Detecting a signal at an unknown frequency is a common task, arising in settings from dark-matter detection to magnetometry. For any detection protocol, the precision achieved depends on the frequency of the signal and can be quantified by the quantum Fisher information (QFI). To study limitations in broadband sensing, we introduce the integrated quantum Fisher information and derive inequality bounds that embody fundamental trade-offs in any sensing protocol. Our inequalities show that sensitivity in one frequency range must come at the cost of reduced sensitivity elsewhere. For many protocols, including those with small phase accumulation and those consisting of $\pi$ pulses, we find that the integrated quantum Fisher information scales linearly with $T$. We also find protocols with substantial phase accumulation that can have integrated QFI that grows quadratically with $T$ and prove that this scaling is asymptotically optimal. These protocols may allow the very rapid detection of a signal with unknown frequency over a very wide bandwidth. We discuss the implications of these results for a wide variety of contexts, including dark-matter searches and dynamical decoupling. Thus we establish fundamental limitations on the broadband detection of signals and highlight their consequences.

• Received 18 April 2022
• Revised 22 September 2022
• Accepted 16 November 2022

DOI:https://doi.org/10.1103/PhysRevApplied.19.014029