  Laser stabilization sits at the heart of many precision scientific
experiments and applications, including quantum information science, metrology
and atomic timekeeping. These systems narrow the laser linewidth and stabilize
the carrier by use of Pound-Drever-Hall (PDH) locking to a table-scale,
ultra-high quality factor (Q), vacuum spaced Fabry-Perot reference cavity.
Integrating these cavities, to bring characteristics of PDH stabilization to
the chip-scale, is critical to reduce their size, cost, and weight, and enable
a wide range of portable and system-on-chip applications. We report a
significant advance in integrated laser linewidth narrowing, stabilization and
noise reduction, by use of a photonic integrated 4.0-meter-long coil resonator
to stabilize a semiconductor laser. We achieve a 36 Hz 1/{\pi}-integral
linewidth, an Allan deviation (ADEV) of 1.8x10^{-13} at 10 ms measurement time,
and a 2.3 kHz/sec drift, to the best of our knowledge the lowest integral
linewidth and highest stability demonstrated for an integrated reference
cavity. Two coil designs, stabilizing lasers operating at 1550 nm and 1319 nm
are demonstrated. The resonator is bus coupled to a 4.0-meter-long coil, with a
49 MHz free spectral range (FSR), a mode volume of 1.0x10^{10} {\mu}m^3 and a
142 million intrinsic Q, fabricated in a CMOS compatible, ultra-low loss
silicon nitride waveguide platform. Our measurements and simulations show that
the thermorefractive noise floor for this particular cavity is reached for
frequencies down to 20 Hz in an ambient environment with simple passive
vibration isolation and without vacuum or thermal isolation. The TRN limited
performance is estimated to be an 8 Hz 1/{\pi} integral linewidth and ADEV of
5x10^{-14} at 10 ms, opening a stability regime that heretofore has only been
available in fundamentally un-integrated systems.
