https://doi.org/10.1140/epjd/s10053-022-00574-5
Regular Article – Cold Matter and Quantum Gases
n-body correlation of Tonks–Girardeau gas
1
Institute of Theoretical Physics and Department of Physics, University of Science and Technology Beijing, 100083, Beijing, People’s Republic of China
2
Institute of Theoretical Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, 030006, Taiyuan, People’s Republic of China
Received:
7
October
2022
Accepted:
30
November
2022
Published online:
7
December
2022
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 
-order Toeplitz determinant, whose element is the integral dependent on 2 sign functions and can be computed analytically. By reducing the integral on domain ![$$[0,2\pi ]$$](/articles/epjd/abs/2022/12/10053_2022_Article_574/10053_2022_Article_574_tex_eq3.png)
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.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
