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One-jump span in the nine-dimensional lowering model

Statement

Consider the purely finite-matrix model on $C^{9}$ with the specified orthonormal basis $e_{J, m}$ , $J = 0, 1, 2$ , $m = - J, \dots, J$ , using the finite operator and adjoint conventions. Let $γ > 0$ , $Δ \in R$ , and $E_{0}, E_{1}, E_{2} \in R$ . For $J = 1, 2$ , let $b_{J, m, q}$ be the normalized downward coefficients of the low-spin support lemma. Define $H e_{J, m} = (E_{J} + Δ J m) e_{J, m}$ . For each $q \in {- 1, 0, 1}$ , define $L_{q}$ by $L_{q} e_{J, m} = \sqrt{γ} b_{J, m, q} e_{J - 1, m + q}$ when $J \in {1, 2}$ and $∣ m + q ∣ \leq J - 1$ , and by $L_{q} e_{J, m} = 0$ otherwise, including $J = 0$ ; extend these basis assignments complex-linearly. Put $G = - i H - \sum_{q = - 1}^{1} L_{q}^{†} L_{q} / 2$ , $D (u) = e^{u G}$ . At every $t > 0$ , the complex linear span of $D (t - s) L_{q} D (s)$ for $q = - 1, 0, 1$ , $0 < s < t$ , has dimension six if $Δ = 0$ , and twelve if $Δ \neq 0$ .

Statement references

Finite operator and ket conventions

Nonvanishing downward coefficients for spin one and two

Distinct complex exponentials have full interval Gram rank

Finite no-jump and ordered-jump expansion

The exponential series of a bounded operator

Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S

Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

Facts & Assumptions

[F1]

The coefficients are nonzero at every allowed downward entry and their squared sums equal one. Nonvanishing downward coefficients for spin one and two

[F2]

Distinct complex exponential profiles on any positive-length interval are independent. Distinct complex exponentials have full interval Gram rank

[F3]

The one-jump amplitude in the finite GKSL expansion is $D (t - s) L_{q} D (s)$ . Finite no-jump and ordered-jump expansion

[F4]

The specified orthonormal coordinates identify operators with complex matrices; the adjoint is the conjugate transpose, and products use matrix multiplication. Finite operator and ket conventions

[F5]

An inner product on a complex vector space induces the norm $∥ x ∥ = \sqrt{⟨ x, x ⟩}$ . The induced length is a norm

[F6]

A real or complex normed space admitting an ordered basis of finite length is Banach. Every finite-dimensional normed space is Banach

[F7]

A linear map between normed spaces over the same scalar field is bounded when its domain admits an ordered basis of finite length. A linear map from a finite-dimensional normed space is bounded

[F8]

For a bounded linear operator $A$ on a real or complex Banach space, $e^{u A} = \sum_{n = 0}^{\infty} u^{n} A^{n} / n!$ converges absolutely in operator norm for real $u$ , and these exponentials satisfy $e^{u A} e^{v A} = e^{(u + v) A}$ and $e^{0 A} = I$ . The exponential series of a bounded operator

[F9]

In a complex vector space, the span of a family is the smallest complex linear subspace containing it. The dimension over $C$ of a space with a finite basis is the number of vectors in such a basis. Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

[F10]

For $z \in C$ , write $e^{z} = \exp z = \sum_{n = 0}^{\infty} z^{n} / n!$ , using the complex exponential defined by its everywhere-convergent power series. The complex exponential by its power series

[F11]

If $\sum a_{n} = A$ and $\sum b_{n} = B$ converge absolutely in $C$ , their Cauchy product with coefficients $c_{n} = \sum_{k = 0}^{n} a_{k} b_{n - k}$ converges absolutely to $A B$ . The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums

Proof

Given: The finite data and hypotheses in the Statement.

Goal: Establish the asserted complex span dimensions.

1.1givenF1F3F4algebra

In the matrix conventions [F4], the zero assignments exclude every forbidden transition. Orthogonality of the nonzero outputs at fixed $q$ and [F1] give $\sum_{q} L_{q}^{†} L_{q} = γ P_{e x c}$ , where $P_{e x c}$ projects onto $J = 1, 2$ . Thus $G e_{J, m} = g_{J, m} e_{J, m}$ , where $g_{0, 0} = - i E_{0}$ and $g_{J, m} = - γ / 2 - i (E_{J} + Δ J m)$ for $J = 1, 2$ . The real diagonal entries of $H$ make it Hermitian, and the three $L_{q}$ are finite matrices, as required by [F3].

2.1F4F5F6F7F8F10F11step 1.1algebra

Give $X = C^{9}$ the induced norm $∥ x ∥_{2} = \sqrt{\sum_{J, m} ∣ x_{J, m} ∣^{2}}$ . This is a complex norm by [F4] and [F5], and the specified nine-vector ordered basis makes $X$ Banach by [F6]. Matrix multiplication makes $G : X \to X$ complex-linear, so [F7] makes it bounded. Apply [F8] with $A = G$ . Since $G^{n} e_{J, m} = g_{J, m}^{n} e_{J, m}$ for every $n \geq 0$ , evaluating its partial sums on $e_{J, m}$ and passing to the limit gives $D (u) e_{J, m} = e^{u g_{J, m}} e_{J, m}$ for real $u$ ; evaluation is continuous because $∥ B e_{J, m} ∥_{2} \leq ∥ B ∥$ in the induced operator norm. For the scalar series, apply [F5]--[F8] to $C$ with basis $(1)$ and the complex-linear map $M_{z} (w) = z w$ . Since $∥ M_{z}^{n} ∥ = ∣ z ∣^{n}$ , [F8] at real parameter $1$ gives absolute convergence of $\sum z^{n} / n!$ , and evaluation of the operator partial sums at $1$ identifies their limit with $e^{z}$ in [F10]. The group identity for $M_{z}$ at real parameters $1, - 1$ gives $e^{z} e^{- z} = 1$ . For arbitrary complex $z, w$ , both scalar series converge absolutely by this specialization, so [F11] applies with $a_{n} = z^{n} / n!$ and $b_{n} = w^{n} / n!$ . Its coefficient is $c_{n} = \sum_{k = 0}^{n} z^{k} w^{n - k} / (k! (n - k)!) = (z + w)^{n} / n!$ by finite binomial expansion. Therefore $e^{z} e^{w} = \sum_{n = 0}^{\infty} (z + w)^{n} / n! = e^{z + w}$ by [F10].

3.1givenF1F3F4step 1.1step 2.1algebra

Each allowed $(J, m) \to (J - 1, m + q)$ coordinate of [F3] is $\sqrt{γ} b_{J, m, q} e^{g_{J - 1, m + q} t} e^{(g_{J, m} - g_{J - 1, m + q}) s}$ , with nonzero prefactor. Indeed step 2.1 evaluates the two diagonal propagators, its scalar product identity combines their factors, and [F1] together with $γ > 0$ makes the remaining coefficient nonzero.

4.1F1F2F4F9step 1.1step 3.1algebra∎

For a fixed $q$ there is one $J = 1$ entry, namely $m = - q$ , with exponent of real part $- γ / 2$ . There are three $J = 2$ entries, with $m = - 1 - q, - q, 1 - q$ , each with exponent $- i (E_{2} - E_{1} + Δ (m - q))$ of real part zero. At $Δ = 0$ the latter three share one exponent, distinct from the $J = 1$ exponent by its real part, giving dimension two by [F2]. At $Δ \neq 0$ their imaginary parts are distinct, and all four exponents are distinct, giving dimension four. Different $q$ have disjoint matrix-unit supports: the union of bases for their spans is independent and spans the full family. Its size gives dimension six or twelve in the sense of [F9].

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