Coherent State, Unitary Operators, Harmonic Oscillator

Consider the operator:

$$O = e^{\theta(a^\dagger b - b^\dagger a)}$$

where $\theta$ is a constant.

$O$ is a unitary operator.

$a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two harmonic oscillators.

A normalized coherent state is defined as:

$$\lvert\alpha\rangle = e^{-\lvert\alpha\rvert^2/2} e^{\alpha a^\dagger} \lvert 0\rangle$$

where $\lvert0\rangle$ is the ground state of the harmonic oscillator.

I'm trying to see how $O$ acts on the coherent states by calculating $O \lvert\psi\rangle = O\lvert\alpha\rangle\lvert\beta\rangle$ in terms of coherent states.

Also, how does $O$ act on $\alpha$ and $\beta$?

I'm trying to use

$$O a O^\dagger = a \cos(\theta) + b \sin(\theta)$$

and

$$O b O^\dagger = -a \sin(\theta) + b \cos(\theta).$$

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There are many ways to go around this. You can start from the coherent states and apply the unitary $\hat{O}$ directly on them. That will not be that simple because you will get a term $\hat{O}e^{\alpha\hat{a}^\dagger}e^{\beta\hat{b}^\dagger}$. Now, the typical approach would be to exchange the order of the operators to get something like $e^{\alpha\hat{a}^\dagger}e^{\beta\hat{b}^\dagger}\hat{O}$ (up to some extra term due to the commutator). This not really a simple task but once you are done, you can Taylor expand the operator $\hat{O}$ and keep only the zeroth order (all other terms contain annihilation operators acting on vacuum). I am not going to dig into the calculation in more detail, there are many ways to do it and none of them is really pleasant.

But there is a better way. You can define a displacement operator by the action $\hat{D}(\alpha)|0\rangle = |\alpha\rangle$ and then you have $\hat{D}(\alpha)\hat{a}\hat{D}^\dagger(\alpha) = \hat{a}+\alpha$. You can combine this with the formulas for $\hat{O}\hat{a}\hat{O}^\dagger$, $\hat{O}\hat{b}\hat{O}^\dagger$ to see how the annihilation operators are transformed. What you should get is a beam-splitting of the two coherent states, i.e., $$|\alpha,\beta\rangle\to|t\alpha+r\beta,t\beta-r\alpha\rangle$$, where $t = \cos\theta$, $r = \sin\theta$.

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Could you explain how to combine $\hat{D}(\alpha)\hat{a}\hat{D}^\dagger(\alpha)$ with $\hat{O}\hat{a}\hat{O}^\dagger$ and $\hat{O}\hat{b}\hat{O}^\dagger$ to get the result? I don't see how to do this. –  Randy May 7 '13 at 11:15
Hi, I would still like some clarification on the above step if possible. –  Randy May 8 '13 at 4:24
I'll get to it in a while, having a busy week. Please be patient. –  Ondřej Černotík May 8 '13 at 11:28
Okay, Thank you so much! –  Randy May 8 '13 at 12:02

Let us change OP's notation $a\to a_1$ and $b \to a_2$. We write collectively the two annihilation operators as a column two-vector

$$\tag{1} \vec{a}~:=~\begin{bmatrix} a_1 \\ a_2 \end{bmatrix}.$$

We have the Heisenberg algebra

$$\tag{2} [a_i,a_j^{\dagger}] ~=~\delta_{ij} {\bf 1}\qquad i,j~\in~\{1,2\}.$$

and the vacuum state

$$\tag{3} a_i | 0\rangle ~=~0,$$

Define un-normalized coherent states

$$\tag{4} |\vec{\alpha} )_a ~:=~ e^{ a^{\dagger}_i \alpha_i} | 0\rangle .$$

The idea is now to diagonalize the ${\cal O}$ operator. Define unitary matrix

$$\tag{5} U ~:=~\frac{\sqrt{2}}{2} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}~=~ \exp\left[i\frac{\pi}{4}\sigma_x \right] .$$

Define new operators

$$\tag{6} b_i ~:=~ U_{ij} a_j, \qquad [b_i,b_j^{\dagger}] ~=~\delta_{ij} {\bf 1},$$

and new coherent continuous labels

$$\tag{7} \beta_i ~:=~ U_{ij} \alpha_j.$$

Define un-normalized coherent states

$$\tag{8} |\vec{\beta} )_b ~:=~ e^{ b^{\dagger}_i \beta_i} | 0\rangle~=~ e^{ a^{\dagger}_i \alpha_i} | 0\rangle~=~|\vec{\alpha} )_a .$$

Note that the operator becomes diagonal

$$\tag{9} {\cal O}~:=~ \exp\left[i\theta a^{\dagger}_i (\sigma_y)_{ij} a_j\right] ~=~ \exp\left[i\theta b^{\dagger}_i (\sigma_z)_{ij} b_j\right] ~=~ \exp\left[i\theta (n_1-n_2)\right],$$

$$\tag{10} n_i~:=~b^{\dagger}_i b_i \qquad\text{(no sum over i).}$$

Next deduce the commutation relations

$$\tag{11} \exp\left[i\theta n_i\right]\exp\left[b^{\dagger}_i \beta_i \right]~=~\exp\left[b^{\dagger}_i \beta_i e^{i\theta} \right]\exp\left[i\theta n_i\right] \qquad\text{(no sum over i).}$$

We conclude from (11) that

$$\tag{12} {\cal O}|\beta_1, \beta_2 )_b ~=~|\beta_1e^{i\theta}, \beta_2e^{-i\theta} )_b.$$

We leave it as an exercise to translate (12) back to normalized $a$-coherent states.

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