Finite-mode light–matter models and detection statistics — Examples

Given: Resonance ${\omega }_0=\omega >0$, real $g$, $\mathrm{\hslash }>0$, time $\tau \ge 0$, initial state $\mathrm{\mid }e,0⟩$, and the closed Jaynes–Cummings model.

The adopted single-mode, two-level rotating-wave model and its AC/CC assumptions are those of Jaynes–Cummings invariant blocks and Rabi frequency; no continuum bath or experiment is included.

Assume AC and ${\mathrm{A}\mathrm{C}}_{\omega }$ as mathematical premises carried by the model and Fock-domain suppliers. Adopt the isolated autonomous dynamics postulate P4 for the self-adjoint Hamiltonian supplied by the Jaynes–Cummings theorem; this is a model assumption, not an empirical claim (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned energy observable, unitary flow and the two pictures of autonomous evolution).

On resonance, the invariant block ${\mathcal{K}}_0=\mathrm{span}\{\mathrm{\mid }e,0⟩,\mathrm{\mid }g,1⟩\}$ has matrix $H_0=\frac{\mathrm{\hslash }\omega }{2}I+\mathrm{\hslash }g{\sigma }_x$ in that ordered orthonormal basis, where ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ (Jaynes–Cummings invariant blocks and Rabi frequency).

For the self-adjoint assigned energy $H_{\mathrm{J}\mathrm{C}}$, P4 defines the model evolution on all states by $U(\tau )=\exp (-i\tau H_{\mathrm{J}\mathrm{C}}\mathrm{/}\mathrm{\hslash })$ through its spectral calculus (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned energy observable, unitary flow and the two pictures of autonomous evolution).

For the excited-state projection $P_e=\mathrm{\mid }e⟩⟨e\mathrm{\mid }\otimes I$, the assigned probability of its outcome in a unit state $\psi$ is $\mathrm{\parallel }{P}_e\psi {\mathrm{\parallel }}^2$ (Declared ideal measurements and instruments of the framework).

Given: The number state $\mathrm{\mid }1⟩$, the ideal single-mode count PVM, the vacuum state in the unused port, a passive two-mode unitary with $0\le \eta \le 1$, and the assigned finite-mode model.

The finite-mode postulate supplies an ideal number-count PVM but makes no claim about a laboratory detector or a measured count record (Finite-mode dipole and detector assignment).

Assume AC and ${\mathrm{A}\mathrm{C}}_{\omega }$ as mathematical prerequisites of the finite-mode passive implementer and Fock suppliers; these are set-theoretic assumptions, not physical postulates (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

For the declared number distribution, $g^{(2)}(0)={\sum }_{n}n(n-1)p_n\mathrm{/}({\sum }_{n}n{p}_n{)}^2$ when the denominator is finite and positive (Ideal single-mode normally ordered second factorial statistic).

The ladder actions give $a\mathrm{\mid }1⟩=\mathrm{\mid }0⟩$, $a\mathrm{\mid }0⟩=0$, $a^{\ast }a\mathrm{\mid }1⟩=\mathrm{\mid }1⟩$, and $\mathrm{\mid }1⟩,\mathrm{\mid }0⟩$ are orthonormal (Fixed-mode number law).

A one-quantum input and vacuum at the second port have joint output state $t\mathrm{\mid }1,0⟩+r\mathrm{\mid }0,1⟩$ and ideal count probabilities $\mathrm{\mid }t{\mathrm{\mid }}^2=\eta$, $\mathrm{\mid }r{\mathrm{\mid }}^2=1-\eta$ (Passive beam-splitter photon statistics).

Ideal number-count effects are PVM projections with probabilities given by the Born rule; they do not imply a physical device or recorded data (Declared ideal measurements and instruments of the framework).