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Wigner Function of Observed Quantum Systems

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arxiv 2501.03885 v2 pith:X6M6ARCC submitted 2025-01-07 quant-ph

Juan Camilo López Carreño This is my paper · ORCID
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The Wigner function was introduced as an attempt to describe quantum-mechanical fields with the tools inherited from classical statistical mechanics. In particular, it is widely used to describe the properties of radiation fields. In fact, a useful way to distinguish between classical and nonclassical states of light is to ask whether their Wigner function has a Gaussian profile or not, respectively. In this paper, we use the basis of Fock states to provide the closed-form expression for the Wigner function of an arbitrary quantum state. Thus, we provide the general expression for the Wigner function of a squeezed Fock, coherent and thermal states, with an arbitrary squeezing parameter. Then, we consider the most fundamental quantum system, Resonance Fluorescence, and obtain closed-form expressions for its Wigner function under various excitation regimes. With them, we discuss the conditions for obtaining a negative-valued Wigner function and the relation it has with population inversion. Finally, we address the problem of the observation of the radiation field, introducing physical detectors into the description of the emission. Notably, we show how to expose the quantumness of a radiation field that has been observed with a detector with finite spectral resolution, even if the observed Wigner function is completely positive.

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    (21), in which only two elements are nonzero: 𝜌0 0= 𝛾𝜎 Γ𝜎 and 𝜌1 1= 𝑃𝜎 Γ𝜎 , (A3) so that, following Eq

    Wigner function of the 2LS with incoherent driving In this case, the steady-state density matrix of the 2LS is given in Eq. (21), in which only two elements are nonzero: 𝜌0 0= 𝛾𝜎 Γ𝜎 and 𝜌1 1= 𝑃𝜎 Γ𝜎 , (A3) so that, following Eq. (A1), the Wigner function is given by 𝑊(𝛼)= 𝜌0 0𝑊0 0(𝛼)+ 𝜌1 1𝑊1 1(𝛼), (A4) which,usingthecoefficientsspelledoutinEq.(A2),becomes ...

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    (23), so its elements can be written as 𝜌0 0=1− 𝑛𝜎 𝜌1 0= ⟨𝜎⟩ (A7a) 𝜌0 1= ⟨𝜎⟩∗ 𝜌1 1=𝑛𝜎

    Wigner function of the 2LS with coherent driving In this case, the steady-state density matrix of the 2LS is given in Eq. (23), so its elements can be written as 𝜌0 0=1− 𝑛𝜎 𝜌1 0= ⟨𝜎⟩ (A7a) 𝜌0 1= ⟨𝜎⟩∗ 𝜌1 1=𝑛𝜎. (A7b) Following Eq. (A1), the Wigner function is then given by 𝑊(𝛼)= 𝜌0 0𝑊0 0(𝛼)+ 𝜌1 0𝑊0 1(𝛼)+ 𝜌0 1𝑊1 0(𝛼)+ 𝜌1 1𝑊1 1(𝛼), = 2 𝜋𝑒−2𝑟2 (1− 𝑛𝜎)+ 2 𝜋𝑒−2𝑟...

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