Single-domain Bose condensate magnetometer achieves energy resolution per bandwidth below ħ

We present a magnetic sensor with energy resolution per bandwidth E_R < ħ. We show how a ⁸⁷Rb single-domain spinor Bose–Einstein condensate, detected by nondestructive Faraday rotation probing, achieves single-shot low-frequency magnetic sensitivity of 72(8) fT measuring a volume V = 1,091(30) μm³ for 3.5 s, and thus, E_R = 0.075(16)ħ. We measure experimentally the condensate volume, spin coherence time, and readout noise and use phase space methods, backed by three-dimensional mean-field simulations, to compute the spin noise. Contributions to the spin noise include one-body and three-body losses and shearing of the projection noise distribution, due to competition of ferromagnetic contact interactions and quadratic Zeeman shifts. Nonetheless, the fully coherent nature of the single-domain, ultracold two-body interactions allows the system to escape the coherence vs. density trade-off that imposes an energy resolution limit on traditional spin precession sensors. We predict that other Bose-condensed alkalis, especially the antiferromagnetic ²³Na, can further improve the energy resolution of this method.