Machine learning assisted vector atomic magnetometry.

Multiparameter sensing such as vector magnetometry often involves complex setups due to various external fields needed in explicitly connecting one measured signal to one parameter. Here, we propose a paradigm of indirect encoding for vector atomic magnetometry based on machine learning. We encode the three-dimensional magnetic-field information in the set of four simultaneously acquired signals associated with the optical rotation of a laser beam traversing the atomic sample. The map between the recorded signals and the vectorial field information is established through a pre-trained deep neural network. We demonstrate experimentally a single-shot all optical vector atomic magnetometer, with a simple scalar-magnetometer design employing only one elliptically-polarized laser beam and no additional coils. Magnetic field amplitude sensitivities of about 100 [Formula: see text] and angular sensitivities of about [Formula: see text] (for a magnetic field of around 140 nT) are derived from the neural network. Our approach can reduce the complexity of the architecture of vector magnetometers, and may shed light on the general design of multiparameter sensing.

Retrodiction beyond the Heisenberg uncertainty relation.

In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of "hypothetical future measurements", and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. Here, we study what happens under such circumstances with an atomic ensemble containing 1011 rubidium atoms, initiated nearly in the ground state in the presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, [Formula: see text] and [Formula: see text] that satisfy [Formula: see text]. Quantum non-demolition measurements of [Formula: see text] before and of [Formula: see text] after time t allow precise estimates of both observables at time t. By means of the past quantum state formalism, we demonstrate that outcomes of measurements of both the [Formula: see text] and [Formula: see text] observables can be inferred with errors below the standard quantum limit. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.

Spin squeezing of 1011 atoms by prediction and retrodiction measurements.

The measurement sensitivity of quantum probes using N uncorrelated particles is restricted by the standard quantum limit1, which is proportional to [Formula: see text]. This limit, however, can be overcome by exploiting quantum entangled states, such as spin-squeezed states2. Here we report the measurement-based generation of a quantum state that exceeds the standard quantum limit for probing the collective spin of 1011 rubidium atoms contained in a macroscopic vapour cell. The state is prepared and verified by sequences of stroboscopic quantum non-demolition (QND) measurements. We then apply the theory of past quantum states3,4 to obtain spin state information from the outcomes of both earlier and later QND measurements. Rather than establishing a physically squeezed state in the laboratory, the past quantum state represents the combined system information from these prediction and retrodiction measurements. This information is equivalent to a noise reduction of 5.6 decibels and a metrologically relevant squeezing of 4.5 decibels relative to the coherent spin state. The past quantum state yields tighter constraints on the spin component than those obtained by conventional QND measurements. Our measurement uses 1,000 times more atoms than previous squeezing experiments5-10, with a corresponding angular variance of the squeezed collective spin of 4.6 × 10-13 radians squared. Although this work is rooted in the foundational theory of quantum measurements, it may find practical use in quantum metrology and quantum parameter estimation, as we demonstrate by applying our protocol to quantum enhanced atomic magnetometry.

Sub-Hertz resonance by weak measurement.

Weak measurement (WM) with state pre- and post-selection can amplify otherwise undetectable small signals and thus has potential in precision measurement applications. Although frequency measurements offer the hitherto highest precision due to the stable narrow atomic transitions, it remains a long-standing interest to develop new schemes to further escalate their performance. Here, we demonstrate a WM-enhanced correlation spectroscopy technique capable of narrowing the resonance linewidth down to 0.1 Hz in a room-temperature atomic vapour cell. The potential of this technique for precision measurement is demonstrated through weak magnetic-field sensing. By judiciously pre- and post-selecting frequency-modulated input and output optical states in a nearly orthogonal manner, a sensitivity of 7 fT Hz-1/2 at a low frequency near DC is achieved using only one laser beam with 15 µW of power. Additionally, our results extend the WM framework to a non-Hermitian Hamiltonian and shed new light on metrology and bio-magnetic field sensing.