This review provides a theoretical basis for understanding the current-phase relation (C⌽R) for the stationary (dc) Josephson effect in various types of superconducting junctions. The authors summarize recent theoretical developments with an emphasis on the fundamental physical mechanisms of the deviations of the C⌽R from the standard sinusoidal form. A new experimental tool for measuring the C⌽R is described and its practical applications are discussed. The method allows one to measure the electrical currents in Josephson junctions with a small coupling energy as compared to the thermal energy. A number of examples illustrate the importance of the C⌽R measurements for both fundamental physics and applications.

We have realized controllable coupling between two three-junction flux qubits by inserting an additional coupler loop between them, containing three Josephson junctions. Two of these are shared with the qubit loops, providing strong qubit-coupler interaction. The third junction gives the coupler a nontrivial current-flux relation; its derivative (i.e., the susceptibility) determines the coupling strength J, which thus is tunable in situ via the coupler's flux bias. In the qubit regime, J was varied from approximately 45 (antiferromagnetic) to approximately -55 mK (ferromagnetic); in particular, J vanishes for an intermediate coupler bias. Measurements on a second sample illuminate the relation between two-qubit tunable coupling and three-qubit behavior.

Laser cooling of the atomic motion paved the way for remarkable achievements in the fields of quantum optics and atomic physics, including Bose-Einstein condensation and the trapping of atoms in optical lattices. More recently superconducting qubits were shown to act as artificial two-level atoms, displaying Rabi oscillations, Ramsey fringes, and further quantum effects 1,2,3 . Coupling such qubits to resonators 4,5,6,7 brought the superconducting circuits into the realm of quantum electrodynamics (circuit QED). It opened the perspective to use superconducting qubits as micro-coolers or to create a population inversion in the qubit to induce lasing behavior of the resonator 8,9,10,11 . Furthering these analogies between quantum optical and superconducting systems we demonstrate here Sisyphus cooling 12 of a low frequency LC oscillator coupled to a near-resonantly driven superconducting qubit. In the quantum optics setup the mechanical degrees of freedom of an atom are cooled by laser driving the atom's electronic degrees of freedom. Here the roles of the two degrees of freedom are played by the LC circuit and the qubit's levels, respectively. We also demonstrate the counterpart of the Sisyphus cooling, namely Sisyphus amplification.For red-detuned high-frequency driving of the qubit the low-frequency LC circuit performs work in the forward and backward part of the oscillation cycle, always pushing the qubit up in energy, similar to Sisyphus who always had to roll a stone uphill. The oscillation cycle is completed with a relaxation process, when the work performed by the oscillator together with a quantum of energy of the high-frequency driving is released by the qubit to the environment via spontaneous emission. For blue-detuning the same mechanism creates excitations in the LC circuit with a tendency towards lasing and the characteristic line-width narrowing. In this regime "lucky Sisyphus" always rolls the stone downhill. Parallel to the experimental demonstration we analyze the system theoretically and find quantitative agreement, which supports the interpretation and allows us to estimate system parameters.The system considered is shown in the inset of Fig. 1. It consists of a three-junction flux qubit 13 , with the two qubit FIG. 1: (a) The energy levels of the qubit as a function of the energy bias of the qubit ε(fx) = 2Φ0Ipfx. The sinusoidal current in the tank coil, indicated by the wavy line, drives the bias of the qubit. The starting point of the cooling (heating) cycles is denoted by blue (red) dots. The resonant excitation of the qubit due to the high-frequency driving, characterized by ΩR0, is indicated by two green arrows and by the Lorentzian depicting the width of this resonance. The relaxation of the qubit is denoted by the black dashed arrows. The inset shows a schematic of the qubit coupled to an LC circuit. The high frequency driving is provided by an on-chip microwave antenna. (b) SEM picture of the superconducting flux qubit prepared by shadow evaporation technique.

The key issue for the implementation of a metamaterial is to demonstrate the existence of collective modes corresponding to coherent oscillations of the meta-atoms. Atoms of natural materials interact with electromagnetic fields as quantum two-level systems. Artificial quantum two-level systems can be made, for example, using superconducting nonlinear resonators cooled down to their ground state. Here we perform an experiment in which 20 of these quantum meta-atoms, so-called flux qubits, are embedded into a microwave resonator. We observe the dispersive shift of the resonator frequency imposed by the qubit metamaterial and the collective resonant coupling of eight qubits. The realized prototype represents a mesoscopic limit of naturally occurring spin ensembles and as such we demonstrate the AC-Zeeman shift of a resonant qubit ensemble. The studied system constitutes the implementation of a basic quantum metamaterial in the sense that many artificial atoms are coupled collectively to the quantized mode of a photon field.

We compare the results of ground state and spectroscopic measurements carried out on superconducting flux qubits which are effective two-level quantum systems. For a single qubit and for two coupled qubits we show excellent agreement between the parameters of the pseudospin Hamiltonian found using both methods. We argue that by making use of the ground state measurements the Hamiltonian of N coupled flux qubits can be reconstructed as well at temperatures smaller than the energy level separation. Such a reconstruction of a many-qubit Hamiltonian can be useful for future quantum information processing devices.