Jump correlators and bath-memory parameters

PDF p. 9: Eq. (S2.2) and following paragraph: coupling convention; PDF p. 10: Eqs. (S2.7)–(S2.10)

Assume the Axiom of Choice and Countable Choice, and use the bath correlation and spectral-density convention in which ${\mathcal{J}}_{\alpha \beta }(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}{\mathcal{J}}_{\alpha \beta }(t)e^{i\omega t}dt$. Here $\alpha ,\beta =1,\dots ,n$ label a finite positive number $n$ of bath coupling channels. Assume the traces and entrywise integrability required for this spectral density, and suppose that $\mathcal{J}(\omega )$ is Hermitian positive semidefinite for every real $\omega$.

The frequency-domain jump correlator is $g(\omega )=[\mathcal{J}(\omega )\mathrm{/}(2\pi ){]}^{1\mathrm{/}2}$, the unique Hermitian positive semidefinite square root. If its entries are Lebesgue integrable, its time-domain jump correlator is $g(t)={\int }_{\mathbb{R}}g(\omega )e^{-i\omega t}d\omega$, entrywise. For an $n\times n$ matrix $M$, define the mixed norm $\mathrm{\parallel }M{\mathrm{\parallel }}_{2,1}={\sum }_{b=1}^n({\sum }_{\alpha =1}^n\mathrm{\mid }{M}_{\alpha b}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$.

Let $\gamma >0$ be the coupling parameter in $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=\sqrt{\gamma }{\sum }_{\alpha =1}^n{X}_{\alpha }\otimes B_{\alpha }$, in units $\mathrm{\hslash }=1$, for a finite-dimensional system and Hermitian coupling operators with system operator norms $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$. The $B_{\alpha }$ are the same bath operators used to construct $\mathcal{J}$ above. For unbounded bath operators, a concrete Hamiltonian application requires an appropriate common domain and self-adjoint realization. This uses the interaction convention of the system–bath postulate; its Gaussianity, preparation and short-memory assumptions are additional conditions for a ULE approximation, not requirements for defining the quantities below.

Assume $0<I_0<\mathrm{\infty }$ and $I_1<\mathrm{\infty }$, where $I_0={\int }_{\mathbb{R}}\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$ and $I_1={\int }_{\mathbb{R}}\mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$. Define $\mathrm{\Gamma }=4\gamma I_0^2$ and ${\tau }_B=I_1\mathrm{/}{I}_0$. Equivalently, the numerator of ${\tau }_B$ is ${\int }_{\mathbb{R}}\mathrm{\parallel }tg(t){\mathrm{\parallel }}_{2,1}dt$. The article interprets $\mathrm{\Gamma }$ as an upper bath-induced evolution rate and ${\tau }_B$ as a characteristic bath-correlation time; its notation distinguishes ${\tau }_B$ from checking time, the interval used for error checks. These interpretations do not by themselves establish a dynamical bound.

ULE system–bath coupling and short bath memory

Bath correlation matrix and Fourier convention

Finite support square roots and generalized inverses

Stationary correlation spectra and ULE kernel regularity

The Axiom of Choice

The Axiom of Countable Choice (AC_ω)

Given: The bath and choice assumptions, positive semidefinite spectral matrix, entrywise integrability of its square-root kernel, and the stated moment conditions.

Goal: Show that the displayed correlators and parameters are well defined under these assumptions.

1.1 By [F1], the spectral matrix is defined entrywise. At each real frequency it is finite-dimensional and positive semidefinite, so [F2] supplies its basis-independent positive square root after division by the positive scalar $2\pi$. The assumed entrywise integrability and [F3] then give a finite matrix $g(t)$ for every real $t$. [given, A1, F1, F2, F3]

2.1 By [F4], $\gamma$ is the positive parameter of the stated interaction convention. The finite sums in the mixed norm are defined for every such matrix. By the assumed moment conditions, $I_0$ is finite and strictly positive and $I_1$ is finite; hence $\mathrm{\Gamma }$ and ${\tau }_B$ are finite and the latter has a nonzero denominator. Since $t$ is real, direct substitution into the mixed norm gives $\mathrm{\parallel }tg(t){\mathrm{\parallel }}_{2,1}=\mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}$, establishing the equivalent numerator. [given, F3, F4, step 1.1, algebra] ∎