We propose a four-level scheme for generating a uniform imaginary magnetic field for ultracold atoms using Raman coupling and state-dependent particle losses. Eliminating two lossy auxiliary states and subsequently projecting onto one of the two remaining Raman dressed atomic states, the atomic motion is affected by an imaginary vector potential with a constant curl corresponding to a spatially uniform imaginary synthetic magnetic field. We analytically derive the Gaussian wavepacket dynamics for atoms in such a synthetic magnetic field and study the width-dependent transport, which is unique to non-Hermitian systems. We find numerical agreement between the two-level non-Hermitian Hamiltonian and the adiabatic projection description, confirming the applicability of the latter adiabatic approach.
We propose a four-level scheme for generating a uniform imaginary magnetic field for ultracold atoms using Raman coupling and state-dependent particle losses. Eliminating two lossy auxiliary states and subsequently projecting onto one of the two remaining Raman dressed atomic states, the atomic motion is affected by an imaginary vector potential with a constant curl corresponding to a spatially uniform imaginary synthetic magnetic field. We analytically derive the Gaussian wavepacket dynamics for atoms in such a synthetic magnetic field and study the width-dependent transport, which is unique to non-Hermitian systems. We find numerical agreement between the two-level non-Hermitian Hamiltonian and the adiabatic projection description, confirming the applicability of the latter adiabatic approach.
Ultracold gases are remarkably versatile and controllable systems with untapped potential to emulate a variety of quantum phenomena [1]. Atoms are, however, electrically neutral and therefore it is highly desirable to make them behave as charged particles. This capability has recently been demonstrated in a series of recent experiments [2–4]. Although many challenges remain ahead, atoms exposed to synthetic gauge field offer great opportunities for the realization of new classes of quantum materials with custom-designed properties, and with unforeseen applications. The first implementation of an effective magnetic field was accomplished by rotating an atomic cloud [5]. In the rotating frame of reference, an atom experiences the Coriolis force that emulates the Lorentz force experienced by a charged particle in a uniform magnetic field. Large vortex lattices have been created in this way, however, so far the strength of the effective magnetic field achieved by rotation is too weak to lead to new physics. An alternative approach invokes the vector potential, A, that induces a geometric phase as a charged particle moves around a closed loop. This notion of geometric phase can be adapted to a neutral atom that slowly moves in the presence of laser fields. If the lasers are chosen to ‘dress’ the internal atomic levels, so that the dressed eigenstates become position dependent, the center of mass adiabatically evolves according the operator p-A, instead of just p = −iℏ∇. The atom thus effectively moves as a charged particle in a magnetic field B = ∇ × A (see Fig. 1a). Exploiting this idea series of experiments have implemented artificial magnetic and electric fields [2–4]. The idea has also been implemented in the presence of a superimposed periodic optical lattice potential (see Fig. 1b). In this case, the length scale imposed by the lattice competes with the one associated with the magnetic field and can lead to frustration, manifested as an energy spectrum with fractal characteristics [2,3]. Light induced synthetic gauge fields: (a) A gas of neural atoms (green oval) can behave as a gas of charged particles when it is illuminated by a pair of laser beams (red arrows) that imparts momentum to the atoms and couples their internal (.|↑.〉, .|↓.〉) and motional degrees of freedom. The Raman beams that induce transitions between internal states also transfer two units of their linear momentum 2kL. This leads to the formation of dressed states (superpositions of states with different internal levels and center of mass momenta) which experience a modified kinetic energy, with an effective vector potential. If the effective vector potential has nonzero curl, the atoms move as charged particles in a magnetic field. (b) In the presence of an optical lattice potential, Raman beams can also induce an effective vector potential. In this case, the lattice is engineered so that along one direction atoms can tunnel between adjacent lattice sites (Jlight) only by absorbing and emitting photons from the beams. The imparted momentum from the lasers produces path-dependent phases that resemble the magnetic flux accumulated by a charged particle as it moves in a closed loop. The flux accumulated by the atoms when they move around a single unit cell, ϕ, can be significant (equivalent to the one generated by a real magnetic field of thousands tesla in an electronic system) and induce a fractal energy spectrum. Generalizations to emulate non-Abelian gauge potentials, associated in many cases to spin-orbit coupling can emerge when the geometric phase is replaced by a matrix acting on the internal Hilbert space [2–5]. This happens if there is a manifold of degenerate dressed states, such that even for arbitrarily slow motion, after traveling around a contour the particle ends up in a different internal state. Although various proposals for generating non-Abelian potentials exist, they remain to be implemented. On the other hand, Abelian spin-orbit coupling, arising from the momentum kicks imparted on the atom by the laser photons that drive internal transitions, has been realized in several laboratories using different laser configurations in alkali-metal (one outer electron) and alkaline-earth (two outer electrons) atoms [2–5]. There is a great vista ahead. For example, the inclusion of collisions or dipolar interactions will open a window to engineer fractional Hall liquids, unconventional superfluids and interacting topological phases. The creation of dynamical gauge fields important in many areas of physics such as quantum chromodynamics is also an open direction. The time is just ripe for investigating rich and unexplored physics with synthetic gauge fields for ultracold matter.
Synthetic gauge fields for ultracold neutral atoms-engineered using the interaction between laser fields and the atoms' internal 'spin' degrees of freedom-provide promising techniques for generating the large (synthetic) magnetic fields required to reach the fractional quantum Hall (FQH) limit in quantum gases, bosonic or fermionic alike.Because neutral atoms can move in a nearly disorder-free environment and they have extremely simple contact interactions, the resulting FQH states would be revealed in their most essential form.Moreover, bosonic FQH states represent a new frontier and have never been seen in any setting.Going beyond electromagnetismʼs conventional scalar gauge field, it is possible to create more general non-Abelian gauge potentials.When these are spatially uniform, they are equivalent to spin-orbit coupling familiar in material systems, and can lead to cold atom analogs of topological insulators and topological superconductors.In this tutorial, we introduce basic concepts underlying these gauge fields, making connections to the Aharonov-Bohm phase and geometric phase.We focus on the system of neutral atoms 'dressed' by multiple laser beams, where the eigenstates of the resulting Hamiltonian are known as dressed states.Synthetic gauge potentials arise from the unitary transformation required to express these dressed states in terms of the laser-free eigenstates.We discuss stability of laser-dressed atoms corresponding to the adiabatic condition and the probability of non-adiabatic transitions.Adopting both the semiclassical and quantum mechanical approaches, we demonstrate they agree in the suitable limit.We also analyze using both the conventional adiabatic picture and exact picture, where the kinetic energy is neglected in the former and retained in the latter picture.
We use Raman coupling between magnetic sublevels of a 87 Rb BEC to create an effective vector gauge field for the neutral atoms. They behave as if they were charged particles in a magnetic vector potential. With appropriate spatial gradients and time derivatives of this vector potential we produce synthetic magnetic and electric fields for the atoms. This paper is a brief summary of the presentation made by the first author at ICAP 2010 in Cairns, Australia. It is not intended as a complete report on the research described, nor does it attempt to be a fully referenced documentation of that research and the relevant literature. Here, we provide references to the published work where a more complete description and more complete references to related literature may be found.
We describe a method to generate a synthetic gauge potential for ultracold atoms held in an optical lattice. Our approach uses a time-periodic driving potential based on quickly alternating two Hamiltonians to engineer the appropriate Aharonov-Bohm phases, and permits the simulation of a uniform tunable magnetic field. We explicitly demonstrate that our split-driving scheme reproduces the behavior of a charged quantum particle in a magnetic field over the complete range of field strengths, and obtain the Hofstadter butterfly band structure for the Floquet quasienergies.
We propose a scheme to create an effective magnetic field for ultracold atoms in a planar geometry. The setup allows the experimental study of classical and quantum Hall effects in close analogy to solid-state systems including the possibility of finite currents. The present scheme is an extention of the proposal in Phys. Rev. Lett. 93, 033602 (2004), where the effective magnetic field is now induced for three-level $\ensuremath{\Lambda}$-type atoms by two counterpropagating laser beams with shifted spatial profiles. Under conditions of electromagnetically induced transparency the atom-light interaction has a space-dependent dark state, and the adiabatic center-of-mass motion of atoms in this state experiences effective vector and scalar potentials. The associated magnetic field is oriented perpendicular to the propagation direction of the laser beams. The field strength achievable is one flux quantum over an area given by the transverse beam separation and the laser wavelength. For a sufficiently dilute gas the field is strong enough to reach the lowest Landau level regime.
The adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields creating a dark state can give rise to effective gauge potentials. We show that these potentials lead to effective magnetic fields if the light fields possess a relative orbital angular momentum. This allows to study quantum-Hall like effects in ultra-cold atomic gases in various geometries. If the atom-light interaction creates several degenerate adiabatic eigenstates, the associated gauge potentials are non-Abelian. A pair of such degenerate dark states emerges e.g. if laser fields couple three internal states of an atom to a fourth common one under pairwise two--photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.