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arXiv:2204.09205 · Journal of Physics B Atomic Molecular and Optical Physics

Energetics and Efimov states of three interacting bosons and mass-imbalanced fermions in a three-dimensional spherical harmonic trap

A. D. Kerin, A. Martin

Published Jan 30, 2023• 2 citations
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Abstract

Abstract We consider a system of three particles, either three identical bosons or two identical fermions plus an impurity, within a three-dimensional isotropic trap interacting via a contact interaction. Using two approaches, one using an infinite sum of basis states for the wavefunction and the other a closed form wavefunction, we calculate the allowable energy eigenstates of the system as a function of the interaction strength, including the strongly and weakly interacting limits. For the fermionic case this is done while maintaining generality regarding particle masses. We find that the two methods of calculating the spectrum are in excellent agreement in the strongly interacting limit. However the infinite sum approach is unable to uniquely specify the energy of Efimov states, but in the strongly interacting limit there is, to a high degree of accuracy, a correspondence between the three-body parameter required by the boundary condition of the closed form approach and the summation truncation order required by the summation approach. This specification of the energies and wavefunctions forms the basis with which thermodynamic variables such as the virial coefficients or Tan contacts, or dynamic phenomena like quench dynamics can be calculated.

DOI
10.1088/1361-6455/acb74a

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Abstract We consider a system of three particles, either three identical bosons or two identical fermions plus an impurity, within a three-dimensional isotropic trap interacting via a contact interaction. Using two approaches, one using an infinite sum of basis states for the wavefunction and the other a closed form wavefunction, we calculate the allowable energy eigenstates of the system as a function of the interaction strength, including the strongly and weakly interacting limits. For the fermionic case this is done while maintaining generality regarding particle masses. We find that the two methods of calculating the spectrum are in excellent agreement in the strongly interacting limit. However the infinite sum approach is unable to uniquely specify the energy of Efimov states, but in the strongly interacting limit there is, to a high degree of accuracy, a correspondence between the three-body parameter required by the boundary condition of the closed form approach and the summation truncation order required by the summation approach. This specification of the energies and wavefunctions forms the basis with which thermodynamic variables such as the virial coefficients or Tan contacts, or dynamic phenomena like quench dynamics can be calculated.
We consider a system of two identical fermions of general mass interacting with a third distinguishable particle via a contact interaction within an isotropic three-dimensional harmonic trap. We calculate time-dependent observables of the system after it is quenched in sss-wave scattering length. To do this we use exact closed-form mass-imbalanced hyperspherical solutions to the static three-body problem. These exact solutions enable us to calculate two time-dependent observables, the Ramsey signal and particle separation, after the system undergoes a quench from noninteracting to the unitary regime or vice versa.
We investigate the effects of Efimov states on the postquench dynamics of a system of three identical bosons with contact interactions in a spherically symmetric three-dimensional harmonic trap. The quench we consider is in the sss-wave contact interaction, and we focus on quenches from the noninteracting to strongly interacting regimes and vice versa. The calculations use the hyperspherical solutions of the three-body problem and enable us to evaluate the semianalytical results of the Ramsey and particle separation, postquench. In the case where the interactions are quenched from the noninteracting to strongly interacting regime we find convergent aperiodic solutions for both the Ramsey signal and the particle separation. In contrast, for quenches from the strongly interacting regime to the noninteracting regime both the Ramsey signal and particle separation are periodic functions. However, in this case we find that the solutions for the particle separation diverge, indicating that in such a system large oscillations may be observable.
The mass-imbalanced fermionic mixture is studied, where $N\ensuremath{\le}5$ identical fermions interact resonantly with an impurity, a distinguishable atom. The shell structure is explored, and the physics of a dynamic light-impurity is shown to be different from that of the static heavy-impurity case. The energies in a harmonic trap at unitarity are calculated and extrapolated to the zero-range limit. In doing so, the scale factor of the ground state, as well as of a few excited states, is calculated. In the $2\ensuremath{\le}N\ensuremath{\le}4$ systems, pure (N+1)(N+1)(N+1) Efimov states exist for large enough mass ratio. However, no sign for a six-body Efimov state in the (5+1)(5+1)(5+1) system is found in the mass ratio explored, $M/m\ensuremath{\le}12$.
We solve the three-body bound-state problem in three dimensions for mass imbalanced systems of two identical bosons and a third particle in the universal limit where the interactions are assumed to be of zero range. The system displays the Efimov effect and we use the momentum-space wave equation to derive formulas for the scaling factor of the Efimov spectrum for any mass ratio assuming either that two or three of the two-body subsystems have a bound state at zero energy. We consider the single-particle momentum distribution analytically and numerically and analyze the tail of the momentum distribution to obtain the three-body contact parameter. Our findings demonstrate that the functional form of the three-body contact term depends on the mass ratio, and we obtain an analytic expression for this behavior. To exemplify our results, we consider mixtures of lithium with either two caesium or rubidium atoms which are systems of current experimental interest.
Abstract For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix ρB\rho _\mathrm{B}ρB​ ρ B in terms of center of mass and relative coordinates of the particles. The deviation from ρF\rho _\mathrm{F}ρF​ ρ F , the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained ρB\rho _\mathrm{B}ρB​ ρ B allows us to derive a closed form expression of the corresponding momentum distribution nB(p)n_\mathrm{B}(p)nB​(p) n B ( p ) . We show how the result deviates from the noninteracting fermionic case, the deviation being associated with the short-range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum 1/p41/p^4 1/p4 1 / p 4 tail with the appropriate Tan’s coefficient. Numerical results support our findings.
Abstract Strongly interacting particles in one dimension subject to external confinement have become a topic of considerable interest due to recent experimental advances and the development of new theoretical methods to attack such systems. In the case of equal mass fermions or bosons with two or more internal degrees of freedom, one can map the problem onto the well-known Heisenberg spin models. However, many interesting physical systems contain mixtures of particles with different masses. Therefore, a generalization of the recent strong-coupling techniques would be highly desirable. This is particularly important since such problems are generally considered non-integrable and thus the hugely successful Bethe ansatz approach cannot be applied. Here we discuss some initial steps towards this goal by investigating small ensembles of one-dimensional harmonically trapped particles where pairwise interactions are either vanishing or infinitely strong with focus on the mass-imbalanced case. We discuss a (semi)-analytical approach to describe systems using hyperspherical coordinates where the interaction is effectively decoupled from the trapping potential. As an illustrative example we analyze mass-imbalanced four-particle two-species mixtures with strong interactions between the two species. For such systems we calculate the energies, densities and pair-correlation functions.