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Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation

Physical scope

The conditional number and energy statistics of a declared normalisable mode of the adopted radiation sector: the n-quanta mode state has sharp count n; the mode-coherent state has the Poisson count law of dimensionless parameter |z|^2 with mean and variance |z|^2; the energy means and variances of these preparations are computed from the mode profile, are strictly positive in variance for n>=1 and z!=0 (the vacuum cases have zero variance), and the full-sector Gibbs operator e^(-beta H_f) is not trace class for beta>0 in J^-1. This excludes only its trace-normalized Gibbs density form; other preparations are outside the theorem. The result is conditional on P1, P2 and the adopted radiation model, uses the ideal counting model only, reports no measured count and asserts no thermalisation or temperature identification.

Statement

Adopt the state and composition postulate P1 and the observable and probability postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), the radiation extension postulate (Radiation extension of the framework: transverse mode and photon assignments), the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments and the declared ideal counting model The declared ideal counting, energy-window and analyser detection models of the radiation sector. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice). Fix a unit mode vector $φ \in h$ , a dimensionless label $z \in C$ , and write $U : = U_{φ}$ for the fixed-mode isometry of The fixed-mode isometry of Fock space and the transfer of number and ladder operators. Then:

(a) for every $n \in N$ the $n$ -quanta mode vector $U e_{n} = φ^{\otimes n}$ is a unit vector of the $n$ -photon sector; the count law of the declared ideal total-number detector in the preparation of its ray is $p_{m} = δ_{m n}$ , and the number readout is $n$ with probability one;

(b) the count law of the mode-coherent preparation $U ψ_{z}$ of label $z$ is the Poisson law $p_{n} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!} (n \in N),$ with mean and variance $\sum_{n \in N} n p_{n} = ∣ z ∣^{2}, \sum_{n \in N} n^{2} p_{n} - (\sum_{n \in N} n p_{n})^{2} = ∣ z ∣^{2};$ in particular the mode-coherent preparation is not a number state unless $z = 0$ ;

(c) if $φ \in D (ε_{h})$ , with unit-mode coordinates $φ (k)$ in $m^{3 / 2}$ under $d^{3} k$ as in clause (b) of The radiation sector of the framework: modes and the number, energy and momentum assignments, and writing $\overset{ˉ}{ε} : = ⟨ ε_{h} φ, φ ⟩ = ℏ c \int ∣ k ∣ ∥ φ (k) ∥^{2} d^{3} k$ , the energy means of the assigned free-field energy observable are $⟨ H_{f} ⟩_{U e_{n}} = n \overset{ˉ}{ε}, ⟨ H_{f} ⟩_{U ψ_{z}} = ∣ z ∣^{2} \overset{ˉ}{ε};$

(d) under the same condition $φ \in D (ε_{h})$ , writing $\overline{ε^{2}} : = ∥ ε_{h} φ ∥^{2}$ , the energy variances are ${Var}_{U e_{n}} (H_{f}) = n (\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2}), {Var}_{U ψ_{z}} (H_{f}) = ∣ z ∣^{2} \overline{ε^{2}};$ the first is strictly positive for every $n \geq 1$ and the second is strictly positive for every $z \neq 0$ . Thus $U e_{n}$ has non-sharp energy for $n \geq 1$ and $U ψ_{z}$ has non-sharp energy for $z \neq 0$ . At $n = 0$ or $z = 0$ the corresponding preparation is the vacuum $Ω$ , with $H_{f} Ω = 0$ and energy law $δ_{0}$ ;

(e) full-sector Gibbs limitation: let $β > 0$ have SI unit $J^{- 1}$ , so that $β H_{f}$ is dimensionless. The bounded operator $e^{- β H_{f}}$ is not compact and hence not trace class; therefore it does not define a density operator by trace normalization on the full transverse continuum sector. This conclusion rules out only that trace-normalized full-sector Gibbs form. Other mode-selected, regularized or otherwise separately defined preparations are outside this theorem; no identification of $β$ with an inverse thermodynamic temperature and no thermal interpretation of any state is asserted here;

(f) all assertions are conditional on P1, P2 and the adopted radiation model and use only the declared ideal counting model. No measured count, frequency, efficiency or temperature is asserted, and no sampling model is introduced.

Facts & Assumptions

Given: the postulates P1, P2 and the adopted radiation model with transverse space $h$ , Fock space $F (h)$ , sectors $H_{n}^{+}$ , projections $Π_{n}$ , number operator $N$ , energy $H_{f}$ with $ε (k) = ℏ c ∣ k ∣$ , a unit mode vector $φ$ with coordinate units $m^{3 / 2}$ under $d^{3} k$ , a dimensionless label $z \in C$ and the isometry $U = U_{φ}$ .

[F1]

Fixed-mode isometry. $U e_{n} = φ^{\otimes n} \in ran P_{+}^{(n)}$ , $(U e_{m}, U e_{n})$ is orthonormal, $∥ U ψ ∥ = ∥ ψ ∥$ , $N U ψ = U N^{'} ψ$ for $ψ \in D (N^{'})$ , $A_{-} (φ) U ψ = U A_{-} (1) ψ$ for $ψ \in D (A_{-} (1))$ , and for the transverse case with $φ \in D (ε_{h})$ , $⟨ H_{f} U e_{n}, U e_{n} ⟩ = n ⟨ ε_{h} φ, φ ⟩$ and $⟨ H_{f} U ψ, U ψ ⟩ = (\sum_{n} n ∣ ψ_{n} ∣^{2}) ⟨ ε_{h} φ, φ ⟩$ for $\sum_{n} n^{2} ∣ ψ_{n} ∣^{2} < \infty$ (The fixed-mode isometry of Fock space and the transfer of number and ladder operators clauses (a),(b),(d)).

[F2]

Fock sectors and number atoms. $Π_{n}$ is the orthogonal projection onto the coordinate- $n$ subspace, the ranges are pairwise orthogonal with strong sum $I$ , and for the number operator $N$ one has $N ι_{n} (u) = n ι_{n} (u)$ for every $u \in H_{n}^{+}$ ; the count law of the declared ideal total-number detector is $p_{n} (ρ) = tr (ρ Π_{n})$ (Fock sector projections, Number operator domain clause (b), The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (b)).

[F3]

Coherent vector coefficients. $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e_{n}$ is unit with $⟨ e_{n}, ψ_{z} ⟩ = e^{- ∣ z ∣^{2} / 2} \frac{{\overset{ˉ}{z}}^{n}}{\sqrt{n!}}$ and $∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!}$ (Fixed-mode number law clauses (b),(c)).

[F4]

Energy moments on mode sectors. For a unit $φ \in D (ε_{h})$ , put $\overset{ˉ}{ε} = ⟨ ε φ, φ ⟩$ and $\overline{ε^{2}} = ∥ ε φ ∥^{2}$ . Then $Φ_{n} = φ^{\otimes n} \in D (T_{n})$ , $⟨ T_{n} Φ_{n}, Φ_{n} ⟩ = n \overset{ˉ}{ε}$ and $∥ T_{n} Φ_{n} ∥^{2} = n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}$ . The last is the spectral second moment, not an assertion of membership in $D (T_{n}^{2})$ (The fixed-mode isometry of Fock space and the transfer of number and ladder operators clause (d)).

[F5]

Exponential series. For every $x \geq 0$ , $\sum_{n} \frac{x^{n}}{n!} = e^{x} < \infty$ , and the series converges absolutely (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument).

[F6]

Non-compactness of the free Gibbs operator. For every $β > 0$ read with SI unit $J^{- 1}$ , the operator $e^{- β H_{f}}$ is not compact and in particular not trace class (Transverse continuum free Fock Hamiltonian clause (d)).

Proof

1.1F1F2algebra

[Number states] By [F1] each $U e_{n} = φ^{\otimes n}$ is a unit vector of the sector $ran P_{+}^{(n)}$ ; by [F2] $N$ acts on it as $n$ times itself, so the number readout is $n$ with probability one by the atom law of P2 (Atoms of the spectral measure and eigenvalues of a self-adjoint operator), and the count law of the declared detector is $p_{m} = tr (T_{U e_{n}, U e_{n}} Π_{m}) = ⟨ Π_{m} U e_{n}, U e_{n} ⟩ = δ_{m n}$ . This proves (a).

1.2F1F2F3algebra

[Poisson law] For the mode-coherent preparation $U ψ_{z}$ the count law is $p_{n} = ⟨ Π_{n} U ψ_{z}, U ψ_{z} ⟩ = ⟨ U Π_{n}^{'} ψ_{z}, U ψ_{z} ⟩ = ⟨ Π_{n}^{'} ψ_{z}, ψ_{z} ⟩ = ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!},$ using [F1] for the sector transfer and the isometry, and [F3] for the coefficient.

2.1F3F5step 1.2algebra

[Poisson moments] Put $x : = ∣ z ∣^{2} \geq 0$ . By [F5], $\sum_{n \geq 1} n \frac{x^{n}}{n!} = x \sum_{m \geq 0} \frac{x^{m}}{m!} = x e^{x}$ and $\sum_{n \geq 2} n (n - 1) \frac{x^{n}}{n!} = x^{2} e^{x}$ , so the mean is $e^{- x} x e^{x} = x$ and the second moment is $e^{- x} (x^{2} + x) e^{x} = x^{2} + x$ ; hence the variance is $x$ . For $z \neq 0$ the law is supported on more than one value, so the preparation is not a number state. This proves (b).

3.1F1F2F4step 2.1algebra

[Energy means] The sector diagonality of $H_{f}$ and orthogonality of the sectors give, for the coherent preparation, $⟨ H_{f} U ψ_{z}, U ψ_{z} ⟩ = \sum_{n \in N} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} ⟨ T_{n} φ^{\otimes n}, φ^{\otimes n} ⟩ = (\sum_{n} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} n) \overset{ˉ}{ε} = ∣ z ∣^{2} \overset{ˉ}{ε},$ where the middle identity is [F1] and [F4] and the last is the Poisson mean of step 2.1; the number-state value is the $n$ -th instance of [F1]. The Poisson second number moment is finite by step 2.1, so [F1] places $U ψ_{z}$ in the actual domain of $H_{f}$ and justifies the graph-truncation passage. The mean exists because both sides are finite: $\overset{ˉ}{ε} < \infty$ by $φ \in D (ε_{h})$ and Cauchy-Schwarz, and $\sum_{n} n ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = x < \infty$ . This proves (c).

4.1F1F4step 2.1step 3.1algebra

[Energy variances] For the number state, $∥ H_{f} U e_{n} ∥^{2} = ∥ T_{n} φ^{\otimes n} ∥^{2} = n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}$ by [F4], so the variance is $n (\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2})$ , which is strictly positive for $n \geq 1$ by clause (d) of Restricted momentum and energy multipliers of the transverse one-photon space (the energy multiplier has no eigenvalues, so $\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2} = {Var}_{φ} (ε) > 0$ ). For the coherent preparation, its Poisson second moment and [F1] give $U ψ_{z} \in D (H_{f})$ ; the orthogonal-sector norm expansion therefore gives $∥ H_{f} U ψ_{z} ∥^{2} = \sum_{n} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} (n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}) = x \overline{ε^{2}} + x^{2} {\overset{ˉ}{ε}}^{2},$ using the Poisson first and second factorial moments of step 2.1; subtracting the square of the mean $x \overset{ˉ}{ε}$ from step 3.1 gives $x \overline{ε^{2}}$ , which is strictly positive for $x = ∣ z ∣^{2} > 0$ because $\overline{ε^{2}} = ∥ ε_{h} φ ∥^{2} > 0$ : $ε_{h} φ$ vanishes only where $φ$ vanishes off the null set ${k = 0}$ , so a unit $φ$ cannot satisfy $ε_{h} φ = 0$ in $h$ . For $n = 0$ and $z = 0$ , respectively, $U e_{0} = U ψ_{0} = Ω$ ; the vacuum sector has $T_{0} = 0$ , hence $H_{f} Ω = 0$ , and the atom/eigenvalue law of P2 gives the sharp energy law $δ_{0}$ . This proves (d), including its vacuum exceptions.

5.1F6step 1.1step 1.2step 2.1step 3.1step 4.1∎

[Full-sector Gibbs limitation and assembly] A trace-normalized density operator proportional to $e^{- β H_{f}}$ would require that operator to be trace class, which [F6] excludes for every stated $β$ . This proves only the full-sector Gibbs limitation in (e). Steps 1.1 and 1.2 prove (a) and (b), steps 3.1 and 4.1 prove (c) and (d), and no step introduces a detector response, a sampling model or an empirical premise, which is (f).

Remarks

Poisson versus geometric. The mode-coherent count law is Poisson, with variance equal to its mean; the declared single-mode Gibbs-type preparation of Number statistics of a single declared mode: sharp, Poisson and geometric preparations ↗ has a geometric count law with larger variance at the same mean. Both are conditional model computations; distinguishing them in a laboratory requires a declared detector, preparation and sampling model and reported data.
No temperature identification. The parameter $β$ in (e) has unit $J^{- 1}$ so that $β H_{f}$ is dimensionless. Identifying it with $(k_{B} T)^{- 1}$ is an additional physical assumption not made here.

Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation

Statement

Facts & Assumptions

Proof

Remarks

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