In cavity quantum electrodynamics (QED) 1-3 , light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED 4-13 , where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both in cavity and circuit QED, but the design flexibility and engineering potential of the latter allowed for increasing the ratio between the atom-cavity coupling rate g and the cavity transition frequency ωr above the percent level 8,14,15 . While these experiments are well described by the renowned Jaynes-Cummings model 16 , novel physics is expected when g reaches a considerable fraction of ωr. Promising steps towards this so-called ultrastrong coupling regime 17,18 have recently been taken in semiconductor structures 19,20 . Here, we report on the first experimental realization of a superconducting circuit QED system in the ultrastrong coupling limit and present direct evidence for the breakdown of the Jaynes-Cummings model. We reach remarkable normalized coupling rates g/ωr of up to 12 % by enhancing the inductive coupling of a flux qubit 21 to a transmission line resonator using the nonlinear inductance of a Josephson junction 22 . Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of the ultrastrong interaction between light and matter.In the strong coupling regime, the atom-cavity coupling rate g exceeds the dissipation rates κ and γ of both, cavity and atom, giving rise to coherent light-matter oscillations and superposition states. This regime was reached in various types of systems operating at different energy scales [1][2][3][23][24][25] . At microwave frequencies, strong coupling is feasible due to the enormous engineerability of superconducting circuit QED systems 4,5 . Here, small cavity mode volumes and large dipole moments of artificial atoms 26 enable coupling rates g of about 15 1 % of the cavity mode frequency ω r . Nevertheless, as in cavity QED, the quantum dynamics of these systems follows the Jaynes-Cummings model, which describes the coherent exchange of a single excitation between the atom and the cavity mode. Although the Hamiltonian of a realistic atom-cavity system contains so-called counterrotating terms allowing the simultaneous creation ior annihilation of an excitation in both atom and cavity mode, these terms can be safely neglected for small normalized coupling rates g/ω r . However, when g becomes a significant fraction of ω r , the counterrotating terms are expected to manifest, giving rise to exciting effects in QED.The ultrastrong coupling regime is difficult to reach in traditional quantum optics, but was recently realized in a solid-stat...

We show how to simulate numerically the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium. The method applies to both finite and inhomogeneous systems, and it is based on two ideas: (a) a representation for density operators which extends that of matrix product states to mixed states; (b) an algorithm to approximate the evolution (in real or imaginary time) of matrix product states which is variational.

We measure the dispersive energy-level shift of an LC resonator magnetically coupled to a superconducting qubit, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of ≈ 0.82 GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50 MHz Bloch-Siegert shift when the qubit is in its symmetry point, fully consistent with our analytical model.

We study the quantum dynamics of a two-level system interacting with a quantized harmonic oscillator in the deep strong coupling regime (DSC) of the Jaynes-Cummings model, that is, when the coupling strength g is comparable or larger than the oscillator frequency ω (g/ω > ∼ 1). In this case, the rotating-wave approximation cannot be applied or treated perturbatively in general. We propose an intuitive and predictive physical frame to describe the DSC regime where photon number wavepackets bounce back and forth along parity chains of the Hilbert space, while producing collapse and revivals of the initial population. We exemplify our physical frame with numerical and analytical considerations in the qubit population, photon statistics, and Wigner phase space.The interaction between a two-level system and a harmonic oscillator is ubiquitous in different physical setups, ranging from quantum optics to condensed matter and applications to quantum information. Typically, due to the parameter accessibility of most experiments, the rotating-wave approximation (RWA) can be applied producing a solvable dynamics called the Jaynes-Cummings (JC) model [1]. In this case, Rabi oscillations inside the JC doublets or collapses and revivals of the system populations [2] are paradigmatic examples of the intuitive physics behind the JC dynamics. To achieve these and other phenomena in the lab, the strong coupling (SC) regime is required, that is, the qubit-oscillator coupling has to be comparable or larger than all decoherence rates. This model accurately describes the dynamics of cavity QED [3,4], trapped ion experiments [5], and several setups in mesoscopic physics, where the qubit-oscillator model is essential in modeling superconducting qubits [6] with either coplanar transmission lines [7][8][9][10] or nanomechanical resonators [11,12]. Nowadays, solid-state semiconductor [13] or superconductor systems [14][15][16][17][18][19][20] have allowed the advent of the ultrastrong coupling (USC) regime, where the coupling strength is comparable or larger than appreciable fractions of the mode frequency: g/ω > ∼ 0.1. In this regime, the RWA breaks down and the model becomes analytically unsolvable, although some limits can be explored [21][22][23][24][25]. Confident of the impressive fast development of current technology, one could explore further regimes where the rate between the coupling strength and oscillator frequency could reach g/ω > ∼ 1, here called deep SC (DSC) regime. This unusual regime, yet to be experimentally explored, is the focus of our current efforts. In this letter, we introduce a rigorous and intuitive description of the DSC regime of the JC model, providing an insightful picture where photon number wavepackets propagate coherently along two independent parity chains of states. In this way, the Hilbert space splits in two independent chains, exhibiting a comprehensible collapse-revival pattern of the system populations.We consider the Jaynes-Cummings Hamiltonian without the RWA, also called Rabi Hamiltonian, desc...

The study of light-matter interaction has led to many fundamental discoveries as well as numerous important technologies. Over the last decades, great strides have been made in increasing the strength of this interaction at the single-photon level, leading to a continual exploration of new physics and applications. Recently, a major achievement has been the demonstration of the so-called strong coupling regime [1, 2], a key advancement enabling great progress in quantum information science. Here, we demonstrate light-matter interaction over an order of magnitude stronger than previously reported, reaching the nonperturbative regime of ultrastrong coupling (USC). We achieve this using a superconducting artificial atom tunably coupled to the electromagnetic continuum of a one-dimensional waveguide. For the largest coupling, the spontaneous emission rate of the atom exceeds its transition frequency. In this USC regime, the description of atom and light as distinct entities breaks down, and a new description in terms of hybrid states is required [4, 8]. Our results open the door to a wealth of new physics and applications. Beyond light-matter interaction itself, the tunability of our system makes it a promising tool to study a number of important physical systems such as the well-known spin-boson [9] and Kondo models [12]. * These authors contributed equally to this work.