We develop a systematic procedure to quantise canonically Hamiltonians of light-matter models of transmission lines coupled through lumped linear lossless ideal nonreciprocal elements, that break time-reversal symmetry, in a circuit QED set-up. This is achieved through a description of the distributed subsystems in terms of both flux and charge fields. We prove that this apparent redundancy is required for the general derivation of the Hamiltonian for a wider class of networks. By making use of the electromagnetic duality symmetry in transmission lines (waveguides), we provide unambiguous identification of the physical degrees of freedom, separating out the nondynamical parts. This doubled description can also treat the case of other extended lumped interactions in a regular manner that presents no spurious divergences, as we show explicitly in the example of a circulator connected to a Josephson junction through a transmission line. This theory enhances the quantum engineering toolbox to design complex networks with nonreciprocal elements.
The classical description of these devices is well established. The fundamental nonreciprocal device, the two-port gyrator, introduces a π-phase shift to input signals in only one of the two directions of information flow. The more sophisticated N-port circulator, which can be described in terms of a network of gyrators, allows for one-way routing of information, e.g. energy flow entering from port N leaves from port N + 1. The existing nonreciprocal devices for microwave circuits are relatively bulky and operate in a classical regime. Recently, there have been several proposals for constructing these nonreciprocal devices to integrate them with the rest of the superconducting quantum technology working at mK, thus in a quantum regime. There is therefore a need to have quantum theoretical descriptions and tools that can serve the introduction of all these quantum nonreciprocal devices and the corresponding technology.
In this work, we include systematically ideal nonreciprocal elements (generalized gyrators and circulators) in canonical Hamiltonian models when they are coupled to transmission lines. To achieve that goal, we start from the telegrapher’s Lagrangian written in terms of flux and charge fields, i. e. a double configuration space, with a magnetic coupling between them. The apparent redundancy introduced in the description becomes the cornerstone for the correct expansion of fields when the lines are connected to nonreciprocal devices. The correct identification of independent modes is explicitly carried out in the Hamiltonian through a symplectic transformation. We illustrate this theory by applying it to a pedagogical example of a Josephson junction, from which superconducting qubits are constructed, capacitively coupled to a 3-port circulator through a transmission line. Due to the correct basis expansion, the quantized Hamiltonian inherits convergent multi-mode properties, such as Lamb shifts or multi-partite effective couplings.
A direct consequence of our study is that the canonical quantization of networks with ideal nonreciprocal N-port devices described in terms of scattering matrices can be easily carried out in terms of a combination of flux and charge variables. This is the outcome of the duality symmetry in Maxwell’s equations, which, in the present context of one-dimensional circuits, simplifies to the telegrapher’s equations.
To summarise, our work provides a common ground for the canonical quantization of superconducting distributed networks with nonreciprocal devices, sets up a general framework for their correct modeling in the quantum regime, and increases the toolbox of all researchers in quantum superconducting circuit technologies, as well as of those in quantum information processing and simulation.
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