Progress in quantum metrology and applications for optical atomic clocks

Abstract

Quantum entanglement offers powerful opportunities for enhancing measurement sensitivity beyond classical limits, with optical atomic clocks serving as a leading platform for such advances. This chapter introduces the principles of entanglement-enhanced quantum metrology and explores their applications to timekeeping. We review the theoretical framework of quantum phase estimation, comparing frequentist and Bayesian approaches, and discuss paradigmatic entangled states such as spin-squeezed and GHZ states. Particular emphasis is placed on the challenges posed by decoherence, which constrain the practical advantages that can be realized in large-scale devices. The discussion then turns to frequency estimation in atomic clocks, highlighting how experimental constraints shape the translation of abstract quantum limits into real performance gains. Finally, we outline emerging directions of contemporary quantum metrology. Together, these developments underscore the increasingly close interplay between quantum information processing and precision metrology.

Figures (14)

• Figure 1: Functional dependence of the mean measurement outcome and its variance for a coherent spin state. Shown is the expectation value of the observable $\braket{\mathcal{M}}{\phi}$ (blue line) and the corresponding measurement variance $\Delta^2{\mathcal{M}{\phi}} = \braket{\mathcal{M}^2}{\phi} - \braket{\mathcal{M}}{\phi}^2$ (blue shaded region), where $\mathcal{M} = J_y$ and the phase is encoded through the generator $J_z$. A measured sample mean $\frac{1}{r} \sum \mu_k$ can be converted into to a phase estimate $\varphi(\bm{\mu})$ by inverting the function $\braket{\mathcal{M}}_{\phi}$. In the limit of many measurement repetitions $r\gg 1$ the distribution of phase estimates is centered around the true encoded phase $\phi_0$ and with a variance $\Delta^2_{\varphi}$ that is proportional to the variance of the of the observable $\Delta^2_{\mathcal{M}_{\phi_0}}$ and the tangent of $\braket{\mathcal{M}}_{\phi}$ (black dashed line) at $\phi_0$.
• Figure 2: (a) Schematic of a likelihood function $p(\bm{\mu}|\phi)$ for $r= 1,\dots,10$ simulated measurement outcomes. As the number of measurements increases, the maximum of the likelihood function (dashed lines) converges, on average over many realizations of the simulated measurements, to the true encoded phase $\phi_0$ (b) Histogram of the distribution of simulated maximum likelihood estimates, compared to a Gaussian distribution centered at the true encoded phase value $\phi_0$ with standard deviation $\sqrt{1/(r\, F)}$, where $F$ is the Fisher information of the conditional probability distribution $p(\mu|\phi)$.
• Figure 3: Comparison of the squared estimation errors for a coherent spin state composed of $N = 16$ atoms, obtained from $r = 1$, $10$, and $100$ randomly sampled measurement outcomes in the $y$-basis, averaged over $10^4$ simulated estimation experiments. For a sensor consisting of uncorrelated atoms and in the absence of additional noise, the estimation errors of the maximum-likelihood estimator and the sample-mean estimator coincide. The gray dotted line indicates the standard quantum limit.
• Figure 4: Comparison of the squared estimation errors for spin-squeezed states with $N = 16$ atoms, based on $r = 10$ measurement repetitions in the $y$-basis and averaged over $1000$ simulated estimation experiments for a sample mean estimator (a) and a maximum likelihood estimator (b). The states correspond to the ground states of the spin-squeezing Hamiltonian [Eq.\ref{['eq:SqueezingParent']}] for $\omega/\chi = 1 / N^2, 1 / \sqrt{N}, (\omega/\chi)^*$, where $(\omega/\chi)^*$ denotes the value at which $\Delta^2_{J_y}=\Delta^2_{J_x}$. Dashed lines indicate the phase-dependent sample mean estimator variance [Eq.\ref{['eq:squeezing_phase']}] for each corresponding state. Dotted lines indicate the Cramér-Rao bound for each spin squeezed state state.
• Figure 5: Comparison of the squared estimation errors obtained using a GHZ state with $N = 16$ atoms, evaluated for $r = 1, 10, 100$ measurement outcomes in the basis of $\Pi_x$ and averaged over 10000 simulated estimation experiments. In the absence of noise, the sample mean estimator and the maximum likelihood estimator show the same performance. Note that the scale of the horizontal axis is compressed by a factor of $N$ relative to Figs. \ref{['fig:CSS_MLE_SME']}, and \ref{['fig:SSS']}. The gray dashed line denotes the Heisenberg limit. In this example, the GHZ state has been rotated to ensure symmetric sensitivity around $\phi = 0$.