Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential
We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose particles in an external potential. We present a method of finding an approximate analytical solution for the spectra of collective excitations of a Bose gas in a linear potential and in a...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential / D.V. Fil, S.I. Shevchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 365-369. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-800522015-04-10T03:03:07Z Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential Fil, D.V. Shevchenko, S.I. Quantum fluids We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose particles in an external potential. We present a method of finding an approximate analytical solution for the spectra of collective excitations of a Bose gas in a linear potential and in a potential of the form u(r)=m-u₀cosh²x/l, where m is the chemical potential. Numerical study shows that the analytical solution corresponds to collective modes localized at the edge or at the low-density region. We investigate the influence of the external potential on a critical velocity of a superfluid flow. It is shown that the effect of strong suppression of the critical velocity takes place in a nonuniform Bose system. We discuss a possibility of Bose-Einstein condensation (BEC) in the systems under investigations at nonzero temperatures and find that in case of a finite number of the particles BEC can emerge. 2001 Article Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential / D.V. Fil, S.I. Shevchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 365-369. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: O3.75.Fi http://dspace.nbuv.gov.ua/handle/123456789/80052 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Quantum fluids Quantum fluids |
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Quantum fluids Quantum fluids Fil, D.V. Shevchenko, S.I. Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential Вопросы атомной науки и техники |
| description |
We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose particles in an external potential. We present a method of finding an approximate analytical solution for the spectra of collective excitations of a Bose gas in a linear potential and in a potential of the form u(r)=m-u₀cosh²x/l, where m is the chemical potential. Numerical study shows that the analytical solution corresponds to collective modes localized at the edge or at the low-density region. We investigate the influence of the external potential on a critical velocity of a superfluid flow. It is shown that the effect of strong suppression of the critical velocity takes place in a nonuniform Bose system. We discuss a possibility of Bose-Einstein condensation (BEC) in the systems under investigations at nonzero temperatures and find that in case of a finite number of the particles BEC can emerge. |
| format |
Article |
| author |
Fil, D.V. Shevchenko, S.I. |
| author_facet |
Fil, D.V. Shevchenko, S.I. |
| author_sort |
Fil, D.V. |
| title |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential |
| title_short |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential |
| title_full |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential |
| title_fullStr |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential |
| title_full_unstemmed |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential |
| title_sort |
collective excitations and superfluid properties of a two-dimensional interacting bose gas in an external potential |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Quantum fluids |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80052 |
| citation_txt |
Collective excitations and superfluid properties of a two-dimensional interacting Bose gas in an external potential / D.V. Fil, S.I. Shevchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 365-369. — Бібліогр.: 15 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT fildv collectiveexcitationsandsuperfluidpropertiesofatwodimensionalinteractingbosegasinanexternalpotential AT shevchenkosi collectiveexcitationsandsuperfluidpropertiesofatwodimensionalinteractingbosegasinanexternalpotential |
| first_indexed |
2025-07-06T03:59:18Z |
| last_indexed |
2025-07-06T03:59:18Z |
| _version_ |
1836868550064603136 |
| fulltext |
COLLECTIVE EXCITATIONS AND SUPERFLUID PROPERTIES OF A
TWO-DIMENSIONAL INTERACTING BOSE GAS IN AN EXTERNAL
POTENTIAL
D.V. Fil1, S.I. Shevchenko2
1Institute for Single Crystals National Academy of Sciences of Ukraine, Kharkov, Ukraine
e-mail: fil@isc.kharkov.com
2B.I.Verkin Institute for Low Temperature Physics and Engineering National Academy of
Sciences of Ukraine, Kharkov, Ukraine
e-mail: shevchenko@ilt.kharkov.ua
We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose
particles in an external potential. We present a method of finding an approximate analytical solution for the spectra
of collective excitations of a Bose gas in a linear potential and in a potential of the form lxuu /cosh)( 2
0−= µr ,
where µ is the chemical potential. Numerical study shows that the analytical solution corresponds to collective
modes localized at the edge or at the low-density region. We investigate the influence of the external potential on a
critical velocity of a superfluid flow. It is shown that the effect of strong suppression of the critical velocity takes
place in a nonuniform Bose system. We discuss a possibility of Bose-Einstein condensation (BEC) in the systems
under investigations at nonzero temperatures and find that in case of a finite number of the particles BEC can
emerge.
PACS: O3.75.Fi
1. INTRODUCTION
The influence of an external potential on Bose-
Einstein condensation (BEC) in two-dimensional (2D)
Bose systems is one of challenge problems. In an ideal
Bose gas an external potential may cause BEC at
nonzero temperatures in two dimensions [1-3]. The
question whether it is the case for interacting Bose gases
has not been clearly understood yet. Discovery of BEC
in alkali metal vapours confined in a trap [4-6] revives
the interest to this question. Petrov et al [7] argued that
at 0≠T a true condensate or a quasicondensate with a
fluctuating phase can emerge in an interacting 2D Bose
gas confined in a harmonic trap. A mean field study of a
2D interacting trapped gas was done by Bayindir and
Tanatar [8]. Basing on the similarity in thermodynamic
behavior of ideal 2D Bose systems and those with weak
interactions they conclude that BEC takes place in
systems with a finite numbers of the particles. Mullin
[9,10] demonstrate the absence of BEC in trapped
interacting Bose gases in 2D in the thermodynamic
limit.
In the present paper we address this problem with
reference on two specific forms of the external
potential. We consider the potential
l
xuyxu 2cosh),( −= µ and the linear potential
xyxu αµ −=),( ( µ is the chemical potential). The
first one corresponds to a situation when a low-density
valley is formed in a Bose cloud. Since this potential
pushes the particles away from the centre of the Bose
cloud we call it an “anti-trap” one. The second potential
models a Bose cloud with a linear increase of the
density in a direction perpendicular to the edge. For
such a choice of the potentials the analytical expressions
for the low energy collective excitations can be found
and macroscopic quantum properties of the systems can
the investigated systematically. We should mention that
the third example of the potential for which the
analytical solution can be obtained is the harmonic trap
potential. In 2D this problem was considered by
Stringari [11].
In Sections 2 and 3 we outline our approach. In more
details it was given in Ref. [12]. In Sec. 4 the influence
of the external potential on the critical velocity of a
superfluid flow is studied. In Sec. 5 we discuss the
possibility of BEC in non-uniform systems.
2. BASIC EQUATIONS
We consider a Bose gas with a point interaction
between the particles in an external potential )(ru . The
Hamiltonian of the system has form
},ˆˆˆˆ
2
1
ˆˆ][ˆˆ
2
{
2
2
ΨΨΨΨ
ΨΨuΨΨ
m
rdH
++
++
γ+
µ−+∇∇= ∫
(1)
where Ψˆ is the boson field operator, m , the boson
mass, γ , the interaction constant. To rewrite the
Hamiltonian in terms of elementary excitation creation
and annihilation operators we use the approach of Ref.
[13]. We decompose the Bose field operators as
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 365-369. 109
10
ˆ ˆˆ ρρΨ ϕ += ie , where ϕ̂ is the phase operator and
1ρ̂ , the density fluctuation operator. Expanding the
Hamiltonian into powers of ϕ̂∇ and 1ρ̂ we obtain
+++= 210 HHHH , (2)
where 0H is the operator independent part,
∫
∇
−+−= ]
2
[ˆ
0
0
22
01
2
1 ρ
ρ
γ ρµρ
m
urdH
, (3)
]}ˆ,ˆ[]ˆ,ˆ{[
2
ˆˆˆ
4
1ˆˆˆ[
1
0
0
1
2
0
1
0
1
00
2
2
ϕρ
ρ
ρϕρ
ρ
ρ
ρ
ρϕρϕρ
∇∇−∇∇+
+= ∫
m
i
GTrdH
. (4)
In Eq.(4)
)(
2
ˆ
0
0
2
2
2
ρ
ρ∇
−∇=
m
T
, 02ˆˆ γ ρ+= TG . (5)
We require the vanishing of the linear term in the
Hamiltonian (2). It yields the equation for 0ρ
0
2 0
0
22
0 =
∇
−+−
ρ
ρ
γ ρµ
m
u
. (6)
The value of 0ρ coincides with the density of the
particles at T=0. One should note that our consideration
is valid at temperature mT /0
2 ρ< < .
To reduce the Hamiltonian into a diagonal form we
rewrite the phase and density fluctuation operators as
∑ +−=
ν
νννν ΘΘ
ρ
ϕ )ˆˆ(
2
1ˆ *
0
bb
i , (7)
∑ ++=
ν
ννννρρ )ˆˆ(ˆ *
01 bFbF , (8)
where +bb ˆ,ˆ are the Bose operators. The functions F
and Θ satisfy the equations
νννΘ FET =ˆ , (9)
ννν ΘEFG =ˆ , (10)
They are normalized by the condition 1*2 =∫ ννΘ Frd .
After substitution Eqs. (7,8) into Eq. (4) the quadratic
part of the Hamiltonian reads as
∑ ++=
ν
ννν bbEconstH ˆˆ
2 . (11)
Eq. (11) shows that the operators νν bb ˆ,ˆ+ are the creation
and annihilation operators of the elementary excitations.
The energies of the excitations νE can be found from
the solution of Eqs. (9,10) with the boundary conditions
specified.
3. SPECTRUM OF THE EXCITATIONS
Let us consider an external potential
l
xuu 2
0 cosh)( −= µr . (12)
For the parameters satisfying the inequality
22
0 2/ mlu > > one can neglect the fourth term in the
l.h.s. of Eq. (6). It yields )/(cosh)/( 2
00 lxu γρ = . In
the low-energy approximation one can also omit the
operator part in the quantity Ĝ and reduce the
eigenvalue problem to the following one
ννν
ν θθ
θ 22
22
2
2
2 1cosh Ek
ldx
d
l
x
m
u =
−+−
. (13)
Here we take into account that the system is uniform in
y direction and put )()( xeikyθΘ =r . Eq. (13) is reduced
to a hypergeometric equation and their general solution
is expressed through hypergeometric functions.
For the system finite in the x direction with the rigid
walls at Lx ±= the flow through the walls should
vanish. It determines the boundary conditions
0
0
=
±= Lx
dx
d
ρ
θ
. (14)
In the limiting case of ∞→L instead of Eq. (14) we
should require the fluctuations be finite at ∞→x . In
this case the eigenvalue problem can be solved
analytically. The spectrum has the form
+++
++= 2222
0 111 lknlknEnk ω , (15)
where 2
0
2
0 / mlu=ω and 2,1,0=n .
Comparing the analytical result with the solution
obtained numerically for finite L we find that at
)/( cEk nk > ( c is the sound velocity in the uniform
system with the same average density of the particles)
Eq. (15) approximates the numerical solution with a
good accuracy (Fig. 1).
Fig. 1. Spectrum of the excitations in the “anti-
trap” potential. Solid curves – numerical solution for
L/l=3, Dashed curves – analytical approximation
(Eq. (15))
The spatial dependence of the density fluctuation for
the three lowest modes is shown in Fig. 2. One can see
110
that at small k the whole system is disturbed, while at
large k the modes are localized at the low-density
region. Analytical expression (15) describes the
spectrum of the localized modes.
Fig. 2. The amplitude (in relative units) of the
density fluctuation for the low-energy collective modes
in the “anti-trap” potential. Solid curves, n=0 mode;
dashed curves, n=1, mode, dotted curves, n=2 mode
For a linear potential xu αµ −=)(r the eigenvalue
problem is reduced to a confluent hypergeometric
equation
ννν
ν θθθα 22
22
22
4
1 Ek
xdx
dx
m
=
−+−
. (16)
If the system is confined in a region 0<x<L the solution
should satisfy the boundary condition (14) at x=+L. The
second boundary condition is the requirement for the
solution to be finite at 0→x .
The analytical expression for the excitation energies
can be found in a limit ∞→L . It has the form
)12(||
2
+= nk
m
Enk
α (17)
( 2,1,0=n ). The spectra of the excitations given by
the analytical expression and by the numerical solution
are presented in Fig. 3.
Fig. 3. The spectrum of the excitations in the linear
potential ( mL/2
0 αε = ). Solid curves – numerical
solution, dashed curves – analytical approximation (Eq.
(17))
Here as in the previous case the analytical solution is
valid at )/( cEk nk > . The spatial dependence of the
density fluctuations for three lowest modes is shown in
Fig. 4. One can see that analytical solution (17)
corresponds to the modes localized at the edge.
Fig. 4. The amplitude of the density fluctuation for
the low enegry modes in the linear potential. Solid,
dashed and dotted curves – n=0,1,2 modes,
correspondingly
Eq. (16) was derived in a linear approximation for
the function )(0 xρ . This approximation is correct at
3 2
0 / αmxx => , when the fourth term in the l.h.s. of
Eq. (6) can be neglected. The solutions of Eq. (16)
singular at 0→x can be omitted at kx0<<1. It yields
the condition
3/1
2
=<
mkk c
α . (18)
Inequality (18) establishes the validity of the analytical
as well as the numerical solution.
4. CRITICAL VELOCITIES
In this section we consider the influence of the
external potential on a critical velocity of a superfluid
flow. We specify the system infinity in x direction with
a nonuniform density area of a finite width. Let the
external potential has the form
≥
≤−=
LxU
Lx
l
xuu
at
atcosh2
0µ , (19)
where )/(cosh2
0 lLuU −= µ . In the potential (19) a
Bose cloud has a low-density valley of the width 2L
aligned along the y direction. The valley separates two
uniform density areas. In this case the analytical
solution (15) corresponds to the modes localized in the
valley. The extended modes have higher energies at the
same k ( 22
xext kkcE += ). Using the Landau
criterium we determine the critical velocity for the
superfluid flow along the y direction as
=
k
E
v nk
c
min . (20)
Since the n=0 localized mode has the lowest energy one
should substitute Eq. (15) with n=0 into Eq. (20), taking
into account that Eq. (15) is valid at k1<k<k2. Here k1 is
given by the equation 10 1
kcE k = . Since the low-
energy approximation was used in Eq. (15), the upper
restriction on k emerges. This approximation requires
)0(22/ 0
22 γ ρ<mk . It yields
)/(cosh)/2( 1
2 lLLk c
−= , where cmLc /= is some
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 365-369. 111
length1. At 21 kk < the critical velocity is equal to
20 /
2
kE k . If the opposite inequality 21 kk > is satisfied
the critical velocity is determined by the extended
modes. The last ones yield cvc = . The general formula
for cv is the following
=
cc
c L
L
L
lfcv ,,1min , (21)
where
2
1
22
4
1
22
sech411
sech41
2
1),(
++×
+=
x
yx
x
yx
x
yxf
. (22)
At 1/,/ > >cc LLLl we find )/(cosh 1 lLcvc
−= . In this
limit the presence of the low-density valley results in a
strong suppression of the critical velocity. When the
ratio cLL / becomes smaller the critical velocity grows
up. It is illustrated in Fig. 5.
At cLL < Eq. (21) predicts no suppression of the
critical velocity. Physically, such a case corresponds to
two weakly coupled semi-infinite Bose clouds. Thus,
the quantity cL plays a role of a critical parameter for
the width of the low-density valley. If this width
becomes large then cL2 the critical velocity decreases.
Fig. 5. Critical velocity (in units of c ) in the “anti-
trap” potential. Solid curve – L/Lc=3; dashed curve –
L/Lc=5; dotted curve – L/Lc=7
Let us then consider the potential of the form
≥−
≤−
=
LxL
Lxx
u
at
at
αµ
αµ
. (23)
In such a potential the density of the particles increases
linearly from x=0 to x=L and then at x>L becomes
uniform. For this geometry Eq. (17) yields the spectrum
of collective modes localized at the edge. These modes
have lower energies then extended ones and determine
the critical velocity. The value of the critical velocity is
given by the formula (20) with the energy (17). The
values or k are restricted by the condition (18). They
1 If the density of the particles in the uniform part is fixed, this
length remains constant under variation of the parameters of the
external potential.
should also satisfy the inequality okEck > . If
cokc Eck < (this inequality is equivalent to cLL < ) the
localized modes do not exist and the spectrum of the
extended modes should be put into Eq. (20). The
dependence of the critical velocity on the parameters of
the system has the form
<
>
=
c
c
cc
LL
LL
L
L
cv
at1
at
3/1
. (24)
One can see from Eq. (24), that a suppression of the
critical velocity takes place at L large then the critical
length cL .
We should note, that Eqs. (21,24) should be
understood as estimate expressions. They describe the
situation more qualitatively then quantitatively. Rigorous
consideration should be based on the solution of
Eqs. (6,9,10) without the approximations used in Sec. 3.
5. BOSE-EINSTEN CONDENSATION
A temperature dependence of the density of the
Bose-Einstein condensate can be extracted from the
asymptotic behavior of the one particle density matrix
)'()( rr ΨΨ + . If this quantity remains finite at
∞→− |'| rr then the Bose condensate exists and its
density )(rcn is given by the equation
( ) 2/1
'||
)'()()'()(lim rrrr
rr cc nn=+
∞→−
ΨΨ . (25)
In 2D uniform systems at 0≠T the quantity (25) is
equal to zero. The destruction of the coherence is caused
mainly by the thermally excited phase fluctuations.
Taking into account only the phase fluctuation we
obtain
( ) 2
)',(
2/1
00 )'()()'()(
rr
rrrr
Φ
ρρΨΨ
−+ = e , (26)
where ( ) 2
'ˆˆ)',( rrrr ϕϕΦ −= . It is convenient to
direct the vector 'rr − along the x axis. Then
( )
+
−−×
= ∑
2
1)'(cos1
|)(|
)(
1)',;,( 2
0
T
Enyyk
x
x
yxyx
nk
B
nk
nkθ
ρ
Φ
. (27)
Let us consider a Bose cloud of a rectangle shape
yLL × in the “anti-trap” potential and evaluate the value
(27) at x=0. The answer at lyy >− |'| , 0ω> >T is
0
00
|'|ln
2
ln
2
)',0;,0( Φ
ξπξπ
Φ +−+=
TT
yy
T
Tl
T
Tyy , (28)
where mT 2/)0(0
2
0 ρ= , mT 2/0
2
0 ρ= ,
2
0
2 /)0( mTT γ ρξ = 2
0
2 / mTT ργξ = . The first
term in the r.h.s. of Eq. (28) is caused by the localized
modes, the second one – the acoustic mode and the third
112
one - the zero point fluctuations. The value of 0Φ does
not depend of the temperatures and determine the
density of the condensate 0cn at T=0. While the
quantity (28) depends on |'| yy − , for the system with
lLL y > >~ this dependence can be neglected. In this
case the density of the condensate at 0≠T is equal to
04
0 )0()0(
T
T
T
cc l
nn
πξ
= (29)
It is important to note that our result does not
contradict the general theorems [14,15] about the
absence of true Bose-Einstein condensation in two-
dimensional systems in the thermodynamic limit. In this
limit one should put the total number of the particles N
tends to infinity, keeping the average density
unchanged. Then the quantity L also tends to infinity
while the ratio lL / remains constant. Correspondingly,
if such a definition of the thermodynamic limit is
implied, the parameter l tends to infinity as well. One
can see from Eq. (29) that the density of the condensate
approaches to zero at ∞→l . It is obvious result, since
at ∞→l the system becomes locally uniform at all
scales.
In case of linear potential similar derivation yields
)(4
0
0)(
)()(
xT
T
T
cc x
x
xnxn
πξ
= , (30)
where mxxT 2/)()( 0
2
0 ρ= ,
2
0
2 /)()( mTxxT γ ρξ = , and the inequality
xxT <)(ξ is implied. In the thermodynamic limit we
are interested in a density of the condensate at cx ,
satisfying the condition constxc =)(0ρ at ∞→N .
Substituting into Eq. (30) cxx = and taking into
account that in the thermodynamic limit ∞→cx , we
find that the density of the condensate tends to zero.
One can see, that for the potential considered (as
well as for the harmonic trap potential [9,10]) the
definition of the thermodynamic limit implies that the
form of the potential depends on the total number of the
particles. In practice, the form of the potential is fixed
and under variation of the total number of the particles
the average density is changed. Since the density cannot
be infinitely large, the number of the particles should
remain finite. Thus, our conclusion about the existence
of the Bose-Einstein condensate at nonzero
temperatures is formally applicable only to the systems
with a finite number of the particles. But, in practice,
this limitation is not important, because just such
systems are used in experimental studies of BEC.
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PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 365-369. 113
D.V. Fil1, S.I. Shevchenko2
1Institute for Single Crystals National Academy of Sciences of Ukraine, Kharkov, Ukraine
e-mail: fil@isc.kharkov.com
1. INTRODUCTION
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