## 1 Introduction

Circuit QED describes the interaction between nonlinear superconducting circuits [1-3] and microwave photons stored in resonators or propagating in transmission lines. To minimize Ohmic dissipation, the circuits used in this context are made of superconducting materials. The currents and voltages in these circuit are governed by a quantum Hamiltonian and a corresponding Schrödinger equation. Similar to atoms, superconducting circuits have a ground state and low-lying excitations which form a discrete and anharmonic energy spectrum. In much of the research geared towards solid-state quantum information processing, the energy levels above the lowest two can be ignored and, in slight but common abuse of language, the circuit is referred to as a superconducting qubit. Once coupled to a resonator and external microwave generators as well as detectors, the quantum state of the circuit can be manipulated and read out at will.

Strikingly, such superconducting circuits exhibit quantum coherence on a macroscopic scale: the size of superconducting qubits and transmission line resonators typically ranges from hundreds of micrometers to millimeters. Nowadays, experiments with such circuits are routinely used in producing entangled states and in generating effective photon-photon interactions, rendering them promising building blocks for larger quantum networks. This appeal of electric circuits, their modularity and relative ease of fabrication have greatly boosted the interest in the relatively young field of circuit QED [4-7].

The simplest circuit QED system consists of a single superconducting qubit coupled to the electromagnetic field of a single mode inside a microwave resonator (see Fig. 1, left panel). It represents an open-system realization of the well-known and exactly solvable Jaynes-Cummings model [11] described by the Hamiltonian

Here, creates a photon with energy in a particular resonator mode, are the raising and lowering operators for the superconducting qubit with transition energy , and *g* determines the strength of the qubit-photon coupling. Control and readout of the superconducting qubit can be modeled by incorporating photon leakage and driving into an open-system version of the Jaynes-Cummings model (JCM). Such open-system formulations are based on the reduced density matrix of the system as obtained from a Lindblad master equation or its unravelling in terms of quantum trajectories [12]. Natural extensions of the JCM including the Rabi model [13, 14, 134], the Dicke model [16, 17] and the Tavis-Cummings model [18-21] provide the framework for recent circuit QED experiments involving multiple qubits. At the time of writing, the state-of-the-art in circuit QED entails the implementation of one, two and three-qubit gates on chips with up to four qubits [22-24] and up to five resonators [23]. With coherence times of superconducting qubits now approaching the mark [25, 26], simplified versions of Shor's algorithm and quantum error correction have been demonstrated experimentally [22, 23].

The relatively slow rise of the number of qubits per chip –an increase from 1 to 4 in the last eight years1— does not indicate limitations in fabrication. On the contrary, fabrication of hundreds of qubits on a single chip is well within the realm of existing technology. The slow increase in qubit number rather points to the formidable challenges posed by quantum computation, in particular the need for long coherence times, very high gate fidelities, and individual addressability of qubits. From this viewpoint, building a circuit-based quantum *simulator*[28] rather than a quantum computer, might offer a less ambitious but, arguably, equally exciting goal.

Indeed, well-controlled quantum systems with tunable Hamiltonians and many degrees of freedom might ultimately provide invaluable clues about the nature of different phases and dynamics of strongly correlated many-body systems for which the validity of various approximation schemes not infrequently remains a matter of debate. Spectacular results have already been achieved with quantum simulators based on systems of ultracold atoms [29, 30] and trapped ions [31]. The idea that photon-based systems could likewise serve as quantum simulators goes back to the theoretical proposals by Hartmann [32], Angelakis[33], and Greentree *et al*. [34].

Photons are an unconventional choice for a quantum simulator. As opposed to atoms, their number is not strictly conserved and their interaction is virtually negligible for frequencies easily accessible in the lab. The Jaynes-Cummings model (1) indicates how to cure the latter ailment. In this model, the interaction between photons and matter leads to mediated photon-photon interaction. The circuit QED implementation of a photon-based quantum simulator thus consists of the step from a single Jaynes-Cummings system (see Fig. 1, left panel) to an array of coupled Jaynes-Cummings systems (see Fig. 1, right panel). The physics of the resulting Jaynes-Cummings Hubbard model (JCHM)

thus involves interaction on each site *j* described by the on-site Jaynes-Cummings term , and hopping of photons between nearest-neighbor sites with rate .

We will review both the evident similarities between the JCHM and the Bose-Hubbard model [35] as well as the important differences which could render circuit QED lattices a promising quantum simulator for dissipative phase transitions [36]. Several excellent reviews of the field already exist [37-40] and are recommended to the interested reader. The present article aims to give a synthesis of the most important and most recent results of the field, with an emphasis on ideas advancing the circuit QED-based implementation of interacting photon lattices. The material we present is structured as follows.

In section 'From circuits to lattice Hamiltonians', we introduce the generic model of a circuit QED lattice and show how it reduces to the standard Jaynes-Cummings-Hubbard model. We discuss its validity and the typical energy scales associated with its input parameters and show how the openness of the system is taken into account within a master equation formalism. In section 'Photon blockade', we explain how effective photon-photon interactions arise in circuit QED due to light-matter coupling. We discuss the photon blockade effect, which is the basic underlying mechanism for the generation of strongly correlated photon states. In section 'Small lattice systems', we summarize recent proposals and results for small, finite-size lattices. Finally, section 'Large lattice systems' deals with large arrays and, in particular, focuses on the superfluid-to-Mott-insulator phase transition of polaritons. Our review closes in section 'Conclusions' with a summary and outlook on the future of circuit QED lattices.