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arXiv · 1803.01318

Coherent states for ladder operators of general order related to exceptional orthogonal polynomials

Scott E. Hoffmann·Véronique Hussin·Ian Marquette·Yao-Zhong Zhang
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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, $\tilde c(m)$, of the annihilation operators and their coherent states and calculate the same properties that we investigate for the 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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Scott E. Hoffmann, Véronique Hussin, Ian Marquette, Yao-Zhong Zhang. 2018-03-04. Coherent states for ladder operators of general order related to exceptional orthogonal polynomials. https://doi.org/10.1088/1751-8121/aacb3b

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