Extending the fundamental limit of atomic clock stability

Abstract

Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to $\approx 4.5 \text{ dB}$ compared with the two-level model. Furthermore, we identify an even greater stability enhancement of $\approx 5.4 \text{ dB}$ for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an $^{27}\text{Al}^{+}$ optical ion clock.

Figures (3)

• Figure 1: Three-level atom model. (a) A three level atom with excited level $\left|e\right\rangle$ having a linewidth $\Gamma$ can spontaneously decay to ground levels $\left|g_{a}\right\rangle$ and $\left|g_{b}\right\rangle$ with probabilities $p_{a}$ and $p_{b}$, respectively. (b) The model assumes two projective state measurements: $\text{M}_{1}=\left|g_{b}\right\rangle\left\langle g_{b}\right|$ reveals whether the atom is in state $\left|g_{b}\right\rangle$ without affecting other populated states, while $\text{M}_{2}=\left|e\right\rangle\left\langle e\right|$ detects if the atom is in the excited state. (c) An atom is prepared in a coherent superposition of the states $\left|g_{a}\right\rangle$ and $\left|e\right\rangle$, denoted schematically by a Bloch sphere representation, and where populated levels are depicted by black circles. At later times, the atom state is a mixed state of the three-level system: when no decay occurred, the atom is in a superposition of the states $\left|g_{a}\right\rangle$ and $\left|e\right\rangle$. This superposition acquires a phase proportional to the detuning of the laser probe from the atomic transition, and has lower excited state amplitude due to partial measurement by the atom's environment. When the atom decays, it can end up either in $\left|g_{a}\right\rangle$ or in $\left|g_{b}\right\rangle$.
• Figure 2: Interrogation protocols. All Ramsey-type protocols include a state preparation step resulting in a (possibly unbalanced) superposition, followed by an interrogation wait time that concludes with a closing pulse mapping phase into population, and finally end with a state measurement sequence. In the IDR protocol only a single $\text{M}_{2}$ measurement is applied and successful and unsuccessful interrogation results are averaged together. In the DDR protocol both $\text{M}_{1}$ and $\text{M}_{2}$ are performed at the end of the interrogation. If the $\text{M}_{1}$ decay detection is positive, the result is rejected. Otherwise, a final $\text{M}_{2}$ measurement provides the interrogation result to be averaged. In the DDR with mid-interrogation decay detection the atom is continuously checked for decay throughout the interrogation time. As soon as a decay is detected the result is discarded and the interrogation restarts. If no decay is detected by the interrogation time, a $\text{M}_{2}$ measurement produces the result to be averaged.
• Figure 3: $^{27}$Al$^+$ Distinguishable Decay Ramsey (DDR) protocol. The steps depicted in the timing diagram (top) correspond to the general procedure shown in Fig. \ref{['fig:protocols figure']}. We propose a Ramsey sequence that produces a superposition $c_g|^1\rm{S}_0, m=+5/2\rangle + c_e|^3\rm{P}_0, m=-5/2\rangle$ by combining an initial optical $\pi/x$ pulse with $x\le2$ and a nuclear spin flip pulse. These steps are shown in the energy level diagrams on the left column. To close the Ramsey sequence, two pulses reverse the sequence above using a $\pi/2$ optical pulse, which is optimal for all protocols discussed here. The right column shows the $^{27}\rm{Al}^+$ laser pulses involved in a quantum logic spectroscopy (QLS) procedure, which is designed to distinguish population in states $|^3\rm{P}_0, m= +5/2\rangle$, $|^1\rm{S}_0, m=-5/2\rangle$ and $\{|^1\rm{S}_0, m=+3/2\rangle, |^1\rm{S}_0, +5/2\rangle\}$ corresponding respectively to $|e\rangle$, $|g_a\rangle$ and $|g_b\rangle$ in the general protocol. This process involves excitation of the ion motion from the ground state ($n=0$) to an excited Fock state ($n=1$), dependent on the internal state of the ion schmidt2005spectroscopy. The motional state is then detected using sideband transitions on the qubit ion followed by resonance fluorescence detection. Note that steps in the process involving laser pulses on the qubit ion are omitted here.