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arXiv:cond-mat/0603747 · Physical Review A

Pairing of a harmonically trapped fermionic Tonks-Girardeau gas

Anna Minguzzi, Marvin D. Girardeau

Published Jun 12, 2006• 16 citations
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Abstract

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.

DOI
10.1103/physreva.73.063614

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We investigate some static and dynamic properties of harmonically trapped fermionic Tonks-Girardeau gases in tight de Broglie waveguides with attractive p-wave interactions induced by a Feshbach resonance. An exact solution for the one-body density matrix is analyzed in terms of its natural orbitals, with the surprising result that for odd, but not for even, numbers of fermions the maximally occupied orbital coincides with the ground harmonic oscillator orbital. A dynamical consequence of this is that when the interactions are turned off the on-axis density remains a maximum for an odd number of fermions, whereas a minimum develops for an even number.

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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 ppp-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∕2N∕2N∕2, where NNN is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.
We determine some exact static and time-dependent properties of the fermionic Tonks-Girardeau (FTG) gas, a spin-aligned one-dimensional Fermi gas with infinitely strongly attractive zero-range odd-wave interactions. We show that its two-particle reduced density matrix exhibits superconductive off-diagonal long-range order, and on a ring an FTG gas with an even number of atoms has a highly degenerate ground state with quantization of Coriolis rotational flux and high sensitivity to rotation and to external fields and accelerations. For a gas initially under harmonic confinement, we show that during an expansion the momentum distribution undergoes a "dynamical bosonization," approaching that of an ideal Bose gas without violating the Pauli exclusion principle.
We theoretically investigate an ultracold spin-polarized atomic Fermi gas with resonant odd-channel (p(p(p-wave) interactions trapped in one-dimensional harmonic traps. We solve the Yang-Yang thermodynamic equations based on the exact Bethe ansatz solution, and predict the finite-temperature density profile and breathing mode frequency by using a local density approximation to take into account the harmonic trapping potential. The system features an exotic super fermionic Tonks-Girardeau (super-fTG) phase, due to the large effective range of the interatomic interactions. We explore the parameter space for such a fascinating super-fTG phase at finite temperature and provide smoking-gun signatures of its existence in both breathing mode frequencies and density profiles. Our results suggest that the super-fTG phase can be readily probed at temperatures of about 0.1TF0.1{T}_{F}0.1TF​, where TF{T}_{F}TF​ is the Fermi temperature. These results are to be confronted with future cold-atom experiments with 6Li^{6}\mathrm{Li}6Li and 40K^{40}\mathrm{K}40K atoms.
The fermionic Tonks-Girardeau (FTG) gas is known to exhibit conventional superfluidity, but may also be a vessel for unconventional odd-frequency superfluidity that is not revealed when studying only equal-time pairing correlations. In this thesis, we investigate odd-frequency superfluidity in the FTG gas. Specifically, this system is explored as a possible example of situations where odd-frequency pairing arises alongside conventional, even-frequency superfluidity. Previous works have extensively studied macroscopic quantum phenomena in the FTG gas via reduced density matrices, but this work is the first to investigate the hidden dynamical pairing in this system. We search for even fermion-exchange symmetry in the time-dependent two-body correlation matrix (T2bCM) that was formalised and shown to be synonymous with odd-frequency pair correlations by previous work. The eigenfunctions and eigenvalues of this matrix are studied for the homogeneous and harmonically trapped FTG gas. In the homogeneous case, it is found that the T2bCM eigenfunction is entirely antisymmetric under exchange of fermion coordinates and therefore no symmetric pairing occurs and no odd-frequency superfluidity exists within the system. Furthermore, the conventional macroscopic order present at zero time continues to exist at finite values of the pairing-time difference, but the magnitude of the macroscopic eigenvalue decreases monotonically with this time parameter. In the harmonically trapped case, the T2bCM eigenfunction contains a finite symmetric component that becomes concentrated at the boundary of the system for a large number of particles. It is found that, like the untrapped scenario, the macroscopic eigenvalue is finite and increases with particle number. We then consider a symmetric pairing order parameter and find that it decays with the particle number according to a power law, indicating that the harmonically trapped FTG gas exhibits non-macroscopic interface-localised odd-frequency order.
A harmonically trapped, ultracold, one-dimensional (1D) spin-1 Bose gas with strongly repulsive or attractive 1D even-wave interactions induced by a three-dimensional (3D) Feshbach resonance is studied. The exact ground state, a hybrid of Tonks-Girardeau (TG) and ideal Fermi gases, is constructed in the TG limit of infinite even-wave repulsion by a spinor Fermi-Bose mapping to a spinless ideal Fermi gas. It is then shown that in the limit of infinite even-wave attraction this same state remains an exact many-body eigenstate, now highly excited relative to the collapsed generalized McGuire-cluster ground state, showing that the hybrid TG state is completely stable against collapse to this cluster ground state under a sudden switch from infinite repulsion to infinite attraction. It is shown to be the TG limit of a hybrid super-Tonks-Girardeau (STG) state, which is metastable under a sudden switch from finite but very strong repulsion to finite but very strong attraction. It should be possible to create it experimentally by a sudden switch from strongly repulsive to strongly attractive interaction, as in the recent Innsbruck experiment on a spin-polarized bosonic STG gas. In the case of strong attraction, there should also exist another STG state of much lower energy, consisting of strongly bound dimers, a bosonic analog of a recently predicted STG state which is an ultracold gas of strongly bound bosonic dimers of fermionic atoms, but it is shown that this STG state cannot be created by such a switch from strong repulsion to strong attraction.
The fermionic Tonks-Girardeau gas is an exactly solvable model realizing a conventional macroscopic condensate of p-wave Cooper pairs. Here we demonstrate that, when confined by a trapping potential, it also hosts unconventional odd-frequency pairing, i.e., pair correlations with s-wave symmetry that are only present with a finite time delay t between fermions forming a pair. Quantitative results for fixed particle number are obtained using a generalization of Yang's theory of off-diagonal long-range order to define a t-dependent order parameter that is antisymmetric under combined t inversion and exchange of fermion-pair indices. Odd-frequency pairing in the fermionic Tonks-Girardeau gas is found to be a mesoscopic effect, localized at the system's boundary and therefore nonextensive thermodynamically. Our results show that this elusive hidden order is accessible in systems having fixed particle number, suggesting new avenues towards its experimental realization and further detailed study.
We investigate some static and dynamic properties of harmonically trapped fermionic Tonks-Girardeau gases in tight de Broglie waveguides with attractive p-wave interactions induced by a Feshbach resonance. An exact solution for the one-body density matrix is analyzed in terms of its natural orbitals, with the surprising result that for odd, but not for even, numbers of fermions the maximally occupied orbital coincides with the ground harmonic oscillator orbital. A dynamical consequence of this is that when the interactions are turned off the on-axis density remains a maximum for an odd number of fermions, whereas a minimum develops for an even number.