Higher-order local and non-local correlations for 1D strongly interacting Bose gas
The correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the M -body local correlation functions, g _M , of the one-dimensional...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
IOP Publishing
2016-01-01
|
| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/18/5/055014 |
| _version_ | 1827873578406641664 |
|---|---|
| author | EJKP Nandani Rudolf A Römer Shina Tan Xi-Wen Guan |
| author_facet | EJKP Nandani Rudolf A Römer Shina Tan Xi-Wen Guan |
| author_sort | EJKP Nandani |
| collection | DOAJ |
| description | The correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the M -body local correlation functions, g _M , of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb–Liniger model using the analytical method proposed by Gangardt and Shlyapnikov (2003 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.90.010401 90 http://dx.doi.org/10.1103/PhysRevLett.90.010401 ; 2003 New J. Phys. http://dx.doi.org/10.1088/1367-2630/5/1/379 5 http://dx.doi.org/10.1088/1367-2630/5/1/379 ). In the strong repulsion regime the 1D Bose gas at low temperatures is equivalent to a gas of ideal particles obeying the non-mutual generalized exclusion statistics with a statistical parameter $\alpha =1-2/\gamma $ , i.e. the quasimomenta of N strongly interacting bosons map to the momenta of N free fermions via ${k}_{i}\approx \alpha {k}_{i}^{F}$ with $i=1,\ldots ,N$ . Here γ is the dimensionless interaction strength within the Lieb–Liniger model. We rigorously prove that such a statistical parameter α solely determines the sub-leading order contribution to the M -body local correlation function of the gas at strong but finite interaction strengths. We explicitly calculate the correlation functions g _M in terms of γ and α at zero, low, and intermediate temperatures. For M = 2 and 3 our results reproduce the known expressions for g _2 and g _3 with sub-leading terms (see for instance (Vadim et al 2006 Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.73.051604 73 http://dx.doi.org/10.1103/PhysRevA.73.051604 ; Kormos et al 2009 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.103.210404 103 http://dx.doi.org/10.1103/PhysRevLett.103.210404 ; Wang et al 2013 Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.87.043634 87 http://dx.doi.org/10.1103/PhysRevA.87.043634 ). We also express the leading order of the short distance non-local correlation functions $\langle {{\rm{\Psi }}}^{\dagger }({x}_{1})\cdots {{\rm{\Psi }}}^{\dagger }({x}_{M}){\rm{\Psi }}({y}_{M})\cdots {\rm{\Psi }}({y}_{1})\rangle $ of the strongly repulsive Bose gas in terms of the wave function of M bosons at zero collision energy and zero total momentum. Here ${\rm{\Psi }}(x)$ is the boson annihilation operator. These general formulas of the higher-order local and non-local correlation functions of the 1D Bose gas provide new insights into the many-body physics. |
| first_indexed | 2024-03-12T16:39:05Z |
| format | Article |
| id | doaj.art-07f4eea94d934243959770348f3a5598 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:39:05Z |
| publishDate | 2016-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-07f4eea94d934243959770348f3a55982023-08-08T14:35:06ZengIOP PublishingNew Journal of Physics1367-26302016-01-0118505501410.1088/1367-2630/18/5/055014Higher-order local and non-local correlations for 1D strongly interacting Bose gasEJKP Nandani0Rudolf A Römer1Shina Tan2Xi-Wen Guan3State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics , Chinese Academy of Sciences, Wuhan 430071, People's Republic of China; University of Chinese Academy of Sciences , Beijing 100049, People's Republic of China; Department of Mathematics, University of Ruhuna , Matara, 81000, Sri LankaDepartment of Physics and Centre for Scientific Computing, University of Warwick , Coventry CV4 7AL, UKSchool of Physics, Georgia Institute of Technology , Atlanta, GA, 30332, USA; Center for Cold Atom Physics , Chinese Academy of Sciences, Wuhan 430071, People's Republic of ChinaState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics , Chinese Academy of Sciences, Wuhan 430071, People's Republic of China; Department of Theoretical Physics , Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, AustraliaThe correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the M -body local correlation functions, g _M , of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb–Liniger model using the analytical method proposed by Gangardt and Shlyapnikov (2003 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.90.010401 90 http://dx.doi.org/10.1103/PhysRevLett.90.010401 ; 2003 New J. Phys. http://dx.doi.org/10.1088/1367-2630/5/1/379 5 http://dx.doi.org/10.1088/1367-2630/5/1/379 ). In the strong repulsion regime the 1D Bose gas at low temperatures is equivalent to a gas of ideal particles obeying the non-mutual generalized exclusion statistics with a statistical parameter $\alpha =1-2/\gamma $ , i.e. the quasimomenta of N strongly interacting bosons map to the momenta of N free fermions via ${k}_{i}\approx \alpha {k}_{i}^{F}$ with $i=1,\ldots ,N$ . Here γ is the dimensionless interaction strength within the Lieb–Liniger model. We rigorously prove that such a statistical parameter α solely determines the sub-leading order contribution to the M -body local correlation function of the gas at strong but finite interaction strengths. We explicitly calculate the correlation functions g _M in terms of γ and α at zero, low, and intermediate temperatures. For M = 2 and 3 our results reproduce the known expressions for g _2 and g _3 with sub-leading terms (see for instance (Vadim et al 2006 Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.73.051604 73 http://dx.doi.org/10.1103/PhysRevA.73.051604 ; Kormos et al 2009 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.103.210404 103 http://dx.doi.org/10.1103/PhysRevLett.103.210404 ; Wang et al 2013 Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.87.043634 87 http://dx.doi.org/10.1103/PhysRevA.87.043634 ). We also express the leading order of the short distance non-local correlation functions $\langle {{\rm{\Psi }}}^{\dagger }({x}_{1})\cdots {{\rm{\Psi }}}^{\dagger }({x}_{M}){\rm{\Psi }}({y}_{M})\cdots {\rm{\Psi }}({y}_{1})\rangle $ of the strongly repulsive Bose gas in terms of the wave function of M bosons at zero collision energy and zero total momentum. Here ${\rm{\Psi }}(x)$ is the boson annihilation operator. These general formulas of the higher-order local and non-local correlation functions of the 1D Bose gas provide new insights into the many-body physics.https://doi.org/10.1088/1367-2630/18/5/055014high order correlation functionsgeneralized exclusion statisticsFermi distributionBethe ansatz weave functions |
| spellingShingle | EJKP Nandani Rudolf A Römer Shina Tan Xi-Wen Guan Higher-order local and non-local correlations for 1D strongly interacting Bose gas New Journal of Physics high order correlation functions generalized exclusion statistics Fermi distribution Bethe ansatz weave functions |
| title | Higher-order local and non-local correlations for 1D strongly interacting Bose gas |
| title_full | Higher-order local and non-local correlations for 1D strongly interacting Bose gas |
| title_fullStr | Higher-order local and non-local correlations for 1D strongly interacting Bose gas |
| title_full_unstemmed | Higher-order local and non-local correlations for 1D strongly interacting Bose gas |
| title_short | Higher-order local and non-local correlations for 1D strongly interacting Bose gas |
| title_sort | higher order local and non local correlations for 1d strongly interacting bose gas |
| topic | high order correlation functions generalized exclusion statistics Fermi distribution Bethe ansatz weave functions |
| url | https://doi.org/10.1088/1367-2630/18/5/055014 |
| work_keys_str_mv | AT ejkpnandani higherorderlocalandnonlocalcorrelationsfor1dstronglyinteractingbosegas AT rudolfaromer higherorderlocalandnonlocalcorrelationsfor1dstronglyinteractingbosegas AT shinatan higherorderlocalandnonlocalcorrelationsfor1dstronglyinteractingbosegas AT xiwenguan higherorderlocalandnonlocalcorrelationsfor1dstronglyinteractingbosegas |