Magnetometry via a double-pass continuous quantum measurement of atomic spin

Abstract

We argue that it is possible in principle to reduce the uncertainty of an atomic magnetometer by double-passing a far-detuned laser field through the atomic sample as it undergoes Larmor precession. Numerical simulations of the quantum Fisher information suggest that, despite the lack of explicit multi-body coupling terms in the system's magnetic Hamiltonian, the parameter estimation uncertainty in such a physical setup scales better than the conventional Heisenberg uncertainty limit over a specified but arbitrary range of particle number N . Using the methods of quantum stochastic calculus and filtering theory, we demonstrate numerically an explicit parameter estimator (called a quantum particle filter) whose observed scaling follows that of our calculated quantum Fisher information. Moreover, the quantum particle filter quantitatively surpasses the uncertainty limit calculated from the quantum Cramér-Rao inequality based on a magnetic coupling Hamiltonian with only single-body operators. We also show that a quantum Kalman filter is insufficient to obtain super-Heisenberg scaling, and present evidence that such scaling necessitates going beyond the manifold of Gaussian atomic states.

Advances in quantum metrology

“…For instance, RMSE with scalings of the order ∝ N −k can be obtained when using Hamiltonians that involves k-system interactions between the probes [85], while ∝ 2 −N scaling can be achieved by introducing an exponentially large number of coupling terms [83]. Proposed implementations include scattering in Bose condensates [87,88], Duffing nonlinearity in nano-mechanical resonators [93], two-pass effective nonlinearity with an atomic ensemble [89], Kerr-like nonlinearities [80-82, 95, 96], and nonlinear quantum atomlight interfaces [92].…”

“…Several Authors [80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96] have recently considered the possibility of using nonlinear effects to go beyond the N −1 Heisenberg-like scalings in phase estimation problems. These new regimes have been called "superHeisenberg" scalings in Ref.…”

“…[97]. Ultimately the idea of these proposals is to consider settings where the unitary transformation that "writes" the unknown parameter x into the probing signals, is characterized by many-body Hamiltonian generators which are no longer extensive functions of the number of probes employed in the estimation [83][84][85][86][87][88][89][90][91][92] or, for the optical implementations which yielded the inequality (6), in the photon number operator of the input signals [80][81][82][83][93][94][95][96]. Consequently, in these setups, the mapping (e −ixH ) ⊗n which acts on the input states ρ (n) 0 , gets replaced by a transformations of the form e −ixH (n) which couples the probes non trivially.…”

Dynamically decoupled three-body interactions with applications to interaction-based quantum metrology

“…Scaling ofn −3/2 , wheren is the number of photons, has been experimentally achieved [16] in the detection of atomic magnetization that couples to effective pairwise (k = 2) photon-photon interactions. Theory proposals and analysis exist on performing interaction-based metrology with two-body interactions in a number of systems [10,[17][18][19][20][21][22][23].…”

Quantum metrology with cold atomic ensembles

“…A number of systems have been proposed to observe these anomalous scaling properties, including Kerr nonlinearities [45], cold collisions in condensed atomic gases [6], Duffing nonlinearity in nano-mechanical resonators [46] and a two-pass effective nonlinearity with an atomic ensemble [47].Topological excitations in nonlinear systems may also give advantageous scaling [48].…”