Ground state property of Bose-Einstein gas for arbitrary power low interaction

17 Pages·0.14 MB·English

by  Makoto Hiramoto

Ground state property of Bose-Einstein gas
for arbitrary power low interaction
2
0
0
2
M. Hiramoto1
n
a
J College of Science and Technology, Nihon University,

6
1 Funabashi, Chiba 274-8501, Japan

]
h
c
e
m
Abstract
-
t
a
t
s
t. Westudy Bose-Einstein gas foranarbitrarypower low interaction Cαr−α.
a
m This is done by the Hartree Fock Bogoliubov (HFB) approach at T T and
c
≤
-
d
the mean field approach at T > T . Especially, we investigate the ground
n c
o
c state property of Bose gas interacting through the Van der Waals C6r−6
−
[

plusC r−3 interactions. Weshowthatthegroundstateunderthisinteraction
1 3
v
5 is stable if the ratio of coupling constants is larger than that of the critical
6
2 curve. We find that the C r−3 term plays an important role for the stability
1 3
0
of the ground state when the density of atoms becomes sufficiently large
2
0
/ at low temperature. Further, using the numerical values of C and C , we
t 3 6
a
m
confirm that the ground state of alkali atoms are stable.
-
d
n
o PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 51.30.+i
c
:
v
i
X
r
a
1e-mail: [email protected]
1
1 Introduction
Bose-Einstein condensation (BEC) was first observed in dilute ultracold al-
kali atoms of rubidium [1], lithium [2] and sodium [3]. Furthermore, Mewes
et al. and Ensher et al. measured the condensate fraction and the energy
of rubidium atoms [4, 5]. It is shown that the transition temperature T is
0
shifted not only by the interaction effect but also by the finite size effect by
a few percent, where T denotes the transition temperature of the noninter-
0
acting Bose gas within the external field in the thermodynamic limit. This
means that both the interaction and the finite size effects play an important
role in the real Bose gases. In fact, the number of trapped atoms is typically
N 107, and this may not be sufficiently large to take the thermodynamic
≤
limit.
Ontheotherhand,theseexperimentalfindingshavestimulatedmuchinterest
in the theory of the interacting Bose gas. Since BEC occurs when atoms are
dilute and cold, we can treat the interaction of atoms as the two-body inter-
action. In this case, the s-wave scattering length characterizes the strength
of the two-body interaction. Under these conditions, the Gross Pitaevskii
(GP) equation can well describe the behavior of the interacting Bose gas at
zero temperature [6, 7]. This is a mean field approach for the order parame-
ter associated with the condensate. Using the GP equation, several authors
studied the ground state and the excitation properties of the condensate [8].
To study BEC at finite temperature, the GP equation at zero temperature
was extended by Griffin [9]. It is called the Hartree Fock Bogoliubov (HFB)
theory. In particular, Popov approximation to the HFB theory has been
employed to explain the experimental results [10].
In the realistic point of view, BEC experiments are carried out in magnetic
traps. In this situation, spin-polarized atoms interact through a triplet po-
2
tential. As long as atoms remain polarized, they cannot form molecules. For
alkali atoms, the triplet potential has many bound states which allow them
to recombine into molecules. Since this recombination can only occur in a
three-body scattering, it cannot occur for sufficiently low density of atoms.
Thus, the two-body scattering is dominant. Therefore, spin-polarized atoms
can remain the gas through dipole two-body scattering which flip the spin
to untrapped state, and then atoms can produce BEC. In this sense, it is of
particular importance to find a way to investigate BEC with more realistic
interactions.
In this paper, we study BEC for an arbitrary power low interaction C r−α
α
which is more realistic than the ordinary contact interaction (δ-function).
This interaction plays an important role for alkali atoms [11]. The atomic
interaction V of alkali atoms can be approximately written as [12]
V = V +V +V +V +V , (1)
c d hf Z so
where V is the central force of the interaction. Further, V can be written
c c
in terms of the electron exchange interaction V and the dispersion force
ex
C r−6 C r−8 C r−10 . The magnetic-dipole interaction V and the
6 8 10 d
− − − ···
hyperfine interaction V behave like C r−3, where the coupling constant C
hf 3 3
is the spin part of the interaction. The last two terms, V and V represent
Z so
the Zeeman interaction and the spin-orbit interaction, respectively. These
interactions are important at high density.
Here, we study Bose gas interacting through the Van der Waals C r−6 plus
6
−
C r−3 interactions. We show that the ground state under this interaction is
3
stable if the ratio of the coupling constants is larger than that of the critical
curve. We find that the C r−3 term plays an important role to stabilize the
3
ground state when the density of atoms becomes large at low temperature.
Using the numerical values of C and C , we confirm that the ground state
3 6
3
of alkali atoms are stable.
This paper is organized in the following way. In the next section, we derive
the effective Hamiltonian for the interacting Bose gas. Then, in section
3, we study the ground state stability for the Van der Waals plus C r−3
3
interactions. Section 4 summarizes what we have clarified in this paper.
2 The Effective Hamiltonian for Interacting
Bose Gas
In this section, we derive the effective Hamiltonian for the interacting Bose
gas. Since Bose gas dramatically changes its behavior below the transition
temperature T , we derive the effective Hamiltonian with the HFB approach
c
at T T . On the other hand, we derive the effective Hamiltonian with the
c
≤
mean field approach at T > T .
c
The Hamiltonian for the interacting Bose gas confined in a harmonic oscil-
lator potential can be written as
h¯2 1
Hˆ = drΨˆ†(r) 2 + mωr2 Ψˆ(r)
Z (cid:20)−2m∇ 2 (cid:21)
1
+ dr dr Ψˆ†(r )Ψˆ†(r )V( r r )Ψˆ(r )Ψˆ(r ), (2)
1 2 1 2 1 2 2 1
2 Z | − |
where Ψˆ(r) is the boson field operator. The two-body atomic interaction
V( r r ) is given as
1 2
| − |
C
V( r r ) = α . (3)
| 1 − 2| r r α
1 2
| − |
Now, we write down the second quantized Hamiltonian for Eq. (2) [13]. The
boson field operator for the ideal system can be expanded by plane waves,
but for the general case, the corresponding field operator is a sum over all
normal modes [14]
Ψˆ(r) = aˆ χ (r), (4)
ν ν
Xν
4
whenχ (r)’sareanycompletesetofnormalizedsingle-particlewavefunction,
ν
and a is a bosonic annihilation operator for the single-particle state ν. From
ν
Eq. (4), the Hamiltonian can be written as
1
Hˆ = Tνν′aˆ†νaˆν′ + 2 Vνν′λλ′aˆ†νaˆ†λaˆν′aˆλ′, (5)
Xνν′ νXλν′λ′
with
h¯2
Tνν′ = Z drχ†ν(r)(cid:20)−2m∇2 +Vext(r)(cid:21)χν′(r), (6)
Vνν′λλ′ = Z dr1dr2χ†ν(r1)χ†λ(r2)V(|r1 −r2|)χν′(r2)χλ′(r1). (7)
Here, we expand V( r r ) in terms of the Legendre polynomial P
1 2 l
| − |
∞
V( r r ) = v (r ,r )P (cosθ ),
1 2 l 1 2 l 12
| − |
X
l=0
∞ l
4π
= v (r ,r )Y∗ (θ ,ϕ )Y (θ ,ϕ ), (8)
2l+1 l 1 2 l,m 1 1 l,m 2 2
X X
l=0 m=−l
where v (r ,r ) is given by
l 1 2
2l+1 1
v (r ,r ) = dt V( r r )P (t). (9)
l 1 2 1 2 l
2 Z | − |
−1
In the case of the power low potential V( r r ) = C r r −α, we can
1 2 α 1 2
| − | | − |
integrate v (r ,r );
l 1 2
Γ(1)Γ(l+ α) rl α α 1 3 r2
v (r ,r ) = C 2 2 2 F +l, ,l+ , 2 , (10)
l 1 2 αΓ(l+ 1)Γ(α)rα+l 2 1(cid:18)2 2 − 2 2 r2(cid:19)
2 2 1 1
for r > r , and
1 2
Γ(1)Γ(l+ α) rl α α 1 3 r2
v (r ,r ) = C 2 2 1 F +l, ,l+ , 1 , (11)
l 1 2 αΓ(l+ 1)Γ(α)rα+l 2 1(cid:18)2 2 − 2 2 r2(cid:19)
2 2 2 2
for r > r , where F (a,b,c,x) denotes hypergeometric function. Thus, we
2 1 2 1
obtain the second quantized Hamiltonian for Bose gas interacting through
the power low interaction. From now on, we present the mean field Hamil-
tonian and the HFB Hamiltonian.
5
2.1 Mean Field Theory
First, we present the mean field theory for the interacting Bose gas [15].
This theory is particularly valid for T > T [16]. But for T T , this
c c
≤
theory is inadequate, since the low energy part of the excited states plays an
important role at low temperature. This effective Hamiltonian Hˆ is given
eff
by the diagonal part of the Hamiltonian (5), and can be written as
1
Hˆeff = Tννnˆν + vνν′nˆνnˆν′, (12)
2
Xν Xνν′
where n = a†a and
ν ν ν
v = dr dr χ (r ) 2V( r r ) χ (r ) 2, (13)
νν 1 2 ν 1 1 2 ν 2
Z | | | − | | |
for ν = ν′ and
vνν′ = dr1dr2 χν(r1) 2V( r1 r2 ) χν′(r2) 2
Z | | | − | | |
+Z dr1dr2χ†ν(r1)χ†ν′(r2)V(|r1 −r2|)χν(r2)χν′(r1) (14)
for ν = ν′. We rewrite Eq. (12) in the following way
6
Hˆ = Hˆ +Hˆ′, (15)
eff 0
with
1
ˆ ˜
H0 = Eνnˆν vνν′ρνρν′, (16)
− 2
Xν Xνν′
1
Hˆ′ = vνν′(nˆν ρν)(nˆν′ ρν′), (17)
2 − −
Xνν′
E˜ν = Tνν + vνν′ρν′. (18)
Xν′
Here, we minimize Hˆ′ by taking ρ’s in the following way
Tr n e−(Hˆ0−µNˆ)/kBT 1
ν
ρ = n = = . (19)
ν h νi Tr e−(Hˆ0−µNˆ)/kBT e(E˜ν−µ)/kBT 1
−
6
Then, the thermodynamic potential Ω = pV can be given as
−
Ω = k T lnTr e−(Hˆ0−µNˆ)/kBT,
B
−
1
vνν′ nν nν′ +kBT ln(1 e−(E˜ν′−µ)/kBT). (20)
≃ −2 h ih i −
Xνν′ Xν′
We note that Eq. (19) is also determined by the condition
∂Ω 1
(cid:18)∂hnνi(cid:19)T,µ,V,hnν′i6=hnνi = −Xν′ vνν′hnν′i+Xν′ vνν′e(E˜ν′−µ)/kBT −1 = 0.(21)
This condition means that the thermodynamic potential Ω is an extremum
with respect to any change of T, V and µ in a state of thermal equilibrium.
2.2 HFB Theory
Now, wepresent theHFBtheory. Thistheoryisreliableforthedescription of
the low temperature behavior of the Bose gas. In this theory, the operators
a and a† are replaced by the c-number a ,a† √N . Then, the HFB
0 0 0 0 ≈ 0
Hamiltonian Kˆ = Hˆ µNˆ is given by
−
1
Kˆ = (T µ)N + V00N2
00 − 0 2 00 0
+ Tνν′ −µδνν′ +2N0V00νν′ +2n′λVλλνν′ a†νaν′
νXν′6=0(cid:8) (cid:9)
N
+ 20 Vν0ν0′(aνaν′ +a†νa†ν′), (22)
νXν′6=0
where n′ is the expectation value of the non-condensate density of particles,
λ
and we eliminate the lower power of N and terms which are proportional to
0
a anda†. Now,wedefinenewbosonoperatorsbyBogoliubovtransformation
ν ν
c = u a +v a†, (23)
ν ν ν ν ν
c† = u a† +v a , (24)
ν ν ν ν ν
7
where u and v satisfy u2 v2 = 1. Then, we can write the Hamiltonian in
ν ν ν − ν
terms of the new operator c
ν
1
Kˆ = (T00 −µ)N0 + 2V0000N02 + ενν′vν2 −N0Vν0ν0′uνvν′
νXν′6=0(cid:0) (cid:1)
+ (cid:20) ενν′(uνuν′ +vνvν′)−N0Vν0ν0′(uνvν′ +vνuν′) c†νcν′
νXν′6=0 (cid:8) (cid:9)
1
+(cid:26)−ενν′uνvν′ + 2N0Vν0ν0′(uνuν′ +vνvν′)(cid:27)(c†νc†ν′ +cνcν′)(cid:21), (25)
where ενν′ is given as
ενν′ = Tνν′ −µδνν′ +2 N0Vν0ν0′ +n′λVλλνν′ . (26)
(cid:0) (cid:1)
We can eliminate the terms proportional to c†νc†ν′ + cνcν′ by imposing the
condition
1
−ενν′uνvν′ + 2N0Vν0ν0′(uνuν′ +vνvν′) = 0, (27)
or
ενν′uν′ −N0Vν0ν0′vν′ = Eν′uν′δνν′, (28)
ενν′vν′ −N0Vν0ν0′uν′ = −Eν′vν′δνν′, (29)
where the eigenvalue E is given as
ν
E = ε2 (N V00)2. (30)
ν q νν − 0 νν
Then, we obtain the diagonal Hamiltonian
1
Kˆ = (T µ)N + V00N2 + E c†c . (31)
00 − 0 2 00 0 ν ν ν
X
ν6=0
Here, we note that the first two terms in the right hand side of Eq. (31)
correspond to the GP equation, and the third term in the right hand side of
Eq. (25) represents small correction terms, and therefore we can ignore the
8
third term. From Eq. (31), we can also obtain the thermodynamic potential
Ω = k T lnTre−Kˆ/kBT
B
−
1
(T µ)N + V00N2 +k T ln(1 e−Eν/kBT). (32)
≃ 00 − 0 2 00 0 B −
X
ν6=0
The chemical potential is given by the condition
∂Ω
0 = ,
∂N
0
= T µ+V00N +2 n′V00, (33)
00 − 00 0 ν νν
X
ν6=0
where we ignore the anomalous average.
We note that Eqs. (31) and (33) are similar to results of the HFB approach
to the Popov approximation [9]. In the next section, we will see the ground
state stability of the interacting Bose gas using the GP equation.
3 Ground State Stability
In this section, we study the ground state properties of the Bose gas interact-
ing through the Van der Waals type C r−6 and the C r−3 interactions. We
6 3
−
note that we must introduce a cut off at r = 2 R , since these interactions
h i
diverge at r = 0. Here, we assume the hard core potential inside the atomic
radius R . This is chosen to be the exponential or C r−12 potential [12].
12
h i
Now, we study the ground state stability of Bose gas for the condensate state
at T T . In this case, the HFB theory is reliable for the investigation of the
c
≤
condensate state. To investigate the ground state stability, we employ a vari-
ational method. It is a good approximate scheme to study the ground state
stability. We assume the following Gaussian wave function for the ground
state
N
Ψ(r) = e−r2/2σ2, (34)
rπ3/2σ3
9
where σ represents the variational parameter. This choice is natural when
we take the noninteracting limit.
First, we consider the Van der Waals interaction. Here, we make comments
on the scattering length a for this interaction. This can be analytically given
by [17]
1
Γ(3/4) mC 4 π
6
a = 1 tan φ , (35)
2√2Γ(5/4)(cid:18) h¯2 (cid:19) h − (cid:16) − 8(cid:17)i
where φ is the semiclassical phase calculated at zero energy from the classical
turning point to infinity. This phase depends on the repulsive hard core
potential. Substituting Eq. (34) into Eq. (31), we can obtain the ground
state energy E
g6
3 8N2C
E = Nh¯ω(σ2 +σ−2) 6I ,
g6 4 − πa6 σ6 6
ho
3 1
= Nh¯ω (σ2 +σ−2) g σ−6 , (36)
6
2 (cid:26)2 − (cid:27)
where we rewrite σ as σ = σ/a in units of the harmonic oscillator length
ho
a = h¯/mω and I denotes the dimensionless integral
ho 6
p
∞ ∞ 1 1+s2/t2
I = 2 ds s2e−s2 dt t2e−t2 . (37)
6
Z0 Zs+2hR˜i t6(1−s2/t2)4
The dimensionless coupling constant g is defined by
6
16NC I
6 6
g = . (38)
6 3πh¯ωa6
ho
10