Reinforcement-learning-based matter-wave interferometer in a shaken optical lattice

We demonstrate the design of a matter-wave interferometer to measure acceleration in one dimension with high precision. The system we base this on consists of ultracold atoms in an optical lattice potential created by interfering laser beams. Our approach uses reinforcement learning, a branch of machine learning that generates the protocols needed to realize lattice-based analogs of optical components including a beam splitter, a mirror, and a recombiner. The performance of these components is evaluated by comparison with their optical analogs. The interferometer's sensitivity to acceleration is quantitatively evaluated using a Bayesian statistical approach. We find the sensitivity to surpass that of standard Bragg interferometry, demonstrating the future potential for this design methodology.

Figure 1

Interferometers are composed of (i) a beam splitter, (ii) mirrors, and (iii) a recombiner. Examples shown are (a) an optical Mach-Zehnder interferometer (using half-silvered or conventional mirrors), (b) a Bragg interferometer (using three short light pulses of varying pulse area, $\pi /2$ or $\pi$), and (c) a shaken lattice interferometer. In (c), the shaken lattice mimics the interferometer components by splitting, reflecting, and recombining the atoms through design of a specific shaking function for each case.

Figure 2

Framework of reinforcement learning. The agent chooses an action, the environment responds to the action and gives the next state and a reward as feedback. In our design task, the action is the translation of the optical lattice, the environment evolves according to the Schrödinger equation, the observation is the momentum population distribution, and the reward is a function of the quantum fidelity. Illustrated on the left is an example neural network as the decision-making agent. The neural network takes the state represented on its input layer, passes it through a hidden layer, and generates a vector of Q-values at its output layer.

Figure 3

Shaking function for splitting. The learned lattice phase, $\varphi \left(t\right)$, (top) is allowed to take on one of a discrete set of five possible values that span the range shown. The momentum probability distribution (bottom) is initialized to the ground state of the lattice and at the terminal time well approximates the desired superposition for a beam splitter.

Figure 4

Shaking function for reflection. The learned lattice phase, $\varphi \left(t\right)$, (top) is sinusoidal and the amplitude is allowed to take on any one of a discrete set of five values at each half cycle. The momentum distribution (bottom) is prepared in $-4\hslash k_L$ and well approximates $4\hslash k_L$ at the terminal time.

Figure 5

Time evolution of the matter-wave density throughout the entire interferometry sequence. The white dotted lines separate the plot into five regions. In region I, we apply the splitting protocol. In region II, we allow the matter wave to propagate freely in the lattice. The appearance of two wave packets traveling in opposite directions shows that the beam splitter operates as expected. In region III, we apply the mirror shaking function. The matter wave undergoes free propagation in region IV again, and the two wave packets switch directions, demonstrating the functionality of the mirror. Last, we apply the recombining protocol in region V. Apart from the main closed diamond-shaped paths, we observe that there are auxiliary paths that are fainter but still clearly evident. They arise due to the imperfection of the components and also due to the side peaks arising from the third-excited Bloch state.

Figure 6

(a) Posterior probability density of the acceleration for the first 100 atoms. The true acceleration that we aim to reveal by the measurements is $-3\times {10}^{-3}{\omega }_r{v}_r$ (red line), and the measurements are sampled from the momentum distribution at the end of the interferometry sequence, as shown in the inset. (b) Standard deviation of the acceleration estimated using Bayes theorem for up to ${10}^4$ atoms. We show the results from both the shaken lattice interferometer (SLI) and the Bragg interferometer and conclude that the SLI has a higher sensitivity. The standard deviations ${\sigma }_a$ are roughly inversely proportional to the square root of the number of measurements $N$, for $N>{10}^2$. The black dashed lines are the Cramér-Rao (CR) lower bounds and scale exactly as $1/\sqrt{N}$.

Figure 7

An example topology of the network that we used in the deep Q-algorithm.