About Classical to Quantum Weyl Correspondence

Wünsche, Alfred

doi:10.4236/apm.2017.710034

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…This leads to the representation of (C.7) The same relation was derived in other way in [23] (Equation (5.16)) and it was known also without using Laguerre 2D polynomials.…”

Section: Appendix B Completeness Relation Over Heisenberg-weyl Group mentioning

confidence: 86%

Wigner Quasiprobability with an Application to Coherent Phase States

Wünsche¹

2018

APM

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Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.

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“…This leads to the representation of (C.7) The same relation was derived in other way in [23] (Equation (5.16)) and it was known also without using Laguerre 2D polynomials.…”

Section: Appendix B Completeness Relation Over Heisenberg-weyl Group mentioning

confidence: 86%

Wigner Quasiprobability with an Application to Coherent Phase States

Wünsche¹

2018

APM

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“…The expectation values of symmetrically ordered power operators { } †k k a a  are connected with †k k a a by (see, e.g., Equation (7.6) in [22] …”

Section: Expectation Values Of Powers Of Number Operator and Related mentioning

confidence: 99%

“…W α α is to make first the normal ordering of the operator involved in the representation (4.1) that leads to [22] ( ) …”

Section: Bargmann Representation and Quasiprobabilities For Squeezed mentioning

confidence: 99%

Squeezed Coherent States in Non-Unitary Approach and Relation to Sub- and Super-Poissonian Statistics

Wünsche¹

2017

APM

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Evolution of quantum states via Weyl expansion in dissipative channel*

Rao

Kuang

2019

Chinese Phys. B

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Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.

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About Classical to Quantum Weyl Correspondence

Cited by 3 publications

References 31 publications

Wigner Quasiprobability with an Application to Coherent Phase States

Wigner Quasiprobability with an Application to Coherent Phase States

Squeezed Coherent States in Non-Unitary Approach and Relation to Sub- and Super-Poissonian Statistics

Evolution of quantum states via Weyl expansion in dissipative channel*

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