# Can one make a synthetic dimension “curl around” into a cylinder?

A really cool recent proposal,

Synthetic Gauge Fields in Synthetic Dimensions. A. Celi et al. Phys. Rev. Lett. 112, 043001 (2014), arXiv:1307.8349,

shows how you can simulate a synthetic magnetic field in a fictional 2D lattice by taking a 1D lattice and populating it with atoms that have an internal degree of freedom, usually a hyperfine manifold of ground states. A set of Raman coupling lasers with cleverly engineered phases allows one to set up a synthetic magnetic flux in each lattice plaquette and therefore a full synthetic magnetic field in the whole lattice.

I also know of one experimental realization,

Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. M. Mancini et al. arXiv:1502.02495.

This is limited, of course, by the number of available internal states, which means that the lattice looks more like a thin strip than anything else. Is it possible to alter the strip's topology? In particular, can one couple the maximal $m=M$ and $m=-M$ hyperfine states in a way that will 'knit' the two strip edges together to make the synthetic lattice doubly connected? This would obviously have really nice implications in terms of quantum simulation of "curled up" extra dimensions. Has anything like this been proposed?

The simple answer to this is "yes, go read the paper". Indeed, the OP is somewhat negligent in their post, since the Celi et al. paper plainly states in the introduction that

We also show that by using additional Raman and radio-frequency transitions one can connect the edges in the extra dimension.

Thus, the possibility of engineering synthetic cylinders was on the cards to begin with.

However, later work does propose much fancier topologies for these sorts of schemes. More specifically, in

Quantum simulation of non-trivial topology. O. Boada et al. N. J. Phys. 17 045007 (2015) (open access)

the ICFO group proposes a number of interesting schemes, including

• a circle,

• a cylinder,

• a torus,

• a Möbius strip,

• and a twisted torus,

Of course, whether these schemes can be realized experimentally is another matter. In particular, they require very specific hyperfine couplings to be turned on, while other couplings are not. This is in principle OK but it does require a good bit of fine-tuned control. Additionally, some of these schemes require sites to be individually targeted, which can be very hard, particularly if the links are directly implemented in the microwave domain.

As an example of the difficulties, consider the implementation by Mancini et al.. They used Raman transitions for the hyperfine couplings, as they (i) are generally easier to implement than direct microwave transitions, and (ii) have a much larger spatial phase variation, which is required to implement the large Peierls phases they needed to simulated the synthetic magnetic field. However, Raman transitions consist of two dipole transitions instead of one, which forces $\Delta m=2$ between lattice sites instead of $\Delta m=1$, and therefore cuts in half the available lattice width (i.e. three sites instead of six, with a spin-5/2 system).

An additional point worth mentioning is that often you don't really even want such fancy topologies in the first place. This is the case, for example with the Mancini et al. paper, where they care about chiral edge states. These are cyclotron orbits in a magnetic field, which can only exist in a single direction near a sharp edge:

Thus, for these purposes it is vital to have a sharply defined cutoff, and it is advantageous to have a thin strip (as long as it is wide enough to support the cyclotron orbits), as this maximizes the relative contribution of the edge states.

For other interesting effects, of course, you probably do care about the topology. To see what signatures you might look for, and why you'd care about them, see the Boada et al. paper.