The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting and superconducting systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter-light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators. It has been suggested that an observation of the scaling of the resonant atom-photon coupling strength in the Jaynes-Cummings energy ladder with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom-field interaction in cavity QED. We explore atom-photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication.

In quantum information science, the phase of a wavefunction plays an important role in encoding information. While most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrate the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit, manipulating the qubit geometrically using microwave radiation, and observing the accumulated phase in an interference experiment. We find excellent agreement with Berry's predictions, and also observe a geometry dependent contribution to dephasing.When a quantum mechanical system evolves cyclically in time such that it returns to its initial physical state, its wavefunction can acquire a geometric phase factor in addition to the familiar dynamic phase [1,2]. If the cyclic change of the system is adiabatic, this additional factor is known as Berry's phase [3], and is, in contrast to dynamic phase, independent of energy and time.In quantum information science [4], a prime goal is to utilize coherent control of quantum systems to process information, accessing a regime of computation unavailable in classical systems. Quantum logic gates based on geometric phases have been demonstrated in both nuclear magnetic resonance [5] and ion trap based quantum information architectures [6]. Superconducting circuits [7,8] are a promising solid state platform for quantum information processing [9,10,11,12,13,14], in particular due to their potential scalability. Proposals for observation of geometric phase in superconducting circuits [15,16,17,18,19] have existed since shortly after the first coherent quantum effects were demonstrated in these systems [20].Geometric phases are closely linked to the classical concept of parallel transport of a vector on a curved surface. Consider, for example, a tangent vector v on the surface of a sphere being transported from the North pole around the path P shown in Fig. 1A, with v pointing South at all times. The final state of the vector v f is rotated with respect to its initial state v i by an angle φ equal to the solid angle subtended by the path P at the origin. Thus, this angle is dependent on the geometry of the path P , and is independent of the rate at which it is traversed. As a result, departures from the original path that leave the solid angle unchanged will not modify φ. This robustness has been interpreted as a potential fault tolerance when applied to quantum information processing [5].The analogy of the quantum geometric phase with the above classical picture is particularly clear in the case of a two-level system (a qubit) in the presence of a bias field that changes in time. A familiar example is a spin-1/2 particle in a changing magnetic field. The general Hamiltonian for such a system is H =hR · σ/2, where σ = (σ x , σ y , σ z ) are the Pauli operators, and R is the bias field vector, expressed in units of angular frequency. The qubit dynamics is best visualiz...

We present an ideal realization of the Tavis-Cummings model in the absence of atom number and coupling fluctuations by embedding a discrete number of fully controllable superconducting qubits at fixed positions into a transmission line resonator. Measuring the vacuum Rabi mode splitting with one, two and three qubits strongly coupled to the cavity field, we explore both bright and dark dressed collective multi-qubit states and observe the discrete √ N scaling of the collective dipole coupling strength. Our experiments demonstrate a novel approach to explore collective states, such as the W -state, in a fully globally and locally controllable quantum system. Our scalable approach is interesting for solid-state quantum information processing and for fundamental multi-atom quantum optics experiments with fixed atom numbers.PACS numbers: 42.50. Ct, 42.50.Pq, 03.67.Lx, 85.35.Gv In the early 1950's, Dicke realized that under certain conditions a gas of radiating molecules shows the collective behavior of a single quantum system [1]. The idealized situation in which N two-level systems with identical dipole coupling are resonantly interacting with a single mode of the electromagnetic field was analyzed by Tavis and Cummings [2]. This model predicts the collective N -atom interaction strength to be G N = g j √ N , where g j is the dipole coupling strength of each individual atom j. In fact, in first cavity QED experiments the normal mode splitting, observable in the cavity transmission spectrum [3,4], was demonstrated with on averageN > 1 atoms in optical [5,6] and microwave [7] cavities to overcome the relatively weak dipole coupling g j . The √ N scaling has been observed in the regime of a small mean number of atomsN with dilute atomic beams [7,8,9] and fountains [10] crossing a high-finesse cavity. In these experiments, spatial variations of the atom positions and Poissonian fluctuations in the atom number inherent to an atomic beam [4,8,11] are unavoidable. In a different limit where the cavity was populated with a very large number of ultra-cold 87 Rb atoms [12] and more recently with Bose-Einstein condensates [13,14] the √ N nonlinearity was also demonstrated. However, the number of interacting atoms is typically only known to about ∼ 10% [13].Here we present an experiment in which the TavisCummings model is studied for a discrete set of fully controllable artificial atoms at fixed positions and with virtually identical couplings to a resonant cavity mode. The investigated situation is sketched in Fig. 1 a, depicting an optical analog where three two-state atoms are deterministically positioned at electric field antinodes of a cavity mode where the coupling is maximum. In our circuit QED [15,16] realization of this configuration (Fig. 1 b), three transmon-type [17] superconducting qubits are embedded in a microwave resonator which contains a quantized radiation field. The cavity is realized as a coplanar waveguide resonator with a first harmonic full wavelength resonance frequency of ω r /2π = 6.729 GHz and a ...

We have designed and fabricated superconducting coplanar waveguide resonators with fundamental frequencies from 2 to 9 GHz and loaded quality factors ranging from a few hundreds to a several hundred thousands reached at temperatures of 20 mK. The loaded quality factors are controlled by appropriately designed input and output coupling capacitors. The measured transmission spectra are analyzed using both a lumped element model and a distributed element transmission matrix method. The experimentally determined resonance frequencies, quality factors and insertion losses are fully and consistently characterized by the two models for all measured devices. Such resonators find prominent applications in quantum optics and quantum information processing with superconducting electronic circuits and in single photon detectors and parametric amplifiers.