n-body Correlation of Tonks-Girardeau Gas Yajiang HaoYaling Zhang and Yiwang Liu Institute of Theoretical Physics and Department of Physics

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n-body Correlation of Tonks-Girardeau Gas

Yajiang Hao,∗Yaling Zhang, and Yiwang Liu

Institute of Theoretical Physics and Department of Physics,

University of Science and Technology Beijing, Beijing 100083, China

Li Wang

Institute of Theoretical Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices,

Collaborative Innovation Center of Extreme Optics,

Shanxi University, Taiyuan 030006, P. R. China

(Dated: October 10, 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 (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 [0,2π]

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.

I. INTRODUCTION

Since Glauber generalized the ﬁrst-order correlation

to higher-order correlation in the optical research [1], n-

body correlation has gradually become one of the founda-

tional properties of many-body quantum systems, which

is extremely important not only to the deﬁnition of co-

herence but also to the characterization of the properties

of quantum matter including the quantum phases and

topological states. Quantum correlation is also the cru-

cial resource for quantum information and computation

[2]. The study of correlation has become the driving force

for the development of many research ﬁelds. For exam-

ple, the famous Hanbury Brown and Twiss experiment

[3–5] has ever kept pushing the development of quantum

optics. The calculation of correlation functions played a

pivotal role in the theoretical study of many-body quan-

tum system. It is helpful to reveal and facilitate the

understanding of those exotic quantum eﬀects. For most

systems the analytical calculation of correlation functions

remains intractable, particularly the higher-order corre-

lation functions, although the high-order correlation is

required by rigorous description of the coherence. Exper-

imentally, with the development of cold atom technique

including the single-atom-sensitive detection techniques,

the third-order [6, 7], the fourth order [8] , and even sixth-

order [9, 10] correlation of ultracold Bose atoms is mea-

surable. With the development of the quantum gas mi-

∗Electronic address: haoyj@ustb.edu.cn

croscopes [7, 11, 12], the momentum microscope [13] and

quantum ghost imaging technique [14] the measurement

of the high-order correlation of quantum many-body sys-

tem has become feasible.

To develop the techniques that measure and control the

quantum phase as well as coherence is one of the most

important goals of cold atom research, both of which are

closely related with the quantum correlation. The ap-

plication of optical lattice and Feshbach resonance tech-

nique extremely improves the controllability of the di-

mension [15–17] and interacting regime [16–18] of cold

atom system. The great progress in experiment has made

it a popular platform to investigate the basic problems

of quantum many-body system. One of the remarkable

achievements of the above techniques is the experimen-

tal realization of the strong correlated Tonks-Girardeau

(TG) gas [19, 20], a one-dimensional neutral Bose atom

gas with inﬁnitely strong repulsive interaction. TG gas

has now constituted one important portion of the low

dimensional quantum gas research [21–23].

Theoretically the TG gas was ﬁrst studied as a toy

model [19, 20], its eigen wavefunction can be exactly ob-

tained based on the many-particle wavefunction of polar-

ized fermions utilizing the Bose-Fermi mapping method.

Since its experimental realization the static and dynam-

ical structure factor[24, 25], universal contact [26], noise

correlation [27–29], full counting statistics [30] have been

studied. Although the one-body, two-body and local

three-body correlation functions [31–33], momentum dis-

tribution [34–37] and its dynamics [38–40] have ever been

studied, so far the explicit formula for the nth-order cor-

relation function of TG gases is still lacking.

The motivation of the present paper is to analytically

arXiv:2210.03578v1 [cond-mat.quant-gas] 7 Oct 2022

derive a concise formula of the nth-order correlation func-

tion for TG gas. It is important for the accurate under-

standing of the many-body quantum correlation in the

future reachable experiments. The paper is organized as

follows. In Sec. II, we give a brief review of Bose-Fermi

mapping method and introduce the ground state wave-

function for TG gases. In Sec. III, we present the general

explicit expression of the n-body correlation function. In

Sec. IV, the reduced two-body density matrix and the

reduced three-body density matrix are investigated as

examples. A brief summary is given in Sec. V.

II. GROUND STATE WAVEFUNCTION

As the temperature is low enough and the transver-

sal conﬁnement is strong enough to froze the motion

of atoms in transversal direction, the quantum gas can

only distribute in the longitudinal directions. The system

composed of Ncold atoms with mass mcan be described

by the Hamiltonian

j=1 −~2

∂2

∂x2

+g1DX

j<l

δ(xj−xl),(1)

where atoms interact by δ-potential and the eﬀective in-

teraction strength g1Dcan be tuned to inﬁnite strong

repulsive interaction by Feshbach resonance and conﬁn-

ment induced resonance such that the cold atoms are in

the TG regime. In this situation, the many-body wave-

function Ψ(x1,··· , xN) satisﬁes not only the Schr¨odinger

equation HΨ(x1,··· , xN) = EΨ(x1,··· , xN) but also

the boundary condition Ψ(x1,··· , xN) = 0 for xj=xl.

This condition is equivalent to the constraint on the

wavefunction of identical fermions for the Pauli exclusive

principle. Therefore we can construct the exact wave-

function of TG gas basing on the wavefunction of nonin-

teracting polarized fermions. In the present paper we will

assume atoms distribute in a circular ring with length L.

With the periodical boundary condition the single parti-

cle eigenfunction is formulated as φj(x) = 1

√Leikjxwith

kj= 2jπ/L (j= 0,±1,±2,···). The wavefunction of N

fermions is the Slater determinant of φj(xn)

ΨF(x1,··· , xN) = (LNN!)−1/2det[eiklxj]j=1,...,N

l=1,...,N (2)

with kl= 2π/L(l−(N+ 1)/2). Here Nis assumed to be

odd for simplicity. Using the properties of Vandermonde

determinant formula

det[(xk)j−1]j,k=1,··· ,N =Y

1≤j<k≤N

(xj−xk),

we reformulate the determinant form of Fermi wavefunc-

tion as the simpliﬁed product form

ΨF(x1,··· , xN) = (LNN!)−1/2Qj=1,...,N e−i(N−1)πxj/L

×Q1≤j<l≤N(ei2πxj/L −ei2πxl/L).

This is important to reduce the computation complex-

ity in the later calculation of correlation functions. With

the reduction the computation eﬃciency reduces to N-

scaling from the N2-scaling. In order to obtain the ex-

change symmetrical wavefunction satisﬁed by the identi-

cal bosons the Bose-Fermi mapping method can be uti-

lized with the sign function (x), which is 0, +1, or -1

for x=0, >0 or <0, respectively. The ground state

wavefunction of TG gas is expressed as [19, 20]

Ψ (x1, x2,··· , xN)=(LNN!)−1/2Qj=1,...,N e−i(N−1)πxj/L

×Q1≤j<l≤N(xj−xl)ei2πxj/L −ei2πxl/L.(3)

For simplicity we take the length unit 2π/L in the latter

part so the length of circular ring is 2π. The original

notation will be preserved. Therefore the wavefunction

of ground state for TG gases is

Ψ (x1, x2,··· , xN) (4)

= (√2π)−N(N!)−1/2Y

j=1,...,N

e−i(N−1)xj/2

×Y

1≤j<l≤N

(xj−xl)eixj−eixl.

III. n-BODY CORRELATION FUNCTION

The n-body correlation function is deﬁned as (in unit

of (2π

L)n)

ρn(xN−n+1,··· , xN;x0

N−n+1,··· , x0

=N!

(N−n)! R2π

0dx1···R2π

0dxN−nΨ∗(x1,··· , xN)

×Ψ(x1,··· , xN−n, x0

N−n+1,··· , x0

N),(5)

which is the integral over N−nvariables (x1,··· , xN−n).

The multiple integral is usually diﬃcult to analytically

calculate and have to resort to numerical method such as

Monte Carlo method. In the case of TG gases, the inte-

gral can be calculated analytically and we will obtain its

explicit formula in this section. Substituting the wave-

function Eq. (4) into Eq. (5) and using the following

result for Toeplitz matrix [34, 35, 41–43]

N!QN

l=1 R2π

0dxlg(xl)Q1≤j<k≤N|eixj−eixk|2(6)

= det[R2π

0dtg(t) exp(it(j−k))]j,k=1,...,N ,

the above (N−n)-fold integral transforms into a (N−n)-

order Toeplitz determinant

ρn(xN−n+1,··· , xN;x0

N−n+1,··· , x0

(2π)NY

j=N−n+1,...,N

ei(N−1)(xj−x0

j)/2

N−n+1≤j<l≤N

(xj−xl)(e−ixj−e−ixl)

×(x0

j−x0

l)(eix0

j−eix0

×det[bj−k(xN−n+1,··· , xN;x0

N−n+1,··· , x0

N)]j,k=1,...,N −n

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n-bodyCorrelationofTonks-GirardeauGasYajiangHao,YalingZhang,andYiwangLiuInstituteofTheoreticalPhysicsandDepartmentofPhysics,UniversityofScienceandTechnologyBeijing,Beijing100083,ChinaLiWangInstituteofTheoreticalPhysics,StateKeyLaboratoryofQuantumOpticsandQuantumOpticsDevices,CollaborativeInnovation...

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n-body Correlation of Tonks-Girardeau Gas Yajiang HaoYaling Zhang and Yiwang Liu Institute of Theoretical Physics and Department of Physics

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