due to numerous frequency-dependent effects ([*e.g.*]{}, frequency dispersion and the requisite tolerances to control impedance).

Amplitude spectroscopy is a technique that allows broadband spectroscopic characterization of a quantum system. With amplitude spectroscopy, spectroscopic information is obtained from the system response to driving-field amplitude at a [*fixed*]{} frequency. The resulting spectroscopic interference patterns, “spectroscopy diamonds,” are mediated by multilevel LZS transitions and Mach-Zehnder-type interferometry, and they serve as a fingerprint of the artificial atom’s multilevel energy spectrum (Figs. \[fig:5\]A and \[fig:5\]B). The energy spectrum is then determined by analyzing the atomic fingerprint. In this way, the amplitude spectroscopy technique complements frequency spectroscopy: although a less direct approach, it allows one to probe the energy level structure of a quantum system over extraordinarily large (even practically prohibitive) bandwidths by circumventing many of the challenges associated with a frequency-based approach.

In general, the spectroscopy diamonds arise due to an interplay between the static flux detuning $\delta f_{\rm {dc}}$ and driving amplitude $V$. As described in Sections \[sec:LZS\] and \[sec:MZ\], transitions occur when an avoided crossing is reached for a particular set of values ($\delta f_{\rm {dc}}$,$V$). For example, at a flux detuning $\delta f^*_{\rm {dc}}$, the diamond boundaries occur at $V = V_1, 