Bose 2.2 User Manual download pdf (Page 32)

energy

ext

(

) =

mω

2. In the other limit of small and decreasing radii, Heisenberg’s

uncertainty relation

~/R

leads to a divergence in the kinetic energy, given by

kin

) =

). This energy divergence results in the so-called quantum

pressure, which stabilizes the non-interacting condensate. The energy in the system is

minimal when the contributions from the potential energy and the kinetic energy are the

same, resulting in the condensate radius

~/(mω

)

. Finally, the chemical

potential

in a non-interacting condensate is given by the zero point energy of the

three-dimensional quantum-mechanical harmonic oscillator,

= 3

~ω

2, and is therefore

independent of the atom number.

2.4.2 GPE with Dominant Contact Interactions

The contact interactions contribute with the mean-ﬁeld potential Φ

contact

(

) =

g n

(

)

(see Eq.

(2.16a)

) to the stationary GPE. While in a non-interacting BEC the quantum

pressure stabilizes the system at small radii, it is now the interactions that take over this

role, if they are suﬃciently strong. In this case, the kinetic term in the GPE may be

neglected, and we obtain the GPE in Thomas-Fermi (TF) approximation

µψ(r)

= [V

ext

(r) + g n(r)] ψ(r). (2.21)

Inserting the harmonic trapping potential

ext

(

), given by Eq.

(2.18)

, into Eq.

(2.21)

yields the parabolic density of the BEC:

n(r) = N |ψ(r)|

µ − V

ext

(r)

= n

· max

(

1 −

−

, 0

)

, (2.22)

with

= 15

πR

) the density at the center of the condensate, and with

x,y,z

the TF-radii in the respective directions. The central density may also be expressed in

terms of the mean radius

def

(

)

1/3

, which in the case of a spherically symmetric

trap is given by

= 15

1/5

(

Na/a

)

1/5

. We see that the calculated mean radius of a

contact interacting BEC is only signiﬁcantly larger than the harmonic oscillator length

if the condition

Na/a



1 is fulﬁlled. Therefore, this condition deﬁnes the regime in

which the TF-approximation is valid. For a typical set of parameters in a chromium BEC,

= 20

000

, a

= 100

, ω

= 2

π ·

500

Hz}

, we obtain

Na/a

170 and thus the

system is typically well inside the Thomas-Fermi regime.

In the case of a non-spherically symmetric trap, the harmonic oscillator length

has to be replaced by

the mean harmonic oscillator length

¯a

~/(m¯ω)

, with the mean trap frequency

¯ω

(ω

)

1/3

as we show in appendix A.2. Note, that these formulae hold only for moderate trap frequencies with

~ω

x,y,z

 µ. We discuss the case of an highly oblate trap with ~ω

 µ  ~ω

x,y

in section 4.3.2.

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