In the context of condensed matter physics, what does it mean for time to have two dimensions?

In an online article that describes condensed matter physics for laypersons, the author describes various so-called "designer materials" that have exotic properties, including one in which the path that light travels along is constrained to a two-dimensional plane, and time measurements operate in two dimensions:

But there is also much more to the field than quasiparticles. Physicists can now create materials in which the speed of light is much slower than usual, say 40 miles an hour. They can even create materials in which light moves as if there were two space dimensions and two time dimensions, instead of the usual three dimensions of space and one of time! Normally we think that time can go forward in just one direction, but in these substances light has a choice between many different directions it can go “forward in time.” On the other hand, its motion in space is confined to a plane.

My very basic understanding of time is related to measurements of causality or increase in disorder. Movement of light in that context is measured as distance-over-time, where distance is the length of a path between two points in space ("start" and "end"), and time is some unit (e.g., "seconds").

What does it mean for a light particle or wave to move within such a material along two dimensions of time, as described above?

• The author, John Baez, also posted a version of this article on his blog and clarified in a comment that he's referring to hyperbolic metamaterials. Maybe that tidbit will be useful for someone answering the question. May 3 at 20:21
• I don't know what experiment this is referring to (and google isn't helping). Because light is always in motion at a constant speed (in a material with fixed index), there is often a mapping between position along the direction of travel and time. For example, light traveling along a waveguide can be seen as a 2+1 dimensional system, with position along the waveguide direction equivalent to time. I am guessing that Baez is referring to work using a similar mapping, except in two dimensions, but I don't know the specific reference (oh, note that I wrote the above before seeing Anyon's comment) May 3 at 21:01
• I don't know what exactly he's referring to either, but you may be interested in the analysis of Greg Egan's novel Dichronauts. It gets into how you could have a ++-- metric and still have "linear time". gregegan.net/DICHRONAUTS/00/DPDM.html May 4 at 1:35
• Greg Egan's universe doesn't quite match ours, but it seems like an interesting premise for a book! May 7 at 0:34