But how can just one signal determine the ship's location? Doesn't that just say how far the ship is from Earth?
tl;dr: The atomic clock increases the precision of the measurement and could even enable one-way orbit determination, but navigation is still done with some combination of range, range-rate, Delta-DOR and calculation of gravitational effects using numerical integration, solar system ephemerides, and other force models relevant to the spacecraft trajectory.
Navigation of spacecraft in deep space is done with several tools. Three of the main ones are:
Range-Rate is a measurement of the frequency shifts from signals broadcast from one location on Earth and received by the spacecraft (range-rate as in the rate of change of the range, expressed in units of range divided by a time, typically km/s). It is most commonly done in a two-way fashion whereby the spacecraft will return the same signal in a frequency-coherent way and received back on Earth. An atomic clock enables one-way range-rate measurements because the frequency of the signal generated onboard the spacecraft is extremely precise, and therefore, the measurement of Doppler shift is most entirely due to the relative velocity of the spacecraft compared to the ground station, instead of being due to a poor signal generation onboard the spacecraft. Similarly, an excellent onboard clock allows the spacecraft to precisely discriminate the ranging tones sent from the ground station: this means a very precise reading of the time of flight information. This October 2020 paper details how an onboard atomic clock helps for orbit determination.
A range-rate measurement is often combined with a range measurement, which is the time of flight of a signal. The combination of range and range-rate measurement is often called delay-Doppler in radar systems. You get a range from the delay, and a rate-of-change of range (a 1 dimensional velocity) from the Doppler shift.
Delta-DOR: is a one-way measurement of signals broadcast from a spacecraft and received by two widely separated deep-space ground stations on earth and the difference in the times of signal arrival is precisely measured (and used to calculate a bearing). This is corrected using information about the current delays due to Earth’s atmosphere, obtained by simultaneously tracking (from each ground location) radio signals from a quasar (within 10 degrees of the same direction).
Numerical Integration In spacecraft navigation you would use the above two methods plus extremely accurate simulations of the history of the spacecraft's trajectory based on all known forces, gravitational and otherwise, to build a complete picture of the trajectory of the spacecraft. Masses and positions of all large solar system objects already exist as ephemerides and these are inputs to such a calculation.
- Range-Rate gives a distance and speed without an accurate direction
- Delta-DOR gives an accurate direction
- Numerical integration augments these two and completes the picture, providing more accuracy than possible with the first two alone
How does an onboard atomic clock help interplanetary navigation?
@DavidHammen's excellent answer explains how having an atomic clock aboard a spacecraft allows the range-rate measurement to be done one-way instead of two way. I recommend you read the full answer, but briefly, the spacecraft broadcasts a signal encoded with time information from the atomic clock. These signals are received on Earth and compared to known time here, and the difference, once corrected for relativistic effects gives the distance. Doppler shift of the time signals or the frequency (if locked to the atomic clock) gives the rate or relative velocity.