Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1610.07773 · The European Physical Journal Special Topics

Tan’s contact of a harmonically trapped one-dimensional Bose gas: Strong-coupling expansion and conjectural approach at arbitrary interactions

Guillaume Lang, Patrizia Vignolo, Anna Minguzzi

Published May 1, 2017• 27 citations
arXivHTMLPDF

Abstract

We study Tan’s contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan’s contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan’s contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases.

DOI
10.1140/epjst/e2016-60343-6

Related papers

Symmetries and Correlations in Strongly Interacting One-dimensional Quantum Gases

Jean Decamp

The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical $k^{-4}$ laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.

arXiv:1811.05211•Nov 13, 2018•5 citations

Density-functional theory for one-dimensional harmonically trapped Bose-Fermi mixture

Hongmei Wang, Yajiang Hao, Yunbo Zhang

We present a density-functional theory for the one-dimensional harmonically trapped Bose-Fermi mixture with repulsive contact interactions. The ground-state density distribution of each component is obtained by solving the Kohn-Sham equations numerically based on the local density approximation and the exact solution for the homogeneous system given by Bethe ansatz method. It is shown that for sufficiently strong interaction, a considerable amount of fermions are repelled out of the central region of the trap, exhibiting partial phase separation of Bose and Fermi components. Oscillations emerge in the Bose density curves, reflecting the strong correlation with fermions. For infinitely strong interaction, the ground-state energy of the mixture and the total density are consistent with the scenario that all atoms in the mixture are fully fermionized.

arXiv:1203.6211•Physical Review A•May 22, 2012•13 citations

Breakdown of Tan’s Relation in Lossy One-Dimensional Bose Gases

Isabelle Bouchoule, Jérôme Dubail

In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as ∼1/|p|^{4} at large momentum p. Tan's relation connects the amplitude of that 1/|p|^{4} tail to the adiabatic derivative of the energy with respect to the coupling constant or scattering length of the gas. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution that itself decreases as 1/|p|^{4}. In the momentum distribution, that rapidity tail adds to the usual Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments, do produce such peculiar states. The development of the tail of the rapidity distribution originates from the ghost singularity of the wave function immediately after each loss event. This phenomenon is discussed for arbitrary interaction strengths, and it is supported by exact calculations in the two asymptotic regimes of infinite and weak repulsion.

arXiv:2011.13250•Physical Review Letters•Apr 23, 2021•47 citations

High-momentum tails as magnetic-structure probes for strongly correlated SU(κ) fermionic mixtures in one-dimensional traps

Jean Decamp, Johannes Jünemann, Mathias Albert +3 more

A universal ${k}^{\ensuremath{-}4}$ decay of the large-momentum tails of the momentum distribution, fixed by Tan's contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. Here we consider a repulsive multicomponent Fermi gas under harmonic confinement, as in the experiment of G. Pagano et al. [Nat. Phys. 10, 198 (2014)], realizing a gas with tunable $\text{SU}(\ensuremath{\kappa})$ symmetry. We exploit an exact solution at infinite repulsion to show a direct correspondence between the value of the Tan's contact for each of the $\ensuremath{\kappa}$ components of the gas and the Young tableaux for the ${S}_{N}$ permutation symmetry group identifying the magnetic structure of the ground state. This opens a route for the experimental determination of magnetic configurations in cold atomic gases, employing only standard (spin-resolved) time-of-flight techniques. Combining the exact result with matrix-product-state simulations, we obtain the Tan's contact at all values of repulsive interactions. We show that a local-density approximation (LDA) on the Bethe-ansatz equation of state for the homogeneous mixture is in excellent agreement with the results for the harmonically confined gas. At strong interactions, the LDA predicts a scaling behavior of the Tan's contact. This provides a useful analytical expression for the dependence on the number of fermions, number of components, and interaction strength. Moreover, using a virial approach, we study the Tan's contact behavior at high temperature and in the limit of infinite interactions, and we show that it increases with the temperature and the number of components. At zero temperature, we predict that the weight of the momentum distribution tails increases with interaction strength and the number of components if the population per component is kept constant. This latter property was experimentally observed in G. Pagano et al. [Nat. Phys. 10, 198 (2014)].

arXiv:1607.08744•Physical Review A•Nov 16, 2016•65 citations

Harmonically trapped Fermi gas: Temperature dependence of the Tan contact

Yangqian Yan, D. Blume

Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the $N$-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.

arXiv:1305.5383•Physical Review A•Aug 27, 2013•20 citations

Density-functional theory of two-component Bose gases in one-dimensional harmonic traps

Yajiang Hao, Shu Chen

We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.

arXiv:0905.3207•Physical Review A•Oct 14, 2009•34 citations
We study Tan’s contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan’s contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan’s contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases.
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k−4k^{-4}k−4 laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.
We present a density-functional theory for the one-dimensional harmonically trapped Bose-Fermi mixture with repulsive contact interactions. The ground-state density distribution of each component is obtained by solving the Kohn-Sham equations numerically based on the local density approximation and the exact solution for the homogeneous system given by Bethe ansatz method. It is shown that for sufficiently strong interaction, a considerable amount of fermions are repelled out of the central region of the trap, exhibiting partial phase separation of Bose and Fermi components. Oscillations emerge in the Bose density curves, reflecting the strong correlation with fermions. For infinitely strong interaction, the ground-state energy of the mixture and the total density are consistent with the scenario that all atoms in the mixture are fully fermionized.
In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as ∼1/|p|^{4} at large momentum p. Tan's relation connects the amplitude of that 1/|p|^{4} tail to the adiabatic derivative of the energy with respect to the coupling constant or scattering length of the gas. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution that itself decreases as 1/|p|^{4}. In the momentum distribution, that rapidity tail adds to the usual Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments, do produce such peculiar states. The development of the tail of the rapidity distribution originates from the ghost singularity of the wave function immediately after each loss event. This phenomenon is discussed for arbitrary interaction strengths, and it is supported by exact calculations in the two asymptotic regimes of infinite and weak repulsion.
A universal ${k}^{\ensuremath{-}4}$ decay of the large-momentum tails of the momentum distribution, fixed by Tan's contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. Here we consider a repulsive multicomponent Fermi gas under harmonic confinement, as in the experiment of G. Pagano et al. [Nat. Phys. 10, 198 (2014)], realizing a gas with tunable $\text{SU}(\ensuremath{\kappa})$ symmetry. We exploit an exact solution at infinite repulsion to show a direct correspondence between the value of the Tan's contact for each of the $\ensuremath{\kappa}$ components of the gas and the Young tableaux for the SN{S}_{N}SN​ permutation symmetry group identifying the magnetic structure of the ground state. This opens a route for the experimental determination of magnetic configurations in cold atomic gases, employing only standard (spin-resolved) time-of-flight techniques. Combining the exact result with matrix-product-state simulations, we obtain the Tan's contact at all values of repulsive interactions. We show that a local-density approximation (LDA) on the Bethe-ansatz equation of state for the homogeneous mixture is in excellent agreement with the results for the harmonically confined gas. At strong interactions, the LDA predicts a scaling behavior of the Tan's contact. This provides a useful analytical expression for the dependence on the number of fermions, number of components, and interaction strength. Moreover, using a virial approach, we study the Tan's contact behavior at high temperature and in the limit of infinite interactions, and we show that it increases with the temperature and the number of components. At zero temperature, we predict that the weight of the momentum distribution tails increases with interaction strength and the number of components if the population per component is kept constant. This latter property was experimentally observed in G. Pagano et al. [Nat. Phys. 10, 198 (2014)].
Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the NNN-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.
We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.