Bath correlation matrix and Fourier convention

PDF p. 10: Eqs. (S2.5)–(S2.6)

Assume the Axiom of Choice and Countable Choice. Work in units with $\mathrm{\hslash }=1$. Let $H_B$ be a self-adjoint bath Hamiltonian on a complex Hilbert space, ${\rho }_B$ a stationary bath density operator, and $B_1,\dots ,B_n$ bath coupling operators, with $n$ a finite nonnegative integer. Stationary means ${\rho }_B$ is unchanged by the free bath unitary evolution. A density operator is positive, trace class and has trace one. Write ${\mathrm{Tr}}_B$ for the basis-independent trace of a trace-class operator on the bath space, and define the interaction-picture bath operators by ${\tilde{B}}_{\alpha }(t)=e^{i{H}_{B}t}{B}_{\alpha }{e}^{-i{H}_{B}t}$.

Whenever the products in the following expression have well-defined traces in this trace-class sense and the resulting two-time functions are invariant under a common time translation, the bath correlation matrix is the matrix-valued function of the time difference defined by ${\mathcal{J}}_{\alpha \beta }(t-s)={\mathrm{Tr}}_B[{\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B]$. In particular ${\mathcal{J}}_{\alpha \beta }(t)$ uses the second time $s=0$. If every entry is Lebesgue integrable over $\mathbb{R}$, define its spectral density, with the article's Fourier convention, by ${\mathcal{J}}_{\alpha \beta }(\omega )=(2\pi {)}^{-1}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathcal{J}}_{\alpha \beta }(t)e^{i\omega t}dt$. Here $\alpha ,\beta$ label coupling channels and $\omega$ is real angular frequency. For $n=0$, the correlation and its spectrum are the unique empty $0\times 0$ matrices; there are no entries to integrate.

For unbounded bath operators the trace condition includes the necessary operator domains and trace-class products. Stationarity alone does not imply integrability. If that integrability condition fails, the density formula above supplies no spectral density; any extension beyond it requires a separately specified convention.

ULE system–bath coupling and short bath memory

Time-local interaction-picture convention

Stationary correlation spectra and ULE kernel regularity

Trace class operator

Trace of a trace class operator

Trace is absolutely convergent and basis independent

Hilbert space

Symmetric, self-adjoint and essentially self-adjoint operators

Unbounded linear operators: domain, graph and extension

Integrable real and complex functions, and their integrals

Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ

Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume

Given: The bath data, choice assumptions, trace and common-time-translation conditions, and the entrywise integrability condition where the spectral density is used.

Goal: Verify that the displayed correlation and spectral density are well defined under those conditions.

1.1 By [F2], each bath unitary conjugation is defined on its stated transformed domain. The trace and domain conditions explicitly required in the definition put each displayed product in the trace class, so [F4], under Countable Choice from [A1], gives a basis-independent complex number for each two-time entry. Common-time-translation invariance makes this number depend only on $t-s$: translating both times by $-s$ gives the value at $(t-s,0)$. Thus the definition assigns one value to each time difference. [given, A1, F2, F4, algebra]

2.1 If $n=0$, both matrices are empty and no Fourier supplier is invoked. Otherwise $n$ is a positive integer, so the finite channel set gives a finite matrix of these functions, in the coupling convention of [F1]. Interpret the integrability condition using Lebesgue measure and the complex integral in [F5], under [A1]. When each entry is integrable, all hypotheses of the Fourier-definition part of [F3] hold, and it applies entrywise to make the displayed Fourier integral finite for every real $\omega$, with the stated normalization and sign. The assumptions do not assert that a nonintegrable correlation has an ordinary spectral density. [given, A1, F1, F3, F5, step 1.1] ∎