Coherent State in 2 dimensions

Question

I am looking at a 2D harmonic oscillator $$H=\frac{1}{2m}(p_x^2+p_y^2)+\frac 12m(\omega_x^2x^2+\omega_y^2y^2)$$ Where $\omega_x=5\omega_y$. I am told that the oscillator is prepared in a coherent state with the following qualities: $$\langle x(0)\rangle=x_0$$ $$\langle p_x(0)\rangle=0$$ $$\langle y(0)\rangle=0$$ $$\langle p_y(0)\rangle=p_0$$ I am looking for the time dependent state of the system. My approach:

Seeing as this is a coherent state, the expectation values act as classical variables, this means that the position and momentum will be given as $$q(t)=q(0)\cos(\omega t)+\frac{1}{m\omega}p(0)sin(\omega t)$$ $$p(t)=p(0)\cos(\omega t)-m\omega x(0)\sin(\omega t)$$ Additionally, the coherent state is given as: $$\alpha (t)=\frac{1}{\sqrt{2m\omega\hbar}}(ip(t)+m\omega x(t))$$ So if I want to find the state as a function of time, do I find the alpha for both x and y and add them together?

Answer 1

I would rewrite the Hamiltonian using two independent creation and annihilation operators $$ \begin{align} \sqrt{\frac{\hbar}{2 m\omega_x}}(\hat{a}^{\dagger} + \hat{a}) & = x, \\ i\sqrt{\frac{\hbar m\omega_x}{2}}(\hat{a}^{\dagger} - \hat{a}) & = p_x, \\ \sqrt{\frac{\hbar}{2 m\omega_y}}(\hat{b}^{\dagger} + \hat{b}) & = y, \\ i\sqrt{\frac{\hbar m\omega_y}{2}}(\hat{b}^{\dagger} - \hat{b}) & = p_y. \end{align} $$ The Hamiltonian has the following form $$\hat{\mathcal{H}} = \hbar \omega_x (\hat{a}^{\dagger}\hat{a} + 1/2) + \hbar \omega_y(\hat{b}^{\dagger}\hat{b} + 1/2).$$ Coherent state $|\alpha, \beta\rangle$ is defined as an eigenstate of annihilation operators $$\hat{a}|\alpha, \beta\rangle = \alpha|\alpha,\beta\rangle \ \ \ \hat{b}|\alpha, \beta\rangle = \beta|\alpha,\beta\rangle.$$ In fact $|\alpha,\beta\rangle$ is a product of two independent coherent states $|\alpha,\beta\rangle = |\alpha\rangle \otimes |\beta\rangle$. When you initially start evolution with $|\alpha,\beta\rangle$ you will end up with $|\alpha e^{-i\omega_x t}, \beta e^{-i\omega_y t}\rangle$ at time $t$.

Coherent state

A collection of operators $\{\hat{a},\hat{b},\hat{a}^{\dagger},\hat{b}^{\dagger},1\}$ span the Heisenberg algebra. We follow the definition of Perelomov. General elements of the Heisenberg group can be written like this

$$G(\gamma,\alpha,\beta) = \exp\left(\gamma 1 + \alpha\hat{a}^{\dagger} - \alpha^{*}\hat{a} + \beta\hat{b}^{\dagger} - \beta^{*}\hat{b} \right) = \exp\left(\alpha\hat{a}^{\dagger} - \alpha^{*}\hat{a}\right) \exp\left(\beta\hat{b}^{\dagger} - \beta^{*}\hat{b} \right)\exp(1\gamma).$$ We choose a specific state in the Hilbert space, in our case it will be vacuum $|0\rangle$, and act with $G(\gamma,\alpha,\beta)$ on it $$|\gamma,\alpha,\beta\rangle = G(\gamma,\alpha,\beta)|0\rangle.$$ Because $\exp(1\gamma)$ imprints just a global phase factor it is not a part of the definition of the coherent state. A coherent state is defined like this $$|\alpha,\beta\rangle = \exp\left(\alpha\hat{a}^{\dagger} - \alpha^{*}\hat{a}\right) \exp\left(\beta\hat{b}^{\dagger} - \beta^{*}\hat{b} \right)|0\rangle$$

You can define similar coherent states for n-dimensional harmonic oscillator.