State mapping between atoms and photons, and photon-photon interactions play an important role in scalable quantum information processing. We consider the interaction of a two-level atom with a quantized propagating pulse in free space and study the probability P e (t) of finding the atom in the excited state at any time t. This probability is expected to depend on (i) the quantum state of the pulse field and (ii) the overlap between the pulse and the dipole pattern of the atomic spontaneous emission. We show that the second effect is captured by a single parameter ∈ [0,8π/3], obtained by weighting the dipole pattern with the numerical aperture. Then, P e (t) can be obtained by solving time-dependent Heisenberg-Langevin equations. We provide detailed solutions for both single-photon Fock state and coherent states and for various temporal shapes of the pulses.

We present a method to implement fast two-qubit gates valid for the ultrastrong coupling (USC) and deep strong coupling (DSC) regimes of light-matter interaction, considering state-of-the-art circuit quantum electrodynamics (QED) technology. Our proposal includes a suitable qubit architecture and is based on a four-step sequential displacement of the intracavity field, operating at a time proportional to the inverse of the resonator frequency. Through ab initio calculations, we show that these quantum gates can be performed at subnanosecond time scales, while keeping a fidelity above 99%.Introduction.-With the advent of quantum information science [1], there have been enormous efforts in the design of devices with high level of quantum control and coherence [2]. Circuit QED [3][4][5] has become a leading technology for solidstate based quantum computation and its performance is approaching that of trapped ions [6] and all-optical implementations [7]. Considerable progress has been made in recent circuit QED experiments involving ultrastrong coupling [8,9], two-qubit gate and algorithms [11][12][13][14][15][16], three-qubit gate and entanglement [17][18][19]. Most of the proposed gates are based on slow dispersive interactions or faster resonant gates, and would require operation times of about tens of nanoseconds.To speed up gate operations, the latest circuit-QED technology offers the USC regime of light-matter interactions [8-10, 20, 21], where the coupling strength g is comparable to the resonator frequency ω r (0.1 < ∼ g/ω r < ∼ 1). This should open the possibility to achieve fast gates operating at subnanosecond time scales [22,23] . In this sense, the design of these novel gates becomes a challenge as the rotating-wave approximation (RWA) breaks down and the complexity of the quantum Rabi Hamiltonian emerges [24,25]. Preliminary efforts have been done in this direction involving different configurations of superconducting circuits [26][27][28]. Likewise, in a recent contribution, it has been discussed the possibility of performing protected quantum computing [29].In this Letter, we propose a realistic scheme to realize fast two-qubit controlled phase (CPHASE) gates between two newly designed flux qubits [30], coupled galvanically to a single-mode transmission line resonator (Fig. 1). Our proposal includes: (i) a CPHASE gate protocol operating at times proportional to the inverse of the resonator frequency; (ii) the design of the qubit-resonator system, allowing for high controllability on both the qubit transition frequency and the qubit-resonator coupling, in USC [8,9] and potentially the DSC regime [31] of light-matter interaction. Through ab initio numerical analysis, we discuss the main features of this scheme in detail and show that the fidelity could reach 99.6%. This is an important step in the reduction of resources requirement for fault-tolerant quantum computation [32].Design of a versatile flux qubit.-The junction array is schematically depicted in Fig. 1. It consists of a six-

We report the experimental observation of high-order sideband transitions at the single-photon level in a quantum circuit system of a flux qubit ultrastrongly coupled to a coplanar waveguide resonator. With the coupling strength reaching 10% of the resonator's fundamental frequency, we obtain clear signatures of higher-order redand first-order blue-sideband transitions, which are mainly due to the ultrastrong Rabi coupling. Our observation advances the understanding of ultrastrongly-coupled systems and paves the way to study high-order processes in the quantum Rabi model at the single-photon level.