Colloquium: Quantum limits to the energy resolution of magnetic field sensors

The energy resolution per bandwidth $E_R$ is a figure of merit that combines the field resolution, bandwidth or duration of the measurement, and size of the sensed region. Several different dc magnetometer technologies approach $E_R=\hslash$, while to date none have surpassed this level. This suggests a technology-spanning quantum limit, a suggestion that is strengthened by model-based calculations for nitrogen-vacancy centers in diamond, for superconducting quantum interference device sensors, and for some optically pumped alkali-vapor magnetometers, all of which predict a quantum limit close to $E_R=\hslash$. This Colloquium reviews what is known about energy resolution limits, with the aim of understanding when and how $E_R$ is limited by quantum effects. A survey of reported sensitivity versus size of the sensed region for more than 20 magnetometer technologies is included, the known model-based quantum limits are reviewed, and possible sources for a technology-spanning limit are critically assessed, including zero-point fluctuations, magnetic self-interaction, and quantum speed limits. Finally, sensing approaches are described that appear to be unconstrained by any of the known limits, thus making them candidates to surpass $E_R=\hslash$.

Figure 1

Schematic diagram of a dc SQUID in constant current mode with resistively shunted Josephson junctions. A superconducting loop (thick blue) is interrupted by two Josephson junctions (dashed red). The junctions are shunted by resistances $R$. A constant current $I$ feeds the SQUID, and the generated voltage $V$ is used to infer the flux $\mathrm{\Phi }$ threading the SQUID loop.

Figure 2

Reported magnetic sensitivity $\delta B\sqrt{T}$ for different sensor technologies versus size of the sensitive region. Effective linear dimension $l_{\mathrm{eff}}$ indicates $\sqrt{\mathrm{area}}$ for planar sensors and $\sqrt[3]{\text{volume}}$ for volumetric ones. For pointlike systems such as single spins, $l_{\mathrm{eff}}$ indicates $\sqrt[3]{\text{volume}}$ for a sphere with radius equal to the minimum source-detector distance. For work reporting sensitivity in units of magnetic dipole moment, we convert to field units using the reported sample distance. Excepting RFNVD, noise levels are the lowest reported value at frequency $\le 1\mathrm{kHz}$. An arrow indicates that the value is off the scale. SQUID, superconducting quantum interference device; SQUIPT, superconducting quantum interference proximity transistor; SKIM, superconducting kinetic impedance magnetometer; OPM, optically pumped magnetometer; FCOPM, OPM with flux concentrators; CEOPM, cavity-enhanced OPM; COPM, OPM with cold thermal atoms; BEC, Bose-Einstein condensate; RSC, Rydberg Schrödinger cat; NVD, nitrogen-vacancy center in diamond; RFNVD, radio-frequency NVD; FCNVD, NVD with flux concentrators; YIG, yttrium-aluminum-garnet; GMR, giant magnetoresistance; EMR, extraordinary magnetoresistance; MTJ, magnetic tunnel junction; MEMF, magnetoelectric multiferroic; HALL, Hall-effect sensor; GRA, graphene; PAFG, parallel gating fluxgate; MFM, magnetic force microscope, WGM, whispering-gallery mode magnetostrictive. Line shows $E_R\equiv ⟨\delta B^2⟩T{l}_{\mathrm{eff}}^3/(2{\mu }_0)=\hslash$. Numeric labels refer to Table 1.