On the Mehler formula for Hermite polynomials

Viskov, O. V.

doi:10.1134/s1064562408010018

Cited by 6 publications

(4 citation statements)

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“…We also plot the entanglement of an ideal two-mode squeezed state (39), using the mapping (41) to convert the squeezing parameter r to interaction time. For the ideal two-mode squeezed state, the entanglement increases towards a maximally entangled state, as can be directly seen from (39) where there is an equal superposition of pairs of spin Fock states. Such a state is never attained in our protocol due to the binomial factors as discussed above.…”

Section: Entanglementmentioning

confidence: 97%

“…One difference between the approximated distribution and the atomic entangled state is that probability distri- butions of the former tend to broaden as the number correlations become stronger, while the latter distributions stay within a fixed envelope. This occurs because in the infinitely squeezed limit the mapped two-mode squeezed state (39) is…”

Section: Analytic Approximation Of Probability Densitiesmentioning

confidence: 99%

“…where λ = tanh(r). Using the completeness relation for Hermite functions, the sum over k x can be expressed in terms of Mehler kernels [39], which states that…”

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confidence: 99%

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Light-mediated non-Gaussian entanglement of atomic ensembles

Pettersson

Byrnes

2017

Phys. Rev. A

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We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature 413, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein-Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find this produces a state with similar characteristics as a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the two-mode squeezed state characteristics start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock state probability distributions, Husimi Q distributions, and non-local entanglement between the ensembles. We compare and connect several quantities obtained using the Holstein-Primakoff approach and our exact time evolution methods.

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Section: Entanglementmentioning

confidence: 97%

Section: Analytic Approximation Of Probability Densitiesmentioning

confidence: 99%

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Light-mediated non-Gaussian entanglement of atomic ensembles

Pettersson

Byrnes

2017

Phys. Rev. A

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“…which was proved combinatorially by Foata [97] (see also [98,99]). For several extensions and generalizations of the Mehler formula (39), the reader is referred to [100][101][102][103][104][105]).…”

Section: Generating Functions Of Orthogonal Polynomialsmentioning

confidence: 99%

Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions

Srivástava

2022

Mathematics

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In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.

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