The circularly driven two-level model: exact rotating-frame reduction, Rabi populations, resonance and inversion times

Adopt the framework data and the declared circular two-level drive model of clause (f) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models: $\mathfrak{H}={\mathbb{C}}^2$ with the orthonormal basis $(e_{\uparrow },e_{\downarrow })$ and the Pauli matrices $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ of Finite spin matrices clause (e), real parameters ${\omega }_0,{\mathrm{\Omega }}_1,\omega$ and $H_{\mathrm{c}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1(\cos (\omega t){\sigma }_x+\sin (\omega t){\sigma }_y\left)\right),t\in I=[t_0,t_1],$ read as the model's energy family in the bounded norm-continuous class, with $U_{\mathrm{c}}(t,s)$ its unique unitary propagator (The bounded norm-continuous driven model: unique unitary propagator, Dyson truncation budget and outcome-law error clause (a)). Put $\mathrm{\Delta }:={\omega }_0-\omega$, $b:=\sqrt{{\mathrm{\Delta }}^2+{\mathrm{\Omega }}_1^2}$, let $R(t):=\cos (\omega t\mathrm{/}2)I+i\sin (\omega t\mathrm{/}2){\sigma }_z$ be the rotating frame of The two-level rotating frame: conjugation identities, the transformed families and invariance of the level populations and let $P_{\uparrow },P_{\downarrow }$ be the projections onto $\mathbb{C}{e}_{\uparrow },\mathbb{C}{e}_{\downarrow }$, with populations $p_{\bullet }(t):=⟨P_{\bullet }\psi (t),\psi (t)⟩$ of the evolved vector. 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 (differential assertions apply when $t_0<t_1$, with one-sided endpoint derivatives; on a singleton the propagator identity and population formulas remain valid):

(a) Exact rotating-frame reduction. For every solution $\psi$ of $i\mathrm{\hslash }{\psi }^{\mathrm{\prime }}(t)=H_{\mathrm{c}}(t)\psi (t)$ the rotating-frame vector ${\psi }_R(t):=R(t)\psi (t)$ satisfies exactly $i\mathrm{\hslash }{\psi }_R^{\mathrm{\prime }}(t)=\frac{\mathrm{\hslash }}{2}(\mathrm{\Delta }{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x){\psi }_R(t),$ with the constant matrix $\frac{\mathrm{\hslash }}{2}(\mathrm{\Delta }{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x)$; equivalently $U_{\mathrm{c}}(t,s)=R(t{)}^{\ast }{e}^{-i(t-s)(\mathrm{\Delta }{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x)\mathrm{/}2}R(s)(s,t\in I)\mathrm{.}$

(b) Rabi populations. Let $\psi (s)=e_{\uparrow }$ at time $s$. Define $p_{\bullet }^R(\tau ):=⟨P_{\bullet }{\psi }_R(s+\tau ),{\psi }_R(s+\tau )⟩$ for $\tau \in I-s:=\{t-s:t\in I\}$. Then for $\tau :=t-s$ $p_{\downarrow }(t)=p_{\downarrow }^R(\tau )=\mid ⟨e_{\downarrow },e^{-i\tau (\mathrm{\Delta }{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x)\mathrm{/}2}{e}_{\uparrow }⟩{\mid }^2=\frac{{\mathrm{\Omega }}_1^2}{b^2}{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{b\tau }{2}\right)(b>0),$ and $p_{\uparrow }(t)=1-p_{\downarrow }(t)$; the populations of the rotating-frame vector ${\psi }_R(t)$ are the same numbers.

(c) Resonance and inversion times. At resonance $\mathrm{\Delta }=0$ one has $p_{\downarrow }(t)={\sin }^2({\mathrm{\Omega }}_1\tau \mathrm{/}2)$; when ${\mathrm{\Omega }}_1\mathrm{\ne }0$, the drive inverts the population fully at the pi-pulse time $\tau =\pi \mathrm{/}\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }$ and returns it to the initial level at $\tau =2\pi \mathrm{/}\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }$, whenever the respective evaluation time $s+\tau$ belongs to $I$; for $b>0$ the population obeys $p_{\downarrow }^R(\tau +2\pi \mathrm{/}b)=p_{\downarrow }^R(\tau )$ whenever both elapsed times lie in $I-s$. For ${\mathrm{\Omega }}_1=0$ the populations are constant, $p_{\downarrow }\equiv 0$ and $p_{\uparrow }\equiv 1$.

(d) Scope. The statements are exact for the declared finite model and its declared parameters; the rotating frame is a change of description of the same model and does not change the populations. The theorem asserts no damping or relaxation, no decoherence, no measurement record, no identification of ${\omega }_0,{\mathrm{\Omega }}_1$ with laboratory fields or of the two levels with a species, and no empirical value.

• Why this is exact. The circular drive is special: the drive vector rotates in the same sense as the frame, so the transformed family is exactly time-independent and no rotating-wave approximation is needed. The linearly polarized drive of the companion theorem contains a counter-rotating term; the two theorems therefore have genuinely different hypotheses.
• Populations are not records. The numbers $p_{\uparrow },p_{\downarrow }$ are model outcome probabilities of the two-level projections in the evolved preparation, in the reading of P2; no apparatus, pulse protocol or record is asserted.