Session H4: Spin-orbit Coupling in Ultracold Gases

Topological phase transitions in ultra-cold Fermi superfluids: the evolution from BCS to BEC under artificial spin-orbit fields

We discuss topological phase transitions in ultra-cold Fermi superfluids induced by interactions and artificial spin orbit fields. We construct the phase diagram for population imbalanced systems at zero and finite temperatures, and analyze spectroscopic and thermodynamic properties to characterize various phase transitions. For balanced systems, the evolution from BCS to BEC superfluids in the presence of spin-orbit effects is only a crossover as the system remains fully gapped, even though a triplet component of the order parameter emerges. However, for imbalanced populations, spin-orbit fields induce a triplet component in the order parameter that produces nodes in the quasi-particle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally a fully gapped phase exists, where a crossover from indirect to direct gap occurs, but a topological transition to a gapped phase possessing Majorana fermions edge states does not occur.   

BCS-BEC Crossover and Topological Phase Transition in 3D Spin-Orbit Coupled Degenerate Fermi Gases

We investigate the BCS-BEC crossover in three-dimensional degenerate Fermi gases in the presence of spin-orbit coupling (SOC) and Zeeman field. We show that the superfluid order parameter destroyed by a large Zeeman field can be restored by the SOC. With increasing strengths of the Zeeman field, there is a series of topological quantum phase transitions from a nontopological superfluid state with fully gapped fermionic spectrum to a topological superfluid state with four topologically protected Fermi points (i.e., nodes in the quasiparticle excitation gap) and then to a second topological superfluid state with only two Fermi points. The quasiparticle excitations near the Fermi points realize the long-sought low-temperature analog of Weyl fermions of particle physics. We show that the topological phase transitions can be probed using the experimentally realized momentum-resolved photoemission spectroscopy. Phys. Rev. Lett. 107, 195303 (2011).   

BCS-BEC crossover in 2D spin-orbit coupled degenerate Fermi gases

The recent experimental realization of spin-orbit coupling for ultra-cold atoms has generated much interest in the physics of spin-orbit coupled degenerate Fermi gases. Although recently the BCS-BEC crossover in 3D spin-orbit coupled Fermi gases has been intensively studied, the corresponding 2D crossover physics has remained unexplored. In this talk, we discuss the BCS-BEC crossover physics in 2D degenerate Fermi gases in the presence of spin-orbit coupling. We derive the zero temperature mean field gap and atom number equations suitable for the 2D spin-orbit coupled Fermi gases, from which the dependence of the ground state properties (pairing gap, chemical potential, etc.) on the system parameters (e.g., binding energy, spin-orbit coupling strength) is obtained, both numerically and analytically. We characterize the dependence of the BKT transition temperature as well as the vortex-antivortex lattice melting temperature on the spin-orbit coupling strength and the external Zeeman field.   

One-dimensional Ultracold Fermi Gases with Spin-orbit Coupling

We study one-dimensional ultracold Fermi gases with spin-orbit coupling. We use the Bogoliubov-de Gennes equations to determine pairing order parameter, and find out new phases in addition to fully polarized normal phase, fully paired BCS phase and the FFLO phase. We complete the phase diagram in terms of polarization, interaction parameter and strength of spin-orbit coupling.   

Who is the Lord of the Rings in the Zeeman-spin-orbit Saga: Majorana, Dirac or Lifshitz?

Zeeman, spin-orbit fields and interactions can be tuned in the context of ultra-cold atoms and allow for the visitation of several different phases. For systems with zero Zeeman field, the evolution from BCS to BEC superfluidity in the presence of spin-orbit effects is only a crossover [1]. In contrast, for finite Zeeman fields, spin-orbit coupling induces a triplet component in the order parameter that produces nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type [2]. A fully gapped phase also exists, where a crossover from indirect to direct gap occurs. For spin-orbit couplings with equal Rashba and Dresselhaus strengths the nodal quasi-particles are Dirac fermions that live at and in the vicinity of rings of nodes. Transitions from and to nodal phases can occur via the emergence of zero-mode Majorana fermions at phase boundaries, where rings of nodes of Dirac fermions annihilate [3]. Lastly, we characterize different phases via spectroscopic and thermodynamic properties and conclude that Lifshitz is the ``Lord of the Rings.'' \\[4pt] [1] Li Han, C. A. R. Sa de Melo, arXiv:1106.3613v1.\\[0pt] [2] Kangjun Seo, Li Han and C. A. R. Sa de Melo, arXiv:1108.4068v2.\\[0pt] [3] Kangjun Seo, Li Han and C. A. R. Sa de Melo, arXiv:1110.6364v1.   

Quantum phases of atomic Fermi gases with spin-orbit coupling

We consider a general anisotropic spin-orbit coupling and analyze the phase diagrams of both balanced and imbalanced Fermi gases for the entire BCS-BEC evolution. First we use the self-consistent mean-field theory at zero temperature, and show that the topological structure of the ground-state phase diagrams is quite robust against the effects of anisotropy. Then we go beyond the mean-field description, and investigate the effects of Gaussian fluctuations near the critical temperature. This allows us to derive the time-dependent Ginzburg-Landau theory, from which we extract the effective mass of the Cooper pairs and their critical condensation temperature in the molecular BEC limit.   

Low-density molecular gas of tightly-bound Rashba-Dresselhaus fermions

We study interacting Rashba-Dresselhaus fermions in two spatial dimensions. First, we present a new exact solution to the two-particle pairing problem of spin-orbit-coupled fermions for arbitrary Rashba and Dresselhaus spin-orbit interactions. An exact molecular wave function and the Green function are explicitly derived along with the binding energy and the spectrum of the molecular state. In the second part, we consider a thermal Boltzmann gas of fermionic molecules and compute the time-of-flight velocity and spin distributions for a single fermion in the gas. We show that the pairing signatures can be observed already in the first-moment expectation values, such as time-of-flight density and spin profiles.   

Order by Disorder in Spin-Orbit Coupled Bose-Einstein Condensates

Motivated by recent experiments, we investigate the system of isotropically-interacting bosons with Rashba spin-orbit coupling. At the non-interacting level, there is a macroscopic ground-state degeneracy due to the many ways bosons can occupy the Rashba spectrum. Interactions treated at the mean-field level restrict the possible ground-state configurations, but there remains an accidental degeneracy not corresponding to any symmetry of the Hamiltonian, indicating the importance of fluctuations. By finding analytical expressions for the collective excitations in the long-wavelength limit and through numerical solution of the full Bogoliubov- de Gennes equations, we show that the system condenses into a single momentum state of the Rashba spectrum via the mechanism of order by disorder. We show that in 3D the quantum depletion for this system is small, while the thermal depletion has an infrared logarithmic divergence, which is removed for finite-size systems. In 2D, on the other hand, thermal fluctuations destabilize the system. This work is supported in part by JQI-PFC.   

Exotic 3D Spin-Orbit Couplings

We describe a scheme for creating an isotropic three-dimensional spin-orbit coupling, dubbed Weyl spin-orbit coupling, in systems of ultracold atoms. This coupling is induced by Raman transitions that link four internal atomic states with a tetrahedral geometry. This spin-orbit coupling gives rise to a Dirac point that is robust against environmental perturbations. We then propose a general procedure for generating exotic three-dimensional spin-orbit couplings with degenerate ground states on more complex manifolds. The procedure is applied to produce a spin-orbit coupling with a toroidal ground state manifold. Finally, we discuss the many-body implications of the exotic spin-orbit couplings.   

Cold-atom systems with synthetic SU(3) spin-orbit coupling

Recently, the ability to create and control artificial gauge fields in cold gases has been experimentally demonstrated. Here, we propose a scheme to realize synthetic SU(3) spin-orbit interactions and derive an effective single-particle Hamiltonian, parameterized by the Gell-Mann matrices. We then investigate a many-body system of SU(3)-spin-orbit-coupled bosons and derive and analyze numerically the Gross-Pitaevskii equation to describe the effect of interaction on the possible ground states. The time-of-flight density profiles to probe various many-body states in the rich phase diagram of the system are calculated.   

Superfluidity of Bosons in Optical lattices with Spin-Orbit coupling

Recent experimental and theoretical work has explored artificial spin-orbit coupling induced among two species of boson. Here we examine superfluidity of a cold gas of bosons with spin-orbit coupling in a periodic optical lattice, in the presence of additional short-range interactions. We compute the density distribution after free expansion from the lattice as a probe of superfluidity, and phase transitions, of the trapped gas.   

Spinor Bose-Einstein Condensates Under Synthetic Gauge Field

Due to the recent popularity of synthetic gauge field in ultracold atoms, I will talk about the combined effects of Rashba spin-orbit coupling (SOC) and rotation in spin -1/2 condensates [X.-Q. Xu et al, Phys. Rev. Lett. 107, 200401 (2011)]. Novel features appear in the ground state wave function, such as the existence of a half-quantum vortex or giant vortex, domains of stripe-like phase, suppressed Skyrmion order. Additionally, I will talk about the interesting mapping between pure Rashba BECs and chiral magnets with Dzyaloshinskii-Moriya (DM) interaction.   

BCS-BEC crossover induced by a synthetic non-Abelian gauge field

We investigate the ground state of interacting spin-$half$ fermions(3D) at a finite density ($\rho \sim k_F^3$) in the presence of a uniform non-Abelian gauge field (with magnitude $\lambda$) that generates a generalized Rashba spin-orbit interaction. For a weak attractive interaction in the singlet channel described by a small negative scattering length $(k_F |a_s| \la 1)$, the ground state in the absence of the gauge field ($\lambda=0$) is a BCS superfluid with large overlapping pairs. With increasing $\lambda$, a non-Abelian gauge field engenders a crossover of this BCS ground state to a BEC ground state of bosons even with a weak attractive interaction. For large gauge couplings $(\lambda/k_F \gg 1)$, the BEC attained is a condensate of bosons whose properties are solely determined by the gauge field (and not by the scattering length); we call these bosons ``rashbons.'' In the absence of interactions ($a_s = 0^-$), the shape of the Fermi surface of the system undergoes a topological transition at a critical gauge coupling $\lambda_T$. For high symmetry gauge field configurations we show that the crossover from the BCS superfluid to the rashbon BEC occurs in the regime of $\lambda$ near $\lambda_T$.   

Trapped fermions in a synthetic non-Abelian gauge field

On increasing the coupling strength ($\lambda$) of a non-Abelian gauge field that induces a generalized Rashba spin-orbit coupling, the topology of the Fermi surface of a homogeneous gas of non-interacting fermions of density $\rho \sim k_F^3$ undergoes a change at a critical value, $\lambda_T \approx k_F$ [PRB {\bf 84}, 014512 (2011)]. We analyze how this affects the size/shape of a cloud of fermions trapped in a harmonic potential. We develop an adiabatic formulation, with Pancharatnam-Berry phase terms, for the one particle states in a trap with the gauge field. Local density approximation reveals that the cloud shrinks in a {\em characteristic fashion with increasing $\lambda$} and predicts a spherical cloud for all gauge field configurations. We show, via a calculation of the cloud shape using exact eigenstates, that for certain gauge fields there is systematic anisotropy in the cloud shape that increases with increasing gauge coupling $\lambda$. An important spin-off of our adiabatic formulation is that it reveals exciting possibilities for the cold-atom realization of interesting Hamiltonians (eg. quantum hall spherical geometry) by using a non-Abelian gauge field in conjunction with another potential.   

Effects of the interplay between spin-orbit coupling and interaction on bosons

We show that spin-orbit coupling drastically changes the properties of bosons. The interplay between the spin-orbit coupling and interaction determines the fate of Bose-Einstein condensate, which may even not exist in the presence of isotropic spin-orbit coupling. For anisotropic spin-orbit coupling, condensates survive and are characterized by anisotropic energy spectrum, with a slower sound velocity along the direction of weaker spin-orbit coupling. The spectrum can also be used to distinguish the plane wave phase and the tripe phase.