Exact regulated transition probabilities and controlled coefficients

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), and use the finite regulated model type of Example observable types and scoped meanings. Let $d\ge 1$, let $\tau >0$ be a time interval endpoint in seconds, let $\mathrm{\hslash }>0$ have unit $\mathrm{J}\mathrm{s}$, and let $\lambda \in \mathbb{R}$ be dimensionless. Let $H_0=H_0^{\ast }\in M_d(\mathbb{C})$ have unit joule and let $V:[0,\tau ]\to M_d(\mathbb{C})$ be norm-continuous, Hermitian-valued, and joule-valued. Set $V_I(t)=e^{it{H}_0\mathrm{/}\mathrm{\hslash }}V(t)e^{-it{H}_0\mathrm{/}\mathrm{\hslash }},B={\sup }_{0\le t\le \tau }\mathrm{\parallel }{V}_I(t)\mathrm{\parallel },z=\frac{\mathrm{\mid }\lambda \mathrm{\mid }B\tau }{\mathrm{\hslash }}\mathrm{.}$ Define $S_0(t)=I$ and $S_{n+1}(t)=-\frac{i}{\mathrm{\hslash }}{\int }_0^t{V}_I(r)S_n(r)dr,S(t)={\sum }_{n=0}^{\mathrm{\infty }}{\lambda }^n{S}_n(t)\mathrm{.}$ For $W\in M_d(\mathbb{C})$, unitary means $W^{\ast }W=W{W}^{\ast }=I$. Write $U(\tau )=e^{-i{H}_0\tau \mathrm{/}\mathrm{\hslash }}S(\tau )$ and, for $N\ge 0$, $U_N(\tau )=e^{-i{H}_0\tau \mathrm{/}\mathrm{\hslash }}{\sum }_{n=0}^N{\lambda }^n{S}_n(\tau ),R_N=e^z\frac{z^{N+1}}{(N+1)!}\mathrm{.}$

Let $\psi \in {\mathbb{C}}^d$ be a unit initial vector. A supplied finite projective measurement is a family $P_1,\dots ,P_m\in M_d(\mathbb{C})$, $m\ge 1$, satisfying $P_a=P_a^{\ast }=P_a^2$, $P_a{P}_b=0$ for $a\mathrm{\ne }b$, and ${\sum }_{a=1}^m{P}_a=I$. Equivalently, the one-Kraus-per-outcome data $M_{a,1}=P_a$ form a normalized finite instrument in the sense of Finite bounded Kraus instrument. Adopt the constructed-model Born interpretation of Physical interpretation of a supplied constructed quantum model only for this actually constructed finite model and measurement. For its physical reading, also adopt the explicit preparation/evolution assumption [A3]: the initial density is ${\rho }_0:=T_{\psi ,\psi }$, and the model density at the measurement time is ${\rho }_{\tau }:=U(\tau ){\rho }_{0}U(\tau {)}^{\ast }=T_{U(\tau )\psi ,U(\tau )\psi }$. Here $T_{u,v}$ is the rank-one map $x\mapsto ⟨x,v⟩u$ of Finite bounded instruments normalize measurement probabilities. The Dyson result proves that this matrix $U(\tau )$ is unitary; identifying it with the selected model's state update is an adopted physical assumption, not a conclusion about nature. Define the exact model probability and the truncated squared norm by $p_a=\mathrm{\parallel }{P}_{a}U(\tau )\psi {\mathrm{\parallel }}^2,q_{a,N}=\mathrm{\parallel }{P}_a{U}_N(\tau )\psi {\mathrm{\parallel }}^2\mathrm{.}$ Then $p_a\in [0,1]$, ${\sum }_a{p}_a=1$, and $\mathrm{\mid }{p}_a-q_{a,N}\mathrm{\mid }\le (2\sqrt{q_{a,N}}+R_N)R_N\mathrm{.}$ The nonnegative number $q_{a,N}$ is a truncation-derived square, not generally a probability.

For the exact finite interaction matrix write $S(\tau ;\lambda )=I+i\mathsf{T}(\lambda )$ and $\mathsf{T}(\lambda )=\lambda {\mathsf{T}}_1+{\lambda }^2{\mathsf{T}}_2+\cdots$. Use the standard basis $e_1,\dots ,e_d$ for matrix entries, so ${\mathsf{T}}_{fi}:=e_f^{\ast }\mathsf{T}{e}_i$ for $1\le f,i\le d$. For a matrix $A$, define its Hermitian imaginary part by ${\mathrm{Im}}_{\mathrm{H}}A:=(A-A^{\ast })\mathrm{/}(2i)$. Exact unitarity gives $2\mathrm{Im}{\mathsf{T}}_{ii}(\lambda )={\sum }_{f=1}^d\mathrm{\mid }{\mathsf{T}}_{fi}(\lambda ){\mathrm{\mid }}^2,{\mathsf{T}}_1={\mathsf{T}}_1^{\ast },2{\mathrm{Im}}_{\mathrm{H}}{\mathsf{T}}_2={\mathsf{T}}_1^{\ast }{\mathsf{T}}_1\mathrm{.}$ These are finite-regulator coefficient identities, not continuum optical or scattering theorems.

In the two-level case $H_0=0$, $\lambda =1$, and $V(t)=g_E{\sigma }_x$ with $g_E>0$ in joules and ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, the exact transition probability from $e_1=(1,0)$ to the projector $P=\mathrm{\mid }{e}_2⟩⟨e_2\mathrm{\mid }$ is ${\sin }^{2}z$, where the previously defined $z$ equals $g_E\tau \mathrm{/}\mathrm{\hslash }$ because $V_I=V$ and $\mathrm{\parallel }{\sigma }_x\mathrm{\parallel }=1$; the first-order squared transition coefficient is $z^2$ and may exceed $1$. For all real $z$, $\mathrm{\mid }{\sin }^{2}z-z^2\mathrm{\mid }\le \frac{\mathrm{\mid }z{\mathrm{\mid }}^4}{3}\mathrm{.}$ The exact survival factor is ${\cos }^{2}z$, which is not $e^{-\mathrm{\Gamma }\tau }$ for any $\mathrm{\Gamma }>0$ on all $\tau \ge 0$.

For the occupation truncation with orthonormal basis $e_0,\dots ,e_N$, define $a_N{e}_n=\sqrt{n}{e}_{n-1}$ (zero for $n=0$), $a_N^{\ast }{e}_n=\sqrt{n+1}{e}_{n+1}$ for $n<N$, and $a_N^{\ast }{e}_N=0$. Then $[a_N,a_N^{\ast }]=I-(N+1)\mathrm{\mid }{e}_N⟩⟨e_N\mathrm{\mid }\mathrm{.}$ More generally, no finite-dimensional matrices $A,B$ satisfy $[A,B]=I$. These exact regulator facts assert neither an infinite-dimensional CCR construction nor a continuum limit.