What is the longest coherence length stimulated-emission beam demonstrated?

I think the coherence length depend on the power that you want from that laser, the frequency profile of it and the loss in your optical fiber. Also you can use repeaters to increase the coherence length. Usually in laser industry to increase the coherence length people using the brag grating. It could help us having single frequency with long coherence length in vacuum but as it decrease the power significantly it's not always welcome by laser industry for the optical fibers.

For long distance quantum communication it is better to use entangled photons sources. If you want to know more about the quantum coherency take a look at this article and some of it's references:

https://arxiv.org/abs/0710.1143

An atomic clock consists of a source such as a laser or a microwave source (called a local oscillator) and an atom-based measuring technique. For example, the laser or microwave source is used to stimulate a transition in the atom, or in a cloud of atoms. Depending on the atomic response, the laser or microwave source is then adjusted, so as to keep it in step with the atoms.

The result of all this is that the laser or microwave source becomes frequency-stable on long timescales---that is, timescales larger than those required to probe the atoms. But frequency stability is the same thing as coherence time which in turn gives rise to coherence length. Suppose, for example, that the clock drifts by one second in one hour. This means that its relative frequency stability is approx $3 times 10^{-4}$ and if its frequency is $12$ GHz then its linewidth will be $3$ MHz and coherence length 90 metres.

The best atomic clocks currently attain a relative precision of order $10^{-18}$, using an optical transition of frequency around $10^{15}$ Hz. That translates to a linewidth of 1 millihertz and a coherence length of $3 times 10^{11}$ metres. The linewidth of a microwave-based clock can be lower still (even if its relative precision is not quite so good).

It might be objected that these very long coherence lengths are not quite what the question was about. But I think they do represent a legitimate example of the concept of coherence length. One can send out a signal from a frequency-standard laboratory, a signal such as a laser beam or other oscillation, in such a way that two people separated by $10^{11}$ metres will receive oscillations with a non-random phase relationship. They could, for example, recombine their two signals by sending them on to a third party, and the third party could observe interference between the signals. But since that is the definition of coherence, it means that this is indeed a form of temporal coherence and thus a coherence length.

Having said all this, I remain a little uneasy about whether the concept of coherence length is really the most helpful one in such cases. Orbiting bodies in astronomy remain regular over millions of years. This implies that they emit gravitational waves with coherence length of the order of millions of light years (I think---happy to be corrected!)

"Coherence length" is not necessarily a clearly defined term. A laser source can have phase jitter but still have a very large coherence length, as long as the phase jitter is periodic or very small. A laser whose cavity has a fixed given length may emit multiple wavelengths corresponding to different longitudinal modes, resulting in what appears to be a relatively short coherence length of a few centimeters. However, in a two-path interferometer such as a Mach-Zehnder interferometer, it is easily seen that the fringe contrast (a measure of coherence) is a slowly decaying periodic function of twice the laser's cavity length.

A bit of thought reveals that the absolute coherence length of the laser (the path length difference at which fringe contrast drops to 50%) really depends on the Q of the cavity: the average number of times a photon bounces between the two mirrors of the cavity before it makes its way through one of the mirrors. The longer a typical photon stays in the cavity, the longer the laser's absolute coherence length.

A laser's coherence length can be increased at least two ways: by increasing the reflectivity of the cavity mirrors (to increase the Q), and by increasing the length of the cavity. In graduate school I was involved in a project in which a laser was constructed with a kilometer-length cavity. (It was used to monitor stretching of the Earth's surface.) We never measured its coherence length, but its Q was probably around 100, which would have given it a coherence length on the order of tens of kilometers.

But if coherence length is defined in terms of fringe contrast in an interferometer with unequal path lengths, there is another way to increase coherence length: by phase locking. Two separate lasers can be made mutually coherent by interfering their beams and continuously adjusting the cavity length of one of the lasers to maintain a nearly-zero phase difference between the two beams. The frequency of the pair of lasers may drift, but the difference between their frequencies - and their phases - will be nearly zero.

If the two beams come from the same laser, but in the interferometer one beam traverses a very long path before being combined with the other beam, it is possible to continuously adjust the laser cavity to maintain a zero phase difference, even if there is a slow or periodic drift in the laser's emission frequency. But the combined beam will always produce high contrast fringes in a downstream interferometer, as long as the path length difference in the second interferometer matches that in the first interferometer.

The combined beam from such a laser will behave as if it is emitted by a laser whose cavity length is equal to the difference D between the two paths in the first interferometer: the coherence will be a slowly decaying periodic function of D which can stretch out to many times D. The decay rate of the periodic function depends on how close to zero the phase difference between the two paths can be maintained (in practice, better than 0.00001 radians).

This means that it is possible (in principle) to make a laser whose coherence length is light-years long, in that the phase difference between the light emitted now and the light emitted years ago from the same laser is maintained at zero. In practice, the long path in the first interferometer can be down a coiled length of optical fiber, as long as the attenuation coefficient is small enough that a reliable signal can be received at the other end of the fiber. The lowest attenuation coefficient of an optical fiber is around 0.22 dB/km, which means that a fiber can be as long as 100 km and still conduct a reliable signal. So, a coherence length (per this definition) of around 1000 km is readily achievable.