# Entanglement in coupled harmonic oscillator system

I am considering the problem of two coupled harmonic oscillators. Ignoring factors of two, the Hamiltonian for this system is $$H=p_1^2+p_2^2 +k x_1^2+kx_2^2 +k(x_2-x_1)^2$$ One can do a nice coordinate transformation and gets: $$H=\hat{p}_1^2+\hat{p}_2^2 +k \hat{x}_1^2+3k\hat{x}_2^2$$

This is just two uncoupled harmonic oscillators and the solution of the Schrödinger equation with this Hamiltonian is just that of two independent harmonic oscillators. If I now look at the ground state I get of course two unentangled gaussian wavefunctions. But if I now do the reverse of the coordinate transformation I get unseperable wave functions for the ground state. But how can entanglement be basis dependent? Or does it not make sense to talk about entanglement here because these are only pure states?

EDIT: I want to add another question here. Is the ground state in the basis of the new coordinates the ground state in the original basis? Or is this basis independent. If not is there a way to transform the number operators?

• Mar 1 at 1:19

• @eeqesri absolutely. I always think of NOON states, which are optimal for interferometry. These states are a superposition of being [$N$th excited state of the first mode and ground state of the second mode] and [ground state of the first mode and $N$th excited state of the second mode]. If you are trying to sense a relative phase acquired between these two modes, then NOON states are optimal. But if the relative phase is acquired between the two modes in the transformed coordinates, the original NOON state is no longer optimal! Instead, a NOON state in the new coordinates is optimal Feb 28 at 15:21