The concept of the Wigner function is used to construct a semi-classical kinetic theory describing the evolution of the axial current phase-space density of spin-12 particles in the relaxation time ...approximation. The resulting approach can be used to study spin polarization effects in relativistic matter, in particular, in heavy-ion collisions. An expression for the axial current based on the classical treatment of spin is also introduced and we show that it is consistent with earlier calculations using Wigner functions. Finally, we derive non-equilibrium corrections to the spin tensor, which are used to define, for the first time, the structure of spin transport coefficients in relativistic matter.
Many novel quantum phenomena emerge in non-equilibrium relativistic quantum matter under extreme conditions such as strong magnetic fields and rotations. The quantum kinetic theory based on Wigner ...functions in quantum field theory provides a powerful and effective microscopic description of these quantum phenomena. In this article we review some of recent advances in the quantum kinetic theory and its applications in describing these quantum phenomena.
The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of ...distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten-p-norm of the generator of quantum dynamics. Since computing Schatten-p-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.
Wigner function is an intuitive and powerful tool for understanding quantum systems in terms of functions on phase space. However, it is uniquely defined only for continuous systems, and there is no ...universally accepted Wigner function for arbitrary discrete systems. Several different versions of discrete Wigner functions have been developed, each with its own usage. The basic reason for this phenomenon lies in that for discrete systems, the number-theoretic aspects of the system dimension come into play, and many properties of the system depend crucially on the dimension. The purpose of this work is twofold: First, we present a concise review of continuous and discrete Wigner functions. Second, we establish some connections between continuous and discrete Wigner functions via the Gottesman-Kitaev-Preskill (GKP) encoding, which encodes a qudit into a continuous system and allows us to use the tools and techniques developed for continuous quantum systems to study discrete quantum systems. We formalize GKP encoding from the perspective of linear mapping going beyond the Hilbert space framework, and employ it to define GKP-induced discrete Wigner function. We further reveal its basis properties, and offer a new perspective for understanding discrete Wigner functions in a unified fashion.
In this paper, we show that by partial tracing the pure state in higher dimension one can deduce the mixed state of light field. In this paper the pure state in higher dimension is a three-mode ...squeezed state S23σS31τ000000S31−1τS23−1σ, while the mixed state’s density operator is sech2σsech2τea2†a3†tanhσsech2σtanh2τa3†a30220ea2a3tanhσ, which involves both squeezing and chaotic properties of optical field. We further explain how such kind of light field can be generated by passing a two-mode squeezed light though a single-mode dissipative channel, its Wigner function is discussed by using the coherent state representation. Our work provides an important method, which can be used to theoretically find more optical fields in quantum communication and quantum optics.
•Theoretical tools to describe generation, manipulation and characterization of optical quantum states are provided.•Useful examples and exercises are offered.•Main optical states, beam splitter, ...single- and two-mod squeezing operations are discussed.•The reader is introduced to the normal, antinormal and symmetric operator ordering and the characteristic functions.•A simple but useful model of photodetection, the homodyne detection and the homodyne tomography are described.
In this brief tutorial we provide the theoretical tools needed to describe the generation, manipulation and characterization of optical quantum states and of the main passive (beam splitters) and active (squeezers) devices involved in experiments, such as the Hong-Ou-Mandel interferometer and the continuous-variable quantum teleportation. We also introduce the concept of operator ordering and the description of a system by means of the p-ordered characteristic functions. Then we focus on the quasi-probability distributions and, in particular, on the relation between the marginals of the Wigner function and the outcomes of the quadrature operator measurement. Finally, we introduce the balanced homodyne detection to measure the quadrature operator and the homodyne tomography as a tool for characterizing quantum optical states also in the presence of non-unit quantum efficiency.
•The Kenfack–Życzkowski indicator of quantumness is a unitary invariant.•The indicator of quantumness is sensitive to the choice of representations for the Wigner function.•“Quantum”-“classical” ...transitions in terms of the indicator are smooth.
Following Kenfack and Życzkowski, we consider the indicator of nonclassicality of quantum states for N−level systems defined via the integral of the absolute value of the Wigner function. For these systems, remaining in the framework of Stratonovich-Weyl correspondence, one can construct a whole family of representations of the Wigner functions defined over the continuous phase-space and characterized by a set of (N−2) moduli parameters. It is shown that the nonclassicality indicator, being invariant under the SU(N) transformations of states, turns to be sensitive to the representation of the Wigner function. We analyse this representation dependence computing the Kenfack-Życzkowski indicators for pure and mixed states of a 3-level system using a generic and two degenerate Stratonovich-Weyl kernels respectively. Our calculations reveal three classes of states: the “absolutely classical/quantum” states, which have zero and non-vanishing indicator for all values of the moduli parameters correspondingly, and the “relatively quantum-classical” states whose classicality/quantumness is susceptible to a representation of the Wigner function. Herewith, all pure states of qutrit belong to the “absolutely quantum” states.
•It is found that the Wigner function is the link between quantum and Bohmian mechanics.•The dynamics of the Bohmian trajectories are expressed in terms of the Wigner function.•The previous relations ...are illustrated through the coherent states.
Bohmian mechanics has proven to be very useful in numerical simulations for hard problems, what otherwise would be very time-consuming. Nevertheless, it is also well-known that Bohmian mechanics makes use of additional postulates to justify the Bohmian trajectories. Since the usefulness of the latter has been proven over the time, it is worth finding the link between Bohmian mechanics and conventional quantum mechanics, without the need of additional postulates or even rhetorical reasoning.
In this work, a connection between conventional quantum mechanics and Bohmian mechanics is found through the Wigner formalism. The Bohmian framework can be viewed as a projected aspect of the Wigner function. This confirms the idea of formulating Bohmian mechanics through the use of projections of observables onto continuous representations.
Wigner function of the 4-th rank Perepelkin, E.E.; Sadovnikov, B.I.; Inozemtseva, N.G. ...
Physics letters. A,
10/2023, Letnik:
484
Journal Article
Recenzirano
The Lorentz-Abraham-Dirac differential motion equation is of the third order since it contains the second order acceleration v→¨, which stands for the radiation friction force. To achieve a rigorous ...and comprehensive assessment of the dynamics of such radiation-friction systems it is necessary to expand the phase space {r→,v→} and to introduce higher kinematic values, such as v→˙ and v→¨.
Our study is dedicated to the problem of building of an expanded phase space {r→,v→,v→˙,v→¨}, within which the fourth Vlasov equation for the distribution function f(r→,v→,v→˙,v→¨,t) is considered. Since the second Vlasov equation for the distribution function f(r→,v→,t) in a particular case is shown to transform into the Moyal equation for the Wigner function W(r→,p→,t), the authors have elaborated and present a way to expand the Wigner function on the phase space {r→,v→,v→˙,v→¨} and also propose a new Moyal equation, which this function satisfies.
•The Wigner function W(r→,p→,p→˙,p→¨,t) of the 4th rank for the phase space {r→,p→,p→˙,p→¨} is obtained.•The new extended Moyal equation for the Wigner function W(r→,p→,p→˙,p→¨,t) is obtained.•The method for finding the exact solution of the third Vlasov equation is proposed.
PDF
