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I'm extremely confused by what physicists mean by the spatial mode of light. I am also equally if not more confused by what the spatial mode of a massive particle is. Can anyone help me out by providing an explanation? (I'm also a visual learner, so diagrams will help)

Spatial mode of light makes no sense to me because I don't understand what its referring to!

I'm thinking that the spatial mode of a massive particle is like the eigenstate of an atom. At every energy level, there is a corresponding eigen state and this is the atom's spatial distribution. I'm guessing that taking a particle in general also has eigen states, and to distinguish them from atoms, they call them spatial modes?

I'm reading this paper:

Heaney, Libby. Teleportation of a quantum state of a spatial mode with a single massive particle. Proceedings of the 5th Conference on Theory of Quantum Computation, Communication and Cryptography. Lect. Notes Comput. Sc. 6519 pp. 175-186 (2011). doi:10.1007/978-3-642-18073-6_15, arXiv:1011.3743 [quant-ph].

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A spatial mode is just what it sounds like, a state that is localised in space. For a massive particle this is easy to visualise: it is a localised wavepacket. There is no need for such a wavepacket to be an eigenstate of the Hamiltonian. For example, in section 2 of the paper you linked, Heaney considers a confining potential split into two regions $A$ and $B$. The ground state wavefunction of the Hamiltonian can be considered as a coherent sum of two distinct spatial modes, one localised in $A$ and one localised in $B$.

For states of light I can understand your confusion, since it is obviously not possible to confine light to one point in space. However, in quantum optics it is common to confine a quantum state of light to propagate along a waveguide, for example. States confined to spatially separated waveguides are in separate spatial modes. For a simple example, consider a Mach-Zender interferometer; see diagram here. The beamsplitter splits the incoming light into two spatial modes, labelled SB and RB in the picture I linked.

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