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arXiv:1612.03462 · Physical review. B./Physical review. B

Phase diagram for the trapped p -wave fermionic superfluid with population imbalance

Ammar A. Kirmani, Khandker Quader, Maxim Dzero

Published Apr 6, 2017• 1 citation
arXivHTMLPDF

Abstract

We consider the problem of spin-triplet $p$-wave superfluid pairing with total spin projection ${m}_{s}=0$ in atomic Fermi gas across the Feshbach resonance. We allow for imbalanced populations and take into account the effects due to the presence of a harmonic trapping potential. Within the mean-field approximation for the one- and two-channel pairing models we show that the $p$-wave superfluid state has the lowest energy up to some distance from the trap center, the distance depending on population imbalance and the side of the Feshbach resonance (BCS or BEC) the system is in. Beyond this, the system is in the normal state. Since the system, even in the presence of the harmonic trap, is invariant under O(3) space rotation, the superfluid cores which emerge in the trap are comprised of states that are degenerate with respect to the ``orbital ferromagnet'' state, ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$, as well as the ones in which all angular momentum components of the order parameter are nonzero, provided the modulus of the order parameter maintains a fixed magnitude.

DOI
10.1103/physrevb.95.134503

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We consider the problem of spin-triplet ppp-wave superfluid pairing with total spin projection ms=0{m}_{s}=0ms​=0 in atomic Fermi gas across the Feshbach resonance. We allow for imbalanced populations and take into account the effects due to the presence of a harmonic trapping potential. Within the mean-field approximation for the one- and two-channel pairing models we show that the ppp-wave superfluid state has the lowest energy up to some distance from the trap center, the distance depending on population imbalance and the side of the Feshbach resonance (BCS or BEC) the system is in. Beyond this, the system is in the normal state. Since the system, even in the presence of the harmonic trap, is invariant under O(3) space rotation, the superfluid cores which emerge in the trap are comprised of states that are degenerate with respect to the ``orbital ferromagnet'' state, px\ifmmode±\else\textpm\fiipy{p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}px​\ifmmode±\else\textpm\fiipy​, as well as the ones in which all angular momentum components of the order parameter are nonzero, provided the modulus of the order parameter maintains a fixed magnitude.
We consider a dilute two-component atomic fermion gas with unequal populations in a harmonic trap potential using the mean field theory and the local density approximation. We show that the system is phase separated into concentric shells with the superfluid in the core surrounded by the normal fermion gas both on the weak-coupling BCS side and near the Feshbach resonance. On the strong-coupling BEC side, the composite bosons and left-over fermions can be mixed. We calculate the cloud radii and compare axial density profiles systemically for the BCS, near resonance and BEC regimes.
The problem of a two-component Fermi gas in a harmonic trap, with an imbalanced population and a pairing interaction of zero total momentum, is mapped onto the exactly solvable reduced BCS model. For a one-dimensional trap, the complete ground state diagram is determined with various topological features in ground state energy spectra. In addition to the conventional two-shell density profile of a paired core and polarized outer wings, a three-shell structure as well as a double-peak superfluid distribution are unveiled.
We extend our recent work [Y. Endo et al., Phys. Rev. A 92, 023610 (2015)] for a parity-mixing effect in a model of two-dimensional lattice fermions to a realistic three-dimensional ultracold Fermi gas. Including effects of broken local spatial inversion symmetry by a trap potential within the framework of the real-space Bogoliubov-de Gennes theory at T=0T=0T=0, we point out that an odd-parity p-wavep\text{-wave}p-wave Cooper-pair amplitude is expected to have already been realized in previous experiments on an (even-parity) s-waves\text{-wave}s-wave superfluid Fermi gas with spin imbalance. This indicates that when one suddenly changes the s-waves\text{-wave}s-wave pairing interaction to an appropriate p-wavep\text{-wave}p-wave one by using a Feshbach technique in this case, a nonvanishing p-wavep\text{-wave}p-wave superfluid order parameter is immediately obtained, which is given by the product of the p-wavep\text{-wave}p-wave interaction and the p-wavep\text{-wave}p-wave pair amplitude that has already been induced in the spin-imbalanced s-waves\text{-wave}s-wave superfluid Fermi gas. Thus, by definition, the system is in the p-wavep\text{-wave}p-wave superfluid state, at least just after this manipulation. Since the achievement of a p-wavep\text{-wave}p-wave superfluid state is one of the most exciting challenges in cold Fermi gas physics, our results may provide an alternative approach to this unconventional pairing state. In addition, since the parity-mixing effect cannot be explained as far as one deals with a trap potential in the local density approximation (LDA), it is considered as a crucial example which requires us to go beyond the LDA.
We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes equation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under population imbalance turns into a phase separation (PS) between locally binding superfluid and fully polarized spin domains. We also demonstrate numerical calculations with a large particle number O(10^5), comparable to that observed in recent experiments. The resulting density profile on a resonance yields the PS, which is in good agreement with the recent experiments, while the FFLO modulation exists in the pairing field. It is also proposed that the most favorable location for the detection of the FFLO oscillation is in the vicinity of the critical population imbalance in the weak coupling BCS regime, where the oscillation periodicity becomes much larger than the interparticle spacing. Finally, we analyze the radio-frequency (RF) spectroscopy in the imbalanced system. The clear difference in the RF spectroscopy between BCS and BEC sides reveals the structure of the pairing field and local ``magnetization''.
We study ppp-wave pairing in a two-component Fermi system with unequal population across weak-coupling BCS to strong-coupling BEC regimes. We find a rich ms=0{m}_{s}=0ms​=0 spin triplet ppp-wave superfluid (SF) ground state (GS) structure as a function of population imbalance. Under a phase stability condition, the ``global'' energy minimum is given by a multitude of ``mixed'' SF states formed of linear combinations of $m=\ifmmode\pm\else\textpm\fi{}1,0$ substates of the $\ensuremath{\ell}=1$ orbital angular momentum state. Except for the ``pure'' SF states ($\ensuremath{\ell}=1,m=\ifmmode\pm\else\textpm\fi{}1$), other states exhibit oscillation in energy with the relative phase between the constituent gap amplitudes. We also find states with ``local'' energy minimum that can be stable at higher polarizations, suggesting a quantum phase transition between the ``global'' and ``local'' minima phases driven by polarization. We relate the local and global minimum states to Morse and non-Morse critical points. We calculate the variations of the ground state energy with a relative phase angle, coupling, and polarization, and discuss possible consequences for experiments.
We map out the detailed phase diagram of a trapped ultracold Fermi gas with population imbalance across a wide Feshbach resonance. We show that under the local density approximation, the properties of the atoms in any (anisotropic) harmonic traps are universally characterized by three dimensionless parameters: the normalized temperature, the dimensionless interaction strength, and the population imbalance. We then discuss the possible quantum phases in the trap, and quantitatively characterize their phase boundaries in various typical parameter regions.