First-order transition amplitudes for a bounded control: Dyson remainder, the sinusoidal drive with its co- and counter-rotating parts, and the outcome-probability error

Adopt the framework data and the declared driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models in the bounded-control class of clause (b)(ii): $\mathfrak{H}$ is the model's state space, $H_0$ is self-adjoint on $D(H_0)$, $V:I\to \mathcal{B}(\mathfrak{H})$ is strongly continuous with self-adjoint values, $v={\sup }_{t\in I}\mathrm{\parallel }V(t)\mathrm{\parallel }<\mathrm{\infty }$ and $H(t)=H_0+V(t)$; let $U(t,s)$ be the mild propagator and put $K:=v\mathrm{/}\mathrm{\hslash }$. Let $({\phi }_j{)}_{j\in J}$ be an orthonormal family in $D(H_0)$ with $H_0{\phi }_j=E_j{\phi }_j$ (Orthonormal families, complete orthonormal systems and Hilbert bases) — a complete family is not required — and fix labels $i,j\in J$ and $s\le t$ in $I$, with $\tau :=t-s$ and $\mathrm{\Delta }:=E_j-E_i$. Let $A_{ji}(t,s)=e^{i{E}_j\tau \mathrm{/}\mathrm{\hslash }}⟨U(t,s){\phi }_i,{\phi }_j⟩$ be the transition amplitude of clause (e) of the model and define its first-order term $A_{ji}^{(1)}(t,s):=-\frac{i}{\mathrm{\hslash }}{\int }_s^t{e}^{i\mathrm{\Delta }(r-s)\mathrm{/}\mathrm{\hslash }}⟨V(r){\phi }_i,{\phi }_j⟩dr\mathrm{.}$ Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Then:

(a) Exact amplitude with Dyson remainder. There is a remainder $R_{ji}(t,s)$ with $A_{ji}(t,s)={\delta }_{ij}+A_{ji}^{(1)}(t,s)+R_{ji}(t,s),\mathrm{\mid }{R}_{ji}(t,s)\mathrm{\mid }\le e^{K\tau }\frac{(K\tau {)}^2}{2},$ and if $\eta :I\to [0,\mathrm{\infty })$ is measurable and integrable, $\mathrm{\parallel }V(r)-V_0(r)\mathrm{\parallel }\le \eta (r)$ for $r\in I$, and $V_0:I\to \mathcal{B}(\mathfrak{H})$ has self-adjoint values with integrable scalar coefficient $r\mapsto ⟨V_0(r){\phi }_i,{\phi }_j⟩$, then the corresponding first-order amplitude differs from $A_{ji}^{(1)}$ by at most $\frac{1}{\mathrm{\hslash }}{\int }_s^t\eta (r)dr$.

(b) Sinusoidal control. Let $W\in \mathcal{B}(\mathfrak{H})$ be self-adjoint, $\omega ,\vartheta \in \mathbb{R}$, and suppose $V(r)=W\cos (\omega r+\vartheta )$ for $r\in I$. Put ${\vartheta }_0:=\omega s+\vartheta$ and decompose $A_{ji}^{(1)}=a_{+}+a_{-},a_{\pm }:=-\frac{i}{2\mathrm{\hslash }}⟨W{\phi }_i,{\phi }_j⟩e^{\mp i{\vartheta }_0}\frac{e^{i(\mathrm{\Delta }\mp \mathrm{\hslash }\omega )\tau \mathrm{/}\mathrm{\hslash }}-1}{i(\mathrm{\Delta }\mp \mathrm{\hslash }\omega )\mathrm{/}\mathrm{\hslash }},$ with the convention that the quotient is the limit $\tau$ when $\mathrm{\Delta }\mp \mathrm{\hslash }\omega =0$. Then the co-rotating term $a_{+}$ (the one whose denominator vanishes at $\mathrm{\Delta }=\mathrm{\hslash }\omega$) satisfies $\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨W{\phi }_i,{\phi }_j⟩\mid \frac{\mid \sin ((\mathrm{\Delta }-\mathrm{\hslash }\omega )\tau \mathrm{/}(2\mathrm{\hslash }))\mid }{\mathrm{\mid }\mathrm{\Delta }-\mathrm{\hslash }\omega \mathrm{\mid }},\text{and at }\mathrm{\Delta }=\mathrm{\hslash }\omega :\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨W{\phi }_i,{\phi }_j⟩\mid \frac{\tau }{2\mathrm{\hslash }},$ and the counter-rotating term satisfies $\mathrm{\mid }{a}_{-}\mathrm{\mid }\le \mathrm{\mid }⟨W{\phi }_i,{\phi }_j⟩\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\Delta }+\mathrm{\hslash }\omega \mathrm{\mid }$ whenever $\mathrm{\Delta }+\mathrm{\hslash }\omega \mathrm{\ne }0$.

(c) Outcome probability. The model's transition probability is $p_{ji}(t,s)=\mathrm{\mid }{A}_{ji}(t,s){\mathrm{\mid }}^2$; with $\epsilon :=e^{K\tau }(K\tau {)}^2\mathrm{/}2$ and $A^{(1)}:=A_{ji}^{(1)}$, for distinct labels $i\mathrm{\ne }j$ $\mid p_{ji}(t,s)-\mathrm{\mid }{A}^{(1)}{\mathrm{\mid }}^2\mid \le 2\mathrm{\mid }{A}^{(1)}\mathrm{\mid }\epsilon +{\epsilon }^2,$ and for $i=j$ the survival probability obeys the analogous bound $\mid p_{ii}(t,s)-\mathrm{\mid }1+A^{(1)}{\mathrm{\mid }}^2\mid \le (2+2\mathrm{\mid }{A}^{(1)}\mathrm{\mid })\epsilon +{\epsilon }^2$; in both cases the exact outcome probability is controlled by the first-order term and the explicit Dyson error of clause (a).

(d) Scope. The statements are finite-time and conditional on the bounded control and on the orthonormal eigenfamily; the remainder bound grows like ${\tau }^2{e}^{K\tau }$ and no long-time limit, transition rate, exponential decay, continuum density of states or Fermi golden rule is asserted, and no measured frequency, count or rate is claimed.

• Resonance versus Rabi oscillation. Clause (b) shows that the first-order co-rotating amplitude grows linearly in $\tau$ at exact resonance. That linear growth is the small-coupling, short-time first-order behaviour of the general bounded-control model; the exact bounded oscillation of the two-level circular drive is proved separately on this page for the declared finite model, and the two statements have different hypotheses: the first applies to any bounded control with an orthonormal eigenpair, the second is exact for the declared two-level family.
• What "first order" controls. The bound of clause (a) is uniform in the labels and in the family; it is a bound on the amplitude, not a rate, and the counter-rotating bound of clause (b) shows explicitly how far the physical first-order term is from its co-rotating part.