Writing the creation and annihilation operators in terms of of number ($N$) and phase ($\Theta$): $$a = e^{i\Theta}\sqrt{N}; \qquad a^{\dagger} = \sqrt{N}e^{-i\Theta}$$ Obtain the communtation of $N$ and $\Theta$ and their uncertainty relations.
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Step 1: We know that the number operator $N$ is defined as: $$N = a^{\dagger}a$$ and the phase operator $\Theta$ is defined as: $$\Theta = \frac{1}{2i}(a^{\dagger}a - aa^{\dagger})$$ Show more…
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Among the stationary states of the harmonic oscillator (Equation 2.68 ) only $n=0$ hits the uncertainty limit $\left(\sigma_{x} \sigma_{p}=\hbar / 2\right) ;$ in general, $\sigma_{x} \sigma_{p}=(2 n+1) \hbar / 2,$ as you found in Problem $2.12 .$ But certain linear combinations (known as coherent states also minimize the uncertainty product. They are (as it turns out) eigenfunctions of the lowering operator: $^{42}$ $$a_{-}|\alpha\rangle=\alpha|\alpha\rangle$$ (the eigenvalue $a$ can be any complex number). (a) Calculate $\langle x\rangle,\left\langle x^{2}\right\rangle,\langle p\rangle,\left\langle p^{2}\right\rangle$ in the state $|\alpha\rangle .$ Hint: Use the technique in Example $2.5,$ and remember that $a_{+}$ is the hermitian conjugate of $a_{-} .$ Do not assume $\mathbf{a}$ is real. (b) Find $\sigma_{x}$ and $\sigma_{p}$ show that $\sigma_{x} \sigma_{p}=\hbar / 2$ (c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: $$|\alpha\rangle=\sum_{n=0}^{\infty} c_{n}|n\rangle$$ Show that the expansion coefficients are $$c_{n}=\frac{\alpha^{n}}{\sqrt{n !}} c_{0}$$ (d) Determine $c_{0}$ by normalizing $|\alpha\rangle .$ Answer: $\exp \left(-|\alpha|^{2} / 2\right)$ (e) Now put in the time dependence: $$|n\rangle \rightarrow e^{-i E_{n} t / \hbar}|n\rangle$$ and show that $|\alpha(t)\rangle$ remains an eigenstate of $a_{\rightarrow},$ but the eigenvalue evolves in time: $$\alpha(t)=e^{-i \omega t} \alpha$$ So a coherent state stays coherent, and continues to minimize the uncertainty product. (f) Based on your answers to (a), (b), and (e), find $\langle x\rangle$ and $\sigma_{x}$ as functions of time. It helps if you write the complex number a as $$\alpha=C \sqrt{\frac{m \omega}{2 \hbar}} e^{i \phi}$$ for real numbers $C$ and $\phi .$ Comment: In a sense, coherent states behave quasi-classically. (g) Is the ground state $(|n=0\rangle$ ) itself a coherent state? If so, what is the eigenvalue?
Problem 3.35 Coherent states of the harmonic oscillator. Among the stationary states of the harmonic oscillator $left(|n angle=psi_{n}(x) ight.$, Equation 2.67) only $n=0$ hits the uncertainty limit $left(sigma_{x} sigma_{p}=hbar / 2 ight) ;$ in general, $sigma_{x} sigma_{p}=(2 n+1) hbar / 2$, as you found in Problem 2.12. But certain linear combinations (known as coherent states) also minimize the uncertainty product. They are (as it turns out) eigenfunctions of the lowering operator: $$ a_{-}|alpha angle=alpha|alpha angle $$ (the eigenvalue $alpha$ can be any complex number). (a) Calculate $langle x angle,leftlangle x^{2} ight angle,langle p angle,leftlangle p^{2} ight angle$ in the state $|alpha angle .$ Hint: Use the technique in Example $2.5$, and remember that $a_{+}$ is the hermitian conjugate of $a_{-}$. Do not assume $alpha$ is real. (b) Find $sigma_{x}$ and $sigma_{p}$; show that $sigma_{x} sigma_{p}=h / 2$. (c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: $$ |alpha angle=sum_{n=0}^{infty} c_{n}|n angle $$ Show that the expansion coefficients are $$ c_{n}=frac{alpha^{n}}{sqrt{n !}} c_{0} $$ (d) Determine $c_{0}$ by normalizing $|alpha angle .$ Answer: $exp left(-|alpha|^{2} / 2 ight)$. (e) Now put in the time dependence: $$ |n angle ightarrow e^{-i E_{n} i / Lambda}|n angle . $$ and show that $|alpha(t) angle$ remains an eigenstate of $a_{-1}$, but the eigenvalue evolves in time: $$ alpha(t)=e^{-i omega u} alpha $$ So a coherent state stays coherent, and continues to minimize the uncertainty product.
David M.
The Jaynes-Cummings model is defined by the Hamiltonian Ĥ = ̉ωâ†â + ω_at/2 (1 0; 0 -1) + g/2 (0 â; ↠0), acting on the Hilbert space H = H_at ⊗ F, where F is the single mode Fock space generated by {|n>}, and H_at = C^2 is the Hilbert space of a two level atom model. 1) Show that the operator N̂ := â†â + 1/2 (1 0; 0 -1) commutes with Ĥ. What is the interpretation of the operator N̂? 2) The set {[0; 1] ⊗ |n>, [1; 0] ⊗ |n - 1>, n = 1, 2, ...}, â†â|n> = n|n>, is a basis of the Hilbert space. Show that it is an orthonormal basis. 3) Show that this implies that the dynamics of an initial state [0; 1] ⊗ |n> stays in the subspace H_n generated by [0; 1] ⊗ |n> and [1; 0] ⊗ |n - 1>. 4) Show that the restriction of the Hamiltonian to the subspace H_n can be written as Ĥ_n = ω(n - 1/2)1 + 1/2 (δ g_n; g_n -δ), where δ := ω_at - ω is the detuning and g_n := g∑n.
Sri K.
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