144 MHz omni directional 1/4 wave antennas are about 18 inches long. 160 meter amateur radio antennas need to be several hundred feet long to pick up weak signals. Atomic clocks work with a signal at 60 KHz. Why don't they need antennas miles long?
For the same reason that your AM broadcast radio doesn't need a super-long antenna.
For transmitting, you really want an antenna that's as big as possible, to couple as much of your expensive transmitter power to the aether as possible.
For receiving, you only need an antenna that's big enough so that you have an adequate signal to noise ratio. Even at 160m (to some extent even at 40m) you don't need a "full sized" antenna for a decent receiver's internal noise to be swamped by atmospheric noise.
There's three factors involved in your "atomic" radio having such a small antenna:
- Atmospheric noise dominates over thermal noise at that frequency -- so even if you did have a 5km-long antenna, all you'd do is get more atmospheric noise in lock step with more signal, for no SNR improvement.
- Because of this, the transmitters are way more powerful than used in amateur communications, and the signal is designed to be pulled out of the mud.
- Your receiver is cheap. There may actually be some improvement to be had by a larger antenna -- but you'll probably be well into the land of diminishing returns when you get up to about 1 meter in diameter.
You are referring to a radio-controlled clock (often incorrectly referred to as an atomic clock) that is automatically synchronized to a time code transmitted by a radio transmitter.
For example, WWVB is a time signal radio station near Fort Collins, Colorado and is operated by the National Institute of Standards and Technology (NIST). The 70 kW ERP signal transmitted from WWVB is a continuous 60 kHz carrier wave modulated by the time signal.
High power medium frequency and low frequency transmissions require giant antennas. But small loop antennas / ferrite loopstick antennas are good enough to receive those transmissions.
The rule of receiving antennas is that you should have enough efficiency to "hear the noise floor" — that is, as long as your antenna delivers enough power to the receiver that the background (atmospheric and man-made) noise is above the receiver's sensitivity, then all that matters is the ratio between the signal and that noise; making the antenna more efficient won't make very much difference at all.
The noise floor is pretty high on 60kHz, and modern solid-state receivers are very sensitive, so even an antenna that's only 0.00001% efficient at capturing RF, compared to a full-sized antenna, is good enough for receiving. The transmiter uses a very large antenna (a 400' top-loaded vertical) and 50kW of input power, in order to radiate enough signal to be heard above the noise floor over most of the continent, but the receiver doesn't have any such requirement.
Low-band amateur stations can actually "hear weak signals" using antennas no more than a few meters in size. They use very large antennas to achieve directivity, that is, to hear in one direction while rejecting noise (and interfering signals) from other directions. Those very large antennas are actually still very inefficient in most cases — their low gain in the desired direction is made up for by even lower gain in other directions, which decreases the effective noise floor. While directivity might help improve a radio clock's reception, aesthetic and convenience reasons prevent its use in consumer units. Nobody wants to deal with an external antenna, or be told that their clock will only work if it's installed facing a specific direction.