Coauthors: Dominic W. Berry and Jason Twamley

In this work we present a new method to estimate, with near Heisenberg limited precision, an unknown dynamical phase which is acquired by a two-level atomic spin system over a known time period. We utilise Ramsey interferometry and combine this with adaptive feedback measurements [1], and multiple applications of the unknown phase shift [2]. These methods have been previously used in linear optics to estimate the phase acquired when a photon passes through a phase-shifter. The resulting estimation protocol proposed in this work [2], achieved a precision close to the Heisenberg limit. In our work we adapt this protocol to estimate the phase (rotation about the Z-axis), experienced by a spin within a fixed, known, time duration. Since such a phase may be generated through the spin's interaction with a magnetic field or a local strain field, our phase estimation protocol can be regarded as a high sensitive sensing method. We focus our discussions on estimating the phase associated with an external unknown, temporally constant, magnetic field and one hence could consider our protocol as Heisenberg-limited magnetometry.

Heisenberg-limited phase estimation, which can provide the phase estimate with the uncertainty below standard quantum limit, leads to a better estimation precision than that of standard procedures for a given amount of available resources. Our protocol requires no pre-existing knowledge about the system phase, (or magnetic field in the case of magnetometry), since the final phase estimate is independent to the initial random guess of the phase.

In principle, the proposal can be implemented in any two-level physical system, e.g. atomic vapours [3], or nitrogen-vacancy (NV) centres [4]. The latter are remarkably interesting because they can be individually addressed, optically polarised and measured, and maintain their coherences at room temperature [5]. It has been reported recently that electronic spins associated with NV centres could be used for high-spatial-resolution magnetic field detection and are predicted to possess sensitivities exceeding that of other magnetometry methods (SQUIDs and Hall bar), by an order of magnitude [4]. The realisation of Heisenberg-limited magnetometry in an NV-magnetometry protocol may be feasible in the near future using our protocol and we perform numerical simulations of this to explore the boundaries of the ultimate precision possible using our protocol.

[1] D. W. Berry and H. M. Wiseman, Phys. Rev. A, 63, 013813 (2000). [2] B. L. Higgins, et. al., Nature, 450, 393 (2007). [3] I. M. Savukov, et. al., Phys. Rev. Lett., 95, 063004 (2005). [4] J. M. Taylor et. al., Nat. Phys., 4, 810 (2008). [5] F. Jelezko and J. Wrachtrup, J. Phys. Condens. Matter, R1089 (2004).