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Vacuum limit and cubic free-space spectral density

PDF p. 13: Eq. (S3.18) and surrounding paragraphs

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice and the hypotheses and conventions of the isotropic thermal time-correlator result, with a fixed Borel coupling $κ$ , $c, γ > 0$ , and an inverse temperature $β_{0} > 0$ for which $\int_{0}^{\infty} ∣ κ (ω) ∣^{2} ω^{2} [2 (e^{β_{0} ω} - 1)^{- 1} + 1] d ω < \infty$ . In that supplier $κ = d_{0} E$ , $B_{α} = - d_{0} E_{α} / \sqrt{γ}$ , and the continuum coupling measure includes the chosen mode-density normalization.

For each finite positive-frequency mode regulator, its normalized Gibbs density tends in trace norm to the vacuum projector as $β \to \infty$ ; the normal moment $⟨ a_{i}^{†} a_{j} ⟩_{β}$ tends to zero. For the continuum spectral measures of the thermal spectral-density result, the same limit exists in total variation and has Lebesgue density, almost everywhere,

$J_{0} (x) = \frac{8 π}{3 γ c^{3}} x^{2} ∣ κ (x) ∣^{2} Θ (x) .$

Here $Θ (x) = 1$ for $x > 0$ and $Θ (x) = 0$ for $x < 0$ ; its value at zero is immaterial. Thus the limiting measure is supported on nonnegative frequencies and has no atom at zero. For the article's normalization $γ J_{0}$ , or numerically $γ = 1$ , the prefactor is $8 π / (3 c^{3})$ . If $d_{0}$ is frequency independent and the box field normalization $∣ E_{ω} ∣^{2} = ω / (2 ε_{0} V)$ applies, then $∣ κ (ω) ∣^{2}$ is proportional to $ω$ and the unmodified positive-frequency law is proportional to $ω^{3}$ on the range where that coupling applies. A fixed ultraviolet form factor may alter this law outside that range.

Here trace norm means $∥ T ∥_{1} = \sum_{n \geq 1} s_{n} (T)$ for a compact operator with summable singular values, on the full oscillator Fock space. For finite Borel measures $μ, ν$ on $R$ , total-variation convergence means $∣ μ - ν ∣ (R) \to 0$ , where $∣ σ ∣ (E)$ is the supremum of $\sum_{n \geq 0} ∣ σ (E_{n}) ∣$ over countable measurable partitions of $E$ .

Statement references

Isotropic thermal bath time correlator

Thermal bath spectral density and frequency signs

Trace class operator

The total variation |nu|(E) from countable measurable partitions

Facts & Assumptions

Given: The model, normalization, choice assumptions and integrable envelope in the Statement.

[F1]

For the finite-mode symmetric Fock realization, the bosonic moment contract supplies a complete occupation basis $∣ n ⟩$ , vacuum $∣ 0 ⟩$ , and diagonal Gibbs weights $p_{n} = \prod_{j} (1 - q_{j}) q_{j}^{n_{j}}$ , $q_{j} = e^{- β ω_{j}}$ , with $\sum_{n} p_{n} = 1$ . It also supplies $⟨ a_{i}^{†} a_{j} ⟩_{β} = δ_{i j} (e^{β ω_{j}} - 1)^{- 1}$ and the other stated moments. Finite-mode bosonic Gibbs occupations and moments

[F2]

The thermal continuum spectrum is the sum of masses at $- ω$ and $+ ω$ weighted respectively by $N_{β} (ω)$ and $N_{β} (ω) + 1$ , with common radial factor $8 π ∣ κ (ω) ∣^{2} ω^{2} / (3 γ c^{3})$ , where $N_{β} (ω) = (e^{β ω} - 1)^{- 1}$ . The supplier specifies the positive-exponential Fourier convention and a distributional spectrum. Thermal bath spectral density and frequency signs Stationary correlation spectra and ULE kernel regularity

[F3]

The continuum construction and its normalization use $κ = d_{0} E$ and include the mode-density convention in the coupling measure. Isotropic thermal bath time correlator

[F4]

The photon-field postulate gives $∣ E_{ω} ∣^{2} = ω / (2 ε_{0} V)$ , while the regulated bath postulate distinguishes finite-mode preparations from a continuum state. Photon modes and field normalization Regularized transverse photon dipole bath model

[F5]

Dominated convergence applies to pointwise convergent measurable functions bounded by an integrable majorant. Dominated convergence

[F6]

Trace norm is the sum of the zero-padded singular values of a compact operator. Singular values are the positive eigenvalues, with multiplicity, of its unique positive compact square root $∣ T ∣ = (T^{*} T)^{1 / 2}$ . For an absolutely summable complex sequence $d = (d_{n})$ on $ℓ^{2} (N, C)$ , its diagonal operator $D_{d}$ is trace class and $∥ D_{d} ∥_{1} = \sum_{n} ∣ d_{n} ∣$ , under Choice. Trace class operator Absolute value and singular values of a compact operator Diagonal trace-class operators on ℓ²(ℕ,ℂ)

[F7]

Total variation is the supremum of absolute masses summed over countable measurable partitions, for a signed or complex measure. The total variation |nu|(E) from countable measurable partitions

[F8]

Under Countable Choice, an orthonormal family and square-summable coefficients have a norm-convergent vector sum whose squared norm is the sum of the squared coefficient moduli. For the mathematical Hilbert-space suppliers use the first-variable-linear pairing $(u, v)_{1} = ⟨ v, u ⟩_{2}$ on Fock space; this preserves norms and adjoints. Square-summable orthogonal families have norm-convergent finite sums

[F9]

A nonnegative measurable density on a measure space defines a measure by integration over measurable sets. The indefinite integral of a nonnegative measurable function is a measure

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice as required by the named constructions. The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

Proof

Given: The model, normalization, choice assumptions and integrable envelope in the Statement.

Goal: Establish the finite-mode vacuum limit, the continuum spectral limit, and the stated normalization and frequency dependence.

1.1F1F4F6F8A1constructalgebra

For a finite nonempty mode list, enumerate $N^{m}$ by increasing total occupation and lexicographic order within each finite level, starting at $0$ . By [F1] the corresponding vectors $b_{n}$ form a complete orthonormal basis. By [F8], $U : ℓ^{2} (N, C) \to H_{B}$ , $U z = \sum_{n} z_{n} b_{n}$ , is a linear isometry. Its range is closed because $ℓ^{2}$ is complete by [F6], and contains the dense span of all $b_{n}$ , so it is onto and $U^{*} = U^{- 1}$ . For $T = ρ_{β} - ∣ 0 ⟩ ⟨ 0 ∣$ , the coordinate operator $U^{*} T U = D_{d}$ has entries $d_{0} = p_{0} - 1$ and $d_{n} = p_{n (n)}$ at nonvacuum indices, where $n (n)$ is the enumeration. Since the weights are nonnegative and sum to one, $\sum_{n} ∣ d_{n} ∣ = (1 - p_{0}) + \sum_{n \neq 0} p_{n} = 2 (1 - p_{0}) < \infty$ . Hence [F6] gives $∥ D_{d} ∥_{1} = 2 (1 - p_{0})$ . To transport this to Fock space, unitary conjugation preserves compactness because it maps the unit ball onto the unit ball and preserves convergent subsequences. Also $U ∣ D_{d} ∣ U^{*}$ is positive and compact with square $T^{*} T$ , so uniqueness of the positive compact square root in [F6] gives $∣ T ∣ = U ∣ D_{d} ∣ U^{*}$ . The eigenspaces are carried bijectively by $U$ , preserving their multiplicities; thus the singular-value lists and their sums agree, giving $∥ T ∥_{1} = 2 (1 - p_{0})$ . Now $q_{j} = e^{- β ω_{j}} \to 0$ for every $j$ , so $p_{0} = \prod_{j} (1 - q_{j}) \to 1$ . The normal moment in [F1] also tends to zero. For the empty mode list the bath is one-dimensional by [F4], its Gibbs state already equals the vacuum projector, and there are no normal moments to check.

1.2givenF2F5F7F9algebra

Put $A = 8 π / (3 γ c^{3})$ and $w (ω) = ∣ κ (ω) ∣^{2} ω^{2}$ . For $β \geq β_{0}$ , $0 \leq N_{β} (ω) \leq N_{β_{0}} (ω)$ and $N_{β} (ω) \to 0$ for every $ω > 0$ . The given envelope makes $w N_{β_{0}}$ integrable, and all these functions are Borel. Apply [F5] along $β_{n} = β_{0} + n$ to obtain $I_{β_{n}} = \int_{0}^{\infty} w (ω) N_{β_{n}} (ω) d ω \to 0$ . Since $I_{β}$ decreases in $β$ , this also proves $I_{β} \to 0$ for the full real parameter. By [F9], define $μ_{0}$ to be the positive-frequency measure with density $A w (x)$ on $x > 0$ and zero elsewhere; it is finite since $w$ is bounded in integral by the given envelope. The density formula in [F2] gives a finite positive difference $ν_{β} = μ_{β} - μ_{0}$ , with density $A w (- x) N_{β} (- x)$ on $x < 0$ and $A w (x) N_{β} (x)$ on $x > 0$ , and total mass $2 A I_{β}$ by the point-mass representation in [F2]. Its nonnegative Borel density also defines a measure by [F9]. For every countable Borel partition $(E_{n})$ of $R$ , positivity and countable additivity give $\sum_{n} ∣ ν_{β} (E_{n}) ∣ = \sum_{n} ν_{β} (E_{n}) = ν_{β} (R)$ . Taking the supremum in [F7] yields $∣ μ_{β} - μ_{0} ∣ (R) = 2 A I_{β} \to 0$ . The positive-frequency density from [F2] is exactly the displayed $J_{0}$ almost everywhere. Its extension by zero to the negative half-line and the Lebesgue-null singleton zero proves the stated support and absence of an atom at zero.

2.1F3F4step 1.2algebra∎

By [F3], the factor $γ^{- 1}$ comes from the stated bath-operator normalization; multiplication by $γ$ , or setting $γ = 1$ , yields the article-prefactor convention. By [F4], frequency-independent $d_{0}$ gives $∣ κ (ω) ∣^{2} = ∣ d_{0} ∣^{2} ω / (2 ε_{0} V)$ before the selected mode-density normalization, so the product of $ω^{2}$ and the squared coupling is cubic in frequency wherever that unmodified normalization applies. A frequency-dependent regulator changes this conclusion where it changes the coupling.

Source notes

The article citation attributes Eq. (S3.18). Its box-to-continuum convention absorbs $V / (2 π)^{3}$ into the coupling; the continuum supplier states the corresponding coupling-measure convention explicitly. The globally unmodified cubic weight does not satisfy the integrable envelope above, so this result does not assert removal of the ultraviolet regularization or the separate jump-kernel moment conditions.

Depends on

Used by · 3

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 13, Eq. (S3.18) and surrounding paragraphs