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Another possible experimental signature of the super-Tonks regime can be provided by
the study of collective modes. To this aim, we calculate the frequency of the lowest
compressional mode of a system of particles in a harmonic potential
. We make use of LDA (Sec. 1.6)
which allows us to calculate the chemical potential of the inhomogeneous system
and the density profile from the local equilibrium equation
, and the normalization condition
, where
is the size of
cloud. For densities smaller than the critical density for cluster formation,
is the equation of state of the homogeneous system derived from the fit to
the VMC energies (Fig. 6.1). From the knowledge of the density profile
one can obtain the mean square radius of the cloud
and thus, making use of the result
[MS02]
|
(6.3) |
one can calculate the frequency of the lowest breathing mode. Within LDA,
the result will depend only on the dimensionless parameter
, where
is the harmonic oscillator length. For , i.e. in the case of the LL Hamiltonian, the frequency of the lowest compressional
mode increases from
in the weak-coupling mean-field regime
(
) to
in the strong-coupling TG regime
(
). The results for in the super-Tonks regime are
shown in Fig. 6.4 as a function of the coupling strength. In the regime
, where the HR model is appropriate, we can calculate
analytically the first correction to the frequency of a TG gas (refer to
Table 1.1).
One finds the result
. Fig. 6.4 shows that this expansion accurately
describes the frequency of the breathing mode when
, for
larger values of the coupling strength the frequency reaches a maximum and drops to
zero at
. The observation of a breathing mode with a
frequency larger than would be a clear signature of the super-Tonks
regime.
Figure 6.4:
Square of the lowest breathing mode frequency, ,
as a function of the coupling strength
for the LL Hamiltonian
() and in the super-Tonks regime (). The dashed line is obtained
from the HR expansion (see Table 1.1).
|
Next: Conclusions
Up: Beyond Tonks-Girardeau: super-Tonks gas
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G.E. Astrakharchik
15-th of December 2004