I need to transmit frequencies as low as 1Hz and I know the wavelength is the speed of light but wondered if there has been any breakthroughs in this field? I need a handheld antenna or at least one that can fit in a backpack.
The wavelength at 1 Hz is several times greater than the circumference of the Earth. At these frequencies, one arguably can't even have an "antenna" for terrestrial communication, since any two such antennas would be so close together they would be more like two plates of a capacitor or two windings of an inductor than an antenna.
1 Hz isn't even ELF, which is defined by the ITU as 3 to 30 Hz. And before you say "but 1 is almost the same as 3!" consider difference between the 7 meter band at the 15 meter band is the same magnitude.
ELF transmitters are (exclusively, to my knowledge) the domain of militaries and natural sources, and they involve multiple large facilities, and operate at frequencies significantly higher than you are proposing. Even with their huge size, the antennas are incredibly inefficient, and so the transmit power is well beyond the capabilities of a private individual, and certainly not portable.
While I wouldn't rule out some clever amateurs constructing an ELF station, it's such a technical challenge given the limited space, power, and money available that I imagine it would take hours, if not days, to transmit a single bit. I'm afraid a portable station that can communicate over any large distance is not feasible: the portable size would make the inefficiencies orders of magnitude worse, and there simply isn't any known technology that could make portable enough energy to overcome that.
For your required distance of 10-20 ft, you might consider first testing a dummy load.
An antenna approaching any kind of efficiency will be larger than the space you're looking to cover.
And any antenna that is portable will closely approximate a dummy load... So try a few wire wound resistors and see what happens.
Will be interesting to see what your measured distance per transmitted watt will be, please share!
An electrically resonant circuit or object would be very difficult or extremely large at those low frequencies (possibly county, country, or planet sized).
But mechanically resonant generation of a 1 Hz alternating EM field might be easy.
You can accelerate either a charged object, or a magnet in some cyclic or periodic fashion. Note that according to Maxwell's equations, the electric field and magnetic field are coupled duals, so changing (time varying) either will change the other, and if done in a periodic fashion, will generate EM waves. So you could either wiggle a charged object, or move or spin a magnet, permanent or electromagnetic, at 1 Hz. Although possibly very difficult to detect away from the near field, Maxwell's equations seem to include no lower bound on frequency of EM waves.
I'm halfway tempted to connect a sensitive galvanometer up to one of my magnetic loop antenna's and see how far away I can make the galvanometer needle wiggle by swinging a large permanent magnet on a rope around in a circle (or maybe on an old 78 RPM turntable with a variable speed control) nearby to the loop. Instead of a galvanometer, maybe try a sensitive ADC plus a data logger, feed the data to some SDR code, and demodulate away (e.g. how fast and how large a circle the magnet might be getting swung around).
Range will be limited. Maybe a few steps in distance.
For semi-portable ELF highly wound ferrite rod antenna's, here are some vendors for your frequency range:
Used for geomagnetic observations, etc. The active ones are Rx only, but the passive ones might take a few milliwatts of transmit power. Again, likely very short range. Maybe more similar to a colossal NFC antenna. But likely more range at ELF than a typical amateur radio mag loop or extremely short dipole.