Jaynes–Cummings invariant blocks and Rabi frequency

Take the one-mode, two-level specialization of the finite-mode assignment and use its SI convention with $\mathrm{\hslash }>0$ in joule-seconds. Let $\omega >0$ be the selected angular frequency in seconds${}^{-1}$, let ${\mathcal{H}}_A={\mathbb{C}}^2$ have orthonormal states $\mathrm{\mid }g⟩,\mathrm{\mid }e⟩$, and choose ${\omega }_0>0$ in seconds${}^{-1}$, $d\in \mathbb{R}$ in coulomb-metres, and $\mathcal{E}\in \mathbb{R}$ in volts per metre. Set ${\sigma }_{+}=\mathrm{\mid }e⟩⟨g\mathrm{\mid }$, ${\sigma }_{-}=\mathrm{\mid }g⟩⟨e\mathrm{\mid }$, ${\sigma }_z=\mathrm{\mid }e⟩⟨e\mathrm{\mid }-\mathrm{\mid }g⟩⟨g\mathrm{\mid }$, and ${\sigma }_x={\sigma }_{+}+{\sigma }_{-}$. Choose $H_A=(\mathrm{\hslash }{\omega }_0\mathrm{/}2){\sigma }_z$ and $D=d{\sigma }_x$; then the dipole expression $-\mathcal{E}D\otimes (a+a^{\ast })$ has energy units. Define the real coupling $g:=-\mathcal{E}d\mathrm{/}\mathrm{\hslash }$ in seconds${}^{-1}$. Retaining from the dipole expression just the excitation-exchange terms ${\sigma }_{+}\otimes a$ and ${\sigma }_{-}\otimes a^{\ast }$ gives coefficient $-\mathcal{E}d=\mathrm{\hslash }g$. Thus the rotating-wave Hamiltonian below is a specified specialization of the model assignment, not an independently measured interaction.

Let ${\mathcal{F}}_s(\mathbb{C})$ be the infinite-dimensional one-mode bosonic Fock space with occupation basis $\mathrm{\mid }n⟩$, $n\in {\mathbb{N}}_0$, and operators $a\mathrm{\mid }n⟩=\sqrt{n}\mathrm{\mid }n-1⟩$, $a^{\ast }\mathrm{\mid }n⟩=\sqrt{n+1}\mathrm{\mid }n+1⟩$ on the finite-particle core. Write $I_A$ and $I_F$ for the identity operators on the atomic and field factors, respectively. Define $\mathrm{\Delta }={\omega }_0-\omega$. For this rotating-wave model, whose usual scale $\mathrm{\mid }g\mathrm{\mid }\sqrt{n+1}\ll \omega ,{\omega }_0$ applies only to the photon-number sectors under study and is not uniform over the full Fock space, set $H_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=\mathrm{\hslash }\omega I_A\otimes a^{\ast }a+\frac{\mathrm{\hslash }{\omega }_0}{2}{\sigma }_z\otimes I_F+\mathrm{\hslash }g({\sigma }_{+}\otimes a+{\sigma }_{-}\otimes a^{\ast })$ on the finite-particle core of ${\mathbb{C}}^2\widehat{\otimes }{\mathcal{F}}_s(\mathbb{C})$. The Hamiltonian is the exact operator for the selected rotating-wave model; no error bound relative to the unapproximated dipole interaction is asserted.

The total excitation operator $N_{\mathrm{e}\mathrm{x}\mathrm{c}}:=I_A\otimes a^{\ast }a+{\sigma }_{+}{\sigma }_{-}\otimes I_F$ is conserved on that core. The ground line $\mathbb{C}\mathrm{\mid }g,0⟩$ is invariant. For each $n\in {\mathbb{N}}_0$, the subspace ${\mathcal{K}}_n=\mathrm{span}\{\mathrm{\mid }e,n⟩,\mathrm{\mid }g,n+1⟩\}$ is invariant and the restriction has matrix $H_n=\left(\begin{array}{cc}\mathrm{\hslash }\omega n+\frac{\mathrm{\hslash }{\omega }_0}{2}& \mathrm{\hslash }g\sqrt{n+1}\\ \mathrm{\hslash }g\sqrt{n+1}& \mathrm{\hslash }\omega (n+1)-\frac{\mathrm{\hslash }{\omega }_0}{2}\end{array}\right)\mathrm{.}$ Its eigenvalue splitting is $\mathrm{\hslash }\sqrt{{\mathrm{\Delta }}^2+4g^2(n+1)}$. Define the full Hamiltonian to act on the ground line by $-\mathrm{\hslash }{\omega }_0\mathrm{/}2$ and as the orthogonal direct sum of the $H_n$ on the ${\mathcal{K}}_n$, with squared-block-norm domain $D(H_{\mathrm{J}\mathrm{C}})=\{c\mathrm{\mid }g,0⟩\oplus {⨁}_{n\ge 0}{\psi }_n:c\in \mathbb{C},{\psi }_n\in {\mathcal{K}}_n,{\sum }_{n\ge 0}\mathrm{\parallel }{H}_n{\psi }_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ This direct-sum operator is self-adjoint and extends $H_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$.

Finite-mode dipole and detector assignment

Dimensions, SI unit conventions and quantity types of the framework

Closed adjoint sector domains

Fixed-mode number law

Hilbert tensor construction: inner product, completion and product bases

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The indexed blocks form an orthogonal decomposition by total excitation, not a finite-dimensional approximation of the full Fock space. The RWA inequality above is a model-validity scale for specified sectors, not a uniform bound on omitted counter-rotating terms.

diagram failed to render:
\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center)]
\node[draw,minimum width=2.3cm,minimum height=1.1cm] (g) at (0,0) {$\mathbb C|g,0\rangle$};
\node at (1.8,0) {$\oplus$};
\node[draw,minimum width=3.2cm,minimum height=1.1cm] (k0) at (4.1,0) {$\mathcal K_0=\operatorname{span}\{|e,0\rangle,|g,1\rangle\}$};
\node at (6.2,0) {$\oplus$};
\node[draw,minimum width=3.2cm,minimum height=1.1cm] (k1) at (8.5,0) {$\mathcal K_1=\operatorname{span}\{|e,1\rangle,|g,2\rangle\}$};
\node at (10.5,0) {$\oplus\cdots$};
\end{tikzpicture}

The boxes label mutually orthogonal excitation sectors in $\mathbb{C}\mathrm{\mid }g,0⟩\oplus {⨁}_{n\ge 0}{\mathcal{K}}_n$. Each $\oplus$ denotes the Hilbert direct sum; no transition between blocks is represented. The proof of invariance and self-adjointness is the operator argument above and does not rely on the diagram.