Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

5.1 Density profile, energy and chemical potential 49
Typical values of gn/ER in current experiments can be estimated by taking the density at
the center of the harmonic trap in absence of the lattice and evaluating ER =¯h 2 π2 /2md2 for
the lattice period d used in the respective experiment. In this way, one finds values ranging
from gn =0.02ER to gn =1.1ER in the experiments [79, 73, 71, 87, 66].
For each choice of parameter s and gn/ER, we obtain a different groundstate wave function
ϕ(z) and chemical potential
d/2
µ = ϕ
−d/2
∗
(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin + gnd|ϕ(z)|
∂z2 d
2
ϕ(z)dz (5.7)
and groundstate energy per particle
d/2
ε = ϕ
−d/2
∗
(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin +
∂z2 d
gnd
2 |ϕ(z)|2
ϕ(z)dz . (5.8)
Calculating these quantities for different choices of s and gn/ER allows us to elucidate the
role played by mean field interaction: Under what conditions does it play an important role ?
How does it alter the effects produced by the lattice?
Let us first discuss some solutions for the groundstate wavefunction ϕ(z) or, equivalently,
the density nd |ϕ(z)| 2 . In Fig. 5.1, we report the results obtained at different values of lattice
depth and interaction parameter. When s is increased for fixed gn/ER, the density becomes
more and more modulated by the optical potential. Instead, when s is kept fixed while gn/ER
is increased, the modulation of the density is reduced. Repulsive interactions screen off the
lattice as mentioned above. In fact, at low lattice depth explicit formulas for the effective
potential can be derived [103] and are found to be in agreement with experimental results
[67].
Let us now proceed to the results obtained for the chemical potential and the energy per
particle. In Fig. 5.2, we report the results obtained as a function of lattice depth s for different
values of the interaction parameter gn/ER.
Chemical potential and energy per particle coincide only in the absence of interactions. In
general, they are linked by the relation
µ =
∂ (nε)
∂n
∂ε
= ε + gn . (5.9)
∂ (gn)
The second term is necessarily positive for repulsive interactions and hence the chemical potential
is always larger than the energy per particle (see Fig.5.2).
When plotting the relative difference (µ−ε)/ε as a function of gn at fixed high s, weobserve
that the curve is proportional to gn for sufficiently small gn (see Fig.5.3). The deviation from
the linear behavior (see dotted line) for larger gn is due to the density dependence of ϕ. For
gn small enough, the derivative ∂ε/∂ (gn) is essentially given by its gn =0-limit
∂ε d d/2
≈ |ϕgn=0(z)|
∂ (gn) 2 −d/2
4 dz , (5.10)
where ϕgn=0(z) is the groundstate solution in absence of interactions (gn =0). In the next
section, we will show that the quantity (5.10) is closely related to the compressibility of the
system.