arXiv:1809.01592 · Physical Review A

Universal off-diagonal long-range-order behavior for a trapped Tonks-Girardeau gas

A. Colcelli, Jacopo Viti, Giuseppe Mussardo, Andrea Trombettoni

Published Dec 26, 2018• 17 citations

Abstract

The scaling of the largest eigenvalue ${\ensuremath{\lambda}}_{0}$ of the one-body density matrix of a system with respect to its particle number $N$ defines an exponent $\mathcal{C}$ and a coefficient $\mathcal{B}$ via the asymptotic relation ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}$. The case $\mathcal{C}=1$ corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead $\mathcal{C}=1/2$. Here we investigate the inhomogeneous case, initially addressing the behavior of $\mathcal{C}$ in the presence of a general external trapping potential $V$. We argue that the value $\mathcal{C}=1/2$ characterizes the hard-core system independently of the nature of the potential $V$. We then define the exponents $\ensuremath{\gamma}$ and $\ensuremath{\beta}$, which describe the scaling of the peak of the momentum distribution with $N$ and the natural orbital corresponding to ${\ensuremath{\lambda}}_{0}$, respectively, and we derive the scaling relation $\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}$. Taking as a specific case the power-law potential $V(x)\ensuremath{\propto}{x}^{2n}$, we give analytical formulas for $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ as functions of $n$. Analytical predictions for the coefficient $\mathcal{B}$ are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.

DOI: 10.1103/physreva.98.063633

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Oct 7, 20200 citations

Finite temperature off-diagonal long-range order for interacting bosons

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Pairing of a harmonically trapped fermionic Tonks-Girardeau gas

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Ground-state Properties of Tonks-Girardeau Gas in One Dimensional Periodic Potential

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One-particle density matrix and momentum distribution of the out-of-equilibrium one-dimensional Tonks-Girardeau gas: Analytical results at large N

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arXiv:1809.01592 · Physical Review A

Universal off-diagonal long-range-order behavior for a trapped Tonks-Girardeau gas

A. Colcelli, Jacopo Viti, Giuseppe Mussardo, Andrea Trombettoni

Published Dec 26, 2018• 17 citations

arXiv HTML PDF

Abstract

The scaling of the largest eigenvalue

{\ensuremath{\lambda}}_{0}

of the one-body density matrix of a system with respect to its particle number

N

defines an exponent

\mathcal{C}

and a coefficient

\mathcal{B}

via the asymptotic relation

{\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}

. The case

\mathcal{C}=1

corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead

\mathcal{C}=1/2

. Here we investigate the inhomogeneous case, initially addressing the behavior of

\mathcal{C}

in the presence of a general external trapping potential

V

. We argue that the value

\mathcal{C}=1/2

characterizes the hard-core system independently of the nature of the potential

V

. We then define the exponents

\ensuremath{\gamma}

and

\ensuremath{\beta}

, which describe the scaling of the peak of the momentum distribution with

N

and the natural orbital corresponding to

{\ensuremath{\lambda}}_{0}

, respectively, and we derive the scaling relation

\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}

. Taking as a specific case the power-law potential

V(x)\ensuremath{\propto}{x}^{2n}

, we give analytical formulas for

\ensuremath{\gamma}

and

\ensuremath{\beta}

as functions of

n

. Analytical predictions for the coefficient

\mathcal{B}

are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.

DOI: 10.1103/physreva.98.063633

Related papers

Scaling properties and Floquet engineering for low dimensional Bose gases

A. Colcelli

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).

Oct 7, 20200 citations

Finite temperature off-diagonal long-range order for interacting bosons

A. Colcelli, Nicolò Defenu, Giuseppe Mussardo +1 more

Characterizing the scaling with the total particle number ( $N$ ) of the largest eigenvalue of the one-body density matrix (${\ensuremath{\lambda}}_{0}$) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}{N}^{{\mathcal{C}}_{0}}$, then ${\mathcal{C}}_{0}=1$ corresponds in ODLRO. The intermediate case, $0<{\mathcal{C}}_{0}<1$ , corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in ${\mathcal{C}}_{0}$ (and in the corresponding quantities ${\mathcal{C}}_{k\ensuremath{\ne}0}$ for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that ${\mathcal{C}}_{k\ensuremath{\ne}0}=0$ in the thermodynamic limit. In one dimension it is ${\mathcal{C}}_{0}=0$ for nonvanishing temperature, while in three dimensions it is ${\mathcal{C}}_{0}=1$ ( ${\mathcal{C}}_{0}=0$ ) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$ , studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of ${\mathcal{C}}_{0}$ near the Berezinskii-Kosterlitz-Thouless temperature ${T}_{\text{BKT}}$ is $7/8$ . The dependence of ${\mathcal{C}}_{0}$ on temperatures between $T=0$ (at which ${\mathcal{C}}_{0}=1$ ) and ${T}_{\text{BKT}}$ is studied in the different models. An estimate for the (nonperturbative) parameter $\ensuremath{\xi}$ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a ``double jump'' behavior for ${\mathcal{C}}_{0}$ , and correspondingly of the anomalous dimension $\ensuremath{\eta}$, right below ${T}_{\text{BKT}}$ in the limit of vanishing interactions.Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one-body density matrix (${\ensuremath{\lambda}}_{0}$) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}{N}^{{\mathcal{C}}_{0}}$, then ${\mathcal{C}}_{0}=1$ corresponds in ODLRO. The intermediate case, $0<{\mathcal{C}}_{0}<1$, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in ${\mathcal{C}}_{0}$ (and in the corresponding quantities ${\mathcal{C}}_{k\ensuremath{\ne}0}$ for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that ${\mathcal{C}}_{k\ensuremath{\ne}0}=0$ in the thermodynamic limit. In one dimension it is ${\mathcal{C}}_{0}=0$ for nonvanishing temperature, while in three dimensions it is ${\mathcal{C}}_{0}=1$ (${\mathcal{C}}_{0}=0$) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$, studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of ${\mathcal{C}}_{0}$ near the Berezinskii-Kosterlitz-Thouless temperature ${T}_{\text{BKT}}$ is $7/8$. The dependence of ${\mathcal{C}}_{0}$ on temperatures between $T=0$ (at which ${\mathcal{C}}_{0}=1$) and ${T}_{\text{BKT}}$ is studied in the different models. An estimate for the (nonperturbative) parameter $\ensuremath{\xi}$ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a ``double jump'' behavior for ${\mathcal{C}}_{0}$, and correspondingly of the anomalous dimension $\ensuremath{\eta}$, right below ${T}_{\text{BKT}}$ in the limit of vanishing interactions.

arXiv:2007.01403Physical review. B./Physical review. BNov 17, 20207 citations

Pairing of a harmonically trapped fermionic Tonks-Girardeau gas

Anna Minguzzi, Marvin D. Girardeau

The fermionic Tonks-Girardeau (FTG) gas is a one-dimensional spin-polarized Fermi gas with infinitely strong attractive zero-range odd-wave interactions, arising from a confinement-induced resonance reachable via a three-dimensional $p$ -wave Feshbach resonance. We investigate the off-diagonal long-range order (ODLRO) of the FTG gas subjected to a longitudinal harmonic confinement by analyzing the two-particle reduced density matrix for which we derive a closed-form expression. Using a variational approach and numerical diagonalization we find that the largest eigenvalue of the two-body density matrix is of order $N∕2$ , where $N$ is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.The fermionic Tonks-Girardeau (FTG) gas is a one-dimensional spin-polarized Fermi gas with infinitely strong attractive zero-range odd-wave interactions, arising from a confinement-induced resonance reachable via a three-dimensional $p$-wave Feshbach resonance. We investigate the off-diagonal long-range order (ODLRO) of the FTG gas subjected to a longitudinal harmonic confinement by analyzing the two-particle reduced density matrix for which we derive a closed-form expression. Using a variational approach and numerical diagonalization we find that the largest eigenvalue of the two-body density matrix is of order $N∕2$, where $N$ is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.

arXiv:cond-mat/0603747Physical Review AJun 12, 200616 citations

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Zhao-Liang, Wang, An-Min, Wang, Xuechao Li

The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For $N$ -body systems of large enough, RSPDM and its lowest natural orbital do not vary with $N$ in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with $N$ , which results in that the ONLNO and MD are invariant with $N$ . While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$ , which results in the different behavior of the ONLNO and MD.The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For $N$-body systems of large enough, RSPDM and its lowest natural orbital do not vary with $N$ in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with $N$, which results in that the ONLNO and MD are invariant with $N$. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$, which results in the different behavior of the ONLNO and MD.

arXiv:1201.6019Jan 29, 20120 citations

One-particle density matrix and momentum distribution of the out-of-equilibrium one-dimensional Tonks-Girardeau gas: Analytical results at large N

Stefano Scopa, Paola Ruggiero, Pasquale Calabrese +1 more

In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms $N$ , which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large $N$ , at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms $N$, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large $N$, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.

arXiv:2301.04094Physical Review AJul 31, 202310 citations

Ground-state properties of a Tonks-Girardeau gas in a periodic potential

Bo-Bo Wei, Shi-Jian Gu, Hai-Qing Lin

In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau (TG) gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to $N=265$ bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau (TG) gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to $N=265$ bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.

arXiv:0812.1277Physical Review AJun 24, 20095 citations

The scaling of the largest eigenvalue

{\ensuremath{\lambda}}_{0}

of the one-body density matrix of a system with respect to its particle number

N

defines an exponent

\mathcal{C}

and a coefficient

\mathcal{B}

via the asymptotic relation

{\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}

. The case

\mathcal{C}=1

corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead

\mathcal{C}=1/2

. Here we investigate the inhomogeneous case, initially addressing the behavior of

\mathcal{C}

in the presence of a general external trapping potential

V

. We argue that the value

\mathcal{C}=1/2

characterizes the hard-core system independently of the nature of the potential

V

. We then define the exponents

\ensuremath{\gamma}

and

\ensuremath{\beta}

, which describe the scaling of the peak of the momentum distribution with

N

and the natural orbital corresponding to

{\ensuremath{\lambda}}_{0}

, respectively, and we derive the scaling relation

\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}

. Taking as a specific case the power-law potential

V(x)\ensuremath{\propto}{x}^{2n}

, we give analytical formulas for

\ensuremath{\gamma}

and

\ensuremath{\beta}

as functions of

n

. Analytical predictions for the coefficient

\mathcal{B}

are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.

Characterizing the scaling with the total particle number (

N

) of the largest eigenvalue of the one-body density matrix (${\ensuremath{\lambda}}_{0}$) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}{N}^{{\mathcal{C}}_{0}}$, then

{\mathcal{C}}_{0}=1

corresponds in ODLRO. The intermediate case,

0<{\mathcal{C}}_{0}<1

, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in

{\mathcal{C}}_{0}

(and in the corresponding quantities ${\mathcal{C}}_{k\ensuremath{\ne}0}$ for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that ${\mathcal{C}}_{k\ensuremath{\ne}0}=0$ in the thermodynamic limit. In one dimension it is

{\mathcal{C}}_{0}=0

for nonvanishing temperature, while in three dimensions it is

{\mathcal{C}}_{0}=1

(

{\mathcal{C}}_{0}=0

) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to

D=2

, studying the

XY

and the Villain models, and the weakly interacting Bose gas. The universal value of

{\mathcal{C}}_{0}

near the Berezinskii-Kosterlitz-Thouless temperature

{T}_{\text{BKT}}

7/8

. The dependence of

{\mathcal{C}}_{0}

on temperatures between

T=0

(at which

{\mathcal{C}}_{0}=1

) and

{T}_{\text{BKT}}

is studied in the different models. An estimate for the (nonperturbative) parameter $\ensuremath{\xi}$ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a ``double jump'' behavior for

{\mathcal{C}}_{0}

, and correspondingly of the anomalous dimension $\ensuremath{\eta}$, right below

{T}_{\text{BKT}}

in the limit of vanishing interactions.

p

-wave Feshbach resonance. We investigate the off-diagonal long-range order (ODLRO) of the FTG gas subjected to a longitudinal harmonic confinement by analyzing the two-particle reduced density matrix for which we derive a closed-form expression. Using a variational approach and numerical diagonalization we find that the largest eigenvalue of the two-body density matrix is of order

N∕2

, where

N

is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.

N

-body systems of large enough, RSPDM and its lowest natural orbital do not vary with

N

in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with

N

, which results in that the ONLNO and MD are invariant with

N

. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to

\sqrt{N}

, which results in the different behavior of the ONLNO and MD.

N

, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large

N

, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.

N=265

bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.