Open problems in extending thermal-noise-limited sub-10 mHz cavity lasers to 10^-18-level stability and quantum-correlation-enhanced metrology

Two 1.5 um lasers locked to independent 124 K silicon Fabry-Perot cavities were shown in 2017 to be limited only by the fundamental Brownian thermal noise of their mirror coatings, reaching a flicker instability floor of 4×10^-17, linewidths as small as about 5 mHz, and usable coherence times of 11-60 s. The source paper identified what remains to be done for this performance to be pushed further and actually exploited: implementing the projected thermal-noise reductions (operation of a comparable system near 4 K, where the predicted floor is about 8×10^-18, and/or AlGaAs crystalline mirror coatings with higher mechanical Q) while keeping technical noise below that floor; controlling the long-term instability above roughly 10 s so that a lower thermal-noise floor converts into longer usable coherence; and establishing local-oscillator phase-noise conditions under which quantum-correlation (spin-squeezing) gains in clock interrogation survive. No laser has yet been demonstrated at the projected low-10^-18 thermal-noise floor, no unsteered cryogenic cavity sits at its thermal floor, and no absolute (non-differential) quantum-enhanced clock runs against such a free-running local oscillator. The problem remains open.

Background

Ultrastable continuous-wave lasers are the local oscillators (LOs) of optical atomic clocks and of precision instruments such as optical frequency combs, atom interferometers, and searches for drift of fundamental constants. The state-of-the-art technique locks the laser frequency to a resonance of a rigid Fabry-Perot cavity (two high-reflectivity mirrors separated by a length standard), so laser frequency fluctuations mirror fluctuations of the cavity length. The fundamental limit of such a reference is the Brownian thermal noise of the cavity components: thermally driven random motion of the mirror coatings, substrates, and spacer. This sets a fractional length instability that cannot be removed by better engineering, only reduced by material and geometric choices. Performance is quoted as the fractional frequency instability (Allan deviation) versus averaging time; for noise with flicker (1/f) spectral character one distinguishes the short-term instability, an operational linewidth (from the phase-noise spectrum or from a beat note between two independent lasers), and a usable coherence time (the duration over which the accumulated phase uncertainty stays below an application-dependent threshold).

Single-crystal silicon cavities operated near 124 K, where silicon's coefficient of thermal expansion crosses zero, first supported a laser linewidth below 40 mHz (Kessler et al., 2012). Matei et al. (2017) then reported two independent 1.5 um lasers on 124 K silicon cavities whose instability between 0.8 s and a few tens of seconds was completely determined by the Brownian noise of the mirror coatings: a flicker floor of $4\times 10^{-17}$, individual linewidths as small as 5 mHz at 194 THz, and usable phase-coherence times of 11 s and 60 s for the interrogation protocols analyzed. With this, interrogation-laser stability became comparable to the quantum projection noise (QPN, the statistical limit from measuring a finite number of uncorrelated atoms) of the best optical clocks.

The same paper analyzed what further improvement requires. Lowering the operating temperature to 4 K was predicted to reduce the thermal-noise floor to about $8\times 10^{-18}$ in the modified Allan deviation (a variant of the Allan deviation that averages frequency samples within each averaging window and therefore suppresses white phase noise, making it the standard measure for flicker-limited oscillators), and a comparable noise figure was expected from replacing dielectric (amorphous, ion-beam-sputtered) coatings with AlGaAs-based crystalline coatings, which have a higher mechanical Q factor, i.e. lower mechanical loss and thus lower thermally driven length noise (Cole et al., 2013; Cole et al., 2016). Implementing both was expected to reach the lower half of the $10^{-18}$ range, roughly an order of magnitude below the demonstrated level. The authors stressed that for this improvement to yield longer coherence it is also necessary to reduce the long-term instability (frequency fluctuations and drifts at averaging times above about 10 s); the present short-term instability was already sufficiently small. Finally, they argued that a rigorous linewidth and coherence-time treatment becomes critical when such lasers interrogate classically or quantum-correlated atoms: enhanced stability from quantum correlations such as spin squeezing (entangled atomic states with phase uncertainty reduced below the standard quantum limit, prepared for example by quantum-non-demolition measurement) requires an LO that does not introduce excessive phase noise, since LO phase noise can remove the benefit of the correlations (Kessler et al., 2014).

It is an experimental-physics problem whose completion boundary is a working cavity-stabilized laser operating at (or demonstrably near) its projected $10^{-18}$-level thermal-noise floor at cryogenic temperature with all technical noise below that floor, together with long-term (averaging times above about 10 s) instability controlled well enough that the reduced floor translates into increased usable coherence, and with the LO phase-noise conditions for quantum-correlation-enhanced interrogation established and met experimentally.

Problem Statement

The 2017 sub-10 mHz, 124 K silicon-cavity lasers are at the fundamental thermal-noise limit of their dielectric mirror coatings, with a flicker floor of $4\times 10^{-17}$, linewidths down to about 5 mHz, and usable coherence times of 11-60 s. The unresolved problem, as posed by the source paper, is to convert the identified routes to a lower thermal-noise floor into demonstrated, usable performance, and to exploit the result in quantum-correlation-enhanced metrology. Specifically:

(1) Implement the projected thermal-noise reduction in a real system: operate a comparable silicon-cavity-stabilized laser at $T\approx 4$ K (predicted thermal-noise floor $\approx 8\times 10^{-18}$ in the modified Allan deviation) and/or incorporate AlGaAs-based crystalline mirror coatings (higher mechanical Q, comparable predicted noise figure) into a cryogenic silicon-cavity assembly, with the engineering required to keep all technical noise - cryostat vibration, temperature and residual-gas effects, and the excess birefringent and power-dependent noise mechanisms that later work has uncovered in crystalline coatings - below the reduced floor, so the laser operates at its thermal-noise limit at the floor projected for the implemented configuration - about $8\times 10^{-18}$ with either route alone, or the lower half of the $10^{-18}$ range, roughly an order of magnitude below the 2017 level, when both routes are combined.

(2) Control and reduce the long-term instability - frequency fluctuations and drifts at averaging times above roughly 10 s - of such a cavity system, without continuous steering by an atomic reference, so that the reduced thermal-noise floor actually translates into increased usable coherence times (linewidths, interrogation durations, clock stability) rather than being masked by slow drift.

(3) For quantum-correlation-enhanced interrogation of atoms (e.g. spin-squeezed optical clocks), establish the local-oscillator phase-noise requirements of specific quantum-enhanced protocols and meet them experimentally, so that residual local-oscillator phase noise does not negate the advantage of quantum-correlated interrogation - ultimately demonstrating an absolute (not merely differential, common-mode-noise-cancelling) quantum enhancement of clock stability using a cavity-stabilized local oscillator of this class.

The common objective uniting the three parts is longer usable coherence from a cavity-stabilized laser: parts (1) and (2) lower and expose the coherence-limiting noise floor of the oscillator itself, and part (3) is the condition the source identified for that coherence to yield metrological gain in quantum-enhanced atomic interrogation.

References

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