A key advantage of quantum metrology is the ability to surpass the standard quantum limit (SQL) for measurement precision through the use of nonclassical states. However, there is typically little to no improvement in precision with the use of nonclassical states for measurements whose duration exceeds the decoherence time of the underlying quantum states. Measurements aimed at the ultimate possible precision are thus performed almost exclusively with classical states and, therefore, are constrained by the SQL. Here, we demonstrate that by using the phenomenon of subharmonic excitation, in combination with a recently demonstrated technique of Raman excitation of a harmonic oscillator, the frequency of an electric field can be measured at a resolution below the SQL of the corresponding linear generator. With this method we measure a radio-frequency electrical signal with a fractional frequency uncertainty of $0.56\mathrm{Hz}/80\mathrm{MHz}=7\times {10}^{-9}$, which to our knowledge is the most precise frequency measurement of a radio-frequency electrical signal using a quantum harmonic oscillator. Because the input states can be classical, the coherence time is not degraded by the enhanced decoherence typically associated with nonclassical states, thereby improving the ultimate achievable precision. While we demonstrate this technique using motional Raman subharmonic excitation of a single ${}^{40}{\mathrm{Ca}}^{+}$ ion through engineered Floquet states, this technique is expected to be extendable to other platforms, such as NV centers, solid-state qubits, and neutral atoms, where it can provide metrological gain for sensing across the radio frequency, microwave, and optical domains.

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• Quantum metrology
• Quantum sensing
• Ions