#### 2.1 Circuit quantization

Circuit quantization essentially follows the usual procedure of canonical quantization, consisting of

- Formulation of the classical circuit Lagrangian,
- Legendre transform to the classical Hamiltonian,
- Quantization via ,

Employing these coordinates and their associated velocities, we can specify all kinetic and potential energy contributions and express the Lagrangian in the usual form . The common elementary building blocks of superconducting circuits are capacitors, inductors, and Josephson junctions (JJ). For each such element, when connecting node *n* to , the corresponding energy contributions are

and

The special role of magnetic flux in superconductors produces one additional rule: any superconducting loop (interrupted, at most, by Josephson junctions) is subject to fluxoid quantization and is influenced by any external magnetic flux threading the loop. In the Lagrangian, this constraint is accounted for by selecting one element in each loop to act as the “closure branch,” for which the variables enter as

- (3)

thus including the external flux. After Legendre transformation and canonical quantization, one obtains the quantum Hamiltonian for the circuit. A word of caution is in order regarding “shortcuts” that claim to directly obtain the Hamiltonian: the Legendre transform, in general, requires the inversion of the capacitance matrix and is trivial only for the most simplistic circuits describable by a single generalized coordinate.

**Superconducting qubits**. The circuits used as superconducting qubits [1-3] are prime examples for circuit quantization at work. The Hamiltonians obtained using the above three steps allow for detailed predictions of energy spectra and other observables in good agreement with the host of experimental data on the various flavors of charge [44-46, 9], flux [47-49], and phase qubits [50]. In most cases, the “qubits” actually have excited states beyond the and states which are commonly neglected in the pseudo-spin Hamiltonian

- (4)

Here, denote the Pauli raising and lowering operator for qubit *n*. Higher levels may be important to take into account for qubits with only weak anharmonicity, such as the transmon qubit [8, 9, 51] and recent implementations of phase qubits [53, 54, 52, 55]. In this case, black-box quantization introduced recently by Nigg *et al*. [43], provides an elegant alternative to conventional circuit quantization [15]. Starting from the full admittance function of the linearized circuit (which immediately includes renormalizations of the qubit parameters), the anharmonicity can be treated systematically either by perturbation theory or numerics. In the context of resonator-qubit lattices, a similar approach (starting, however, from the bare qubit parameters) has been presented by Leib et al. [56].

In systems with closely spaced superconducting qubits, additional terms describing direct qubit-qubit interaction need to be included in and could produce interesting “photon-less” lattices of superconducting qubits [1, 57].

For a full-wavelength mode and uniform hopping, , the hopping amplitude

- (9)

carries a positive sign [59]. Whenever half-wavelength modes are involved, the sign change of individual mode functions may induce negative hopping amplitudes and lead to modifications of the photonic spectrum if the lattice contains loops consisting of an odd number of resonators [58]. More advanced schemes with intermediate coupler circuits have been proposed for rendering photon-hopping tunable in-situ and for inducing complex hopping phases associated with broken time-reversal symmetry [58, 60].

**Qubit-photon interaction**. The interaction between qubits and resonator photons typically takes the form

- (10)

assuming that each site of the circuit QED lattice consists of a simple Rabi or Jaynes-Cummings system, i.e., a single resonator coupled to a qubit located inside the resonator. Generalizations of this, such as multiple qubits coupling to the same local resonator, are naturally conceivable and have been demonstrated experimentally [61, 62, 55]. The specific origin and strength of this coupling differs according to qubit type and coupling scheme. For a simple superconducting qubit —namely, a single Josephson junction with capacitive shunt as used for transmon qubits—Devoret *et al*. [63] give a careful and instructive discussion of possible coupling types and their respective strengths. Capacitive coupling of the junction electric charge to the voltage field inside the resonator resembles the dipole coupling of an atom to the electromagnetic field. However, the small mode volume of chip-based transmission line resonators and the large effective dipole moments of superconducting qubits can render the effective coupling strength quite large. (Here, symbols denote α: fine structure constant, β: capacitance ratio of order unity, : Josephson energy, : charging energy.) Values of *g* approaching the ultra-strong coupling regime () are feasible by coupling the qubit directly to the current through the center-pin of a coplanar wave-guide resonator [63-66].

#### 2.2 Open-system description of circuit QED lattices

Experimentally, circuit QED devices are operated at sub-Kelvin temperatures – not only to maintain commonly used metals like aluminum and niobium deep in the superconducting phase but also to minimize spurious population of the microwave resonators with thermal photons. In striking contrast to most other quantum simulators, circuit QED arrays realizing the JCHM have a rather trivial ground state, namely the vacuum state void of photons and with all qubits occupying the ground state. Here, a key difference from ultracold atoms arises from the fact that the grand-canonical ensemble does not exist for photons: coupling to a heat bath alone, i.e., working in the canonical ensemble, already introduces additional photons into the system. (An alternative wording describing the same situation is that the chemical potential for photons always vanishes, see [67].)

A significant host of theoretical work has been based on the assumption that a chemical potential might still be engineered if the time scale for reaching a quasi-equilibrium with given photon number is much shorter than the time scale at which photons escape from the system. The resulting simplicity of equilibrium physics at is highly convenient for the theorist. Whether this regime is accessible in the experiment, however, remains an open question. One way or another, the realistic theoretical description of circuit QED arrays will need to account for the essential openness of this quantum system: the array is driven out of equilibrium with microwave generators to introduce photons, and photons leaking out of the array reveal crucial information about the quantum state inside.

For reference, we summarize the most important parameters for circuit QED arrays along with typical ranges for their values in table 1. Entries do not claim fundamental limitations but rather reflect theoretical proposals and current experimental capabilities. We note that experimental demonstrations of in-situ tunability, so far, typically refer to systems with only a few sites.

Table 1. Circuit QED parameters and typical valuesModel Parameter | | Typical values | In-situ tunability? |
---|

| resonator frequency | 1…15 GHz | Yes – with SQUID terminated resonators [68] |

| qubit frequency | 100 MHz…15 GHz [74] | Yes – e.g., via global or local magnetic flux [61] |

*g* | photon-qubit coupling strength | 0…400 MHz | Yes [64, 69] |

*J* | photon hopping strength | 1…100 MHz [70] | Yes – with additional coupler circuits [58, 60] |

κ | photon escape rate at port | 10 kHz…80 MHz [70] | Yes – with additional coupler circuits [58, 71] |

| intrinsic photon escape/dissipation rate | [72] | No |

γ_{1} | qubit relaxation rate | [25] | Yes – e.g., via Purcell effect [73] |

| qubit dephasing rate | [25] | Yes – by inducing extra noise |