3 0 5 10 15 20 B GLYPH<LParen1> T GLYPH<RParen1> 0 5 10 Ω GLYPH<LParen1> 10$^{9}$m GLYPH<Minus> $^{1}$GLYPH<RParen1> 0 2 4 6 GLYPH<CapDelta> E GLYPH<LParen1> 10 GLYPH<Minus> $^{23}$J GLYPH<RParen1> For the Fermi gas, the reduced single-particle density matrix can be simplified to Calculations show that the energy gap △ E between the first and the second Bloch bands decreases with the de- crease of M , and when M is large, the energy gap △ E is independent of M . The energy gap between the first and the second Bloch bands as a function of magnetic field B and frequency ω are plotted in Fig. 1, respectively. At B = 0, raised by Girardeau [2], the energy gap of TG gas is the smallest (not 0). The energy gap increases as B and ω increase. As the Boltzman Distribution Law notes, n$_{k}$ n$_{k}$$_{′}$ ∝ exp( − △ E kT ), with △ E = E$_{k}$ − E$_{k}$ ′ , n$_{k}$ and n$_{k}$$_{′}$ are particle numbers on the corresponding states. So the system have a bigger probability in the ground state with large B and ω than the case B = 0 when the to- tal number N = M . That is to say, one could realize ground-stated TG system more simple with large q and ω than q = 0 when the total number N = M . ρ $^{F}$( x, x $^{′}$, t ) = N ∫ ψ F $^{∗}$( x, x$_{2}$, · · · , x$_{N}$ , t ) × ψ $^{F}$( x $^{′}$, x$_{2}$ , · · · , x$_{N}$ , t ) d x$_{2}$ · · · d x$_{N}$ = ∑ α = all states occupyed ϕ$_{α}$ ( x, t ) $^{∗}$ϕ$_{α}$ ( x $^{′}$, t ) . (19) FIG. 1: Plots of the energy gap between the first and the second Bloch bands as a function of magnetic induction B and frequency ω . M = 9, m = 1 . 44 × 10 − $^{25}$kg, µ = 9 . 274 × 10 − $^{24}$J · T − $^{1}$. FIG. 2: Density-plots of ρ ( z, z $^{′}$) of the hard-core TG gas ( a , b , c ) and Fermi gas ( a $^{′}$, b $^{′}$, c $^{′}$), M = 7. Abscissa is z$_{1}$ , Ordi- nate is z$_{2}$ . ( a , a $^{′}$) N = 3; ( b , b $^{′}$) N = 5; ( c , c $^{′}$) N = 7. III. PROPERTIES OF THE TG GAS COMPARED WITH THE FERMI GAS In this section, we further study properties of the 1D N magnetized hard-core Bosons in the external magnetic field (7), and then compared with the properties of 1D N magnetized Fermions in the “same” state which hard- core Bosons occupy. Both the reduced single-particle matrices of hard-core bosons and fermions in the ground state in the exter- nal magnetized field (7) are plotted separately in Fig. 2. The multidimensional integral in equation (18) is evalu- ated numerically by Monte Carlo integration using Math- ematica. Because the relationship between x and z is proportional, for the sake of simplicity, we take z as a variable. From the figure we can see the off-diagonal ele- ments of the matrix of both the two kinds of gases decay quickly as N increases; The bright diagonal stripe of TG gas is thinner than Fermi gas under the same condition. This reflects the condensate properties of Bose gas; And The reduced single-particle density matrix with nor- malization ∫ ρ ( x, x ) d x = N is given by A. Reduced single-particle density matrix and Single-particle density ρ $^{B}$( x, x $^{′}$, t ) ≡ N ∫ ψ B $^{∗}$( x, x$_{2}$, · · · , x$_{N}$ , t ) × ψ $^{B}$( x $^{′}$, x$_{2}$ , · · · , x$_{N}$ , t ) d x$_{2}$ · · · d x$_{N}$ . (18)