Coherent states for ladder operators of general order related to exceptional orthogonal polynomials
Bibliographic record
Abstract
We construct the coherent states of general order, m, for the ladder operators c(m) and c(dagger)(m), which act on rational deformations of the harmonic oscillator. The position wavefunctions of the eigenvectors involve type III Hermite exceptional orthogonal polynomials. We plot energy expectations, time-dependent position probability densities for the coherent states and for the even and odd cat states, Wigner functions, and Heisenberg uncertainty relations. We find generally non-classical behaviour, with one exception: there is a regime of large magnitude of the coherent state parameter, z, where the otherwise indistinct position probability density separates into m + 1 distinct wavepackets oscillating and colliding in the potential, forming interference fringes when they collide. The Mandel Q parameter is calculated to find sub-Poissonian statistics, another indicator of non-classical behaviour. We plot the position standard deviation and find squeezing in many of the cases. We calculate the two 'photon' number probability density for the output state when the m = 4, mu = -5 coherent states (where mu labels the lowest weight in the superposition) are placed on one arm of a beamsplitter. We find that it does not factorize, again indicating non-classical behaviour. Calculation of the linear entropy for this beamsplitter output state shows significant entanglement, another non-classical feature. We also construct linearized versions, (c) over tilde (m), of the annihilation operators and their coherent states and calculate the same properties that we investigate for the c(m) coherent states. For these we find similar behaviour to the c(m) coherent states, at much smaller magnitudes of z, but comparable average energies.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".