Can I burn a piece of wood by emitting only one photon per second on it? Can I burn a piece of wood by emitting single photons on it? (for example by emitting only one photon per second or per milisecond etc to the wood). How much should be the rate of emitting single photons? How long does it take for wood to catch fire in this way? Are the energy of photons important in this process?
 A: The short answer is no.  The energy of a photon is $$E=hf$$ where $h$ is Plancks constant and $f$ is frequency.
For visible light $E= 10^{-34}10^{15}$ about $10^{-19}J$.
This doesn't seem high enough to set fire to wood, as the heat energy generated would easily be conducted away before it could build up to a value high enough to cause a fire.
A: Define "burn". If you mean "set on fire", the answer in general is an obvious "no". Remember that even in space, far away from any stars, there is a constant barrage of ubiquitous background radiation. In order to provide an environment where only "single photons" (your "one per second") hit the wood, you need to shield it from the background radiation, radio, cosmic rays etc. with a thick-walled container that itself is cooled down close to 0K in order to avoid black body radiation. I can assure you that wood in near absolute darkness at ~0K would not spontaneously ignite.
If you instead mean "oxidize" the answer is probably "yes". Above a certain frequency threshold photons will create oxygen radicals which are very reactive and will oxidize the carbon and hydrogen molecules making up most of the wood.  A gram of wood contains something like $10^{22}$ or so atoms; if you oxidize one of them every second you will need about $10^{14}$ years to oxidize the wood. If you speed up the process by a factor thousand you only need $10^{11}$ years, which may or may not constitute progress, depending on the observer. If that is any consolation: Versus the end of the experiment you'll not need the lead chamber any longer because you'll be alone with a very dark sky.
A: No. To set fire to a piece of wood you must deliver heat energy to it at a faster rate than it will dissipate. With visible light, the energy of a single photon arriving per second is negligible.
There is an interesting article on Wikipedia https://en.wikipedia.org/wiki/Ultra-high-energy_gamma_ray which illustrates the energy of extremely high frequency gamma ray photons. In theory it is possible for an individual gamma ray to have more energy than a burning match, say, but there are two practical difficulties that would prevent you from igniting wood that way- firstly there is no way to create such high energy photons at present, and secondly they would tend to pass straight through your stick without interacting with it.
A: If you could make sure your system, which includes some oxygen, is perfectly heat insulated from anything else (which includes your emitter or you are somehow able to emit the photon while maintaining the heat insulation), then over a loooooong time your system would heat up enough.
A: Yes. As others have correctly explained, you need to deliver enough energy per photon to ignite the wood. @charles-tucker-3 calculated that an ultra-high energy gamma ray with about 500 Joule would do the trick, which is incredibly high for a single photon energy.
Others argued that the photon will not interact with the wood, this is wrong. While wood is fairly transparent to X-rays, every material is opaque for ultra-high energy gamma rays. Even "empty" space itself is opaque to such gamma rays which react with photons from the cosmic microwave background.
Even if the probability of interacting was small, the probability is not zero. The question was whether it is possible to ignite wood with a flux of 1 photon/second at all. Taken literally, it means we have infinitely many trials (there is no time-limit on the flux). We can just wait until the interaction happens.
Others argued that the photon will do inverse Compton scattering and that the electron will escape the wood without delivering energy. That is unlikely. For such high energy photons, the most likely interaction is e+e- pair production. The electron-positron pair has a chance to escape the wood, but it also has a chance to interact again inside the material. The energy loss per unit length of such high-energetic particles is extremely high, so for sure energy will be dumped into the wood. Even if it is just a small fraction of the energy of the original photon, we can simply increase the energy of the photon a bit more in our gedankenexperiment to compensate for the losses.
In some cases a local bubble of high-energy plasma would be formed by the interaction. The plasma gets absorbed by the surrounding matter which is heated in turn. If this bubble is close enough to the surface so that the heated material has a chance to combust in contact with air, then yes you could ignite the piece of wood with a photon flux of 1 Hz.
A: Let's try to calculate that:
The heat capacity of wood is (Wikipedia): $C=1700 \frac{J}{kg K}$ and the temperature at which wood starts to burn is approx $300^\circ C$. Assuming that the wood has approx $0^\circ C$ (it's winter...) we need to heat it by $\Delta T = 300 K$.
An assumption is needed for the mass of the wood we want to burn, let's assume a very small piece of wood of only $m=0.001$ kg.
The energy $Q$ needed to heat this piece of wood is: $Q = C \cdot m \cdot \Delta T = 510$ J $= 3.2 \cdot 10^{21}$ eV. That is massive, a Photon in the visible spectral range has only about 3 eV.
Anyways, the corresponding wavelength of a photon of that energy is $\lambda = \frac{h \cdot c}{Q} = 3.88 \cdot 10^{-28}$ m $= 3.88 \cdot 10^{-19}$ nm.
Thats really ultra short, much much shorter than X-rays or gamma rays, but still larger than the Planck length, so could exist in theory.
However, that's the mathematical answer. In the real world the energy of the photon would not 1:1 go into heat because in the first place the photon would produce a free electron of large energy that would in a cascade produce other electrons and ions and most of them would just escape from the piece of wood taking their energy with them.
A: Sure, emit them while traveling toward the piece of wood at high velocity and they'll bunch up enough to make fire in any size of timber
