We theoretically study single and two-qubit dynamics in the circuit QED architecture. We focus on the current experimental design [Wallraff et al., Nature 431, 162 (2004); Schuster et al., Nature 445, 515 (2007)] in which superconducting charge qubits are capacitively coupled to a single high-Q superconducting coplanar resonator. In this system, logical gates are realized by driving the resonator with microwave fields. Advantages of this architecture are that it allows for multi-qubit gates between non-nearest qubits and for the realization of gates in parallel, opening the possibility of fault-tolerant quantum computation with superconduting circuits. In this paper, we focus on one and two-qubit gates that do not require moving away from the charge-degeneracy 'sweet spot'. This is advantageous as it helps to increase the qubit dephasing time and does not require modification of the original circuit QED. However these gates can, in some cases, be slower than those that do not use this constraint. Five types of two-qubit gates are discussed, these include gates based on virtual photons, real excitation of the resonator and a gate based on the geometric phase. We also point out the importance of selection rules when working at the charge degeneracy point.

Photons are ideal carriers for quantum information as they can have a long coherence time and can be transmitted over long distances. These properties are a consequence of their weak interactions within a nearly linear medium. To create and manipulate nonclassical states of light, however, one requires a strong, nonlinear interaction at the single photon level. One approach to generate suitable interactions is to couple photons to atoms, as in the strong coupling regime of cavity QED systems [1, 2]. In these systems, however, one only indirectly controls the quantum state of the light by manipulating the atoms [3]. A direct photon-photon interaction occurs in so-called Kerr media, which typically induce only weak nonlinearity at the cost of significant loss. So far, it has not been possible to reach the single-photon Kerr regime, where the interaction strength between individual photons exceeds the loss rate. Here, using a 3D circuit QED architecture [4], we engineer an artificial Kerr medium which enters this regime and allows the observation of new quantum effects. We realize a Gedankenexperiment proposed by Yurke and Stoler [5], in which the collapse and revival of a coherent state can be observed. This time evolution is a consequence of the quantization of the light field in the cavity and the nonlinear interaction between individual photons. During this evolution non-classical superpositions of coherent states, i.e. multi-component Schrödinger cat states, are formed. We visualize this evolution by measuring the Husimi Q-function and confirm the non-classical properties of these transient states by Wigner tomography. The ability to create and manipulate superpositions of coherent states in such a high quality factor photon mode opens perspectives for combining the physics of continuous variables [6] with superconducting circuits. The single-photon Kerr effect could be employed in QND measurement of photons [7], single photon generation [8], autonomous quantum feedback schemes [9] and quantum logic operations [10].A material whose refractive index depends on the intensity of the light field is called a Kerr medium. A light beam traveling through such a material acquires a phase shift φ Kerr = Kτ I [11] where I is the intensity of the beam, τ is the interaction time of the light field with the material, and K is the Kerr constant. The Kerr effect is a widely used phenomenon in nonlinear quantum optics and has been successfully employed to generate quadrature and amplitude squeezed states [12], parametrically convert frequencies [13], and create ultra-fast pulses [14]. In the field of quantum optics with microwave circuits, the direct analog of the Kerr effect is naturally created by the nonlinear inductance of a Josephson junction (specifically the4 term in the Taylor expansion of the cos φ of the Josephson energy relation) [15,16]. This effect has been used to create Josephson parametric amplifiers [17][18][19] and to generate squeezing of microwave fields [20]. However, in both the microwave and optical dom...

We present a theoretical study of a superconducting charge qubit dispersively coupled to a transmission line resonator. Starting from a master equation description of this coupled system and using a polaron transformation, we obtain an exact effective master equation for the qubit. We then use quantum trajectory theory to investigate the measurement of the qubit by continuous homodyne measurement of the resonator out-field. Using the same porlaron transformation, a stochastic master equation for the conditional state of the qubit is obtained. From this result, various definitions of the measurement time are studied. Furthermore, we find that in the limit of strong homodyne measurement, typical quantum trajectories for the qubit exhibit a crossover from diffusive to jumplike behavior. Finally, in the presence of Rabi drive on the qubit, the qubit dynamics is shown to exhibit quantum Zeno behavior.

No abstract

We explore the physics of optomechanical systems in which an optical cavity mode is coupled parametrically to the square of the position of a mechanical oscillator. We derive an effective master equation describing two-phonon cooling of the mechanical oscillator. We show that for high temperatures and weak coupling, the steady-state phonon number distribution is non-thermal (Gaussian) and that even for strong cooling the mean phonon number remains finite. Moreover, we demonstrate how to achieve mechanical squeezing by driving the cavity with two beams. Finally, we calculate the optical output and squeezing spectra. Implications for optomechanics experiments with the membrane-in-the-middle geometry or ultracold atoms in optical resonators are discussed.Comment: 4 pages, 3 figure

No abstract

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].

We employ pulse shaping to abate single-qubit gate errors arising from the weak anharmonicity of transmon superconducting qubits. By applying shaped pulses to both quadratures of rotation, a phase error induced by the presence of higher levels is corrected. Using a derivative of the control on the quadrature channel, we are able to remove the effect of the anharmonic levels for multiple qubits coupled to a microwave resonator. Randomized benchmarking is used to quantify the average error per gate, achieving a minimum of 0.007 ± 0.005 using 4 ns-wide pulse.PACS numbers: 03.67. Ac, 42.50.Pq, The successful realization of quantum information processing hinges upon the ability to perform high-fidelity control (gates) of quantum bits, or qubits. A value of 10 −3 error per gate (EPG) is typically quoted as the necessary threshold for fault-tolerant quantum computation [1]. For any two-level system, the optimal achievable gate performance is set by the ratio of gate time to coherence time. However, quantum information processing systems consist of multiple coupled qubits. Often these qubits are not truly two-level systems, but rather quantum objects with a rich level structure. Thus, one important challenge is to achieve optimized single-qubit gates in a large Hilbert space.Optimal control theory has been previously employed in nuclear magnetic resonance, non-linear optics, and trapped ions to combat specific errors such as alwayson qubit couplings and spatial inhomogeneities [2]. In these systems, even with optimized qubit control, limits to qubit gates are often due to systematic errors rather than decoherence. Superconducting qubit control, however, has primarily been limited by coherence times. With recent progress such as the demonstration of highfidelity single-qubit gates [3,4], two-qubit gates [5][6][7][8], entanglement [9][10][11] and simple quantum algorithms [7], the superconducting qubit architecture is growing into a more complex quantum information testbed, placing the level of qubit control under increased scrutiny. One approach towards improving single-qubit gates is to decrease the total gate time. However, in multi-level qubits such as the transmon [12] or phase qubit [3], the weak anharmonicity sets a lower limit on the gate time. Furthermore, superconducting qubit coupling schemes such as circuit quantum electrodynamics (QED), in which a transmission-line cavity couples multiple qubits [13,14], can make single-qubit control more difficult as a result of many extra levels in the Hilbert space.In this Letter, we implement simple optimal control techniques to improve single-qubit gates in a circuit QED device with two superconducting transmons. The pulseshaping protocols we investigate are guided by the recent theoretical exploration of derivative removal via adiabatic gate (DRAG) in Ref. [15]. We demonstrate the improvement of single-qubit gates on two separate qubits in the same device using the first-order correction in DRAG, by switching from rotations around a single axis induced by Gaussian-modul...