n-body correlation of Tonks–Girardeau gas

For the well-known exponential complexity it is a giant challenge to calculate correlation function for general many-body wave function. We investigate ground state $n$th-order functions of Tonks-Girardeau (TG) gases. Basing on wavefunction free fermions and Bose-Fermi mapping method we obtain exact TG Utilizing properties Vandermonde determinant Toeplitz matrix, formulated as $(N-n)$-order determinant, whose element integral dependent 2$(N-n)$ sign can be computed analytically. By reducing domain $[0,2\pi]$ into summation several independent domains, explicit form matrix ultimately. As applications deduce concise formula reduced two-body density discuss its properties. The corresponding natural orbitals their occupation distribution are plotted. Furthermore, give three-body It shown that in successive second measurements, atoms appear regions where populate with maximum probability first measurement.