Quantum Coherence between Two Atoms beyond $Q={10}^{15}$

We place two atoms in quantum superposition states and observe coherent phase evolution for $3.4\times {10}^{15}$ cycles. Correlation signals from the two atoms yield information about their relative phase even after the probe radiation has decohered. This technique allowed a frequency comparison of two ${}^{27}\mathrm{Al}^{+}$ ions with fractional uncertainty ${3.7}_{-0.8}^{+1.0}\times {10}^{-16}/\sqrt{\tau /\mathrm{s}}$. Two measures of the $Q$ factor are reported: The $Q$ factor derived from quantum coherence is ${3.4}_{-1.1}^{+2.4}\times {10}^{16}$, and the spectroscopic $Q$ factor for a Ramsey time of 3 s is $6.7\times {10}^{15}$. We demonstrate a method to detect the individual quantum states of two ${\mathrm{Al}}^{+}$ ions in a ${\mathrm{Mg}}^{+}\mathrm{\text{−}}{\mathrm{Al}}^{+}\mathrm{\text{−}}{\mathrm{Al}}^{+}$ linear ion chain without spatially resolving the ions.

Figure 1

Illustration of the protocol. (a) The detected states from the previous Ramsey experiment serve as the initial states for the current measurement. (b) First $\pi /2$ pulse, driven by a laser beam whose axis is along the axis of the ion array. (c) The clock state superpositions freely evolve. (d) Spacing adjustment at the end of the free-evolution period to vary the differential phase $\Delta \varphi$, followed immediately by the second $\pi /2$ pulse. At the end of the sequence, ion states are detected to obtain the correlation.

Figure 2

Correlation probabilities $P_c$ versus $\Delta {\varphi }_z$ for various Ramsey times: (a) 0.1 s, 1500 probes; (b) 0.5 s, 600 probes;(c) 1 s, 600 probes; (d) 2 s, 360 probes; (e) 3 s, 300 probes; (f) 5 s, 100 probes. Dots: measurement outcomes; lines: maximum-likelihood fits to the fringes.

Figure 3

(a) Differential phase ${\varphi }_2-{\varphi }_1$ versus Ramsey time $T$. The solid line is a linear fit, with slope $0.84\pm 0.06\mathrm{rad}/\mathrm{s}$. (b) Measurement uncertainty extrapolated to 1 s averaging time as a function of Ramsey time. Dots: measurement results, where the uncertainties are derived from the uncertainties in the contrast $C$; solid line: theoretical lifetime-limited instability, where only phases corresponding to $\Delta \varphi \approx \pm \pi /2$ are probed; dashed line: expected experimental instability, with $\Delta \varphi$ uniformly distributed over [0, $2\pi$). The dashed line is derived from the measured coherence time of 9.7 s, and an approximate overhead of 100 ms per Ramsey measurement, which reduces the duty cycle.

Figure 4

Proposed frequency comparison of remote optical clocks, here based on ${\mathrm{Al}}^{+}$ ions. Paths 1 and 2 that direct the clock laser light to the ions must be phase stabilized. Local frequency fluctuations, such as those caused by magnetic-field noise, should be minimized. The free-evolution periods are synchronized so that the atoms experience the same phase noise in the Ramsey pulses, the effect of which cancels in the protocol.