# Two-mode squeezing and EPR

Consider the two-mode squeezing operator $$S(\xi)=\exp\left(\xi\hat{a}^{\dagger}\hat{b}^{\dagger}-\xi^*\hat{a}\hat{b} \right)$$ with $$\xi=r\exp(\text{i}\phi)$$, and assume that the initial state of the system is the vacuum in A and B given by $$\left|\psi(0) \right>=\left|0,0 \right>$$. This operation entangles A and B, such that $$\left|\xi\right>=\frac{1}{\cosh r}\sum_{n=0}^{\infty}\exp(\text{i}n\phi)(\tanh r)^n \left|n,n \right>$$, so I was wondering how this entangling operation can lead to an state that shares some properties between A and B analogous to the EPR entangled pair like state. Given that the squeezing grows exponential with time, the most entangled state is obtained at large times? Is this state analogous to an EPR pair between A and B ensembles?

In the state that you posted, you decided to expand $$|\xi\rangle$$ in the photon-number basis $$\{|n\rangle \}_n$$. However, as there is no preferred basis per se when describing a quantum state, you can expand it in the position basis: in this basis the EPR-like correlations are revealed. Remember that, in its original formulation, the EPR state was showing continuous variables correlations between the position and momentum variables of the two systems. The wavefunction corresponding to an EPR state is of the type $$\psi(q_1,q_2)\sim\delta(q_1-q_2)$$ in the position basis.
You can rewrite the EPR state in the position basis as $$$$\psi(q_1,q_2) = e^{-\frac{(q_1+q_2)^2}{4s^2}}e^{-\frac{s^2(q_1-q_2)^2}{4}}$$$$ where $$s$$ is related to your squeezing parameter as $$s=e^r$$. Here I'm using the convention in which the variance of the $$q$$ and $$p$$ quadratures of the vacuum state is $$1/2$$.
This wavefunction a 2D gaussian function where the $$(q_1+q_2)/\sqrt{2}$$ variables have a $$s^2/2$$ variance, while the $$(q_1-q_2)/\sqrt{2}$$ have a $$1/(2s^2)$$ variance. Note that if $$r=0$$ then $$s=1$$ and you recover the vacuum state quadrature variance, as you would expect in the case of no squeezing. However, if $$r>0$$, then $$s>1$$ and the $$(q_1+q_2)/\sqrt{2}$$ observable exhibit a variance ABOVE the variance of the vacuum state (that is 1/2). Conversely, the $$(q_1-q_2)/\sqrt{2}$$ exhibit a variance BELOW the one of the vacuum state. If you go to the limit of infinite squeezing, pushing $$r\to\infty$$, you recover the ideal EPR state, for which the variance of $$(q_1-q_2)/\sqrt{2}$$ goes to zero and the gaussian corresponding to the $$(q_1-q_2)/\sqrt{2}$$ mode becomes a Dirac delta. You can do an equivalent reasoning for the momentum as well, only in this case the variable $$p_1+p_2$$ will be the one with reduced noise.