n-body correlation of Tonks–Girardeau gas

Abstract

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

Graphic abstract

Appendix A

The coefficients of the Toeplitz matrix element of the R3BDM:

$$\begin{aligned} c_0= & {} 1, \\ c_1= & {} e^{ix}+e^{iy}+e^{iz}+e^{ix^{\prime }}\\{} & {} +e^{iy^{\prime }}+e^{iz^{\prime }}, \\ c_2= & {} e^{i(x+y)}+e^{i(x+z)}+e^{i(x+x^{\prime })}\\{} & {} +e^{i(x+y^{\prime })}+e^{i(x+z^{\prime })}+e^{i(y+z)}\\{} & {} +e^{i(y+x^{\prime })}+e^{i(y+y^{\prime })}+e^{i(y+z^{\prime })} \\{} & {} +e^{i(z+x^{\prime })}+e^{i(z+y^{\prime })}\\{} & {} +e^{i(z+z^{\prime })}+e^{i(x^{\prime }+y^{\prime })}\\{} & {} +e^{i(x^{\prime }+z^{\prime })}+e^{i(y^{\prime }+z^{\prime })}, \\ c_3= & {} e^{i(x+y+z)}+e^{i(x+y+x^{\prime })}+e^{i(x+y+y^{\prime })}\\{} & {} +e^{i(x+y+z^{\prime })}\\{} & {} +e^{i(x+z+x^{\prime })}+e^{i(x+z+y^{\prime })}\\{} & {} +e^{i(x+z+z^{\prime })}+e^{i(x+x^{\prime }+y^{\prime })} \\{} & {} +e^{i(x+x^{\prime }+z^{\prime })}+e^{i(x+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(y+z+x^{\prime })}+e^{i(y+z+y^{\prime })}+e^{i(y+z+z^{\prime })}\\{} & {} +e^{i(y+x^{\prime }+y^{\prime })}+e^{i(y+x^{\prime }+z^{\prime })}+e^{i(y+y^{\prime }+z^{\prime })} \\{} & {} +e^{i(z+x^{\prime }+y^{\prime })}+e^{i(z+x^{\prime }+z^{\prime })}+e^{i(z+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(x^{\prime }+y^{\prime }+z^{\prime })}, \\ c_4= & {} e^{i(x+y+z+x^{\prime })}+e^{i(x+y+z+y^{\prime })}+e^{i(x+y+z+z^{\prime })}\\{} & {} +e^{i(x+y+x^{\prime }+y^{\prime })}+e^{i(x+y+x^{\prime }+z^{\prime })}\\{} & {} +e^{i(x+y+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(x+z+x^{\prime }+y^{\prime })}+e^{i(x+z+x^{\prime }+z^{\prime })} \\{} & {} +e^{i(x+z+y^{\prime }+z^{\prime })}+e^{i(x+x^{\prime }+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(y+z+x^{\prime }+y^{\prime })}+e^{i(y+z+x^{\prime }+z^{\prime })} \\{} & {} +e^{i(y+z+y^{\prime }+z^{\prime })}+e^{i(y+x^{\prime }+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(z+x^{\prime }+y^{\prime }+z^{\prime })}, \\ c_5= & {} e^{i(x+y+z+x^{\prime }+y^{\prime })}+e^{i(x+y+z+x^{\prime }+z^{\prime })}\\{} & {} +e^{i(x+y+z+y^{\prime }+z^{\prime })}+e^{i(x+y+x^{\prime }+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(x+z+x^{\prime }+y^{\prime }+z^{\prime })}\\{} & {} +e^{i(y+z+x^{\prime }+y^{\prime }+z^{\prime })}, \\ c_6= & {} e^{i(x+y+z+x^{\prime }+y^{\prime }+z^{\prime })}. \end{aligned}$$