What Do Entangled Eigenvectors Indicate in a Quantum Harmonic Oscillator?

[Daniiel]

Hey,

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 each other 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

 

[eljose79]

Hi there,

It sounds like you are working on some interesting research with coupled oscillators. The entangled eigenvectors you are seeing are a result of the mixed pq terms in the Hamiltonian matrix. This means that the eigenvalues and eigenvectors are no longer purely p or q, but a combination of both. This can be difficult to visualize, but essentially it means that the oscillators are not just moving in a single direction, but their motion is a combination of both p and q directions.

As for the free modes, your understanding is correct. A free mode is a mode of motion where the system is not being driven by an external force, but is simply moving on its own. In the example you provided, the first mode is a free mode as the spring and masses are just spinning around the loop without any external force driving them. The second mode, on the other hand, is a normal mode where the system is being driven by the forces of the spring and masses.

As for what the free modes would look like as entangled eigenvectors, it would depend on the specific system and its Hamiltonian. But generally, they would be a combination of p and q directions, with no external force driving them.

I hope this helps clarify things for you. Keep up the great work with your research!