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arXiv:1604.02229 · Atoms

Cavity Optomechanics with Ultra Cold Atoms in Synthetic Abelian and Non-Abelian Gauge Field

Bikash Padhi, Sankalpa Ghosh

Published Dec 25, 2015• 1 citation
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Abstract

In this article we present a pedagogical discussion of some of the optomechanical properties of a high finesse cavity loaded with ultracold atoms in laser induced synthetic gauge fields of different types. Essentially, the subject matter of this article is an amalgam of two sub-fields of atomic molecular and optical (AMO) physics namely, the cavity optomechanics with ultracold atoms and ultracold atoms in synthetic gauge field. After providing a brief introduction to either of these fields we shall show how and what properties of these trapped ultracold atoms can be studied by looking at the cavity (optomechanical or transmission) spectrum. In presence of abelian synthetic gauge field we discuss the cold-atom analogue of Shubnikov de Haas oscillation and its detection through cavity spectrum. Then, in the presence of a non-abelian synthetic gauge field (spin-orbit coupling), we see when the electromagnetic field inside the cavity is quantized, it provides a quantum optical lattice for the atoms, leading to the formation of different quantum magnetic phases. We also discuss how these phases can be explored by studying the cavity transmission spectrum.

DOI
10.3390/atoms4010001

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In this article we present a pedagogical discussion of some of the optomechanical properties of a high finesse cavity loaded with ultracold atoms in laser induced synthetic gauge fields of different types. Essentially, the subject matter of this article is an amalgam of two sub-fields of atomic molecular and optical (AMO) physics namely, the cavity optomechanics with ultracold atoms and ultracold atoms in synthetic gauge field. After providing a brief introduction to either of these fields we shall show how and what properties of these trapped ultracold atoms can be studied by looking at the cavity (optomechanical or transmission) spectrum. In presence of abelian synthetic gauge field we discuss the cold-atom analogue of Shubnikov de Haas oscillation and its detection through cavity spectrum. Then, in the presence of a non-abelian synthetic gauge field (spin-orbit coupling), we see when the electromagnetic field inside the cavity is quantized, it provides a quantum optical lattice for the atoms, leading to the formation of different quantum magnetic phases. We also discuss how these phases can be explored by studying the cavity transmission spectrum.
Ultracold fermionic atoms placed in a synthetic magnetic field arrange themselves in Landau levels. We theoretically study the optomechanical interaction between the light field and collective excitations of such fermionic atoms in synthetic magnetic field by placing them inside a Fabry-Perot cavity. We derive the effective Hamiltonian for particle hole excitations from a filled Landau level using a bosonization technique and obtain an expression for the cavity transmission spectrum. Using this we show that the cavity transmission spectrum demonstrates cold atom analog of Shubnikov‐de Haas oscillation in electronic condensed matter systems. We discuss the experimental consequences for this oscillation for such a system and the related optical bistability. By allowing an ultracold atomic ensemble to interact with a selected mode of a high-finesse cavity it is possible to probe the quantum many-body state of such an ultracold atomic system. The resulting cavity optomechanics or cavity quantum electrodynamics with ultracold atoms has recently witnessed a significant development [1]. The experimental successes include the coupling of collective density excitation of an ultracold bosonic condensate with a single-cavity mode and observation of the coupled dynamics through cavity transmission [2], a strongly coupled cavity mode with a highly localized ultracold atomic condensate trapped inside a single antinode of a cavity field [3], demonstration of strong ultracold atomcavity-coupling-induced optical nonlinearity even at low photon density [4], and selected atom-photon coupling of a single atomic ensemble in a multiensemble system [5 ]t o name a few. In another development, there has been significant experimental and theoretical progress in studying the effect of optically induced artificial or synthetic gauge field [6] on such neutral atoms, making it a playground for quantum simulation of phenomena that occur when an electronic condensed matter system is placed in a real magnetic field. The experimental achievements in this direction include the observation of vortices, Abrikosov vortex lattice in trapped ultracold atomic superfluid initially by achieving the synthetic magnetic field through the rotation of the trap [7], and later by Raman laser induced spatially varying coupling of the hyperfine states of such ultracold atoms [8]. A more recent development in this direction includes the creation of optical flux lattices [9], realization of spin-orbit coupling for such neutral ultracold bosonic [10] and fermionic atoms [11] with the possibility of creating ultracold atomic analogs of topological condensed matter phases [12]. This Letter aims to combine these two developments by considering a system of such ultracold atoms trapped inside a high-finesse Fabry-Perot cavity interacting with
Ultracold fermionic atoms placed in a synthetic magnetic field arrange themselves in Landau levels. We theoretically study the optomechanical interaction between the light field and collective excitations of such fermionic atoms in synthetic magnetic field by placing them inside a Fabry-Perot cavity. We derive the effective Hamiltonian for particle hole excitations from a filled Landau level using a bosonization technique and obtain an expression for the cavity transmission spectrum. Using this we show that the cavity transmission spectrum demonstrates cold atom analog of Shubnikov-de Haas oscillation in electronic condensed matter systems. We discuss the experimental consequences for this oscillation for such a system and the related optical bistability.
We start by reviewing the concept of gauge invariance in quantum mechanics, for Abelian and Non-Ableian cases. Then we idescribe how the various gauge potential and field can be associated with the geometrical phase acquired by a quantum mechanical wave function while adiabatically evolving in a parameter space. Subsequently we show how this concept is exploited to generate light induced gauge field for neutral ultra cold bosonic atoms. As an example of such light induced Abelian and Non Abelian gauge field for ultra cold atoms we disucss ultra cold atoms in a rotating trap and creation of synthetic spin orbit coupling for ultra cold atomic systems using Raman lasers.
Abelian and non-Abelian gauge potentials and quantum gauge theories play central roles in our understanding of Nature. Gauge potentials are also of great significance in condensed matter physics. In fact, minimal coupling of electrons to an Abelian or non-Abelian gauge potential is the essential ingredient for the realization of topological states of matter. Due to the high controllability of ultracold atoms, they are commonly exploited to test fundamental theories of physics, simulate intractable systems, and realize exotic many-body states. Nonetheless, the charge neutrality of atoms places severe constraints on ultracold atomic systems, since neutral particles do not couple to gauge potentials the way charged particles do. That said, coupling a multi-component quantum gas to laser light can lead to the emergence of artificial Abelian and non-Abelian gauge potentials minimally coupled to the center-of-mass momenta of ultracold neutral atoms, paving the way for realizing topological states of matter and simulating gauge theories in quantum gases. All previous proposals for inducing gauge potentials have been developed in the semiclassical regime, where the radiation field is treated classically. I have developed a two-photon Raman scheme in the strong atom-photon coupling (i.e., quantum) regime, based on two counter-propagating modes of a ring cavity, to induce both synthetic magnetic field and spin-orbit coupling for a single neutral atom inside the cavity. The spin-orbit interaction is only weakly dependent on the occupation of the cavity modes, whereas the strength of the magnetic field is proportional to the square of the total number of photons in the cavity and can be made arbitrarily large, which is desirable for realizing the quantum Hall phase. I have then extended this single-atom cavity quantum-electrodynamics scheme to many bosons in the weak atom-photon coupling regime. In addition to inducing spin-orbit coupling for the individual atoms, the cavity fields also mediate infinite-ranged interactions between atoms, whose strengths and signs can readily be tuned experimentally. The interplay between these cavity-mediated interactions and the intrinsic two-body interactions determines the many-body ground state and its elementary excitations, with novel consequences such as the stabilization of an attractive Bose-Einstein condensate which otherwise is unstable.
We study the spin-orbit-coupled ultracold Bose-Einstein condensate placed in a single-mode Fabry-P\'erot cavity. The cavity introduces a quantum optical lattice potential which dynamically couples with the atomic degrees of freedom and realizes a generalized modified Bose-Hubbard model whose zero-temperature phase diagram can be controlled by tuning the cavity parameters. In the noninteracting limit, where the atom-atom interaction is set to 0, the resulting atomic dispersion shows interesting features such as a bosonic analog of Dirac points, a cavity-controlled Hofstadter spectrum which bears the hallmark of pseudospin-1/2 bosons in the presence of Abelian and non-Abelian gauge fields (the latter due to spin-orbit coupling) in a cavity-induced optical lattice potential. In the presence of atom-atom interaction, using a mapping to a generalized Bose-Hubbard model of spin-orbit-coupled bosons in a classical optical lattice, we show that the system realizes a host of quantum magnetic phases whose magnetic order can be detected from the cavity transmission. This provides an alternative approach to detecting quantum magnetism in ultracold atoms. We discuss the effects of cavity-induced optical bistability on these phases and their experimental consequences.
Gauge fields are central in our modern understanding of physics at all scales. At the highest energy scales known, the microscopic universe is governed by particles interacting with each other through the exchange of gauge bosons. At the largest length scales, our Universe is ruled by gravity, whose gauge structure suggests the existence of a particle-the graviton-that mediates the gravitational force. At the mesoscopic scale, solid-state systems are subjected to gauge fields of different nature: materials can be immersed in external electromagnetic fields, but they can also feature emerging gauge fields in their low-energy description. In this review, we focus on another kind of gauge field: those engineered in systems of ultracold neutral atoms. In these setups, atoms are suitably coupled to laser fields that generate effective gauge potentials in their description. Neutral atoms 'feeling' laser-induced gauge potentials can potentially mimic the behavior of an electron gas subjected to a magnetic field, but also, the interaction of elementary particles with non-Abelian gauge fields. Here, we review different realized and proposed techniques for creating gauge potentials-both Abelian and non-Abelian-in atomic systems and discuss their implication in the context of quantum simulation. While most of these setups concern the realization of background and classical gauge potentials, we conclude with more exotic proposals where these synthetic fields might be made dynamical, in view of simulating interacting gauge theories with cold atoms.