Bose-Einstein distribution
Consider the three-dimensional (3D) Gross-Pitaevskii equation (GPE) for modeling BoseEinstein condensate (BEC) with a harmonic plus optical lattice potential:
ih¯ ??(x, t)
?t = –
h¯2
2m
?2?(x, t) + [Vho(x) + Vopt(x)] ?(x, t) + g|?(x, t)|2?(x, t), (1)
where x = (x, y, z)T is the spatial coordinate vector, t is time, ? = ?(x, t) is the macroscopic wave function, m is the atomic mass, ¯ h is the planck constant, g = 4p¯ hm2as describes
the interaction between atoms in the condensate with as the s-wave scattering length
(positive for repulsive interaction and negative for attractive interaction). Vho(x) and
Vopt(x) are external harmonic and optical lattice trapping potentials, respectively and
given as
Vho(x) = m
2 ?x2x2 + ?y2y2 + ?z2z2 ,
with ?x, ?y and ?z trap frequencies in x-, y- and z-direction, respectively, and
Vopt(x) = S0 E0 sin2 2?px 0 + sin2 2?py 0 + sin2 2?pz 0 ,
with ?0 the wavelength of the laser light creating the lattice, and E0 = 2m? p2¯ h22
0
the so-called
recoil energy and S0 a dimensionless constant. The macroscopic wave function ? at time
t = 0 is normalized at
ZR3 |?(x, 0)|2 dxdydz = N, (2)
with N the total number of atoms in the condensate.
1. Dimensionless the above GPE (1) by choosing the dimensionless time unit ts = ?1x
and xs = q m? ¯ h x such that the dimensionless macroscopic wave function is normalized to
unity. Using the two typical experimental parameters in the bottom, find the dimensionless time unit ts, space unit xs and energy unit Es = t¯ hs = mx ¯ h22 s and express the
dimensionless constants in terms of the total number of particle N in the condensate.
2. When N ≫ 1, re-scale the above dimensionless GPE into the semiclassical form.
3. Dimensionless the above GPE (1) by choosing the dimensionless time unit ts =
m?2
0
4p2¯ h and xs = 2?p0 such that the dimensionless macroscopic wave function is normalized
to unity. Again, using the two typical experimental parameters in the bottom, find the
dimensionless time unit ts, space unit xs and energy unit Es = t¯ hs = mx ¯ h22 s and express the
dimensionless constants in terms of the total number of particle N in the condensate.
4. Based on the dimensionless GPE in part 1, prove that the normalization and
energy are conserved.
5. Reduce the 3D dimensionless GPE to a 2D GPE when ?
y ˜ ?x, ?z ≫ ?x and 1D
GPE when ?
y ≫ ?x, ?z ≫ ?x. Summarize the 1D, 2D and 3D GPE in a general form.
1
6. Find the approximate ground state solution of the dimensionless GPE for strong
repulsive interaction (Thomas-Fermi approximation).
7. Design an imaginary time method to compute the ground state solution and
develop a code to compute the ground state in 1D and 2D for different parameter regimes.
Plot the ground state and the corresponding energy and chemical potential. Compare your
numerical results with the Thomas-Fermi approximation. What conclusions can you get?
8. Design a time-splitting spectral method to compute the dynamics of GPE and
develop a code for the the method in 1D and 2D. Use your code to study the dynamics
of the GPE, with different interaction parameters and with different trapping potentials.
What conclusion can you get?
9. In 1D, if we choose the initial data as
?(x, 0) = fg(x – x0) eiax, -8 < x < 8,
with fg(x) the ground state of the original GPE and x0 and a given constants, study
asymptotically and numerically the dynamics of the center of mass and condensate width
defined as
x(t) := hxi(t) = Z-8 8 x|?(x, t)|2 dx, t = 0,
s(t) := qd(t) := sZ-8 8 x2|?(x, t)|2 dx, t = 0.
The following are two sets of physical parameters used in typical BEC experiments:
• BEC experiment with 87Rb
h¯ = 1.05 × 10-34[J s], m = 1.443 × 10-25[kg],
?x = 20 × 2p[1/s], ?y = ?z = 10?x;
as = 5.1[nm], ?0 = 911.8[nm], S0 = 2.
• BEC experiment with 23Na
h¯ = 1.05 × 10-34[J s], m = 3.816 × 10-26[kg],
?x = 20 × 2p[1/s], ?y = ?z = 10?x;
as = 2.6[nm], ?0 = 911.8[nm], S0 = 2.
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