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Handover Constellation in a Quasi-One-Dimensional Bose Gas: From a Single-Offset Crossover to the Five Characteristic Temperatures

Guiru Zhao

Published Sep 26, 2026• 0 citations
Publisher

Abstract

A phase transition is conventionally described by a single critical temperature T_c. We analyze Bose–Einstein condensation of a quasi-one-dimensional interacting gas and find the transition presents itself as a cluster of separately locatable sub-events, quantified by a five-temperature constellation: condensate half-height T_R, peak-interaction handover T_I, cloud-size minimum T_B, momentum-width minimum T_D, and overlap peak T_S. Across 17 interaction strengths and three thermal-cloud models (Boltzmann, Bose, Hartree–Fock), we find: (i) the handover offset ΔT_I(g) is a monotonic crossover—hump at weak coupling (+0.15 T_c at g = 30), zero at g* ≈ 39 (Bose) or 44 (Boltzmann), deepening to −0.10 T_c—with a statistics double valley at the crossover; (ii) the constellation expands (separation 0.21 → 0.53 T_c) and its internal order reverses (T_I crosses T_R near g ≈ 32), these two markers bracketing the crossover region; (iii) the two cloud-sensitive dimensions carry different signatures (T_I: zero-crossing; T_S: deepening −0.13 → −0.18 T_c). The density profile remains center-peaked at all temperatures—an observed central dip would falsify the model family. Ground states use multi-start Krotov minimization (μ(g) strictly monotone, dominant-peak fractions > 0.994). All features map onto laboratory parameters (N ≈ 200–1600 for ⁸⁷Rb) and are falsifiable predictions for Feshbach experiments with simultaneous condensate-fraction and in-situ density measurements.

DOI
10.5281/zenodo.22979805

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A phase transition is conventionally described by a single critical temperature T_c. We analyze Bose–Einstein condensation of a quasi-one-dimensional interacting gas and find the transition presents itself as a cluster of separately locatable sub-events, quantified by a five-temperature constellation: condensate half-height T_R, peak-interaction handover T_I, cloud-size minimum T_B, momentum-width minimum T_D, and overlap peak T_S. Across 17 interaction strengths and three thermal-cloud models (Boltzmann, Bose, Hartree–Fock), we find: (i) the handover offset ΔT_I(g) is a monotonic crossover—hump at weak coupling (+0.15 T_c at g = 30), zero at g* ≈ 39 (Bose) or 44 (Boltzmann), deepening to −0.10 T_c—with a statistics double valley at the crossover; (ii) the constellation expands (separation 0.21 → 0.53 T_c) and its internal order reverses (T_I crosses T_R near g ≈ 32), these two markers bracketing the crossover region; (iii) the two cloud-sensitive dimensions carry different signatures (T_I: zero-crossing; T_S: deepening −0.13 → −0.18 T_c). The density profile remains center-peaked at all temperatures—an observed central dip would falsify the model family. Ground states use multi-start Krotov minimization (μ(g) strictly monotone, dominant-peak fractions > 0.994). All features map onto laboratory parameters (N ≈ 200–1600 for ⁸⁷Rb) and are falsifiable predictions for Feshbach experiments with simultaneous condensate-fraction and in-situ density measurements.
By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasi-condensate regime throughout the dimensional crossover from a purely 1D to an almost 3D gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasi-condensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas. We also perform measurements of the third moment of atom number fluctuations. We find a positive skewness of the atom number distribution in the ideal gas regime and a reduced skewness compatible with zero in the quasi-condensate regime. For our parameters, the third moment is a thermodynamic quantity whose measurement constitutes a sensitive test of the equation of state and our results are in agreement with a modified Yang-Yang thermodynamic prediction. Moreover, we show that the measured skewness reveals the presence of true three body correlations in the system.
By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasi-condensate regime throughout the dimensional crossover from a purely 1D to an almost 3D gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasi-condensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas.
By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasicondensate regime throughout the dimensional crossover from a purely one-dimensional (1D) to an almost three-dimensional (3D) gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasicondensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas.
We analyze the results of a recent experiment with bosonic rubidium atoms harmonically confined in a quasi-two-dimensional (2D) geometry. In this experiment a well-defined critical point was identified, which separates the high-temperature normal state characterized by a single component density distribution, and the low-temperature state characterized by a bimodal density distribution and the emergence of high-contrast interference between independent 2D clouds. We first show that this transition cannot be explained in terms of conventional Bose–Einstein condensation of the trapped ideal Bose gas. Using the local density approximation (LDA), we then combine the mean-field (MF) Hartree–Fock theory with the prediction for the Berezinskii–Kosterlitz–Thouless (BKT) transition in an infinite uniform system. If the gas is treated as a strictly 2D system, the MF predictions for the spatial density profiles significantly deviate from those of a recent quantum Monte Carlo (QMC) analysis. However, when the residual thermal excitation of the strongly confined degree of freedom is taken into account, excellent agreement is reached between the MF and the QMC approaches. For the interaction strength corresponding to the experiment, we predict a strong correction to the critical atom number with respect to the ideal gas theory (factor ∼2). Quantitative agreement between theory and experiment is reached concerning the critical atom number if the predicted density profiles are used for temperature calibration.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.
We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within the mean-field level, and calculate the zero-temperature cloud size and number density distribution. For a model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size for arbitrary detunings. For a model 2 Hamiltonian with renormalized interactions between atoms and dressed molecules, the cloud size deceases from the BCS to BEC side, which is consistent with the picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional Fermi systems, especially on the BEC side of the Feshbach resonance.