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arXiv:2201.00885 · Physical Review Letters

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields

Mehedi Hasan, Chetan Sriram Madasu, Ketan D. Rathod, Chang Chi Kwong, Christian Miniatura, Frédéric Chevy, David Wilkowski

Published Sep 19, 2022• 27 citations
arXivHTMLPDF

Abstract

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

DOI
10.1103/physrevlett.129.130402

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It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.
Gauge fields play a prominent role in modern physics from electromagnetic theory to standard model of particle physics. These are cutting edge mathematical tools to understand fundamental forces and interaction between sub-atomic particles. Though gauge fields are considered natural tools of study in high-energy physics, they can be extended to low-energy condensed matter physics to study the behaviour of quantum particles in special environments like electrons in a magnetic field. Apart from naturally occurring gauge fields, controllable artificial gauge fields can be used to tailor physical effects on quantum systems like neutral atoms, qubits, photons, anions etc. This thesis reports the study of laser-induced artificial non-Abelian gauge fields interacting with an ultracold atomic wave packet in free space. In sharp contrast to the Abelian gauge fields, spatially uniform non-Abelian gauge fields can induce non-inertial motion of a particle. This motion is locked to the pseudo-spin state of the system and induces spin-orbit coupling. These non-Abelian gauge fields are experimentally realized by adiabatic evolution of the state of the atoms in the degenerate subspace of the coupling Hamiltonians. We produce SU(2) and SU(3) non-Abelian gauge fields using doubly degenerate dark state subspace of tripod and triply degenerate dark state subspace of double-tripod atom-light coupling schemes, respectively. Using the SU(2) gauge field in one dimension, we demonstrate the function of an atomtronic Datta-Das transistor (DDT) where spin polarized atoms enter the gate region and exit with their spin orientation controlled by the gate parameter analogous to the gate voltage in conventional field effect transistor. We demonstrate the geometric nature of the spin rotation in our DDT by showing insensitivity of the result to the input velocity and velocity dispersion of the atomic cloud. We have also implemented a Ramsey interferometric sequence to extract the phase of the output state of the DDT showing that the spin rotation by the DDT is coherent. In two dimensions, the dynamics of the wave packet in SU(2) gauge field show Zitterbewegung oscillations revealing spin Hall effect characteristics of the system. Whereas the dynamics of the wave packet in SU(3) gauge field show oscillations with multiple frequencies emulating color-orbit coupling. These frequencies correspond to the energy separation between the eigenstates of the Hamiltonian. Using the SU(3) Hamiltonian, we show two paths of reaching the same final state from the initial state demonstrating the existence of additional ladder operators in the SU(3) system in addition to the well-known raising and lowering operators of angular momentum represented by SU(2) symmetry. With this study, we realize a system that can be used to study the rich physics of SU(3) Hamiltonians on table top experiments.
Non-Abelian gauge field plays an important role in high-energy physics through Yang-Mill theory, in condensed-matter physics, for instance, with anomalous quantum Hall effect, and in quantum information. In its SU(2) symmetry version, Hamiltonians with non-Abelian gauge field reduce to spin-orbit coupling-like systems, taking the general form: H = p 2 /2m+ A.p /m , where m is the mass of the particle, p the momentum and A the non-Abelian gauge field with 2x2 matrix components, decomposed on SU(2) Lie group generators. These class of Hamiltonians can be simulate with ultracold gas [1] using various methods [2] . The topological properties of spin-orbit-coupled system have been explore analysing the energy bands in 2D Fermionic system [3] . Here, the eigenenergies have momentum dependence, leading to Rabi flopping with periodic change of quasi-momentum, even if the non-Abelian gauge field is homogenous. This oscillation is similar to relativistic Zitterbewegung predicted in Dirac equation and simulated in ultracold system [4] .
Abstract Quantum states can acquire a geometric phase called the Berry phase after adiabatically traversing a closed loop, which depends on the path not the rate of motion. The Berry phase is analogous to the Aharonov–Bohm phase derived from the electromagnetic vector potential, and can be expressed in terms of an Abelian gauge potential called the Berry connection. Wilczek and Zee extended this concept to include non-Abelian phases—characterized by the gauge-independent Wilson loop—resulting from non-Abelian gauge potentials. Using an atomic Bose–Einstein condensate, we quantum-engineered a non-Abelian SU(2) gauge field, generated by a Yang monopole located at the origin of a 5-dimensional parameter space. By slowly encircling the monopole, we characterized the Wilczek–Zee phase in terms of the Wilson loop, that depended on the solid-angle subtended by the encircling path: a generalization of Stokes’ theorem. This observation marks the observation of the Wilson loop resulting from a non-Abelian point source.
On increasing the coupling strength ($\ensuremath{\lambda}$) of a non-Abelian gauge field that induces a generalized Rashba spin-orbit interaction, the topology of the Fermi surface of a homogeneous gas of noninteracting fermions of density $\ensuremath{\rho}\ensuremath{\sim}{k}_{F}^{3}$ undergoes a change at a critical value, ${\ensuremath{\lambda}}_{T}\ensuremath{\approx}{k}_{F}$ [Phys. Rev. B 84, 014512 (2011)]. In this paper we analyze how this phenomenon affects the size and shape of a cloud of spin-12\frac{1}{2}21​ fermions trapped in a harmonic potential such as those used in cold atom experiments. We develop an adiabatic formulation, including the concomitant Pancharatnam-Berry phase effects, for the one-particle states in the presence of a trapping potential and the gauge field, obtaining approximate analytical formulas for the energy levels for some high symmetry gauge field configurations of interest. An analysis based on the local density approximation reveals that, for a given number of particles, the cloud shrinks in a characteristic fashion with increasing $\ensuremath{\lambda}$. We explain the physical origins of this effect by a study of the stress tensor of the system. For an isotropic harmonic trap, the local density approximation predicts a spherical cloud even for anisotropic gauge field configurations. We show, via a calculation of the cloud shape using exact eigenstates, that for certain gauge field configurations there is a systematic and observable anisotropy in the cloud shape that increases with increasing gauge coupling $\ensuremath{\lambda}$. The reasons for this anisotropy are explained using the analytical energy levels obtained via the adiabatic approximation. These results should be useful in the design of cold atom experiments with fermions in non-Abelian gauge fields. An important spin-off of our adiabatic formulation is that it reveals exciting possibilities for the cold-atom realization of interesting condensed matter Hamiltonians by using a non-Abelian gauge field in conjunction with another potential. In particular, we show that the use of a spherical non-Abelian gauge field with a harmonic trapping potential produces a monopole field giving rise to a spherical geometry quantum Hall-like Hamiltonian in the momentum representation.
Quantum states can acquire a phase called the Berry phase after adiabatically traversing a closed loop in parameter space. This phase is geometric -- dependent on the path -- not the rate of motion. Wilczek and Zee extended this concept to include non-Abelian phases that can be characterized the Wilson loop, a gauge independent quantity. Here we quantum-engineer a non-Abelian SU(2) gauge field for an atomic Bose-Einstein Condensate (BEC), in a 5-dimensional synthetic parameter space derived from internal atomic degrees of freedom. By slowly encircling a topological monopole, we observed the adiabatic Wilczek-Zee phase that we characterize in terms of the Wilson loop.