Effective Hamiltonian and excitation spectrum of harmonically trapped bosons
An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a
 Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify
 higher (three and four operator) products to the required form. Afte...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2016
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| Cite this: | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons / A. Rovenchak // Физика низких температур. — 2016. — Т. 42, № 1. — С. 49–55. — Бібліогр.: 45 назв. — англ. |
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| citation_txt | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons / A. Rovenchak // Физика низких температур. — 2016. — Т. 42, № 1. — С. 49–55. — Бібліогр.: 45 назв. — англ. |
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| description | An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a
Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify
higher (three and four operator) products to the required form. After the Hamiltonian diagonalization, the expression
for the excitation spectrum is obtained containing in particular temperature-dependent corrections. Numerical
calculations are made for a one-dimensional system. Some prospects towards the extension of the suggested
approach to study binary bosonic mixtures are briefly discussed.
|
| first_indexed | 2025-12-07T18:28:54Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1, pp. 49–55
Effective Hamiltonian and excitation spectrum
of harmonically trapped bosons
Andrij Rovenchak
Department for Theoretical Physics, Ivan Franko National University of Lviv
12 Drahomanov St., Lviv 79005, Ukraine
E-mail: andrij.rovenchak@gmail.com
Received April 3, 2015, revised July 31, 2015, published online November 23, 2015
An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a
Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify
higher (three and four operator) products to the required form. After the Hamiltonian diagonalization, the expres-
sion for the excitation spectrum is obtained containing in particular temperature-dependent corrections. Numeri-
cal calculations are made for a one-dimensional system. Some prospects towards the extension of the suggested
approach to study binary bosonic mixtures are briefly discussed.
PACS: 03.75.–b Matter waves;
05.30.–d Quantum statistical mechanics;
67.85.–d Ultracold gases, trapped gases.
Keywords: Bose-system, excitation spectrum, harmonic traps, approximate second quantization.
1. Introduction
After the discovery of the Bose–Einstein condensation
(BEC) in ultracold alkali gases [1,2], studies of trapped
dilute bosonic systems received a new impulse. The issue
of collective excitations in such systems appeared in the
focus of research immediately [3–5].
BEC in harmonic traps was discussed in particular in
[6–8], with quasi-one-dimensional systems studied in [9].
Reviews on relevant quantum many-body phenomena from
both experimental and theoretical points of view can be
found, for instance, in [10–13].
A description of excitations in trapped bosonic systems
beyond mean-field or Gross–Pitaevskii approximations
cannot be considered a completely solved problem at pre-
sent. Studies of excitation spectra of low-dimensional
trapped bosonic systems was made, e.g., in [14,15] using
also the Lieb–Liniger theory. In the analysis of Bose-
condensates in optical lattices the Bose–Hubbard model is
often applied [16–18]. Mathematical aspects of the
Bogoliubov approximation [19] for excitation spectra of
weakly-interacting bosons are discussed in [20,21]. Theo-
retical description of Bogoliubov-type excitations in
exciton–polariton Bose-condensates is given in [22].
Another possibility to study a trapped Bose-system us-
ing the second quantization approach is addressed in this
work. Certain approximations are suggested to take into
consideration items in the Hamiltonian beyond quadratic in
the creation–annihilation operators. The obtained effective
excitation spectrum contains in particular temperature-
dependent corrections.
Such an approach correlates with the Hartree–Fock–
Bogoliubov (HFB) approximation and its extensions. A
gapeless modification was proposed in [23] for homoge-
nous Bose-condensates. Analysis of transport properties of
Bose-condensates in mesoscopic waveguides was given in
[24]. Bosons in random potentials were studied in [25], see
also [26] for a comprehensive review containing, in partic-
ular, analysis of the HFB approximation in the nonuniform
case and its generalizations. The density-functional theory
to study Bose-condensates at finite temperatures was ap-
plied in [27]. Thermodynamic properties of a Bose-gas
beyond the HFB approximation were analyzed in [28] using
imaginary-time Green’s functions. Hard-sphere bosonic
systems were considered within a self-consistent t-matrix
theory in the HFB approximation [29] and beyond [30].
Recently, the HFB approximation was applied to study
phase separation in two-component Bose-condensates [31].
This approximation was also used to describe a finite-
particle trapped one-dimensional Bose-system [32].
The main idea of the present work is to suggest a math-
ematically simple approximation to account for higher-
order corrections in the Hamiltonian. This is done in a
phenomenological way by linking creation–annihilation
© Andrij Rovenchak, 2016
Andrij Rovenchak
operator products to the respective occupation numbers.
The obtained analytical expressions might be further also
used with results of more thorough derivations.
The paper is organized as follows. In the next Section,
the Hamiltonian of the problem is written in the second
quantization formalism with some simplification made to
account for a macroscopic occupation of the ground state.
The diagonalization procedure leading to an effective
Hamiltonian is described in Sec. 3. Numerical analysis of a
one-dimensional problem is given in Sec. 4. Some pro-
spects of the proposed techniques in application to binary
bosonic mixtures are analyzed in Sec. 5. A brief discussion
in Sec. 6 concludes the paper.
2. Hamiltonian in the second quantization formalism
Let a D-dimensional system of N weakly-interacting
bosons of mass m be confined to a harmonic trap
( )2 2 2 2
1 1 1( , , ) =
2D D D
mV x x x xω + +ω (1)
and the potential of interatomic interaction is given by
1( , , ) = ( ),DU x x gδ x
where the vector 1= ( , , )Dx xx and g is the coupling
constant.
The Hamiltonian of such a system reads:
2
0
=1 1 <
ˆˆ ˆ ˆ= ( ) ( ) = .
2
N
i
i i j
i i j N
H V U H U
m ≤ ≤
+ + − +
∑ ∑p
x x x (2)
Here, ˆ ip is the momentum operator of the ith particle, ix is
its coordinate.
It is possible to develop the second quantization ap-
proach using the eigenfunctions of the operator 0Ĥ , i. e.,
an ordinary D-dimensional harmonic oscillator.
Let †ˆ ˆ, jja a be the creation and annihilation operators for
the state 1| = | , , Dj j〉 〉j , respectively. The correspond-
ing energy levels
( )1 1= .D Dj jε ω + +ωj (3)
In this representation the Hamiltonian is
† † †
, , ,
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ= | | .
2
H a a U a a a aε + 〈 〉∑ ∑j l mkj j
j j k l m
j jk lm (4)
The operators satisfy standard bosonic commutation relations:
†ˆ ˆ, = .a a δ j jkk (5)
Following Bogoliubov [19], we assume the occupation
number of the lowest state 0N is a macroscopic number.
As it is an eigenvalue of the operator †
00ˆ ˆa a , one can treat †
0â
and 0â as c-numbers, so that
† †
0 0 0 0 00 0ˆ ˆ ˆ ˆ= , = 1 ,a a N a a N N+
†
0 0 00ˆ ˆ, .a N a N
Upon singling out items with one, two, three, and four
operators with the zero index, the Hamiltonian becomes:
_____________________________________________________
† † † † †3/2 0
0
0 0 , 0
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ= const { 0 | | 00 00 | | 0 } {4 0 | | 0 | | 00
2
N
H a a N U a U a U a a U a a
≠ ≠ ≠
+ ε + 〈 〉 + 〈 〉 + 〈 〉 + 〈 〉 +∑ ∑ ∑j j j k kj j j j
j j j k
j j j k jk
1/2
† † † † †0
, , 0 , , , 0
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ00 | | } {2 | | 0 2 0 | | } | | ,
2 2
N
U a a U a a a U a a a U a a a a
≠ ≠
+ 〈 〉 + 〈 〉 + 〈 〉 + 〈 〉∑ ∑j k l k l l mk kj j j
j k l j k l m
jk jk l j kl jk lm (6)
________________________________________________
where “const” denotes items of a non-operator nature. We
will not neglect the terms having more than two operators
with non-zero index as it is usually done in the approxi-
mate second quantization, but instead will further try to
take them into account in some effective manner.
It was shown in [33] that items linear in †â or â pro-
duce solely a constant shift of the energy levels, so one can
neglect them in further analysis. Dropping also a constant
term, we obtain the following Hamiltonian:
† † † †0
0 , 0
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ= (4 )
2
N
H a a c a a a a a a
≠ ≠
ε + + + +∑ ∑j j jk k j kkj j j
j j k
† † †1/2
0
, , 0
ˆ ˆ ˆ ˆ ˆ ˆ( )N c a a a a a a
≠
+ + +∑ jkl l k lkj j
j k l
† †
, , , 0
1 ˆ ˆ ˆ ˆ ,
2
c a a a a
≠
+ ∑ jklm l mkj
j k l m
(7)
where, recalling that the harmonic oscillator eigenfunctions
are real-valued,
= 0 | | 0 = | | 00 = 00 | | ,c U U U〈 〉 〈 〉 〈 〉jk j k jk jk (8)
= | | 0 = 0 | | ,c U U〈 〉 〈 〉jkl jk l j kl (9)
= | | .c U〈 〉jklm jk lm (10)
3. Diagonalization procedure
In order to obtain a quadratic form with respect to the
†â , â operators suitable for subsequent diagonalization, the
following procedure might be applied. The occupation
number corresponding to the state | 〉j is
†ˆ ˆ= ,n a a〈 〉j jj (11)
from which we can propose to make a substitution of oper-
ators in items with more than two of them:
50 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Effective Hamiltonian and excitation spectrum of harmonically trapped bosons
† 1ˆ ˆ,x xa n a n −→ →j j jj (12)
with a parameter x to be fixed later.
After this procedure, we arrive at the diagonalization
problem for the quadratic form of a general type
( )† † † *
,
ˆ ˆ ˆ ˆ ˆ ˆ ,jk k jk jk j kj j k
j k
A a a B a a B a a+ +∑ (13)
which has been already addressed in some works [33–36].
A mathematically simpler treatment can be achieved by
introducing further approximations, namely by linking the
creation–annihilation operators with different indices via
some factor,
† †*ˆ ˆ ˆ ˆ= , = .a f a a f ak kj j kjk j (14)
With correlation (12) in mind, the following expression is
consistently derived:
1
*= , = .
x x
n nf f
n n
−
k k
kj kj
j j
(15)
Provided that the occupation numbers jn are real and posi-
tive, so * =f fkj kj and thus = 1/2x . For brevity, the follow-
ing notation will be also used below:
= = .xh n nj j j (16)
Three-operator items are thus substituted in the follow-
ing manner
† † † † † † †ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )
3
c
c a a a a a a a a h a a h a a h+ → + + +jkl
jkl l k l l l k l jk k kj j j j
† †ˆ ˆ ˆ ˆ ˆ ˆ( ),
3
c
a a h a a h a a h+ + +jkl
k l l k k l jj j (17)
while for the four-operator product one has
† † † †ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ(
6
c
c a a a a a a h h a a h h→ + +jklm
jklm l m l m l m j kk kj j
† † † †ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ).a a h h a a h h a a h h a a h h+ + + +
l l lk m m j j m m k lk kj j . (18)
Taking to account the symmetry of the coefficients cjk ,
cjkl , and cjklm over all the indices, after simple transfor-
mations described above the following effective Hamilto-
nian can be written down:
†
eff 0 0
0 0 , 0
4ˆ ˆ ˆ= 2
3
H a a N f c N f c h
≠ ≠ ≠
ε + + +
∑ ∑ ∑j j kj jk kj jkl lj
j k k l
† † 0
, , 0 0 0
1 ˆ ˆ ˆ ˆ( )
3 2
N
f c h h a a a a f c
≠ ≠ ≠
+ + + +
∑ ∑ ∑kj jklm l m j j kj jkj j
k l m j k
0
, 0 , , 0
1 1 .
3 12
N f c h f c h h
≠ ≠
+ +
∑ ∑kj jkl l kj jklm l m
k l k l m
(19)
The proposed transformation can be better illustrated
using the Green’s function formalism. Consider the equa-
tion of motion for the two-time temperature Green’s func-
tion †ˆ ˆ|q qa a〈〈 〉〉:
† †1 ˆˆ ˆ ˆ ˆ| = [ , ] |
2
a a a H aω〈〈 〉〉 + 〈〈 〉〉 =
πq q q q
†† † †
0
0
1 ˆ ˆ ˆ ˆ ˆ ˆ= | (2 | | )
2
a a N c a a a a
≠
+ε 〈〈 〉〉 + 〈〈 〉〉 + 〈〈 〉〉 +
π ∑q q q qk k q qk
k
† † †
0
, 0
ˆ ˆ ˆ ˆ ˆ ˆ(2 | | )N c a a a a a a
≠
+ 〈〈 〉〉 + 〈〈 〉〉 +∑ qkl l q k l qk
k l
† †
, , 0
ˆ ˆ ˆ ˆ| .c a a a a
≠
+ 〈〈 〉〉∑ qklm l m qk
k l m
Using the effective Hamiltonian (19) yields instead:
† † †
eff
1 1ˆˆ ˆ ˆ ˆ ˆ ˆ| = [ , ]| = |
2 2
a a a H a a aω〈〈 〉〉 + 〈〈 〉〉 + ε 〈〈 〉〉 +
π πq q q q q q q
† † †
0
0
ˆ ˆ ˆ ˆ(2 | | )N c f a a f a a
≠
+ 〈〈 〉〉 + 〈〈 〉〉 +∑ qk kq q q kq q q
k
† † †
0
, 0
2 ˆ ˆ ˆ ˆ(2 | | )
3
N c f h a a f h a a
≠
+ 〈〈 〉〉 + 〈〈 〉〉 +∑ qkl kq l q q kq l q q
k l
† † †
, , 0
1 ˆ ˆ ˆ ˆ( | | ).
12
c f h h a a a a
≠
+ 〈〈 〉〉 + 〈〈 〉〉∑ qklm kq l m q q q q
k l m
The proposed operator substitutions would thus lead to a
closed set of two equations of motion (for †ˆ ˆ|a a〈〈 〉〉q q and
† †ˆ ˆ|a a〈〈 〉〉q q ) by means of getting rid of the off-diagonal
functions
† † †
0
0 0
ˆ ˆ ˆ ˆ ˆ ˆ| | ( ) |c a a a a c f G a a
≠ ≠
〈〈 〉〉→〈〈 〉〉 ≡ 〈〈 〉〉∑ ∑qk k q q q qk kq q q
k k
q
as well as decoupling higher-order functions
† † †
, 0
ˆ ˆ ˆ ˆ ˆ| ( ) |c a a a G a a
≠
〈〈 〉〉 → 〈〈 〉〉 +∑ qkl l q l q qk
k l
q
† †
2 ˆ ˆ( ) | , etc.G a a+ 〈〈 〉〉q qq
with coefficients 1,2 ( )G q linked to fkq and hl . The former
relation can be — to a certain extent — related to the so-
called quantum-number conservation condition [26]
† †
,ˆ ˆ ˆ ˆ= ,a a a a〈 〉 δ 〈 〉q k q q qk
while the latter is a mean-field-like approximation. A simi-
lar approach proved to be successful in modeling the spec-
trum of strongly interacting Bose-systems [37].
The following notations will be used below:
0
0
= ;
2
N
f c
≠
γ ∑j kj jk
k
(20)
0
, 0
1= ;
3
N f c h
≠
η ∑j kj jkl l
k l
(21)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 51
Andrij Rovenchak
, , 0
1= .
12
f c h h
≠
ϕ ∑j kj jklm l m
k l m
(22)
The Hamiltonian is thus the quadratic form
† † †
eff
0
ˆ ˆ ˆ ˆ ˆ ˆ ˆ= {( 4 ) ( )}H Q a a Q a a a a
≠
ε + + +∑ j j j j j jj j j
j
(23)
with
= ,Q γ + η + ϕj j j j (24)
which can be diagonalized using the standard Bogoliubov’s
u–v transform.
The operators in each mode (state) are expressed as
† † †* *ˆ ˆ ˆ ˆˆ ˆ= , = .a u b b a u b b+ +j j j j j j jj j jv v (25)
From the bosonic commutator
†ˆ ˆ, = 1a a
j j (26)
requiring that off-diagonal products ˆ ˆb bj j and † †ˆ ˆb bj j are
eliminated in the final expressions, the Hamiltonian
diagonalizes to the form
†
eff
0
ˆ ˆˆ = ,H E b b
≠
∑ j jj
j
(27)
where the elementary excitation spectrum is
2 2= 8 12 .E Q Qε + ε +j j j j j (28)
Evaluation of the above expression cannot be done without
specifying the functional dependence of f j and hj. On the
other hand, the c coefficients (matrix elements of the po-
tential energy operator) can be rewritten as follows:
1 1 1 11
= , = ,j k j k j k l j k lD D D D D
c c c c c cjk jkl
1 11 1
= j k l m j k l mD D D D
c c cjklm (29)
because the D-dimensional harmonic oscillator eigenfunctions
are factorized into the one-dimensional ones.
4. A one-dimensional problem
To provide an example of calculations with the sug-
gested effective Hamiltonian, a one-dimensional problem
can be considered. For the harmonic oscillator of mass m
and frequency ω, the eigenfunctions are
1/4 2 /21| = e ,
2 !
m x
jj
m mj H x
j
− ω ω ω 〉 π
= 0, 1, 2, ,j (30)
where ( )jH ξ is the Hermite polynomial.
Using the harmonic oscillator length
ho =a
mω
(31)
matrix elements (8)–(10) are obtained in the form
22
ho
1 1= e ( ) ( )
2 ! !
jk j kj k
gc H H d
a j k
∞
− ξ
+ −∞
ξ ξ ξ =
π ∫
( )/2
ho
1 ( 1) 1= .
22 ! !
j kg j k
a j k
−− + + Γ π
(32)
22
ho
1 1= e ( ) ( ) ( )
2 ! ! !
jkl j k lj k l
gc H H H d
a j k l
∞
− ξ
+ + −∞
ξ ξ ξ ξ =
π ∫
2
ho
1 1 1 1=
2 2! ! !2
g j k l j k l
a j k l
+ − + − + + Γ Γ ×
π
1 ,
2
k j l− + + ×Γ
(33)
ho
1 1=
2 ! ! ! !
jklm j k l m
gc
a j k l m+ + +
×
π
22e ( ) ( ) ( ) ( )j k l mH H H H d
∞
− ξ
−∞
× ξ ξ ξ ξ ξ =∫
( )/2
ho
1 ( 1)=
2 ! ! ! !
k j m lg
a j k l m
− + −−
×
π
1 1
2 2
1
2
j k l m j k l m
j k l m
+ + − + − + + + Γ Γ
× ×
− + − + Γ
3 2
1, , ;
2 1 .
1 1, ;
2 2
k j m lj l
F
m j k l k j l m
− + − + − −
×
− − − + − − − +
(34)
The integration can be made in a closed form as discussed,
for instance, in [5,38–40]. In the above expressions, ( )zΓ
is Euler’s gamma-function, and 3 2 1 2 3 1 2( , , ; , ; )F a a a b b z is
the hypergeometric function. Note that indices in the c
coefficients ( j k+ , j k l+ + , and j k l m+ + + ) must sum
up to an even number otherwise the respective integrals are
zero due to the parity property of the Hermite polynomial
products.
Models for kjf and jh use the occupation numbers jn :
= , = .k
kj j j
j
n
f h n
n
(35)
Generally, for harmonic oscillators
1 /
1= ,
e 1
j j Tn
z− ω −
(36)
52 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Effective Hamiltonian and excitation spectrum of harmonically trapped bosons
and fugacity z can be approximately put unity at low tem-
peratures corresponding to the BEC regime.
In the limit of Tω >> ,
/e .j T
jn − ω
(37)
In most experiments with cold atomic gases an opposite
condition is satisfied, Tω << , cf. [3,41]. It leads to
.j
Tn
jω
(38)
With this simple expression we will make numerical calcu-
lations below. Dependences of kjf and jh become thus:
= , = .kj j
j Tf h
k jω
(39)
This means in particular that there will be no temperature
dependence in jγ but corrections jη and jϕ are propor-
tional to T and T , respectively.
At BEC, the occupation of the lowest level 0N ap-
proaches the total number of particles N , so we put
0 =N N is expressions for jγ and jη yielding
ho
= ,j j
gN
a
γ γ (40)
ho
= ,j j
g NT
a
η η
ω
(41)
ho
= ,j j
g T
a
ϕ ϕ
ω
(42)
where jγ , jη , and jϕ are dimensionless coefficients ob-
tained by inserting Eqs. (32)–(34) with kjf and jh from
(35) into one-dimensional analogs of Eqs. (20)–(22).
Spectrum (28) in one dimension with =j jε ω be-
comes
2
= 1 8 12 .j j j j j j
jE j
j j
γ + η + ϕ γ + η + ϕ
ω + + ω ω
(43)
Upon introducing dimensionless parameters,
ho
1= , = ,gN Tx y
a Nω ω
(44)
the spectrum yields
=jE
2
= 1 8 12 .j j j j j jx x y y x x y y
j
j j
γ + η + ϕ γ + η + ϕ
ω + +
(45)
Numerical values of jγ , jη , and jϕ are shown in Table 1.
Dependences of the energies of the first three levels
= 1, 2, 3j on the values of x and y parameters are shown
in Fig. 1.
Note that when x reaches values above 2, an intersec-
tion of the energies 2E and 3E occurs. A similar behavior
is possible for some other levels since jγ and jη can have
both positive and negative values.
Obviously, the validity of the approximation decreases
for items containing jη and even more for jϕ terms be-
cause of number of simplifications applied, in comparison
with the jγ term.
Table 1. Values of dimensionless coefficients in Eqs. (40)–(42)
depending on the level number j
j jγ jη jϕ
1 0,074987 0,020906 0,041237
2 0,049719 0,047775 0,073939
3 –0,067169 0,045062 0,087680
4 –0,044015 –0,022696 0,081353
5 0,041606 –0,044378 0,056191
6 0,030211 0,005830 0,036522
7 –0,023169 0,030029 0,039952
8 –0,018589 0,000509 0,049381
9 0,012314 –0,017838 0,045257
10 0,010780 –0,001987 0,035160
11 –0,006394 0,009961 0,033071
12 –0,006034 0,001827 0,036096
13 0.003288 –0,005385 0,031684
14 0,003311 –0,001339 0,031684
15 –0,001702 0,002872 0,029970
16 –0,001815 0,000955 0,030215
17 0,000929 –0,001549 0,029776
18 0.001035 –0,000784 0,028471
19 –0,000590 0,000883 0,027376
20 –0,000678 0,000844 0,026641
25 0,000843 –0,000360 0,023545
30 0,001743 –0,004430 0,023388
50 0,007818 0,000009 0,025675
100 –0,000160 0,000125 0,000214
Fig. 1. (Color online) Energies of the first three excited levels
1 2 3, ,E E E (vertical axis, in ω units) given by Eq. (45) as func-
tions of the x and y parameters (44). Red surface is 1E , green
sufrace is 2E , and blue surface is 3E . Grey-shaded planes corre-
spond to noninteracting oscillator energies 1 4, ,ε ε .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 53
Andrij Rovenchak
5. A binary Bose mixture
Weakly-interacting Bose mixtures confined in harmonic
traps became a subject of experimental and theoretical
studies recently, e. g., mixture of ytterbium bosonic iso-
topes [42,43] or rubidium with some other elements [44],
see also [45].
Let the system contain two boson species, a and b , be-
ing characterized by coupling constants ag , bg , and abg ,
the latter corresponds to the inter-species interaction. The
one particle problems remain the same for each species:
1 0 1 0
ˆ ˆ ˆ ˆ= ; = ,a a a b b bH H U H H U+ + (46)
where the non-interacting oscillator Hamiltonians are
0 0
ˆ ˆ ˆ ˆ= ; =a a a b b bH K V H K V+ + (47)
with trapping potentials
( )2 2 2 2
1 1( ) = ;
2
a
a D D
m
V x xω + +ωx
( )2 2 2 2
1 1( ) =
2
b
b D D
m
V x xω + +ωx (48)
and the interaction items are
= ( ); = ( ); = ( ).a a b b ab abU g U g U gδ δ δx x x (49)
In the occupation number formalism we obtain:
1 1
ˆ ˆ ˆ= =a b abH H H U+ +∑ ∑ ∑
† †ˆ ˆˆ ˆ= ,n n n nn na a b ba bn na a
a a b b Uε + ε +∑ ∑ (50)
where the unidexed sums run over all the relevant particles
and U is the total potential energy in the system. The fol-
lowing Hamiltonian is thus of special interest:
( † † † † *ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ= mn m n mn m n mn m n mn m n
mn
H A a a B b b C a a C a a′ + + + +∑
† † * † † *ˆ ˆ ˆ ˆ ˆ ˆˆ ˆmn m n mn m n mn m n mn m nD b b D b b F a b F a b+ + + + +
)† * †ˆ ˆˆ ˆ ,mn m n mn m nG a b G b a+ + (51)
which is obtained by getting rid of three- and four-
operator products as described in Sec. 3. A diagonalized
Hamiltonian
† ( ) † ( ) †ˆ ˆ ˆ ˆˆ ˆ= =j j m m m n n nj
j m n
H E E Eα β′′ ξ ξ α α + β β∑ ∑ ∑ (52)
сan be obtained by a direct application of the diagonalization
procedure similar to that discussed in papers [33–36]. It
is technically complicated and involves nonlinear equa-
tions for infinite-size matrices; a perturbative solution
might be a good alternative.
On the other hand, the procedure from Sec. 3 can be
applied leading to
ˆˆ = ,j
j
H h′′ ∑ (53)
where the diagonalization of the quadratic form in each
mode ˆ
jh is nothing but the problem reducing to a bi-
quadratic equation.
For a simplified expression representing a single mode
† †ˆ ˆ ˆˆ ˆ= a bh a a b bε + ε +
† † † † † †ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ(4 ) (4 )a ba a a a aa b b b b bb+λ + + + λ + + +
† †ˆ ˆˆ ˆ( ),ab a b ab+λ + (54)
where aλ , bλ , and abλ are linked to interaction parameters
ag , bg , and abg as well as contain some effective contribu-
tions from simplifications of the original Hamiltonian, one
can obtain easily two branches of the excitation spectrum:
(1,2) 2= /2 /4E p p q± − (55)
with
2 2 2 2 2= ( 4 ) ( 4 ) 2a a b b a b abp ε + λ + ε + λ −λ −λ + λ (56)
and
2 2= ( 4 ) ( 4 )a a a aq ε + λ ε + λ −
2 2 2 2 2 2( 4 ) ( 4 )a b b b a a a b−λ ε + λ −λ ε + λ −λ λ −
2 2 42 ( 4 )( 4 ) 2 .ab a a b b a b ab ab− λ ε + λ ε + λ − λ λ λ −λ (57)
6. Discussion
In summary, an approach was proposed for obtaining
an effective Hamiltonian for bosonic systems in a har-
monic trap. An approximation consisting in the substitu-
tion of the off-diagonal creation–annihilation operator
products, † †ˆ ˆ ˆ ˆa a f a a→j kj jk j and alike, gave the possibility to
diagonalize the Hamiltonian in a mathematically simple
fashion and to calculate the elementary excitation spec-
trum. The factors fkj were related to the occupation num-
bers. Other models, beyond a simple dependence in the
T >> ω limit considered here, could be tested in subse-
quent works.
A more consistent approach to derive the expressions
for hj and fkj would require solving a variational problem
similar to the Gross–Pitaevskii one but being of a higher
order. The most straightforward way is to make a decom-
position of the operator â into a non-operator part h and a
small correction α̂, that is ˆˆ =a h +α. It was not addressed
in this paper due to significant mathematical complica-
tions, which were in contradiction with the idea of intend-
ed effective simplification of the diagonalization problem.
Such a systematic analysis requires a separate study, while
the results of the present paper allow for making computa-
tions with different expressions for hj and fkj.
54 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Effective Hamiltonian and excitation spectrum of harmonically trapped bosons
Numerical calculations of the excitation spectrum for
the one-dimensional systems will be further extended to
higher-dimensional systems as all the analytical expres-
sions were derived in this paper. Binary bosonic mixtures
might be also treated using the proposed approach.
The obtained excitation spectrum makes it possible to
calculate the properties of the system, like energy, specific
heat, and condensate fraction, using standard thermody-
namic methods [33]. Carrying out such computations is
more a technical task and requires specification of the sys-
tem parameters, including the trap geometry for two- or
three-dimensional systems.
I am grateful to the anonymous Referees for useful
comments and criticisms, which open a possibility to devel-
op the method of the present paper on a more rigorous level.
The paper is based on the research provided by the
grant support of the State Fund For Fundamental Research
of Ukraine, Project F-64/41-2015 (No. 0115U004838).
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 55
1. Introduction
2. Hamiltonian in the second quantization formalism
3. Diagonalization procedure
4. A one-dimensional problem
5. A binary Bose mixture
6. Discussion
|
| id | nasplib_isofts_kiev_ua-123456789-128448 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:28:54Z |
| publishDate | 2016 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Rovenchak, A. 2018-01-09T15:39:26Z 2018-01-09T15:39:26Z 2016 Effective Hamiltonian and excitation spectrum of harmonically trapped bosons / A. Rovenchak // Физика низких температур. — 2016. — Т. 42, № 1. — С. 49–55. — Бібліогр.: 45 назв. — англ. 0132-6414 PACS: 03.75.–b, 05.30.–d, 67.85.–d https://nasplib.isofts.kiev.ua/handle/123456789/128448 An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a
 Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify
 higher (three and four operator) products to the required form. After the Hamiltonian diagonalization, the expression
 for the excitation spectrum is obtained containing in particular temperature-dependent corrections. Numerical
 calculations are made for a one-dimensional system. Some prospects towards the extension of the suggested
 approach to study binary bosonic mixtures are briefly discussed. I am grateful to the anonymous Referees for useful
 comments and criticisms, which open a possibility to develop
 the method of the present paper on a more rigorous level.
 The paper is based on the research provided by the
 grant support of the State Fund For Fundamental Research
 of Ukraine, Project F-64/41-2015 (No. 0115U004838). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Бозе-эйнштейновская конденсация Effective Hamiltonian and excitation spectrum of harmonically trapped bosons Article published earlier |
| spellingShingle | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons Rovenchak, A. Бозе-эйнштейновская конденсация |
| title | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons |
| title_full | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons |
| title_fullStr | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons |
| title_full_unstemmed | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons |
| title_short | Effective Hamiltonian and excitation spectrum of harmonically trapped bosons |
| title_sort | effective hamiltonian and excitation spectrum of harmonically trapped bosons |
| topic | Бозе-эйнштейновская конденсация |
| topic_facet | Бозе-эйнштейновская конденсация |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/128448 |
| work_keys_str_mv | AT rovenchaka effectivehamiltonianandexcitationspectrumofharmonicallytrappedbosons |