Heisenberg-Limited Atom Clocks Based on Entangled Qubits

We present a quantum-enhanced atomic clock protocol based on groups of sequentially larger Greenberger-Horne-Zeilinger (GHZ) states that achieves the best clock stability allowed by quantum theory up to a logarithmic correction. Importantly the protocol is designed to work under realistic conditions where the drift of the phase of the laser interrogating the atoms is the main source of decoherence. The simultaneous interrogation of the laser phase with a cascade of GHZ states realizes an incoherent version of the phase estimation algorithm that enables Heisenberg-limited operation while extending the coherent interrogation time beyond the laser noise limit. We compare and merge the new protocol with existing state of the art interrogation schemes, and identify the precise conditions under which entanglement provides an advantage for clock stabilization: it allows a significant gain in the stability for short averaging time.

Figure 1

The proposed clock operation scheme employs $M$ different groups of clock atoms prepared in correlated states of varying size to interrogate the relative phase ${\mathrm{\Phi }}_{\mathrm{LO}}$ of the LO field. A single group $j$ contains $n_0$ independent instances of GHZ-like states, each entangling $2^j$ qubits, and therefore accumulating a phase ${\mathrm{\Phi }}_j=2^j{\mathrm{\Phi }}_{\mathrm{LO}}\mathrm{mod}[-\pi ,\pi ]$ during a single cycle. Each group is then used to measure this phase, which gives a direct estimate on the digit $Z_{j+1}$ in a binary representation of the LO phase $({\mathrm{\Phi }}_{\mathrm{LO}}+\pi )/2\pi =(0.Z_1{Z}_2{Z}_3\dots )$, subsequently used to feedback the LO frequency.

Figure 2

Allan deviation ${\sigma }_y$ for different protocols as a function of averaging time $\tau$, normalized to the standard quantum limit, for ${\gamma }_{\mathrm{LO}}/{\gamma }_{\mathrm{ind}}={10}^3$. The solid black line corresponds to the standard scheme using a single uncorrelated ensemble. It fails to reach the fundamental noise floor set by the atomic transition linewidth (cf., Eq. (5), broken line). A more sophisticated classical scheme which uses exponentially increasing Ramsey times in each cycle [20, 21] allows us to extend the regime of linear scaling with $1/\tau$ up to the point where the bound (5) is met. In comparison, the proposed cascaded GHZ protocol (blue solid curves) enables an $\sim N$ times faster convergence. For short averaging times the stability is enhanced by a factor $\sqrt{N}$ as compared to classical protocols.