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Commutator and spontaneous-plus-stimulated emission contribution

PDF p. 12: Eqs. (S3.9)–(S3.11)

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice, and adopt the finite-mode thermal-bath, regularized photon-mode and real-polarization dipole-coupling postulates. Let $j = 1, \dots, m$ , $m \geq 1$ , index independent bosonic modes of frequencies $ω_{j} > 0$ , and let $β > 0$ and $γ > 0$ . Write $a_{j}, a_{j}^{†}$ for the annihilation and creation operators, $∣ n ⟩$ for the occupation basis, $H_{B} = \sum_{j} ω_{j} a_{j}^{†} a_{j}$ , $Z = \prod_{j} (1 - e^{- β ω_{j}})^{- 1}$ , and $ρ_{B} = Z^{- 1} e^{- β H_{B}}$ . Put $N_{j} = (e^{β ω_{j}} - 1)^{- 1}$ and $P_{α j} = {\hat{e}}_{α} \cdot ϵ_{j}$ , where ${\hat{e}}_{α}$ is a Cartesian unit vector and $ϵ_{j}$ is a real transverse polarization vector. Let $κ (ω_{j})$ be the finite-mode coupling coefficient, and let $J_{α α} (t) = Tr [{\tilde{B}}_{α} (t) B_{α} ρ_{B}]$ for $B_{α} = - d_{0} E_{α} / \sqrt{γ}$ .

Use the maximal occupation-basis mode operators, identified with their finite-mode symmetric Fock realization as in the thermal-correlator expansion. For this correlator, the field operators are the extensions to $D = ⋂_{j = 1}^{m} (D (a_{j}) \cap D (a_{j}^{†}))$ , with $B_{α} = - γ^{- 1 / 2} \sum_{j} P_{α j} [κ (ω_{j}) a_{j} + κ (ω_{j})^{*} a_{j}^{†}]$ on $D$ . Its free conjugate has the same domain. The Gibbs-weighted two-field products are defined on the whole bath space and trace class by that expansion result.

The antinormal contraction is $Tr (a_{j} a_{k}^{†} ρ_{B}) = δ_{j k} (1 + N_{j})$ . Its contribution to the correlator, meaning the part containing annihilation at time $t$ and creation at time zero, is, for real $t$ ,

$J_{α α} (t) {a a^{†}} = \frac{1}{γ} \sum_{j = 1}^{m} P_{α j}^{2} ∣ κ (ω_{j}) ∣^{2} e^{- i ω_{j} t} (1 + N_{j}) .$

The factor $1$ is conventionally called the spontaneous contribution and $N_{j}$ the stimulated contribution. In the article's continuum notation the corresponding formal expression is $\int d^{3} k \sum_{ε} [{\hat{e}}_{α} \cdot ε (k)]^{2} ∣ κ (ω) ∣^{2} e^{- i ω t} [1 + (e^{β ω} - 1)^{- 1}]$ . The article's equations (S3.9)–(S3.10) display the occupation sum and insert $a_{1} a_{2}^{†} = δ_{12} + a_{2}^{†} a_{1}$ before its cancellation of partition factors. No continuum limit is established here; absorbing the finite expression's $1 / γ$ requires a consistent redefinition of the coupling or correlator.

Statement references

Photon-number expansion of the diagonal bath correlator

Thermal contractions and absorption contribution

Bath correlation matrix and Fourier convention

Thermal photon bath and partition function

Dimensionless dipole and ULE coupling identification

Regularized transverse photon dipole bath model

Finite-mode bosonic Gibbs occupations and moments

Facts & Assumptions

Given: The choice assumptions, positive finite-mode data and physical postulates in the Statement.

[F1]

The regulated bath has finitely many independent positive-frequency modes, and its thermal preparation has the occupation basis, Hamiltonian and partition function stated above. Regularized transverse photon dipole bath model Thermal photon bath and partition function

[F2]

The correlator and dipole coupling use the definition and normalization stated above. Bath correlation matrix and Fourier convention Dimensionless dipole and ULE coupling identification

[F3]

The finite-mode Gibbs products of two ladder operators are trace class; the normal contraction is $Tr (a_{k}^{†} a_{j} ρ_{B}) = δ_{j k} N_{j}$ , and the antinormal contraction is $δ_{j k} (1 + N_{j})$ . Finite-mode bosonic Gibbs occupations and moments

[F4]

In the regulated correlator's absolutely convergent occupation expansion, the coefficient of $a_{j}$ at time $t$ is $P_{α j} κ (ω_{j}) e^{- i ω_{j} t} / \sqrt{γ}$ , up to the common minus sign of the bath operator, and the coefficient of $a_{k}^{†}$ at time zero is its corresponding conjugate. Photon-number expansion of the diagonal bath correlator

[F5]

The normal occupation sum factors into the selected mode's weighted number sum and the other modes' partition functions, giving $N_{j}$ after normalization. Thermal contractions and absorption contribution

[F6]

In the finite-mode Fock realization, write $e_{j}$ for the standard orthonormal mode vectors, so $a_{j} = a (e_{j})$ and $a_{j}^{†} = a^{†} (e_{j})$ . The common invariant finite-particle core $D_{f i n}$ contains every occupation vector. On this core the bosonic identity is $a_{j} a_{k}^{†} = δ_{j k} I + a_{k}^{†} a_{j}$ , since $⟨ e_{j}, e_{k} ⟩ = δ_{j k}$ . Dimensionless dipole and ULE coupling identification Bosonic and fermionic Fock operators with exact domains

[F7]

Under Countable Choice, a trace-class operator has an absolutely convergent diagonal sum in any supplied Hilbert basis, equal to its trace. Apply the trace theorem to the occupation basis and to each product in [F3]. To match its first-variable-linear convention with the Fock convention, use $(u, v)_{1} = ⟨ v, u ⟩_{2}$ ; then $(T u, u)_{1} = ⟨ u, T u ⟩_{2}$ . Trace is absolutely convergent and basis independent

[F8]

The thermal-correlator expansion identifies the occupation and symmetric Fock realizations, defines these field extensions on $D$ , proves invariance of that domain under free evolution, and proves that the Gibbs-weighted two-field product is defined on the whole space and trace class. Photon-number expansion of the diagonal bath correlator

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice for the cited 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 choice assumptions, positive finite-mode data and physical postulates in the Statement.

Goal: Establish the antinormal contraction, its partition-factor cancellation, and the finite-mode correlator contribution.

1.1F6A1algebra

Each occupation vector lies in $D_{f i n}$ , where [F6] gives $a_{j} a_{k}^{†} = δ_{j k} I + a_{k}^{†} a_{j}$ . Thus $⟨ n ∣ a_{j} a_{k}^{†} ∣ n ⟩ = δ_{j k} + ⟨ n ∣ a_{k}^{†} a_{j} ∣ n ⟩$ for every $n$ . This uses only the common-core identity, without asserting equality on unrestricted product domains.

2.1F1F3F7A1step 1.1algebra

By [F1], $ρ_{B} ∣ n ⟩ = p_{n} ∣ n ⟩$ , where $p_{n} = Z^{- 1} e^{- β \sum_{l} n_{l} ω_{l}}$ , and $\sum_{n} p_{n} = 1$ . Multiply the diagonal identity in step 1.1 by $p_{n}$ and sum. The two ladder products composed with $ρ_{B}$ are trace class by [F3], so [F7] identifies the absolutely convergent diagonal sums with their traces. Consequently $Tr (a_{j} a_{k}^{†} ρ_{B}) = δ_{j k} + Tr (a_{k}^{†} a_{j} ρ_{B}) = δ_{j k} (1 + N_{j})$ , using the normal contraction in [F3].

3.1F1F5step 2.1algebra

Explicitly, put $q_{l} = e^{- β ω_{l}}$ . For $j = k$ , the identity part of the occupation sum is $Z^{- 1} \prod_{l} \sum_{n_{l} \geq 0} q_{l}^{n_{l}} = 1$ by [F1]. The remaining normal part factors and equals $N_{j}$ by [F5]. For $j \neq k$ the Kronecker factor and normal contraction both vanish. This is the finite-mode partition-factor cancellation corresponding to the article's two displayed occupation sums.

4.1F2F4F8step 2.1step 3.1algebra∎

Use the common-domain field realization in [F8]; its trace-class product is the correlator being expanded. Select annihilation at time $t$ and creation at time zero in the finite expansion [F4]. The two common minus signs cancel, and the coefficient multiplying $Tr (a_{j} a_{k}^{†} ρ_{B})$ is $γ^{- 1} P_{α j} P_{α k} κ (ω_{j}) κ (ω_{k})^{*} e^{- i ω_{j} t}$ . Step 2.1 leaves only $j = k$ , yielding the displayed finite sum. The real polarization factors give $P_{α j}^{2}$ , while the coupling product gives $∣ κ (ω_{j}) ∣^{2}$ . The calculation uses no continuum interchange or regulator-removal limit.

Source notes

The article's continuum expression and its labels for the two contributions are recorded for attribution. The finite-mode result follows from the stated suppliers and retains the normalization specified by the bath operator; the article citation is not used as its proof.

Depends on

Used by · 1

Isotropic thermal bath time correlator

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 12, equations (S3.9)–(S3.11)