Recall the energy levels of this system, and its ground state wave function φ0r.. We wish to obtain an upper bound on this ground state energy by the variational method.. a Using perturb
Trang 1Properties of a Bose–Einstein Condensate
By cooling down a collection of integer spin atoms to a temperature of less than one micro-Kelvin, one can observe the phenomenon of Bose–Einstein condensation This results in a situation where a large fraction of the atoms are in the same quantum state Consequently, the system possesses remarkable
coherence properties We study here the ground state of such an N particle
system, hereafter called a condensate We will show that the nature of the system depends crucially on whether the two-body interactions between the atoms are attractive or repulsive
19.1 Particle in a Harmonic Trap
We consider a particle of mass m placed in a harmonic potential with a fre-quency ω/2π The Hamiltonian of the system is
ˆ
H = pˆ2
2m+
1
2mω
2ˆr2 ,
where ˆr = (ˆ x, ˆ y, ˆ z) and ˆ p = (ˆ p x , ˆ p y , ˆ p z) are respectively the position and
momentum operators of the particle We set a0=
¯
h/(mω).
19.1.1 Recall the energy levels of this system, and its ground state wave
function φ0(r).
19.1.2 We wish to obtain an upper bound on this ground state energy by
the variational method We use a Gaussian trial wave function:
ψ σ (r) = 1
(σ2π) 3/4 exp(−r2/(2σ2)) with σ > 0 (19.1)
The values of a relevant set of useful integrals are given below.
By varying σ, find an upper bound on the ground state energy Compare
the bound with the exact value, and comment on the result
Trang 2194 19 Properties of a Bose–Einstein Condensate
Formulas:
|ψ σ (r)|2dx dy dz = 1
|ψ σ (r)|4dx dy dz = 1
(2π) 3/2
1
σ3
x2|ψ σ (r)|2dx dy dz = σ
2
2
∂ψ σ (r)
∂x
2dx dy dz = 1
2σ2
19.2 Interactions Between Two Confined Particles
We now consider two particles of equal masses m, both placed in the same
harmonic potential We denote the position and momentum operators of the two particles by ˆr1, ˆr2 and ˆp1, ˆp2
19.2.1 In the absence of interactions between the particles, the Hamiltonian
of the system is
ˆ
H = ˆp21
2m+
ˆ
p22
2m+
1
2mω
2ˆr21+1
2mω
2ˆr22 .
(a) What are the energy levels of this Hamiltonian?
(b) What is the ground state wave function Φ0(r1, r2)?
19.2.2 We now suppose that the two particles interact via a potential v( r1−
r2) We assume that, on the scale of a0, this potential is of very short range
and that it is peaked around the origin Therefore, for two functions f (r) and g(r) which vary appreciably only over domains larger than a0, one has
f (r1) g(r2) v(r1− r2) d3r1d3r2 4π¯ h
2a m
f (r) g(r) d3r (19.2)
The quantity a, which is called the scattering length, is a characteristic of the
atomic species under consideration It can be positive (repulsive interaction)
or negative (attractive interaction) One can measure for instance that for sodium atoms (isotope 23Na) a = 3.4 nm, whereas a = −1.5 nm for lithium
atoms (isotope7Li)
(a) Using perturbation theory, calculate to first order in a the shift of the
ground state energy of ˆH caused by the interaction between the two
atoms Comment on the sign of this energy shift
(b) Under what condition on a and a0is this perturbative approach expected
to hold?
Trang 319.3 Energy of a Bose–Einstein Condensate
We now consider N particles confined in the same harmonic trap of angular frequency ω The particles have pairwise interactions through the potential
v(r) defined by (19.2) The Hamiltonian of the system is
ˆ
H =
N
i=1
ˆ
p2i 2m+
1
2mω
2rˆ2i
+ 1 2
N
i=1
N
j=1
j=i v(ˆ r i − ˆr j )
In order to find an (upper) estimate of the ground state energy of the system,
we use the variational method with factorized trial wave functions of the type:
Ψ σ (r1, r2, , r N ) = ψ σ (r1) ψ σ (r2) ψ σ (r N ) ,
where ψ σ (r) is defined in (19.1).
19.3.1 Calculate the expectation values of the kinetic energy, of the potential
energy and of the interaction energy, if the N particle system is in the state
|Ψ σ :
Ek(σ) = Ψ σ |
N
i=1
ˆ
p i2 2m |Ψ σ Ep(σ) = Ψ σ |
N
i=1
1
2mω
2rˆi2|Ψ σ
Eint(σ) = Ψ σ |1
2
N
i=1
N
j=1
j=i v(ˆ r i − ˆr j)|Ψ σ
We set E(σ) = Ψ σ | ˆ H |Ψ σ .
19.3.2 We introduce the dimensionless quantities ˜E(σ) = E(σ)/(N ¯ hω) and
˜
σ = σ/a0 Give the expression of ˜E in terms of ˜ σ Cast the result in the form
˜
E(σ) = 3
4
1
˜
σ2 + ˜σ2
+ η
˜
σ3
and express the quantity η as a function of N , a and a0 In all what follows,
we shall assume that N 1.
19.3.3 For a = 0, recall the ground state energy of ˆ H.
19.4 Condensates with Repulsive Interactions
In this part, we assume that the two-body interaction between the atoms is
repulsive, i.e a > 0.
Trang 4196 19 Properties of a Bose–Einstein Condensate
19.4.1 Draw qualitatively the value of ˜E as a function of ˜ σ Discuss the variation with η of the position of its minimum ˜ Emin
19.4.2 We consider the case η 1 Show that the contribution of the kinetic
energy to ˜E is negligible In that approximation, calculate an approximate
value of ˜Emin
19.4.3 In this variational calculation, how does the energy of the
conden-sate vary with the number of atoms N ? Compare the prediction with the
experimental result shown in Fig 19.1
19.4.4 Figure 19.1 has been obtained with a sodium condensate (mass m =
3.8 × 10 −26 kg) in a harmonic trap of frequency ω/(2π) = 142 Hz.
(a) Calculate a0 and ¯hω for this potential.
(b) Above which value of N does the approximation η 1 hold?
(c) Within the previous model, calculate the value of the sodium atom scat-tering length that can be inferred from the data of Fig 19.1 Compare the
result with the value obtained in scattering experiments a = 3.4 nm Is it
possible a priori to improve the accuracy of the variational method?
Fig 19.1 Energy per atom E/N in a sodium condensate, as a function of the
number of atoms N in the condensate
19.5 Condensates with Attractive Interactions
We now suppose that the scattering length a is negative.
19.5.1 Draw qualitatively ˜E as a function of ˜ σ.
19.5.2 Comment on the approximation (19.2) in the region σ → 0.
Trang 519.5.3 Show that there exists a critical value ηc of |η| above which ˜ E no
longer has a local minimum for a value ˜σ = 0 Calculate the corresponding size σc as a function of a0
19.5.4 In an experiment performed with lithium atoms (m = 1.17 ×
10−26 kg), it has been noticed that the number of atoms in the condensate never exceeds 1200 for a trap of frequency ω/(2π) = 145 Hz How can this
result be explained?
19.6 Solutions
Section 19.1: Particle in a Harmonic Trap
19.1.1 The Hamiltonian of a three-dimensional harmonic oscillator can be
written
ˆ
H = ˆ H x+ ˆH y+ ˆH z ,
where ˆH i represents a one dimensional harmonic oscillator of same frequency
along the axis i = x, y, z We therefore use a basis of eigenfunctions of ˆ H of the form φ(x, y, z) = χ n x (x) χ n y (y) χ n z (z), i.e products of eigenfunctions of
ˆ
H x , ˆ H y , ˆ H z , where χ n (x) is the nth Hermite function The eigenvalues of ˆ H can be written as E n = (n + 3/2)¯ hω, where n = n x + n y + n zis a non-negative integer
The ground state wave function, of energy (3/2)¯ hω, corresponds to n x = n y=
n z = 0, i.e
φ0(r) = 1
(a2π) 3/4exp[−r2/(2a2)]
19.1.2 The trial wave functions ψ σ are normalized In order to obtain an upper bound for the ground-state energy of ˆH, we must calculate E(σ) =
ψ σ | ˆ H |ψ σ and minimize the result with respect to σ Using the formulas
given in the text, one obtains
ψ σ | 2m pˆ2 |ψ σ
= 3 ¯h
2
2m
1
2σ2
ψ σ |12mω2r2|ψ σ
= 3 mω
2
2
σ2
2 and
E(σ) = 3
4¯hω
a2
σ2 +σ
2
a2
This quantity is minimum for σ = a0, and we find Emin(σ) = (3/2) ¯ hω In
this particular case, the upper bound coincides with the exact result This is due to the fact that the set of trial wave functions contains the ground state wave function of ˆH.
Trang 6198 19 Properties of a Bose–Einstein Condensate
Section 19.2: Interactions between Two Confined Particles
19.2.1 (a) The Hamiltonian ˆH can be written as ˆ H = ˆ H1+ ˆH2, where ˆH1
and ˆH2 are respectively the Hamiltonians of particle 1 and particle 2 A basis
of eigenfunctions of ˆH is formed by considering products of eigenfunctions
of ˆH1 (functions of the variable r1) and eigenfunctions of ˆH2 (functions of
the variable r2) The energy eigenvalues are E n = (n + 3)¯ hω, where n is a
non-negative integer
(b) The ground state of ˆH is:
Φ0(r1, r2) = φ0(r1) φ0(r2) = 1
a3π 3/2 exp"
−(r2
1+ r22)/(2a20)#
.
19.2.2 (a) Since the ground state of ˆH is non-degenerate, its shift to first order in a can be written as
∆E = Φ0|˜v|Φ0 =
|Φ0(r1, r2)|2
v(r1− r2) d3r1d3r2
4π¯ h
2a m
|φ0(r)|4d3r = 4π¯ h
2a m
1
(2π) 3/2
1
a3
therefore
∆E
¯
hω =
2
π
a
a0 . For a repulsive interaction (a > 0), there is an increase in the energy of the system Conversely, in the case of an attractive interaction (a < 0), the ground
state energy is lowered
(b) The perturbative approach yields a good approximation provided the
energy shift ∆E is small compared to the level spacing ¯ hω of ˆ H Therefore,
one must have|a| a0, i.e the scattering length must be small compared to the spreading of the ground state wave function
Section 19.3: Energy of a Bose–Einstein Condensate
19.3.1 Using the formulas provided in the text, one obtains:
Ek(σ) = N 3
4
¯
h2
mσ2 Ep(σ) = N 3
4 mω
2σ2
Eint(σ) = N (N − 1)
2
2
π ¯hω
aa2
σ3 Indeed, there are N kinetic energy and potential energy terms, and N (N −1)/2
pairs which contribute to the interaction energy
Trang 719.3.2 With the change of variables introduced in the text, one finds
˜
E(σ) = 3
4
1
˜
σ2 + ˜σ2
+N √ − 1 2π
a
a0
1
˜
σ3
so that
η = N √ − 1 2π
a
a0 .
19.3.3 If the scattering length is zero, there is no interaction between the
particles The ground state of the system is the product of the N functions
φ0(r i ) and the ground state energy is E = (3/2)N ¯ hω.
Section 19.4: Condensates with Repulsive Interactions
19.4.1 Figure 19.2 gives the variation of ˜E(˜ σ) as a function of ˜ σ for increas-ing values of η The value of the function for a given value of ˜ σ increases as η
increases For large ˜σ, the behavior of ˜ E does not depend on η It is dominated
by the potential energy term 3˜σ2/4.
The minimum ˜Emin increases as η increases This minimum corresponds
the point where the potential energy term, which tends to favor small values
of σ, matches the kinetic and interaction energy terms which, on the contrary, favor large sizes σ Since the interactions are repulsive, the size of the system
is larger than in the absence of interactions, and the corresponding energy is also increased
Fig 19.2 Variation of ˜E(˜ σ) with ˜ σ for η = 0, 10, 100, 1000 (from bottom to top)
19.4.2 Let us assume η is much larger than 1 and let us neglect a priori
the kinetic energy term 1/˜ σ2 The function (3/4)˜ σ2+ η/˜ σ3 is minimum for
˜
σmin= (2η) 1/5where its value is
˜
Emin=5
4(2η)
2/5
One can check a posteriori that it is legitimate to neglect the kinetic energy
term 1/˜ σ2 In fact it is always smaller than one of the two other contributions
to ˜E:
Trang 8200 19 Properties of a Bose–Einstein Condensate
• For ˜σ < ˜σmin, one has 1/˜ σ2 η/˜σ3
• For ˜σ > ˜σmin, one has 1/˜ σ2 ˜σ2
19.4.3 For a number of atoms N 1, the energy of the system as calculated
by the variational method is
E
N =
5
4¯hω
2
π N
a
a0
2/5
This variation of E/N as N 2/5is very well reproduced by the data In Fig 19.3
we have plotted a fit of the data with this law One finds E/N αN 2/5with
α = 8.2 × 10 −33 Joule.
Fig 19.3 Fit of the experimental data with an N 2/5law
19.4.4 (a) One finds a0= 1.76µm and ¯hω = 9.4 10 −32 Joule.
(b) Consider the value a = 3.4 nm given in the text The approximation
η 1 will hold if N 1300 This is clearly the case for the data of Fig 19.1.
(c) The coefficient α = 8.2 × 10 −33 Joule found by fitting the data leads to
a = 2.8 nm This value is somewhat lower than the expected value a = 3.4 nm This is due to the fact that the result (19.3), E/(N ¯ hω) 1.142 (Na/a0)2/5, obtained in a variational calculation using simple Gaussian trial functions, does not yield a sufficiently accurate value of the ground state energy With more appropriate trial wave functions, one can obtain, in the mean field
ap-proximation and in the limit η 1: Egs/(N ¯ hω) 1.055 (Na/a0)2/5 The fit
to the data is then in agreement with the experimental value of the scattering length
Trang 9Section 19.5: Condensates with Attractive Interactions
19.5.1 The function ˜E(˜ σ) is represented in Fig 19.4 We notice that it has
a local minimum only for small enough values of η For η < 0, there is always
a minimum at 0, where the function tends to−∞.
-10 -5 0 5
10 ( )
σ
Fig 19.4 Plot of ˜E(˜ σ) for η = 0; η = −0.1; η = −0.27; η = −1 (from top to bottom)
19.5.2 The absolute minimum at σ = 0 corresponds to a highly compressed
atomic cloud For such small sizes, approximation (19.2) for a “short range” potential loses its meaning Physically, one must take into account the forma-tion of molecules and/or atomic aggregates which have not been considered here
19.5.3 The local minimum at ˜σ = 0 disappears when ˜ E(˜ σ) has an inflexion point where the derivative vanishes This happens for a critical value of η
determined by the two conditions:
d ˜E
d˜σ = 0
d2E˜ d˜σ2 = 0
This leads to the system
0 =− 1
˜
σ4 + 1− 2η
˜
σ5
0 = 3
˜
σ4 + 1 + 8η
˜
σ5
from which we obtain
|ηc| = 2
55/4 0.27 σ˜c= 1
51/4 0.67
Trang 10202 19 Properties of a Bose–Einstein Condensate
or σc 0.67 a0 If the local minimum exists, i.e for|η| < |ηc|, one can hope
to obtain a metastable condensate, whose size will be of the order of the minimum found in this variational approach On the other hand, if one starts with a value of |η| which is too large, for instance by trying to gather too
many atoms, the condensate will collapse, and molecules will form
19.5.4 For the given experimental data one finds a0= 3.1µm, and a critical number of atoms:
Nc=√ 2π ηc a0
|a| ∼ 1400 ,
in good agreement with experimental observations
19.7 Comments
The first Bose–Einstein condensate of a dilute atomic gas was observed in Boulder (USA) in 1995 (M.H Anderson, J.R Ensher, M.R Matthews, C.E
Wieman, and E.A Cornell, Science 269, 198 (1995)) with rubidium atoms.
The experimental data shown in this chapter for a sodium condensate come from the results published by M.-O Mewes, M.R Andrews, N.J van
Druten, D.M Kurn, D.S Durfee, and W Ketterle, Phys Rev Lett 77, 416
(1996) The measurement of the energy E/N is done by suddenly switching
off the confining potential and by measuring the resulting ballistic expansion The motion of the atoms in this expansion essentially originates from the conversion of the potential energy of the atoms in the trap into kinetic energy The experimental results on lithium have been reported by C Bradley,
C.A Sackett, and R.G Hulet, Phys Rev Lett 78, 985 (1997).
The Nobel prize 2001 has been awarded to E Cornell, W Ketterle, and
C Wieman for the achievement of Bose-Einstein condensation in dilute gases
of alkali atoms, and for early fundamental studies of the properties of the condensates