Can you explain what happens if two possible photon paths interfere with each other but are different in their length? When does the interference pattern appear? I suggest it happens after the time for passing through the longer path passed. Is this true? I am asking this question in the light of Wheeler delayed choice tought experiment when a photon passes on both sides of a galaxy and then interferes with itself on the semireflecting mirror at the detector on Earth.
Think about it this way. If you had a radioactive atom it might decay and emit a photon.
And you don't know when. So there is a whole range of times for when a nearby photon detector might go off.
But there is also an issue of coherence. Which is a specific technical term, not a word people use for emotional purposes.
If you have a short coherence length then effectively after a short time the object has either decayed in that time or it hasn't. If you have a longer coherence length, then if the detectors are all far enough away then for that same time interval, it might not be the case that it decayed or didn't decay.
There are things you can do to increase the coherence, and usually it's about isolating something. Though there are limits to how much something can be isolated. For instance a spontaneous decay can sometimes be thought of as a decay stimulated by the quantum vacuum.
But don't expect that you can forget to isolate something and expect that later you can start to isolate it at the last minute and hope it can not decohere. That's not a good idea.
I don't understand very well what will happen if the path difference is significant but the coherence length is greater than the path difference.
You write as if there is some special length that is intrinsically significant. There are the lengths of the arms, the wavelengths, and the coherence length. If all of them were scaled up or down, would the physics really be different?
Does it mean that the interference from both path will produce an interference pattern
The interference pattern happens spatially. Some places will get hit more often than others when you repeat the experiment. And it will take some time from the creation of the emitter until a detector a ways away goes off.
It's not like you'll see some splotch and then see it turn into bands some time later. You see a hit or not every moment. And a pattern builds up over repeated experiments.
If you specifically want to see a correlation between the time a detector goes off and where the detector is then you have to deal not only with the emitter but the detector. You can't resolve a monochromatic beam into a perfect spot at a perfect time. If you start to focus better on when it arrives, you'll affect the wavelengths the detector reacts to, and no source is perfectly monochromatic at a fixed frequency. For instance if it is an atom decaying, there is a doppler shift from the motion of the atom and the motion and the location aren't unrelated. So the wavelength and the location of the atom are related. So the phase of the photon from different motions of the atom matters.
The coherence matters. It might be good to step back and ask yourself what you want to learn, what you want to understand, and what you want to predict. The universe never promised to be simple, so if your goal was to hear a simple story, think carefully.
There are entire textbooks on quantum optics, and they are trying to make it as simple as it can be, they don't write them in a complicated to hide understanding from lay people. They just address the issues that matter for making precision experiments. And the emitter and the detector and their details and the coherence of the interaction all matter.