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I'm doing a vacation scholarship at my university where I am helping a masters student with some of his research.

We have a 3x3 lattice of coupled oscillators which we have determined the Hamiltonian of and applied the squeeze operator.

We constructed a 18x18 conical Hamiltonian matrix, which is just the matrix form of the quadratic form Hamiltonian. The matrix also consists of mixed pq terms within the matrix, not just pure p,q's and pq's. This results in eigenvalues which contain an entanglement of p's and q's.

For example an eigenvector of pure q's would be

{1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0}

We are getting 3 vectors similar to this but the rest are mixed, like this for example

{1,1,1,0,0,1,1,0,1,0,0,1,1,0,1,1,0,1}

(These arn't the actual vectors just examples)

We used a Bogolubov Transformation to determine the eigenvectors.

I just wanted to ask if anyone knows what these entangled eigenvectors mean? I've been trying to picture it but its kind of difficult.

We were also told that there should be no Free modes present. From my understanding a free mode is when the system is in motion, so for example if there were a loop, with a spring around it and two masses attached to the spring, the first mode is a free mode, when the spring and the masses simply spin around the loop, and the second mode is a normal mode when both masses meet together at one side of the loop then repel eachother to meet at the other side.

Is that a correct way of describing a free mode?

What would the free modes look like as entangled eigenvectors?

Thanks