Creating Schrödinger cat states with trapped ions

[Marioweee]

Homework Statement

We will consider that the ion is in an harmonic trap. The ion has two internal states |g⟩ and |s⟩ and it interacts with a laser that induces a state-dependent force. The quantum dynamics is governed by the Hamiltonian

$$H = H_R + H_f$$
$$H_R = Ω(|s><g| + |g><s|)$$
$$H_f = g(|s><s| − |g><g|)(a^\dagger + a)$$

HR induces Rabi oscillations between internal states. Hf describes an optical force that depends on the state of the
ion. Note that the operator $(a^\dagger + a)$ is proportional to the position of the ion. The force’s sign depends on the internal state of the ion, such that

$$|s⟩ → Hf = g(a^\dagger + a)$$
$$|g⟩ → Hf = −g(a^\dagger + a)$$

1. Imagine the ion is initially in the ground state
|ψ1⟩ = |g⟩|0⟩, where |g⟩ is the electronic ground state and |0⟩ is the ground state of the vibrational degrees of freedom (zero phonons). We consider that there is no applied force and the Rabi term HR acts for a time $t=\frac{\pi}{4\Omega}$ . After that, the state of the ion is
|ψ2⟩ = (a|g⟩ + b|s⟩)|0⟩ Calculate a and b.

2. Now we turn off the Rabi term (Ω = 0) and let the state-dependent force act for a time τ .Show that the state of the ion is
|ψ2⟩ = a|g⟩|α⟩ + b|s⟩| − α⟩
and calculate the value of α.

Relevant Equations

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I have already solved question number 1 by applying the schrödinger equation obtaining that

$$\ket{\psi_2}(t) = \cos(\Omega t)\ket{g} - i \sin (\Omega t)\ket{s}$$

and therefore in $t=\frac{\pi}{4\Omega}$

$$\ket{\psi_2}(t) = \dfrac{1}{\sqrt{2}}(\ket{g} - i \ket{s})$$

I have some doubts about the second exercise. What I have tried is the following

$$|ψ(t)⟩ = e^{-iH_f t}\dfrac{1}{\sqrt(2)}(|g⟩ -i |s⟩)\ket{0} = \dfrac{1}{\sqrt(2)}(e^{it}|g⟩ -i e^{-it} |s⟩)\otimes e^{-igt(a^\dagger + a)}\ket{0}$$

I think that

$$e^{-ig(a^\dagger + a)}\ket{0}$$

should be something proportional to $\ket{\alpha} = \ket{-igt}$ (because of the displacement operator). But the displacement operator proportional to $a^\dagger - a$ so im a bit confused. Moreover, the $e^{it}$ and $e^{-it}$ terms make me think something is wrong.

 

[DrClaude]

I think that

$$e^{-ig(a^\dagger + a)}\ket{0}$$

should be something proportional to $\ket{\alpha} = \ket{-igt}$ (because of the displacement operator).

It is a harmonic oscillator, so $a$ and $a^\dagger$ are ladder operators.