Time-dependent atomic magnetometry with a recurrent neural network

Maryam Khanahmadi and Klaus Mølmer

Phys. Rev. A 103, 032406 – Published 10 March 2021

Abstract

We propose to employ a recurrent neural network to estimate a fluctuating magnetic field from continuous optical Faraday rotation measurement on an atomic ensemble. We show that an encoder-decoder architecture neural network can process measurement data and learn an accurate map between recorded signals and the time-dependent magnetic field. The performance of this method is comparable to Kalman filters while it is free of the theory assumptions that restrict their application to particular measurements and physical systems.

Received 3 September 2020
Revised 7 December 2020
Accepted 24 February 2021

DOI:https://doi.org/10.1103/PhysRevA.103.032406

Physics Subject Headings (PhySH)

Atom optics Biological neural networks

Atomic, Molecular & OpticalNetworks

Authors & Affiliations

Maryam Khanahmadi ^1,2,* and Klaus Mølmer^3,†

¹Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan 45137, Iran
²Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK 8000 Aarhus C, Denmark
³Center for Complex Quantum Systems, Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK 8000 Aarhus C, Denmark

^*m.khanahmadi@phys.au.dk
^†moelmer@phys.au.dk

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Vol. 103, Iss. 3 — March 2021

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Images

Figure 1
(a) Schematic of the physical model. The atomic spins are polarized along the $x$ direction and interact with a stochastic magnetic field directed along the $y$ axis. The interaction of the system and the magnetic field causes a Larmor rotation towards the $z$ axis. The system is probed by a laser beam propagating along $z$ and the polarization rotation, represented by a canonical field quadrature variable $x_{ph}$ , is measured (see text). Panel (b) shows an example of such a simulated Faraday rotation signal $x_{ph}$ . Panel (c) shows the noisy magnetic field $B (t)$ that causes the atomic precession and gives rise to the optical signal in panel (b). We note that the data in panel (b) are compatible with (minus) the integral over time of the data in panel (c)
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Figure 2
Schematic of the encoder-decoder RNN to process the sequence to sequence translation (measurement data to magnetic-field estimate). First, each sequence of the recorded signal ${x_{0}, x_{τ}, ..., x_{T}}$ is fed to the encoder. The encoder sequentially updates its hidden state $h_{t}$ , such that the final state holds information about the entire signal from $t = 0$ to $T$ . The final hidden state $h_{T}$ is subsequently treated as the initial state of the decoder, which sequentially extracts the predicted magnetic field as its output. Both the encoder and decoder are LSTM cells (see the Appendix). They have different nonlinear layers which are followed in the decoder by a fully connected layer with one neuron that outputs the predicted value of the field. As shown in the figure, the input of the decoder at each time step $t$ is the previous output estimate $B_{t - τ}$ and the hidden state $h_{t - τ}^{'}$ .
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Figure 3
(a) The loss function (7) is reduced according to the number of epochs of the gradient descent procedure (6) applied on the training data. After 30 epochs, the loss function has converged, i.e., it changes by less than or the order of $10^{- 6}$ in the last two epochs. (b) The time-dependent mean-squared error (8) between the predicted and the true magnetic field attains a constant low value inside the probing interval, while increasing at both ends of the time interval, where the lack of prior and posterior measurement data, respectively, causes larger uncertainty. Panels (c) and (d) show the true magnetic field with red dots together with the value inferred by the RNN, which is shown by the blue curve. The optical signal leading to the inferred magnetic field $B (t)$ in panel (c) is shown in Fig. 1. The simulations are made with the physical parameters $κ^{2} = 18 {(ms)}^{- 1}, μ = 90 {(ms)}^{- 1}, N_{at} = 2 \times 10^{12}$ , and $N_{ph} = 5 \times 10^{9}$ during each time interval $τ = 0.01$ ms. The magnetic field is sampled with $σ_{b} = 2 (p T^{2} / ms)$ and $γ_{b} = 1 {(ms)}^{- 1}$ .
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Figure 4
Schematic diagram of the LSTM cell. Each LSTM cell has three inputs where $c_{t - τ}, h_{t - τ}$ represent the hidden and cell state at the previous time step $t - τ$ and $r_{t}$ is the input to the cell. Rectangles represent LSTM layers and circles are entrywise operations; see Eqs. (A1, A2, A3, A4, A5). For the encoder, the input is the recorded signals and we retain the final hidden and cell state $h_{T}, c_{T}$ as the initial internal state for the decoder by which the output, magnetic field, is predicted. Bifurcations point out the copy operation.
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