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arXiv:2505.05186 · Physical Review A

Expansion of one-dimensional spinor gases from power-law traps

Ovidiu I. Pâţu, Gianni Aupetit-Diallo

Published Jul 17, 2025• 0 citations
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Abstract

Free expansion following the removal of axial confinement represents a fundamental nonequilibrium scenario in the study of many-body ultracold gases. Using the stationary phase approximation, we analytically demonstrate that for all one-dimensional spinor gases with repulsive contact interactions, whether bosonic or fermionic, the asymptotic density and momentum distribution can be directly determined from the quasimomentum distribution (Bethe rapidities) of the trapped gas. We efficiently obtain the quasimomentum distribution numerically by solving the integral equations that characterize the ground state of the integrable system within the local density approximation. Additionally, we derive analytical solutions for both weakly and strongly interacting regimes. Unlike in bosonic gases, where rapidity distributions and density profiles vary significantly across interaction regimes, fermionic gases maintain similar profiles in both weakly and strongly interacting limits. Notably, the gas expands self-similarly only when released from a harmonic trap. For other power-law trapping potentials, the asymptotic density profile is strongly influenced by the initial confinement geometry. Our results extend readily to Bose-Fermi mixtures and finite temperatures.

DOI
10.1103/clxm-5yd9

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Free expansion following the removal of axial confinement represents a fundamental nonequilibrium scenario in the study of many-body ultracold gases. Using the stationary phase approximation, we analytically demonstrate that for all one-dimensional spinor gases with repulsive contact interactions, whether bosonic or fermionic, the asymptotic density and momentum distribution can be directly determined from the quasimomentum distribution (Bethe rapidities) of the trapped gas. We efficiently obtain the quasimomentum distribution numerically by solving the integral equations that characterize the ground state of the integrable system within the local density approximation. Additionally, we derive analytical solutions for both weakly and strongly interacting regimes. Unlike in bosonic gases, where rapidity distributions and density profiles vary significantly across interaction regimes, fermionic gases maintain similar profiles in both weakly and strongly interacting limits. Notably, the gas expands self-similarly only when released from a harmonic trap. For other power-law trapping potentials, the asymptotic density profile is strongly influenced by the initial confinement geometry. Our results extend readily to Bose-Fermi mixtures and finite temperatures.
We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the Lieb-Liniger equations in the thermodynamic limit. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the self-similar scaling solutions known from the hydrodynamic approach. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence of the density profile evolution on the trap anharmonicity. We also characterize dynamical fermionization of the expanding cloud in terms of correlation functions describing phase and density fluctuations.
We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the thermodynamic Bethe ansatz equations. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the known scaling solutions of the hydrodynamic approach corresponding to self-similar expansion. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence on the trap anharmonicity of the shape variation of the density profile during evolution. We also characterize dynamical fermionization of an expanding cloud in terms of its first- and second-order coherences describing phase and density fluctuations.
Following the removal of axial confinement the momentum distribution of a Tonks-Girardeau gas approaches that of a system of noninteracting spinless fermions in the initial harmonic trap. This phenomenon, called dynamical fermionization, has been experimentally confirmed in the case of the Lieb-Liniger model and theoretically predicted in the case of multicomponent systems at zero temperature. We prove analytically that for all spinor gases with strong repulsive contact interactions at finite temperature the momentum distribution after release from the trap asymptotically approaches that of a system of spinless fermions at the same temperature but with a renormalized chemical potential which depends on the number of components of the spinor system. In the case of the Gaudin-Yang model we check numerically our analytical predictions using the results obtained from a nonequilibrium generalization of Lenard's formula describing the time evolution of the field-field correlators.
One of quantum physics' fundamental, but largely unsolved, problems is the computation of the correlation functions in many-body systems. We address this problem in the case of one-dimensional spinor gases with repulsive contact interactions in the presence of a confining potential. We take advantage of the fact that in the strong-coupling limit, the wave function factorizes with the charge degrees of freedom expressed as a Slater determinant of spinless fermions and the spin sector described by a spin chain of Sutherland type with exchange coefficients that depend only on the trapping potential. This factorization is also present in the expressions for the correlation functions. Still, analytical and numerical investigations were hindered by the fact that the local exchange coefficients and the charge component of the correlators are expressed as $N\ensuremath{-}1$ multidimensional integrals, with NNN the number of particles, which are notoriously hard to compute using conventional methods. We introduce a different approach to calculating these integrals that is extremely simple, scales polynomially with the number of particles, and is several orders of magnitude faster than the previous methods reported in the literature. This allows us to investigate the static and dynamic properties, temperature dependence, and nonequilibrium dynamics for systems with a larger number of particles than previously considered and discover new phenomena. We show that contrary to natural expectations, the momentum distribution of strongly interacting trapped spinor gases becomes narrower as we increase the temperature and derive simple determinant representations for the correlators in the spin incoherent regime that is valid for both equilibrium and nonequilibrium situations. For homogeneous systems of impenetrable particles at zero temperature, we analytically compute the large distance asymptotics of the correlators, finding that the leading term of the asymptotic expansion is proportional to ${e}^{\ensuremath{-}x{k}_{F}ln\ensuremath{\kappa}/\ensuremath{\pi}}$, with $\ensuremath{\kappa}$ the number of components of the system, which precludes any singularity of the momentum distribution.
After release from the trap the momentum distribution of an impenetrable gas asymptotically approaches that of a spinless noninteracting Fermi gas in the initial trap. This phenomenon is called dynamical fermionization and, very recently, has been experimentally confirmed in the case of the Lieb-Liniger model in the Tonks-Girardeau regime. We prove analytically and confirm numerically that following the removal of axial confinement the strongly interacting Bose-Fermi mixture exhibits dynamical fermionization and the asymptotical momentum distribution of each component has the same shape as its density profile at t=0t=0t=0. Under a sudden change of the trap frequency to a new nonzero value the dynamics of both fermionic and bosonic momentum distributions presents characteristics which are similar to the case of single component bosons experiencing a similar quench. Our results are derived using a product representation for the correlation functions which, in addition to analytical considerations, can be implemented numerically very easily with complexity which scales polynomially in the number of particles.
Dynamical fermionization refers to the phenomenon in Tonks-Girardeau gases where, upon release from harmonic confinement, the gases' momentum density profile evolves asymptotically to that of an ideal Fermi gas in the initial trap. This phenomenon has been demonstrated theoretically in hardcore and anyonic Tonks-Girardeau gases and was recently experimentally observed in a strongly interacting Bose gas. We extend this study to a one-dimensional spinor gas of arbitrary spin in the strongly interacting regime and analytically prove that the total momentum distribution after the harmonic trap is turned off approaches that of a spinless ideal Fermi gas, while the asymptotic momentum distribution of each spin component takes the same shape of the initial real space density profile of that spin component. Our work demonstrates the rich physics arising from the interplay between the spin and the charge degrees of freedom in a spinor system.