Wigner Operator Research Articles

Generalized Wigner Operators and Relativistic Gauge Fields

Generalized Wigner Operators and Relativistic Gauge Fields

Normal product form of two-mode Wigner operator

In the context of normal product, we use the method of the integration within an ordered product (IWOP) of operators to derive three representations of the two-mode Wigner operator: SU(2) symmetric description, SU(1,1) symmetric description and polar coordinate form. We find that two-mode Wigner operator has multiple potential degrees of freedom. As the physical meaning of the selected integral variable changes, Wigner operator shows different symmetries. In particular, in the case of polar coordinates, we reveal the natural connection between the two-mode Wigner operator and the entangled state representation.

Operator Transpose within Normal Ordering and its Applications for Quantifying Entanglement

AbstractPartial transpose is an important operation for quantifying entanglement. In this study, the (partial) transpose of any single (two‐mode) operator is investigated. Using the Fock basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacing with , rather than the c‐number within the normal ordering form. The transpose of the displacement and Wigner operators is also investigated, from which the relation of the Wigner function, characteristics function, and average values such as covariance matrix is constructed between the density operator and transposed density operator. These observations can be further extended to multi‐mode cases. As for the application, partial transpose of the two‐mode squeezed operator and the entanglement of the two‐mode squeezed vacuum through a laser channel is considered.

Massless particles in five and higher dimensions

We describe a five-dimensional analogue of Wigner's operator equation Wa=λPa, where Wa is the Pauli-Lubanski vector, Pa the energy-momentum operator, and λ the helicity of a massless particle. Higher dimensional generalisations are also given.

Phase-space elementary information content of confined Dirac spinors

Reporting on the Wigner formalism for describing Dirac spinor structures through a covariant phase-space formulation, the quantum information quantifiers for purity and mutual information involving spin–parity (discrete) and position–momentum (continuous) degrees of freedom are consistently obtained. For Dirac spinor Wigner operators decomposed into Poincaré classes of $$\mathrm{SU}(2)\otimes \mathrm{SU}(2)$$ spinor couplings, a definitive expression for quantum purity is identified in a twofold way: firstly in terms of phase-space positively defined quantities and secondly in terms of the spin–parity traced-out-associated density matrix in the position coordinate representation, both derived from the original Lorentz-covariant phase-space Wigner representation. Naturally, such a structure supports the computation of relative (linear) entropies, respectively, associated with discrete (spin–parity) and continuous (position–momentum) degrees of freedom. The obtained theoretical tools are used for quantifying (relative) purities, mutual information as well as, by means of the quantum concurrence quantifier, the spin–parity quantum entanglement, for a charged fermion trapped by a uniform magnetic field which, by the way, has the phase-space structure completely described in terms of Laguerre polynomials associated with the quantized Landau levels. Our results can be read as the first step in the systematic computation of the elementary information content of Dirac-like systems exhibiting some kind of confining behavior.

Development in Phase Space Quantum Mechancis for Constructing Different Masses’ Entangled State Representation

For developing the phase space quantum mechanics we explore how to construct different masses’ entangled state representation with the use of tthe Weyl-Wigner correspondence. We shall adopt a new approach, which uses the Radon transform of two-mode Wigner operator in normally ordering form to realize our goal. As an application of the entangled state representation the new normally ordered generalized squeezing operator is derived. The technique of integration within ordered product of operators is fully used.

Optical complex integration-transform for deriving complex fractional squeezing operator**Project supported by the National Natural Science Foundation of China (Grant No. 11775208) and Key Projects of Huainan Normal University, China (Grant No. 2019XJZD04).

We find a new complex integration-transform which can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.

Time evolution law of Wigner operator in diffusion channel

As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the <i>P</i> representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.

Evolution of quantum states via Weyl expansion in dissipative channel**Project supported by the National Natural Science Foundation of China (Grant No. 11664017), the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034), the Degree and Postgraduate Education Teaching Reform Project of Jiangxi Province, China (Grant No. JXYJG-2013-027), and the Science Fund of the Education Department of Jiangxi Province,

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.

A New Approach for Calculation of Time-Dependent Wigner Operator

The solution to the master equation obtained using the ket-vector representation of the density operator usually appears as the Kraus operator sum representation. Another useful form of such solutions is the time-dependent Wigner operator. We derive the time-dependent Wigner operator from these solutions by a direct and simple method. Application of this method is easy when any Kraus operator maps a coherent state into another coherent state.

The Effects of an Oscillating External Field on the Damping Behavior of the Single-Mode Cavity in the Thermal Bath

By using the solution of the equation for density operator of the single-mode cavity driven by an oscillating external field in the thermal bath (Master equation) and the coherent representation of the initial Wigner operator we derive time evolution of Wigner operator with a direct approach. For an initial state known as quasi-classical state, we extract time evolution of some quasi-probability functions. In order to study the effects of external field on the behavior of the single-mode cavity we analyze these distribution functions and their relation with the nature of damping in the thermal bath.

Localizability, gauge symmetry and Newton–Wigner operator for massless particles

The notion of position operator for massless spinning particles is discussed in some detail. The noncommutativity of coordinates is related to the gauge symmetry following from the freedom in definition of standard state in Wigner’s construction of induced representations of Poincare group. The counterpart of Newton–Wigner operator is discussed. It is explained why the Newton–Wigner construction works only for helicity |λ|≤1∕2.

Optical Tomograms of Multiple-Photon-Added Gaussian States via the Intermediate State Representation Theory

Optical tomogram of a quantum state can serve as an alternative to its density matrix since it contains all the information on this state. In this paper we use the explicit expression of the intermediate state |q, f, g〉(that is, an eigenvector of the coordinate and momentum operator fQ + gP) and the Radon transform between the Wigner operator Δ(p, q) and the pure-state density operator |q, f, g〉〈q, f, g| to calculate the specific tomograms for multiple-photon-added coherent states (MPACSs), thermal states (MPATSs) and displaced thermal states (MPADTSs). Their analytical tomograms are actually the finite-summation representations related to single-variable Hermite polynomials. Besides, the numerical results indicate that the larger photon-addition number can lead to the multi-peak tomographic distributions for these three states, and the measurable tomograms of MPACSs and MPADTSs exhibit π-periodic features but MPATSs can’t.

Time Evolution of the Wigner Operator as a Quasi-density Operator in Amplitude Dessipative Channel

For developing quantum mechanics theory in phase space, we explore how the Wigner operator ${\Delta } (\alpha ,\alpha ^{\ast } )\equiv \frac {1}{\pi } :e^{-2(\alpha ^{\ast } -\alpha ^{\dag })(\alpha -\alpha )}$ :, when viewed as a quasi-density operator correponding to the Wigner quasiprobability distribution, evolves in a damping channel. with the damping constant κ. We derive that it evolves into $$\frac{1}{T + 1}:\exp \frac{2}{T + 1}[-(\alpha^{\ast} e^{-\kappa t}-a^{\dag} )(\alpha e^{-\kappa t}-a)]: $$ where T ≡ 1 − e− 2κt. This in turn helps to directly obtain the final state ρ(t) out of the dessipative channel from the initial classical function corresponding to initial ρ(0). Throught the work, the method of integration within ordered product (IWOP) of operators is employed.

New relationship between quantum state’s tomogram and its wave function

By searching for - and -ordering form of the coordinate-momentum intermediate representation (formed by the eigenvector of with eigenvalue being x, ), and noting that is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ’s Gaussian integration transformation with the parameter and another is ’s Gaussian integration transformation with the parameter . In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of and , which can be either viewed as - and -ordered operator correspondence of the classical function .

A Space-Time Wigner Function Approach to Long Time Schrödinger--Poisson Dynamics

A long time description of electrostatic Schrodinger--Poisson states, satisfying $ i \partial_t \psi = -\frac{1}{2} \Delta_x \psi + ( \frac{C}{|x|} \ast_x |\psi|^2 + \frac{1}{2} |x|^2 ) \psi, $ is provided in terms of a non-Markovian Wigner formalism through the choice of the simplest charge-preserving scale group, $\psi_{\varepsilon}(t,x) = \psi({\varepsilon}^{-1}t, x)$, in which the position variable $x \in \mathbb R^3$ remains unscaled while time $t \in \mathbb R^+$ is sent to infinity as $\varepsilon \to 0$. Typically, the introduction of this group of scale transformations leads to high frequency, time oscillatory states that may not converge in such a good topology as to deal with the nonlinear term. To overcome this drawback, the sequence of wavefunctions is Wignerized via the action of the so-called $\tau$-convoluted space-time Wigner transform. The main goal of this extended Wigner operator consists in producing an attenuating effect on the temporal oscillations as time grows, which in turn allow...

Generalized Schrödinger cat states and their classical emulation

We demonstrate that superpositions of coherent and displaced Fock states, also referred to as generalized Schr\"odinger cats cats, can be created by application of a nonlinear displacement operator which is a deformed version of the Glauber displacement operator. Consequently, such generalized cat states can be formally considered as nonlinear coherent states. We then show that Glauber-Fock photonic lattices endowed with alternating positive and negative coupling coefficients give rise to classical analogs of such cat states. In addition, it is pointed out that the analytic propagator of these deformed Glauber-Fock arrays explicitly contains the Wigner operator opening the possibility to observe Wigner functions of the quantum harmonic oscillator in the classical domain.

Studying Weyl prescription by virtue of [formula omitted] and [formula omitted] technique of operators

Q-ordering and P-ordering for the coordinate and momentum operators are introduced, some fundamental identities of Q-ordering and P-orderingP-ordering are derived. the Q-ordered and P-ordered form of the Wigner operator are deduced and Weyl prescription is studied. Moreover, Weyl prescription is developed and perfected.

An integral-transformation corresponding to quantum mechanical fundamental commutative relation and its application in deriving Wigner function

In this paper, it can be found that there is a type of integra-transformation which corresponds to a quantum mechanical fundamental commutative relation, with its integral kernel being 1/exp[2i(q-Q)(p-P)], here denotes Weyl ordering, and Q and P are the coordinate and the momentum operator, respectively. Such a transformation is responsible for the mutual-converting among three ordering rules(P-Q ordering, Q-P ordering and Weyl ordering). We also deduce the relationship between this kernel and the Wigner operator, and in this way a new approach for deriving Wigner function in quantum states is obtained.

经高斯函数光滑的Wigner算符的性质

经高斯函数光滑的Wigner算符的性质