  The combination of optical time transfer and optical clocks opens up the
possibility of large-scale free-space networks that connect both ground-based
optical clocks and future space-based optical clocks. Such networks promise
better tests of general relativity, dark matter searches, and gravitational
wave detection. The ability to connect optical clocks to a distant satellite
could enable space-based very long baseline interferometry (VLBI), advanced
satellite navigation, clock-based geodesy, and thousand-fold improvements in
intercontinental time dissemination. Thus far, only optical clocks have pushed
towards quantum-limited performance. In contrast, optical time transfer has not
operated at the analogous quantum limit set by the number of received photons.
Here, we demonstrate time transfer with near quantum-limited acquisition and
timing at 10,000 times lower received power than previous approaches. Over 300
km between mountaintops in Hawaii with launched powers as low as 40 $\mu$W,
distant timescales are synchronized to 320 attoseconds. This nearly
quantum-limited operation is critical for long-distance free-space links where
photons are few and amplification costly -- at 4.0 mW transmit power, this
approach can support 102 dB link loss, more than sufficient for future time
transfer to geosynchronous orbits.
