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arXiv:0704.3275 · Physical Review A

Matter-wave interference in s -wave and p -wave Fermi condensates

Wei Zhang, C. A. R. Sá de Melo

Published Jul 31, 2007• 4 citations
arXivHTMLPDF

Abstract

We discuss the time evolution and matter-wave interference of Fermi condensates on the Bose-Einstein condensate side of Feshbach resonances for $s$- and $p$-wave superfluids, upon release from harmonic traps. In $s$-wave systems, where the order parameter is a complex scalar, we find that the interference patterns depend on the relative phase of the order parameters of the condensates. In $p$-wave systems involving the mixture of two hyperfine states, we show that the interference pattern exhibits a polarization effect depending on the relative orientation of the two vector order parameters. Lastly, we also point out that $p$-wave Fermi condensates exhibit an anisotropic expansion, reflecting the spatial anisotropy of the underlying interaction between fermions and the orbital nature of the vector order parameter. Potential applications of our results include systems of ultracold atoms that exhibit $p$-wave Feshbach resonances, such as $^{6}\mathrm{Li}$ or $^{40}\mathrm{K}$.

DOI
10.1103/physreva.76.013627

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We discuss the time evolution and matter-wave interference of Fermi condensates on the Bose-Einstein condensate side of Feshbach resonances for sss- and ppp-wave superfluids, upon release from harmonic traps. In sss-wave systems, where the order parameter is a complex scalar, we find that the interference patterns depend on the relative phase of the order parameters of the condensates. In ppp-wave systems involving the mixture of two hyperfine states, we show that the interference pattern exhibits a polarization effect depending on the relative orientation of the two vector order parameters. Lastly, we also point out that ppp-wave Fermi condensates exhibit an anisotropic expansion, reflecting the spatial anisotropy of the underlying interaction between fermions and the orbital nature of the vector order parameter. Potential applications of our results include systems of ultracold atoms that exhibit ppp-wave Feshbach resonances, such as 6Li^{6}\mathrm{Li}6Li or 40K^{40}\mathrm{K}40K.
We discuss matter wave interference of Fermi condensates in strongly coupled s-wave and p-wave channels and the time evolution of a single cloud upon release from trap. In s-wave systems, where the order parameter is a complex scalar, we find that the interference patterns depend on the relative phase of the order parameters of the condensates. In p-wave systems involving the mixture of two-hyperfine states, we show that the interference pattern exhibits a polarization effect depending on the relative orientation of the two vector order parameters. However, in p-wave systems involving a single hyperfine state, we show that this angular effect reduces to an overall phase difference between the two interfering clouds, similar to s-wave. Lastly, we also point out that p-wave Fermi condensates exhibit an anisotropic expansion, reflecting the spatial anisotropy of the underlying interaction between fermions and the orbital nature of the vector order parameter.
We discuss the time-of-flight expansion of dilute p-wave Fermi condensates on the BEC side of Feshbach resonances, as a way to extract information about the order parameter symmetry for superfluidity. We show that the cloud profile is in general sensitive to the interaction strength between fermions, the magnitude and direction of external magnetic fields, and to the angular momentum projection of the order parameter. In particular, due to the anisotropic nature of p-wave interactions we show that the time-of-flight expansion of a p-wave superfluid is anisotropic even if the superfluid is confined to a completely isotropic trap, unlike the case of Bose or s-wave Fermi condensates, which under the same circumstances expand isotropically. Furthermore, we demonstrate that expanding p-wave superfluids released from axially symmetric traps experience anisotropy inversions, where the aspect ratio between the axial and radial directions change during expansion, while the axial symmetry can also be lost reflecting the spatial anisotropy of the underlying interaction.
For a Bose-condensed gas in a combined potential consisting of an axially-symmetric harmonic magnetic trap and one-dimensional (1D) optical lattice, using the mean-field Gross–Pitaevskii (G-P) equation and the propagator method, we obtain the analytical result of the order parameter for matter wave interference at any time. The evolution of the interference pattern under a variation of the relative phase Δϕ between successive subcondensates trapped on an optical lattices is also studied. For Δϕ = π, the interference pattern is symmetric with two sharp peaks, which are symmetrically located on a straight line on both sides of a vacant central peak and moving apart from each other. This work is in agreement with available experimental results.
We study the interference patterns of a superfluid Fermi gas released from optical lattices below and above Feshbach resonance based on a simple phenomenological approach. We first solve the order-parameter equation valid for the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) to obtain an initial distribution of subcondensates formed in an optical lattice. Then we investigate the coherent evolution of the subcondensates when both harmonic oscillator and optical lattice potentials are switched off. The interference patterns of the superfluid Fermi gas along the BCS-BEC crossover during a nearly ballistic expansion are calculated by means of Feynman propagator method combined with numerical simulations. The result obtained agrees with the recent experimental observation reported by the MIT group [J. K. Chin et al., Nature (London) 443, 961 (2006)].
We review the recent developments which had taken place in the domain of quasi one dimensional BECs from the viewpoint of integrability. To start with, we consider the dynamics of scalar BECs in a time independent harmonic trap and observe that the scattering length (SL) can be suitably manipulated either to compress the bright solitons to attain peak matter wave density without causing their explosion or to broaden the width of the condensates without diluting them. When the harmonic trap frequency becomes time dependent, we notice that one can stabilize the condensates in the confining domain while the density of the condensates continue to increase in the expulsive region. We also observe that the trap frequency and the temporal SL can be maneuvered to generate matter wave interference patterns indicating the coherent nature of the atoms in the condensates. We also notice that a small repulsive three body interaction when reinforced with attractive binary interaction can extend the region of stability of the condensates in the quasi-one dimensional regime. On the other hand, the investigation of two component BECs in a time dependent harmonic trap suggests that it is possible to switch matter wave energy from one mode to the other confirming the fact that vector BECs are long lived compared to scalar BECs. The Feshbach resonance management of vector BECs indicates that the two component BECs in a time dependent harmonic trap are more stable compared to the condensates in a time independent trap. The introduction of weak time dependent Rabi coupling rapidly compresses the bright solitons which however can be again stabilized through Feshbach resonance or by finetuning the Rabi coupling while the spatial coupling of vector BECs introduces a phase difference between the condensates which subsequently can be exploited to generate interference pattern in the bright or dark solitons.
We have observed the interference between two Bose–Einstein condensates of weakly bound Feshbach molecules of fermionic 6 Li atoms. Two condensates are prepared in a double-well trap and, after release from this trap, overlap in expansion. We detect a clear interference pattern that unambiguously demonstrates the de Broglie wavelength of molecules. We verify that only the condensate fraction shows interference. With increasing interaction strength, the pattern vanishes because elastic collisions during overlap remove particles from the condensate wave function. For strong interaction, the condensates do not penetrate each other as they collide hydrodynamically.