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Ideal one-mode intensity correlations separate model states

Physical scope

Exact model predictions for coherent-mode, fixed-number and geometric one-mode preparations under ideal wavepacket number counting. The geometric distribution is the specified single-mode thermal occupation law; no universal classification of radiation or empirical outcome is asserted.

Statement

Assume the ideal wavepacket counting postulate for a normalized mode $v$ , and let $p_{n}$ be the resulting count law of a prepared state. Write $μ : = \sum_{n = 0}^{\infty} n p_{n}, F_{2} : = \sum_{n = 0}^{\infty} n (n - 1) p_{n}, g_{v}^{(2)} (0) : = F_{2} / μ^{2}$ when $0 < μ < \infty$ and $F_{2} < \infty$ . Let $J_{v} : F_{s} (C v) \hat{\otimes} F_{s} (v^{⊥}) \to F_{s} (H_{T})$ be the one-mode Fock unitary, let $∣ n ⟩_{v} = v^{\otimes n}$ with $∣ 0 ⟩_{v} = 1$ , and let $Ω_{v^{⊥}}$ be the vacuum in $F_{s} (v^{⊥})$ . Put $ψ_{n} : = J_{v} (∣ n ⟩_{v} \otimes Ω_{v^{⊥}})$ . For a unit vector $ψ$ , $g_{v}^{(2)} (0; ψ)$ denotes the postulate's value for the pure density operator $∣ ψ ⟩ ⟨ ψ ∣$ . For $α \in C ∖ {0}$ put $λ = ∣ α ∣^{2}$ , $Z_{λ} = \sum_{n = 0}^{\infty} λ^{n} / n!$ , and define the coherent-mode vector $ψ_{α} : = Z_{λ}^{- 1 / 2} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} ψ_{n} .$ For $m \geq 1$ , the number preparation is $ψ_{m}$ . For $0 < r < 1$ , the geometric one-mode state is $ρ_{r} : = (1 - r) \sum_{n = 0}^{\infty} r^{n} ∣ ψ_{n} ⟩ ⟨ ψ_{n} ∣ .$ Then the ideal model predictions are $g_{v}^{(2)} (0; ψ_{α}) = 1, g_{v}^{(2)} (0; ψ_{m}) = 1 - \frac{1}{m}, g_{v}^{(2)} (0; ρ_{r}) = 2.$ The final state has the specified single-mode geometric occupation law; these conclusions do not classify all radiation as coherent, number, or thermal.

Statement references

Ideal wavepacket number counting

One-mode photon number operator and count PVM

Density operators and scalar spectral probability

The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$

The real exponential function and the number $e$ by a power series

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Glauber (1963)

Steck, Quantum and Atom Optics

Facts & Assumptions

Given: The ideal count law $(p_{n})$ of the postulate, a normalized selected mode $v$ , and the stated finite-moment hypotheses.

[A1]

The ideal wavepacket postulate assigns $p_{n} = tr (ρ P_{n} (v))$ , defines $g_{v}^{(2)} (0) = F_{2} / μ^{2}$ , and identifies $F_{2}$ with the spectral sum for the normally ordered factorial correlation. Ideal wavepacket number counting

[F1]

The one-mode number projections are $P_{n} (v)$ , the unit occupation vector $ψ_{n}$ has $P_{j} (v) ψ_{n} = 1_{j = n} ψ_{n}$ , and $N_{v} (N_{v} - I) = a^{†} (v)^{2} a (v)^{2}$ on the finite-particle core. One-mode photon number operator and count PVM

[F2]

For real $λ$ , $\exp (λ) = \sum_{n = 0}^{\infty} λ^{n} / n!$ ; for $λ > 0$ this positive series is finite and at least $1$ . The real exponential function and the number $e$ by a power series

[F3]

Factorials satisfy $n! = n (n - 1)!$ for $n \geq 1$ and $n (n - 1) / n! = 1 / (n - 2)!$ for $n \geq 2$ . The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$

[F4]

For $0 < r < 1$ , $\sum_{n = 0}^{\infty} r^{n} = 1 / (1 - r)$ . For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges

[F5]

Nonnegative sums on countable product sets may be regrouped by Tonelli's theorem. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F6]

A unit vector defines a pure density operator, and a countable weighted sum of orthogonal rank-one states with nonnegative weights summing to one is a density operator; its PVM probabilities are the corresponding count law. We use only the supplier's density and scalar-probability clauses, which require Countable Choice. Density operators and scalar spectral probability

Proof

1.1A1F1F2F3F6algebra

The coefficients of $ψ_{α}$ are square-summable because their squared norms sum to $Z_{λ}^{- 1} \sum_{n} λ^{n} / n! = 1$ by [F2]; the orthogonal one-mode sectors therefore make $ψ_{α}$ a unit vector and [F6] makes its rank-one state a density operator. Its count probabilities are $p_{n} = Z_{λ}^{- 1} λ^{n} / n!$ by [A1] and [F1]. Shifting indices in the nonnegative exponential series gives $μ = Z_{λ}^{- 1} \sum_{n \geq 1} \frac{n λ^{n}}{n!} = λ, F_{2} = Z_{λ}^{- 1} \sum_{n \geq 2} \frac{n (n - 1) λ^{n}}{n!} = λ^{2},$ so $g_{v}^{(2)} (0) = 1$ . The same calculation shows both required moments are finite.

1.2A1F1algebra

For $ψ_{m}$ , [F1] gives $p_{n} = 1_{n = m}$ , hence $μ = m$ and $F_{2} = m (m - 1)$ . Since $m \geq 1$ , the denominator is positive, and substitution into [A1] gives $g_{v}^{(2)} (0) = m (m - 1) / m^{2} = 1 - 1 / m$ .

2.1A1F4F5F6algebra∎

The weights $(1 - r) r^{n}$ are nonnegative and sum to one by [F4], so [F6] makes $ρ_{r}$ a density operator and [A1] gives the geometric count law $p_{n} = (1 - r) r^{n}$ . By Tonelli [F5], ${(\sum_{j \geq 0} r^{j})}^{2} = \sum_{i, j \geq 0} r^{i + j} = \sum_{n \geq 0} (n + 1) r^{n} .$ Therefore $\sum_{n \geq 0} n r^{n} = (1 - r)^{- 2} - (1 - r)^{- 1} = r / (1 - r)^{2}$ . Applying Tonelli to the triple product and counting the $(i, j, k) \in N_{0}^{3}$ with $i + j + k = n - 2$ gives $r^{2} {(\sum_{j \geq 0} r^{j})}^{3} = \sum_{n \geq 2} \frac{n (n - 1)}{2} r^{n},$ because that finite set has $\sum_{i = 0}^{n - 2} (n - 1 - i) = n (n - 1) / 2$ elements. Hence $\sum_{n \geq 0} n (n - 1) r^{n} = 2 r^{2} / (1 - r)^{3}$ . Multiplying by $1 - r$ gives $μ = r / (1 - r) > 0$ and $F_{2} = 2 r^{2} / (1 - r)^{2} < \infty$ , so [A1] yields $g_{v}^{(2)} (0) = 2$ .

Ideal one-mode intensity correlations separate model states

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources