The coupled nonlinear dynamics of ultracold quantum matter and electromagnetic field modes in an optical resonator exhibits a wealth of intriguing collective phenomena. Here we study a Lambda-type, three-component Bose-Einstein condensate coupled to four dynamical running-wave modes of a ring cavity, where only two of the modes are externally pumped. However, the unpumped modes play a crucial role in the dynamics of the system due to coherent backscattering of photons. On a mean- field level we identify three fundamentally different steady-state phases with distinct characteristics in the density and spatial spin textures: a combined density and spin-wave, a continuous spin spiral with a homogeneous density, and a spin spiral with a modulated density. The spin-spiral states, which are topological, are intimately related to cavity-induced spin-orbit coupling emerging beyond a critical pump power. The topologically trivial density-wave-spin-wave state has the characteristics of a supersolid with two broken continuous symmetries. The transitions between different phases are either simultaneously topological and first-order, or second-order. The proposed setup allows the simulation of intriguing many-body quantum phenomena by solely tuning the pump amplitudes and frequencies, with the cavity output fields serving as a built-in nondestructive observation tool.The coupled nonlinear dynamics of ultracold quantum matter and electromagnetic field modes in an optical resonator exhibits a wealth of intriguing collective phenomena. Here we study a Lambda-type, three-component Bose-Einstein condensate coupled to four dynamical running-wave modes of a ring cavity, where only two of the modes are externally pumped. However, the unpumped modes play a crucial role in the dynamics of the system due to coherent backscattering of photons. On a mean- field level we identify three fundamentally different steady-state phases with distinct characteristics in the density and spatial spin textures: a combined density and spin-wave, a continuous spin spiral with a homogeneous density, and a spin spiral with a modulated density. The spin-spiral states, which are topological, are intimately related to cavity-induced spin-orbit coupling emerging beyond a critical pump power. The topologically trivial density-wave-spin-wave state has the characteristics of a supersolid with two broken continuous symmetries. The transitions between different phases are either simultaneously topological and first-order, or second-order. The proposed setup allows the simulation of intriguing many-body quantum phenomena by solely tuning the pump amplitudes and frequencies, with the cavity output fields serving as a built-in nondestructive observation tool.

Jan 30, 201914 citations

Many-body phases of a planar Bose-Einstein condensate with cavity-induced spin-orbit coupling

Stefan Ostermann, Helmut Ritsch, Farokh Mivehvar

We explore the many-body phases of a two-dimensional Bose-Einstein condensate with cavity-mediated dynamic spin-orbit coupling. With the help of two transverse noninterfering, counterpropagating pump lasers and a single standing-wave cavity mode, two degenerate Zeeman sub-levels of the quantum gas are Raman coupled in a double-$\mathrm{\ensuremath{\Lambda}}$-configuration. Beyond a critical pump strength the cavity mode is populated via coherent superradiant Raman scattering from the two pump lasers, leading to the appearance of a dynamical spin-orbit coupling for the atoms. We identify three quantum phases with distinct atomic and photonic properties: the normal ``homogeneous'' phase, the superradiant ``spin-helix'' phase, and the superradiant ``supersolid spin-density-wave'' phase. The last exhibits an emergent periodic atomic density distribution with an orthorhombic centered rectangular-lattice structure due to the interplay between the coherent photon scattering into the resonator and the collision-induced momentum coupling. The transverse lattice spacing of the emergent crystal is set by the dynamic spin-orbit coupling.We explore the many-body phases of a two-dimensional Bose-Einstein condensate with cavity-mediated dynamic spin-orbit coupling. With the help of two transverse noninterfering, counterpropagating pump lasers and a single standing-wave cavity mode, two degenerate Zeeman sub-levels of the quantum gas are Raman coupled in a double-$\mathrm{\ensuremath{\Lambda}}$-configuration. Beyond a critical pump strength the cavity mode is populated via coherent superradiant Raman scattering from the two pump lasers, leading to the appearance of a dynamical spin-orbit coupling for the atoms. We identify three quantum phases with distinct atomic and photonic properties: the normal ``homogeneous'' phase, the superradiant ``spin-helix'' phase, and the superradiant ``supersolid spin-density-wave'' phase. The last exhibits an emergent periodic atomic density distribution with an orthorhombic centered rectangular-lattice structure due to the interplay between the coherent photon scattering into the resonator and the collision-induced momentum coupling. The transverse lattice spacing of the emergent crystal is set by the dynamic spin-orbit coupling.

arXiv:2009.06475Physical Review AFeb 2, 202111 citations

Cavity-Quantum-Electrodynamical Toolbox for Quantum Magnetism

Farokh Mivehvar, Helmut Ritsch, Francesco Piazza

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas-cavity systems. Based on this platform, we introduce a generic scheme for the implementation of long-range quantum spin Hamiltonians composed of various types of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. Our model is composed of an effective two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double Λ scheme. Cavity photons mediate the long-range spin-spin interactions with spatially modulated coupling coefficients, where the latter ones can be tuned by modifying spatial profiles of the pump lasers. As experimentally relevant examples, we demonstrate that by properly choosing the spatial profiles of the pump lasers achiral domain-wall antiferromagnetic and chiral spin-spiral orders emerge beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss extensions of our scheme for the implementation of other classes of spin Hamiltonians.The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas-cavity systems. Based on this platform, we introduce a generic scheme for the implementation of long-range quantum spin Hamiltonians composed of various types of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. Our model is composed of an effective two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double Λ scheme. Cavity photons mediate the long-range spin-spin interactions with spatially modulated coupling coefficients, where the latter ones can be tuned by modifying spatial profiles of the pump lasers. As experimentally relevant examples, we demonstrate that by properly choosing the spatial profiles of the pump lasers achiral domain-wall antiferromagnetic and chiral spin-spiral orders emerge beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss extensions of our scheme for the implementation of other classes of spin Hamiltonians.

arXiv:1809.09129Physical Review LettersMar 22, 201969 citations

Dynamical Spin-Orbit Coupling of a Quantum Gas

Ronen M. Kroeze, Yudan Guo, Benjamin Lev

We realize the dynamical 1D spin-orbit coupling (SOC) of a Bose-Einstein condensate confined within an optical cavity. The SOC emerges through spin-correlated momentum impulses delivered to the atoms via Raman transitions. These are effected by classical pump fields acting in concert with the quantum dynamical cavity field. Above a critical pump power, the Raman coupling emerges as the atoms superradiantly populate the cavity mode with photons. Concomitantly, these photons cause a backaction onto the atoms, forcing them to order their spin-spatial state. This SOC-inducing superradiant Dicke phase transition results in a spinor-helix polariton condensate. We observe emergent SOC through spin-resolved atomic momentum imaging and temporal heterodyne measurement of the cavity-field emission. Dynamical SOC in quantum gas cavity QED, and the extension to dynamical gauge fields, may enable the creation of Meissner-like effects, topological superfluids, and exotic quantum Hall states in coupled light-matter systems.We realize the dynamical 1D spin-orbit coupling (SOC) of a Bose-Einstein condensate confined within an optical cavity. The SOC emerges through spin-correlated momentum impulses delivered to the atoms via Raman transitions. These are effected by classical pump fields acting in concert with the quantum dynamical cavity field. Above a critical pump power, the Raman coupling emerges as the atoms superradiantly populate the cavity mode with photons. Concomitantly, these photons cause a backaction onto the atoms, forcing them to order their spin-spatial state. This SOC-inducing superradiant Dicke phase transition results in a spinor-helix polariton condensate. We observe emergent SOC through spin-resolved atomic momentum imaging and temporal heterodyne measurement of the cavity-field emission. Dynamical SOC in quantum gas cavity QED, and the extension to dynamical gauge fields, may enable the creation of Meissner-like effects, topological superfluids, and exotic quantum Hall states in coupled light-matter systems.

arXiv:1904.08388Physical Review LettersOct 18, 201967 citations

A cavity-QED toolbox for quantum magnetism.

Farokh Mivehvar, Helmut Ritsch, Francesco Piazza

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas $-$ cavity systems. Based on this platform, we introduce a generic scheme for the implementation of quantum spin Hamiltonians composed of various kinds of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. These interactions with spatially modulated coupling coefficients are mediated by cavity photons. Specifically, our model is comprised of a two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double $\Lambda$ scheme. We exemplify the emergence of an achiral antiferromagnetic domain-wall phase and a chiral spin-spiral order beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss straightforward extensions of our scheme for the implementation of other classes of spin Hamiltonians.The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas$-$cavity systems. Based on this platform, we introduce a generic scheme for the implementation of quantum spin Hamiltonians composed of various kinds of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. These interactions with spatially modulated coupling coefficients are mediated by cavity photons. Specifically, our model is comprised of a two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double $\Lambda$ scheme. We exemplify the emergence of an achiral antiferromagnetic domain-wall phase and a chiral spin-spiral order beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss straightforward extensions of our scheme for the implementation of other classes of spin Hamiltonians.

Sep 24, 20180 citations

Topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in cavity

Kashif Ammar Yasir, Lin Zhuang, Wu-Ming Liu

Abstract We investigate topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in a cavity. The cavity is driven by a pump laser and a weak probe laser. Both lasers excite Bose-Einstein condensate, in the presence of standard Raman process for spin-orbit coupling, to an intermediate storage level. We theoretically show that the quantum interference at the transitional pathways of dressed atomic states results in different types of optical transparencies, which get completely inverted in atomic damping induced gain regime. The synthetic pseudo-spin states also implant different phases in the probe field forcing modes in probe transparencies to form gapless Dirac cones, which become gapped in presence of Raman detuning. These features get interestingly enhanced in gain regime where the amplified part of probe transparencies appear as gapless topological edge-like states between the probe bulk modes and cause non-trivial phase transition. We illustrate that the nonlinear interactions of the pseudo-spin states also enhance the slow light features in probe transmission. The manipulation of dressed states for topological optical transparencies in our findings could be a crucial step towards topological photonics and their application in quantum computation.Abstract We investigate topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in a cavity. The cavity is driven by a pump laser and a weak probe laser. Both lasers excite Bose-Einstein condensate, in the presence of standard Raman process for spin-orbit coupling, to an intermediate storage level. We theoretically show that the quantum interference at the transitional pathways of dressed atomic states results in different types of optical transparencies, which get completely inverted in atomic damping induced gain regime. The synthetic pseudo-spin states also implant different phases in the probe field forcing modes in probe transparencies to form gapless Dirac cones, which become gapped in presence of Raman detuning. These features get interestingly enhanced in gain regime where the amplified part of probe transparencies appear as gapless topological edge-like states between the probe bulk modes and cause non-trivial phase transition. We illustrate that the nonlinear interactions of the pseudo-spin states also enhance the slow light features in probe transmission. The manipulation of dressed states for topological optical transparencies in our findings could be a crucial step towards topological photonics and their application in quantum computation.

arXiv:1803.03815npj Quantum InformationSep 9, 202212 citations

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas

-

cavity systems. Based on this platform, we introduce a generic scheme for the implementation of quantum spin Hamiltonians composed of various kinds of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. These interactions with spatially modulated coupling coefficients are mediated by cavity photons. Specifically, our model is comprised of a two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double

\Lambda

scheme. We exemplify the emergence of an achiral antiferromagnetic domain-wall phase and a chiral spin-spiral order beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss straightforward extensions of our scheme for the implementation of other classes of spin Hamiltonians.