We propose a device architecture capable of direct quantum electro-optical conversion of microwave to optical photons. The hybrid system consists of a planar superconducting microwave circuit coupled to an integrated whispering-gallery-mode microresonator made from an electrooptical material. We show that by exploiting the large vacuum electric field of the planar microwave resonator, electro-optical (vacuum) coupling rates g0 as large as ∼ 2π O(10 − 100) kHz are achievable with currently available technology -a more than three order of magnitude improvement over prior designs and realizations. Operating at millikelvin temperatures, such a converter would enable high-efficiency conversion of microwave to optical photons. We analyze the added noise, and show that maximum quantum coherent conversion efficiency is achieved for a multi-photon cooperativity of unity which can be reached with optical power as low as O(1) mW.The interconversion of microwave and optical signals is of practical relevance in a broad range of electronic applications, from optical and wireless communications to timing. The spectacular advances of the past decade in manipulating the quantum states of the microwave field [1, 2] has increased interest in techniques to convert them to optical fields, since the latter can be propagated via optical fiber at room temperature while preserving their quantum state. In the long term, converting quantum states between microwave and optical photons may enable long distance quantum communication [3,4], and in the near term, it provides a path towards realizing single photon detectors of the microwave field that may find use in quantum science and metrology, radio astronomy and technology alike. For these reasons, hybrid systems for such microwave to optical interfaces have recently attracted significant experimental efforts. Several approaches have been investigated [5,6]: optomechanical and electromechanical devices [7][8][9][10] as well as cold atoms [11] and spin ensembles [12,13]. Indeed, a bi-directional and efficient link has been established recently using a mechanical oscillator coupled to both optical and microwave modes. Alternatively, it has been proposed that the parametric coupling of an LC circuit to an optical cavity via an electro-optical crystal would realize an effective optomechanical-type interaction [14]. Such a system could convert states from the microwave to the optical domain by driving sideband cooling transitions [15][16][17]. Similar to optomechanical systems, the interaction requires large vacuum coupling rates and the resolved-sideband regime [15][16][17] to be efficient as well as a optical cavity decay rate that greatly exceeds the microwave decay rate. Despite interest in the scheme, to date, it has not been realized. Previous demonstrations attained vacuum coupling rates of ∼ 2π O(1 − 10)Hz insufficient for an efficient transfer. In addition, several previous schemes operated with a microwave dissipation that was larger than the optical one, preventing efficient transfer...
We review recent developments regarding non-equilibrium quantum dynamics and many-body physics with light, in superconducting circuits and Josephson analogues. We start with quantum impurity models addressing dissipative and driven systems. Both theorists and experimentalists are making efforts towards the characterization of these non-equilibrium quantum systems. We show how Josephson junction systems can implement the equivalent of the Kondo effect with microwave photons. The Kondo effect can be characterized by a renormalized light-frequency and a peak in the Rayleigh elastic transmission of a photon. We also address the physics of hybrid systems comprising mesoscopic quantum dot devices coupled to an electromagnetic resonator. Then, we discuss extensions to Quantum Electrodynamics (QED) Networks allowing to engineer the Jaynes-Cummings lattice and Rabi lattice models through the presence of superconducting qubits in the cavities. This opens the door to novel many-body physics with light out of equilibrium, in relation with the Mott-superfluid transition observed with ultra-cold atoms in optical lattices. Then, we summarize recent theoretical predictions for realizing topological phases with light. Synthetic gauge fields and spin-orbit couplings have been successfully implemented with ultra-cold atoms in optical lattices -using time-dependent Floquet perturbations periodic in time, for exampleas well as in photonic lattice systems. Finally, we discuss the Josephson effect related to Bose-Hubbard models in ladder and two-dimensional geometries. The Bose-Hubbard model is related to the Jaynes-Cummings lattice model in the large detuning limit between light and matter (the superconducting qubits). In the presence of synthetic gauge fields, we show that Meissner currents subsist in an insulating Mott phase. arXiv:1505.00167v2 [cond-mat.mes-hall] 15 Jan 2016 be either Rydberg atoms (cold atoms) or trapped ions [3, 4] for example. A step towards the realization of manybody physics has also been made through the realization of model Hamiltonians such as the Dicke Hamiltonian [5], where the associated super-radiant quantum phase transition has been observed in non-equilibrium conditions [6]. A solid-state version of cavity quantum electrodynamics, related to circuit quantum electrodynamics, built with superconducting quantum circuits [7,8], is also a very active field both from experimental and theoretical points of view. Theorists have predicted novel emergent quantum phenomena either in relation with strong light-matter coupling [9] or non-equilibrium quantum physics [10,11].The goal here is to review developments, both theoretical and experimental, towards realizing many-body physics and quantum simulation in circuit QED starting from small networks to larger ensembles of superconducting elements in the microwave limit. An experimental endeavor has been accomplished towards the realization of larger arrays in circuit QED [12,13,14,15] and towards controlling trajectories in small systems [16,17]. This research is c...
We show the emergence of topological Bogoliubov bosonic excitations in the relatively strong coupling limit of an LC (inductance-capacitance) one-dimensional quantum circuit. This dimerized chain model reveals a Z2 local symmetry as a result of the counter-rotating wave (pairing) terms. The topology is protected by the sub-lattice symmetry, represented by an anti-unitary transformation. We present a method to measure the winding of the topological Zak phase across the Brillouin zone by a reflection measurement of (microwave) light. Our method probes bulk quantities and can be implemented even in small systems. We study the robustness of edge modes towards disorder.Topological Bloch bands are characterized by a topological number which is manifested in the appearance of protected edge states. For non-interacting fermions this results in the celebrated integer quantum Hall effect [1], conducting surface states of topological insulators [2,3] and semimetals [2,4]. Topological properties can also be accessed with bosonic systems such as cold atoms [5], photons [6][7][8] and polaritons [9].The Su-Schrieffer-Heeger (SSH) model defined on the dimerized one-dimensional lattice with two sites per unit cell is one of the simplest models demonstrating topological properties [10,11]. The edge states are protected by the bipartite nature of the system (particles can hop only between the two sublattices). Arbitrary long range interaction which respect the bipartite nature of the SSH model may change the topological number but not the robustness of the edge states to disorder [12]. The topology of the SSH model is described by the Zak phase [13], which was measured in cold atoms by introducing an artificial gauge field and mimicking Bloch oscillations [5], in photonic quantum walk [14] and in photonic crystals [15]. The midgap edge stated were observed in dielectric resonators [16], polariton systems [17] and classical LC chains [18], and their wave function was explored in Ref. [18,19]. A two-leg ladder of SSH chains supports fractional excitations and shows a rich topology with two types of corner edge states [20,21]. The ladder is a onedimensional version of a two-dimensional quadrupole insulator [22] realized in Refs. [23][24][25]. In the non-linear regime a dimerized chain shows topologically enforced bifurcations [26].Here, we study topology in the strong coupling regime of quantum circuits in which the rotating wave approximation (RWA) is not applicable, leading to the appearance of counter rotating (pairing) terms. Such strong coupling limit also leads to the evolution of the JaynesCummings model towards the Rabi model when describing the qubit-cavity interaction [7,27], and to the super radiant phase in Dicke model with a macroscopic number of photons in the ground state [28,29] but has not been studied in the framework of topological systems. We start by showing that the counter rotating terms do not change the topology of the system although they modify the nature of the excitations from pure particles to Bogoliubov...
We consider a Floquet triple-layer setup composed of a two-dimensional electron gas with spinorbit interactions, proximity coupled to an s-wave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a time-oscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a second-order topological superconducting phase with a pair of localized zero-energy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective time-reversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder and parameter variations and are within experimental reach.
The realization of synthetic gauge fields has attracted a lot of attention recently in relation with periodically driven systems and the Floquet theory. In ultra-cold atom systems in optical lattices and photonic networks, this allows to simulate exotic phases of matter such as quantum Hall phases, anomalous quantum Hall phases and analogs of topological insulators. In this paper, we apply the Floquet theory to engineer anisotropic Haldane models on the honeycomb lattice and two-leg ladder systems. We show that these anisotropic Haldane models still possess a topologically non-trivial band structure associated with chiral edge modes (without the presence of a net unit flux in a unit cell), then referring to the quantum anomalous Hall effect. Focusing on (interacting) boson systems in s-wave bands of the lattice, we show how to engineer through the Floquet theory, a quantum phase transition between a uniform superfluid and a BEC (Bose-Einstein Condensate) analog of FFLO (FuldeFerrell-Larkin-Ovchinnikov) states, where bosons condense at non-zero wave-vectors. We perform a GinzburgLandau analysis of the quantum phase transition on the graphene lattice, and compute observables such as chiral currents and the momentum distribution. The results are supported by exact diagonalization calculations and compared with those of the isotropic situation. The validity of high-frequency expansion in the Floquet theory is also tested using time-dependent simulations for various parameters of the model. Last, we show that the anisotropic choice for the effective vector potential allows a bosonization approach in equivalent ladder (strip) geometries.
We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half-filling resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counter-propagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. We propose an alternative definition of the topological invariant on the Bloch sphere.Topological systems have attracted a considerable attention these last decades [1, 2] as they show robust gapless edge modes which are relevant for quantum information purposes [3]. The Haldane model [4] on the honeycomb lattice, which has been realized in ultra-cold atoms [5,6], graphene [7], quantum materials [8], and photonic topological systems [9-13] now appears as a paradigmatic model in the topological classification of Bloch energy bands. For spinless fermions, the bulk state is insulating at half-filling and characterized by a topological invariant, the first Chern number, while the edges of the system reveal a one-dimensional gapless chiral mode by analogy with the quantum Hall effect [14][15][16][17][18][19]. Topological proximity effects induced by a topological band insulator [20][21][22] have also started to gain interest as a generalization of the proximity effect induced from a superconductor onto a metallic system [23,24]. In this Letter, we study the proximity effect when tunnel coupling a Haldane model with a layer of graphene [25][26][27]. We assume spinless particles in both layers and the tunnel process couples the same sublattice in the two layers. Particle-hole processes at the interface open a gap as a result of pseudospin effects, inducing an inverse topological order in the graphene system when both layers are half-filled.The Haldane model and graphene layers are described through the same pseudospin-1/2 representation in momentum space, as a result of the two sublattices of the honeycomb lattice [28], allowing us to describe the proximity effect in the same torus representation of the first Brillouin zone and fiber bundle approach on the Bloch sphere. We address different geometries and protocols to describe the bulk-edge correspondence and the Berry curvatures [29] of Bloch bands which could be equivalently probed for fermions and bosons at the one-particle level. We draw an analogy with the Kane-Mele model [30][31][32] and with ...
Recently, the frustrated XY model for spins 1/2 on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultracold atom setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using bosonic dynamical mean-field theory. Focusing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do find an emergent chiral spin state in the intermediate frustration regime. Using exact diagonalization we study more closely the physics of the effective frustrated XY model and the properties of the chiral spin state. This gapped phase displays a chiral order, breaking time-reversal and parity symmetry, but is not topologically ordered (the Chern number is zero).
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