Phase Locking a Clock Oscillator to a Coherent Atomic Ensemble

The sensitivity of an atomic interferometer increases when the phase evolution of its quantum superposition state is measured over a longer interrogation interval. In practice, a limit is set by the measurement process, which returns not the phase but its projection in terms of population difference on two energetic levels. The phase interval over which the relation can be inverted is thus limited to the interval $[-\pi /2,\pi /2]$; going beyond it introduces an ambiguity in the readout, hence a sensitivity loss. Here, we extend the unambiguous interval to probe the phase evolution of an atomic ensemble using coherence-preserving measurements and phase corrections, and demonstrate the phase lock of the clock oscillator to an atomic superposition state. We propose a protocol based on the phase lock to improve atomic clocks limited by local oscillator noise, and foresee the application to other atomic interferometers such as inertial sensors.

Popular Summary

Atomic interferometers and atomic clocks are widely used as high-precision sensors for defining time in geophysics and navigation or in high-accuracy tests of fundamental physics. Their principle of operation can be understood by considering matter-wave interferometry in analogy with optical interferometry: An incoming de Broglie wave is split, evolves over different paths, and is finally recombined. The phase difference accumulated along the two paths, which increases with the baseline of the interferometer, leads to a sinusoidal output signal. As a consequence, unambiguous measurements can only be achieved if the phase difference stays well below $\pm \pi /2$, when the interferometer baseline is kept short (to the detriment of the instrument sensitivity).

Several proposals to overcome this limit for atomic clocks have been proposed either by combining the information from several atomic ensembles or by repeated nondestructive measurements of the same atomic ensemble and feedback. Here, we implement a measurement-and-correction scheme to phase lock a clock oscillator to an atomic ensemble of half a million ${}^{87}\mathrm{Rb}$ atoms in a superposition state between two hyperfine levels. We demonstrate how this protocol allows for longer interrogation intervals in a model atomic clock by avoiding ambiguous phase readouts; we experimentally verify a nearly fivefold increase in the clock stability. This novel approach will enable the improvement of the best-reported atomic clocks for which the main limitation is represented by the quality of the local oscillator.

Our work represents a key step in quantum metrology toward improved sensitivity in applications such as fundamental physics tests and absolute inertial navigation. The correction protocol that we develop can be extended to other atomic sensors such as gyroscopes or accelerometers, where the fast variations of rotation and acceleration can hinder the measurements for long-baseline interferometers.

Figure 1

Experimental scheme. The evolution of the LO phase ${\phi }_{\mathrm{LO}}$ is compared to the phase ${\phi }_{\mathrm{at}}$ of an atomic ensemble in a superposition state using coherence-preserving measurements in a Ramsey-spectroscopy sequence. The relative phase is obtained from the readout of the population difference and is used to implement the phase lock between the two oscillators by applying a feedback correction phase ${\phi }_{\mathrm{F}\mathrm{B}}$ on the LO output using a phase actuator. The light shift induced by the optical trap and by the probe has been engineered to have a homogeneous measurement of the atomic ensemble [28].

Figure 2

Bloch-sphere representation of the phase lock between the LO and the atomic superposition state. The phase lock between the classical oscillator and the atomic spin is obtained using repeated, time-correlated Ramsey interrogations and feedback. The sequence begins by preparing the atomic CSS in the $|\downarrow ⟩$ state via optical pumping (step 1). The measurement of the relative phase between the CSS and the LO starts when a $\pi /2$ microwave pulse around the $y$ axis brings the CSS into a balanced superposition of the $|\downarrow ⟩$ and $|\uparrow ⟩$ states, depicted as a vector on the equatorial plane of the Bloch sphere (step 2). The $x$ axis is chosen to represent the phase of the local oscillator and $\phi$ the relative phase between the LO and the atomic superposition that evolves because of the LO noise (step 3). After an interrogation time $T$, the projection of $\phi$ is mapped onto a population difference by a projection $\pi /2$ pulse around the $x$ axis and read out with the coherence-preserving detection (step 4). The CSS is rotated back to the equatorial plane by a reintroduction $\pi /2$ pulse around the $x$ axis (step 5), and feedback is applied on the phase of the LO (step 6). The PLL between the LO and the atomic ensemble consists in the repetition of the steps from 3 to 6, potentially until the atomic ensemble shows a residual coherence.

Figure 3

Coherence-preserving measurement of the relative phase between the LO and the atomic superposition. Real-time measurement of the normalized population difference to which the relative phase $\phi$ between the local oscillator and the atomic superposition state is periodically mapped via microwave rotation pulses. The phase precession is induced by setting the LO frequency 100 Hz off the nominal atomic transition frequency. The experimental points are fitted with a sinusoidal evolution, damped because of the decoherence induced by the $J_z$ measurement and the residual differential light shift on the clock transition.

Figure 4

Phase lock between the LO and the atomic superposition state. The evolution of the LO-atom relative phase is reconstructed from the $J_z$ signal in Fig. 3 (solid red line) and when feedback is applied on the phase of the LO after each measurement (solid blue line). In the open-loop case, the points are reported at the values given by the fit in the previous figure. In the closed-loop case, the relative phase is always smaller than $\pi /2$ and does not enter in the region represented in gray, where it cannot be univocally determined from the measurement.

Figure 5

Correction of phase jumps between the LO and the atomic ensemble. Evolution of the LO-CSS phase when periodic phase jumps of $\pi /3$ are applied back and forth on the LO: The points are obtained from the measured population difference, whereas the solid line represents the LO phase measured after the correction phase actuator. The controller, implemented using coherence-preserving measurements and feedback on a phase actuator, maintains the relative phase close to 0. Above, the system is operated in an open loop, below in a closed loop. In the latter case, the phase corrections are applied $150\mu \mathrm{s}$ after each measurement.

Figure 6

Protocol for atomic clock with PLL. Operation of the clock: At each cycle of duration $T_C$, the relative phase is repeatedly measured in a coherence-preserving way during the phase-lock interval (above: shaded gray areas). Each interrogation, represented in the inset by a light red peak between the manipulation $\pi /2$ pulses, is followed by a phase correction ${\phi }_{\mathrm{FB}}^{(i)}$ on the LO, represented in the inset by a light blue peak on the right after the reintroduction of the spin on the equatorial plane of the Bloch sphere. The final phase readout ${\phi }_f$ (dark red peak in the inset), whose SNR is set by the residual coherence, together with the previously applied phase shifts on the LO, provides the total phase drift $\phi$ experienced during the extended interrogation interval $T_{\mathrm{tot}}=N\times T$. The interrogation sequence ends with the application of a frequency correction on the LO (dark blue peak in the inset); then, a new atomic ensemble is prepared in the dead-time interval $T_D$ for the next cycle.

Figure 7

Atomic clock implementing a PLL. Top: Allan-frequency standard deviation ${\sigma }_1$ for a normal Ramsey clock with interrogation time $T=1\mathrm{ms}$ (red line) and ${\sigma }_9$ for a clock implementing the phase lock between the LO and the atomic superposition state for nine successive, correlated interrogations on the same atomic ensemble, for a total interrogation time of $9\times T=9\mathrm{ms}$ (blue line). The dead time is in both cases $T_D=1.9\mathrm{s}$. The red and blue lines are fits to the data, with a slope set to ${\tau }^{-1/2}$. The continuous black line lies a factor 9 below the red curve and represents the best achievable level of the phase-lock sequence for the same number of interrogations. The dashed black line lies a factor 3 below the red curve and is the optimum level for nine consecutive uncorrelated Ramsey measurements with duration $T$ each and the same total cycle time. Bottom: The gray triangles represent the ratio of the Allan deviation for the two clocks; the solid and dashed lines correspond to the factors 9 and 3 from the top plot.