We study the nonlinear dynamics of an ensemble of cold trapped atoms with a hyperfine transition magnetically coupled to a resonant microwave cavity mode. Despite the minute single atom coupling one obtains strong coupling between collective hyperfine qubits and microwave photons enabling coherent transfer of an excitation between the long lived atomic qubit state and the mode. Evidence of strong coupling can be obtained from the cavity transmission spectrum even at finite thermal photon number. The system makes it possible to study further prominent collective phenomena such as superradiant decay of an inverted ensemble or the building of a narrowband stripline micromaser locked to an atomic hyperfine transition.
We have fabricated an atom chip device which combines the circuitry for magnetic trapping of cold atoms with high-finesse optical resonators suitable for cavity QED in the single-atom strong coupling regime. Fabry-Perot optical resonators with finesse F > 2 X 10^5 were formed between a micropatterned on-chip planar mirror with lateral dimension of < 100 um and a curved mirror suspended above the chip. The strong and rapid thermal coupling between on-chip electrical and optical elements was utilized to stabilize the cavity mirror separation with servo bandwidth exceeding 100 kHz during simulate…
We experimentally demonstrate optical spectroscopy of magnetically trapped atoms on an atom chip. High resolution optical spectra of individual trapped clouds are recorded within a few hundred milliseconds. Detection sensitivities close to the single atom level are obtained by photoionization of the excited atoms and subsequent ion detection with a channel electron multiplier. Temperature and decay rates of the trapped atomic cloud can be monitored in real time for several seconds with only little detection losses. The spectrometer can be used for investigations of ultracold atomic mixtures an…
We present evidence of cavity quantum electrodynamics from a sparse density of strongly quantum-confined Pb-chalcogenide nanocrystals (between 1 and 10) approaching single-dot levels on moderately high-Q mesoscopic silicon optical cavities. Operating at important near-infrared (1500-nm) wavelengths, large enhancements are observed from devices and strong modifications of the QD emission are achieved. Saturation spectroscopy of coupled QDs is observed at 77K, highlighting the modified nanocrystal dynamics for quantum information processing.
We suggest a method to simulate lattice compact Quantum Electrodynamics (cQED) using ultracold atoms in optical lattices, which includes dynamical Dirac fermions in 2+1 dimensions. This allows to test dynamical effects of confinement as well as 2d flux loops deformations and breaking, and to observe Wilson-loop area-law.
We propose a method for simulating 2+1-d compact lattice quantum-electrodynamics (QED), using ultracold atoms in optical lattices. In our model local Bose-Einstein condensates' phases correspond to the electromagnetic vector-potential, and the fluctuations of the local number operators represent the conjugate electric field. The gauge invariant Kogut-Susskind Hamiltonian is obtained as an effective low energy theory. The field is then coupled to external static charges. We show that in the strong coupling limit this gives rise to 'electric flux-tubes' and to confinement. The effect can be obse…
Superconducting atom chips have very significant advantages in realizing trapping structures for ultracold atoms compared to conventional atom chips. We extend these advantages further by developing the ability to dynamically tailor the superconducting trap architecture. Heating the chosen parts of a superconducting film by transferring optical images onto its surface we are able to modify the current density distribution and create desired trapping potentials. This method enables us to change the shape and structure of magnetic traps, enabling versatile applications in atomtronics.
Studies of ultracold atoms in optical lattices link various disciplines, providing a playground where fundamental quantum many-body concepts, formulated in condensed-matter physics, can be tested in much better controllable atomic systems, e.g., strongly correlated phases, quantum information processing. Standard methods to measure quantum properties of Bose-Einstein condensates (BECs) are based on matter-wave interference between atoms released from traps which destroys the system. Here we propose a nondestructive method based on optical measurements, and prove that atomic statistics can be m…
We consider the single photon transistor in coupled cavity system of resonators interacting with multilevel superconducting artificial atom simultaneously. Effective single mode transformation is used for the diagonalization of the the hamiltonian and impedance matching in terms of the normal modes. Storage and transmission of the incident field are described by the interactions between the cavities controlling the atomic transitions of lowest lying states. Rabi splitting of vacuum induced multiphoton transitions is considered in input/output relations by the quadrature operators in the absenc…
We demonstrate a novel atom chip trapping system that allows the placement and high-resolution imaging of ultracold atoms within microns from any <100 um-thin, UHV-compatible material, while also allowing sample exchange with minimal experimental downtime. The sample is not connected to the atom chip, allowing rapid exchange without perturbing the atom chip or laser cooling apparatus. Exchange of the sample and retrapping of atoms has been performed within a week turnaround, limited only by chamber baking. Moreover, the decoupling of sample and atom chip provides the ability to independently t…
We present a complete fabrication study of an efficiently-coupled microring optical circuit tailored for cavity quantum electrodynamics (QED) with trapped atoms. The microring structures are fabricated on a transparent membrane with high in-vacuum fiber edge-coupling efficiency in a broad frequency band. In addition, a bus waveguide pulley coupler realizes critical coupling to the microrings at both of the cesium D-line frequencies, while high coupling efficiency is achieved at the cesium 'magic' wavelengths for creating a lattice of two-color evanescent field traps above a microring. The pres…
We describe the fabrication and construction of a setup for creating lattices of magnetic microtraps for ultracold atoms on an atom chip. The lattice is defined by lithographic patterning of a permanent magnetic film. Patterned magnetic-film atom chips enable a large variety of trapping geometries over a wide range of length scales. We demonstrate an atom chip with a lattice constant of 10 $μ$m, suitable for experiments in quantum information science employing the interaction between atoms in highly-excited Rydberg energy levels. The active trapping region contains lattice regions with square …
We consider a system of ultracold atoms in an optical lattice as a quantum simulator for electron-positron pair production in quantum electrodynamics (QED). For a setup in one spatial dimension, we investigate the nonequilibrium phenomenon of pair production including the backreaction leading to plasma oscillations. Unlike previous investigations on quantum link models, we focus on the infinite-dimensional Hilbert space of QED and show that it may be well approximated by experiments employing Bose-Einstein condensates interacting with fermionic atoms. The calculations based on functional integ…
Considering ultracold atoms traversing a high-Q Fabry-Perot cavity, we theoretically demonstrate a quantum nondemolition measurement of the photon number. This fully quantum mechanical approach may be understood utilizing concepts as effective mass and group velocity of the atom. The various photon numbers induce a splitting of the atomic wave packet, and a time-of-flight measurement of the atom thereby reveals the photon number. While repeated atomic measurements increase the efficiency of the protocol, it is shown that by considering long interaction times only a few atoms are needed to reso…
We study dressed Bose-Einstein condensates in an atom chip radio-frequency trap. We show that in this system sufficiently strong dressing can be achieved to cause the widely used rotating wave approximation (RWA) to break down. We present a full calculation of the atom - field coupling which shows that the non-RWA contributions quantitatively alter the shape of the emerging dressed adiabatic potentials. The non-RWA contributions furthermore lead to additional allowed transitions between dressed levels. We use RF spectroscopy of Bose-Einstein condensates trapped in the dressed state potentials …
We present a method to enact a deterministic, measurement-free, optically generated controlled-phase gate on two qubits defined by single electrons trapped in large-area quantum dots in a planar microcavity. This method is robust to optical quantum dot inhomogeneity, requires only a modest-$Q$ planar cavity, employs only a single laser pulse, and allows the integration of many entangled qubits on one semiconductor chip. We present the gate in the contexts of both adiabatic evolution and geometric phases, and calculate the degradation of performance in the presence of both spontaneous emission …
When an incident light beam is scattered off a sample of ultracold atoms trapped in an optical lattice, the statistical properties of the retro-reflected field contain information about the quantum state of the atoms, and permit for example to distinguish between atoms in a superfluid and a Mott insulator state. This paper extends our previous analysis of this problem to include the effects of cavity damping. We use a Monte Carlo wave function method to determine the two-time correlation function and time-dependent physical spectrum of the retro-reflected field in the case of a simple two-well…
We describe a chip-based, solid-state analogue of cavity-QED utilizing acoustic phonons instead of photons. We show how long-lived and tunable acceptor impurity states in silicon nanomechanical cavities can play the role of a matter non-linearity for coherent phonons just as, e.g., the Josephson qubit plays in circuit-QED. Both strong coupling (number of Rabi oscillations ~ 100) and strong dispersive coupling (0.1-2 MHz) regimes can be reached in cavities in the 1-20 GHz range, enabling the control of single phonons, phonon-phonon interactions, dispersive phonon readout of the acceptor qubit, …
Ultracold $^{87}$Rb atoms are delivered into a high-finesse optical micro-cavity using a translating optical lattice trap and detected via the cavity field. The atoms are loaded into an optical lattice from a magneto-optic trap (MOT) and transported 1.5 cm into the cavity. Our cavity satisfies the strong-coupling requirements for a single intracavity atom, thus permitting real-time observation of single atoms transported into the cavity. This transport scheme enables us to vary the number of intracavity atoms from 1 to $>$100 corresponding to a maximum atomic cooperativity parameter of 5400, t…
We propose that condensed matter phenomena involving the spontaneous emergence and dynamics of crystal lattices can be realized in the setting of ultracold Bose-condensed atoms coupled to multimode cavities. Previously, it was shown that in the case of a transversely pumped single-mode cavity, the atoms self-organize at either the even or the odd antinodes of the cavity mode, given sufficient pump intensity, and hence spontaneously break a discrete translational symmetry. Here, we demonstrate that in multimode cavities the self-organization brings the additional feature of continuous translati…
We propose a method for using ultracold atomic Bose-Einstein condensates, to form an analog model of a relativistic massive field that carries "charge" and interacts with an external nondynamical gauge field. Such a "scalar QED" analog model, may be useful for simulating various effects of QFT involving charged particles, such as the Schwinger pair-creation of charged phonons in a constant external field, and vacuum instability.
We study cavity optomechanics of a mixture of ultracold atoms with tunable nonlinear collisions. We show that atomic collisions provide linear couplings between fictitious condensate oscillators, leading to possibilities of achieving a globally coupled quantum optomechanical network with an integrated atom chip. Potential applications range from simulating collective nonequilibrium dynamics in fields well past physics to probing unique properties of quantum mixtures.
We study an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-$Q$ cavity mode. In the limit of very low temperatures, cavity field and atomic dynamics require a quantum description. Starting from a cavity QED single atom Hamiltonian we use different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters. In the limit of large enough cavity damping the different models agree. Compared to free space optical lattices, quantum uncertainties of the potential and the possibility of atom-fiel…
We theoretically propose a scheme for a spin quantum bit based on a double quantum dot contacted to ferromagnetic elements. Interface exchange effects enable an all electric manipulation of the spin and a switchable strong coupling to a superconducting coplanar waveguide cavity. Our setup does not rely on any specific band structure and can in principle be realized with many different types of nanoconductors. This allows to envision on-chip single spin manipulation and read-out using cavity QED techniques.
Entanglement-based technologies, such as quantum information processing, quantum simulations, and quantum-enhanced metrology, have the potential to revolutionise our way of computing and measuring and help clarifying the puzzling concept of entanglement itself. Ultracold atoms on atom chips are attractive for their implementation, as they provide control over quantum systems in compact, robust, and scalable setups. An important tool in this system is a potential depending on the internal atomic state. Coherent dynamics in this potential combined with collisional interactions allows entanglemen…
Cavity quantum electrodynamic (QED) is studied for two strongly-coupled charge qubits interacting with a single-mode quantized field, which is provided by a on-chip transmission line resonator. We analyze the dressed state structure of this superconducting circuit QED system and the selection rules of electromagnetic-induced transitions between any two of these dressed states. Its macroscopic quantum criticality, in the form of ground state level crossing, is also analyzed, resulting from competition between the Ising-type inter-qubit coupling and the controllable on-site potentials.
We introduce a general method for designing tailored lattices of magnetic microtraps for ultracold atoms, on the basis of patterned permanently magnetized films. A fast numerical algorithm is used to automatically generate patterns which provide optimal atom confinement while respecting desired lattice symmetries and trap parameters. The algorithm can produce finite and infinite lattices of any plane symmetry; we focus specifically on square and triangular lattices which are of interest to future experiments. Typical trap parameters are discussed as well as the impact of realistic imperfection…
Atom chips provide a versatile `quantum laboratory on a microchip' for experiments with ultracold atomic gases. They have been used in experiments on diverse topics such as low-dimensional quantum gases, cavity quantum electrodynamics, atom-surface interactions, and chip-based atomic clocks and interferometers. A severe limitation of atom chips, however, is that techniques to control atomic interactions and to generate entanglement have not been experimentally available so far. Such techniques enable chip-based studies of entangled many-body systems and are a key prerequisite for atom chip app…
A chiral photonic interface is a quantum system that has different probabilities for emitting photons to the left and right. An on-chip compatible chiral interface is attractive for both fundamental studies of light-matter interactions and applications to quantum information processing. We propose such a chiral interface based on superconducting circuits, which has wide bandwidth, rich tunability, and high tolerance to fabrication variations. The proposed interface consists of a core that uses Cooper-pair boxes (CPBs) to break time-reversal symmetry, and two superconducting transmons that conn…
Quantum photonic integrated circuit (QPIC) is a promising tool for constructing integrated devices for quantum technology applications. In the optical regime, silicon photonics empowered by complementary-metal-oxide-semiconductor (CMOS) technology provides optical components useful for realizing large-scale QPICs. Optical nonlinearity at the single-photon level is required for QPIC to facilitate photon-photon interaction. However, to date, realization of optical elements with deterministic( i.e., not probabilistic) single-photon nonlinearity by using silicon-based components is challenging, de…
We present an atom-chip-based realization of quantum cavity optomechanics with cold atoms localized within a Fabry-Perot cavity. Effective sub-wavelength positioning of the atomic ensemble allows for tuning the linear and quadratic optomechanical coupling parameters, varying the sensitivity to the displacement and strain of a compressible gaseous cantilever. We observe effects of such tuning on cavity optical nonlinearity and optomechanical frequency shifts, providing their first characterization in the quadratic-coupling regime.
We discuss and model an experiment to study quasi-condensate growth on an atom chip. In particular, we consider the addition of a deep dimple to the weak harmonic trap confining an ultracold one-dimensional atomic Bose gas, below or close to the characteristic temperature for quasi-condensate formation. The subsequent dynamics depends critically on both the initial conditions, and the form of the perturbing potential. In general, the dynamics features a combination of shock-wave propagation in the quasi-condensate, and quasi-condensate growth from the surrounding thermal cloud.
A high-resolution projection and imaging system for ultracold atoms is implemented using a compound silicon and glass atom chip. The atom chip is metalized to enable magnetic trapping while glass regions enable high numerical aperture optical access to atoms residing in the magnetic trap about 100 microns below the chip surface. The atom chip serves as a wall of the vacuum system, which enables the use of commercial microscope components for projection and imaging. Holographically generated light patterns are used to optically slice a cigar-shaped magnetic trap into separate regions; this has …
The generation and control of quantum states of light constitute fundamental tasks in cavity quantum electrodynamics (QED). The superconducting realization of cavity QED, circuit QED, enables on-chip microwave photonics, where superconducting qubits control and measure individual photon states. A long-standing issue in cavity QED is the coherent transfer of photons between two or more resonators. Here, we use circuit QED to implement a three-resonator architecture on a single chip, where the resonators are interconnected by two superconducting phase qubits. We use this circuit to shuffle one- …
Hybrid quantum systems that unite laser-cooled trapped ions and ultracold quantum gases in a single experimental setup have opened a rapidly advancing field of study, including Quantum chemistry, polaron physics, quantum information processing and quantum simulations. We present a fully developed and tested ion trap chip and propose a flat chip trap that can be placed beneath the ion trap. This design substantially addresses the difficulties specific to hybrid traps and features well-aligned chips that allow for independent adjustment of the depth of the atomic trap and the confinement and pos…
Matter-wave interference experiments enable us to study matter at its most basic, quantum level and form the basis of high-precision sensors for applications such as inertial and gravitational field sensing. Success in both of these pursuits requires the development of atom-optical elements that can manipulate matter waves at the same time as preserving their coherence and phase. Here, we present an integrated interferometer based on a simple, coherent matter-wave beam splitter constructed on an atom chip. Through the use of radio-frequency-induced adiabatic double-well potentials, we demonstr…
Quantum simulations of High Energy Physics, and especially of gauge theories, is an emerging and exciting direction in quantum simulations. However, simulations of such theories, compared to simulations of condensed matter physics, must satisfy extra restrictions, such as local gauge and Lorentz invariance. In this paper we discuss these special requirements, and present a new method for quantum simulation of lattice gauge theories using ultracold atoms. This method allows to include local gauge invariance as a fundamental symmetry of the atomic Hamiltonian, arising from natural atomic interac…
In this work we consider the advantages and challenges of using free-standing two-dimensional electron gases (2DEG) as active components in atom chips for manipulating ultracold ensembles of alkali atoms. We calculate trapping parameters achievable with typical high-mobility 2DEGs in an atom chip configuration, and identify advantages of this system for trapping atoms at sub-micron distances from the atom chip. We show how the sensitivity of atomic gases to magnetic field inhomogeneity can be exploited for controlling the atoms with quantum electronic devices and, conversely, using the atoms t…
Arrays of trapped atoms are the ideal starting point for developing registers comprising large numbers of physical qubits for storing and processing quantum information. One very promising approach involves neutral atom traps produced on microfabricated devices known as atom chips, as almost arbitrary trap configurations can be realised in a robust and compact package. Until now, however, atom chip experiments have focused on small systems incorporating single or only a few individual traps. Here we report experiments on a two-dimensional array of trapped ultracold atom clouds prepared using a…
We report a technique that uses clouds of ultracold atoms as sensitive, tunable, and non-invasive probes for microwave field imaging with micrometer spatial resolution. The microwave magnetic field components drive Rabi oscillations on atomic hyperfine transitions whose frequency can be tuned with a static magnetic field. Readout is accomplished using state-selective absorption imaging. Quantitative data extraction is simple and it is possible to reconstruct the distribution of microwave magnetic field amplitudes and phases. While we demonstrate 2d imaging, an extension to 3d imaging is straig…
We report on a class of configurations of permanent magnets on an atom chip for producing 1D and 2D periodic arrays of magnetic microtraps with non-zero potential minima and variable barrier height for trapping and manipulating ultracold atoms and quantum degenerate gases. We present analytical expressions for the relevant physical quantities and compare them with our numerical results and with some previous numerical calculations. In one of the configurations of permanent magnets, we show how it is possible by changing the angle between the crossed periodic arrays of magnets to go from a 1D a…
We report a process for fabricating high quality, defect-free spherical mirror templates suitable for developing high finesse optical Fabry-Perot resonators. The process utilizes the controlled reflow of borosilicate glass and differential pressure to produce mirrors with 0.3 nanometer surface roughness. The dimensions of the mirrors are in the 0.5-5mm range making them suitable candidates for integration with on-chip neutral atom and ion experiments where enhanced interaction between atoms and photons are required. Moreover the mirror curvature, dimension and placement is readily controlled a…
We study the reflection of two counter-propagating modes of the light field in a ring resonator by ultracold atoms either in the Mott insulator state or in the superfluid state of an optical lattice. We obtain exact numerical results for a simple two-well model and carry out statistical calculations appropriate for the full lattice case. We find that the dynamics of the reflected light strongly depends on both the lattice spacing and the state of the matter-wave field. Depending on the lattice spacing, the light field is sensitive to various density-density correlation functions of the atoms. …
We study an atom-cavity system in which the cavity has several degenerate transverse modes. Mode-resolved cavity transmission spectroscopy reveals well-resolved atom-cavity resonances for several cavity modes, a signature of collective strong coupling for the different modes. Furthermore, the experiment shows that the cavity modes are coupled via the atomic ensemble contained in the cavity. The experimental observations are supported by a detailed theoretical analysis. The work paves the way to the use of interacting degenerate modes in cavity-based quantum information processing, where qubits…
Quantum transmission lines are a central to superconducting and hybrid quantum computing. Parallel to these developments are those of left-handed meta-materials. They have a wide variety of applications in photonics from the microwave to the visible range such as invisibility cloaks and perfect flat lenses. For classical guided microwaves, left-handed transmission lines have been proposed and studied on the macroscopic scale. We combine these ideas in presenting a left-handed/right-handed hybrid transmission line for applications in quantum optics on a chip. The resulting system allows circuit…
Hybrid quantum systems involving cold atoms and microwave resonators can enable cavity-mediated infinite-range interactions between atomic spin systems and realize atomic quantum memories and transducers for microwave to optical conversion. To achieve strong coupling of atoms to on-chip microwave resonators, it was suggested to use atomic Rydberg states with strong electric dipole transitions. Here we report on the realization of coherent coupling of a Rydberg transition of ultracold atoms trapped on an integrated superconducting atom chip to the microwave field of an on-chip coplanar waveguid…
We present a compact atomic clock interrogating ultracold 87Rb magnetically trapped on an atom chip. Very long coherence times sustained by spin self-rephasing allow us to interrogate the atomic transition with 85% contrast at 5 s Ramsey time. The clock exhibits a fractional frequency stability of $5.8\times 10^{-13}$ at 1 s and is likely to integrate into the $1\times10^{-15}$ range in less than a day. A detailed analysis of 7 noise sources explains the measured frequency stability. Fluctuations in the atom temperature (0.4 nK shot-to-shot) and in the offset magnetic field ($5\times10^{-6}$ r…
Transitions between quantum states by photon absorption or emission are intimately related to symmetries of the system which lead to selection rules and the formation of dark states. In a circuit quantum electrodynamics setup, in which two resonant superconducting qubits are coupled through an on-chip cavity and driven via the common cavity field, one single-excitation state remains dark. Here, we demonstrate that this dark state can be excited using local phase control of individual qubit drives to change the symmetry of the driving field. We observe that the dark state decay via spontaneous …
Although the study of ultracold quantum gases trapped by light is a prominent direction of modern research, the quantum properties of light were widely neglected in this field. Quantum optics with quantum gases closes this gap and addresses phenomena, where the quantum statistical nature of both light and ultracold matter play equally important roles. First, light can serve as a quantum nondemolition (QND) probe of the quantum dynamics of various ultracold particles from ultracold atomic and molecular gases to nanoparticles and nanomechanical systems. Second, due to dynamic light-matter entang…
Starting from the first principles of nonrelativistic QED we have developed the quantum theory of the interaction of a two-component ultracold atomic ensemble with the electromagnetic field of vacuum and laser photons. The main attention has been paid to the consistent consideration of dynamical dipole-dipole interactions in the radiation field. Taking into account local-field effects we have derived the system of Maxwell-Bloch equations. Optical properties of the two-component Bose gas are investigated. It is shown that the refractive index of the gas is given by the Maxwell-Garnett formula. …
We theoretically study a circuit QED architecture based on a superconducting flux qubit directly coupled to the center conductor of a coplanar waveguide transmission-line resonator. As already shown experimentally [Abdumalikov et al. Phys. Rev. B 78, 180502 (2008)], the strong coupling regime of cavity QED can readily be achieved by optimizing the local inductance of the resonator in the vicinity of the qubit. In addition to yielding stronger coupling with respect to other proposals for flux qubit based circuit QED, this approach leads to a qubit-resonator coupling strength g which does not sc…
We propose an approach for studying quantum information and performing high resolution spectroscopy of rotational states of trapped molecular ions using an on-chip superconducting microwave resonator. Molecular ions have several advantages over neutral molecules. Ions can be loaded into deep (1 eV) RF traps and are trapped independent of the electric dipole moment of their rotational transition. Their charge protects them from motional dephasing and prevents collisional loss, allowing 1 s coherence times when used as a quantum memory, with detection of single molecules possible in <10 ms. An a…
Continuous Narrow-Linewidth Superradiance in Waveguide QED
University of Innsbruck（因斯布鲁克大学）；Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences（奥地利科学院量子光学和量子信息研究所）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Department of Physics, Harvard University（哈佛大学物理系）
Ultracold and quantum degenerate gases held near conductive surfaces can serve as sensitive, high resolution, and wide-area probes of electronic current flow. Previous work has imaged transport around grain boundaries in a gold wire by using ultracold and Bose-Einstein condensed atoms held microns from the surface with an atom chip trap. We show that atom chip microscopy may be applied to useful purpose in the context of materials exhibiting topologically protected surface transport. Current flow through lithographically tailored surface defects in topological insulators (TI)---both idealized …
In cavity quantum electrodynamics (QED), light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED, where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both i…
