Supplementary Information
Tonks-Girardeau gas in an optical lattice – B. Paredes, A. Widera et al.
Supplementary Figure 1 Momentum profiles of the one-dimensional quantum gases
for different axial lattice depths a-l. The experimental data (blue dots) are displayed
together with our theoretical predictions for a fermionized gas (black line), an ideal
Bose gas (green dotted line) and an ideal Fermi gas (yellow dashed line). In order to
emphasize the linear part of the momentum profiles an auxiliary straight line with the
corresponding slope is shown in each plot. For all plots an atomic distribution
characterized by an atom number N0,0 =18 in the central tube is assumed (see
methods). The temperatures for the Tonks and ideal Fermi gas have been obtained in
the same way as for Fig. 3 (see text). The temperatures for the ideal Bose gas have
been derived again assuming conservation of entropy for increasing axial lattice depths.
In this case, the initial temperature at Vax=0 has been obtained using an ideal Bose gas
fit to low momenta for this momentum profile, where the ideal Bose gas is a good
description of the system. In plot d the momentum profiles for the ideal Bose gas
(green lines) are also displayed for different temperatures and particle numbers in the
central tube of kBT/J=3.7, N0,0 =16 (dash-dotted green line) and kBT/J =0.75, N0,0 =20
(dashed green line). The lattice depths and the slopes  of the linear part of the
momentum profiles are summarized in the table below together with the calculated
temperatures for a Tonks gas, an ideal Fermi gas and an ideal Bose gas.
Figure
Lattice depth
Slope
Tonks and Ideal
Fermi Gas (kbT/J)
(Er)
a
b
c
d
e
f
g
h
i
j
k
l
4.6
5.6
6.5
7.4
8.3
9.3
10.2
11.1
12.0
12.9
13.9
18.5
Ideal Bose Gas
(kbT/J)
1.9
1.73
1.58
1.44
1.31
1.17
1.05
0.95
0.84
0.75
0.66
0.59
0.5
0.56
0.62
0.69
0.77
0.87
0.99
1.13
1.29
1.49
1.76
3.93
1.1
1.25
1.41
1.58
1.77
2.02
2.27
2.56
2.90
3.29
3.82
8.0