Qm Driven Dynamics And Adiabatic Limits — Examples

Adopt P1, P2 and P4 in the readings of the declared driven model (The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (a)) and the circular two-level drive model of its clause (f) with ${\mathbb{C}}^2$, the basis $(e_{\uparrow },e_{\downarrow })$ and the Pauli matrices of Finite spin matrices clause (e). All angular frequencies below are declared model parameters read in ${\mathrm{s}}^{-1}$ (Dimensions, SI unit conventions and quantity types of the framework); no value is attributed to a laboratory system. For the numerical substitutions below, frequency symbols denote their numerical values in ${\mathrm{s}}^{-1}$ and $t,s$ denote numerical time values in seconds; thus a displayed elapsed time $t-s=\pi \mathrm{/}2$ means a physical duration $(\pi \mathrm{/}2)\mathrm{s}$, and the displayed sine arguments are dimensionless. Define ${\sigma }_{\pm }:=({\sigma }_x\pm i{\sigma }_y)\mathrm{/}2$. Three declared parameter sets are used:

1. Resonant set. ${\omega }_0=\omega =1$, ${\mathrm{\Omega }}_1=2$, so $\mathrm{\Delta }=0$ and $b=2$;
2. Detuned set. ${\omega }_0=1+\sqrt{3}$, $\omega =1$, ${\mathrm{\Omega }}_1=1$, so $\mathrm{\Delta }=\sqrt{3}$ and $b=2$;
3. Undriven set. ${\mathrm{\Omega }}_1=0$ and arbitrary ${\omega }_0,\omega$.

In each set prescribe the displayed circular energy family for all real physical times. For $s,t\in \mathbb{R}$, define $U_{\mathrm{c}}(t,s)$ by the model's unique propagator on any compact interval $J$ containing $s,t$. These restrictions agree: on a larger compact interval containing two choices, uniqueness of the norm Dyson integral equation identifies both restrictions. Thus this definition extends the compact-interval model consistently to all real evaluation times. Let $U_{\mathrm{c}}(t,s)$ be that propagator and let $p_{\downarrow }(t)=⟨P_{\downarrow }{U}_{\mathrm{c}}(t,s)e_{\uparrow },U_{\mathrm{c}}(t,s)e_{\uparrow }⟩$ be the down-population of the initially up state. Then:

(a) for the resonant set, $p_{\downarrow }(t)={\sin }^2(t-s)$, so $p_{\downarrow }=1\mathrm{/}2$ at the quarter-inversion time $t-s=\pi \mathrm{/}4$, $p_{\downarrow }=1$ at the pi-pulse time $t-s=\pi \mathrm{/}2$, and $p_{\downarrow }=0$ again at $t-s=\pi$;

(b) for the detuned set, $p_{\downarrow }(t)=\frac{1}{4}{\sin }^2(t-s)$, so $p_{\downarrow }=1\mathrm{/}4$ at $t-s=\pi \mathrm{/}2$ and the population is exactly $\pi$-periodic in $t-s$;

(c) for the undriven set, $p_{\downarrow }(t)\equiv 0$ and $p_{\uparrow }(t)\equiv 1$ for all $t$, while the transverse coherences $⟨{\sigma }_{\pm }⟩(t):=⟨{\sigma }_{\pm }{U}_{\mathrm{c}}(t,s)e_{\uparrow },U_{\mathrm{c}}(t,s)e_{\uparrow }⟩$ rotate with phase $e^{\pm i{\omega }_0(t-s)}$: the general coherence transforms by this phase at the splitting frequency ${\omega }_0$, while the specified initial up state has both transverse coherences zero;

(d) in every set the population is the outcome probability of the projection $P_{\downarrow }$ in the evolved preparation in the reading of P2; no damping, decoherence, measurement record or laboratory identification is asserted.

• Why the checks are exact. Each population in clauses (a)-(c) is the square of a sine at a rational multiple of $\pi$, so the values $1\mathrm{/}2$, $1$, $0$ and $1\mathrm{/}4$ are exact rational outputs of the model, not numerical approximations or measurements.
• Declared parameters, no measurements. The three sets are declared frequency parameters with the numerical-unit convention above; no frequency, energy or time of any laboratory species is claimed, and no source-reported measured value is used.

Adopt P1, P2 and P4 in the readings of the declared driven model (The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (a)) and the linear two-level drive model of its clause (f) with ${\mathbb{C}}^2$, the basis $(e_{\uparrow },e_{\downarrow })$ and the Pauli matrices of Finite spin matrices clause (e). Declare angular-frequency parameters $\omega >0$, $\mathrm{\Delta }={\omega }_0-\omega =0$ (resonance) and ${\mathrm{\Omega }}_1=\omega \mathrm{/}50$, so that the dimensionless frequency ratio is ${\mathrm{\Omega }}_1\mathrm{/}\mathrm{\mid }\omega \mathrm{\mid }=1\mathrm{/}50$; all angular frequencies are declared model parameters read in ${\mathrm{s}}^{-1}$ (Dimensions, SI unit conventions and quantity types of the framework), and no value is attributed to a laboratory system. Fix an initial time $s$ and use a compact interval $I$ containing $s$ and every displayed evaluation time $t=s+\tau$, where $\tau \ge 0$; for example $I=[s,s+200\mathrm{/}\omega ]$ contains them all, since $100\pi \mathrm{/}3<200$. Let $p_{\downarrow }(t)$ be the exact down-population of the initially up state under the model's propagator, $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2(\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\tau \mathrm{/}4)$ the averaged-model Rabi population of the page's error contract, and $\epsilon (\tau )=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\mathrm{\mid }\omega \mathrm{\mid }}[1+\tau (\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}2+3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\mathrm{/}4)]$. Then:

(a) the rotating-wave error contract reads $\epsilon (\tau )=\frac{1}{100}(1+\frac{3\omega \tau }{200})$, so that $\epsilon =1\mathrm{/}100$ at $\omega \tau =0$, $\epsilon =5\mathrm{/}200=1\mathrm{/}40$ at $\omega \tau =100$, and $\epsilon =1\mathrm{/}25$ at $\omega \tau =200$; the corresponding population-difference bounds $2\epsilon$ are $2\mathrm{/}100$, $1\mathrm{/}20$ and $2\mathrm{/}25$;

(b) at the declared elapsed time $\omega \tau =100\pi \mathrm{/}3$ one has ${\mathrm{\Omega }}_1\tau \mathrm{/}4=\pi \mathrm{/}6$ and the averaged-model population is exactly $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}=1\mathrm{/}4$;

(c) for the circular drive of the companion theorem with the same ${\omega }_0,\omega ,{\mathrm{\Omega }}_1$ the model's exact population obeys the same Rabi-type formula with no rotating-wave error, so the non-zero bound of (a) is specific to the linear polarization;

(d) the numbers in (a)-(b) are outputs of the declared model and its error contract; they are not measured values, and clause (a)'s bound is conservative (at $\tau =0$ the two propagators coincide while the operator-norm bound is $\epsilon (0)=1\mathrm{/}100$ and the population-difference bound is $2\epsilon (0)=1\mathrm{/}50$).

• Reading the budget. The error contract is a uniform bound, not a prediction of the actual difference; the example shows the two faces of that: the bound is exact arithmetic (rational numbers at the declared times), and it is conservative at the initial time, where the true difference vanishes.
• No physical numbers. The example deliberately uses only declared ratios and dimensionless elapsed times; no frequency, field amplitude or relaxation parameter of a laboratory species appears.

Adopt P1, P2 and P4 in the readings of the declared driven model (The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (a)) and the basis $(\mathrm{\mid }\mathrm{E}⟩,\mathrm{\mid }\mathrm{G}⟩)$ of a declared two-level inversion model: two orthogonal unit vectors with the Pauli matrices of Finite spin matrices clause (e) acting on their span, so that ${\sigma }_z=\mathrm{diag}(1,-1)$ distinguishes the two levels. Declare real parameters ${\mathrm{\Delta }}_0>0$ (half the inversion splitting, joules), $\mu >0$ (a dipole-like coupling, in $\mathrm{C}\mathrm{m}$), $E_0>0$ (a field amplitude, in $\mathrm{V}{\mathrm{m}}^{-1}$) and a drive frequency $\omega >0$ in ${\mathrm{s}}^{-1}$, and declare the model's energy family $H(t):={\mathrm{\Delta }}_0{\sigma }_z+2\mu E_0\cos (\omega t){\sigma }_x,t\in \mathbb{R},$ read as the energy of the declared model, with $\mathrm{\hslash }$ the reduced Planck constant of Dimensions, SI unit conventions and quantity types of the framework; populations are the diagonal entries of the two projections onto $\mathbb{C}\mathrm{\mid }\mathrm{E}⟩$ and $\mathbb{C}\mathrm{\mid }\mathrm{G}⟩$ in the evolved preparation. Fix the coupling ratio $x:=\mu E_0\mathrm{/}{\mathrm{\Delta }}_0$ (dimensionless). Then:

(a) Identification with the declared linear drive. $H(t)=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega t){\sigma }_x)$ with ${\omega }_0=\frac{2{\mathrm{\Delta }}_0}{\mathrm{\hslash }},{\mathrm{\Omega }}_1=\frac{4\mu E_0}{\mathrm{\hslash }},{\omega }_0\in {\mathrm{s}}^{-1},{\mathrm{\Omega }}_1\in {\mathrm{s}}^{-1},$ so the model is an instance of the declared linear two-level drive model of clause (f) with $\mathrm{\Delta }={\omega }_0-\omega$; the level splitting is $2{\mathrm{\Delta }}_0$.

(b) Resonance and averaged inversion. At resonance $\omega ={\omega }_0$, i.e. $\mathrm{\hslash }\omega =2{\mathrm{\Delta }}_0$, the averaged model of the page's error contract predicts that the initial $\mathrm{\mid }\mathrm{E}⟩$ population is transferred to $\mathrm{\mid }\mathrm{G}⟩$ with probability $p_{\mathrm{G}}^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\mu E_0\tau }{\mathrm{\hslash }}\right),$ which equals $1$ at the inversion time $\tau =\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$; equivalently ${\mathrm{\Omega }}_1\tau \mathrm{/}4=\pi \mathrm{/}2$ at that time.

(c) Error contract at inversion. At the inversion time $\tau =\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$ the exact $\mathrm{\mid }\mathrm{G}⟩$ population $p_{\mathrm{G}}$ satisfies $\mid p_{\mathrm{G}}-1\mid \le 2x(1+\frac{3\pi }{2}),$ so with the declared ratio $x=1\mathrm{/}100$ the contract gives $p_{\mathrm{G}}\ge 1-\frac{2+3\pi }{100}$;

(d) status. The model, its parameters and the time $\tau$ are declared model data; no measured transition frequency, no laboratory molecule, cavity or field, no damping or relaxation and no measurement record appears, and the averaged formula is accompanied by the explicit contract of clause (c), not asserted as exact.

• What is and is not asserted. The example asserts a conditional model statement: the declared two-level family has a resonance at $\mathrm{\hslash }\omega =2{\mathrm{\Delta }}_0$ at which the averaged model inverts the population at the time $\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$, with an explicit error contract. It does not assert that any molecule, cavity or field realizes these parameters, and it records no measured frequency; the source's reported numerical frequency is not used as a premise.
• Why the contract is needed. The source treatment drops the counter-rotating term; the contract of clause (c) is what makes that step quantitative, and the ratio $x=\mu E_0\mathrm{/}{\mathrm{\Delta }}_0$ is exactly the small parameter controlling it.

Adopt P1, P2 and P4 in the readings of the declared driven model (The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (a)) and the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (b), (c) with parameters $m,\omega ,\mathrm{\hslash }>0$, length scale $\mathrm{\ell }=\sqrt{\mathrm{\hslash }\mathrm{/}(m\omega )}$, energy observable $(H_{\mathrm{o}\mathrm{s}\mathrm{c}},\mathrm{J})$ and position and momentum observables $(Q,\mathrm{m})$, $(P,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$; adopt the coherent-state family $({\mathrm{\Psi }}_z{)}_{z\in \mathbb{C}}$ of The coherent states of the oscillator model and the driven oscillator family $H(t):=H_{\mathrm{o}\mathrm{s}\mathrm{c}}-f(t)Q,t\in [a,b],$ with a declared drive $f\in C^1([a,b];\mathbb{R})$ read in $\mathrm{J}{\mathrm{m}}^{-1}$, as an instance of the protocol's common-domain model of The common-domain driven model: strong propagator with domain invariance, and the driven oscillator as its declared instance clause (c). Let $U(t,s)$ be the unique strong propagator of that family (which preserves $Y=D(H_{\mathrm{o}\mathrm{s}\mathrm{c}})$), and let $(q(\cdot ;s),p(\cdot ;s))$ be the classical forced response with zero data at $s$: $q^{\mathrm{\prime }}=\frac{p}{m},p^{\mathrm{\prime }}=-m{\omega }^{2}q+f,q(s)=p(s)=0.$ Then for every $z\in \mathbb{C}$ and $s,t\in [a,b]$, with $⟨A⟩(t):=⟨AU(t,s){\mathrm{\Psi }}_z,U(t,s){\mathrm{\Psi }}_z⟩$:

(a) Exact classical mean position and momentum. $⟨Q⟩(t)=\sqrt{2}\mathrm{\ell }\mathrm{Re}\left(z{e}^{-i\omega (t-s)}\right)+q(t;s),⟨P⟩(t)=\frac{\sqrt{2}\mathrm{\hslash }}{\mathrm{\ell }}\mathrm{Im}\left(z{e}^{-i\omega (t-s)}\right)+p(t;s);$

(b) Stationary fluctuations. the preparation spreads of position and momentum in the evolved coherent preparation are $\mathrm{\Delta }Q=\frac{\mathrm{\ell }}{\sqrt{2}},\mathrm{\Delta }P=\frac{\mathrm{\hslash }}{\mathrm{\ell }\sqrt{2}},\mathrm{\Delta }Q\mathrm{\Delta }P=\frac{\mathrm{\hslash }}{2},$ independent of $t$ and of $f$;

(c) Undriven check. if $f\equiv 0$ then $q\equiv p\equiv 0$ and the means reduce to the free classical rotation $(\sqrt{2}\mathrm{\ell }\mathrm{Re}(z{e}^{-i\omega (t-s)}),\frac{\sqrt{2}\mathrm{\hslash }}{\mathrm{\ell }}\mathrm{Im}(z{e}^{-i\omega (t-s)}))$ of the coherent label;

(d) status. The statements are exact for the declared model and drive; no laboratory oscillator, drive source, damping or measurement record is asserted.

• Why the response is exact. The drive is linear in $Q$, so the displacement operator converts it into the classical forced oscillator without approximation; the quantum fluctuations ride along unchanged. This is a property of the quadratic model and is not asserted for other drives or potentials.
• No classical limit is taken. The means obey the classical equations exactly for every coherent preparation, including small $\mathrm{\mid }z\mathrm{\mid }$; the statement is a theorem of the declared linear model, not an approximation in a limiting parameter.

Worked Rabi populations of the circularly driven two-level model: quarter-inversion, pi-pulse and detuned checks checks the exact Rabi populations of the declared circular two-level model at rational multiples of the Rabi period, including the quarter-inversion value $1\mathrm{/}2$, the pi-pulse value $1$, the return value $0$ and a detuned value $1\mathrm{/}4$; Worked rotating-wave error budget for the linearly driven two-level model: dimensionless instantiation and averaged Rabi check evaluates the rotating-wave error contract of The linearly driven two-level model: rotating-wave approximation with an explicit finite-time error contract and the averaged Rabi populations at declared dimensionless times, including one exact rational averaged population and the observation that the contract is conservative at the initial time; The two-level inversion model in an oscillating field: resonance, averaged Rabi inversion and its error contract matches the declared linear model onto an inversion-doublet family with an oscillating field, computes the averaged inversion probability and its pulse time and evaluates the error contract there for a declared coupling ratio; and The driven oscillator: exact classical response of a coherent preparation and stationary fluctuations checks the exact classical mean response and the stationary fluctuations of a coherent preparation of the driven oscillator of The common-domain driven model: strong propagator with domain invariance, and the driven oscillator as its declared instance clause (c).

The examples contain no measured frequency, energy, population, rate or lifetime; no apparatus, detector, calibration, sampling model or uncertainty record; no damping, relaxation, decoherence or environmental coupling; no identification of a laboratory spin, molecule, cavity or oscillator with the declared models or of a laboratory field with the declared drive parameters; and no claim that any declared parameter value is realized in nature. They also contain no Fermi golden rule, no continuum density of states and no long-time limit, in agreement with the scope of the companion A page (Scope and empirical status of the page: what is adopted, what is proved, and what is excluded).

All four examples are conditional computations inside declared models: their inputs are declared parameters, the cited mathematics and the postulates P1, P2 and P4. The sources cited are mathematical and textbook treatments, not experimental reports, so no empirical premise and no statistical inference enters any statement of the page. Comparing any of these model outputs with a laboratory oscillation or pulse would require a separately declared preparation and apparatus model with its own uncertainty and would test the combined model and apparatus assumptions; it is a tested identification in the sense of P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), not a consequence of the computations here.