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

12.4 Tight binding regime 167
3D thermodynamic limit
An analytic expression for (12.57) can be found in the thermodynamic limit employing the
continuum approximation
px,py,q
→ L3
(2π) 3
1
¯h 2
dpx
The quantum depletion in the tight binding regime then becomes
∆Ntot
=
Ntot
1 L
Ntot
3
(2π) 3
1
2 ×
¯h
⎡
π/d ∞ 1
× dq 2πp⊥dp⊥ ⎣
−π/d 0
2 p
(
2 ⊥
2κ−12m
π/d
dpy dq . (12.58)
−π/d
p 2 ⊥
2κ −1 2m
+1/2+ δ
κ −1 sin 2 ( qd
2 )
δ + κ−1 sin2 ( qd
2 ))(
p2 ⊥
2κ−12m The integral can be solved analytically and the final result reads
∆Ntot
Ntot
= π 1
4 nd3 κ −1
ER
+ δ
κ −1 sin 2 ( qd
2 )+1)
δ
G
κ−1
, (12.60)
where the function G(b), givenby
G(b) = 1
2 −
√
b b 1
+ −
π 2 π arctan(√b)(1 + b)
is depicted in Fig.12.1.
(12.61)
From Eq.(12.60), one recovers the result (12.49) in the limit δ/κ−1 →∞, reflecting the
fact that the case of a shallow lattice is approached by increasing δ. Yet, this limit is only
of academic interest, since in the tight binding regime where (12.60) applies, the ratio δ/κ−1 becomes large only if interactions are vanishingly small. For example, for gn =0.02ER and
s = 10 one still finds δ/κ−1 ≈ 1. Thus, in the tight binding regime the ratio δ/κ−1 is
usually small. It goes to zero as s →∞. Moreover, the incompressibility κ−1 approaches the
expression κ−1 =˜gn. Hence, in the large-s limit the depletion (12.60) converges to
∆Ntot
Ntot
= ã
, (12.62)
d
where, as previously, we have defined an effective scattering length ã through the relation
˜g =4π¯h 2 ã/m. Note that the dependence on the interaction strength is stronger in this case
than in a shallow lattice (see Eq.(12.49)), since the depletion scales like ã rather than ã 3/2 .
The result (12.62) coincides with the thermodynamic limit quantum depletion (12.33) of a
disc shaped uniform system with strong axial harmonic confinement, freezing the wavefunction
to a gaussian of width σ. The link between the result for the disc (12.33) and the lattice result
(12.62) is established by noting that within the gaussian approximation to the Wannier function
in a deep lattice we find ã = ad/ √ 2πσ (see section 6.2, in particular Eq.(6.54)). Thus, the
⎤
− 1⎦
.
(12.59)