Orientation-Dependent Entanglement Lifetime in a Squeezed Atomic Clock

We study experimentally the application of a class of entangled states, squeezed spin states, to the improvement of atomic-clock precision. In the presence of anisotropic noise, the entanglement lifetime is strongly dependent on squeezing orientation. We measure the Allan deviation spectrum of a clock operated with a phase-squeezed input state. For averaging times up to 50 s the squeezed clock achieves a given precision 2.8(3) times faster than a clock operating at the standard quantum limit.

Figure 1

(a) Setup: A standing-wave dipole potential confines ${}^{87}\mathrm{Rb}$ atoms within an optical resonator where they interact with probe light via their state-dependent index of refraction. The probe light is used for cavity feedback squeezing [10, 17] and final state readout. (b) Pulse sequences: A standard Ramsey protocol (top bar) consists of optical pumping into the spin-down CSS, rotation about $S_y$ into the equatorial plane using a microwave pulse (solid red arrow), free precession time (dotted black arrow), and conversion of the accrued phase into a measurable population difference by a rotation about $S_x$. We can shear the CSS into a squeezed state using cavity feedback (dashed blue arrow), then orient the narrow axis of the squeezed state in the phase direction (middle bar) or in the $S_z$ direction (bottom bar). (c) Effect of phase noise: The ideal CSS (shown schematically at top left) is perturbed by classical phase fluctuations accrued during the free precession, increasing its variance in one direction (top right). The same fluctuations are detrimental to phase-squeezed states (middle row) but much less noticeable in number-squeezed states (bottom). The broadened distributions are offset vertically and horizontally in the figure to avoid overlap.

Figure 2

Metrological squeezing parameter $\zeta$ as a function of time. The phase variance for an initial CSS (black solid circles, dotted line) or phase-squeezed state (red solid squares, dashed line) increases due to classical frequency noise. The number variance of a number-squeezed state (green open circles) or the phase variance of a phase-squeezed state protected by a spin echo (blue open squares) can remain below the SQL (solid black line) substantially longer. Error bars show the statistical uncertainty of the variance determination. Inset: Signal contrast $C$ for the CSS (black solid circles) and squeezed states (red solid squares).

Figure 3

Allan deviation of a squeezed clock operating below the SQL. The solid red line with error bars was measured using a squeezed input state. The dotted red line indicates $\sigma (\tau )=1.1\times {10}^{-9}{\mathrm{s}}^{1/2}/\sqrt{\tau }$. The open black circles were measured with a traditional clock using an uncorrelated input state. The dashed black line at $1.85\times {10}^{-9}{\mathrm{s}}^{1/2}/\sqrt{\tau }$ indicates the SQL.