We propose a realizable architecture using one-dimensional transmission line resonators to reach the strong coupling limit of cavity quantum electrodynamics in superconducting electrical circuits. The vacuum Rabi frequency for the coupling of cavity photons to quantized excitations of an adjacent electrical circuit (qubit) can easily exceed the damping rates of both the cavity and the qubit. This architecture is attractive both as a macroscopic analog of atomic physics experiments and for quantum computing and control, since it provides strong inhibition of spontaneous emission, potentially leading to greatly enhanced qubit lifetimes, allows high-fidelity quantum non-demolition measurements of the state of multiple qubits, and has a natural mechanism for entanglement of qubits separated by centimeter distances. In addition it would allow production of microwave photon states of fundamental importance for quantum communication.

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We have spectroscopically measured the energy level separation of a superconducting charge qubit coupled non-resonantly to a single mode of the electromagnetic field of a superconducting on-chip resonator. The strong coupling leads to large shifts in the energy levels of both the qubit and the resonator in this circuit quantum electrodynamics system. The dispersive shift of the resonator frequency is used to non-destructively determine the qubit state and to map out the dependence of its energy levels on the bias parameters. The measurement induces an ac-Stark shift of 0.6 MHz per photon in the qubit level separation. Fluctuations in the photon number (shot noise) induce level fluctuations in the qubit leading to dephasing which is the characteristic back-action of the measurement. A cross-over from lorentzian to gaussian line shape with increasing measurement power is observed and theoretically explained. For weak measurement a long intrinsic dephasing time of T2 > 200 ns of the qubit is found.We have recently demonstrated that a superconducting quantum two-level system can be strongly coupled to a single microwave photon [1]. The strong coupling between a quantum solid state circuit and an individual photon, analogous to atomic cavity quantum electrodynamics (CQED) [2], has previously been envisaged by many authors, see Ref. 3 and references therein. Our circuit quantum electrodynamics architecture [3], in which a superconducting charge qubit, the Cooper pair box [4], is coupled strongly to a coplanar transmission line resonator, has great prospects both for performing quantum optics experiments [5] in solids and for realizing elements for quantum information processing [6] with superconducting circuits [7].In this letter we present spectroscopic measurements which demonstrate the non-resonant (dispersive) strong coupling between a Cooper pair box and a coherent microwave field in a high quality cavity. The quantum state of the Cooper pair box is controlled using resonant microwave radiation and is read out with a dispersive quantum non-demolition (QND) measurement [3,8,9]. The interaction between the Cooper pair box and the measurement field containing n photons on average gives rise to a large ac-Stark shift of the qubit energy levels, analogous to the one observed in CQED [10]. As a consequence of the strong coupling, quantum fluctuations in n induce a broadening of the transition line width, characterizing the back action of the measurement on the qubit.In our circuit QED architecture [3], see Fig. 1a, a split Cooper pair box [4], modelled by the two-level hamiltonian H a = −1/2 (E el σ x + E J σ z ) [11], is coupled capacitively to the electromagnetic field of a full wave (l = λ) transmission line resonator, described by a harmonic oscillator hamiltonian H r =hω r (a † a + 1/2). In the Cooper pair box, the energy difference E a =hω a = E 2 el + E 2 J between the ground state |↓ and the first excited state |↑ , see Fig. 1b, is determined by its electrostatic energy E el = 4E C (1 − n g ) and its Josephson ...

Figure 5a in the Letter presents the ac Stark shift of a Cooper pair box inside a coplanar waveguide resonator as a function of microwave probe power driving the resonator. Within lowest order perturbation theory, the corresponding cavity photon number is linear in the drive power. The photon number fit to the data using Eq. (1) in the Letter is shown in Fig. 5a. Lowest order perturbation theory is expected to break down [1] on the scale of the critical photon number n c 2 =4g 2 82. This breakdown is not visible in Fig. 5a which shows the ac Stark shift to be almost perfectly linear in probe power even slightly beyond n c . We pointed out in the Letter that this was a result of the compensation of the two most important nonlinear effects beyond lowest order perturbation theory. Approximate modeling of these nonlinearities [2] shows that the cavity photon number begins to be superlinear in probe power even below n c . The photon number scale in Fig. 5a is thus low by about 50% at the largest power. However, the ac Stark shift per photon is correspondingly sublinear in photon number leading to the accidental cancellation. As a consequence the photon number scale in Fig. 5b at high powers is low by the same amount. A version of this plot taking into account the higher order corrections is presented in Fig. 3 of Ref.[2].