This chapter explores various aspects of the Dynamical Casimir Effect (DCE) and its implications in the context of circuit quantum electrodynamics (cQED). We begin by reviewing theorigin and fundamental properties of the DCE, including three equivalent mathematical frameworks that offer complementary perspectives on the phenomenon. These formulations will serve as a foundation for the subsequent analyses. We then turn our attention to the practical realization of the DCE in cQED-based architectures, discussing how modern superconducting circuits can be engineered to exhibit this inherently quantum effect. Building on this, we examine how the presence of the DCE influences the performance of a quantum thermal machine operating with a quantum field, shedding light on the interplay between quantum fluctuations and thermodynamic processes. Further, we demonstrate how the DCE can be harnessed to implement a controlled-squeeze gate within a cQED platform, opening a path toward advanced quantum control and quantum information processing. The chapter concludes with a synthesis of the main results and a discussion of potential future directions.
In this work, we studied photon generation due to the Dynamical Casimir Effect (DCE) in a one dimensional (1+1) double superconducting cavity. The cavity consists of two perfectly conductingmirrors and a dielectric membrane of infinitesimal depth that effectively couples two cavities. The total length of the double cavity L, the difference in length between the two cavities ΔL, and the electric susceptibility χ and conductivity v of the dielectric membrane are tunable parameters. All four parameters are treated as independent and are allowed to be tuned at the same time, even with different frequencies. We analyzed the cavity’s energy spectra under different conditions, finding a transition between two distinct regimes that is accurately described by kc=v/χ‾‾‾√. In particular, a lowest energy mode is forbidden in one of the regimes while it is allowed in the other. We compared analytical approximations obtained through the Multiple Scale Analysis method with exact numeric solutions, obtaining the typical results when χ is not being tuned. However, when the susceptibility χ is tuned, different behaviours (such as oscillations in the number of photons of a cavity prepared in a vacuum state) might arise if the frequencies and amplitudes of all parameters are adequate. These oscillations can be considered as adiabatic shortcuts where all generated photons are eventually destroyed. Finally, we present an equivalent quantum circuit that would allow to experimentally simulate the DCE under the studied conditions.
Superconducting circuits reveal themselves as promising physical devices with multiple uses. Within those uses, the fundamental concept of the geometric phase accumulated by the stateof a system shows up recurrently, as, for example, in the construction of geometric gates. Given this framework, we study the geometric phases acquired by a paradigmatic setup: a transmon coupled to a superconductor resonating cavity. We do so both for the case in which the evolution is unitary and when it is subjected to dissipative effects. These models offer a comprehensive quantum description of an anharmonic system interacting with a single mode of the electromagnetic field within a perfect or dissipative cavity, respectively. In the dissipative model, the non-unitary effects arise from dephasing, relaxation, and decay of the transmon coupled to its environment. Our approach enables a comparison of the geometric phases obtained in these models, leading to a thorough understanding of the corrections introduced by the presence of the environment.
We present a quantum open-system approach to analyze the nonunitary dynamics of a superconducting qubit when it evolves under the influence of external noise. We consider the presenceof longitudinal and transverse environmental fluctuations affecting the system’s dynamics and model these fluctuations by defining their correlation function in time. By using a Gaussian-like noise correlation, we can study low- and high-frequency noise contribution to decoherence and implement our results in the computation of geometric phases in open quantum systems. We numerically study when the accumulated phase of a solid-state qubit can still be found close to the unitary (Berry) one. Our results can be used to explain experimental measurements of the Berry phase under high-frequency fluctuations and design experimental future setups when manipulating superconducting qubits.