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Isotropic Lamb shift preserves magnetic degeneracy

PDF p. 11: Opening continuation of S2

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Use the finite-emitter angular-momentum and regulated photon model of Angular-momentum emitter, photon bath, and dipole approximation, with the isotropic thermal bath and continuum-limit hypotheses of Off-diagonal correlators vanish in an isotropic photon bath. Also impose all system, bath, preparation, weak-Markov, Fourier-integrability and principal-value hypotheses of Static ULE formulas and principal-value kernel, with three Cartesian channels and a common coupling normalization. In particular, use a pointwise nonnegative scalar spectral-density representative at every frequency required by that ULE formula.

Let the emitter space be a finite direct sum of complete spin blocks, and let $U_{k} (t) = \exp (- i t J_{k})$ be its axis rotations, as in Rotational scalars on multiplicity-resolved spin blocks. Suppose $H_{S}$ commutes with these rotations and the three Hermitian system coupling operators $X_{i}$ transform as a Cartesian vector: for each axis rotation $U$ there is a real orthogonal matrix $R$ such that $U X_{i} U^{- 1} = \sum_{j} R_{j i} X_{j}$ . Assume the resulting ULE Lamb-shift operator $Λ$ is Hermitian.

Then $Λ$ commutes with every axis rotation. On the sum of copies of spin $J$ , it has the form $A^{(J)} \otimes I_{2 J + 1}$ , where $A^{(J)}$ acts on the copy labels. In the article's multiplicity-one model this reduces to $Λ ∣_{V_{J}} = λ_{J} I$ , with real $λ_{J}$ , so it does not split the magnetic labels $m$ . Adding it to $H_{S}$ replaces $η_{J}$ by $η_{J} + λ_{J} / ω_{0}$ ; the shifted level factors need not remain monotone.

Facts & Assumptions

[F1]

Under its stated hypotheses, the static ULE supplier gives $Λ = \sum_{l, m, n, i, j} f_{i j} (E_{l} - E_{m}, E_{n} - E_{l}) X_{m l}^{(i)} X_{l n}^{(j)} ∣ m ⟩ ⟨ n ∣$ , with $f (E_{1}, E_{2}) = - 2 π γ PV \int_{- \infty}^{\infty} ω^{- 1} g (ω - E_{1}) g (ω + E_{2}) d ω$ and $g = [J / (2 π)]^{1 / 2}$ . Static ULE formulas and principal-value kernel

[F2]

The atomic model has complete angular-momentum blocks and $H_{S} = \sum_{J, m} ω_{0} η_{J} ∣ J, m ⟩ ⟨ J, m ∣$ , with one specified block for each $J$ and $ω_{0} > 0$ . The dipole interaction is a model assumption. Angular-momentum emitter, photon bath, and dipole approximation

[F3]

Under the regulated isotropic thermal-bath hypotheses, the Cartesian spectral matrix has an almost-everywhere scalar representative; for a chosen nonnegative representative its positive jump kernel is $g (x) = s (x) I_{3}$ , where $s (x) = \sqrt{J (x) / (2 π)}$ . The additional ULE regularity and pointwise conditions are separately assumed. Off-diagonal correlators vanish in an isotropic photon bath

[F4]

A contraction $\sum_{i} D_{i} F D_{i}$ commutes with all axis rotations when the $D_{i}$ transform as a Cartesian vector and $F$ commutes with those rotations. An operator commuting with the rotations has a unique matrix on each spin-copy space tensored with the identity on the magnetic space. Rotational scalars on multiplicity-resolved spin blocks

[A1]

Adopt the Axiom of Choice, Countable Choice and Dependent Choice required by the stated supplier interfaces. 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, covariance, Hermiticity, choice and ULE regularity hypotheses in the Statement.

Goal: Prove rotational invariance of the Lamb shift and its stated action on magnetic labels.

1.1givenF1F3

By [F3], $g (x) = s (x) I_{3}$ at the frequencies used in [F1]. Matrix multiplication in the kernel therefore gives $f_{i j} (a, b) = δ_{i j} h (a, b)$ , where $h (a, b) = - 2 π γ PV \int ω^{- 1} s (ω - a) s (ω + b) d ω$ . The assumed principal values make these expressions meaningful.

2.1givenF1step 1.1algebra

Let $P_{E}$ denote the orthogonal projection onto the eigenspace of $H_{S}$ with eigenvalue $E$ . Grouping the finite sums in [F1] by the three energy eigenvalues and applying step 1.1 yields $Λ = \sum_{E, E^{'}, F} h (F - E, E^{'} - F) P_{E} (\sum_{i} X_{i} P_{F} X_{i}) P_{E^{'}}$ . Because $H_{S}$ commutes with each rotation, the rotation maps every eigenspace of $H_{S}$ to itself; its inverse does so as well. Hence each $P_{E}$ commutes with each rotation.

3.1givenF4step 2.1

The assumed vector transformation of $X_{i}$ , the commutation of $P_{F}$ , and [F4] show that each $\sum_{i} X_{i} P_{F} X_{i}$ commutes with every axis rotation. The projections on its left and right commute with those rotations too, so the finite sum in step 2.1 commutes with them.

4.1givenF2F4step 3.1algebra∎

Apply the block conclusion of [F4] to $Λ$ . It has the form $A^{(J)} \otimes I_{2 J + 1}$ on each spin sector, with no coupling between different spins. In the multiplicity-one model of [F2], $A^{(J)}$ is a scalar $λ_{J}$ ; its reality follows from the assumed Hermiticity of $Λ$ . Adding $λ_{J}$ to the common unperturbed energy $ω_{0} η_{J}$ gives the displayed shifted level factor and leaves all magnetic states in that block degenerate. No ordering of shifted energies follows.

Source notes

The article attributes the scalar-block conclusion to Wigner–Eckart. The local argument instead uses the completed rotational-scalar block supplier and the static ULE formula. Isotropy of the bath alone does not establish the separately stated covariance of the matter coupling or rotational invariance of $H_{S}$ .

Depends on

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 11, opening continuation of Supplement S2