the important ingredient of the Bogoliubov theory.
Using the fact that at very low temperature crystalline
order of
4
He atoms suppresses their individual displace-
ments in favor of their coherent collective movements
(which are nothing more than sound waves) Bogoliubov
showed that for low-energy excitations (in harmonic ap-
proximation) spectrum of corresponding hamiltonian of the
microscopic system of atoms can be calculated exactly.
Bogoliubov pointed an attention to the fact that de-
spite a long coherence in the condensate (in other words,
existence of nonzero off-diagonal elements in condensate
density matrix) the elementary excitations in the Bose-
gas exists because of the mobility of individual atoms in
the condensate (―quasiparticles‖ are real particles in this
case). Thus, accepting important role of BEC in under-
standing of superfluidity phenomena, Bogoliubov pro-
posed to find solution of quantum-mechanical problem of
bosons interaction that can give an energy spectrum,
which would correspond to Landau spectrum. It means
that interaction between BEC particles can transform
separate excitations in the Bose-gas into collective excita-
tions, which are observed in superfluid helium as spec-
trum of elementary excitations.
The main advantage of Bogoliubov theory is the depar-
ture from standard perturbative methods, which based on
series expansions over a small interaction constant. Exis-
tence of intensive BEC, which density is close to the total
density of the superfluid system is the basic idea of this
theory. As a consequence, for Bose-particles with zero
momentum and energy one can neglect the noncommuta-
tivity of the creation and annihilation operators and em-
ploying the so-called linear canonical Bogoliubov trans-
formations diagonalize the initial Hamiltonian of the
system and find an expression for the renormalized mo-
mentum of quasiparticles. Note, that main Bogoliubov's
suggestion — the approximation of operators by c-num-
bers, it is a fundamental hypothesis about the existence of
BEC in one specific mode in the He II.
While Bogoliubov's theory was a great step forward in
the understanding of the low-lying spectrum of interacting
bosons, the theory is certainly not applicable to strongly
interacting systems like superfluid
4
He. However, the
theory paved the way for many important developments,
e.g., the concept of symmetry breaking [57], which plays
such a prominent role in modern theoretical physics.
Especially should be noted that gauge symmetry break-
ing plays a very important role in the understanding of the
infrared structure of the Bose-system. A very lucid discus-
sion of the concept of symmetry breaking and its implica-
tions is given by Anderson in Ref. 58.
Bogoliubov theory has been improved further (see, for
instance, Ref. 59) and in the most of these improved mod-
els of superfluidity, using the arbitrary selection of parame-
ters, one can achieve a good agreement between theoretical
and experimental spectrum of elementary excitations in
superfluid
4
He for a certain range of the momentum. But
such agreement is more coincidental, because, as it was
mentioned above, and also it was shown in the experimen-
tal studies (physically different types!) [18,60], a part of
single-particle BEC in the superfluid
4
He is small and
ranges from 2 to 10% in any case, which is the opposite to
Fig. 1. Phonon–roton dispersion curve [42,48].
The nature of superfuidity and Bose-Einstein condensation: from liquid
4
He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 941
a condition of weak nonideality of the Bose-gas in the Bo-
goliubov theory.
Therefore, in order to give an adequate microscopic de-
scription of the Bose-liquid superfluid properties the most
promising is many-body field theory approach and the
Greens function method, which for the first time was used
in the papers of Beliaev [53,54] and widely developed in
the future. Beliaev's approach forms the basis of the syste-
matic application of the quantum field theory methods to
boson systems with condensate, including anomalous
propagators, which representing two particles going into or
out of the condensate. The Green function method let one
find the energy of the system, its equation of state and the
spectrum of quasiparticles for multiparticle Bose-systems.
It allows to obtain a system of equations (Dyson–Beliaev
equations), which express the normal 11G and anomalous
12G single-particle boson Green functions in terms of the
corresponding self-energy parts 11 and 12 . Since the
BEC acts as a particle reservoir from which particles can
be created or into which particles may be lost, it was ne-
cessary to introduce ―anomalous'' Green's functions in or-
der to describe such processes. Therefore, the Dyson equa-
tion for the Green's functions
0 0=G G G G (1)
is turned into a 2 2 matrix equation. The diagonal ele-
ments in this equation correspond to the conventional
Green's functions, and the off-diagonal elements in G and
are the anomalous Green's functions and self energies,
respectively.
But the main difficulty of the microscopic description
of the superfluid state of a Bose-liquid with a nonzero BEC
is the fact that direct application of perturbation theory
leads to divergences and nonanalyticities at small energies
0 and momenta 0q and, as a consequence, to
erroneous results in the calculations of various physical
quantities.
Thus, for example, for a Bose-system with weak inte-
raction, when the ratio of the mean potential energy
3
0 0( )V q q 0(q being a typical momentum transfer) to the
corresponding kinetic energy 2
0 / 2q m of the bosons is
small, the zeroth-approximation polarization operator
( , )q and the density–density response function
( , )q calculated to the first order in the small interac-
tion parameter 0 0= ( ) 1,mq V q are logarithmically
divergent at 0,q 0, whereas the exact values
(0,0) and (0,0) are finite:
2 2 2
(0,0) = = ; (0,0) = .
( )B
n n n
mc m c c
(2)
Here n is the total concentration of bosons, is the
chemical potential of the quasiparticles, 0= /Bc nV m is
the sound velocity in the Bogoliubov approximation for a
weakly nonideal Bose-gas, 0 (0)V V is the zero Fourier
component of the potential, and c is the speed of sound in
the 0q limit for the spectrum of elementary excitations
( ) | |q c q in the Beliaev theory.
Handling of these divergences and contradictions in a
satisfactory manner is necessary in order to understand the
infrared response of a Bose-system and requires a suitable
renormalization procedure. Unfortunately, this proves to be
a rather difficult problem.
Beliaev theory was reanalyzed and extended by Hugen-
holtz and Pines [55] in 1959. Using gauge invariance ar-
guments and a careful analysis of the perturbation series
they showed that the quasiparticle spectrum of superfluid
helium is gapless. In particular, they obtained that
11 12(0) (0) = (3)
(Hugenholtz–Pines theorem). Really, inserting this theo-
rem into Eq. (1) it is easy to see that the spectrum is gap-
less. It is now tempting to obtain the infrared behavior of
the Green's functions directly from the Eq. (1). Assuming
the self-energy parts 11 and 12 to be analytic at small
momenta one obtains
11 12
11 2 2 2 2 2 2
(0) (0)
( , = =
( ) ( )
G
B c B c
q)
q q
(4)
where B and c are constants involving derivatives of the
self-energies at = 0.q Within this approach one also finds
that 12(0) 0 . The argument just presented is given in
more detail in Ref. 61, and it indeed leads to a linear spec-
trum. However, it assumes analyticity of the self-energies
and Green's functions at zero momentum, which appears to
be erroneous as was recently shown in [62].
The quantum-field theory for Bose-systems as devel-
oped by Beliaev and Hugenholtz and Pines was further
thoroughly reexamined in 1963 by Gavoret and Nozières
[56]. They showed that the singular character of interaction
in the Bose-systems with condensate remarked by Bogoli-
ubov manifests itself in divergences of perturbation theory
at small momenta (infrared divergences). They showed
that infrared divergences cancel out in all physical quanti-
ties and the energy gap in elementary excitation spectrum
vanishes to all orders of perturbation theory. In other
words, Gavoret and Nozières established the phonon cha-
racter of the spectrum up to all orders in perturbation
theory. Furthermore, they successfully related the sound
parameter of the field theoretical propagator with the ma-
croscopic sound velocity c given by
2 = /c dp d where
p is the pressure and is the mass density of the Bose-
system. The theory of Gavoret and Nozières effectively
sums up perturbation theory to infinite order, but it does
not solve the problem with infrared divergences. It yields
an anomalous self-energy 12 (0) 0.
A first satisfactory attempt to handle the infrared diver-
gences of the bosonic field theory was undertaken by A.
Nepomnyashchii and Yu. Nepomnyashchii (NN) [63].
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
942 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
Their calculations are rather involved and entail a partial
summation of the perturbation series. If this resummation
of diagrams is done correctly then infrared divergences
disappear from the theory. As an important consequence of
this diagrammatic analysis one obtains that the long-
wavelength behavior of the anomalous self-energy is ac-
tually nonanalytic at = 0q ,
12
0
1
( 0, 0) .
ln ( / )q q
q (5)
Here, = ( / , )q c q and 01/q is a length of the order of
the interparticle distance. Equation (5) is a very important
result, which makes the Green's functions 11G and 12G
also behave nonanalytically at ( , ) = 0.q Obviously,
Eq. (5) leads to 12(0) = 0, which contradicts Eq. (4). NN
confirmed, that the spectrum remains acoustic despite the
nonanalytic behavior of the correlation functions.
The method applied by NN in order to remove the
infrared divergences from the bosonic field theory is cer-
tainly not very transparent. It would be desirable to be able
to construct a perturbation theory where infrared diver-
gences are eliminated from the outset. Such a perturbation
expansion has been suggested by Popov [64,65] starting
from a functional integral approach. This approach yields
the same perturbation expansion as the conventional field
theoretical approach but suggests a more convenient me-
thod to eliminate infrared divergences. Popov's method is
based on a separation of the bosonic fields into ―fast‖ and
―slow‖ components with respect to a certain momentum
0q . Integrating out the ―fast‖ fields, Popov was able to
construct an effective action for the ―slow‖ fields only.
Representing the ―slow‖ fields by their amplitude and
phase one obtains an effective hydrodynamic action. The
diagram technique obtained from this action is free of
infrared divergences. It is then straightforward to calculate
the infrared structure of the various correlation functions.
Popov and Serednyakov [66] were able to obtain Eq. (5),
which was first derived by NN, from the effective hydro-
dynamic action. It implies that the Green's functions ob-
tained in Bogoliubov's theory are not correct despite the
fact that an acoustic spectrum is obtained.
Should be noted that Bogoliubov theory as well as fur-
ther developments by Beliaev, Hugenholtz, Pines, Gavoret,
Noziéres, NN were restricted to the zero temperature. The
first finite-temperature calculations was attempted in 1957
by Lee and Huang [67], for a gas of hard spheres. It was
shown that for 0T K the thermally induced depletion of
condensate take place, so it may spoil the validity of the
Bogoliubov theory. To avoid the ―mismatching‖ one has to
treat the condensate in some consistent fashion and it was
Popov [64,65], who proposed in 1965 a generalization of
the Bogoliubov theory for 0T K that gives the elemen-
tary excitation spectrum similar to that for = 0T K but
now with temperature-dependent condensate.
While the method employed by Popov contains the es-
sential ideas of modern renormalization group theory, it still
contains a number of phenomenological elements. In particu-
lar, the sharp separation of the fields into ―slow‖ and ―fast‖
components at a given momentum 0q appears to be some-
what artificial and, furthermore, the parameter 0q is not real-
ly well defined. A full-fledged renormalization group analy-
sis of the infrared behavior of the Green's functions of a
Bose-system was undertaken only recently by Pistolesi et al.
[68,69] in the frame of renormalization-group approach to
the infrared behavior of a zero-temperature Bose-system.
Within this theory it was explicitly shown that the effective
hydrodynamic action proposed by Popov is indeed the cor-
rect infrared fixed point of the renormalization group flow
which starts at the ―bare‖ action of strongly interacting bo-
sons. In order to expose the effects of the broken gauge
symmetry on the Green's functions, the Bose fields were
separated into longitudinal and transverse components. The
gauge symmetry is broken in the longitudinal component
only. Using this formulation it is particularly easy to set up
Ward identities, which relate various vertices to each other.
To show this, in [62] Popov’s theory was developed further.
Namely, using an analogous separation of the fields into
longitudinal and transverse components and Ward identities
were then used in order to obtain the vertices for the calcula-
tion of the density–density and current–current correlation
functions. So, in this work was shown that if a general form
of the action were written down in terms of ―running‖ coupl-
ings, and were found that all the couplings, that are present
in the ―bare‖ interaction (but not in the hydrodynamic ac-
tion) flow to zero or are irrelevant. This analysis confirms all
results obtained in the NN and Popov approaches.
Should be emphasized that vanishing of the anomalous
self-energy 12 (0) at zero momentum has a definite phys-
ical origin and is not just a peculiar mathematical result. In
the framework of broken symmetry, it is consistent with
the general picture proposed by Patashinskii and Pokrovs-
kii [70] where divergences, which arise in transverse corre-
lation functions connected with a Goldstone mode (zero
mass phonon), drive a divergence in the longitudinal prop-
agators due to the continuously broken symmetry. From
this point of view, the divergence of the Green's functions
at zero momentum due to the vanishing of 12 is an im-
mediate consequence of the Goldstone mode.
Of course, need to note that besides the quantum field-
theoretical methods, there are various other methods that
attempt to provide a basis for Landau's quasiparticle con-
cept. A successful picture of superfluid
4
He has been de-
veloped using correlated basis functions or similar ap-
proaches [71,72]. A very good quantitative description of
the response of superfluid
4
He at long and intermediate
wavelength is obtained using numerical quantum Monte
Carlo (for a review see Ceperley Ref. 73). The hydrody-
namic formulation by Hohenberg and Martin [74] de-
scribes the infrared response without the problem of spu-
The nature of superfuidity and Bose-Einstein condensation: from liquid
4
He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 943
rious infrared divergences, but it does not obtain the im-
portant result that 12(0) = 0. Finally, a very complete
picture of the excitations of Bose-systems is obtained using
a method based on seminal work by Feynman [75] and
Feenberg [76], which is also at the root of a recent theory
of superfluid
4
He by Vakarchuk [77].
However, all these methods do not solve the problem of
an ab initio calculation of the quasiparticle spectrum in the
superfluid
4
He Bose-liquid. Nowadays could be concluded
that, despite above mentioned big progress and achieve-
ments at the same time, the physical origin of superfluidity
on the microscopic level still remains obscure. The analy-
sis of experimental and theoretical publications indicates
that investigations of the unique phenomenon of super-
fluidity of liquid helium are far from being completed (see
for example [78–82]).
In fact, there are some big contradictions between the
theory and the experiment and next questions remain un-
solved.
1. As was mentioned above, applicability of the Landau
criterion of superfluidity for the determination of the criti-
cal velocity dissipationless flow in superfluid helium re-
mains unclear.
2. The origin of the roton minimum in the spectrum of
elementary excitations — an ab initio computation of the
spectrum of elementary excitations in the superfluid
4
He
Bose-liquid remains an actual problem nowadays, despite
certain recent successes in that direction, like an excellent
agreement with experimental data in the region of the roton
minimum obtained by the Monte Carlo method making use
of the so-called ―shadow wave function‖ [83] or by the
correlation basic function method [71] employing modern
interatomic potentials for
4
He. However, the physical rea-
son behind the appearance of the roton minimum in the
quasiparticle spectrum still remains unclear.
3. Nonlinear dependence of the velocity of the first sound
at small value of momentum. Numerous precise data, ob-
tained from the scattering of the cold neutron [84–86], clear-
ly indicates that so-called phonon part of the quasiparticle
spectrum decays.
As was shown in series of works [87–90], this character
of the spectrum of elementary excitations in superfluid
helium leads to the interaction and mutual transformation
of low-frequency and high-frequency phonons, and also
leads to the nonlinear dependence of the velocity of the
first sound from the momentum. Investigation of this prob-
lem is still continue.
4. Also remains open more general question about the in-
teraction of original atoms in superfluid
4
He and emergent
collective fluid volume elements (so-called fluid particles in
the Lagrangian description). Nowadays searches for the
most suitable collective degrees of freedom in superfluids
are still continue. And so the question how the formation
and stability of the volume element of
4
He as a continuous
medium can be explained from first principles remains a
subject of active studies (more details see [91,92]).
5. The quantum mechanical structure of the superfluid
component of the
4
He Bose-liquid below the -point, at
< = 2.17T T K. As was mentioned above, according to
numerous precise experimental data on neutron inelastic
scattering [93–95] and experiments on quantum evaporation
of
4
He atoms [18], the maximal density 0 of the single-
particle Bose-Einstein condensate in the
4
He Bose-liquid
even at very low temperatures T T does not exceed 9%
of the total density of liquid
4
He, whereas the density of
the superfluid component s at 0 K.T
Such low density of the BEC (suppressed BEC) is an
indication of the fact that the quantum structure of the part
of the superfluid condensate in He II carrying the ―excess‖
density 0 0( )s , which calls for a more thorough
investigation.
3. Self-consistent microscopic theory of superfluidity
4
He for the case = 0T K
For the case = 0T K part of above mentioned contradic-
tions between the theory and the experiment were partially
resolved in [96–98]. In this papers authors discussed both
the quantum structure of the superfluid state in a Bose-liquid
at = 0T K and the self-consistent calculation of the spec-
trum ( )E q of elementary excitations in the framework of
renormalized field perturbation theory [63–66]. The ap-
proach is based on the Pashitskii and Yu. Nepomnyashii
[99] microscopical model of superfluidity of a Bose-liquid
with a suppressed BEC and intensive pair coherent conden-
sate (PCC), which can appear due to sufficiently strong ef-
fective attraction between bosons in some domains of mo-
mentum space and is analogous to the Cooper condensate in
a Fermi liquid with attraction between fermions near the
Fermi surface. Should be noted that in the frame of Bogolu-
bov approach the model of superfluidity with two types of
condensates were considered earlier (see, for instance, ar-
ticle of Shevchenko [100]).
In our approach as a small parameter we use the ratio of
the single-particle BEC density to the total Bose-liquid den-
sity 0( / ) 1,n n unlike in the Bogoliubov theory of a non-
ideal Bose-gas, where the small parameter is the ratio of the
number of supracondensate excitations to the density of the
intensive BEC 0 0( )/ 1.n n n Because of this, superfluid
state within presented model can be described by a ―trun-
cated‖ self-consistent system of Dyson–Beliaev equations
for the normal and anomalous self-energy parts ( , ),ij k
where the diagrams of second and higher orders in the BEC
density was neglected. For this renormalized field perturba-
tion theory [64], which is built on combined field variables
[65,66] were used. In this case, the superfluid component
s is a superposition of the ―weak‖ single-particle BEC and
an intensive ―Cooperlike‖ PCC with coinciding phases
(signs) of the corresponding order parameters.
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
944 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
Without any doubts, choice of the pair interaction po-
tential, which will be used during the calculation of quasi-
particle spectrum, is very important step, and that step
should be discussed more carefully. To describe interaction
of helium atoms in real space, various semiempirical po-
tentials are conventionally used. All of them describe
strong repulsion at small distances and weak Van der
Waals attraction at large distances [101–103]. However,
those model potentials fail to take into account the fact that
at distances less than the quantum radius of the helium
electron shell 0 = 1.22r Å, the Coulomb repulsion between
the nuclei 2( ) /Ze r (partially screened by bound electrons)
sets in. The following simple approximation for the
4
He
interatomic potential, diverging as 1r at 0,r could be
suggested:
2
12 6
4
(1 )exp( / ), ,
( ) =
, > .
c
c
e
r r r r
r
V r
r r
r r
(6)
Such a potential has a finite Fourier component with an
oscillating sign-changing momentum dependence (Fig. 2,
dashed curve).
The Fourier component will look like any other potential
of the form (6) in which the interaction at > cr r is deter-
mined by any of the modern
4
He potentials [101–103].
However, those Fourier components are analytically
very complicated and it is technically very difficult to use
in the actual calculations. To be able to go forward, while
retaining the crucial features of the interaction, one should
employ a model potential, characterized by the same sign-
changing Fourier component as the one of Eq. (6), but with
a simpler analytic expression. For example, it is possible to
choose the simple finite repulsive potential of the ―semi-
transparent spheres‖ model, 3
0( ) = ( /4 ) ( )V r V a a r (
is the step function), whose Fourier component
0 3
sin ( ) cos ( )
( ) =
( )
pa pa pa
V p V
pa
(7)
is an oscillating sign-changing function of momentum
transfer p (Fig. 2, curve 1). It is necessary to emphasize
that the existence of negative values of the Fourier compo-
nent ( ) < 0V p is not directly associated with Van der
Waals forces. These oscillations arise even in the absence
of attraction in real space, and are an implication of quan-
tum mechanical diffraction effects of mutual scattering of
the particles.
Should be also noted that the same behavior is characte-
ristic for the Fourier components of more realistic poten-
tials that diverge not faster than 2r at 0r and possess
an inflection points in the radial dependence.
For the calculation of the spectrum of elementary exci-
tations in the superfluid
4
He the system of Dyson–Beliaev
equations were used. These equations allow one to express
the normal 11G and anomalous 12G renormalized single-
particle boson Green functions in terms of the respective
self-energy parts 11 and 12 . As was shown in Ref. 99
for a Bose-liquid with sufficiently strong interaction be-
tween particles when BEC is strongly suppressed, one can,
defining ( , )ik p in the form of a sequence of irreducible
diagrams that contain condensate lines, restrict oneself,
with good precision, to the lowest terms in the expansion
over the small BEC density 0( ).n n As a result for a
Bose-liquid, leaving the terms of the first order in small
parameter 0 / 1,n n one gets ―truncated‖ system of equa-
tions for ik :
11 0 1 11( , ) = ( , ) ( , ) (0) ( , ) ;n V n Vq q q q (8)
12 0 12( , ) = ( , ) ( , ) ( , ) ,n Vq q q q (9)
where
1( , ) = ( )[1 ( ) ( , )] .V V q V qq q (10)
Here ( )V q the Fourier component of the input pair interac-
tion potential; ( , )V q is the renormalized (―screened‖),
due to multiparticle collective effects, Fourier component
of the nonlocal interaction; ( , )q is the boson polariza-
tion operator that takes into accounting the multiparticle
collective effects:
3
113
( , ) = ( , , , ){ ( , )
2(2 )
d d
i G
k
q q k k
11 12 12( , ) ( , ) ( , )};G G Gk q k k q (11)
( , , , )q k is the vertex part, which describes multipar-
ticle correlations; ( , ) = ( , ,0,0) = (0,0, , ),q q q and
1n is the number of supracondensate particles 1 0( ) ,n n
Fig. 2. The Fourier components of: potential (6) (dashed curve);
model potential (7) (curve 1); the corresponding renormalized po-
tential (10), with account for the momentum dependence (11) of the
polarization operator (inset) on the ―mass shell‖ (curve 2).
The nature of superfuidity and Bose-Einstein condensation: from liquid
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He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 945
which is determined from the condition of conservation of
the total number of particles.
In addition, as it was also shown by Pashitskii and
Y. Nepomnyaschii in [99], only residues at the poles of sin-
gle-particle Green functions were taken into account, whe-
reas, the contributions of the poles of the functions
( , , , )q k and ( , )V q , which do not coincide with the
poles of ( , ),ijG q were neglected.
As a result, the functions ( , )ij q on the ―mass shell‖
= ( )E q have the following form (at = 0 K) :T
____________________________________________________
3
11 3
1 ( , ( ))
( , ( )) = ( , ( ); , ( )) ( , ( ) ( )) 1 ;
2 ( )(2 )
d A E k
E q E q E k V E q E k
E k
k k
q q k q k (12)
3
0 12
12 3
( , ( )) ( , ( )) ( , ( ))1
( , ( )) = ( , ( ) ( )) ( , ( ); , ( )) ,
2 ( )(2 )
n E k V E k E kd
E q V E q E k E q E k
E k
k k kk
q q k q k (13)
where
2 2 1/2
11 11 0 12
1
( ) = [ ( , ( )) ( , ( ))] ( , ( )) [ ( , ( )) ( , ( )) ( , ( ))] ,
2
{ }E q E q E q A E q n E q V E q E qq q q q q q (14)
2
0 12 11 11 11
1
( , ( )) = ( , ( )) ( , ( )) (0,0) (0,0) [ ( , ( )) ( , ( ))] .
2 2
q
A E q n E q V E q E q E q
m
q q q q q (15)
_______________________________________________
In this case, the total quasiparticle concentration is deter-
mined by the relation
3
0 3
1 ( , ( ))
= 1 .
2 ( )(2 )
d A E k
n n
E k
k k
(16)
From Eqs. (14) and (15) it follows that the quasiparticle
spectrum, because of the analyticity of the functions
( , ),ij q is acoustic at 0,p and its structure at 0q
depends essentially on the character of the renormalized
interaction of pair of bosons.
Note that in the absence of a BEC 0( = 0),n Eq. (13)
becomes homogeneous and degenerate with respect to the
phase of the function 12 ( ).p It is then become analogous
to the Bethe–Goldstone equation for a pair of particles in
momentum space
3
3
( )
( ) = ( ) ,
2 ( )(2 )
d
V
E k
k k
q q k
with zero binding energy = 0, which has a nontrivial solu-
tion only in the case of attraction ( ) < 0.V q This analogy
allows one to treat 12 ( )q at 0 = 0n as a PCC order para-
meter [99], which describes boson pair condensation in mo-
mentum space (identical to the Cooper condensate of fermion
pairs). Equation (13) being degenerate over the phase of
12 ( )q at 0 0n allows one to meet the condition of sta-
bility of the phonon spectrum
2 *
1 12= (0)/ > 0c m by
choosing the appropriate sign of the pair order parameter
12 (0) > 0. Since at = 0 KT the density s of the super-
fluid component, on the one hand, coincides with the total
density = mn of the Bose-liquid and, on the other hand, is
proportional to 12 (0), which plays the role of the superflu-
id order parameter, one gets the following relations:
12
0 0
(0)
= = [ (1 ) ]
(0) (0)
s s m m n
V
(17)
where
2
2
2
0
1
[ ( ) ( )] ,
( )(2 ) (0) (0)
k dk
k V k
E kV
2
122
0
1
( ) ( ) ( ) ,
( )(2 ) (0) (0)
k dk
k V k k
E kV
and is a certain dimensionless constant.
Since the density of the single-particle BEC is equal to
0 0= ,mn we obtain
1= (1 ) . This means that the
density of the ―Cooperlike‖ PCC is
1= = ,
(1 )
s
m
mn (18)
the concentration 1 0=n n n can be determined from rela-
tion (16), and for liquid
4
He at 0 K,T in accordance
with the experimental data, it should be approximately
90% of the full concentration
22= 2.17 10n cm
–3
. Thus,
the superfluid component of the Bose-liquid at = 0 KT in
this model is an effective coherent condensate [99], which
is a superposition of the weak single-particle BEC and the
intensive PCC.
The key point in the behavior of the Fourier component
of the screened potential ( , ( ))V E qq is that, as long as the
quasiparticle spectrum ( )E q satisfies the condition of sta-
bility with respect to decay into a pair of quasiparticles —
( ) < ( ) ( ),E q E k E q k the real part of the polarization
operator is negative: ( , ( )) < 0E qq on the ―mass shell”
(Fig. 2, inset). As a result strong suppression of repulsion
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
946 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
in the region where ( ) > 0V q and strong enhancement of
attraction in the region where ( ) < 0V q are take place
(compare curves 1 and 2 on Fig. 2).
In order to calculate quasiparticle spectrum ( ),E q one
has to calculate the polarization operator (11) and the re-
normalized retarded interaction (10) on the ―mass shell‖
= ( )E q as well as = ( ) ( ).E q E k At the same time is
necessary to solve the nonlinear integral Eqs. (12) and (13)
for the functions ( , ( )).ij E qq The only parameter va-
ried in order to ensure the best coincidence of ( )E q with
the experimental
4
He quasiparticle spectrum exp ( )E q was
the amplitude 0V of the initial potential (7) (for 3
0 / =V a
1552 K at = 2.44a Å). The BEC concentration was
given, in accordance with the experimental data, as
21
0 = 9% =1.95 10n n cm
–3
. Figure 2, curve 2 depicts the
momentum dependence of the renormalized retarded inte-
raction (10). On Fig. 3 solid line is the theoretical quasipar-
ticle spectrum ( )E q (14), dots are the experimental spec-
trum [93–95], points beyond the roton minimum shown as
stars ( = 0.6T K) [47]. Note that the phase velocity of qua-
siparticles 0[ ( )/ ] | ,qE q q obtained within this model,
coincides with the speed of hydrodynamical sound
1 236c m/s in liquid
4
He. Satisfactory agreement of
( )E q with exp ( )E q at 3.5q Å 1 is evident.
The self-consistency of the model is confirmed by the
following argumentation. On the one hand, theoretical value
of the full particle density from equation (16) th =n
= 2.14·10
22
cm
–3
is close to the experimental
4
He density;
on the other hand, the density 1n of supracondensate par-
ticles from Eq. (18) for the indicated parameter values is
above 90%n, which also agrees with experiment if taking
into account that BEC density is determined to be up to
about 9%n. The main result, obtained within this model of
the superfluid state, is the conclusion that the roton mini-
mum in the spectrum ( )E q and maximum in the structure
factor ( , ( ))S q E q is associated with the first negative min-
imum of the Fourier component of the renormalized
(―screened‖) interaction potential.
However, in case 0T contradictions between the
theory and the experiment, that we mentioned above, re-
mained unsolved and consistent microscopic superfluid
theory of Bose-liquid still not created. It is necessary to em-
phasize that resolving of these contradictions very closely
connected with the problem of quantum mechanical struc-
ture of superfluid
4
He component below the -point.
The answer of this question is crucial for the creation of
the consistent microscopic superfluid theory of Bose-liquid
for the cases of zero and nonzero temperatures.
In any case it is obvious that phenomenon of BEC plays
the key role for the understanding of physical mechanism of
superfluidity, and it seems like that at least two types of
condensates may appear and coexist simultaneously in
4
He.
It is appropriate to mention here that the phenomenon
of BEC (macroscopic occupation of the ground state due to
the saturation mechanism) is rather subtle and fascinating
question. In the next section will be discussed a recent
study of the Bogoliubov model for weakly interacting
Bose-gas, which shows that under certain conditions this
phenomenon manifests at least two kinds of BECs.
4. Conventional and nonconventional Bose-Einstein
condensations. Binary mixtures of Bose-Einstein
condensates
Shortly after discovering the phenomenon of BEC,
London has pointed out that this phenomenon implies
long-range coherence properties of the condensate and
Uhlenback [104] found that thermodynamic limit is very
important for the sharp manifestation of transition into
condensed phase. Later, Casimir [105] has pointed out the
importance of distinguishing between thermodynamic and
coherence properties of BEC in perfect Bose-gas.
Structure of the single-particle density of states (in the
thermodynamic limit) implies that for a given temperature
T there is a critical particle density, which corresponds to
maximal total particle density ( ).p
c T Under condition
that the particle density in the ground state 0 ( ) = 0
p
T and
increasing of total density beyond the saturation threshold
( ),p
c T it leads to macroscopic accumulation of particles
in the ground state, i.e., 0 ( ) = ( ).
p p
cT T This satura-
tion mechanism of condensation does not change the aver-
age kinetic–energy or pressure, which also reach their
maximal values at the critical density ( ).p
c T
Girardeau [106] was the first who introduced the con-
cept of generalized (conventional) BEC on the basis of the
observation that in one-dimensional model of impenetrable
bosons there is a sort of generalized condensation but no
macroscopic occupation of the ground state. Theoretical
importance of this concept was shown by Van den Berg
and Lewis [107–109], who have proposed next classifica-
Fig. 3. Theoretical quasiparticle spectrum E(p) obtained by a self-
consistent calculation (solid line); the experimental 4He excitation
spectrum (); the spectrum beyond the roton minimum () [47].
The nature of superfuidity and Bose-Einstein condensation: from liquid
4
He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 947
tion of Bose condensations: the condensation is called the
type I, when a finite number of single-particle levels are
macroscopically occupied; it is of type II, when an infinite
number of the levels are macroscopically occupied; it is
called the type III, or the nonextensive condensation, when
none of the levels are macroscopically occupied. In work
[109] was given an example of these different condensa-
tions and were demonstrated that three types of BEC can
be realized in the case of the perfect Bose-gas in an aniso-
tropic rectangular box or in a prism of volume V with
sides , , ,a b cV V V where a b c and =1.a b c If
< 0.5,c then for sufficiently large density the Bose-
Einstein condensation of type I appears; if = 0.5c one
gets a condensation of type II, whereas for > 0.5c a con-
densation of type III could be caused. Also, in [108] it was
shown that the type III condensation can be caused in the
perfect Bose-gas by a weak external potential or by a spe-
cific choice of the boundary conditions and geometry. In
[110,111] it was given another example of nonextensive
condensation for repulsive interacting bosons, which are
included in an isotropic box, and which are spread out the
conventional BEC of type I.
Also Bose-condensates could be divided on classes by
their mechanisms of formation. ―Conventional Bose-Ein-
stein condensate‖ — the condensation is due to saturation
of the total particle density (originally were discovered by
Einstein in the Bose-gas without interactions).
Since bosons are very sensitive to attraction, there is
another kind of condensation induced by this interaction.
Important results were obtained in [79,111], where exis-
tence of the phenomenon of condensation that was induced
by interaction has been proved. It may concern the model
of Huang–Yang–Luttinger [112] or full diagonal models
[113], since they both contain attractive interactions, and
also in the case of the Bogoliubov weakly imperfect Bose-
gas [114]. This condensate is called nonconventional Bose-
condensate. This kind of condensation might appear when
the total particle density (or chemical potential) becomes
larger than some critical value, but this is an attractive inte-
raction (and not simply Bose-statistics) which defines the
magnitude of the condensate and its behavior.
The difference between conventional and nonconven-
tional condensations consists in the difference of the me-
chanism of their formation. The conventional BEC is a
consequence of the balance between entropy and kinetic
energy, whereas the nonconventional condensation follows
from the balance between entropy and interaction energy.
Especially important that, as it has been shown in [111],
the nonconventional condensation does not exclude the
appearance of the BEC when the total density of particles
grows and exceeds some saturation limit. To escape the
collapse an attractive interaction in a boson system should
be stabilized by a repulsion, therefore, conventional and
nonconventional condensation may coexist. The possibility
of emergence of two kinds of condensates in two stages
was discussed at the first time in [107,111] in the frame-
work of a so-called ―pair Hamiltonian model‖ and was
shown that single-particle and two-particle states could
appear simultaneously due to off-diagonal interaction
terms. In weakly imperfect Bose-gas these condensates can
occur in two stages. For an interval 0[ ,0] of negative
chemical potentials up to = 0 , one has a macroscopic
occupation 0 ( ) of the mode with momentum = 0k due
to effective attraction of bosons in this mode [115]. Since
the weak interacting Bose-gas can exist only for 0 and
0 ( ) as well as the total density ( ) attain their max-
imal values at = 0 for > ( = 0) one gets (due to the
well-known saturation mechanism) a kind of condensation,
which occurs despite of effective two-bosons repulsion in
the weakly interacting Bose-gas for 0.k Moreover, in
[116] was demonstrated that in so-called ―perturbed mean-
field model with a Gaussian interaction kernel‖ there is no
Bose condensation for negative chemical potentials, but
condensation appears for [0, ] and then again disap-
pears for [ , ], where 2 .
So, binary mixtures of BECs are interacting quantum
systems of the macroscopic scale, which exhibit rich phys-
ics not accessible for a single-component degenerate quan-
tum gas. The key difference between multi-component and
single-component BECs is the intercomponent interaction.
In view of their unique properties binary mixtures of BECs
open up intriguing possibilities for a number of important
physical applications, including quantum simulation [117],
quantum interferometry [118], and precision measurements
[119,120]. As it will be shown later, intercomponent inte-
raction in binary mixtures of Bose-Einstein condensates
leads to appearance of different kinds of excitations in such
system. Especially interesting that some of these excita-
tions are very similar to the excitations in superfluid
4
He.
Recently, fundamental 2D soliton–soliton pairs were
investigated in two-component BECs with attractive intra-
component interactions [121]. General properties of vector
solitons and their stability were studied variationally and
numerically for both attractive and repulsive intercompo-
nent interactions and there were found different types of
soliton–soliton pairs including phase-separated pairs,
where one component is pushed outwards and forms a
ring-like shell and the other component is compressed due
to repulsive intercomponent interactions. It turns out that
for some values of the chemical potentials 1 2, phase-
separated steady-states coexist with collocated states cha-
racterized by bell-shaped density distributions in both
components.
In paper [121] was performed a linear stability analysis
of small azimuthal perturbations, which was checked by an
extensive series of numerical simulations. For attractive
intercomponent interactions matter-wave bright vector
solitons were demonstrated to be stable throughout the
existence domain. For BEC components, which repel each
other, various unstable evolution scenarios including col-
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
948 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
lapse and azimuthal symmetry-breaking instabilities were
observed. The instabilities, as a rule, lead either to separa-
tion of the condensed phases and then a collapse of the
stronger supercritical cr( > )N N component or a periodic
relative motion of the subcritical cr( < )N N solitonic
components backwards and forwards near the bottom of
trapping potential. Nevertheless, there are conditions
where complete stabilization of vector solitons is observed
even in the case of repulsive intercomponent interactions.
Consideration of a mixture of two BECs with atoms of
equal masses 1 2=M M and scattering lengths 11 22= ,a a
21 12=a a in axial symmetric harmonic trap ext ( ) =V r
2 2 2 2 2( )/2 /2zM x y M z allows to investigate it in
nearly two-dimensional case .z At the limit
0 KT in mean-field approximation the system can be
described by set of Gross–Pitaevskiy equations [122]:
2 2 2
,3 3( | | | | ) = 0,
j
j j j j ji r b
t
(19)
where = 1, 2,j 2 2= ,r x y 2 2 2 2= / /x y is
the 2D Laplacian. Here ( , ) ( , )/ ,x y x y l /2,t t
/j j jC is the dimensionless variables, where
22 4/2
= , = , .
| |
jjz
j jj
jj
al
l C g
M g M
Dimensionless coupling parameter is defined as: =
12 11 21 22/ | | = / | |,g g g g 2
12 21 12where = = 4 / .g g a M
Let us discuss in more details the case when internal in-
teractions are attractive while intercomponent can be re-
pulsive or attractive.
Stationary soliton solutions were given as follows:
( , ) = ( ) e ,
i tj
j jt rr (20)
where = 1, 2,j 2 2= ,r x y j are chemical poten-
tials, and real functions ( )j r satisfy the set of equations:
2 2 2
3
1
[ ] = 0.j j j j j j j jr
r
(21)
At fixed strength of cross-interaction we obtain two-
parameter family (with parameters 1 and 2 ) of vector
soliton solutions.
To find region of existence of soliton–soliton pairs and
study some general properties we investigate solutions of
stationary equations. Equations (21) were solved numerical-
ly. To gain a better insight into the properties of the vector
solitons one has to perform also the variational analysis of
the stationary vector fundamental solitons. Typical radial
profiles were found numerically and are shown in Fig. 4.
Stability of the stationary solutions was tested by three
methods. Some results were obtained by means of varia-
tion analysis and were found that soliton–soliton pair is
stable with respect to radial-symmetric collapse if both of
components have number of particles cr< ,N N where
crN is number of particles on Townes' soliton. Stability
Fig. 4. (Color online) Examples of radial profiles 1 (dashed curves) and 2 (solid curves) for = –0.5 and = –2.0 found numerically.
The nature of superfuidity and Bose-Einstein condensation: from liquid
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He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 949
with respect to different azimuthal modes has been investi-
gated by linear analysis. Both of these results were tested
by direct numerical simulations, for which we used split-
step Fourier transform method. Different scenarios of unst-
able and robust evolution have been observed. Examples of
unstable evolution are shown in Figs. 5 (b) and (d).
To summarize, the region of existence and stability of
soliton–soliton pairs was found and were shown that for
attractive cross-interaction region of stability coincides
with region of existence. In case of repulsive intercompo-
nent interaction region of stability is situated near exis-
tence boundary (Figs. 5 (a) and (c)).
5. Conclusions and future challenges
The main scope of the present review is devoted to the
role of the BEC in the superfluidity phenomena. We
present a review of large volume of recent literature that
has been developed over the last decades in this forefront
of research, interfacing between quantum and nonlinear
physics. It is worth to close this paper briefly recalling the
most important ideas that laid in the groundwork of the
progress in understanding of the nature of superfluidity.
Opposite to the Landau's statement that superfluidity
phenomena and BEC have nothing in common, Bogoliubov,
Onsager, Penrose, Feynman, and Yang clearly elucidated the
very close relation between these phenomena. Numerous
experimental evidences of existence of the BEC in the liquid
superfluid
4
He reaffirms this theoretical assumption and
now it is well established that the best estimate of the BEC
fraction in the superfluid
4
He is about 9% at = 0T K. From
the other hand, it is well known that the density of the super-
fluid component practically coincides with total density of
4
He, i.e., s at 0 K.T Such big discrepancy
creates the question about the quantum structure of the su-
perfluid part of He II and indicates the fact that the structure
of ―excess‖ density 0 0( )s is more complicated.
Taking into account the fact that Van den Berg, Levis,
Bru, Zagrebnov, Dorlas and Pulé showed that two different
kinds of BEC (conventional and nonconventional) of dif-
ferent physical nature can appear and coexist simulta-
neously in a weakly nonideal Bose-gases, it is tempting to
assume that the superfluid component is created from the
superposition of these condensates. This means that the
observed quasiparticle spectrum in
4
He follows from the
Fig. 5. (Color online) Stability region in the μ1, μ2 plane (green) as obtained by numerical simulations: = –0.5 (a), = –2.0 (c). Time
evolution of the density distributions 2
1| | (upper rows) and 2
2| | (lower rows) of perturbed vector solitons for = –0.5, μ1 = –1.5,
μ2 = 2 (b), and = –0.8, μ1 = –5, μ2 = –5 (d).
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
950 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
interaction of these condensates. For the case of zero tem-
perature such an assumption enables one to create self-
consistent microscopic theory of superfluidity, which is
expected to explain practically all observed properties su-
perfluid helium. However, for the nonzero temperature
such theory does not exist yet, despite a big efforts. As
though it may sound paradoxical, now a successful solu-
tion to this problem lies rather in the plane of the experi-
mental than theoretical. It is important to understand which
kind of the BEC is observed in superfluid helium (pure
condensate, which, again, is no more than a tenth part of
the superfluid component) and what is the nature of super-
fluid component. What is the building block of the super-
fluid component? As the matter of fact, it is very challeng-
ing problem from experimental point of view, however
after experimental realization of BEC in an atomic Bose-
gas (groups of Cornell and Wieman, Ketterle, and of Hulet
in 1995) has made it feasible to investigate the relation
between superfluidity and BEC on the experimental basis
of ultracold atomic gases, as these systems offer an un-
precedented level of control on the interaction strength,
density and temperature. Really, such close mutual relation
was confirmed and clarified by the investigations on ultra-
cold atomic gases: in 1999 superfluidity was demonstrated
through vorticity in a dilute Bose-gas (Cornell, Wieman,
Dalibard, Ketterle); in 2003–2005 BEC of pairs, and sub-
sequently superfluidity as again characterized through vor-
tices, was realized in Fermi-gases (Zwierlein, Ketterle,
Jin). Future experiments will probe such aspects of super-
fluidity as the appearance of vortices (related to phase co-
herence) and critical velocities (related to the Landau crite-
rion). Without any doubts the versatility of quantum dilute
gases would allow in the nearest future to find out an an-
swer about nature of the different aspects of the phenome-
non of superfluidity
4
He, especially the quantum mechani-
cal structure of the superfluid component of the
4
He below
the -point, at < .T T The answer of this question will
play key role for the consistent microscopic theory of
quantum Bose-liquid has yet to be created.
The anniversary review paper [123], devoted to the his-
tory of superfluidity in XX century, is ending with the
words ―... at least at this time, the evidence for superfluidi-
ty (in atomic BEC) is still quite circumstantial‖. However,
nowadays there are many fixed experimental facts that
demonstrate superfluid properties of cold atomic gases. In
this paper we have reviewed the milestones of this way.
But it is definitely not the end of the story. In the future
one can expect novel intriguing findings in this fast devel-
oping subject. On several examples let us consider an out-
look of our findings concerning superfluidity of atomic
BECs in toroidal optical traps such as stability of persistent
current in spinor BEC and novel type of vortices, which
combine both known types of singularities of vortex mo-
tion: vortex ring and vortex line.
Persistent currents are a hallmark of superfluidity and
superconductivity, and have been studied in liquid helium
and solid state systems for decades. Toroidally trapped
BECs are attractive both for fundamental studies of super-
fluidity and for applications in interferometry for precise
measurements and atomtronics.
There are different methods to create a toroidal trap: us-
ing a Laguerre–Gaussain laser beams [125], combined
magnetic trap and standing wave of light [126]. The expe-
rimental observation of persistent flow in toroidal BEC is
reported in [124,127–129]. The persistent current in 2D
spinor = 1F condensates of Na has been investigated in
Ref. 130, however spin degree of freedom are not investi-
gated in this work ( = 0).sg The point is that the whole
condensate is assumed to be in external magnetic field thus
the spin degree of freedom are frozen. The scheme to real-
ize persistent current using optical vortices is proposed in
Ref. 131 for spinor BEC and in [132] for single-component
BEC. Investigations of rotating toroidal trap is of interest
for precise rotation measurement using Sagnac effect.
Two-component BECs in a 1D ring trap in a rotating frame
are considered in Ref. 133. To generate a ring current it
was proposed the stirring mechanism considered in 3D
geometry [134]. The remarkable manifestation of non-
Abelian magnetic field (the generalization of the nonuni-
form magnetic field characterized by matrix potential A )
is presented the Ref. 135, where two-component 1D BEC
are investigated in the presence of exotic magnetic field
considering the currents and vortex states. One-com-
ponent BECs in 1D ring potential are considered in
Refs. 136,137. In Ref. 138 superfluid 1D ring in the pres-
ence of periodic scattering length modulation along the
ring is investigated. The two-component BECs in 1D and
2D toroidal traps are investigated in [139–144] A system
of two-component 1D BEC in ring potential is considered
also in Ref. 145.
Previous experiments on persistent currents in atomic
BECs were limited to spinless, single-component conden-
sates. Extending such studies to multicomponent systems,
in particular those involving more spin states, is essential
for understanding superfluids with a vectorial order para-
meter and for applications in atom interferometry. In very
recent experimental work [124] the stability of supercur-
rents in a toroidal two-component gas consisting of
87
Rb
atoms in two different spin states has been studied. As was
pointed out in Ref. 124, none of the existing theories is
quantitatively applicable to their experiments, since they
are limited to the simplified cases of reduced dimensionali-
ty and very weak interactions. Let us review the prelimi-
nary results of our investigations of superflow in toroidally
trapped spinor Bose-Einstein condensate of
87
Rb. We have
performed a series of computer simulations of the experi-
ments, presented in [124]. As is seen from Fig. 6 our re-
sults turn out to be in agreement with the experiments: the
two-component vortices with equal number of atoms in
The nature of superfuidity and Bose-Einstein condensation: from liquid
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He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 951
each component decay soon (see Fig. 6(b)), while the pure
one-component vortex survives for very long time (see
Fig. 6(c)). Our results turn out to be in agreement with ex-
periments. However, this issue deserves a more detailed
investigation and will be reported elsewhere.
Atoms of
87
Rb have spin = 1F and the state of this
condensate is ferromagnetic. In experiment 5= 10N atoms
of
87
Rb were loaded into optical trap (see Fig. 6(a)). Trap
was obtained by ―sheet‖ laser beam with a trapping fre-
quency = 350z Hz and ―tube‖ laser beam of radius
min =12 .r m Its depth is about 0 / = 1.2V h kHz, where h
is Planck's constant. As a tube beam authors used La-
guerre–Gauss beam, which carries orbital angular momen-
tum 3 . Hence they obtained vortex of charge = 3m in
effectively two-dimensional ring trap. After this there was
obtained state with two components with different spin-
projections. Authors found that stability of the current de-
pends on ratio between number of particles in different
spin states. This ratio can be presented by spin-polarization
= ( )/( )zP N N N N . The main result that we test is
the presence of critical spin-polarization zP 0.6–0.7
below which supercurrent rapidly decays.
We simulate the trap by sum of harmonic potential,
which models sheet beam, and radial Laguerre–Gauss po-
tential:
22 2 2
0 2
min min
( ) = exp 1 ,
2
m
zM z r r
V V m
r r
r
(22)
where 2 2= ,r x y minr is the radius of the trap (or
coordinate of the minimum), = 3m is the vortex charge,
0V is the trap depth. This useful form of Laguerre–Gauss
potential was obtained in the same way as in [146].
In Appendix we present derivation of GP equations for
spinor condensate in general form. Only first two equations
remains since 0 0 in experiment [124]. We can simplify
these equations providing dimensionless variables and using
symmetry of the system. We look for solutions in form
( , ) = ( , , ) ( , ),j jt x y t z tr where
1/2 2 21
( , ) = ( ) exp /
2 2
z z z
i
z t l t z l
Fig. 6. Schematic illustrations of experimental creation of optical toroidal trap. The trapping potential is created by intersecting a horizontal
―sheet‖ laser beam with a vertical ―tube‖ LGm,0 beam with m = 3 (a). Snapshots of the density distribution in (x, y) plane: numerical simula-
tion of the unstable evolution of the two-component spinor BEC with equal number of atoms: N+ = N– (b); stable single-component super-
flow with m = 3 the upper row is experimental results [124], lower row is numerical simulation (c).
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
952 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
and = /( )z zl M . After integrating out the longitudinal
coordinates we obtain two-dimensional system of GPE
equations:
2 2ˆ= { ( ) | | ( ) | | } ,s a s ai L
t
(23)
2 2ˆ= { ( ) | | ( ) | | } ,s a s ai L
t
(24)
where ext
ˆ = ( /2) ( ),L V r and dimensionless wave-
function and parameters of interaction are ,0 =
,0 / ,C 2 = 2 / | |,z nC l g = / | |a s ng g and
= sgn( ).s ng Here we provided dimensionless coordi-
nates min/ ,r r r / ,zz z l and time ,rt t where
2= /( ).r MR The external axially-symmetric potential
2 2
0( ) = exp( ( 1))mV V r m rr corresponds to Laguerre–
Gauss trap.
Dynamical equations (24) have the following integrals
of motion: Number of atoms
2= ,N n d r
Energy 2= ,E d r where
2 2
ext
1 1
= ( )
2 2
V r n
2 2 2( 2 ).
2 2
s an n n n n
v v
(25)
To define our dimensionless parameters we calculated
them using experimental values: effective radial frequency,
2= / = 4.79 Hzr mR , depth of the potential, 0V
2 2
0 / =mR V 1574.31, number of particles, N
2/ 2 =n zmNg l 5165.4. To find initial conditions for
subsequent dynamics simulation we used imaginary time
propagation method. This method allows to find ground
states of the system with fixed topology. The imaginary
time propagation have been started from the initial guess
obtained in Thomas–Fermi approximation.
To illustrate our findings we present two examples of
computer simulation of the superflow with parameters fit-
ted to the experimental setup reported in Ref. 124. As is
seen from Fig. 6(b) two-component superflow decays fast
for the case when number of atoms in each component are
equal. However, the single-component persistent current
demonstrates a stable evolution up to two minutes as was
demonstrated in experiments (compare experimental plots
the upper row from Fig. 6(c) and our computer simulations
given in the bottom row of Fig. 6(c)).
Quantization of vorticity is a remarkable manifestation
of superfluid properties of BECs.
Different kinds of vortex structures have been theoreti-
cally predicted and observed experimentally in atomic
BECs (see, e.g., [34] and references therein): single vortex
lines, vortex-antivortex pairs, vortex arrays, and vortex
rings, solitary waves moving along the straight vortex line
[147] (which are similar to ―hoop‖ structures known in
field theory [148]).
In contrast to vortex line (which only terminate at the
superfluid boundary) vortex ring have a closed-loop core.
Consequently, a vortex rings in 3D have a lower energy,
since their energies do not diverge with system size.
Hence, these excitations plays a crucial role in any decay
of superflow compared to just a simple vortex line. Several
schemes to create a vortex ring in atomic BEC have been
proposed: using dynamical instabilities in the condensate
to make a dark soliton decay into vortex rings, two-
component BECs with different relative velocity, drag of a
moving object through the condensate [149], space-
dependent Feshbach resonance [150], or by phase imprint-
ing methods [151].
Dynamics of vortex line in 2D trap and vortex ring in
spherical trap was addressed in [149]. It was found that a
core of the vortex undergoes oscillatory motion around a
circle of maximum energy. Due to dissipation the vortex line
as well as vortex ring drift to the edge of the condensate and
decay eventually. Note that in nonuniform light beam optical
vortex also exhibits radial drift and rotation due to back-
ground gradients of phase and intensity, respectively [152].
An interesting vortex complex with vortex ring in the
first component and vortex line in the second component
has been predicted in two-component BEC [151]. It usually
referred to as a skyrmion. Figure 7 (a) illustrates its struc-
ture: the ring vortex core of one component is filled by the
superflow of the other. This topological soliton in two-
component BEC can be identified as a particle-like skyr-
mion, closely resembles cosmic vortons. Its topology is
defined by the fact that far from the compact skyrmion the
density of BEC is not perturbed [153]. Such a structure can
be energetically stable even for multiply quantized vortex
lines [154]. While for some specific conditions the numeri-
cal simulations predict stability of a skyrmion [153], never-
theless it turns out to be rather fragile object. Skyrmion
stability appears to be very sensitive to the strength of the
intercomponent and intracomponent interactions. Moreo-
ver, it requires spatial separation of BEC components,
which is not possible in stable regime without additional
tuning of the scattering lengths using an optical Feshbach
resonance. Thus, experimental observation of the skyr-
mions remains to be rather challenging issue.
This brings up the following questions: is it possible to
stabilize both vortex ring and vortex line in an experimen-
tally available BEC? Probably coexisting vortex line and
vortex ring could stabilize each other, and if so, how to
The nature of superfuidity and Bose-Einstein condensation: from liquid
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He to dilute ultracold atomic gases
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 953
create such a vortex complex? It is of interest to find out
the conditions for existence of stable vortex ring-on-line
(VRL) in toroidal BEC. The VRL appears as the result of
simultaneous poloidal and toroidal flows of atoms in a sin-
gle-component toroidal BEC [see Fig. 7(b)]. It seems rea-
sonable to suggest that a VRL should be completely stable
because all precession motions of the both vortex cores are
expected to be suppressed. Indeed, the centrifugal barrier
caused by toroidal motion around a vortex line prevents a
vortex ring from radial shrinking. At the same time, the
optical toroidal trap with radial trapping by Laguerre–
Gaussian beams not only gives rise to the VRLs by toroid-
al stirring of the trapped vortex ring, but also it saves the
vortex line from a radial drift. The recent experiments
[124,129] with persistent currents of BECs in toroidal opti-
cal traps demonstrates a stable circulation of the superfluid
which corresponds to a multicharge vortex line. The life-
time of the superflow in toroidal trap reached few minutes
and it was restricted only by decay of the BEC itself. The
life-time of the VRLs is expected to be limited only by
dissipative effects. These issues deserves further studies,
and this work now in progress and the results will be pub-
lished elsewhere.
Appendix
Let us consider many-body system of particles with
hyperfine spin =1.F
Existence of three components with different spin states
occurs to complicating of interaction potential (compared
to simple BEC) [33]. The interaction potential for two
atoms with = 1F may be written in the form
2
int 1 2 1 2
=0,2
4 ˆ( ) = ( ) ,f f
f
V a P
m
r r r r (A.1)
where fa is the scattering length and ˆ
fP is the projection
operator on state with total spin equal to f. Note, that for
identical bosonic atoms with spin = 1F in a state of rela-
tive motion only states with total angular momentum
= 0f or = 2f can couple together, since the possibility
to have unit angular momentum is ruled out by the re-
quirement that wave function has to be symmetric under
exchange of two bosons. The interaction via low-energy
collision is invariant under rotations, and therefore it is
diagonal in the total angular momentum of the two atoms.
We could rewrite this expression as linear combination
of identity operator and multiplication of spin operators:
int 1 2 1 2 1 2
ˆ ˆ( ) = ( )( ).n sV g g F Fr r r r (A.2)
where we introduce the analogue of Pauli-matrices for
spin-1 bosons:
0 1 0 0 1 0 1 0 0
1ˆ ˆ ˆ= 1 0 1 ; = 1 0 1 ; = 0 0 0 .
2 2
0 1 0 0 1 0 0 0 1
x y z
i
F F F
(A.3)
Let us rewrite the projection operators on spin states
with = 0f and = 2f as follows:
0 1 2
1ˆ = (1 )
3
P F F , (A.4)
2 1 2
1ˆ = (2 )
3
P F F . (A.5)
Indeed
2 2
1 2= ( ) ,F F F thus 1 1( 1) = ( 1)f f F F
2 2 1 2( 1) 2 .F F F F That is why the eigenvalue f of
the scalar product 1 2( ) | = |ff fF F is determined by
Fig. 7. (Color online) Two-component BEC skyrmion. Schematic illustration of density isosurfaces for the ring component and line
component. The circulation directions of the corresponding flows is indicated by arrows (a). Schematic illustration of vortex ring-on-line
in toroidal trap. We show a constant surface of the trapping potential. A superflow (green line, 3) with toroidal and poloidal components
creates simultaneously two types of phase singularities: vortex line (red, 1) surrounded by vortex ring (black, 2) (b).
S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, and A.V. Chumachenko
954 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
the total spin f as follows: 0 = 2, 2 =1. We obtain
the coupling constants as follows:
2
0 224
=
3
n
a a
g
m
, (A.6)
2
2 04
=
3
s
a a
g
m
. (A.7)
Coupling constants describe interaction between densities
of components and spin interaction, respectively. Hamilto-
nian of such system may be written as follows [155]:
(1) (2) (2)ˆ ˆ ˆ ˆ= ,n sH H H H (A.8)
where
†(1) ˆˆ ˆ ˆ= ( ) ( )i ii
i
H d hr r r (A.9)
† †(2)
,
ˆ ˆ ˆ ˆ ˆ= ( ) ( ) ( ) ( )
2
n
n j ii j
i j
g
H dr r r r r . (A.10)
(2)ˆ =
2
s
s
g
H dr
† †
, ,
, , ,
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )( ) ( ) ( ) ( ) ,i k j l j li j
i j k l
F Fr r r r (A.11)
where
2 2
ˆ = ( )
2
ih V
m
r (A.12)
is one-particle Hamiltonian, ( )V r is trapping potential and
, = 0, 1i j are spin indices. As may be seen, Hamiltonian
consist of three components: one-body interaction part
(Eq. (A.9)), two-body part, which does not depend on spin
and describes interaction of densities (Eq. (A.10)), and two-
body part, which characterize spin interaction (Eq. (A.11)).
We can obtain equations of motion using Heisenberg eq-
uation ˆ ˆ ˆ= [ , ]i ii H
t
and mean-field approximation:
____________________________________________________
2
*
ext 0
ˆ = ( ) ( 2 )
2
n s si V g n g n n g n
t m
r . (A.13)
2
*
0 ext 0 0 0
ˆ = ( ) ( ) 2 ,
2
n s si V g n g n n g
t m
r (A.14)
_______________________________________________
where 0=n n n n , 2=| |j jn .
The spin-1 BEC in the absence of an external magnetic
field has two phases: ferromagnetic (
87
Rb, / =s ng g
34.66 10 ) and polar (
23
Na,
2/ = 3.14 10 )s ng g .
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|
| id | nasplib_isofts_kiev_ua-123456789-118741 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:08:37Z |
| publishDate | 2013 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Vilchynskyy, S.I. Yakimenko, A.I. Isaieva, K.O. Chumachenko, A.V. 2017-05-31T06:18:04Z 2017-05-31T06:18:04Z 2013 The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) / S.I. Vilchynskyy, A.I. Yakimenko, K.O. Isaieva, A.V. Chumachenko // Физика низких температур. — 2013. — Т. 39, № 9. — С. 937–957. — Бібліогр.: 155 назв. — англ. 0132-6414 PACS: 03.75.Lm, 03.75.Mn https://nasplib.isofts.kiev.ua/handle/123456789/118741 We present a brief overview of crucial historical stages in creation of superfluidity theory and of the current
 state of the microscopic theory of superfluid
 ⁴He. We pay special attention to the role of Bose-Einstein condensates
 (BECs) in understanding of physical mechanisms of superfluidity and identification of quantum mechanical
 structure of
 ⁴He superfluid component below -point, in particular — the possibility that at least two types of
 condensates may appear and coexist simultaneously in superfluid
 ⁴He. In this context we discuss the properties
 of the binary mixtures of BECs and types of excitations, which may appear due to intercomponent interaction in
 such binary mixtures of condensates. We also discuss current status of investigations of persistent currents in toroidal
 optical traps and present an outlook of our recent findings on this subject. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Сверхтекучесть и бозе-эйнштейновская конденсация The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) Article published earlier |
| spellingShingle | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) Vilchynskyy, S.I. Yakimenko, A.I. Isaieva, K.O. Chumachenko, A.V. Сверхтекучесть и бозе-эйнштейновская конденсация |
| title | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) |
| title_full | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) |
| title_fullStr | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) |
| title_full_unstemmed | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) |
| title_short | The nature of superfuidity and Bose-Einstein condensation: from liquid ⁴He to dilute ultracold atomic gases (Review Article) |
| title_sort | nature of superfuidity and bose-einstein condensation: from liquid ⁴he to dilute ultracold atomic gases (review article) |
| topic | Сверхтекучесть и бозе-эйнштейновская конденсация |
| topic_facet | Сверхтекучесть и бозе-эйнштейновская конденсация |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118741 |
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