We demonstrate the emergence of the non-Abelian geometric potentials and thus the three-dimensional (3D) spin-orbit coupling (SOC) for ultracold atoms without using the laser beams. This is achieved by subjecting an atom to a periodic perturbation which is the product of a position-dependent Hermitian operator $\hat{V}\left(\mathbf{r}\right)$ and a fast oscillating periodic function $f\left(ωt\right)$ with a zero average. To have a significant spin-orbit coupling (SOC), we analyze a situation where the characteristic energy of the periodic driving is not necessarily small compared to the driving energy $\hbarω$. Applying a unitary transformation to eliminate the original periodic perturbation, we arrive at a non-Abelian (non-commuting) vector potential term describing the 3D SOC. The general formalism is illustrated by analyzing the motion of an atom in a spatially inhomogeneous magnetic field oscillating in time. A cylindrically symmetric magnetic field provides the SOC involving the coupling between the spin $\mathbf{F}$ and all three components of the orbital angular momentum (OAM) $\mathbf{L}$. In particular, the spherically symmetric monopole-type synthetic magnetic field $\mathbf{B}\propto\mathbf{r}$ generates the 3D SOC of the $\mathbf{L}\cdot \mathbf{F}$ form, which resembles the fine-structure interaction of hydrodgen atom. However, the strength of the SOC here goes as $1/r^{2}$ for larger distances, instead of $1/r^3$ as in atomic fine structure. Such a longer-ranged SOC significantly affects not only the lower states of the trapped atom, but also the higher ones. Furthermore, by properly tailoring the external trapping potential, the ground state of the system can occur at finite OAM, while the ground state of hydrogen atom has zero OAM.
We propose a practical protocol to generate and observe a non-Abelian geometric phase using a periodically driven Raman process in the hyperfne ground state manifold of atoms in a dilute ultracold gas. Our analysis is based upon recent developments and application of Floquet theory to periodically driven quantum systems. The simulation results show the non-Abelian gauge transformation and the non-commuting property of the SU(2) transformation operators. Based on these results, we propose a possible experimental implementation with an ultracold dilute Bose gas.
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V\left(\boldsymbolλ\left(t\right)\right)$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (non-commuting) geometric phases appearing when the slowly varying parameter $\boldsymbolλ=\boldsymbolλ\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the …
We propose a method for the emulation of artificial spin orbit coupling in a system of ultracold, neutral atoms trapped in a tight-binding lattice. This scheme does not involve near-resonant laser fields, avoiding the heating processes connected to the spontaneous emission of photons. In our case, the necessary spin dependent tunnel matrix elements are generated by a rapid, spin dependent, periodic force, which can be described in the framework of an effective, time averaged Hamiltonian. An additional radio frequency coupling between the spin states leads to a mixing of the spin bands.
Anomalous multi-gap topological phases in periodically driven quantum rotors
Institute of Science and Technology Austria（奥地利科学技术研究院）；TCM Group, Cavendish Laboratory, Department of Physics, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom（英国剑桥大学卡文迪许实验室物理系）
We study the single-particle dispersion of a spin-orbit coupled (SOC) Bose-Einstein condensate (BEC) under the periodical modulation of the Raman coupling. This modulation introduces a further coupling of the SOC dressed eigenlevels, thus creating a second generation of modulation-dressed eigenlevels. Theoretical calculations show that these modulation-dressed eigenlevels feature a pair of avoided crossings and a richer spin-momentum locking, which we observe using BEC transport measurements. Furthermore, we use the pair of avoided crossings to engineer a tunable Stueckelberg interferometer th…
It is found that all the {\em singlet orbital pairing} instabilities are {\em absent} in a class of spin-polarized multi-orbital systems with quadratic band touching, which opens the way for {\em triplet orbital pairing} order. The ground states are found to be {\em non-Abelian} states with p-wave orbital pairing in checkerboard (away from 1/2 filling) and kagome (above 1/3 filling) lattices with {\em isotropic} attractive interaction which can be realized in ultracold multi-orbital optical lattices. The special property of such systems is generalized to more classes of multi-orbital systems, …
We provide an explicit way to implement non-Abelian spin-orbital-angular-momentum (SOAM) coupling in spinor Bose-Einstein condensates using magnetic gradient coupling. For a spherical surface trap addressable using high-order Hermite-Gaussian beams, we show that this system supports various degenerate ground states carrying different total angular momenta $\mathbf{J}$, and the degeneracy can be tuned by changing the strength of SOAM coupling. For weakly interacting spinor condensates with $f=1$, the system supports various meta-ferromagnetic phases and meta-polar states described by quantized …
The Higgs-like excitations, which distinguish from the Higgs amplitude mode in many-body system, are single-particle excitations in system with non-Abelian gauge potential. We investigate the Higgs-like excitations of cold atom system in artificial non-Abelian gauge potential. We demonstrate that when a non-Abelian gauge potential is reduced to Abelian potential, its Abelian part constructs spin-orbit coupling, and its non-Abelian part emerges Higgs-like excitations. The Higgs-like excitations induce a mass of the non-Abelian gauge field, which offsets the defect of massless of the gauge theor…
In the strongly interacting limit of the Hubbard model localized double-occupancies form effective hard-core bosonic excitations, called a doublons, which are long-lived due to energy conservation. Using time-dependent density-matrix renormalisation group we investigate numerically the dynamics of doublons arising from the sudden expansion of a spatially confined band-insulating state in one spatial dimension. By analysing the occupation scaling of the natural orbitals within the many-body state, we show that doublons dynamically quasicondense at the band edges, consistent with the spontaneous…
Time periodic driving serves not only as a convenient way to engineer effective Hamiltonians, but also as a means to produce intrinsically dynamical phases that do not exist in the static limit. A recent example of the latter are 2D chiral Floquet (CF) phases exhibiting anomalous edge dynamics that pump discrete packets of quantum information along one direction. In non-fractionalized systems with only bosonic excitations, this pumping is quantified by a dynamical topological index that is a rational number -- highlighting its difference from the integer valued invariant underlying equilibrium…
We propose an experiment to directly probe the non-Abelian statistics of Majorana fermions by braiding them in an s-wave superfluid of ultracold atoms. We show different orders of braiding operations give orthogonal output states that can be distinguished through Raman spectroscopy. Realization of Majorana bound states in an s-wave superfluid requires strong spin-orbital coupling and a controllable Zeeman field in the perpendicular direction. We present a simple laser configuration to generate the artificial spin-orbital coupling and the required Zeeman field in the dark state subspace.
Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating non-trivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micro-motion. This framework allows for the identification of driving schemes, based on general N-step modulations, which lead to configurations relevant…
We theoretically introduce and study a three-dimensional Hofstadter model with linearly varying non-Abelian gauge potentials along all three dimensions. The model can be interpreted as spin-orbit coupling among a trio of Hofstadter butterfly pairs since each Cartesian surface ($xy$, $yz$, or $zx$) of the model reduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating the commutativity among arbitrary loop operators around all axes, we derive its genuine (necessary and sufficient) non-Abelian condition, namely, at least two out of the three hopping phases should be neither 0 no…
Chiral d-wave superfluid is a preliminary example of topological matters that intrinsically encodes interaction effects. It exhibits fascinating properties including a finite Chern number in the bulk and topologically protected edge states, which have been invoking physicists for decades. However, unlike s-wave superfluids prevalent in nature, its existence requires a strong interaction in the d-wave channel, a criterion that is difficult to access in ordinary systems. So far, such an unconventional superfluid has not been discovered in experiments. Here, we present a new principle for creatin…
We analytically calculate phase boundaries of tunable spin-orbit coupled BECs with effective two-body interactions by using variational method. Phase diagrams for periodically driving $^{87}\rm{Rb}$ and $^{23}\rm{Na}$ systems are presented, respectively, which display several characteristic features contrast with those of undriven systems. In the $^{87}\rm{Rb}$ BECs, the critical density $n_{c}$ (density at quantum tricritical point) can be dramatically reduced in some parameter regions, thus the prospect of observing this intriguing quantum tricritical point is greatly enlarged. Moreover, a s…
Motivated by the recent experiment realizing bidirectional spin-orbit-coupled Bose-Einstein condensates (BEC), we theoretically explore the properties of repulsive fermions in the two-dimensional (2D) optical lattice with such non-Abelian gauge fields. Within the mean-field level, we find a novel phase of topological antiferromagnetic (TAFM) order which incorporates both the non-trivial topology due to spin-flip hopping and spontaneous symmetry breaking (SSB) for the in-plane spin order. We argue that the appearance of such a phase is generic for repulsive fermions in Chern bands achieved thro…
The Hofstadter model, well-known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment [Science, 365, 1021 (2019)], we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and sym…
We study the Feshbach resonance of spin-1/2 particles in the presence of a uniform synthetic non-Abelian gauge field that produces spin orbit coupling along with constant spin potentials. We develop a renormalizable quantum field theory that includes the closed channel boson which engenders the Feshbach resonance, in the presence of the gauge field. By a study of the scattering of two particles in the presence of the gauge field, we show that the Feshbach magnetic field, where the apparent low energy scattering length diverges, depends on the conserved centre of mass momentum of the two partic…
We consider the simulation of non-abelian gauge potentials in ultracold atom systems with atom-field interaction in the $Λ$ configuration where two internal states of an atom are coupled to a third common one with a detuning. We find the simulated non-abelian gauge potentials can have the same structures as those simulated in the tripod configuration if we parameterize Rabi frequencies properly, which means we can design spin-orbit coupling simulation schemes based on those proposed in the tripod configuration. We show the simulated spin-orbit coupling in the $Λ$ configuration can only be of a…
We discuss the creation of parity violating Fermi superfluids in the presence of non-Abelian gauge fields involving spin-orbit coupling and crossed Zeeman fields. We focus on spin-orbit coupling with equal Rashba and Dresselhaus (ERD) strengths which has been realized experimentally in ultra-cold atoms, but we also discuss the case of arbitrary mixing of Rashba and Dresselhaus (RD) and of Rashba-only (RO) spin-orbit coupling. To illustrate the emergence of parity violation in the superfluid, we analyze first the excitation spectrum in the normal state and show that the generalized helicity ban…
